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Full text of "Principia mathematica"

BOUGHT WITH THE INCOME 
PROM THE 

SAGE ENDOWMENT FUND 

THE GIFT OF 

1891 



A^2mx5 i.5J(/l/i. 



MATHEMATICa 



97>4 



Cornell University Library 
QA 8.4.W59 
V.3 

Principla matheinatica, 




3 1924 001 575 244 




m, 



Cornell University 
Library 



The original of this book is in 
the Cornell University Library. 

There are no known copyright restrictions in 
the United States on the use of the text. 



http://www.archive.org/details/cu31924001575244 



PRINCIPIA MATHEMATICA 



CAMBRIDGE UNIVERSITY PRESS 

EonSon: FETTER LANE, E.C. 

C. F. CLAY, Manager 




(EIlinfiutBt) : loo, PRINCES STREET . 

Setlin: A. ASHER AND CO. 

ILeyijig : F. A. BROCKHAUS 

Jleta Imft: G. P. PUTNAM'S SONS 

iBombsj anlj Calcutta: MACMILLAN AND CO., Ltd. 



All rights reserved 



PRINCIPIA MATHEMATICA 



BY 



ALFRED NORTH WHITEHEAD, Sc.D., F.R.S. 

Fellow and late Lecturer of Trinity College, - Cambridge • 



AND 



BERTRAND RUSSELL, M.A., F.R.S. 

Lecturer and late Fellow of Trinity College, Cambridge 



VOLUME III 



Cambridge 
at the University Press 



1913 

EV- 



etBmbrilige: 

PRINTED BY JOHN CLAY, M.A. 
AT THE UNIVERSITY PRESS 



PREFACE TO VOLUME III 

nnHE present volume continues the theory of series begun in Volume II, 
and then proceeds to the theory of measurement, Geometry we have 
found it necessary to reserve for a separate final volume. 

In the theory of well-ordered series and compact series, we have followed 
Cantor closely, except in dealing with Zermelo's theorem (*257 — 8), and in 
cases where Cantor's work tacitly assumes the multiplicative axiom. Thus 
what novelty there is, is in the main negative. In particular, the multi- 
plicative axiom is required in all known proofs of the fundamental proposition 
that the limit of a progression of ordinals of the second class {i.e. applicable 
to series whose fields have ^{o terms) is an ordinal of the second class (cf *265). 
In consequence of this fact, a very large part of the recognized theory of 
transfinite ordinals must be considered doubtful. 

Part VI, on the theory of ratio and measurement, on the other hand, 
is new, though it is a development of the method initiated in' Euclid Book V 
and continued by Burali-Forti*. Among other points in our treatment of 
quantity to which we wish to draw attention we may mention the following. 
(1) We regard our quantities as in a generalized sense "vectors," and 
therefore we regard ratios as holding between relations. (2) The hypothesis 
that the vectors concerned in any context form a group, which has generally 
been made prominent in such investigations, sinks with us into a very 
subordinate position, being sometimes not verified at all, and at other times 
a consequence of other more fruitful hypotheses. (3) We have developed 
a theory of ratios and real numbers which is prior to our theory of measure- 
ment, and yet is not purely arithmetical, i.e. does not treat ratios as mere 
couples of integers, but as relations between actual quantities such as two 
distances or two periods of time. (4) In our theory of "vector families," 
which are families of the kind to which some form of measurement is 

* Cf. Peano's Formulaire, i. (1895), pp. 28—57. 



VI PREFACE 

applicable, we have been able to develop a very large part of their properties 
before introducing numbers; thus the theory of measurement results from 
the combination of two other theories, one a pure arithmetic of ratios and 
real numbers without reference to vectors, the other a pure theory of vectors 
without reference to ratios or real numbers. (5) With a view to geometrical 
applications, we have devoted a special Section to cyclic families, such as the 
angles about a given point in a given plane. 

The theory of measurement developed in Part VI will be required in the 
next volume for the introduction of coordinates in Geometry. 

We have to thank various friends for their kindness in bringing to our 
notice mistakes and misprints noted in the Errata, both in this and in 
previous volumes. 

A. N. W. 

B. R. 



15 February 1913 



CONTENTS OF VOLUME III 



PART V. SERIES (continued). 

Section B. Well-ordered Series 



*250. 
*251. 
*252. 
*253. 
«254. 
*255. 
*256. 
*257. 
*258. 
*259. 



Elementary properties of well-ordered series 

Ordinal numbers 

Segments of well-ordered series . 
Sectional relations of well-ordered series 
Greater and less among well-ordered series 
Greater and less among ordinal numbers 
The series of ordinals .... 
The transfinite ancestral relation . 

Zermelo's theorem 

Inductively defined correlations . 



Section E. Finite and Infinite Series and Ordinals 

*260. On finite intervals in a series 

«261. Finite and infinite series 

«262. Finite ordinals 

«263. Progressions 

'ii'264. Derivatives of well-ordered series 

'»265. The series of alephs 



Section F. Compact Series, Rational Series, and Continuous Series 

*270. Compact series 

4^271. Median classes in series 

«'272. Similarity of position 

')!'273. Rational series 

*274. On series of finite sub-classes of a series 

«'275. Continuous series 

#276. On series of infinite sub-classes of a series .... 



PART VI. QUANTITY. 

Summary of Part VI 

Section A. Generalization op Number 

*300. Positive and negative integers, and numerical relations 
*301. Numerically defined powers of relations 

*302. On relative primes 

*303. Ratios 



PAGE 



4 
18 
27 
32 
44 
58 
73 
81 
96 
102 

108 

109 
118 
131 
143 
156 
169 

179 

180 
186 
191 
199 
207 
218 
221 



233 

234 

235 

244 
251 
260 



VUl 



CONTENTS 



^04. The series of ratios 

«305. Multiplication of simple ratios 

«306. Addition of simple ratios 

■i(307. Generalized ratios . 

*308. Addition of generalized ratios 

*309. Multiplication of generalized ratios 

*310. The series of real numbers . 

*311. Addition of concordant real numbers 

'»312. Algebraic addition of real numbers 

*313. Multiplication of real numbers 

-^314. Beal numbers as relations . 

Section B. Vector-Families 



♦330. Elementary properties of vector-families 

♦331. Connected families .... 

♦332. On the representative of a relation in a 

♦333. Open families 

*334. Serial families 

♦335. Initial families .... 

*336. The series of vectors .... 

4f337. Multiples and submultiples of vectors . 

Section C. Measurement .... 



*350. Ratios of members of a family 

♦351. Submultipliable families 

♦352. Rational multiples of a given vector 

♦353. Rational families .... 

♦354. Rational nets .... 

♦356. Measurement by real numbers 

♦359. Existence-theorems for vector-families 



family 



PAGE 

278 
283 
289 
296 
299 
309 
316 
320 
327 
333 
336 

339 

350 
360 
367 
376 
383 
390 
393 
403 

407 

412 
418 
423 
431 
436 
442 
452 



Section D. Cyclic Families .... 

♦370. Elementary properties of cycUc families 

♦371. The series of vectors .... 

♦372. Integral sections of the series of vectors 

♦373. Submultiples of identity 

♦374. Principal submultiples . 

♦375. Principal ratios 



457 

462 
466 
470 
475 
485 
487 



ERRATA TO VOLUME III 

p. 3, line ^%for" that there is " read " that this is." 
p. 23, *251-371. add "[*251-37 .*170101 .*171101]." 
p. 25, lines 2 and 3, for " 18311 and *185-11 " read "*183-14 and *185-12." 
p. 25, line 6, for " [*251-61 . *183-11 . *185-11] " read 
" Bern. 

l-.*151-65.*182-05-162. D h . J I' O'P e ( _| JP) sinor P n Rl'smor (1) 

h . (1) . *151-162 . D h : Hp . D . a ! [( J ;P)s"mbr ( J > Q)] n Rl'smor (2) 

h . (1) . *251111 . *18216 . D h : Hp . D . 0' J 5P C fi . J 'P, J 5 Q e Rel*excl (3) 

I- . (2) . (3) . *251-62 . *183-14 . *185-12 . D h . Prop " 

p. 33, line 6, /or "jSXoj" read " a x /3." 

p. 44, line 12, for first "P" read " Ps." 

p. 59, line 20, for "*120-413" read "*120-51." 

p. 71, line 6 from below, /or "Nr'G'P" read "Nc'G'P." 

— > — > 

p. 89, line 4 from below, for " Hp . z e QnxV " Tead " Hp .yea-.ze QbJv" 

p. 90, *257-211, first line of Dem.,for " Hp . " read " Hp . D . ". 

p. 99, *258-221,/or "p'lc = (22*Q)'/<; " read "p'lc = limax (Q/ea)'«-" 

p. 112, line 8, /or " xP^z" read " xP^z^." 

p. 120, line 9 from below, /or "a well-ordered" read "an infinite well-ordered." 

p. 177, *265-48-49,/or "Itp'Q" read "Mp'G'Q." 

p. 194, *272-161, second line of Dem., for "ze D'T " read "H^.ze T>'T." 

p. 195, *272-221, add "z,we DT" to hypothesis. 

p. 196, *272-321, for " z " read " w." 

T TOP 

p. 198, *272-42, second line oi Dem., for " „" read " my-Q-" 

p. 210, *274-12, fourth line of Dem., for " (g^^) " read " (g^:) .zea-p." 
p. 217, *274-31, for " P e Ser n comp " read " P, e Ser n comp." 

p. 224, *276-2, Dem., line 4, /or " 7 = ^S w P'z " 

-» <— «- 

read " 7 = (a n P^'z) w P'minp'(a n P'z)." 

p. 224, *276-2, Dem., line 5, for "ynP'z = l3 nP'z = ar\P'z .zea-y " 

<— -* <——><— 

read " minp'(a n P'z) e a - 7 . a n P'minp'(a n P'^) = 7 n P'minp'(a n P'z) . 

Z€y-^.ynP'z = anP'z = ^nP'z." 
p. 224, *276-2, Dem., line 6, deZeie " *170-16." 
p. 244, line 12, for " ( | i2) 1 1 (-^^ 1) " read " {( \B)\{ (-, 1)}." 
p. 316, line 16,/o7' "limit" read "limit or maximum." 



X ERRATA 

p. 317, line 15, end, add "including zero." 

p. 320, last line but one, for " rationals " read " real numbers." 

p. 347, line 9 from below, for first " D " read " d." 

p. 347, line 8 from helow, for " x + y = a; + 2" read " y + x = z + x." 

p. 379, *333-24, enunciation, /or " g ! v n t"R " read " g ! (y +„ 1) <-« «"P." 

p. 379, line 8 from below, /or "•a^lv nt^'Ris veC'Ul 1?'R " 

read " a ! (i' +« 1) « t^'P isvea'Ul f'P." 
p. 379, *333-25, enunciation, /or " g ! i/ n t"L " read " g; ! (v +c 1) a P'L." 
p. 395, line 14, /or "greater and less" read "greater and less among magni- 
tudes." 
p. 433, *353-22, line 4 of Bern., far " Hp " read « Hp . i? e \g." 

ADDITIONAL ERRATUM TO VOLUME I 
p. 574, line 8, /or "p'a"(8\R)" read "p'(l"Vot'{S\R)r 

ADDITIONAL ERRATA TO VOLUME II 

p. 11, line 13 from below, /or " Nc'a r> t^'a. " read " Nc'a n t't^'a." 
p. 71, last line, /or " Nc'oNc'p = Noc'c^""'^ Df" 

read" (Nc'a)" = (Noc'a)' Df and fj.^<^P = ^^"^'^ Df." 
p. 90, line 21, /or "mutually exclusive classes of /i" 

read " classes of /i mutually exclusive." 
p. 152, margin of figure, /or "D'R = 'D'M'z" read "B'R = I>'M'w:- 

p. 152, last line, /or " e^'^'y" read "€^'T"y." 

p. 279, line 11, for "*124-2-34" read "*124-23-25-252." 

p. 334, line 10 from below, /or "series" read " well-wdered series." 

p. 347, line 7 from below, /or "the series a" read "the class a." 

p. 348, line 5 from below, /or "Cls'excl" read "Cls^excl." 

p. 366, line 8, for "Re G'Q " read " M e G'Q." 

p. 366, line 9, /or "such relations as M" read "such relations as R'M." 

p. 403, footnote, for " mathematischphysischen " 

read " mathematisch-physischen." 
p. 519, *200-21, for " Te Cls -» 1 " read " Te Cls -* 1 . P G J." 
p. 561, last line, /or "D " read "Q." 
p. 570, first line, delete "which is used in *263'11." 
p. 606, line 23, for "*208-4 " read " *208-41." 
p. 606, line 24,/or " S, TePslnor Q . D . ^f= T" 

read "Psmor Q.D. (Psinor Q)e 1." 
p. 614, line 7, for "*214-31 " read "*214-32." 
p. 710, Summary, line 2, for " P" read " Q." 
p. 710, Summary, line 3, for " Q " read " P." 
p. 753, footnote, /or "reallen" read "reellen." 



SECTION D. 

WELL-ORDERED SERIES. 

Summary of Section D. 

A " well-ordered " series is one which is such that every existent class 
contained in it has a first term, or, what comes to the same thiag, one which 
is such that every class which has successors has a sequent. We will call a 
relation in general well-ordered if every existent class contained in its field 
has one or more minima. Then a well-ordered series is a series which is a 
well-ordered relation. 

Well-ordered series have many important properties not possessed by 

series in general. A well-ordered series is Dedekindian, except for the fact 

that it may have no last term; i.e. every section having a last term is 

Dedekindian. A well-ordered series which is not null has a first term, and 

every term of the series (except the last, if there is one) has an immediate 

successor. A very important property of well-ordered series is that they 

obey an extended form of mathematical induction, which we shall call 

" transtinite induction," namely the following : If o- is a class such that the 

sequent (if any) of any class contained in <r and in the series is a member of 

<r, then the whole series is contained in a. (It will be observed that A is 

contained in <t, and therefore, by *206"14, B'P is a member of a.) This 

differs from ordinary mathematical induction by the fact that, instead of 

dealing with the successors of single terms, it deals with the successors 

of classes. A closely analogous property, which holds for all well-ordered 

relations, whether serial or not, is the following : If o- is a class such that, 

— > 
whenever P'x C <r, where x is any member of G'P, x itself belongs to <r, then 

C'P Co-. If P is well-ordered, this property holds for all <r's ; and conversely, 

if this property holds for all a'a, P is well-ordered. Hence this property 

is equivalent to well-orderedness. 

If P is a well-ordered series, minp selects one term out of each member 
of CI ex'C'P. Hence C'P, which is minp"Cl bk'C'P, is a member of the 
multiplicative class of CI ex'G'P ; hence the multiplicative class of CI ex'G'P 
exists, and therefore the multiplicative class of any class contained in 
CI ex'G'P exists (by *88*22). It follows that if s'k can be well-ordered, and 
A ~ e «, the multiplicative class of « exists ; and that, if every class can be 

R. & W. IIL 1 



2 SERIES [part V 

well-ordered, the multiplicative axiom holds. The converse of this latter 
proposition also holds, as has been proved by Zermelo (cf. *258). 

Another important set of properties of well-ordered series results from 
*20841 ff. Two ordinally similar well-ordered series can only be correlated 
in one way ; and no proper section of a well-ordered series is ordinally 
similar to the whole series. (A "proper" section is a section not the 
whole.) 

From the uniqueness of the correlator of two similar well-ordered series, 
it follows that all the uses of the multiplicative axiom in *164 can be avoided 
if the fields of the relations concerned consist of well-ordered series. I.e. 
taking *164"45, which is the fundamental proposition in this subject, we 
have, without assuming the multiplicative axiom, 

P,Qe Rel" excl . D : g ! P smor Q n Rl'smor . = . P smor smor Q, 

whenever C'P and G'Q consist of well-ordered series. Hence, under this 
hypothesis, the multiplicative axiom disappears from the hypotheses of all 
the consequences of *164"45. 

Ordinal numbers (*251) are defined as the relation-numbers of well- 
ordered series. (This definition is in accordance with usage: otherwise, there 
would be no special reason against defining " ordinal numbers " as the 
relation-numbers of series in general. The relation-numbers of series will 
be called serial numbers) Sums of an ordinal number of ordinal numbers 
are ordinal numbers, but products of an ordinal number of ordinal numbers 
are not in general ordinal numbers. The product of an ordinal number of 
serial numbers is a serial number, and the product of an ordinal number (not 
zero) of ordinal numbers other than zero is not zero, i.e. a product of ordinal 
numbers, in which the number of factors is an ordinal number, does not 
vanish unless one of the factors vanishes. (For relations in general, the 
corresponding proposition requires the multiplicative axiom.) If v is an 
ordinal number, and /* is any serial number, /xexprZ/ {i.e. fi" as it would 
naturally be called) is a serial number ; but if /* > 1, fi exp, v is not an 
ordinal number unless v is finite. 

The theory of sections and segments (*252, *2.53) is much simplified for 
well-ordered series, owing to the fact that every proper section has a sequent. 
Proper sections are identical with proper segments, and both are identical 

with P"G'P. The series of sections, s'P*, is P'P-\*G'P. The series of 

segments, s'P, is P'>P or P''P-\* C'P according as there is or is not a last 

— > 
term of C'P. The series of sectional relations, P,, is Pl.'P'PlQ.'P-\* P ; 

its domain is Pl"P"C'P, and its field is P ^"P"C7'P u t'P. If 

— > 
xeC'P, P^P'x is never similar to P. 



SECTION D] well-ordered SERIES 3 

The theory of greater and less among well-ordered series and ordinal 
numbers is dealt witi* in *254 and *255. Cantor has proved, by means of 
segments, that of any two different ordinal numbers one must be the greater. 
This is proved by showing that of any two well-ordered series which are not 
similar, one must be similar to a segment of the other. We define an 
ordinal number a as less than another ^ if series P and Q can be found such 
that P is an a and Q is a yS and P is similar to some relation contained in Q, 
but not to Q. It can be proved that all the ordinals less than Nr'Q belong, 
one each, to the proper segments of Q. Hence to say that the ordinal 
number of P is less than that of Q is equivalent to saying that there is a 
proper segment of Q to which P is similar. 

When two series have the same ordinal, they also have the same cardinal, 
in virtue of *15118, but the converse does not hold. When the cardinal 
number of one series is greater than that of the other, so is the ordinal 
number. When two classes can be well-ordered, any well-ordering will make 
the one class similar to a part of the other, or the other similar to a part of 
the one, in virtue of the properties of segments of well-ordered series. Hence 
of two different cardinals each of which is applicable to classes which can be 
well-ordered, one must be the greater — a property which cannot be proved 
concerning cardinals in general. 

In *256 we deal with the series of ordinals in order of magnitude. We 
show that there is a well-ordered series, and that the series of all ordinals of 
a given type has an ordinal number which is greater than any of the ordinals 
of the given type. This constitutes the solution of Burali-Forti's paradox 
concerning the greatest ordinal : there is no greatest ordinal in any one 
type, and all the ordinals of a given type are surpassed by ordinals of higher 
types. 

*257, *258 and *259 deal with " transfinite induction " and its appli- 
cations, of which the most important is Zermelo's theorem, nanaely, 

*258-34. f-:./i~6l.D:(Se e^'Cl ex'/* . = . 

(gP) . P e fl . O'P = /i . /Sf = mini. I' CI ex'/* 

where O is the class of well-ordered series. This proposition leads to the 
following : 
*?58-36. h : /i e Cil w 1 . = . g ! e^'Cl ex'/* 

I.e. a class can be well-ordered or is a unit class when, and only when, a 
selection can be made from its existent sub-classes. Hence we arrive at 
*258-37. h : Mult ax . = . (7"fl u 1 = Cls 

I.e. the multiplicative axiom is equivalent to the assumption that every class 
can be well-ordered or consists of a single member. 

The proof of Zermelo's theorem uses an extension to transfinite induction 
of the ideas of «90 and j|e91, which is explained in «257. 

1—2 



*250. ELEMENTARY PROPERTIES OF WELL-ORDERED SERIES. 

Summary of *250. 

A relation is called " well-ordered " when every existent sub-class of its 
field has one or more minima. A well-ordered series is defined as a well- 
ordered relation which is a series. We shall denote the class of well-ordered 
relations by " Bord," which is an abbreviation for " bene ordinata " or " bien 
ordonnee." The class of well-ordered series will be denoted by 12. Thus 
our definitions are 

Bord = P(Clex'C'PCa'minp) Df, 

n = Ser A Bord Df. 

Well-ordered relations other than series will be seldom referred to after the 
present number. 

By applying the definition of " Bord " to unit classes, it appears that a 
well-ordered relation must be contained in diversity (*250104!). A well- 
ordered relation is one whose existent upper sections all have minima 
(*250102). Hence by *21117, 

*250103. ViPe Bord . = .P^e Bord 
Hence by *250-104, 

*250105. hiPe Bord . D . Pp„ G J 

By considering couples, it can be shown (*250111) that a well-ordered 
relation in which no class has more than one minimum is connected ; hence 
by *20416 and *250"105, it is a series. Thus we have 

*250125. I- : P eXl . = . E !! minp"Cl ex'C'P, 

I.e. a well-ordered series is a relation such that every existent sub-class 
of the field has a unique minimum. This might have been taken as the 
definition of fi. 

By the definition of SI we have 
*250121. f- :. P e 12 . = : P e Ser : a C C'P . a ! a . D. . E ! miup'a : 
^ : P e Ser : a ! a n C'P . D. . E ! minp'o 

Applying this to G'P we have 
*25013. h:Pef2-t'A.D.E!5'P 



SECTION d] elementary PROPERTIES OF WELL-ORDERED SERIES 5 

We have also 

*25017. 1- :. P, Q 6 n - t'A . D : P smor Q. = .Pt Q'Psmor Q I a'Q 

This proposition justifies the subtraction of 1 from the beginning, and is 

useful in the theory of segments of well-ordered series. 

We have next (*250'2 — '243) an important set of propositions on Pj when 

P e O. The most useful of these is 

*250-21. h : P 6 ft . D . D'P = D'P, 

I.e. in a well-ordered series every term except the last (if any) has an 
immediate successor. (It is not in general the case that every term except 
the first has an immediate predecessor.) Another useful proposition is 

*250-242. h:P6l2.D.P = P,c;Pi|P 

The next set of propositions (*250"3 — •362) is concerned with "trans- 
finite induction." We have 

«250-33. h . fi = connex nP{aC C'P na^.D^. seqp'a C <r : D, . C'P C a] 

I.e. a well-ordered series is a connected relation P such that the whole field 
of P is contained in every class a- which is such that the sequent (if any) of 
every sub-class of G'P^r\ o- is a member of a. 

*25035. 1- . Bord = P {« e C'P . P'a; C o- . D^, . a; e o- : D, . O'P C o-} 

I.e. a well-ordered relation is a relation P whose field is contained in every 
class <T which contains every member of C'P whose predecessors are all 
contained in <x. We may say that a property is " transfinitely hereditary " 
in P if it belongs to the sequents of all classes composed of members of C'P 
which possess the property. In virtue of *250"33, if P is well-ordered> 
every transfinitely hereditary property belongs to every member of C'P, and 
conversely. 

Our next set of propositions (*250"4 — '44) is concerned with A and 
couples. We prove that A e fl (*250"4) and that x^y ."^ .x^yeH 
(*250-41). 

*250'5 — "54 are concerned with selections. We have 

*250-5. h : P e 12 . D , 

minp 1^ CI ex'O'P e e^'Cl ex'C'P . t'O'P = Prod'Cl ex'C'P 
whence 

*25051. I- : a e (7"0 . D . a ! e^'Cl ex'a 

Observe that G"£l is the class of those classes that can be well-ordered. 
From *250'51 we deduce 

*250-54. h : (7"fl u 1 = Cls . 3 . Mult ax 

The converse, which is Zermelo's theorem, is proved in *258. 



6 SERIES [part V 

*250"6 — "67 are concerned with consequences of *208. We show that 
two well-ordered series cannot have more than one correlator (*250'6) ; that 
if P is a well-ordered series, and j8 is contained in a proper section of P, 
P C y3 is not similar to P (*250-65) ; and that if P is any well-ordered 
relation, and a is any class such that there are terms in C'P which are later 
than any member of a r> C'P, then P is not similar to P ^ a (*250-67). 



*25001. Bord = P(Clex'0'PCa'minjp) Df 

*25002. n = SernBord Df 

*250-l. \-:Pe Bord . = . CI ex'O'P C Q'minp [(*250-01)] 

*250101. h :. P 6 Bord . = : g ! a n C'P . D, . g ! minp'a [*2501 . *20515] 

*250102. h : P e Bord . = . sect'P - I'A C a'minj. 
Dem. 

I-.*2501. DhiPeBord.D. sect'P- t'ACQ'minp (1) 

h . *20519 . D I- . liihi (Ppo)'a = i^n (Ppo)'P*"a 

[*205-68] = minp'P*"o (2) 

h . *90-331 . *21113 . D h : a ! a n C'P . D . P^"a e sect'P - t'A (3) 

h . (3) . D I- :. sect'P - I'A C Q'minp . D : a ! o n C'P . D. . g ! minp'(P#"a) . 

[(2)] D„.a!i]^n(Pp„)'a. 

— » 
[*205-26] 3« . a ! minp'a : 

[*250-101] DiPeBord (4) 

I- . (1) . (4) . D h . Prop 

*250103. l-:P6Bord.s.Pp„6Bord [*250102 . *211-17] 

*250104. l-.BordCRl'J 

Dem. 

— » 
I- .*250-l .Dh:PeBord . a; e C'P . D . « e minp't'a; . 

[*205-194] D . ~ {a>Pa;) Oh. Prop 

*250105. f-iPeBord.D.PpoCJ" [*250103104] 
*25011. h :: P 6 connex . D :. P e Bord . = : a '■ a « G'P . D. . E ! minp'a : 

= :aCC'P.a!a.3..E!minp'a 
[*250-l-101 . *205-32] 

«250'111. h :. P 6 Bord . D : P e connex . = . minp e 1 -♦ Cls 

Dem. 
h.*2501.*7l-l.D 

h ::P e Bord. minp e 1 -> Cls . D :. a;, y e G'P . D : (i'ob u t'y) - P"{i'x w I'y) e 1 : 

[*54-4] D : t'a; w I'y - P"{i'x w I'y) = i'x.v . 

I'lesj I'y- P"{i'x u i'y) = i'y (1) 



SECTION D] elementary PROPERTIES OF WELL-ORDERED SERIES 1 

I- . (1) . D h :. P 6 Bord . minp el—* Cls . as, ye G'F .x^y.D: 

• ye P"(l'x u I'y) .v.xe P"(l'x w I'y) : 

[*250-104] D : wPy . v . yPx (2) 

h . (2) , *202103 . D f- : P e Bord . minp e 1 -» Cls . 3 . P e connex (3) 

h.(3).*205-31.Dh.Prop 

*250112. h : P e connex r. Bord . = . E !! minp"Cl ex'O'P 
Dem. 

\- . *2501111 . D 

h : P e connex n Bord . = . minp e 1 — > Cls . CI ex'C'P C Q'minp . 
[*71-16] = . E 1! minp"a'minp . CI ex'G'P C Q'minp . 

[*205-1516] = . E !! minp"Cl ex'C'P : D h . Prop 

«250-113. h . connex n Bord = fl 
Dem. 

I- . *204-l . (*25002) . D h . II C connex n Bord (1) 

h . *250105 . D I- : P e connex rt Bord .D.Pe connex .P^QJ. 
[*204-16] 3 . P e Ser (2) 

h . (2) . (*250-02) .D\-:Pe connex n Bord . D . P e fi (3) 

h . (1) . (3) . D I- . Prop 

*250-12. h:P6«. = .PeSernBord [(*25002)] 

*250121. h : . P e 12 . = : P 6 Ser : a C C'P . a ! a . D. . E ! minp'a : 

= : P e Ser : a ! a A C'P . 3a . E ! minp'a [*2501211] 

*250122. I- :. P 6 n . = : P 6 Ser : a ! C'P n p'P"(o n C'P) . 3. . E ! seqp'o 
Dem. 

l-.*206-13.*250121.D 

h :. P e fl . D : P e Ser : a ! C'P o p'lp'^a r^ C'P) . D. . E ! seqp'a (1) 

l-.*204-62.D 

h : P 6 Ser . a ! a « C'P . D . a ! C'P np'P"2j'^'(a n O'P) . 

[*40-62] D.a!C'Prti3'P"{C'Pn2)'P"(anO'P)} (2) 

l-,(2).*10-l.D 

h :. P 6 Ser : a ! G'P n^'P"(o n C'P) . D. . E ! seqp'a : D : 

a ! a n C'P . Da . E ! seqp'{C'P a p'P"(o a C'P)} . 
[*206-131-54] 3„ . E ! minp'a : 

[*250-121]D:Pen (3) 

h.(l).(3).,DH.Prop 



8 SERIES [part V 

*250123. h :. P e n - I'A . = : P e Ser : a 1 ^'P"(o n G'P) . D. . E ! seq^'a 
Dem. 
l-.*250-122.D 

I- : . P € Ser : a ! p'P"(a n C'P) . 3, . E ! seqp'a : D . P e fl (1) 

V . *40-6 . *24-52 . D 

I- :. a !iJ'P"(a n O'P) . D, . E ! seqp'a : D . E ! seqp'A . 

[*20618] D - a ! -P (2) 

h . *250-122 . *40-62 . D 

I- i.Pefi . D :PeSer : a ! a n O'P. a !i>'P"(a " O'P) . D. ■ E ! seqp'o (3) 

h .*206-14 . D h : a n O'P = A . D . ^qp'a = B'P 

[*205-12] =mmp'C"P (4) 

h . *33-24 . *250121 . D I- : P e fl - t'A . D . E ! minp'C'P (5) 

h . (4) . (5) . D h : P 6 n - t'A . a n O'P = A . D . E ! seqp'o (6) 

h.(3).(6).D ^ 

H :. P e n - I'A . D : P e Ser : a !i3'P"(a n O'P) . D. . E ! seqp'a (7) 

h . (1) . (2) . (7) . D h . Prop 

*250124 I- : P e li . = . P e Ser . sect'P - I'C'P C Q'seqp 

Dem. 
h . *250122 . *211-703 . D h : P e fl . D . P e Ser . sect'P - I'C'P C Q'seqp (1) 
h.*211-7. D h:.P 6 Ser. sect'P- I'C'P C Q'seqp. D: 

/S e sect'P - I'A . Dp . E ! seqp'((7'P - yS) . 
[*211-723] Dp . E ! minp'iS : 

[*250102-12] DrPen (2) 

h . (1) . (2) . D h . Prop 

*250125. I- : P e n . = . E !! minp"Cl ex'O'P . [*250-112113] 

The above proposition might be demonstrated, independently of 
*250-112-113, as follows: 

(a) If E!!minp"Clex'(7'P, it follows that a; e C'P . D . E ! minp't'a;, 
whence w e G'P . D . ~ (xPx), whence PQ.J. 

(b) If E !! minp"Cl ex'O'P, it follows that 

x,ye G'P . a; 4= y • D . E ! minp'(t'a! w I'y), 
whence it follows that 

xPy . ~ Q/Px) . V . yPx . ~ (xPy). 
Hence P e connex .P'dJ. 

(c) If E !! minp"Cl ex'O'P, it follows that 

xPy . yPz . D . E ! minp'(t'a; w I'y w I'z), 



SECTION D] elementary PROPERTIES OF WELL-ORDERED SERIES 9 

whence xPy . yPz . D . ~ {zPx), 

and by P" G J^ (whioMhas just been proved) 

xPy . yPz ."^ .x^z. 
Hence, since, by (6), P e connex, we must have 

xPy . yPz . D . xPz, i.e. P e trans. 
Hence E ! ! minp"Cl ex'O'P . D . P e Ser. 

Hence the above proposition is obvious. 

*250126. h : P e f2 . E ! maxp'o . ~ E ! seqp'a . D . B'P e a . B'P = maxp'a 
Dem. 

h . *250123 . Transp . D h : Hp . D . ~ g ! p'P"(a n G'P) . 
[*205-65] D . ~ a ! P'maxi.'a . 

[*33-4] D . maxp'a ~ e D'P - 

[*93-103] D . maxp'a e B'P . 

[*202-52] D . maxp'a = B'P Oh. Prop 

*25013. l-:PeXl-t'A.D.E!5'P 

Dem. 

h . *33-24 . D I- : Hp . D . a ! (7'P . 

[*250121] D . E ! minp'C'P . 

[*20512] D.E!5*P:Dh.Prop 

*250131. l-:.Pen.D:a!P. = .E!JS'P 

Dem. 

h . *93102 . *33-24 . 3 h : E ! B'P . 3 . g ! P (1) 

h.(l).*250-13.DI-.Prop 

*25014. hiPeBord.D.Rl'PCBord 

Dem. 
h . *250-l . *205-26 . D 

h : P 6 Bord . Q G P . D . CI ex'G'P C a'minp . miup [ CI ex'O'Q C min<, . (1) 
[*60-42.*35-64] D . CI ex'O'Q C CI ex'O'P . a'minp n CI ex'O'Q C a'mine (2) 
I- . (1) . (2) . *22-44-621 . D h : P e Bord . Q G P . D . CI ex'O'Q C Cl'min^ . 
[*2501] D.Qe Bord Oh. Prop 

*250141. hiPefl.D.P^aefl [*250-14 . *204-4] 

*250142. h : P 6 Bord . D . Rl'P n connex C fl 
Dem. 

h . *250'14 . D h : Hp . D . Rl'P n connex C Bord n connex 
[*250113] CnOh.Prop 



10 SERIES [PABT V 

*25015. l-:Pen.E!5'P.D.PeDed 

Dem. 

I- .*250-101 . D h :. Hp . D : a ! a A O'P. D. . a ! minp'a (1) 

h.*206-14. DI-:.Hp.D:anO'P = A.D..a!precp'« (2) 

— » -♦ 

I- . (1) . (2) . D I- : Hp . 3 . (a) . a ! (minp'a w precp'a) . 

[*2141] D.PeDed. 

[*214-14] D . P 6 Ded : D h . Prop 

*250151. V-.PeH. xeOT . D . P ^ P^^'a; eDed 
Dem. 

h.*250-141.Dh:Hp.D.P^P*'a;6n (1) 

I- . *205-41 . D h : Hp . D . 5'Cnv'(P l P^'x) = vaaxp'P^'x 

[*205-197] =t'«. 

[*53-3] D . E ! 5'Cnv'(P l ^'«) (2) 

h.(l).(2).*250-15.Dh,Prop 

*250152. h.nC semi Ded [*214-7 . *2o0-124] 

*25016. f-:Pefl.a!a"0'P.3. P'minp'a = p'P"(a n G'P) 
[*205-65 . *250-121] 

*25017. h :. P,Q 6 fl - I'A . D : P smor Q . = . P t Q'P smor Q ^Q'Q 
[*204-47.*250-13] 

This proposition is useful in connection with the series of segmental 
relations in a well-ordered series, for the series of proper segmental relations 
in a well-ordered series is (as will be proved later) 

Pl'^'^Pia'P, 

and this is ordinally similar to P ^ Q'P. Hence, by the above proposition, 
two well-ordered series which are not null are ordinally similar when, and 
only when, the series of their segmental relations are ordinally similar. 

*250-2. I- : P e Bord . D . D'P = D'(P-^P^) 
Dem. 

l-.*33-4. Dh:a!6D'P. = .a!P'a; (1) 

I- . *2.501 . *20516 . D h :. P 6 Bord . 3 : a ! P'« • = • a ! mfnp'P'a; . 
[*205-251] =.xe D'(P^P^) (2) 

h . (1) . (2) . D h . Prop 



SECTION D] elementary PROPERTIES OP WELL-ORDERED SERIES 11 

*250-21. h:Pen.D.D'P = D'Pi [*201-63 . *250-2] 

In virtue of this proposition, every term of a well-ordered series (except 
the last, if any) has an immediate successor. 

*250-22. h : Pe Ser n Ded . D'P = D'Pj . 3 . Pefl- t'A 
Dem. 

I- . *214101 . D h : Hp . ~ E ! maxp'a . D . E I seq^'o (1) 

h . *206-45 . D I- : Hp . maxp'a e D'P . D . E ! seqjs'maxp'a . 
[*206-46] D . E ! seqp'a (2) 

I- . (1) . (2) . D h :. Hp . D : ~ (maxp'a = B'P) . D» . E ! seqp'o : 
[*93-118] D : ~ {B'P e a) . D. . E ! seqp'a : 

[*202-511.*214-5] D : a ! p'P"{a. n G'P) . D, . E ! seqp'o : 

[*250123] DiPefl-i'A:. Dh.Prop 

*250-23. h : Pefl .E!B'P.= .PeSern Ded .D'P = D'Pi 
Dem. 

h . *250-22 .*214-5 . D h : P e Ser n Ded . D'P = D'P^ . D . P e II . E ! fi'P (1) 

h . *25015-21 . DI-:P6n.E!£'P.D.P6SerftDed.D'P = D'Pi (2) 

h.(l).(2).Dh.Prop 

*250-24 h:Pen.D.P»|P, = PDD'P 
Bern. 
h . *2011 . *1312 . D h : . Hp . xP^z . D : yPx . D . yP'^ : y = a; . D . yP^z : 
[Transp] 3 : ~ (yP^z) . D . ~ (yPa;) -y^x: 

■ [*201-63.*202103] DzyP^z.-^ .xPy (1) 

l-.(l).*201-63. Dy:B.^.xP''z.zP^y.D.xPy.x,yeT>'P (2) 

h . *250-21. D I- : Hp . x,yeD'P . xPy . D . (g^r) . yP^z . 

[*201-63] D . (a^) . 2/P^ . zhy • 

[*341] D.a;(P''|P,)y (3). 

h . (2) . (3) . D h . Prop 

*250-241. l-:P6i2.D.P,|P== (d'Pi) 1 P [Proof as in *250-24] 

*250-242. l-:P6n.D.P = P.c;P,|P 

Dem. 
h . *201-63 . D h :: Hp . D :. xPy . = : xP^y . v . xP'y : 
[*250-21] = : xPiy . v . (g^r) . xP^z . xP'y : 

[*250-241] =:xPry.v. (gi) .xP^z. zPy : : D h . Prop 



12 SERIES [part V 

*250-243. \-:Pen,.D.Pt d'Pi = (CI'Pi) 1 (Pi *a P i -Pi) 
[Proof as in *250-242] 

The following propositions deal with the extended form of mathematical 
induction which is characteristic of well-ordered series. 

*250-3. H : . P 6 Bord : o C O'P n a . D. . seq^'a CaiD.G'PCa 
Dem, 

h . *250-101 . D h : P e Bord . g ! C'P - o- . D . g ! mmp'((7'P - a) . 

[*205-14] D . (a*) .xeC'P-a.P'xCff. 

[*206-4.*250104] D . (ga;) .x eC'P -a . P'xCa- .a; aeqp (P'o;) . 

-» — > 

[*13-195] D . (ga;, a) . a = P'ar . a C C'P n o- . a; e seqp'a - a . 

[*10-24] 3 , (ga) . o C C'P n o- . g ! seqp'a - a (1) 

h . (1) . Transp . D J- . Prop 

«250-301. h : P 6 connex . ~ g ! minp'r . <r = C'P - P"t . a C <r . D . seqp'a C a 
Dem. 

h . *205122 . *202-501 . 3 h : Hp . D . <r C^'P"t . 

[*40-6r] D.tC^'P"o- (1) 

F . *206-134 . D I- : Hp . a;seqp a . D . P'a; C -p'P"a 

[*4016] C-2>'P"(j- 

[(1)] C-T. 

[*37-462] D.a!~6P"T. 

[*20618.Hp] D.a;eo-:DI-.Prop 

«250-31. l-::Pe connexs. a C O'Pno-.D.. seqp'a Co-O,. C'P Co-:.D. Pen 
Bern. 

f- . *250-301 . D 

— » « 

h :. P e connex . g ! C'P n t . ~ g ! minp'r . a = C'P - P"t - D : 

a C o- . Da . seqp'a C o- : g ! C'P - o- (1) 

h . (1) . *10-28 . D 

h :. P 6 connex : (gr) . g ! C'P n t . ~ g ! minp'r : D : 

— » 
(go-) : a C o- . 3. . seqp'a C o- : g ! C'P - a (2) 

h . (2) . Transp . D 

I- :: P € connex :. a C <r - D. . seqp'a C o- : D„ . C'P C o- :. D : 

g ! C'P n r . Dt . g ! minp'r : 
[*250101] 0:Pe Bord (3) 

F.(3).*250113.Dh.Prop 



SECTION D] elementary PROPEBTIES OF WELL-OEDERED SERIES 13 

*250-32. I- ::. P e connex .'^•.-.Pe Bord . = :. 

a CK'P n o- . D. . seqp'a C o- : D, . C"P C <r [*250-3-31] 

*250-33. h . fl = connex n P {a C C'P r» o- . 3. . seqp'a C o- : D, . C'P C o- j 
[*250-32113] 

*250-34. h:.PeBord:a!€(7'P.P'a;C(T.D«.a;eo-: D.O'PCff 
Dem. 

V . *25011 .31-: P eBord . g ! C'P- o- . D . g ! iimip'(C'P - o-) . 

[*205-14] D.(a«).a;eO'P-<r.P'a!C«j- (1) 

h . (1) . Transp . D h . Prop 

*250a41. 1- : : a; e (7'P . P'ic C ff . D« . a; 6 (7 : 3, . C'P C o- : . D . P 6 Bord 
Dem. 
I-.*205'122.*37-462.D 
l-:a!a'PnT.~a'.minp'T.<T = G'P-P"T.a;e(7'P.P'«C<7.D. 

a!,>.6P"7-.a!C"P-o-. 
[Hp] D,a;€o-.a!(7'P-o- (1) 

h . (1) . *10-28 . D h :. (gr) , g ! G'P n t . ~a ! minp'r . D : 

(aff):a;eC"P.P'a:C<r.D«.a!eo-:a!C"P-ff (2) 
h . (2) . Transp . 3 h :. Hp . D : g ! G'P n t . 3, . g ! minp'r : 
[*250101] 3 : P 6 Bord : . D I- . Prop 

*250-35. V . Bord = P {a; e O'P . P'a; C o- . D^ . a; e o- : 3, . O'P C o-} 
[*250-34-341] 

*250-36. h :.P6n:\C«7.g!XnC"P.D^.seqp'\Co-:D.P"o-Ca 
Z)em. 

h .*250121 . D I- : Pefl .g ! P"o--o-. D . E ! mmp\P"a-a) (1) 

l-.*20514.*37-46.D 

h : a; = minp'(P"(r - ff) . 3 . g ! o- A P'a; . P'a; rt (P"o- - 0-) = A . 

[*24-31 1] D . g ! o- A P'a! . P'a; - <7 C - P"(r (2) 

h.(2).*202-501.D 

h : P e Ser . a; = minp'(P"o- - o-) . D . g ! o- a P'a; . P'a; - o- C^'P"((r a C'P) . 

[*4016] p . g ! ff A P'a; . P'a; - o- Zp'P"(a a ^a;) . 

[*40-61] D . P'a; - <r C P"(<r a P'a;) (3) 

I- . (3) . D h : Hp (3) , D . P'a; C (o- A "P'o;) w P"(ff aP'o;) . 



14 SERIES [PABT V 

-♦ 
[*206-l71] D . a; = seqp'(o- « P'oo) . 

-* —* — » — » 

[(2)] D . a ! o- ft P'x .<Tr\P'xC(T.~ {seqp'(o- rt P'x) C <t] - 

[*10-24] D.(a\).\Co-.a!\AO'P.~(seqp'\Co-) (4) 

h . (4) . Transp . D I- : Hp . D . ~ E ! minp'(P"o- - a) . 

[(l).Transp] D . P"a - o- = A : D h . Prop 

*250-361. I- :. P 6 n . Pj"o- C<r:\Co-.a!(\n (7'P) . D^ . limaxp'X C <r : D . 

P"(7C<r 
Dem. 

-» «- 

h . *206-46-43 .Dh:Hp.\Co-.E! maxp'X . D . seqp'X = P,'maxp'X . 

— » 
[Hp] D . seqp'X C a (1) 

— > — > 

I- .*207-4.D V : Hp. \C o-.g! (X « C'P). ~ E ! maxp'X .D.seqp'X = limaxp'X. 

— » 
[Hp] D.seqp'XCo- (2) 

h . (1) . (2) . D h :. Hp . D : X C ff . a ! (X A C'P) . Da • seqp'X C a : 

[*250-36] D : P"o- C o- : . D h . Prop 

*250-362. I- :. P e n . Pi"<r Co-iXCo-.glXft C'P . D^ . liminp'X C o- : D . 

P"o- C o- 

r*250-361 p . *121-26l 

*250-4 h . A e n 

Dem. 

h.*60-33. DI-.Clex'O'ACa'min(A) (1) 

h . (1) . *2.501 . D h . A e Bord (2) 

h.(2).*204-24.Dh.Prop 

*250-41. \-:x:^y.'2.xlyen 
Dem. 
h.*60-39. ^\-.Clex'C'{xly) = i'i'!c\Ji'i'y^Ji'{i'xsji'y) (1) 

h . *20518 . D h : Hp . P = a; ^ y . D . minp't'a; = x . miap'i'y = y (2) 
h . *205-181 . D h : Hp (2) . D . minp'(t'a: w t'y) = x (3) 

h . (1) . (2) . (3) . D h : Hp (2) . D . CI ex'G'(x iy)C a'taiup . 
[*2501] D.a;4yeBord (4) 

I- . (4) . *204-25 . 3 h . Prop • 



SECTION D] elementary PROPERTIES OF WELL-ORDERED SERIES 



15 



*250-42. 
Dem. 



I- : P 6 n - I'A . D . E ! 2p . 2p=P,'B'P. P'2p=i'B'P . P I P'2p=k 

(1) 



V . *12113 . D f- : « = 2p . = . «= P^'B'P 

V . *250-13 . D h : Hp . D . E ! B'P . 
[*250-21 .*2047] D . E ! Pi'5'P 

h . (1) . (2) . D h : Hp . D . E ! 2p . 2p = P/JB'P 
[*204-71] D.P'2p = t'£'P 

[*200-35] D . P p P'2p = A 

I- . (3) . (4) . (5) . D h . Prop 



(2) 
(3) 
(4) 

(5) 



«250-43. 
Dem. 



h.Or = flnC"0 



h.*56-104.Dl-:PeO,.s.P = A. 



«250-44. 
Dem. 



[*250-4.*33-241] 
[*71-37.*54-l] 

|-.2, = OnO"2 



= .Pea.C'P=A. 

= . Pell r.O"0:Dh. Prop 



l-.*56-11.3h:.P62, 
[*250-41] 
[*56-ll-38] 
[*20414] 



(•S^ie,y).x=^y.P=xly: 
PeD.:{'^x,y).x^y.P = xiy: 
Pen,nC"2.PriP = Ai 
P6QftC'"2:.DI-.Prop 

*250-5. h : P e n . D . minp p 01 ex'O'P e e^'Cl ex'O'P . 

I'O'P = Prod'Cl ex'CP [*205-33 . *250-l . *115-17], 
This proposition is of great importance, since it gives the existence- 
theorem for selections from any class of existent classes whose sum can be 
well-ordered (cf. *250"53, below). Observe that " ae CD, " means " a is a 
class which can be well-ordered." 

*250-51. l-:aeO"O.D.a!6A'Clex'a [*250-5] 

*250-52. h:a6C"n./8Ca.D.a!€4'Clex'/3 [*88-22-2 . *250-51] 
h : s'/e 6 G"n . A ~ 6 « . D . a ! es'ic 



«250-53. 

Dem. 



«250 54. 
Dem. 



h . *60-23-57 . D h : Hp . D . /e C 01 ex's'/e . 
[*88-22.*250-51] D . g ! 6a'k : D I- . Prop 

I- : Cn w 1 = Cls . D . Multax 

h . *25063 . *83-4 . D I- :. Hp . 3 : A ~ e « . D, . g ! e^'* 
[*88-37] D : Mult ax :.DI-. Prop 



16 SERIES [part V 

The above proposition states that if every class which is not a unit class 
is the field of some well-ordered series, then the multiplicative axiom holds. 
The converse of this proposition has been proved by Zermelo (cf. *25847). 

*250-6. hiP.Qefl.PsmorQ.D.PslnOTQel [*208-41 . *250121] 

This proposition is very useful, since it enables us, when two similar 
series of similar well-ordered series are given, to pick out the correlators of 
all the pairs without assuming the multiplicative axiom. I.e. given 
P,QeRel^excl./SePsinorQ./SGsmor, if NeG'Q, the correlator of S'N 

and N will be i'(S'N) smor If if S'N.NeD,. This enables us to dispense 
with the multiplicative axiom in the hypotheses of *164*44 and its con- 
sequences, whenever the relations concerned have fields whose members are 
well-ordered series. 

*250-61. h : P e n . D . P smor P = t'(/ [ G'P) [*208-42] 

*250-62. l-:PeBord.Secror'P.D.->..(a[a;).(S'ar)Pa; [*208-43] 

*25063. l-:PennCnv"n.D.Rl'PftNr'P = t'P [*208-45] 

This proposition will be useful in showing that a finite series is not 
similar to any proper part of itself, and is a series which is well-ordered and 
has a converse which is also well-ordered. 

*250-64. h:PeBord.iS6cror'P.D.a'PA;)'P"D'S=A [*208-46] 

In virtue of this proposition, a part of a well-ordered series can only be 
similar to the whole if the part extends to the end of the series. Thus e.g. 
no proper section of a well-ordered series can be similar to the whole. 

*250-65. h : P e ft . a e sect'P - I'G'P . /3 C a . D . ~ {P smor P ^ /3} 
Dem. 

V . *4016 . D 1- : Hp . D . p'P"C'(P I a) C p'*P"G'{P ^ /3) (1) 

1- .*211-133 . D I- : Hp . a~e 1 . D . a= (7'(PDa) . 
[*211-703] D . a ! p'P"G'(P I a) . 

[(1)] D.a!i>'P"(7'(-PD/8) (2) 

I- .(2).*40-6-62 .DI-:Hp.a~el.a!P.D.a! G'P f^ p'P"G'(P I ^) . 

[*208-47] :^.r^{P smor (P 1 0)} (3) 

I- .*211-1 . *24-13 . D h :P = A . D . sect'P - t'G'P = A (4) 

h . (4) . Transp . 3 h : Hp . D . g ! P (5) 

h . *200-36 . *250-104 . D h : Hp . g ! P . a e 1 . D . ~ {P smor (P I /3)} (6) 
h . (3) . (5) . (6) . D 1- . Prop 

#250651. h : P e n . D . Nr'P a P ^"(sect'P - I'C'P) = A [*250-65] 



SECTION D] elementary PROPERTIES OF WELL-ORDERED SERIES 17 

*250-652. I- : P e Bord . QQP .'S^IC'F n p'*P"G'Q . D . ~ (P smor Q) 
[*208-47] • 

*250-653. h : P 6 Bord .•g^lCFn p'F"(oL n C'P) .D.^{P smor P ^ a) 

Dem. 

h . *37-41 . D h . G'(P ^ a) C a n G'P . 

[*40-16] D h .p'P"(a n C'P) C p'P"(7'(P I a) (1) 

h , (1) . D h : Hp . D . a ! C'P r. p^"C'(P I a) . 

[*250-652] :).r^{P smor (P p a)} : D f- . Prop 

*250-66. l-:Pef2.aesect'P.Psinor(Pta).D.«=C'P [*2.50-65 . Traasp] 

*250-67. h : P 6 n . a; e C'P . D . ~ {P smor (P tP'so)} 
Dem. 

\- . *211-302 , D h : Hp . D . P'a;esect'P (1) 

l-.*200-52. Df-:Hp.D.P'a;=|=C'P (2) 

F . (1) . (2) . *2o0-65 . D h . Prop 

*250-7. bi.Pen.^ixeG'P.D^.P [^ P^'aJeO s PeSer 
Bern. 

h. *250-141. 3 f-:.P 6 fl.D:a;e C'P. D^.PpP^'ajefl (1) 

I- . *250-121 . D 

I- : .- a; 6 C'P . D:, . P p P^'x e fi : = : a; e C'P . g ! a n C'(P t P*'a.') . D:», . ■ 

E!miii(PtP*'a;)'a: 
i;*202o5] D : a; 6 Q'P r> a . D^,, . E ! mia (P ^ P*'a;)'a : 
[*205-27] D^,„ . E ! minp'a : 

1*10-23] D:a!a'P/^«.D. .Eiminp'a (2) 

i- . *20518 . *202-52 . D F : P e Ser . a = £'P . 3 . E ! minp'a (3) 

1- . (2) . (3) . D h : . a; e C'P . D^ . P P P^'a; e li : P e Ser : D : 

a ! a n C'P . Da . E ! minp'a : 
[*250-121] :i:PeD, (4) 

h . (1) . (4) . D 1- . Prop 

This proposition is used in proving that the series of ordinals in order of 
magnitude is well-ordered (*256'3). We prove first that if P e £2, the 
ordinals up to and including Nr'P are well-ordered; thence, by the above 
proposition, it follows that the whole series of ordinals is well-ordered. 



R. & W. III. 



*251. ORDINAL NUMBERS. 

Summary of *251. 

The name " ordinal numbers " is commonly confined to the relation- 
numbers of well-ordered series, and will be so confined in what follows. The 
relation-numbers of series in general are commonly called "order-types*." 
Thus a is an order-type if o e Nr"Ser, and a is an ordinal number if a e Nr'Tl. 
In the present number we shall be concerned with a few of the simpler 
properties of ordinal numbers and^ofJ.he sums, products, and powers of well- 
ordered series. 

We put NO = Nr"n Df, 

where " NO " stands for " ordinal number." 

We prove in this number that any relation similar to a well-ordered 
relation is well-ordered (*251"11), and therefore any relation similar to a 
well-ordered series is a well-ordered series (*2ol"lll). We prove 

*251132142. l-:aeN0. = .a-i-leN0.= .l-f-a6N0 

*2511516. l-.0„2,eNO 

*25r24. f-:a,;8eN0.D.a + /86N0 

We prove that if P is a well-ordered series of mutually exclusive well- 
ordered series, 2'P is a well-ordered series (*251'21) ; that if P is a well- 
ordered series of series, Il'P is a series (*251'3) ; that if P is a series and Q 
is a well-ordered series, P'^ and P exp Q are series (*2ol'42) ; that if P, Q are 
well-ordered series, so is P x Q (*251'55), and therefore the product of two 
ordinal numbers is an ordinal number (*251o6). 

In virtue of the uniqueness of the correlator of two well-ordered series, 
we have 

*251-61. 1- : . P, Q e Rel^ excl . (7'P C O . D : 

a ! (P smor Q) n Rl'smor . = . P smor smor Q 
whence, without assuming the multiplicative axiom, 

♦ We shall also speak of them as " serial numbers." 



SECTION D] ordinal NUMBERS 19 

*251-621. h : O'P C X2 . a ! (Pimof Q) n El'smor . D , 

• SNr'P=SNr'Q.nNr'P=nNr'Q 

*251-65. hraeNO-i'A.ySeNR.Pe^.C'PCa.D. 

SNr'P = /3 X a . HNr'P = a exp, /3 

Finally, we have propositions (*251'7'7l) showing that the esfistence of an 
existent II in any type is equivalent to the existence of 2^ in that type, and 
therefore holds for every type of homogeneous relations, except (possibly, so 
far as our primitive propositions can show) in the type of relations of 
individuals to individuals. 



*25101. NO = Nr"n Df 

*25ri. h:aeNO.H.(aP).Pen.a = Nr'P [(*25101)] . . 

*25111. |-:PeBord.PsmorQ.D.QeBord 

Bern. ', 

H . *205-8 . *2501 . *37-431 . D 

h :. P e Bord .SeP smor Q . D : a C C'P . a ! a . D. . g ! minQ'^"a : 
[*37-63-431] D : ;8 e ;S'"01 ex'C'P . a ! /3 . D^ . a ! min^'^ : 

[*71-491] D : yS 6 CI ex'B"G'P . 3^ . a ! mrii<j'/3 : 

[*151-ll-13i.*37-25] D : /3 e CI ex'O'Q . Dp . a ! mniQ'^g : 

[*250-l] D : Q € Bord : . D h . Prop 

*251-111. h : P 6 a . P smor Q.D.QeCl [*25111 . *204-21] 

*251-12. h : P e Bord . D . Nr'P C Bord [*2ol-ll] 

*251121. h : P 6 ft . D . Nr'P C O [*251111] 

*251122. hiaeNO.D.aCn [*251121-1] 

*251-13. l-:PeBord.^~eO'P.= .P-t*^6Bord 
Dem. 

h. *205-83. *250-l.Dl-:Hp. a !C"P A a. D. a !min(P-h>0)'a (1) 

l-.*205-831. Dh:Hp.O'(P-f*^)na=t'^.D.a!mfn(P-t*^)'a (2) 

1-.*161-14. DI-:.Hp.a!C"(P-+*0)na.D: 

a ! C"P n a . V . O'P na=A.'3^lt'zna: 
[*161-14] D:a!C"Pna.v.(7'(P-f*.^)na = t'^ (3) 

l-.(l).(2).(3).D _^ 

1- :. Hp . D : a ! C/'(P-b^) « a • 3a . a ! min (P-f> zYa (4) 

f- . (4) . *2o0101 . 3h:PfiBord.«~ea'P.3.P-h>^eBord (5) 

h.*25014104.*200-41.Dh:P-b^eBord.3.P€Bord.a^~ea'P (6) 

h. (5). (6). 3 h. Prop 

2—2 



20 SERIES [part V 

*251-131. l-:Pen.0~eC'P. = .P-f*0ef2 [*204-51 . *251-13] 
*251132. l-iaeNO.s.a+ieNO 

Dem. 

h.*251-lll.*181-12.Df-:P6n.=.4,A^H;Pen. 

[*18111.(*18r01).*251-131] =.P4»a;en. 

[*181-3.*251-1] = . Nr'P+ 1 e NO (1) 

h . (1) . *2511 . D h . Prop 

*251-14. V'.Pe Bord . ^ ~ e C'P . = . ^ «f P e Bord 

Dem. 

l-.*20o-832.*161-12.D 

— » -* 

\- :. Hp . D : 2 ~ e a . D . min {z «f P)'a = minp'a : 

[*250-101] D : a ! (a n C'P) . « ~ e a . D . g ! mia (^ <f P)'a (1) 

h.*205-833.*161-12.D 

h:Hp.06a.a:!P.D.a!min(0«f P)'a (2) 

h.(l).(2).D ^ 

I- : . Hp . a ! P . D : a ! a n C"(^ <^- P) . D. . a ! min (^ «f P)'a : 
[*250-101] D : «f P 6 Bord (3) 

h . *161-201 . *250-4 . D h :P = A . D.a*f PeBord (4) 

h.(3).(4). Dh:PeBord.^~6C'P.D.^*fPeBord (5) 

h . *250-14-104 . *200-41 . D h : ^ *(- P e Bord ."^.Pe Bord .z^eG'P (6) 
i- . (5) . (6) . D h . Prop 
*251141. h:PeD,.Z'^eC'P. = .z<]-Pen [*204-51 . *251-14] 

*251142. h:oeN0. = .l-t-a6N0 [Proof as in *251-132] 

*25115. h.O^eNO [*250-4.*15311] 

*25116. h . 2, e NO [*250-41 . *153-211] 

*25ri7. \-:x=^y.a!=^z.yJi=z.D.a;ly-{^zen [*251-131 . *250-41] 
*251171. l-.2, + ieN0 [*251-16-132] 

*251-2. I- : P e Rel^ excl n Bord . C'P C Bord .■:i.X'Pe Bord 
Dem. 

h . *162-23 . D I- : a ! a « 0'2'P . D . a ! « '^ J?'"a'P . 

[*37-264] D.'^lC'Pn F"a (1) 

h . *37-46 . *33-5 . D h : Q 6P"a . D . a ! « n C^'Q (2) 

h.(l).(2).*250-101.D 

I- :. Hp . D : a ! a '^ C'X'P . D . (aQ) ■ Q minpP"a . a ! rdn^'a . 
[*205-85] D . a ! min (2'P)'a , (3) 

I- . (3) . *250-101 . D h . Prop 



SECTION d] ordinal NUMBERS 21 

*251-21. f-:PeRePexclnn.O'PCn.D.S'Pen [*204-52 . *251-2] 

*251211. h : Nr'P e NO . Nr"0'P C NO . D . 2 Nr'P e NO 
Dem. 
V.*18216-162. DI-:Hp.D. Nr'TjPeNO-IjPeRePexcI (1) 

h . *1820511 . *151-65 . D h : Hp . D . Nr"C" J JP C NO (2) 

f . (1) . (2) . *251-122 . h : Hp . D . J JP eRel^ excl n fl . C J 'PC Q . 

[*251-21] D.t'l'PeD,. 

[*251-1.(*18301)] D . t Nr'P e NO : D h . Prop 

*251-22. h:P,Qe Bord .C'P n 0'Q= A.O . P^Q € Bord 

Dem. 
h . *162-3 . *163-42 . 3 h : Hp . ~ (P - A . Q = A) . D . 

PlQe Bord . C"(P J, Q) C Bord . P J, Q e ReP excl . 

[*251-2] D.P^LQeBord (1) 

h . *160-21 . *250-4 .Dh:P = A.Q=A.D. P^Q e Bord (2) 

h . (1) . (2) . D h . Prop 

*251-23. \-:P,QeD,.G'PnG'Q = A.D.P^Qe£l [*204-5 . *251-22] 

*251-24. l-:a,j8eNO.D.a+/3 6NO 

Dem. 

h . *251-111 . *180-12-11 . 3 

h : P.^eft . D . 4, (A n G'Q)H'P e O . (A n 0'P)4, UJQe H . 

(7' 4, (A r. G'QYh'P n G'iA n G'P) i H'Q = A . 

[*251-23.(*181-01)] D .P + QeD,. 

[*180-3.*251-1] D . Nr'P -i- Nr'Q e NO (1) 

f . (1) . *251-1 . D f- . Prop 

*251'25. h : P^Qeil . = .P,Q eil . G'P nG'Q = A 

Dem. 
|-.*204-5. Oh:P^Qen.D.P,QeBer.G'PnG'Q = A (1) 

i- . (1) .*205-84 . D H :. P^QeQ,. D : g ! G'Pn a . Da . a ! minp'a : 
[*250-ll] DrPeBord (2) 

h .(1) . *205-841 . D f :. P4.Q e n . D : 

3 ! a - O'P n 0'(P4^Q) . D, . g ! mfnQ'(a - O'P) : 



22 SERIES [part V 

[*160-14.(1)] D : a ! a n C'Q . D. . a ! r^ng'Ca - G'P) . 

[*205-15.(l)] Da . a ! miDQ'a : 

[*250101] D:QeBord . (3) 

l-.(l).(2).(3).Dh:P4iQen.D.P,Qen.C"PnC"Q = A (4) 

h . (4) . *251-23 . 3 h . Prop 

*251-26. h:a,/3eNO-i'A. = .a+/36NO-t'A [*251-25] 

*2513. l-:Pefl.(7'PCSer.D.n'P6Ser [*204-57 . *250-l] 

*251-31. I- : E !! B"G'P .-^.B^CPe Fi,'C'P 
Dem. 

h .*71-571 . D h : Hp . D .P r CPel -»Cls . a'(S [^ (7'P) = O'P (1) 
l-.*93-103.DI-.PGP (2) 

h . (1) . (2) . *80-14 . D h . Prop 

*251-32. h : E !! B"G'P .±IP .0 . B[C'P = B'U'P 

Bern. 

h . *ir2-162 . D h : Hp . D . P'H'P = B^'C'P 

[*82-21] = i'{B [ C'P) : D h . Prop 

*25i-33. h:C"Pcn-i'A.a!P.D.a[!n'P.ppO'P=P'n'P 

[*25013.*251-32] 

*251-34. h : P 6 ReP excl . C"P C fl - I'A . D . a ! e^'C'G'P 
Bern. 

\- . *251-33 . *173-16 .Df-:Hp.a!P-3.a! Prod'P .. 
[*173-161] D . a ! Prod'C'CP . 

[*1151] D.a!6A'0"0'P (1) 

h . *8315 . D t- : P= A . D . a ! e^'C'G'P (2) 

I- . (1) . (2) . D h . Prop 

*251-35. l-::P6n.D:. 

aP,il3.= :tt,^eGVG'P:(-^z).zea-0.afv'p'z=.0nr^'z 
Dem. 

I-.*170-2.D 

f- :. a, y8 e GVG'P : (a^) .zea-^.an^'z =^ n P'^ : D . aP„,/3 (1) 

|-.*170-231.*250121.D 

f- :: Hp . D :. aPeijS . D : a,/3eCl'C"P : (a^) .zea-^ .an'P'z = fin'F'z. (2) 

1- . (1) . (2) . Dh . Prop 



SECTION; D] ordinal NUMBERS 23 

*251-351. h :: P eX2 . 3 :. oPje/S . = : 

a, /3 eCl'O'P : (■^z).ze 13- a.an^'z = ^n*P'z [*251-35 . *1 70101] 

*251-36. l-:P6Xl.D.P„6Ser 

Dem. 
|-.*l7017.DI-.P,iGJ (1) 

1- . *251-35 . D h :: Hp . D :. aP„i/3 . jSP^iy . D : 

('^z,w).Z6a-^.W€0-y.anP'z = ^f\P'2.^nP'w = yr^P'w (2) 

h . *20114 . D 

— » -^ -* -» 

h:. Hp.zea-;S.we;S-7.anP'^:=;SnP'^.y8nP'w = 7nP'w.D: 

— > — > 

2^Pw .0 . zea — y .an P'z = y n Pfz (3) 

h. *201-14. D h :. Hp (3). DiwP^.D. we a- 7. a nP'w = 7 nP'w (4) 

h . (2) . (3) . (4) . *202-104. #251-35. D l-:.Hp.D:aP„,/3 . /3P,iy. D . aPdV (5) 
h . *250-121 . D 
1- : Hp . a, /8 6 Cl'G'P . a =|=/3 . D . (3^) . = minp'{(« - ^8) u (^ - a)} . 

[*205-14] D . (a^) .Z6{(oL-0)vj(l3-a)}.ttnP'z = l3nP'z. 

[*2ol-35] D.a(P„ic;P<,);8 (6) 

h . (1) ..(5) . (6) . D h . Prop 

*251-361. h : P 6 n . D . Pi„ 6 Ser [*251-36 . *170-101] 

*25r37. h:P6fl.D,P„i = Pdf [*251-35 . *171-2] 

*251-371. h : P 6 Xi . D . P,c = Ph 

*251-4. h : P 6 Rel= arithm r. Bord . C'P C Bord . G't'P C Bord . D . 

X'S'P 6 Bord 
Dem. 

h , *251-2 . D : Hp . D . S'P e Rel^ excl n Bord . C't'P C Bord . 
' [*251-2] D . S'S'P 6 Bord : D h . Prop 

*251-41. t- : P e ReP arithm « £1 . C'P C fl . O'S'P C Xi . D . 1,'X'P e il 
[*204-54,*251-4] 

*251-42. h:PeSer.Qefl.3.P«(.PexpQ)6Ser [*204-59 . *250-l] 

*251-43. I- : a 6 NR . a C Ser . /3 e NO . D . (a exp^/3) e NR . (a exp^^S) C Ser 
[*186a3 . *251-42] 

*251-44. h : a e NO - t'Or . /8 6 NO - t'O^ . 3 . a exp,/? + 0^ 
Dem. 

t-.*165-27.D 
l-:Hp.Pea.Qe/3.D.P4,5Q6a-l'A.C"Pjt^;QCn-i'A. 

[*25r33.*1761] D.a[!(PexpQ) (1) 

t- . (1) . *186-13 . D h . Prop 



24 SERIES [part V 

*251-5. h-.'^lP.QeBoTd.D.Pl'QeBovd [*165-25 .*25M1] 

*251-51. [■z'^lP.Qen.D.Pl'QeD, [*165-25 . *204-21 . *251-5] 

*251-52. f-iPeBord.D.C'PiJQCBord [*16o-26 .*25112] 

•> 

*251-53. h-.Pen.D.G'Pi'Qen, [*16.5-26 . *204-22 . *25r52} 

"> 

*25r54. I- :P,Q 6 Bord.D.PxQeBord 

Dem. 

l-.*165-21.*251-.5-52.D 

h : Hp . rj ! Q . D . Q 1 5P 6 Rel'' excl n Bord . (7'Q 1 JP C Bord . 

[*251-2.*1661]D.PxQ6Bord (1) 

h . *166-13 . *250-4 .Dh:Q = A.D.PxQ6 Bord (2) 

h . (1) . (2) . D h . Prop 

*25r55. h:P, Qeli.D.PxQeO [*25r54 . *204-55] 

*251-56. l-:«,y86NO.D.aX/8eNO [*184-13 . *251-5.51] 

*251-6. I- : P, Q 6 Eel'' excl .O'PCn.SeP smof Q n Rl'smor . 

fi = X {(giV) .NeG'Q.X = (S'N) slSof iV} . D . 

i\' fie e^fi . s'i"fi e P smor smof Q 
Dem. 

I- . *250-6 . *251-in . D h : Hp . D . /i C 1 . 

[*83-43] D. tl'yiieeAV- (1) 

[*164-43] D . s'i"/i e P smof sniof Q (2) 

F . (1) . (2) . D f- . Prop 

*251-61. \-:.P,Qe ReP excl .C'PCD,.D: 

3 ! (P smor Q) n Rl'smor . = . P smor smor Q 
Bern. 

f- . *251"6 . D h : Hp . g; ! (P smor Q) n Rl'smor . D . P smor smor Q (1) 
h . (1) . *164-17 . D h . Prop 

*251-62. h : Hp *251-61 . a ! P smof Q n Rl'smor . D . 

S'P smor S'Q . H'P smor U'Q . 
SNr'P = SNr'Q . DNr'P = HNr'Q 



Bern. 



h . *1 64-151 . *251-61 . D h : Hp . D . S'Psmor 2'Q (1) 

t-.*l72-44.*251-61. D I- : Hp . D . n'PsmorH'Q (2) 

l-.(l).*18313. DI-:Hp.D.2Nr'P = SNr'Q (3) 

h . (2) . *1851 . Df-:Hp.D.nNr'P=nNr'Q (4) 

F . (1) . (2) . (3) . (4) . D h . Prop 



SECTION D] ordinal NUMBERS 25 

In the above proposition,. the hypothesis " P, QeRePexcl " is unnecessary 
for 2Nr'P = 2Nr'Q atd nNr'P = nNr'Q, as appears from *18311 and 
*18o'll. Thus we have 

*251-621. h : O'P C fl . a ! (P smof Q) n Rl'smor . D . 

S Nr'P = 2 Nr' (g . n Nr'P = H Nr'Q 
[*251-61 . *183-11 . *185-11] 

*251-63. t- : aeNO- t'A .ySeNR . P eRePexcl .Pe/S . C'P Ca.D. 

2'P6/3><a.2Nr'P = /8xa 
Dem. 
t-.*164-47.*165-2r-21.D 
h:Hp.Q6a.«=|=0,.D.QJ,;P6^.0'Q_i;PCa.P,QpP6RePexcl. 

[*164-47] D . a l(Q I ;P) sTSof P n Rl'smor . P, Q J, JP e ReP excl . 

[*251-61] D.(QJ,;P)smorsmorP. 

[*1 64-151. *166-1] D.(PxQ) smor 2'P . 

[*184-13] 0.t'Pej3xa (1) 

|-.(l).DI-:Hp.a+0^.D.S'P6y8xa (2) 

h . *162-42 . Transp . D h : Hp . a = 0, . D . 2'P = A . 

[*184-16] D.2'Pe/3xa (3) 

l-.(2).(3).DI-:Hp.D.2'P6j8)^a (4) 

[*183-13] D.2Nr'P = /3xa (5) 

h . (4) . (5) . D h . Prop 

*251-64. f- : Hp *251-63 . D . H'P e (a exp, y8) . U Nr'P = a exp, /3 
[Proof as in *251-63] 

*251-65. f- : a e NO - I'A . iS 6 NR . Pe /3 . G'PC a . D . 

2Nr'P = /3 X a . n Nr'P = a exp, /3 

h.*182-1^6.*183-231.D 

l-iHp.Qea.D. IJPeRePexcl. TjPeNr'P . C'iJPCNr'Q. (1) 

[*251-63] 3.2Nr'I;P = Nr'PxNr'Q. 

[*18314] D.2Nr'P = ?rr'P);cNr'Q 

[*152-45] =^xa (2) 

I- . (2) . *10-^3 . DI-:Hp.:3.2Nr'P = ;8xa (3) 

I- . (1) . *25r64 . D I- : Hp . Q,e « . D . nNr' i ;P = (Nr'Q) exp^ (Nr'P) . 

[*185-ri2] "^ D.nNr'P = (Nr'Q)exp,(Nr'P) 

[*152-45] ' =aexpr/3 (4) 

h . (4) . *10-23 . DI-:Hp.. D.nNr'P = aexpr/3 ' (5) 

h.(3>.(5)'.DI-.Prop 



26 SERIES [part V 

In virtue of the above proposition, the usual relations of addition to 
multiplication, and of multiplication to exponentiation, when the summands 
or the factors are all equal, can be established without the multiplicative 
axiom, provided the summands, or the factors, are ordinal numbers. 

*251-7. 1: : a ! n - t'A n f„„'a . = . a ! 2^ n ioo'a • = ■ a ! 2 n i'a . = . a ! 2„ 

Bern. 
|-.*64.-55. Dh:'3^ia.-L'Ar^too'a. = .{'^P).Peil-i'A.C'PCto'a (1) 
1- . *200-l 2.DI-:P6n-i'A.D. (a^;, y).x,y eC'P .x^y . 
[*153-201.*55-3] D . a ! 2,. n El'P (2) 

I- . (1) . (2) . D h : a '• ii - t'A n t^'a. . D . (aP) . C'P C t„'a . a ! 2,. r. Rl'P . 
[*33-26o] D . (aQ) .Qe2r.G'QC t,'a . 

[*64-55] D . a ! 2,. n t^'a. (3) 

V . *251-16-122 . D h : a ! 2^ n «„o'a . D . a ! n - I'A n «„„'a (4) 

l-.(3).(4). DI-:a!n-i'AnCa. = .a!2,nCa {5} 

h . *64-55 . D h : a ! 2,. n i„/a . = . (aa;, y).x^y .x,y e to'a . 

[*63-62] = , (a*', y).x^y .i'x\j i'y e t'a . 

[*54-26] = . a ! 2 r. «'« (6) 

h . (5) . (6) . (*65-01) . D F . Prop 

*251 71. h . a ! n - t'A n t^'Ch . a ! n - t'A n t^'^el 
[*251-7 . *101-42-43] 



*252. SEGMENTS OF WELL-ORDEKED SERIES. 

Summary of *252, 

The properties of sections and segments are greatly simplified in the case; 
of series which are well-ordered, owing to the fact that every proper section 

has a sequent, whence it follows that the class of proper sections is P"C'P ;. 
and this is also the class of proper segments. Hence also the series of proper 

sections or of proper segments is the series P'>P (*252"37). The series of all 

sections is P>P-{*G'P {*2o2S8); hence (*252-381) 

Nr'5'P5,5 = Nr'P + i. 

The most useful propositions in this number are (apart from the above) 

*25212. h-.Peil.D. 

sect'P - I'C'P = D'Pe - I'G'P ='P"G'P . sect'P = P"(?'P w I'C'P 

*25217. h : P e II - I'A . D . sect'P - I'A = "P^a'P w i'C'P 

*252171. 1- : P 6 n . D . sect'P - t'A - I'C'P = P"a'P 

*252-372. h :. P e n . D : s'P e n : E ! 5'P . D . Nr's'P = Nr'P : 

~ E ! £'P . D . Nr's'P = Nr'P + 1 

*252-4. hiPen.XCsect'P.glX.D.p'XeX 



*2521. l-:Pen.a6sect'P-i'C'P.D.E!s^qp'a [*2o0a24] -; 

*25211. I- : P e X2 . D . sect'P - I'C'P = sect'P n d'seq? 

Bern. 

h .,*206-18-2 . D h . O'P ~ 6 Q'seqp (1) 

h . (1) . *252-l . D 1- . Prop 



28 SERIES [part V 

*25212. h-.Pen.D. 

sect'P - i'G'P='D'Pe - I'G'P ='P"G'P . sect'P ='P"G'P o I'G'P 

Bern. 

f-.*211-24.*25211. DhiHp.aesect'P-i'C'P.D.aeD'Pe (1) 

h.*211-15. DF-jHp.aeD'Pe-i'C'P.D.aesect'P-i'O'P (2) 

l-.(l).(2). DI-:Hp. D. sect'P -i'a'P = D'Pe-i'G'P (3) 

V . *211-302 . *2o2-ll . D I- : Hp . D . sect'P - I'C'P ='P"G'P (4) 

h . (3) . (4) . *211-26 . D h . Prop 

In dealing with sections and segments of well-ordered series, it is necessary 

to distinguish series with a last term from such as have no last term. If 

a series has no last term, C'P = P"G'P, so that C'PeD'Pe. But if a 

— > 
series has a last term, (7'P~6D'Pc; in this case, D'Pe=P"C'P. Thus 

— » 
D'Pe is either P"C'P or sect'P, according as there is or is not a last 

term. In either case, 

sect'P = P"0'P u I'O'P, 
as has been already proved in *252'12. 

*25213. h : P 6 12 . E ! B'P . D . sect'P - I'G'P = D'Pe =^"G'P . 

sect'P = D'Pe u I'G'P = P"G'P w t'C'P 
Dem. 

h . *250-21 . *211-36 . D h : Hp . D . sect'P - D'Pe = I'G'P . 
[*24-492.*211-15] D . sect'P - I'C'P = D'Pe (1 ) 

[*252-12] =P"G'P (2) 

l-.(l).(2).*211-26.Dh.Prop 

*252-14 1- : P 6 n . ~ E ! B'P . D . sect'P = D'Pe = P"C'P u t'C'P 
[*250-21 . *211-;361 . *252-12] 

*25215. I- : P 6 n . D . D'Pe = P"D'P u I'D'P 
Bern. 

h . *25213 . D h : Hp . E ! B'P . D . D'Pe = P"D'P u I'P'B'P 

[*202o24] =P"D'P u I'D'P (1) 

h .*25214. D h : Hp.~E!5'P.D.D'Pe = P"D'P o I'D'P (2) 

l-.(l).(2).Dh.Prop 

*25216. F : P 6 n - 2^ . D . D'Pe = sect'(P ^ D'P) 
Dem. 
}■ . *204-27l . D 1- : Hp . D . D'P~ e 1 . 
[*202-55] D . G'(P I D'P) = D'P . 

[*250-141.*25212] D . sect'(P I D'P) = P^D'P"D'P u I'D'P 
[*37-42-421] = P"D'P u I'D'P 

[*252-15] = D'Pe: 31". Prop 



SECTION D] segments OF WELL-ORDERED SERIES 29 

*25217. l-:Pen-i'A.D.sect'P-t'A = P"a'Pwt'C'P .. 
Dem. 

h . *252-12 . D h : Hp . 3 . sect'P- t'A = (P"G'P- fc'A) u t,'C'P 

[*33-41] = P"(I'P u I'G'P : D I- . Prop 

*252171. h : P 6 a . D . sect'P - t'A - i'C'P = P"a'P 
Dem, 

1- . *252-12 . D I- : Hp . D . (sect'P - l'O'P) - I'A = lP"C'P - I'A 
[*33-41] = P"a'P : D h . Prop 

*252-3. h : P 6 n . D . D's'P* = P"a'P [*21217l . *2o2-12] 

*252-31. hzPeH.glP.D. C's'P* = ^"G'P u I'G'P 
[*212-172 . *252-12] 

*252-311. hzPen.glP.D. tt's'P* = P"a'P w t'C'P 
[*212-171 . *25217] 

*252-32. h : P e n . D . D's'P = P"D'P [*212132 . *25215] 

*25233. t- : P e n - t'A . D . C's'P = P"D'P u I'B'P 
[*212133 . *252-15] 

*252-34. h : P e O . E ! 5'P . D . G's'P = P"i7'P 
i>em. 

I- . *202-524 . D h : Hp . 3 . 'P'B'P = D'P - 

[*252-33J D . C's'P ='P"G'P : D h . Prop 

*252-35. h : P 6 O - t'A . ~ E ! 5'P . D . G'<i'P='P"G^P u t'C'P 
[*212-133 . *252-14] . 

*252-36. t-:P6n.E!P'P.D.s'P = P5P 
i)em. 

V . *212-25 , *252-34 . D h : Hp . D . P'^P = (s'P) D {G'<i'P) 
[*36-33] = 9'P : D F . Prop 

*25237. f- : P 6 fi . D . (s'P) t: (- t'C^'P) ='P'^P 
Bern. 

h . *36-3 . 3 h . (s'P) D (- t'G'P) = (s'P) p (O's'P - I'G'P) > 

[*212-133134] = (s'P) t (D'Pe - I'C'P) (1) 

1- . (1) . *252-12 . D h : Hp . D . (s'P) D (- I'G'P) - (s'P) I {P"C'P) 
[*212-25] =p;P:DF. Prop 



30 SERIES [PABT V 

*252-371. h : P e n . ~ E ! 5'P . 3 . s'P = P JP-f»C"P 

Dem. 
h.*212-25.*252-32. D h : Hp. D.P;P = (s'P)C(DVP) (1) 

h.*212133. DI-:Hp.a!P.D.C"P = £'Cnv's'P (2) 

|-.*252-32. DI-:Hp.D.DVP = P"C'P. 

[*20012.*204-34] D . D's'P ~ e 1 (3) 

h . (1) . (2) . (3) . *204-461 . D h : Hp . a ! P . D . P;P-t*G'P= s'P (4) 

h.*212-134.*161-2. Dh:Hp.P = A.D.s'P = A.P;P-t*C"P=A (5) 

h . (4) . (5) . D 1- . Prop 

*252-372. I- :. P 6 fl . D : s'P e n : E ! £'P . D . Nr's'P = Nr'P : 

~ E ! 5'P . D . Nr's'P = Nr'P + 1 
Dem. 

V . *2.52-36 . *204-35 . D h : Hp . E ! B'F . D . s'P smor P . 

[*251-111.*152-321] D.s'Pen.NrVP=Nr'P (1) 

1- . *252-37l . *204-35 . *200-52 . D 

I- : Hp . ~ E ! 5'P . D . Nr's'P = Nr'P + 1. (2) 

[*251132] D.s'Pefl (3) 

1- . (1) . (2) . (3) . D 1- . Prop 

*252 38. h : P e n . D . <i'P^ = P''P-\*G'P 

Dem. 
l-.*2o212.*212-24.D 

I- :: Hp . D :. a(s'P^)^ . = :a,pe'P"C'P u I'C'P . aC;8 . a + yS : 
[*37-6.*200-52] 

= : {■3_x, y) .x,y eC'P .a = P'x . ^^ P'y .'P'xC'P'y .'P'x^'P'y .yf . 
(a*) .xeG'P.a = P'x.^ = C'P: 
[*204-33-34] = : (ga;, jf) . xPy .a = P'x.^ =~P'y . v . 

('3x).xeC'P.a = P'x.^ = C"P: 
[*150-5-22] = : a (P'^P) /3.v.ae G'P'P .^ = C'P: 
[*16111] = : a (P'P-[*G'P) ^ :: D 1- . Prop 

*252-38L h : P 6 n . D . s'P* e il . Nr's'P* = Nr'P -i- 1 
[*252-38 . *200-52 . *204-35 . *25M31] 



SECTION D] segments OF WELL-ORDERED SERIES 31 

*252-4. I- : P e n . \ C sect'P . g ! \ . D . ^j'X e \ 

Bern. • 

f- . *211-44-l . D h : Hp . P = A . 3 . \ = I'A . 

[*53-01] D.p'\e\ (1) 

h . *212ir2 . Dh:Hp.g[!P.D.XC C's'P* . g ! \ . 

[*252-381.*250-121] D . E ! min (s'P*)'X . 

[*210-222.*211'67-66] D.p'XeX (2) 

I- . (1) . (2) . D I- . Prop 

*252'41. (-iPen.XCsect'P.gJX.D.s'XeX [Proof as in *252-4] 

*252-42. hi.Peil. (Cnv's'P*)i"«r C a : 

A, C o- . a ! X n O's'P* . Dx . s'{\ n C's'P*) 6 ff : D . 

(Cnv's'P*)"o- C o- 
[*250-361 . *252381 . *212-322] 

*252-43. l-:.Pen.(s'P*)i"o-Co-: O , 

X C o- , a ! X n O's'P* . Da . i3'(X n C'.s'P*) e o" : D . (s'P*)"o- C <r 

I-.*212-181. D t- . (Gnv VP*) smor (s'P*) ^(1) 

h . (1) . *252-381 . D f- : Hp . D . (Jnv's'P* e O (2) 

h . (2) . *212-34 . *250-362 . D h . Prop 



*253. SECTIONAL KELATIONS OF WELL-ORDERED SERIES. 

Sumimary of *253. 

In the present number we shall consider the properties of the relation 
Ps (defined in *213) when P e n. The relation Ps has great importance in 
this case, owing to the fact (to be proved later) that Nr"D'Ps is the class of 
all ordinals less than Nr'P, and that, if P, Q are any two well-ordered series, 
either P is similar to a member of G'Qs, or Q is similar to a member of 
G'Ps, whence it follows that of any two unequal ordinals one must be the 
greater. 

The present number consists merely of the more elementary properties of 
Ps when P e 12. The interesting properties connected with greater and less 
will be treated in the following number. 

The most useful propositions of the present number are the following : 

*25313. h : P e n . D . D'Ps = P I "'p^a'P = P t ''P^'CP 

*25318. hzPea.D. G'P, C P I ''P^'G'P u I'P . G'Ps C fi 

Instead of O'Ps C P p "P"a'P w I'P we shall have equality, unless 
P = A (*253-15). 

*253-2. \-:Pen-2r.D. 'Nr'P, = Nr'(P I Q'P) 4- 1 

The case when P e 2,. has to be excluded, because then P I d'P = A. 

*253-21. 1- : P e fl . D . 1 + Nr'Ps = Nr'P 4- 1 

This proposition involves Nr'Pj = Nr'P when P is finite, but when P is 
infinite it involves Nr'Ps = Nr'P 4- 1 (cf. *261-38). 

*253-22. h : P e O . D . Ps t D'Ps smor P I Q'P 

*253-24. b-.Pen.D.PseO, 

*253-4. \-'.Pen-i'A.D. 

G'Ps = Q Kai2) •P=Q4^B.y.{-^x) . P = §+>«;} 

*253-421. 1- : P 6 n . Q e D'Ps . 3 . ~ (Q smor P) 

*253-44. l-:a,^eNO-t'A./34=0,.D.a4-/3=|=a 



SECTION D] sectional RELATIONS OF WELL-ORDERED SERIES 33 

This proposition marks a difference between ordinals and cardinals. An 
ordinal is always increased by the addition of anything at the end, whereas 
this is (often if not always) not the case with a cardinal if it is reflexive 
and greater than the addendum. The above proposition ceases to be true 
if we add ^ at the beginning instead of the end: /S -i- « = « will be true if a is 
infinite and /S X &> is not greater than «. (For the definition of «, cf. *263.) 

*253-45. h:aeNO-t'A-t'0,.D.a4-l4=a 

Similar remarks apply to this proposition as to *253'44. 
*253-46. V'.PeD..Q,Re C'P, . Qsmor R.O.Q = R 

I.e. no two different sections of a well-ordered series are similar. 

It follows from *253'46 that the series of the ordinals of proper sections 
of a well-ordered series P is similar to the series of proper sections, and 
therefore, by *253-22, to the series P with its first term omitted (*253'4!63). 

We have next a set of propositions (*253'5 — ".574) on the circumstances 
under which Nr'Ps = Nr'P and those under which Nr'Ps = Nr'P + 1. As 
a matter of fact, the former holds when P is finite, the latter when P is 
infinite. But the distinction of finite and infinite will not be introduced till 
the next section. In the present number, we prove that (assuming P e li) 

Nr'P, = Nr'P if Q'Pj = Q'P . E ! B'P, and if not, then Nr'P, = Nr'P -j- 1 
(*253'56). This is proved by using P, as a correlator. (Pi as a correlator 
moves every term one place down, except the first, which disappears.) For, 
if Pen, we have Pi;P=PtD'P(*253-5); hence we prove Ppa'PiSmorP^D'P 
(*253-502), and hence, if a'P,= O'P, we obtain P I Q'Psmor P ^ B'P (*253-503). 
Hence by *253"2 (with special consideration of the case when P e 2,) we have 
the two propositions 

*253-51. h : P e fl . G'P, = G'P , E ! fi'P . 3 . Nr'P, = Nr'P 

*253-511. h : P 6 n . Q'Pj = Q'P . ~ E ! B'P . D . 

Nr'P, = Nr'P -i- 1 . Nr'P ^ Q'P = Nr'P 

But if there is a term, say x, belonging to Q'P — Q'Pi, use Pj as a correlator 
for the predecessors of a; ; we thus find that, in this case, P smor P p Q'P. 
Hence, by *253-2, Nr'P, = Nr'P + 1. 

The hypothesis Q'Pj = d'P . E ! B'P means that there is a last term, and 
every other term has an immediate successor. This, as we shall prove later, 
and as is indeed obvious, is equivalent to the assumption that P is finite but 
not null. 

From the above propositions it results immediately that 

*253-573. h :. P 6 fl . D : Q'P, = Q'P . E ! £'P . = . 1 -f- Nr'P =f= Nr'P 

Hence it will follow that finite ordinals other than 0^ are those which are 
increased by the addition of 1 at the beginning. We have also 

E. <feW. Ill 3 



34 SERIES [part V 

*253-574. h r.Pefi-t'A. D : a'P, = a'P .ElB'P. = .i + Nr'P = Nr'P + 1 
Whence it will follow that finite ordinals are those for which the addition 
of 1 is commutative. 

*2531. l-:.Pen.D:QPsi2. = . 

(ga, /3) . o, y8 6'P"a'P yj I'G'P .r^l ^-a.Q = P^ol. R = Pl ^ 
Dem. 
I- . *213-1 . *25217 . D h :. Hp . a ! P . D : QP,E . = . 

('3_0L,^).a,^e'P"a'P»l'G'P.^\^-a.Q = Pla.R = Pt^ (1) 

\- . *33-241 . D I- :. P = A . D : P"a'P w I'G'P = I'A : 

[*24-53] 3 : ~ (ga, 0).a,0 eP"a'P u I'C'P . g ! /3 - a : 

[*213-3] D : QP,R . = . 

(a«,^).«,/S6P"a'Pwi'a'P.a!/3-a.Q = Ppa.ii = P^/3 (2) 
h . (1) . (2) . D h . Prop 

*25311. \-::Pea.D:.QP,R. = : 

(^oo,y) .xea'P. xPy .Q^P^'p'x .R = P ^P'y . v . 

{'^x).xe(l'P.Q = PlP'x.R^P 
Dem. 

l-.*33-152. 3h:a=(7'P./36P"a'Put'(7'P.D.~a!y8-a (1) 

h . *200 52 . (1) . DI-:Hp.a6P"a'P./3 = C'P.D.a!/3-a (2) 

h . (1) . (2) . *2.531 . D h :: Hp . D :. QP,R . = : 

(a«./8).a,^eP"a'P.a!/3-a.Q = Pta.i2 = Pt;8.v. 

(a«,;8) . a6P"a'P . /3= G'P . Q = P r « . E = P r /3 : 
[*37-6.*36-33] 

= :('^x,y) .x.yeQ'P .'^IP'y- P'x .Q = P^P'o! . R = Pl.'p'y .V . 
('3.«>)-«!ea'P.Q = Pl^'x.R=P: 
[*211-61.*210-1] 

= :('3.x,y).x,yea'P.P'xCP'y.'P'x^'p'y.Q = Pl'p'x.R = Pl'P'y.v. 
(^x). xea'P. Q = Pl,p''x.R = P: 
[*204-33-34]= : (^_x,y).x,yea'P.xPy.Q=Pl'p'x.R = Pl^'y.v . 

(•^x). xea'P. Q = Pl^'x.R = P (3) 

I- . (3) . *33-14 . D h . Prop 

*25312. l-:P6fl.P~e2,.D.P, = (P^;!p;ppa'P)+»P 

Dem. 
\- . *204-272 . D I- : Hp . D . a'P~ e 1 . 

[*202-55.*213-151] D , P ^"P"a'P = C'P l^P'P p Q'P (i) 



Section d] sectional relations of well-oedered series 35 

H . (1) . *25311 . D h :: Hp . D :. QP,i2 . = : 

[*161-11] = : Q {{P I fpiP I a'P)-bP} i2 :: D h . Prop . . -1 

*253121. f : P e n . D . P ~ e G'P ^ ;P;P t <I'^ 
Dem. 

' h . *200-52 . D I- : Hp . D . C'P ~ 6 P"a'P . 

[*36-25] D . P ~ 6 CP^'PiPt a'P : D h . Prop 

*25313. h:P6n.D.D'P5 = P^"P"a'P = Pp"'P"C'P 
Bern. 

h.*213-141.*252l7l.Dh:Hp.D.D'Ps = Pt"P"CI'P (1) 
h . *37-22 . *25013 . D 

h : Hp . a ! P . D. P l''P"C'P = P p"P"a'P u I'P t P'5'P 

[*33-41.Transp] = P ^'^P'^a'P u I'A (2) 

l-.*250-42.DI-:Hp.a!P.D.AePt"P"a'P (3) 

h . (2) . (3) . D I- : Hp . a ! P . D . P I'^P^'CP = P ^"P"a'P (4) 

I- . *33-241. .D(-:P = A.D.PD ''P"G'P = A . P t"P"g'P = A (5) 

h . (4) . (5) . 3 I- : Hp . D . P l''P"C'P = P D"P"(I'P (6) 

h . (1) . (6) . D t- . Prop 

*25314. hiPen.D. 

a'P, = (P ^"P"a'P u I'P) - I'A = (P P"P"C"P w t'P) - I'A 

Dem. 

h . *213162 . D h : Hp . D . O'Ps = P ^"sect'P - I'A 

[*252-12.*36-33] = (P t"'P"G'P u I'P) - t'A (1) 

[*2.53-13] . =(Pp"P"a'Pui'P)-t'A (2) 

h . (1) . (2) . D h . Prop 

*25315. h : P e fl - t'A . D . O'Ps = P t "P"a'P w t'P = P ^ "P'^O'P u I'P 
[*253-13-14] 

*25316. h : P e a - I'A . 3 . JS'Ps = A . B'P, = P [*213-15515^ . *25013] 

*25317. l-:P6fi.D.PsCD'P5 = Pt;P5PDa'P 

t 2)em. - . 1 

|-.*2.53-ll.D 

|-::Hp.D:.QPsi2.2,:Q(PD5?5Pp<I'P)i2.v.Q6Pt"P"a'P.iJ = P:. 

[*253121] D :. ^(P, p D'Ps)i2 -= • Q(P D'P^P D Q'-P) -R " 3 I" • Prpp 

3—2 



36 SERIES [PART V 

*25318. h : P 6 fl . D . O'P. C P t"P"a'P u I'P . G'F, C H 
Dem. 

I- . *25311 . D 

h::Hp.D:.Qea'Ps.D:(aa;).a;6a'P.Q = PDi''«-v.Q = P: 
[*37-6] 0:QeP l"'P"a'P u t'P (1) 

I- . (1) .*250-141 . D 1- : Hp . D . O'Ps C fi (2) 

h . (1) . (2) . D h . Prop 

*253181. h : P € n . D . C'P, C D'P, u I'P [*2531813] 

*253-2. h : P 6 O - 2, . D . Nr'Ps = Nr'(P t CI'P) 4- i 
Dem. 
h . *253-12-12l . D I- : Hp . D . Nr'P^ = Nr'P ^ JPJP ^ Q'P + i 
[*213151.*252-171] = Nr'PJP ^a'P+i 

[*204-34] = N r'(P t d'P) + l:Dh. Prop 

*253-21. h:P6n.D.l+Nr'Ps = Nr'P + l 
Dem. 
h . *263-2 . 3l-:Hp.P~e2,.D.l-i- Nr'P, = 1 4- Nr'(P p O'P) + 1 
[*204-46-272] =Nr'P + i (1) 

h.*213-32.Dh:P62,.D.i4-Nr'Ps = i + 2, 
[*161-211] =2r+i 

[Hp] =Nr'P + i (2) 

h . (1) . (2) . D I- . Prop 

It would be an error to infer from the above proposition that 
Nr'Ps = Nr'P, since addition of ordinals is not in general commutative. 
When Peil, Nr'Ps = Nr'P holds when G'P is finite, but not otherwise. 
When O'Pis not finite, 1 + Nr'Ps = Nr'Ps, so that Nr'Ps = Nr'P + 1 ; but 
Nr'P 4= Nr'P + 1. 

*253-22. h : P 6 il . D . Ps t D'Ps smor P p Q'P 
[*253-l7 . *213-151 . *252-l7l . *204-34] 

*253-23. 1- : . P e ii . D : Nr'P = Nr'Q . = . Nr'Ps = Nr'Qs : 

P smor Q . = . Ps smor Qs 
Bern. 

h . *181-33 . D h : Nr'P = Nr'Q . = . Nr'P + 1 = Nr'Q 4- 1 (1) 

|-.(1).*253-21.D 

h :. Hp . D : Nr'P = Nr'Q . = . 14- Nr'Ps = 1 4-Nr'Qs . 

[*181-33] = . Nr'Ps = Nr'Qs :.Dh. Prop 



SECTION d] sectional RELATIONS OF WELL-ORDEREI) SERIES 37 

*253-24. hrPefl.D.Psefi 
Dem. • 

h . *253-2 . *250-141 . *251-132 . D h : Hp . P ~ e 2, . D . Nr'Ps e NO (1) 
h . *213-32 . *251-16 . D h : P e 2, . D . Nr'Ps e NO (2) 

h.(l).(2). Dh:Hp.D.Nr'PseNO. 

[*251122] D . Ps 6 n : D h . Prop 

*253-25. h :. P, Q e fl - I'A . D : Ps t D'Ps smor Qs ^ D'Qs . = . P smor Q 
[*253-22 . *25017] 

*253-3. h : P e 12 . D . P/P = P ^"P"a'P = P I'^P'^CP = D'Ps 
[*213-243 . *253-13] 

*253-31. h :. P 6 n . D : QP^P . = . P e P t"P"C"P ^I'P.QeR 1"'R"C'R 

Dem. 
I- . *213-245 . *25313 . D 

I- :. Hp . 3 : QPsP . = . P e O'P, . Q6Rl"R"C'R . 

[*33-24.*213-3] =.Re G'Ps . g ! P . Q e P ^ "R"G'R . 

[*253-15] =.R6P^''P"C'P^i'P.'3,lP.QeRl"R"G'R (1) 

f- . *37-29 . *33-24 .Dh-.QeR t"P"C"P . D . g ! P : (2) 

[*13-12] :ih:QeRl"R"C'R.R = P.D.'a^lP (3) 

h.(2)^. DI-:P6PD"P"C'P.D.a!P (4) 

I- . (3) . (4) . D h : P 6 P 1"'P"G'P yji'P.QeR l"R"G'R . D . g !P (5) 

h . (1) . (5) . D f- . Prop 

*253-32. [-.Pen. Re G'P, . D . Ps'P = R l"R"C'R = D'Ps 
[*213-246 . *25313] 

*253-33. b -..Peil.D ■.Q(Ps^B'P,)R.~ .RePl''P"G'P.QeRl"'R"G'R 
[*213-247 . *25313] 

If a is any ordinal number, and Pea, the ordinal numbers of the 
sectional relations of P are all those ordinals which can be made equal 
to a by being added to, i.e. all ordinals /3 such that, for a suitable y, 
a = y3 + 7. (Here 7 must be an ordinal or i.) Further, in virtue of *250"67, 
no member of D'Ps is similar to P; hence, if a is an ordinal, and a = /84-7, 
where 7=}= Or. it follows that a=f=/8. (Observe that a^7 does not follow from 
^=|=0r-« = ;8 + 7.) These and kindred propositions, which are important in 
the theory of ordinals, are now to be proved. 

*253-4. l-:P6n-t'A.D.a'Ps = 0Ka-R)--P = Q^-K.v.(aa;).P = Q-|*«} 
[*213-41 . *250-13] 



38 SERIES [PART V 

*253-401. hiPeli.D. 

P l"'P"G'P yJi'P = Q {(gi?) . P = Q 4l i2 . V . (aa;) . P = Q+>a!} 
Bern. : '■ 

h . *253-4-15 .DhsHp.glP.D. 

P ^"P"C"P w t'P = Q {(giJ) . P = Q 4. iJ . V . (a*) . P = (34»^} ' (1) 
I- . *37-29 . D f- : P = A . D . P l"'P"C'P u I'P = I'A (2) 

h . *1 60-14 . *33-241 . D I- :. P = A . D : P = Q 4. i? . = . ^ =-A . -B = A : 
[*10-281] D:(ai?).P=Q4:E. = .Q = A (3) 

1- . *161-13 . *33-241 .DI-:.P = A.D:P= Q-\*x . = . Q = A : 
[*10-24-23] D:('s^sc).P=Q-i^x. = .Q = A (4) 

h . (3) . (4) . 3 h : : P = A . D :. (^R) .P=Q^R.v. (a«) ..P = Q-\*x: = .Q=A. 
[(2)] =.Q6P^"P"(7'P«t'P (5) 

h.(l).(5).Dh.Prop i 

*253-402. hzPen-i'K.D. 

D'i's = {(a^) ■ -R 4= A . P = Q 4l P . V . (ga;) . P = Q-f*a;} 
i)em. 
h . *253-16-4 . D 

l-::Hp.D:.Q6D'Ps.s:Q4=P:(aP).P = Q4^P.v.(aa!).P = Q-+*« (1) 
h . *161-14 . *200-41 . D h : Hp . P = Q4*a; . D . a; e (7'P . a; ~ e O'Q . 
[*13-14] 3.e + P (2) 

I- . *160-21 .Dh:Q+P.P = Q4LE.D.a!P (S^) 

I- . *160-14 . *200-4 . D 

l-:Hp.P = Q4iP.a!iJ.D.a!0'PnC'i?.~a!0'QnC"P. - • 
[*13-14] 3.P+Q (4) 

l-.(3).(4).D 

l-::Hp.D:.Q + P:(ai2).P=Q4^P: = .(aP).P + A.P = Q4:E (5) 

I- . (1) . (2) . (5) . D h :: Hp . D :. Q e D'P, . = : 

(aP) . P + A . P = Q 4:P . V . (aa;) . P = Q-|*fl; :: D h . Prop 

*253-41. I- :. P 6 n . Q e G'P, . D : 

(aa) . a e NO . Nr'P = Nr^Q + a . v . Nr'P = Nr'Q + 1 i; 

l-.*213-3.Dl-:.Hp.D:P + A: 

[*253-4] D:(aii).P = Q4^P.v.(a«).P = (2-|*«: 

[*211-283.*200-41] > ,; 

D:(aP).P=Q4^P.a'QnC"P = A.v.(aa;).P = Q+>a!.«~e'0'^- 
[*180-32.*181-32] D : (aP) . Nr'P = Nr'Q + Nr'P . v . Nr'P = Nr'§ + 1 : 
[*251-26] D : (aa) . « e NO . Nr'P = Nr'Q -i- a . v . Nr'P = Nr'Q + 1 :. D h . Prop 



SECTION D] sectional RELATIONS OF WELL-OEDEBED SERIES 3.9 

*253-42. h : P e fl . D . Nr'P n B'P, = A [*250-651 . *213141] 

*253-421. l-:Pea.QeI^Ps.D.~(QsmorP) [*253-42] 

*253-43. I- :. Pe n . x,yea'P . D : P ^ P'a; smor P ^ P'y. = .x = y 
Bern. 

I- . *25311 . D h : Hp . a!Py . D . (P llP'x)P, (PtlP'y) . 

[*213-245] D . P t P'a; € D'(P ^ IP'y), . 

[*253-421] D . ~ {(P tP'«!) smor (P ^ P'j/)} (1) 

Similarly h : Hp. yPa;. D.~{(P ^P'a;) smor (P^P'^)} (2) 

h . (1) . (2) . D h :. Hp . D : (P pP'a;) smor (P^P'y) . D .'^(xPy) .--(yPa!) ■ 
[*202-103] 0.x = y (3) 

t-.(3).*151-13.Dl-.Prop 

*253-431. l-:P4LQ6n.a!Q.D.Nr'P=t=Nr'(P4.Q) 
i)em. 

h . *253-402 . D h : Hp . D . P 6 D'(P:^Q)s (1) 

f-.(l).*253-421.DF.Prop 
*253-432. h : P4>a; e O . g ! P . D . Nr'P + Nr'(P-f>«) [*253402-421] 

*253-44. l-:a,/36NO-t'A./3=t=0,.D.a + ;8 + a 

Dem. 
h.*25ri.*155-34.D 

H : Hp . D . (gP.Q) . P,QeIl . a= N„r'P . /3= N„r'Q . a ! Q. 
[*180-3] 
D.(aP,(2).P,Q6n.a = N„r'P./3 = N„r'Q.a[!Q.a + /3 = Nr'(P + Q) (1) 

h . *18012 . *253431 . (*180-01) . 3 

h:P,QeIl.a!Q.D.Nr'(P + + Nr'P. 

[*155-16] 3 . Nr'(P + Q) + N„r'P (2) 

l-.(l).(2).D " 

1- : Hp . D . (aP, Q) . P, Q 6 X2 . a = N„r'P . /3 = N„r 'Q . a + /3 4= Noi 'P . 

[*13-195] 3 . a + /3 + a : D h . Prop 

*253-45. H:aeNO-t'A-t'0,.D.a + i=t=a 

[Proof as in *253-44, using *253-432 instead of *253-431] 

*253-46. }-:Pen.Q,BeG'Ps.QsmorR.D.Q = B 
Bern. 

f- . *253-421-16 . D I- : Hp . Q = P . D . E = Q (1) 

h.*263-16 . D h : Hp. Q=t=P.i2 + P. D . Q,-BeD'Ps • 

[*253-13] D.('^a;,y).x.y6a'P.Q = PtP''»-R = PtP'y' 
[*253-43.Hp] O.Q = R (2) 

I- . (1) . (2) . D 1- . Prop 



40 SERIES [part V 

*253-461. h : P € n . D . Nr f G'P, e 1-*1 
Dem. 
1- .*253-46 . 3 h : Hp . Q,ReG'P, . Nr'Q = Nr'i2 .0.Q = R:0\-. Prop 

*253-462. h:Pefl.D. 

Nr I (P I) \P [ a'P 6 1^1 . NrJP ^'P^P D <^'P smor P ^ Q'P 
[*253-43] 

*253-463. h : P 6 n . D . 

NrJ (Ps ^ D'PO smor Ps ^ D'Ps • NrJ (Ps p D'P ) smor P t d'P 
[*2o3-462-l7-22] 

*253-47. h : P 6 n - I'A . D . 

Nr"a'Ps = a {(a^) . a + /3 = Nr'P . v . a + 1 = Nr'P j [*253-4] 

*253-471. hiPefl.D. 

Nr"(D'Ps u I'P) = a {(gyS) . a + /8 = Nr'P . v . a 4- 1 = Nr'P) 
[*253-401-13] 

The following propositions are concerned in proving that Nr'Ps is either 
Nr'P or Nr'P 4-1- This is proved by using Pj as a correlator. The 
methods employed anticipate the discussion of finite and infinite series ; 
in fact, when P is finite, Nr'Ps = Nr'P, and when P is infinite, 
Nr'Ps = Nr'P -i- 1. But it is important at this stage to know that Nr'Ps is 
either equal to or greater than Nr'P, and the propositions are therefore 
inserted here. 

*253-5. h:P6n.D.Px;P = PDD'P 
Dem. 

b . *201-63 . *25-411 . D h :: Hp . D :. P = Pi a P"" :. 
[*150-11] 0:.x(PJP)w. = : (gy, z) : xP^y : yP^z . v . yP'z : wP^z : 
[*204-7] = : {^z) . xP^w . wPj,z . v . (gy, z) . xP^y . yP^z . wP^z : 

[*250-21 -24] = : a;P, w . w e D'P . V . (gy) .xP^y.y.we D'P . yPw : 

[*33-14.*34-l] = : a; (P, w Pi I P) w . w e D'P : 
[*3314.*250-242] = :x,W€ D'P . xPw :: D h . Prop 

*253-501. h : P 6 n . D . P,;P = P p G'Pj 
Bern. 

h . *260-242 . D f- : Hp . D . Pi I P = Pi I Pi (a Pi I Pi I P 
[*7l-191.*204-7] =![ a'Pi o (a'Pi) 1 P . 

[*1 50-1 .*50-65] D . Pi ;P = (O'Pi) 1 Pi a (a'P,) 1 P | Pi 

[*250'243] =Pt a'P, : D h . Prop 



SECTION D] sectional RELATIONS OF WELL-ORDERED SERIES 4.1 

*253-502. J- : P e fl . D . P t a'P^ smor P l D'P 
Dem. • 

H . *253-5 . *150-36 . 3 h : Hp . D . P ^ D'P = P,'> (P ^ d'Pi) (1) 

h . *151-21 . *204-7 . D I- : Hp . D . Pi? (P p Q'P.) smor P ^ a'P, (2) 
h.(l).(2).Di-.Prop 

*253-503. h : P 6 n . Q'P, = Q'P . D . P t CI'P smor P ^ D'P [*253-502] 

This proposition shows that if P is a well-ordered series in which every 
term except the first has an immediate predecessor, the series obtained by 
omitting the last term (if any) is similar to that obtained by omitting the 
first term. The converse also holds, as will be shown later. The hypothesis 
Pen.(l'Pi=(l'P is equivalent to the hypothesis that P is finite or a pro- 
gression. (Here a progression is not what was defined as " Prog " in *121, but 
what Cantor calls m; i.e. ii Re Prog, Ppo is a progression in our present sense.) 

*253-51. h : P 6 ft . O'P, = Q'P . E ! P'P . 3 . Nr'Ps = Nr'P 
Bern. 

t- . *253-2 . D h : Hp.P~e 2, . D . Nr'Ps = Nr'(P I O'P) 4- 1 
[*253-503] = Nr'(P C D'P) + 1 

[*204-461-272] = Nr'P (1) 

h . *213-32 . D I- : P 6 2^ . 3 . Nr'Ps = Nr'P (2) 

h . (1) . (2) . D h . Prop 

*253-511. h-.PeD,. a'P, = O'P . ~ E ! 5'P . D . 

Nr'Ps = Nr'P 4- 1 ■ Nr'P t Q'P = Nr'P 
Dem. 

h . *93103 . *202-52 . D h : Hp . 3 . P^ D'P = P . 
[*253-503] 3 . Nr'P ^ Q'P = Nr'P . (1) 

[*253-2] 3. Nr'Ps = Nr'P -I- 1 ' (2) 

I- . (1) . (2) . 3 I- . Prop 

*253-52. f- : P 6 ft . « = minp'(a'P - Q'P,) . 3 . 

Q'P n'p'x C a'P, . P^"'P'a; = P'a . P^"P'x = P'x - I'B'P 



Bern. 



1-.*20514. 3f-:Hp.3.a'PnP'a;Ca'P, (1) 

h . *250-242 . 3 h : Hp . 3 . P'a; = P.'ai u P,"P'x 
[*33-41.Hp] =P."P'a5. (2) 

[*72-501 .*204-7] 3 . P,"^'^ = ?'« " Q'-Pi (3) 

h . (1) . 3 h : Hp . 3 . Q'P n P'a; = a'P a P'x n G'P, 

[*121-305] =a'P^r^'p'x (4) 

h . (3) . (4) . 3 h : Hp . 3 . P/'P'* = 'P'"' « Q'-P 
[*33-15.*202-52] =P'x-i'B'P (5) 

I- . (1) . (2) . (5) . 3 h . Prop 



42 SERIES [PART V 

*253-521. \-:P€il.oo6a'P-a'P,.O.P'x,a'P'^el 
Bern. 

h.*201-66. Dh:Pen.P'a!el.D.a;€a'Pi (1) 

h.(l).Traiisp.DI-:Hp.D.P'a;~el (2) 

h.*201-662. Dh:Hp.D.a'P~6l (3) 

I- . (2) . (3) . D I- . Prop 

*253-522. 1- : P e fl . a; = minp'(a'P - Q'PO .S = P^ [P'x yj I [ P^'x . D . 

S''{Pl(l'P) = P 
Dem. 

h . *34-25-26 . *50-5-51 . D 

h : Hp . D . 5f;(P ^ a'P) = (Pi 1^ P'a;);P p Q'P o (/ T^'^)'-? '^^ 

(Pi I' P'a;) I P I / 1^ P*'a; va 7 [^ P^'a; | P |T'a; ^ P^ 

[*50-6-61.*150-36.*35-452] = (P, ['P'xY'P w P ^ P^'a; c; P, [^ P'a; | P I^ P*'a! c; 

P*'^1_Pr^'a;|P:^ _^ ^ 

[*74-141.*253-52.*200-381]= (P^ p "P'^Y^P w P ^ P^'a; o P'a; 1 P, | P T P^'x 

[*250-242.Hp] = (P, p P'a!)5P c; P t P*'a; c; P'a;1 P I' P*'* 

[*150-36] = (P.'P) i P,''P'x \J P l*P^'x vj'P'x 1 P f P*'a; 

[*253-5-52] =Pl'P'x^JPl %'x va P'a; 1 P I' p-^'x 

[*35-413.*200-381] =P^P'a;up!^'a;) 

[*202-101] =P:DH. Prop 

*253-53. h : P 6 fl , a; = mmp'(a'P - Q'P,) . 3 . 

Pi 1^ P'a; o / 1^ Pj^'a; e {P s^ior (P ^ Q'P)} 
Dem. 

I- . *204-7 . *200-381 . D h : Hp . D . P, T P'a; w / 1^ P^'a; e 1^1 (1) 
h . *253-52 . *50-5-52 . D 

h : Hp . D . a'(Pi [lP'x^JI t^^'x) = (P'x - I'B'P) w ^^'x 

[*202-101] =C'P-i'B'P 

[*93103] ^ =a'P 

[*202-55.*253-521] = G'(P I a'P) (2) 

I- . *253-522 . 3 f- : Hp . 3 . (Pi f P'a; u / [*P^'x)l{P I Q'P) = P (3) 
h . (1) . (2) . (3) . *151-11 . D h . Prop 

*253-54. t-iPefi.aia'P-a'Pi.D.PsmorPpa'P 
Dem. 

h . *250121 . D I- : Hp . D . E ! minp'(a'P - Q'Pi) (1) 

h.(l).*253-53.Dh.Prop 



SECTION D] sectional RELATIONS OF WELL-ORDERED SERIES 43 

*253-55. h : P e n . a ! Q'P - Q'P, . D . Nr'Ps = Nr'P + 1 

Dem. 

h . *253'521 . *204-272 . D h : Hp . D . P ~ e 2^ (1) 

h.(l).*253-54-2.Dh.Prop 

*253-56. I- :. P € n . D : Q'P^ = O'P . E ! 5'P . 3 . Nr'P^ = Nr'P : 
~ (Q'P, = a'P . E ! S'P) . D . Nr'Ps = Nr'P + 1 
[*253-51-511-55] 

*253-57. h:Pea.a'Px = a'P.E!P'P.D. 

1 + Nr'P = Nr'P + 1 . i -i- Nr'P + Nr'P 
Dem. 

I- . *253-51 . D h : Hp . D . Nr'Ps = Nr'P . 

,, t*253-21] D.i4-Nr'P = Nr'P + i. (1) 

[*253-45] D. 14- Nr'P =^= Nr'P (2) 

. h.(l),(2).DI-.Prop 

*253-571. l-:P6n.~(a'P, = a'P.E!5'P).D.l + Nr'P = Nr'P '• '' •"^" 
Dem. 

h . *253-56 . D h : Hp . D . Nr'Ps = Nr'P + 1 . "'/ / 

' 'I ; '[*253-21] D . 1 + Nr'P + 1 = Nr'P + 1 .' 7^ '. ; 

[*181-33] D. 1 -i- Nr'P = Nr'P :Dh. Prop ; „ ; ^J 

*253-572. I- : P 6 n - 1' A . ~ (a'Pi = a'P . E ! P'P> . 3.1+ Nr'P 4=*Nr'P +1 
[*253-671-45] '_ ■- ' '■' ''■'•' ^^ 

*253-573. I- :. P e fi . 3 : Q'Pj = O'P . E ! P'P . = . 1 + Nr'P + Nr'P 
[*253-57-571] ,, ;■ {' 

*253-574. h :. P e fl - I'A . 3 : O'Pj = fl'P . E IB'P . = . 1 + Nr'P = Nr'^P + 1 
[*253-57-572] z';^ 



*254. GREATER AND LESS AMONG WELL-ORDERED SERIES. 

Summary of *254. 

In the present number we have to prove that of any two well-ordered 
series one must be similar to a sectional relation of the other. From this it 
will follow that of any two unequal ordinals one must be the greater. The 
propositions of the present number are due to Cantor*. 

Our procedure is as follows. We define a relation " RP^^Q" meaning 
"R is a proper section of P, and is similar to Q" i.e. 
RP^Q . = .Re D'Ps . R smor Q. 

In virtue of *253-46, if P, Q e H, P3^ e 1 -> CIs (*254-22) and 
Pauitli'Qsel->l (*254-222). Thus if S is any proper section of Q which 
is similar to some proper section of P, the proper section of P to which 
it is similar is Pam'S. It is easy to prove that Psm'Qs ^ D'Qs is a section of 
P ; and if D'Pj C.d'Qg^, i.e. if every proper section of P is similar to some 
proper section of Q, we shall have (*254261) 

PsDD'Ps=p,^;(2sDD'Q». 

Hence it follows (*254-27) that if, further, T>'Q, C d'P^^, we shall have 

PstD'PssmorQspD'Qs, 
i.e. by *25S-25, PsmorQ (*254-31). 

Thus (A) if every proper section of P is similar to some proper section of Q, 
and vice versa, then P is similar to Q. 

Consider next the case in which every proper section of P is similar 
to a proper section of Q (i.e. D'Pj C d'Qgm). but not vice versa, so that 
a ! D'Qj - Q'Pgm- It is easy to prove that, under this hypothesis, if 
8eJ)'Q, - a'P,m, then D'Ps C d'S^ (*254-32). But if S is the minimum 
(in the order Q,) of the class D'Qs - d'P^^, then T>'8, C Q.'P^. Hence, 
by (A), 

8 smor P (*254-321). 

Thus (B) if every proper section of P is similar to a proper section of Q, but 
not vice versa, then P is similar to a proper section of Q (*254"33). 

• Math. Annalen, Vol. 49. 



SECTION D] greater AND LESS AMONG WELL-ORDERED SERIES 45 

From (B), by transposition, we find that if every proper section of P is 
similar to a proper sectio|^ of Q, but P itself is not similar to any proper 
section of Q, then every proper section of Q is similar to a proper section 
of P, whence, by (A), P is similar to Q (*254"34). Hence, if there are 
proper sections of P which are not similar to any proper section of Q, the 
smallest of such sections (say P') must be similar to Q, since it is not itself 
similar to any proper section of Q, but all its proper sections are similar to 
proper sections of Q. Hence (C) if there are proper sections of P which are 
not similar to any proper section of Q, then there is a proper section of P 
which is similar to Q, i.e. 

h : P, Q e n . a ! D'Ps - d'Q,^ . D . Q e a'P^^ (*254-35). 

Thus either (1) g ! D'Ps- Q'Q.^, in which case Q e a'P^^, or 
(2) a ! D'Qs - a'P,„, in which case P e^'Q,^, or (3) D'PsCa'Q,^ and 
D'QsCa'Pg^, in which case, by (A), PsmorQ. Thus (D) if P and Q are 
any two well-ordered series, either they are similar or one is similar to a 
proper section of the other (*254"37). 

We now proceed to define one well-ordered series P as less than another 
well-ordered series Q if P is similar to a part of Q, but not to Q, i.e. we put 

less = PQ {P, Q e n . a ! Rl'Q n Nr'P . ~ (P smor Q)] Df. 
(Observe that we have El'Q in this definition, not D'Qs.) 

It follows from (D) that, P and Q being well-ordered series, if P and Q are 
not similar, one must be less than the other (*254'4). It follows also from 
*25065 that if P is similar to a proper section of Q, Q cannot be less than 
P (*2o4"181). Hence P is less than Q when, and only when, P is similar to 
a proper section of Q, i.e. 

P less Q . = . P, Q e 12 . P 6 Q'Q.^ (*254-41). 

Hence if each of two well-ordered series is similar to a parb of the other, the 
two series are similar (*254"45) ; and in any other case, one of them is similar 
to a proper section of the other. 

From the above results we easily obtain the following propositions, which 
are useful in the ordinal theory of finite and infinite. 

*254-51. I- : Pless Q. = . P.Qefi . Rl'Pn Nr'Q = A 

I.e. one well-ordered series is less than another when, and only when, no 
part of it is similar to the other. 

*254-52. . f : P 6 XI . a C C'P . a ! C'P n p'P"oi . D . P ^ a less P 

//.'e. anySart of a well-ordered series which stops short of the end is less 
thf ^ii the wiole series. 



46 SERIES [PiRT V 

*254-55. i-:.QlessP. = :P,Q6fl:(ai2).i2smorQ.i2GP.a!(7'Pftp'P"a'E 
I.e. one well-ordered series is less than another when, and only when^ it is 
similar to a part of the other which stops short of the end. 



*25401. less = PQ{P,Qefi.a!Rl'QnNr'P.~(PsmorQ)} Df 

*25402. P3^ = (D'Ps)1smor Df 

*254-l. l-:PlessQ. = .P,Qefl.a!Rl'QANr'P.~(PsmorQ) [(*254'01)] 

*254101. h : P, Q £ fl . P G Q . ~ (P smor Q) . D . P less Q [*254-l] 

*25411. y:RP^J^. = .Re'D'P,.RsmorQ [(*254-02)] 

.*254111. l-.P,^'Q = D'PsnNr'Q [*254-ll] 

*25412. I- : Q e d'P^^ • = . a ! D'Ps n Nr'Q [*254-l 1 1] 

*254-121. h.D'PjCa'P,^ • [*254-12.*152-3] 

*25413. h :. P smor P'. Q smor Q'.'H-.P less Q. = .P' less Q' 
[*151-15 . *152-321 . *254-l] 

*254-14. V:Se D'Qs .TeP s"mof Q.D.T'Se T>'P, n Nr'/S 
Dem. 
h . *213-141 . D h : Hp . D . (gyS) . /3 e sect'Q - I'A - t'C'Q .S=Q10 (1) 
1- . *150-37 . Df-:Hp./Sf = Qt/8.3. T'S = (T>Q)l T"0 
[*151-11] =PIT"^ (2) 

K.*212-7. Dh:Hp./3esect'Q.D.T"/36sect'P (3) 

F.*37-43. Dl-:Hp./3esect'Q-i'A.D.a!r"/3 (4) 

1- . *150-22 . D h : Hp . T"^ =^C'P.D. T"^ = T"C'Q : 
[*7248li DI-:Hp.T"/3=C'P.jS6sect'Q.D./3=C'Q: 
[Transp] D I- : Hp . ;8 6 seet'Q - t'C'Q . D . ^"/S + C'P (5) 

l-.(3).(4).(5).D 

I- : Hp . /3 6 sect'Q - I'A - t'C'Q . D . T";8 e sect'P - t'A - t'C'P (6) 

h . (1) . (2) . (6) . D h : Hp . D . (ga) . a e sect'P - I'A - t'C'P . nS = P\^ a . 
[*213-141] O.T'SeB'P, , (7) 

h.*151-21. DI-:Hp.D.(r;*S)smor/S (8) 

I- . (7) . (8) . D h . Prop 

*254141. h : P smor Q . D . D'Qs C a'P^ . D'P, C Q'Q.^ 
Bern. 

h .*254-12-14 . D h :. Hp . D iSeB'Qs . D . S'eQ'P.^ (1) 

h . (1) . *151-14 . D h . Prop 



\ 



*254142. I- : -B 6 G'P^ . 3 . iJs^ C ^sm 

I- . *213-241 . 3 I- : Hp . D . D'Ps C D'Ps ' \'i) 

h.(l).*2541i;Dh.Prop 



SECTION d] greater AND LESS AMONG WELL-ORDERED SERIES 47 

*254143. hzQe a'P,„ . D . G'Q, C Q'P,, 
Dem. m 

I- . *25412 . D h : Hp . D . (gi?) .ReB'P^.R smor Q . 
[*2.54-141] D . (gii) . R e I>'P, . J)'Q, C a'R,^ . 

[*254-142] D.D'Q.Ca'P,^. 

[*213-16.Hp] D . (3 C"(sect'Q - t'A) C a'P,^ . ; 

[*213-1] D.O'QsCa'P,^:DI-.Prop 'j 

*254-144. I-:P = A.D.P3„ = A [*213-3 . *254-ll] 

*25415. h :. Qp„ e J" . a ! £'P . Pp„ e J- . D : Q e Q'P.^ . = . C'Q, C G'P,^ 

Dem. 
h . *254-143 . D h : Q 6 a'P,„ . D . C'Q, C a'P,„ (1) 

h . *213-142 . *211-26 . D h i.JHp . g ! Q . D : Q e G'Qs : 

[*22-441] D:C'Q,Ca'P,„.D.Q6a'P3^ .(2) 

h . *211-18 . D h : Hp . D . a ! sect'P r. 1 . 

[*200-35] D . A e P t "(sect'P - t'A) . 

[*213-16] D.AeD'P,. 

[*254-121] D.AeQ'P.^ ■ (3) 

f-.(2).(3). Dh:.Hp.D:C"Q,Ca'P,^.D.Qea'P,^ (4) 

I- . (1) . (4) . D h . Prop 

*25416. h :. Qsmor Q'. 3 :'P,^'Q = P,^'Q' : Q^a'P,^ . = . Q'eQ'P.^ 
Bern. 

h . *254111 . *152-321 . D h :. Hp . D : 'p.jQ = ^„'Q' : (1) 

[*13-12] :>:a!Psm'Q- = -a!PB.'Q': 

[*33-41] D : Q 6 Q'P.^ . = . Q' e a'P,„ (2) 

h , (1) . (2) . D h . Prop 

*254161. h : P smor P' . D . a'P,^ = a'P',„ 

-Dem. 
I- . *254-14 . D I- : TePslnof P'.SeD'P'^n'Nr'Q . D . T'SeB'P, n Nr'Q : 

[*254-12] Dh:^6PslnorP'.Qea'P',^.D.Q6a'Ps„: 

[*151-12] Dh:PsmorP'.D.a'P',^Ca'P,„ (1) 

f- . (1) . *15114 . D h : P smor P'. D . a'P^n. C d'P',^ (2) 

h . (1) . (2) . D h - Prop 

*254162. h :. P smor P'.Q smor Q'. D : QeQ'P,^. = . Q'eQ'P'^^ 
[*254-16161] 

*254163. \-:Rea'Q,^.:i.a'R,^ca'Q,^ ; 

Dem. ^ 

h.*254-12.Df-:Hp.D.(a/Sf).i2smor/S.*Sf6D'Qs. ! 

[*254i6i-i42] 3 . (g^) . a'R,^ = a's,^ . a's,^ c a'Q^ . : 

[*13195] D.a'R,raCa'Q^:D\-.Fiop j 



48 SERIES [PAET V 

*254164. h : D'Ps C a'Q^ . D . B'P, = P^"(D'Q, r^ a'F,J = P^J'D'Q, 
Dem. 

V . *254-ll . 3 h : Hp . E 6 D'P, . 3 . (gS) . S e B'Q, . R smor 5f . 
[*254-ll] D.{'sS).Se'D'Qs.RP,„,S. 

[*371] D.i2 6P3^"D'Qs (1) 

[-.*25411.DI-.P,„"D'QsCD'P5 (2) 

h . (1) . (2) . D h : Hp . D . B'P, = P^"'D'Qs 
[*37-26] = Psm"(D'Qs '^ a'-Pam) = ^ I" • Prop 

*254-17. h : Pe O . Q eD'Ps . P C Q . D . ~ (PsmorP) 
i)em. 
I- . *204-21 .DhiPeii.PGP.P smor P . D . P e Ser . 
[*204-41] D.R = PtG'B (1) 

I- . *250-65 . Transp . D 

h : Peil . PsmorP . P = Pt C'R . D . ~(aa) . a e sect'P - t'C'P .C'RCa. 
[*211-133-44] D . ~ (aQ) . Q 6 P t "(sect'P - l'G'P) .RdQ. 
[*213-14i] D . ~ (aQ) . Q e D'Ps .RGQ (2) 

l-.(l).(2).3l-:P6fl.PsmorP.PGP.D.~(aQ).Q6D'Ps.PGQ (3) 
h . (3) . Transp . D f- . Prop 

*254-18. l-:Q6D'Ps.D.~(PlessQ) [*254-17-l] 

*254181. V:Qe G'P,^ . D . ~ (P less Q) 

Bern. 

V . *254-1812 , D I- : Hp . D .(gP) . R smor Q . ~ (P less P) . 
[*254-13] D. ~ (P less Q) Oh. Prop 

*254 182. h : P 6 n . Q 6 T>'P, . D . Q less P [*254-101 . *253-421-18] 

*254-2. l-:P6Q.Qea'P,„.D.QlessP 

Dem. 

I- . *25411 . D h : Hp . D , (gP) . P e D'Ps . P smor Q . 
[*254-182] D. (aP).P less P. P smor Q. 

[*2.54-13] D . Q less P : D I- . Prop 

*254-21. I- : P e n . Q e d'Psu. .RCQ.ReD,.'^ .R less P 

Pern. 
I- . *254-12 . D h : Hp . D.(a/S, TJ.SeD'P, . Te-Sfs-SorQ . 
[*151-21.*150-31] :).('3,S,T).8eT)'P,.Te8sE:oiQ.T'yRsmorR.T'R(lS. 
[*254-l7] D.(ar).T5PsmorP.T;PGP.~(r;PsmorP). 

[*15117] D.(ar). r;P smor P. r;PG P. ~(P smor P). 

[*2541] D . P less P : D h . Prop 



SECTION D] greater AND LESS AMONG WELL-ORDERED SERIES 49 

*254-22. h:P6n.D.P,„el^01s 
Dem. 

V . *25411 . D h :. RP^Ji . SP^^Q , D : E, ^ e D'Ps . R smor 8 : 
[*253-46] D:P6n.D.i? = /S (1) 

h . (1) . Comm . D h . Prop 

*254-221. h : P e n . D . a'P,„ C O 

Dewi. 

I- . *254-12 . *25313 . D 

h:Hp.Qea'P,^.D.(a-K,a).P = Pta.PsmorQ. 

[*250-141.*251111] D . Q e n : D I- . Prop 

*254-222. h:P,QeXl.D.P,„,rD'Qs6l-*l 
Dem. 
h . *254-ll .Ohi.R (P«^ r D'QO S . R (P,„ r D'QO -S' ■ ^ = 

/Sf, /Sf'e D'Qs . R smor )S . R smor /S' : 
[*253-46] ' D:Q6n.D.S=S' (1) 

h . (1) . Comm . D I- : Hp . D . P,^ T D'Qs e Cls -* 1 (2) 

I- . (2) . *254-22 . D i- . Prop 

*254-223. 1- . Cnv'iP,^ [ D'Qs) = Q,^ [ D'P, 
Dem. 

h .*254-ll . D I- :R(P,^[I>'Qi)S. = . P e D'Ps . >Sf e D'Qs . Psmor^f. 
[*151-14] = .8e D'Qs . R e D'Ps . S smor R . 

[*254-ll] =.S(Q,^rD'-P0^:3'--Prop 

*254-224. h : Q e n . E ! P^^'fif . ;Sf e D'Qs . D . >S = Q^^'P^^'^ 
2)em. 

I- . *254-223 . D h : . Hp . D : ;SfQ,„ (P,„,'<Sf) . = . (P,^'8) P,^S (1) 
I- . (1) . *30-32 . *254-22 . D h . Prop 

*254-23. h : P 6 n . Q 6 G'P,^ . D . P,„.'Q = ^'(D'Ps n Nr'Q) [*254-22111] 

*254 24. \-:P,Qeil.Re D'Ps n Q'Q.^ . 8 e Rl'P a D'Ps . D . -S e Q'Q.^ 

Dem. 

h . *213-24 . D h : Hp . D . ;S e D'Ps . 

[*254-143.Hp] D . >Sf 6 a'Q,^ : D h . Prop 

E.&W. III. * 



50 SERIES [part V 

*254-241. 1- :. P 6 n . Q, i? e C'P, .:^:R€ a'Q,^ . = .Re D'Q, 
Dem. 
h . *254121 .0\-:Re D'Q, .D.Be a'Q,^ (1) 

h . *254.142 . D h : Hp . Q e C'R, . D . Qsm G -Rem (2) 

h.*253-42. Oh:Ren.D.Rr^ea'R^^ (3) 

t-.(2).(3). D\-:np.QeG'R,.D.R'^€a'Q,^ (4) 

h . (4) . Transp . (3) . D h : Hp . i? e Q'Q.^ . D . Q ~ e C'iis . Q 4= i2 . 
[*213-24.5] D . ~ (QPsR) .Q^R. 

[*213153.Hp] D.RPsQ. 

[*213-245] D . E e D'Qs (5) 

h . (1) . (5) . D h . Prop 

*254-242. 1- : Q 6 Xi . Te P iSor Q . /Sf e D'Qs . 3 . T'>S = P,^'S 

Dem. 

1- . *254-14 . D I- : Hp . D . T''SeI>'P, n Nr'^f . 

[*254-ii] D.(r;,s')P,„^f. 

[*2o4-22.*2511 11] D . TJ/S = P,„'5' : D t- . Prop 

*254-243. h : Q 6 Xl . -Se D'Qs . T e P imor S .S'Q,S.D. T>S' = P,^'S' 
Dem. 

\- . *213-243 .*253-18 . D h : Hp . D . /SeO . 8' eD'S, . 
[*254-242] D . TiS' = P,^'S' : D h . Prop 

*254-244. h : P, Q 6 n . /S 6 D'Qs n a'P,„ . Te {P^'S) imof /S . S'Q,S . 3 . 

T'8 = P3^'>S . y56" = P,JS'. (T-^8') P, (TiS) 
Dem. 

I- . *254-243 . D I- : Hp . iZ = P,^'8 . D . r5>S' = R,^'8' (1) 

h.*25411. Dh:Hp(l).D.i2eD'P5. (2) 

[*254142] D.i2,„CP,„ (3) 

h . (1) . (3) . *254-22 . D h : Hp (1) . D . T'S' = P,„'«' (4) 

l-.*151-ll. DI-:Hp(l).D.i2 = r;fif. (5) 

[(2)] D.r;S6D'P» (6) 

h . (1) . (5) . *254-l 1 . D h : Hp (1) . D . r;6f' e I>'(Ti8) (7) 

I- . (6).(7).*213-244 . D I- : Hp (1) . D . (T'S') P, (T'8) (8) 

f-.(6). Df-:Hp.D.r;S = P3„'^f (9) 
h . (9) . (4) . (8) . D h . Prop 

*254-245. \-:P,Qen.8e I>'Q, n a'P,„ . 8'Q,S . D . (P,^'8')P, (P,^'8) 

Dem. 

h . *254-22-ll . D h : Hp . D . (P^^'S) smor 8 (1) 

h . (1) . *254-244 . D h . Prop 



SECTION D] greater AND LESS AMONG WELL-ORDERED SERIES 31 

*254-25. \-:.P,Qea.S,S'e-D'Q,na'P,^.D:8'Q,8. = .(P,^'S')P,(P,^'S) 
Dem. • 

h . *254-245 . D h :. Hp . D : S'Q,8 . D . (P^^'S') P, (P,JS) (1) 

P '8 P 'SI' P Q 

h :. Hp . D : (P.^'S') Ps (P.^'S) . D . {Q,^'P,^'8') Qs (Q,^'P,^'8) . 
[*254-224] D . fif'Q.fif (2) 

h . (1) . (2) . D 1- . Prop 

*254-26. I- : P, Q 6 ri . D . Q, t; (D'Q, n O'P,^) = Q,„;(^» D D'P,) 
i>em. 

h . *254-25 . D F :: Hp . D :. £f' {Q^^ (D'Qs n Q'P^^)) ^. = : 
8, 8' e D'Q. n a'P,, . (P,^'S') P^ (P,^'8) : 



[*254-22] = 
[*254-223] = 
[*15011] = 



8, 8' 6 T>'Qs : (^R, R) . RP,^S . R'P,^8' . R'P,R : 
(ai2, R') . SQ,^R . 8'Q,^R'. R, R' e D'Ps . R'P^R : 
S'{Q,J(PstJ)'Ps)}S::D\-.FToi, 



*254-261. t- : P, Q 6 n . D'Q. C a'P,„ .D.Qst T>'Q, = Q^l{P, I D'P.) 
[*254-26] 

*254-27. hzP.Qen. D'Ps C Q'Q^ . D'Qs C Q'P,^ . 3 . 

Qs-n r C"(-Ps D D'-PO e (Q, t Jy'Qs) iHor (Ps t B'P,) 
Bern. 

h . *254-222 . 3 h : Hp . O . 0,^ C C'{P, l B'P,) e 1 ^ 1 (1) 

h . *37-41 . 3 I- : Hp . D . G'(Ps I D'P.) C a'Q,^ (2) 

I- . (1) , (2) . *254-261 . *151-22 .Oh. Prop 

In virtue of the above proposition, we have, when its hypothesis is 
realized, 

(QstD'QOsmor(P,tD'Ps), 

whence, by *253'25, Q smor P. 

This proposition is the converse of *254"141. 

In the above proposition we take Qam [ G'(Ps I D'Ps) as the correlator, 
rather than Q^^ f" D'Ps, so as not to have to make an exception for the case 
when P e 2,. For if P e 2,, D'Ps e 1, but Ps I D'Ps = A. Thus Q,^ [ D'Ps is 
not a correlator in this case. 

The following propositions, down to the end of the present number, are 
important, and give the foundations of the theory of inequality between well- 
ordered series and between ordinals. 

4—2 



52 SERIES [part V 

*254-31. h : P, Q e fl . B'Ps C Q'Q.^ . D'Qs C a'P,„ . D . P smor Q 
Dem. 

V . *254-27 . D h : . Hp . D : (Ps t D'P,) smor (Qs p D'Qs) : 

[*253-25] D:a!P.a[!Q.D.PsmorQ (1) 

h . *254144 .DI-:Hp.P = A.D. D'Q, = A . 

[*213-302] D.Q = A. 

[*153-101] D.PsmorQ (2) 

Similarly h : Hp . Q= A . D . Psmor Q (3) 

h . (1) . (2) . (3) . D h . Prop 

*254-311. h :. P, Q 6 fl . D : D'Ps C a'Q^ . B'Q, C a'P,^ . = . P smor Q 
[*254-31-141] 

*254-32. \-:P,Qen. B'P, C Q'Q.^ . 8 e D'Q, - a'P,^ . D . D'Ps C a'S,^ 

Pern. 
l-.*254-24. Di- -.n^ . B,8' eB'Qs . S' (LR.Rea'P,^.D. S'ea'P,^ (1) 
h . (1) . Transp . D h : Hp . E e D'Q, a a'P,„ . D . ~ (^f G i?) . 
[*213-21] D.RQsS. 

[*254-22-ll.*213-245] D . (Pem'^) smor i2 . i? e D'Ss . 

[*25412] D . (P^JR) e a'8,,, (2) 

h . (2) . *37-61 . D h : Hp . D . P,r^"(D'Q, n Q'P, J C Q'/Sf.^ . 
[*254-164] D . D'Ps C a'/S^n, : D I- . Prop 

*254-321. h:P,(3en.D'Ps C a'Q,^ . 8=mm{Q,y(D'Qs - a'P,J . D.SsmorP 
Dem. 

h . *205-14 . D h : Hp . D .^,'8 C a'P,„ . 
[*213-246] D . D';Ss C G'P,^ (1) 

I- . *254-32 . D I- : Hp . D . D'Ps C a'S,^ (2) 

h . (1) . (2) . *254-31 . D I- . Prop 

*254-33. \-:P,Qea. D'Ps C a'Q,„ . a ! D'Qs - Q'P.^ . D . P e Q'Q.^ 
Pern. 

I- . *253-24 . D h : Hp . 3 . E ! min (Qs)'(D'(3s - a'P,„) . 
[*254-321] D . (g/S) . 5f e D'Qs . S smor P . 

[*25411] D.Pea'Q3„:Dl-.Prop 

*254-34. h : P, Q e fl . P ~ e O'Q.^ . D'Ps C a'Q,„ . D . P smor Q 

h . *254-33 . Transp . D h : Hp . D . D'Qs C d'P,^. D'Ps C a'Q,„ . 
[*254-31] D . P smor Q : D h . Prop 



SECTION D] greater AND LESS AMONG WELL-ORDERED SERIES 53 

*254-35. F : P, Q e n . a ! D'Q, - Q'P^ . 3 . P e a'Q,^ 
Bern. • 

h . *253-24 . D h : Hp . D . E ! min {Q,y(D'Qs - Q'P, J . 

[*20514] D . (aS) . -S 6 D'Qs - Q'P^ . 'q,'8 C O'P,^ . 

[*213-246] D.{^S).Se B'Q, - a'P,„ . D',S, C Q'P.^ . 

[*254-34] D . (a^S) .SeB'Q^.S smor P . 

[*254-ll] D.Pea'Q^^iDh.Prop 

*254-36. h : P, Q e O . a ! D'Q, - a'P„„ . D . G'Ps C Q'Q,^ [*254-35143] 

*254-37. l-i.P.Qen.DiPsmorQ.v.Pea'Qs^.v.Qea'P^^ 
Bern. 
h . *254-31 . D h : Hp . D'P, C a'Q,„ . D'Q, C Q'P,^ . D . P smor Q (1) 
h . *254-35 . D F : Hp . a ! T>'Q, - a'P,^ .D.Pe a'Q,^ (2) 

I- . *254-35 . D h : Hp . a ! 'D'P, - a'Q,^ . D . Q e O'P^^ (3) 

h . (1) . (2) . (3) . D h . Prop 

This proposition is the most important on the relations of two well- 
ordered series to each other's segments. It shows that of every two 
well-ordered series which are not similar, one must be similar to a segment 
of the other. 

*254-4. \-:.P,Qen.':i:P less Q.v.P smor Q.v.Q less P 
Bern. 

l-.*254-2. DhiHp.Pea'Qg^.D.PlessQ (1) 

h.*254-2. DI-:Hp.Qea'P,^.D. QlessP (2) 

I- . *254-37 . D f- : Hp , P~ eO^Q.^ . Q~ea'Pa„ . D . Psmor Q (3) 
1- . (1) . (2) . (3) . D h . Prop 

*254-401. h :. P, Q 6 n . 3 : less'P = less'Q . = . P smor Q 
Bern. 

h . *254-l . D h : Hp . less'P = less'Q . 3 . ~ (P less Q) . ~ (Q less P) . 
[*254-4] D.PsmorQ (1) 

h . *2o4-13 . D f- : Hp . P smor Q . D . less'P = less'Q (2) 

I- . (1) . (2) . D h . Prop 

*254-41. h : P less Q . = . P, Qe fl . PeQ'Q.^ . = . Qefl . Pe a'Q,„ 
Bern. 

h.*254-2. 0\-:Qea.Pea'Q,^.:>.Ples8Q (1) 

I- . #254-181 . Oh-.Qe Q'P,^ . D . ~ (P less Q) (2) 

l-.*253-421 . DI-:Q6n.Jt:eD'Qs.Psmori?.D.~(PsmorQ): 

[*254-ll] Dh:Qefl.Pea'P,„.D.~(Psmor(3) (3) 

|-.(2).(3).*254-4.DI-:Q6n.Pea'Ps„.D.PlessQ (4) 

h.(l).(4). DI-:PlessQ. = .Q€n.P6a'Q,„. 

[*254-l] =.P,Qen.Pea'Q,„:DH.Prop 



54 SERIES [part V 

*254-42. h . less G J . less'' G less 
Bern. 

I- .*254-l . D V : Pless Q . 3 . ~ (Psmor Q) . 
[*151-13] 3.-P+Q (1) 

I- . *254163 .Oh: Re Q'Q.^ . S e a'R,^ .O.Se a'Q,^ : 
[*254-41] Db-.R less Q . S less R.D.S less Q (2) 

I- . (1) . (2) . D I- . Prop 

The relation "less" fails to generate a series, because it is not connected, 
two similar well-ordered series being neither greater nor less than each other. 
On the other hand, the relation NrUess is serial, since two similar well- 
ordered series both contribute the same term to the field of Nr'less, and 
therefore connection does not fail. The relation Nr'less will be dealt with in 
the next number. 

*254-43. hzQen-i'A.D.AlessQ [*2541 .*250-4 .*1.52-11] 

*254-431. I- . a'less = O - I'A . C'less C D, 
Dem. 

l-.*2.54-43. DhsQefi-i'A.D.AlessQ (1) 

h . *254-l . *25-13 . DI-:Q = A.D.Q~6a'less (2) 

h . *254-l . D h . Cless C H (3) 

I- . (3) . (2) . Transp . D h . Q'less C fi - t'A (4) 

l-.(l).(4). D t- . a'less = a - t'A (5) 

h . (3) . (5) . D I- . Prop 

In order to obtain C'less = li, we need, as appears from (1) in the above 
proof, a ! O - I'A. In virtue of *251-7, this requires a ! 2. By *101"42-43, 
this holds if " less " has its field defined as belonging to a class-type or a 
relation-type. If, however, " less " has its field defined as composed of 
individuals, the primitive propositions assumed in the present work do not 
enable us to prove g ! 2, nor therefore to prove g; ! less. 

. It should be observed that "less," like "sm" and "smor," is significant when 
it is not homogeneous ; but " (7'less " is only significant for homogeneous typical 
determinations of " less," because only homogeneous relations have fields. 

*254432. 1- : a ! 2a . H . a ! less h t^'a f i^'a . = . a ! fi- t'A n t^'a 
Bern. 

V . *251-7 . D h : a ! 2„ . = . a ! ii - t'A n i„„'a . (1) 

[*254-43] = . (aQ) . Q 6 fl - t'A n «(K,'a . A less Q . 

[*55-37] D . (aQ) . A less Q .k^QQ.t^'a'^ t^'a . 

[*55-3] 3 ■ a ! less n ^^'a f t^'a (2) 



SECTION d] greater AND LESS AMONG WELL-ORDERED SERIES 55 

h . *35-103 . D h : a ! less A t^'a t «„„'« . D . (gP, Q) . P less Q.P,Qe t^'a . 
[*254-431] • D . a ! n - t'A ft foo'a • 

[(1)] D . a ! 2„ (3) 

h . (1) . (2) . (3) . D h . Prop 

*254-433. h . a ! less n t^'Ch f tJCla . a ! less n <„o'Rel f <oo'Rel 
[*254-432 . *101-42-43] 

*254-434. 1- : a ! less . = . C'less =0,. = . B'less = A 

Dem. 

h . *250-4 . *33-24 . D h : a'less = n . D -a ! less (1) 

h . *93-102 . *33-24 . D h : 5'less = A . D . a ! less (2) 

|-.*254-43. DHiQefi-t'A.D.AlessQ (3) 

l-.(3). DhiaSn-i'A.D.AeD'less. 

[*254-431] D . A = £'less (4) 

l-.(4).*254-431. . Dh:a!f^-i'A.D.O'less=fl (5) 
l-.(l).(2).(4).(5).Dh.Prop 

*254-44. h : P 6 G'less . D . Cless = less'P u Nr 'P w less' P 

Z)em. 
|-.*25413. DI-:Hp.D.Nr'PCC"less (1) 

I- . (1) . *33-152 . D I- : Hp . 3 . less'P u Nr'P w less'P C O'less (2) 

I- . *254-l . D h . O'less C 0, . 

i*254-4] D h :. P e Class . D : Q e G'less .D.Qe less'P w Nr'P u less'P (3) 
I- . (2) . (3) . 3 F . Prop 

*254-45. \-■.P,Qen.^^lB,]'Pr^'Nr'Q.'3_l'RVQf^m'P.:^.PsmorQ 

Bern. 

h . *254-42 . D h : P less Q . D . ~ (Q less P) (1) 

|-.*2541. Dt-:P,Q6i:2.a!iy'QftNr'P.~(PsmorQ).D.Ple3sQ. 
[(1)] D.~(QlessP). 

[*254-l.Transp] D . ~ a ! ^1'^ '^ Nr'Q (2) 

h . (2) . Transp . 3 h . Prop 

This proposition is the analogue, for ordinals, of the Schroder-Bernstein 
theorem. 



56 SERIES [part V 

*254-46. h : P less Q . = . P, Q e fl . g ! Rl'Q n Nr'P . ~ g ! Rl'P n Nr'Q 
Z)em. 

I-.*152-11.*61-34.D 
h : P, Q e fi . a ! Rl'Q n Nr'P . ~ g ! Rl'P n Nr'Q . D . 

P,Qen.a!Rl'QnNr'P.~(PsmorQ). 
[*254-l] D.PlessQ (1) 

|-.*2541-45.Transp. D 

h : Pless Q. D . P, Qell . g ! Rl'Q n Nr'P . ~ g ! Rl'P n Nr'Q (2) 

I- . (1) . (2) . D h . Prop 

*254-47. I- : Pen. D.P5 = less ^C'Ps 
Bern. 
I- .*213-245 . D I- :. Hp . D : RP,Q . = .PeD'Qs . QeC'P, . 
[*254-121] D.Rea'Q,^. 

[*254-41] D.ElessQ (1) 

h . *254-181 . Transp . D h : Hp . Q, P e G'P, . R less Q . D . Q ~ e a'i?,„. . 
[*254-121] D . Q ~ e D'P, (2) 

I- . (2) . *213-25 . *254-42 . D h : Hp . Q, P e C'Ps . R less Q.D.Re B'Qs . 
[*213-245] D . PPsQ (3) 

h . (1) . (3) . D I- . Prop 

*254-5. l-:.P,QeIi.D: 

Rl'P n Nr'Q = A . = . a ! Rl'Q n Nr'P . ~ (P smor Q). = .P less Q 
Pern. 

l-.*254-46. Dh:Hp.Rl'PnNr'Q = A.D.~(QlessP) (1) 

h . *61-34 . *15211 . D h : P smor Q.D.Pe Rl'P n Nr'Q (2) 

f- . (2) . Transp . D h : Rl'Pn Nr'Q= A. D .~(Psmor Q) (3) 

h . (1) . (3) . *254-4 . D I- : Hp . Rl'P n Nr'Q = A . D . P less Q (4) 

t-.*254-46. Dh:PlessQ.D.Rl'PftNr'Q = A (5) 

h . (4) . (5) . D h :. Hp . D : Rl'P n Nr'Q = A . = . Pless Q . 

[*254-l] = . g ! Rl'Q n Nr'P . ~ (P smor Q):.-^^-. Prop 

*254-51. h:PlessQ. = .P,Qen.Rl'PftNr'Q=A [*254-5-l] 

*254-52. 1- : P 6 n . a C O'P . g ! C'P n 2j'p""o . D . P ^ a less P 

Pern. 

l-.*250-141.Dh:Hp.D.P^a6fl (1) 

I- . *250-653 . D h : Hp . 3 . ~(P ^ asmorP) (2) 

h . (1) . (2) . *254-101 . D h . Prop 



SECTION D] greater AND LESS AMONG WELL-ORDERED SERIES 57 

*254-53. I- : P, Qeil . QGP. a ! C'P np^"G'Q . D . QlessP 

Bern. 

|-.*250-652.3l-:Hp.D.~(QsmorP) (1) 

f- . (1) . *254-101 . D F . Prop 

*254-54. h:P,Qen.R smor Q . iJ C P . a ! O'P np'^'O'R . D . Q less P 
[*254-53-13] 

*254-55. l-:.QlessP.-=:P,Qen:(ai?).i?smorQ.iJGP.a!0'Pnp'P"C'jB 
Dem. 
h . *254-41 . D h :. QlessP . D : P, Q efi : (gi?) . iismor Q . ReB'Rs : 

[*21318] D : P, Q 6 fi : (gii) . R smor Q . i2 G P . g ! O'P n p'P"G'R (1) 
h . (1) . *254-54 . D h . Prop 



*255. GREATER AND LESS AMONG ORDINAL NUMBERS. 

Summary of *255. 

If P and Q are well-ordered series, we say that Nr'P is less than Nr'Q if 
P is less than Q. Thus if fj. and v are ordinal numbers, we say that /i is less 
than V if there are well-ordered series P, Q, such that /i = Nr'P and i/ = Nr'Q 
and P is less than Q. In order to exclude the case where, in the type 
concerned, we have Nr'P = A or Nr'Q = A, we assume ^ = Nor'P and 
j' = N„r'Q. Thus we put 

/i < i; . = . (gP, Q) . ^ = N„r'P . 1/ = N„r'Q . P less Q, 

i.e. we put <S=Nor'less Df. 

In order to be able to speak of Nr'P (where the type of "Nr" is left 
ambiguous) as greater or less than ^r'Q, we put 

fi < Nr'P . = . yii < N„r'P Df, 

Nr'P < /i . = . Nor'P < /* Df 

The treatment of types proceeds, mutatis mutandis, as in *117, to which, 
together with the prefatory statement in Vol. Ii, the reader is referred for 
explanations. 

In virtue of *254'46 and *117'1, there is a close analogy between cardinal 
and ordinal inequality. That is to say, most of the properties of cardinal 
inequality have exact analogues for ordinal inequality, and these analogues 
have analogous proofs. (In the present number, when a proposition is 
analogous to the proposition with the same decimal part in *117, and has 
an analogous proof, we shall omit the proof) But ordinal inequality has a 
good many properties which have no analogues for cardinal inequality. The 
chief of these, upon which most of the rest depend, is 

^255*112. \- :. /i,ve NjO . D : /i < v . v . /* = smor"i^ . v . v < /i 

where " NjO " stands for " homogeneous ordinals," i.e. NO n NoR. We have 
also, what is often important, 

*25517. (- : Nr'P> Nr'Q . = .Q less P . = . P, Q e fl . Q e Q'P.^ . 

= . P, Q 6 n . a ! D'Ps r^ Nr'Q 



SECTION D] greater AND LESS AMONG ORDINAL NUMBERS 59 

SO that 
*255171. f- :. P 6 n . D : •< Nr'P .= .^,e Nr"D'Ps - I'A 
and more generally, 

*255172. l-r.PeXl.D: 

/i < Nr'P . = . (ga) . a C C'P . g ! C'P n p'*P"a . /* = Nr'P f a . g ! /* 

As in cardinals, /j, is greater than v if (and only if) fi is the sum of v and 
an ordinal other than zero, including 1 except when v = Or (*255-33). But it 
is necessary to the truth of this proposition that the addendum should come 
after v, not before it ; i.e. p + zt^v unless ot = 0, (*255'32-321), but •sr + vis 
often equal to v. 

If a, /3, y are ordinals, and a •> yS, we shall have 

7 + a > 7 4- /8 (*255-561), 

aX;S>/8 if a + O^.^ + Or (*255'571), 

«X7>/SX7if7=|=0, (*255-58), 

7 X j8 > 7 if 7 is of the form 8+1 (*255-573), 

7 X a > 7 X )S if 7 is of the form B+i (*255-582). 

From the above propositions it follows that if a, ^, 7 are ordinals, 

y + a = 7 + 73.D.a = j8 

(*2o5'565, where /3 may be substituted for smor"/8 whenever significance 
permits; cf. note to *120'413), which gives the uniqueness of subtraction 
from the end (subtraction from the beginning is not unique); 

aX7 = /3X7.3.a = /8 unless 7 = 0^ (*255-59), 
which gives the uniqueness of division by an end-factor ; 

7Xa = 7X/3.D.a = /3 if 7 = S+i (*255-591), 
which gives the uniqueness of division by a beginning-factor of the form 

B + i. 

We do not have generally 

tt,/3,ye NqO . a < /S . D . a exp^ 7 < yS exp, 7, 
because aexprj ^^^ /Sexpy7 are in general not ordinal numbers, since series 
having these numbers are in general not well-ordered. Thus the theory of 
ordinal inequality has only a restricted application to exponentiation. This 
subject cannot be adequately dealt with until we have considered finite and 
infinite series. 

If a is an ordinal, C"a is the corresponding cardinal, i.e. the cardinal 
number of terms in a series whose ordinal number is a. Thus the cardinal 
numbers of classes which can be well-ordered are C'"NO, i.e. 

*255-7. h . Nc"a"Ii = C"'NO 



60 SERIES [part V 

It is evident that 
*255-71. h : P less Q . D . Nc'O'P < Nc'O'Q 

whence, by *254'4, 
*255-73. l-:.P,Q6fl.D: 

Nc'C"P< Nc'O'Q . V . Nc'C'P = Nc'C'Q . v . Nc'C'P > Nc'G'Q 

whence also 

*255-74. h:.a,^e C""NO -t'A.D:a</3.v.a>/3 

Thus if two classes can both be well-ordered, they either have the same 
cardinal, or the cardinal of one is less than that of the other. 

We have 
*255-75. h : P, Q 6 n . Nc'C'P < Nc'O'Q .D.P less Q 

or, what comes to the same thing, 
*255-76. h : a, ;e 6 NO . G"a < C'/S .D.a<^ 

The converse of this proposition only holds for finite ordinals. If a is an 
infinite ordinal, a + 1 always exists and is greater than a, but G"a= C"{a. + 1). 
(The existence of a -i- i is deduced from that of a by taking a member of a, 
and removing its first term to the end. The result is a series whose number 
is a -i- i, in virtue of *253-503-54.) 



*25501. <=N„rness Df 

*25502. > = Cnv'< Df 

*25503. NoO = NOnN„B, Df 

Thus "NoO" means "homogeneous ordinals." In virtue of *155'34"22, 
this is the same as "ordinals other than A." It is not, however, strictly 
correct to put N„0 = NO - I'A, because if the " NO " on the right is derived 
from an ascending Nr, it will not contain all the ordinals in the type to which 
it takes us, but only those which are not too big to be derived from the lower 
type from which "Nr" starts. Thus in this case NoO will be a larger class 
than NO - t'A. If, however, the " Nr " from which the " NO " on the right 
is derived is homogeneous or descending, we shall have 

N„0 = NO - I'A. 
*25504. ^ = <c;smore^NoO Df 

This definition leads to the usual meaning of " less than or equal to." We 
want the relation " less than or equal to " to hold only between numbers of 
the sort in question (cardinal or ordinal), and we want " equal to " to hold 
between two numbers which are merely different typical determinations of a 
given number, provided neither of these typical determinations is A. That 
is, if fi is an ordinal which is not A, smor"/A is to be reckoned equal to fi in 
every type in which it is not A. Thus ii v = smor"/*, i.e. if v = smoie'fi, we 



SECTION D] greater AND LESS AMONG ORDINAL NUMBERS 61 

shall reckon v equal to fi if both are ordinals and neither is A, i.e. in virtue of 
*155"34-22, ii fi,ve NoO. «This leads to the above definition. 

*25505. ^ = Onv'^ Df 

*25506. itt < Nr'P . = .//,< Nor'P Df 

On this definition, compare the remarks on *117'02. 

*255-07. Nr'P < /i . = . Nor'P < /* Df 

The following propositions (down to *255'108) merely re-state the above 
definitions. 

*255-l. h : /i <j' . = . (gP, Q).fJ.= N„r'P . v - N„r'Q . P less Q 

*255101. 1- : At < Nr'Q . = .^ < N„r'Q 

*255102. h : Nr'P <!/. = . N„r'P < v 

*255103. \-:fi>v.= .v<fi 

«255'104. \-:./i^v. = :/Ji<iV.v.fi,ve NoO . /a = smor"j/ 

*255105. \- :./i ^v . = :v^/j,: = :v<.fi.v./j,,ve NoO . /ij= smor'S 
[*25^^4 . (#2550^) . *1 55;^4] 

*255106. h : Nr'P -^ Nr'Q . = . Nor'P < Nor'Q [*255--101102f 

*255107. h : Nr'P ^ Nr'Q . = . Nor^P ^ Nor'Q 

*25510a h :. Nr'P ^ Nr'Q . = : Nor'P < Nor'Q . v . Nr'P = Nr'Q .Peil 
[*255-107-104 . *155-16 . *152-53] 

*25511. h :/*<!/. = .(aP,g).P,Q€n.^ = Nor'P.i/ = Nor'Q. 

a ! Rl'Q n Nr'P . ~ a ! Rl'P n Nr'Q [*255-l . *254-46] 

*255-lll. f- : ^ > v . = . (aP, Q).P,Qen.fi = Nor'P .v = N„r'Q . 

a ! RHP r. Nr'Q . ~ a '■ Rl'Q ^ Nr'P [*255-ll-103] 
This proposition is exactly analogous to *117"1, except for the addition 
P, Q e fl. Hence except where this addition is relevant, the analogues of the 
propositions of *117 follow by analogous proofs. Such analogues will be 
given without proof in what follows, and will have the same decimal part 
as the corresponding propositions in *117. Where proofs are given, there 
are no analogues in *117, or else the method of proof is not analogous. 

*255112. I- :. n, v eNoQ . 3 : /i < v . v .fji=smor"v .v.v<fi 

Dem. 
h . *255-l . *254-4 . D h :. Hp . D : 

/i < 1/ . V . K < M . V . (aP, Q) ■ -P> ;@ e ^ ■ /^ = Nor 'P . I' = N„r 'Q . P smor Q : 

[*155-4.*152-321] 

D : /* < I' . V . j; < /. . V . (aP, Q) . /x -^ N„r'P . Nr'P = Nr'Q . Nr'Q = smor"!/ : 

[*155-16] 

D :/.<»/. V . 1/ </*. V . (aP, Q) ./* = Nor'P . Nor'P = Nr'Q . Nr'Q = smor"z; : 

[*1317] D:/i<v.v.j/</i.v.A' = smor"i; :. 3 h .Prop 



62 SERIES [part V 

*255-113. I- :. P, Q e n . D : Nr'P < Nr'Q . v . Nr'P = Nr'Q . v . Nr'^ < Nr'P 

Bern. 
l-.*255112-106.DI-:.Hp.D: 

Nr'P < Nr'Q . v . N„r'P = smor"N„r'Q . v . Nr'Q < Nr'P : 
[*155-4-16] D : Nr'P < Nr'Q . v . Nr'P = Nr'Q . v . Nr'Q < Nr'P :. D h . Prop 
«255-114. \-:.fi,ve N„0 .D:/A^i'.v.z/<;a:/i^v.v.i'>/i 
[*255112-104-105-103] 

*255115. h :. P, Q 6 n . D : Nr'P^^ Nr'Q .v. Nr'Q < Nr'P: 

Nr'P ^ Nr'Q. V. Nr'Q > Nr'P [*255-113108] 

*25512. l-:./i>i'. = i/tji/eNoO : 

P 6 /i . Q 6 1/ . Dp,Q . a ! Rl'P n Nr'Q . ~ a ! Rl'Q n Nr'P 

*255-121. f-:./i>i'. = :/i,i'6 NjO :• 

Pefj,.Dp. (aQ) . Q 6 1/ . a ! Rl'P -^ Nr'Q . ~ a ! Rl'Q <^ Nr'P 
*255-13. f- : Nr'P > Nr'Q . = . P, Q e fl . a ! Rl'P ^ Nr'Q . ~ a ! Rl'Q <^ Nr'P 

*255131. h : Nr'P > Nr'Q . = . Nr'P ^ Nr'Q . Nr'P =j= Nr'Q 

[*25o-13 . *254-4i5] 
*25514. h : /i > V . = . (aP, Q) ■ P, Q 6 n . /Lt = N„r'P . i; = N„r 'Q . Nr'P > Nr'Q 
*255141. \- : ij,>>v . = . fM^v . fj,^smoi"v [*255-131-14] 
*25515. h :/i> y. = ./*, i/eNoO. a!s'Rl"/"^smor"j/. ~a!s'Rl"z/nsmor"/i 

*25516. l-:./i,«/eN„O.D: 

fi'>v . = . smov" fi •> i; . = . /i •> smor"!/ . = . smor"/i •> smor"!/ 

*255-17. h : Nr'P > Nr'Q . = . QlessP . = .P, Q efl . QeQ'P.^ . 

= .P,Qeft.a!D'P.'^Nr'Q 
Dem. 

h . *25513 . *2o4-46 . D h : Nr'P > Nr'Q . = . Q less P . (1) 

[*254-41] =.P,Q6ft.Q6a'P3^. (2) 

[*25412] H.P,Qen.a!D'PsnNr'Q (3) 

1- . (1) . (2) . (3) . D h . Prop 

*255171. \-:.PeD,.':):iJ.< Nr'P . = ./*£ Nr"D'Ps - I'A 
Dem. 
H .*26514 . D h :. Hp . D : /i < Nr'P . = . (aQ) ■/* = N„r'Q . Nr'Q < Nr'P . 
[*255-l7] = . (aQ) . M = Nor'Q .QeCl.^lD'F, n Nr'Q . 

[*152-1] = . (aQ, R)./M= N„r'Q .Qen.Q smor R.ReD'P,. 

[*lo2-35.*15516] H . (aii) . fi = Nr'i? .Reil.Re D'Ps . a ! /* • 
[*25318.*37-6] = . /t e Nr"D'Ps - t'A :. D h . Prop 



SECTION D] greater AND LESS AMONG ORDINAL NUMBERS 63 

*255172. hz.Pea.O: 

/t < Nr'P . = .tan) .aCC'P.^lG'F np''p"oi . /* = Nr'P^ « ■ 3 ! M 
Bern. 

h . *211-703 . *213-141 . D 

V iQeD'Ps . D . (aa) . a C C'P . a ! O'P n_p'P"a . Q = Pp a (1) 
I- . (1) . *255171 . D K: Hp . ;u, < Nr'P . D . 

(aa) . a C O'P . a ! C'P np'^'a . /* = Nr'Pf a . a ! /* (2) 
h . *250-653 . *254-47 . D 

h : Hp . a C C'P . a ! G'Pnp'p'"a . D . P^ aless P . 
[*255-l7] 3 . Nr'P I a < Nr'P (3) 

h . (2) . (3) . 3 f- . Prop 
*255173. hi.PeXl.D: 

Nr'Q < Nr'P . = . (aa) . « C C'P . a ! G'P n jo'P"a . Q smor (P f a) 
i)em. 

l-.*255-172-102.*155-22.D 

h :.Hp.D:Nr'(3< Nr'P. = .(aa).« C (7'P. a! C''Pnp'P"a . N„r'Q=Nr'Pp a . 
[*152-35.*155-22] =.(aa). a C C'P. a ! C"P n^'P""a.Q smor (Pp a) : D h. Prop 
*255174. h : Nr'Q < Nr'P . = . P e fl . Nr'Q e Nr"D'Ps 
Dem. 

h.*255-iril02-13.D 



F:.Nr'Q<Nr'P. = ; 
[*37-6.*155-22] = ; 
[*lo5-16] 
[*37-6] 



: P 6 n . N„r'Q 6 Nr"D'Ps - I'A : 
i : Pen : (ai?) .-ReD'Ps . N„r'Q = Nr'ii : 
:Peil: (-^R) . R e 'D'P, . Nr'Q = Nr'P : 
: : P e n . Nr'Q e Nr"D'Ps :. D h . Prop 
*255175. l-:Nr'Q^Nr'P.s.P6Xl.Nr'QeNr"(D'Psui'P) [*255174-108] 
*255176. h :. a ! -P ■ 3 : Nr'Q ^ Nr'P . = . P e O . Nr'Q e Nr'C'Ps 

[*213-158 . *255-l75] 
*255-21. l-:Nr'P<Nr'Q. = .P,Q6n.Rl'PnNr'Q=A [*254-51 .*255-17] 
This proposition has no analogue in cardinals, because it depends upon 
*254-4. In cardinals, if Cl'anNc'yS =A, it does not follow that g! Cl'/SoNc'a, 
so that Nc'a may be neither less than, nor equal to, nor greater than Nc'/S. 

*255211. 1- :. P, Q e n . D : a ! B.\'Pn Nr'Q . a ! Bl'Q r^ Nr'P. = . Nr'P= Nr'Q 
[*254-45] 

This proposition is the ordinal analogue of the Schroder-Bernstein theorem. 
If P and Q are series which may be not well-ordered, the proposition fails. 
Thus e.g. the series of rationals is like the series of proper fractions, which is 



64 SERIES [part V 

a part of the series of rationals > and ^ 1, and this latter series is part of 
the series of rationals, but is not similar to the series of rationals, since it has 
a last term, which the series of rationals has not. 

*255-22. t- : P, Q 6 n . a ! Rl'P n Nr'Q . = . Nr'P ^ Nr'Q 

*255-221. I- :. Nr'P ^ Nr'Q . = : P, Q e li : (gj?) .RQP.R smor Q 

*255-222. y-.QCP.P.Qeil.-D. Nr'P ^ Nr'Q 

*255-23. h : Nr'P ^ Nr'Q . Nr'Q ^ Nr'P . = . P, Q e fl . Nr'P = Nr'Q 

*255-24. \-:fjL^v. = . (gP, Q) . /t = N„r'P . v = N„r'Q . Nr'P ^ Nr'Q 

*255-241. l-:/i^i/.s.(aP,Q).yci = N„r'P.z. = N„r'Q.P,Q6fi.a!Rl'PnNr'Q 

*255-242. h :./i,i;eNO . D : ^^1/ . = . (gP.Q) . Pe/t . Qey . g ! Rl'P n Nr'Q 

*255-243. t-:.ya^i/. = : 

(gP, Q) : P, Q eXi . /i = N„r'P. i; = N„r'Q : (gi?) .RCP.R smor Q 

*255-244. I- :. /i, i; e N„0 . D : 

/4 ^ i; . = . smor"/i ^ y . = . /^t ^ smor"i/ . = . smor"/ii ^ smor"!/ 

^255°25. hi/x^j'.i/^/i.s./i, i/e NqO . smor"/i = smor"v 

*255-27. H : Nr'P < Nr'Q . = . Nr'P ^ Nr'Q . Nr'P 4= Nr'Q 

*255-28. h : Nr'P > Nr'Q . = . Nr'P ^ Nr'Q . ~ (Nr'Q ^ Nr'P) . 

= . P, Q 6 XI . ~ (Nr'Q^ Nr'P) [*255-13-22-21] 

*255-281. t-:/i>z/.H.y(i^i'.~(z/^/i). = .yLi,i;6N„0.~(i/^/i) [*255-114] 
*255-29. h : Nr'P < Nr'Q . = . Nr'P ^ Nr'Q . ~ (Nr'Q ^ Nr'P) . 

H . P, Q 6 Xi . ~ (Nr'Q ^ Nr'P) [*255-115] 

*255-291. f-:y[t< v. = ./t^i'.~(i/^/i). = ./x,,i/eNoQ.~(z/^/i) [*255-114] 

In the following proposition, we employ an abbreviation which is justified 
by its convenience, namely we put 

(gw) . in- e NO u I'i . Nr'P = Nr'Q -i- la- 
in stead of 

(at!7) . tB- 6 NO . Nr'P = Nr'Q + ti7 . v . Nr'P = Nr'Q + 1. 

In virtue of *51'239, these two expressions would be equivalent if 1 had any 
independent meaning; but as 1 is only significant as an addendum, *51-239 
cannot be applied. We will, however, adopt the following definitions : 

*255-298. (g[CT).OTe«wi'l./(/i + i!j). = :(aii7).CT6«:./(/:i4-cr).v./(;t4.1) Df 

*255-299. ■Bre«wt'i.D^./(/i + 'sr). = :OTeK.D^./(/i + i!r):/(/i + i) Df 

These definitions enable us to state many propositions, in which 1 occurs 
as though 1 were an ordinal number. 



SECTION D] greater AND LESS AMONG ORDINAL NUMBERS 65 

*255-3. t-:.Nr'P^Nr'Q. = :P,Q e f2:(aOT) .^ eNO u ta.Nr'P=Nr'Q+ w 

i)em. • 

l-.*255-175.*253-471.D 

h :. Nr'P^ Nr'Q . = : Pefi : (3^,7) . Nr'Q + t!r= Nr'P . v . Nr'Q + 1 = Nr'P : 
I*251-132-26] = : P e fl : (gw) . Nr'Q, w e NO . Nr'Q + tir = Nr'P . v . 

Nr'Q 6 NO . Nr'Q + 1 = Nr'P : 
[*251-1111] =:P,Qeil: (aisr) . lii e NO . Nr'Q + •sr = Nr'P - v . 

Nr'Q-i-l = Nr'P: 
[(*255-298)] = : P, Q eO : (gtir) . ^ eNO u t'l . Nr'P = Nr'Q + i!r :.D h . Prop 

«255-31. f- :./i^i;.= :/i, i/eNoO : (307). we NO u t'l . /* = !; + «■ 
[*255-3-14] 

*255-32. h :. I/, 1SS- e NjO .D:i; + w>J'. = .'SJ- + Or 
Dem. 

l-.*253-44. DhiHp.OT + O^.D.i' + i + i' (1) 

1-.*255'31. DI-:Hp.D.i/ + iiJ-^i' (2) 

h . (1) . (2) . *255141 . D h : Hp . OT + 0, . D . i/+ii7 > j; (3) 

l-.*255'141 . Dl- :Hp . v + ■oi' > v . D .i; + i3-=t=smor"v. 

[*180-6] D. ■574=0, (4) 

h . (3) . (4) . D 1- . Prop 

*255-321. 1- :. 1/ 6 N„0 .D:z/4=0,. = .i' + i>i' 

l-.*253-4.5,. Dt-:Hp.i'=t=0, .D.v-i-l + i' (1) 

|-.*255-31. Df:Hp.D.i/-t-l^i/ <2) 

I- . (1) . (2) . *255-141 . D I- : Hp . v + 0, . D . v 4- i > v (3) 
f-.*255-141. DF:Hp.i/ + i > i/ . D . 1/ + 1 =t= smor"i/ . 

[*161-2] D.i' + O, (4) 

h . (3) . (4) . 3 h . Prop 

*255-33. \-:.fi>>v. = : 

/jL,ve NoO : (aw) .in-eNO-t'O, ./i = i'-i-'iB-.v.z/4=0,. ./t = j'+i 

F . *255-31 . D 

\-:./j,->v.= :/i, i'eNoO:(a'n7).'5reNO./ii=i' + '=r./i> j/ . V . /t = v + i./*>i' : 

[*255-32-321] 

= :/i,ve NoO : (gw) . w e NO - t'O,. ./j. = v + -sT.v.v^Or.fi = v + i:.':> l-.Prop 

E. &W. III. 5 



66 SERIES [PART V 

*255'4. hi/Li^i/.z/^OT.D./i^OT 

*25541. \-: /jb^v.v^TiT.D.fi^iiT 

*255-42. 1- . ~ (/i > /i) : ~ (/A < /i) 

*255-43. I- : /* ^ v . ~ (/i ^ ot) . D . ~ (i/ ^ -st) 

*255-431. h : yu, ^ 2/ . •ST 6 NoO . c^ {/j,^ ■sr) . . ■ur •> v [*255-43-114] 

*255'44. I- : v ^ OT . ~ (/* ^ •ar) . D . ~ (/tt ^ i*) 

*255-441. \-:v^'!S7./jL6 NoO . ~ (/a ^ tsr) . D . v > /a [*255-44-114] 

*255'45. t-:|U.^z'.z/>t3-.D./A>'!ir 

*25546. hi/A^j/.z/^OT.D./i^OT 

*255'47. l-:;ii'>i'.z/>-OT.D./i>'57 

*255-471. l-:yu.<z/.j;<i!7.D./i<z!7 

*255-482. l-:yit^i'.= ./i, z/6 N„0 . ~ (i/ > /t) 

*255-483. [-:^<z/. = ./t,i/6NoO.~(z'</i) ■ 

*255-5. \-:fie'NoO. = .fi^Or 
Bern. 

V . *255-31 .Dh:./j.^Or. = ■./jlb'N.O : (aw) . ot eNO u I'l . ^= 0,4-^ : 
[*180-61] = : /i e NoO :. D h . Prop 

*255-51. l-:/i,6N„O-t'0^. = .M>0^ [*255-141-5 . *15315] 

*255-52. l-:Pefi-t'A.s.Nr'P^2^ 
Dem. 

h . *25013 . D I- : P e O - t'A . D . E ! 5'P . 

[*93-101] :>.('3,y).(B'F)Py.B'P^y. 

[*56-ll.*55-3] D . (ay) . (B'P) iy62rn Rl'P . 

[*13-195] D . a ! 2, n Rl'P . 

[*255-22] D.Nr'P^2^ (1) 

I- . *255-22 . D h : Nr'P ^ 2, . D . P e £1 . a ! 2, n Rl'P . 

[*6r361] D.PeJQ-i'A (2) 

F . (1) . (2) . D h . Prop 

*255-53. \-:fj,el!ioO-i'Or.= .fi^2r [*255-52] 

*255-54. h:.2,^/i. = :/* = 0y.v./i = 2y 

I- . *255-53 . Transp . *265-281 . D h : 2^ > /^ . = . /tt = 0^ (1) 

1- . (1) . *25o-105 . D h . Prop 



SECTION D] greater AND LESS AMONG ORDINAL NUMBERS 67 

*255-55. h:fi>>2r.= .iJ.e N„0 - I'O, - 1% 
Dem. • 

V . *255-54 . Transp . *255-281 . D 

[-:/[*> 2^.= ./*6NoO./tt4=Or./i=f=2y:DI-.Prop 

*255-56. I- : E e O . Nr'P > Nr'Q . D . Nr'E + Nr'P > Nr'i? + Nr'Q 

Dem. 
h .*255-3 . D h :. Hp . D : P, Q,i2en : (gisr) . ^eNO u t'l . Nr'P = Nr'Q + =7 : 
[*180-56] 

D : P, Q, i2 e fl : (gisr) . ST 6 NO u I'l . Nr'P -i- Nr'P = (Nr'E + Nr'Q) + ^ : 
[*255-31.*251-26] D : Nr'P + Nr'P > Nr'P + Nr'Q :. D h . Prop 

*255-561. h:7eNo0.a>;8.D.7 + a>7 + /3 [*255-56] 

*255-562. F : P e O . Nr'P ^ Nr'Q . D . Nr'P + Nr'P ^ Nr'P + Nr'Q 

Dem. 
V . *180-3 . D h : Nr'P = Nr'Q . D . Nr'P + Nr'P = Nr'P + Nr'Q (1) 

h . (1) . *255-108-56 . D 

h :. Hp . D : Nr'P + Nr'P > Nr'P + Nr'Q . v . Nr'P + Nr'P = Nr'P + Nr'Q : 
[*255-108] D : Nr'P + Nr'P ^ Nr'P + Nr'Q : . 3 h . Prop 

*255-563. l-:76N„0.a^j8.D.7-t-a^7 + /3 [*255-562] 

*255-564. h : P, Q, P 6 n . Nr'P + Nr'P = Nr'P + Nr'Q . D . Nr'P = Nr'Q 
Bern. 

h . *255-42 . D h : Hp . D . ~ (Nr'P + Nr'P > Nr'P + Nr'Q) . 
[*255-56.Transp] D . ~ (Nr'P > Nr'Q) (1) 

Similarly h : Hp . D . ~ (Nr'Q > Nr'P) (2) 

I- . (1) . (2) . *255113 . D h . Prop 

This proposition establishes the uniqueness of subtraction from the end. 
Owing to the fact that ordinal addition is not commutative, we have to 
distinguish " subtraction from the end " from " subtraction from the 
beginning." They may be called terminal and initial subtraction re- 
spectively. Thus by the above proposition, terminal subtraction among 
ordinals is unique. This does not hold in general for initial subtraction 
among ordinals. 

*255-565. l-:a,/3,7eNo0.74-a = 7+jS.3.a=smor"/3 [*255-564] 

The above proposition is still true if we put a = yS instead of a = smor"/S 
in the conclusion, but in that case it is only significant when a and fi are of 
the same type, whereas in the above form it is free from this limitation. 

5—2 



68 SERIES [part V 

*255-57. h : P, Q 6 fl - t'A . D . Q less (PxQ). Nr'Q < Nr'P X Nr'Q 

Bern. 
h.*250-13. Df-:Hp.D.E!5'P. (1) 

[*165-251] D . Q smor Q J, (S'P) (2) 

h . (1) . *1661 . Dh-:Hp.D.QJ,(5'P)GPxQ (3) 

1- . (1) . *93-101 . D h : Hp . D . (g*) . (5'P) Px (4) 

l-.*166-113.Dl-:(S'P)Pa;.R6G'QJ,(£'P).2/eO'^.D.E(Pxe)(2/J,a;) (5) 
h . (5) . (4) . *33-24 . *166-12 . *113-106 . D 

h:.Sp.D:('^x,yy.ReG'Qi{B'P).DR.R(PxQ)(yix):yixeG'(PxQ) (6) 
h.(2).(3).(6).Dh:Hp.D. 

Q i (B'P) smor Q.Qi (B'P) G P x Q . g ! 0'(P xQ)n p''p^"C'Q i (B'P) . 
[*254-54] D . Q less (P x Q) (7) 

F.(7).*255-17.DI-.Prop 

*255-571. l-:a,yS6NoO-t'0,.D./3<a><;8 [*255-57] 

*255-572. h : P, Q e n - t'A . E ! P'P . D . P less (PxQ). Nr'P < Nr'P x Nr'Q 

Dem. 
y . *25013 . D h : Hp . D . E ! 5'Q . (1) 

[*166111] 0.(B'Q)i>P(iPxQ (2) 

h . *151-64 . (1) . D h : Hp . D . (B'Q) 1 5P smorP (3) 

h . *202-511 . D h :. Hp . D : B'P ep'P"D'P : 

[*166-111] D:a;6D'P.2/6a'Q.D.{(5'Q)|a;}(PxQ){y;(5'P)} (4) 

I- . *202-511 . D h :. Hp . D : B'Qep'Q"a'Q : 

[*166-111] D:x = B'P.yea'Q.D.{{B'Q)ix}(PxQ){yi(B'P)} (5) 

h .i4>) .(5) .Dh -..Rp .:> : xeG'P . y ea'Q.:>.{(B'Q) ix](PxQ){y i(B'P)} : 
[*150-22] D:M€ G'(B'Q) i''P .yea'Q.D .M(P xQ)[y i (B'P)} : 

[Hp.*33-24.*166111] 

:>:('^N):NeG'iPxQ):M6G\B'Q)iiP.:>M.M(PxQ)N- (6) 
I- . (2) . (3) . (6) . *254-54 . D F : Hp . D . P less (P x Q) (7) 

h . (7) . *255-l7 . D h . Prop 

*255-573. 1- :. a,)8eN„0-i'0,: (37) . veNO-t'O^wt'l . a = 7 + i0.a<a>:y8 
Pern. 
I- . *204-483 . D h : Hp . D . (gP, Q) . a = N„r'P . y8 = N„r'Q . g ! 5'P (1) 
|-.(l).*255-572.DI-.Prop 



SECTION D] greater AND LESS AMONG ORDINAL NUMBERS 69 

*255-58. h:7eN„O-i'0^.a>jS.D.a><7>/S><7 

Dem. * 

}■ . *255-31 . D 

l-:.Hp.D:(ai!r).,B-6NO-i'0r.a = ;84-^.v./34=0^.a=^ + l (1) 

h.*184-35. D[-:a = ^ + ^.D.a-ky = {^Xy) + (-BTXy) (2) 

h.*184-16. DI-:Hp.i3- + 0^.D.w>C7=t=0, (3) 

1- . (2) . (3) . *255-32 . D 1- : Hp . ct e NO - I'O,. .a = /3 + w.D.a><7>/3><7 (4) 
H.*184-41. Dh:Hp.a = |8 + l.D.aX7=(;8X7) + 7. 

[*255-32] D.a><7>/3X7 (5) 

h . (1) . (4) . (5) . D h . Prop 

*255-581. \-:Pen.ElB'P.Q\essR.D. 

PxQ less PxR. Nr'P X Nr'Q < Nr'P X Nr'S 
Dem. 

h . *254-55 . D F : Hp . D . (gS) . 5fsmor Q . ,S C P . g ! G'Bnp'R"G'S (1) 

t-.*16611.3H:S'GE.D.PxSGPxE (2) 

f-.*lC6-23.Dh:/SfsmorQ.D.PxSsmorPxQ (3) 

f- . *202-524 .■*40-53 . D h :. Hp . ^ e O'P . w e C/Sf . y e G'E n p'B"G'S . D : 

zP(B'P) .v.z = B'P: wRy : 

[*166-113] D:(M;J,0)(Pxi?){2/4,(5'P)) (4) 

I- . (4) . *166111 . D f- :. Hp . 2/ 6 G'Rnp'R"C'S . D : 

ilf e 0\P x8}.DM-M{PxR){yi (B'P)] (5) 

I- . (5) . *10-28 . D h :. Hp . a ! C'i2 A p^"G'S . D : 

(giT) : iVe 0'(P x R) iMeG'iP x /S) . D^f ■ -^(^ x •«) -^ (6) 

1- . (2) .'(3) . (6). D 1- :. Hp . ;Sfsmor Q.SQR .'^IG'R np'R"G'8 . D : 

(P X S)smor(P xQ).Px8QPxR.'^lG'(Px R)r^p''pVR"G'(P x S) : 

[*254-54] D . P X <3 less PxR (7) 

l-.(l).(7).DF:Hp.D.PxQlessPxE (8) 

I- . (8) . *255-l7 . D h . Prop 

*255-582. H :. aeN„0 : (aS) . S e NO - t'O^ w I'l ,a = S-i-i:/3<7:3. 

a X ;8 < a X 7 [*255-581 . *204-483] 

*255-59. l-:a,j8,7eN„O.74=0,.a>iC7 = /3>C7.3.« = smor"y3 
Dem. 

\- . *255-58 . Transp . D I- : Hp . D . ~ (a > /3) . ~ (a < j8) . 
[*255-ll 2] 3 . a = smor"/3 : D h . Prop 



70 SERIES [part V 

This proposition establishes the uniqueness of terminal division, i.e. 
division by an end-factor. Initial division {i.e. division by a beginning- 
factor) is only unique if the divisor is of the form S-i-l. 

*255-591. h :. a, /3, 7 e N„0 : (gS) . S e NO - tU u I'l . a = S-j- 1 : 

aX|S = aX7:D.;8 = smor"7 [*255-582-112] 

*255-6. V : Nr'P > Nr'Q .3.1 + Nr'P > 1 + Nr'Q 
Dem. 

I- .*265-33 . D 1- :. Hp. 3 : (gisr) . -sreNO- I'O^. Nr'P = Nr'Q-i-i!7 . v . 

Nr'P + 0^ . Nr'P = Nr'Q -j- 1 : 
[*181-55] D : (gw) . w e NO - t'O, . 1 4- Nr'P = (1 + Nr'Q) + ^ . v . 

Nr'P + Or . 1 + Nr'P = (i -i- Nr'Q) 4- 1 : 
[*255-33] D : 1 -j- Nr'P > 1 -i-Nr'Q :. D h . Prop 

*255-601. t- : Nr'P > Nr'Q . = . 1 -f- Nr'P > 1 -(- Nr'Q 
Dem. 

h . *255-6 ^ . *255-103 . D 

|-:Nr'P<Nr'Q.D.l + Nr'P<l-f-Nr'Q (1) 

I- . (1) . *255-108 . D h : Nr'P ^ Nr'Q . D . 1 + Nr'P ^ 1 + Nr'Q (2) 

h . (2) . Transp . *251-142 . D 
I- : i -i-Nr'P, i -i-Nr'QeNO . ~ (1 + Nr'P ^ 1 -j- Nr'Q) . D . 

Nr'P, Nr'Q e NO . ~ (Nr'P ^ Nr'Q) (3) 
h . (3) . *255-281 . D I- : i -i- Nr'P > 1 + Nr'Q . D . Nr'P > Nr'Q (4) 

F . (4) . *255-6 . D h . Prop 

*255-61. b:Q,Rea. Nr'P = Nr'Q + Nr'P . Q'iJ, = Q'P . E ! P'^ . D . 

Nr'P+i>Nr'Q-i-l 
Dem. 

f- . *253-57 . D h : Hp . D . Nr'P -j- 1 = Nr'Q -i- 1 + Nr'P . 
[*255-32] D . Nr'P -j- 1 > Nr'Q + i:0\-. Prop 

*255-62. h : Q, P 6 f2 . Nr'P = Nr'Q + Nr'P . Nr'P + 0^ . 

~(a'Pi=a'P.E!P'P).D. 

Nr'P > Nr'Q 4- 1 . Nr'P + 1 > Nr'Q -{- 1 
Dem. 

h . *253-571 . D I- : Hp . D . Nr'P = Nr'Q 4- 1 4- Nr'P . 

[*255-32] D . Nr'P > Nr'Q 4- 1 . (1) 

[*255-321] D . Nr'P 4- i > Nr'Q 4- 1 (2) 

t- . (1) . (2) . D h . Prop 



[*255-483] 


D 


h.*2oll32. 


DI-:~(P,QeO).D 


[*255-12] 


D 


l-.(3).(4). 


DI-:~(Nr'P>Nr'Q).D 



SECTION D] greater AND LESS AMONG ORDINAL NUMBERS 71 

*255-63. H:Nr'P>Nr'Q.D.Nr'P-i-i>Nr'Q-f-i 
Bern. • 

h . *255-33 . D h :. Hp . D : (gi?) . Nr'i2=|=0, . Nr'P = Nr'Q + Nr'i? . v . 

Nr'Q 4= 0, . Nr'P = Nr'Q -i- 1 : ■ 
[*255-62 -321] D : Nr'P + l > Nr'Q + 1 :. D h . Prop 

*255-64. I- : Nr'P > Nr'Q . = . Nr'P + 1 > Nr'Q + 1 

Dem. 
h . *255-63-103 . DH:Nr'P<Nr'(3.D.Nr'P-i-i<Nr'Q-(-i (1) 

h . *181-31 . D h : Nr'P = Nr'Q . D . Nr'P + 1 = Nr'Q + 1 (2) 

I- . (1) . (2) .*255-113 . D h : P, Q efi . ~(Nr'P > Nr'Q) . D . 

Nr'P-fi=^Nr'i3 + l. 
,~(Nr'P + l>Nr'Q-i-i) (3), 
, ~ (Nr'P + 1, Nr'Q-f- i e NR) . 
,~(Nr'P-i-i>Nr'Q + i) (4) 
,~(Nr'P+l>Nr'Q-i-l) (5) 
l-.(5).*255-63.Df-.Prop : 

*255-65. f- :. /i 6 N„0 - t'O^ ."^-.v^ ft,. = .v^fi + i 

Dem. 
h . *255-33 . 3 I- :. J/ > /i . D : (gw) . ot e NO - t'O, . z/ = /^ + ra- . v . k = /a + i (1) 
H . *255o3-31 . 3 

h :. Hp . OT 6 NO - I'O, , J/ = /i-i- tir . D : (g/)) . p e NO u t'l . v = /i + 2 -f-p : 
[*181-56] D:(aio)./>eNOut'l.i; = /i + i + l+/3: 

[(*255-298)] D:i' = ;a + l-i-i.v.z/ = /i-i-i + i + i.v. 

(a/o) . /oeNO- i'0^.i; = yit-i-i-fi -i-p : 
[*255-33] D:i/>/i-i-i (2) 

|-.(1).(2). Dh:v>/*.D.i/^/i-i-i (3) 

h . *2o5-45-321 .DI-:Hp.z/^/x-|-i.I>-i'>A' (4) 

h . (3) . (4) . D h . Prop 

The following propositions are concerned with the relations of ordinals to 
the corresponding cardinals, i.e. to the cardinals of the fields of well-ordered 
series having the given ordinals. If P is a well-ordered series whose ordinal 
is a, 0"a = Nr'CP, so that G"a is a cardinal whose members can be well- 
ordered. Such cardinals have the property that of any two which are not 
equal, one must be the greater. 

If the cardinal number of one series is greater than that of another, so 
is the ordinal number ; but the converse does not hold except for finite 
numbers. 



72 SERIES [fart V 

*255-7. f-.Nc"C'^'X^ = 0"'NO [*152-7 . (*25101)] 

*255-701. h . Nc"a"ll - t'A=a'"(NO-i'A)=a"'N'0-i'A [*255-7 .*37-45] 

*255-71. h : P less Q . 3 . Nc'C'P < Nc'C^'Q 

Bern. 

h .*254-l . D I- : Hp . D . a ! Rl'Q a Nr'P . 

[*154-1] D . a ! CFG'Q n Nc'C'P . 

[*1 17-22] D . Nc'C'P < Nc'C'Q : D h . Prop 

*255-711. h : Nr'P ^ Nr'Q . 3 . Nc'C'P < Nc'O'Q 
[Proof as in *253-7l, using *255-22] 

*255-72. h : a ^ j8 . D . (?"« < C"^ 

h . *255-24 . D h : Hp . 3 . (gP, Q) . a = N„r'P . ^ = N„r'Q . Nr'P ^ Nr'Q . 
[*255-7ll] D . (gP, Q) . « = N„r'P . yS = N„r'Q . Nc'O'P ^ Nc'O'Q . 

[*1527] D . C"a < (7"/3 : 3 h . Prop 

*255-73. l-:.P,Q6fl.3: 

Nc'C'P < Nc'C'Q . V . Nc'O'P = Nc'C'Q . v . Nc'(7'P> No'C'Q 
i)em. 

h . *255-7ll . 3 h : Hp . Nr'P ^^ Nr'Q . 3 . Nc'C'P < Nc'C'Q (1) 

I- . *255-7l . 3 1- : Hp . Nr'Q < Nr'P . 3 . Nc'CQ < Nc'CP (2) 

h . (1) . (2) . *255-115 .31-. Prop 

*255-74. V :.a,y8 e(7"'N0 -i'A.3:a<^.v.«>/3 

Dem. 
V . *255-701 . 3 F : Hp . 3 . a, ;S 6 0"'(N0 - t'A) . 

[*1 55-34] 3 . (aP Q).P,QeD..a= G"l>!,r'P . /3 = 6«'N„r'Q . 

[*152-7] 3.(aP,Q).P,Q6fl.« = N„c'C"P./3 = N„c'C"Q (1) 

F- . *255-73 . *117-106-107-108 . 3 

h :. P, Q e fi . 3 : N„c'C'P < N„c'C"Q . v . N„c'0'P > Noc'O'Q (2) 

h . (1) . (2) . 3 I- . Prop 

*255-75. h : P, Q 6 n . Nc'C'P < Nc'C'Q . 3 . P less Q 

Bern. 

h . *117-291 . 3 h : Hp . 3 . ~ (Nc'C'Q < Nc'C'P) . 
[*255-711.Transp] 3 . ~ (Nr'Q s^ Nr'P) . 

[*255-29] 3 . Nr'P < Nr'Q . 

[*255-l 7] 3 . P less Q : 3 f- . Prop 

*255-76. h : a, y8 6 NO. C"a<C"^, 3. a </3 [*255-75 . *152-7] 



*256. THE SERIES OF OEDINALS. 

Summary of *256. 

In the present number, we have to consider the series of ordinals in order 
of magnitude. Propositions on this subject deserve close attention, because 
it is in this connection that Burali-Forti's paradox* arises. This paradox, as 
we shall show in the present number, is avoided by the doctrine of types. 
But before discussing the paradox, it will be well to explain various propo- 
sitions which raise no difficulty. 

For convenience of notation, we shall, in the present number, employ the 
letter M for the relation " •< ". (This letter is chosen as the initial of 
" minor.") Thus " oM^ " means that a and /3 are ordinals of which a is less 

than /S. Jlf'/3 will be the class of ordinals less than ;8, ifi'/S will be 13 + 1, 
and il/i'jS, when it exists, will be such that either ifZ/S + 1 = yS, or 

|8 = 2, . Mi'/S = Oy. Thus Q'ilfi is the class of ordinals having immediate 

— > 
predecessors, and B'Mi is the class of ordinals not having immediate pre- 
decessors. 

We have (*256-12) 

1- :. ailf;8 . = : a, /3 6 NoO : (37) . 7 6 NO - t'Or w I'l . /3 = a-h7, 

that is, one ordinal is less than another when something not zero can be 
added to the first to make it equal to the second ; 

*25611. h : P 6 fl . D . M'Nr'P = Nr"D'Pj 

I.e. the numbers less than that of P are the numbers of the proper 
segments of P. Also, if P e fl, 

if p-irNr'P = N„r;(Ps ^ D'Ps) . N„r p B'P, e 1 ^ 1 (*256-2-201), 

so that (*256'202) the series of ordinals less than that of P is similar to the 
series of the proper segments of P, i.e. to P ^ d'P (in virtue of *253-22). 
It follows (*256-22) that every section of M is well-ordered, and therefore 
that M is well-ordered (*256-3), i.e. that the ordinals in order of magnitude 
form a well-ordered series. 

* "Una questione sui numeri transflniti," Rendiconti del circolo matematico di Palermo, 
Vol. XI. (1897). 



T4 SERIES [past V 

For the purposes of the present number, it is convenient to include Ig 
(of. *153) in the series of ordinals ; we therefore get 

N=Ms:iOrilsV>(i%)^a'M Dft [*256]. 

The effect of this definition is merely to insert Ig in the series M between 
0^ and 2r. We then have (*256-42) 

Nr'iV=i-i-Nr'ilf. 

Now if PeD,, P[.(I'P (as we have just seen) is similar to a proper 
segment of M, so that if we omit to mention types we obtain 

h : Pefl . D . Nr'P I Q'P < Nr'Jlf. 

Hence Nr'P, which is 1 + Nr'P ^ Q'P, is less than l-j-Nr'ilf (by *255-63), 
i.e. is less than N. Hence 

l-:PeXi.D.Nr'P<Nr'i\r. 

Nevertheless iV e 11, so that it might seem as if Nr'iV must be less than 
itself, which is impossible by *255'42. Hence we are led to Burali-Forti's 
paradox concerning the ordinal number of all ordinals. 

Burali-Forti's own statement of his paradox, which is somewhat different 
from the above, may be summarized as follows. Assuming 

a,/36N„O.D:a<;ff.v.a = /3.v.a>;8 (A), 

we shall have a e NjO . D . a <• a + 1- 

But we also have a e NjO . D . a ^ Nr'iV. 

Hence Nr'iV < Nr'iV-(- 1 . Nr'iV-i- i =^ Nr'iV, 

which is impossible. The conclusion drawn by Burali-Forti is that the 
above proposition (A) is false. This, however, cannot be maintained in view 
of Cantor's proof, reproduced above (*255"112, depending on *254'4). The 
solution of the paradox must therefore be sought elsewhere. 

With regard to Burali-Forti's statement of the paradox, it is to be 
observed that " a < « -j- 1 " only holds if g ! a + 1, i.e. if (gP) . P e a . C'P + V. 
This will always hold if a exists and is infinite, because then, if Pea, 
P ^ d'P -f> B'P 6 a -}- 1. But if a is finite, this method fails, since 

Pia'P-i^B'Pea. 
Thus if the total number of entities in the universe (of any one type) is 
finite, ''a<a-i-l" fails when C"a = t'Y, which is just the crucial case for 
Burali-Fprti's proof. Hence as it stands, his proof is only applicable if we 
assume the axiom of infinity ; it might, therefore, be regarded as a reductio 
ad absurdum of the axiom of infinity, i.e. as showing that the total number 
of entities of any one type is finite. 

In order to make it plain that the paradox does not depend upon the 
axiom of infinity, we have above stated it in a form independent of this 



SECTION D] the series OF ORDINALS 75 

axiom. The paradox, stated simply, is as follows : The ordinal number of 
the series of ordinals from«0^ (including Ij) to any ordinal a is a-i-1 ; hence 
«+l exists, and is therefore > a. But the ordinal o is similar to the 
segment of the series of ordinals consisting of the predecessors of a, and is 
therefore less than the ordinal number of all ordinals. Hence the ordinal 
number of all ordinals is greater than every ordinal, and therefore than itself, 
which is absurd ; moreover, though the greatest of all ordinals, it can be 
mcreased by the addition of 1, which is again absurd. 

In order to dispel the above paradox, it is only necessary to make the 
types explicit. In the proposition 

Pea.D.PlessN (B), 

upon which the paradox depends, the relation " less " is not homogeneous. 
N is of the same type as M, which is defined as Nr'less, where G'leas = D,. 
Thus Nr'P e C'N. Thus N, as it occurs in (B), should really be iV f i'Nor'P, 
i.e. Nlt't'P, i.e. N{P,P), according to the definition *65-12. We have 
therefore 

*256-53. h : P 6 fl . D . P less JSf I t'^o^'P 
but this does not allow the inference 

N t «'N„r'P less N I i'Nor'P, 
which is what would be required in order to elicit a paradox. The correct 
inference is, substituting for N^fN^T^P the equivalent form N{P,P), 
N(P,P)lessN'{N'(P,P),N(P,P)], or, more generally, 

*256-56. H . (iyr p X) less [N [. (tH^'X)} 

Thus in higher types there are greater ordinals than any to be found in 
lower types. This fact is what gave rise to the paradox, as the corresponding 
fact in cardinals gave rise to the paradox of the greatest cardinal. 

*25601. 
*25602. 
*2561. 

Dem. 



M=<. 




Dft [*256] 




N=MKlOrih^{l'l.)\ 


a'ilf 


Dft [*256] 




f-.i/eSer.O'ilfCNoO 








l-.*255-42. 




Dh.TlfCJ" 


(1) 


1- . *255-47l . 




D 1- .ilf 6 trans 


(2) 


h . *255-12 . 




Dh.O'J/CNoO 


(3) 


l-.(3).*255-112.*15 


5-43 . 


, D h . ATeconnex 


(4) 



I- . (1) . (2) . (3) . (4) . D h . Prop 
The above proposition assumes that M is homogeneous, since otherwise 
" CM " is not significant. But M is significant even when it is not homo- 
geneous. Thus the conditions of significance in the above proposition impose 
a limitation upon M which is not always imposed upon M. 



76 SERIES [part V 

*256-101. f- : a ! ilf . D . CM = N„0 . 0^ = B'M : N„0 - t'O, = a'M 
Bern. 

|-.*200-12.*256-l.Dl-.O'M~6l (1) 

h . (1) . *51-4 . Dhia'.M.D.g! G'M- I'Or . 

[*2561] D.g!N„O-i'0^ (2) 

h.*255-51. DI-:/t6N„O-t'0^. = .0,Jl//* (3) 

l-.(3). Dh.'N.O-i'OrCa'M.Or'^ea'M (4) 

F.(2).(3). Dhz'^lM.D.Ore'D'M (5) 

f- . (4) . *256-l . l^h.a'MCN.O-L'Or (6) 

h . (4) . (5) . (6) . D 1- . Prop 

The hypothesis g; ! M will fail in the lowest type for which M is 
significant, if the universe contains only one individual. Under any other 
circumstances, g ! M must hold. 

*256102. h : a ! N„0 - t'O,. .D .±1 M 

Bern. 

h . *256-101 . D h : Hp . D . a ! a'M (1) 

1- . (1) . *33-24 . D h . Prop 
*25611. I- : P 6 12 . D . M'Nr'P = Nr"D'Ps [*225-174] 

*25612. h : . aM^ . = : a, ^ e N„0 : 

(37) . 7 6 NO - I'Or ./8 = a-i-7.v.a=f0, ./3 = a+i [*255-33] 
*256 2. I- : P e n . D . 

M^ (M^'m'P) = N„r;Ps . M I, (M'Nr'P) = N„r;(Ps ^ D'Ps) 

1- . *256-101 . D h : Hp . P 6 0, . D . Jf [: ^*'Nr'P = A . ikf ^ (i^'Nr'P) = A (1) 

l-.*213-3. DI-:Hp.P6 0^.D.N„r;Ps = A.N„r;(PsDD'Ps) = A (2) 

h . *25611 . *213-158 . D h : Hp . P ~ e 0^ . D . J^^'Nr'P = Nr^'^'P, (3) 
h . (3) . *255-l7 • 3 I- :. Hp . P~ e IV . D : o {i/ 1 A'Nr'P)} jS . = , 

(aQ, P) . a = N„r'Q . ^ = N„r'P . Q, P e C'Ps . Q less P . 
[*254-47] = . (aQ. P) . a = N„r'Q . ^ = Nor'P . QPsP . 

[*150-4] =.a(N„r;P0/3 (4) 

Similarly l-:.Hp.P~60^.D:a{ilf p(i^Nr'P)}/3.= .a{N„r;(Ps^D'Ps)}(8 (5) 
h . (1) . (2) . (4) . (5) . D h . Prop 

*256-201. h : P e n . D . N„r I- D'P, e {ilf ^ (M'Nr'P)} g-mor (P^ ^ D'Pj) . 

N„r [ C'Ps 6 {M I (M^'m'P)] smoi P, [*253-461 . *256-2] 



SECTION D] the series OF ORDINALS 77 

*256-202. I- : P 6 12 . D . Nr'{ilf p (i^'Nr'P)} = Nr'(Ps D D'Ps) = Nr'(P ^ CI'P) 
[*256-201 . *25?22] 

*256-203. h : P 6 ft . D . Nr'{M I (J^'Nr'P)} = Nr'Ps [*256-201] 

*256-204. h : a e N„0 - 1% . D . 1 + Nr'(ilf I M'a) = a 



h . *2o5-101 . *256-202 . D 

h :. P 6 ft . a = N„r'P . D : Nr'{Jf p i^'a} = Nr'(P ^ Q'P) : 

[*204-46-272] D : P ~ e 2, . D . 1 + Nr'(j¥ ^ J^a) = Nr'P : . D I- . Prop 

*256-21. f-:/t6N0.P6/z.D.MV = Nr"D'Ps [*25611] 

*256-211. h : /i e NO - I'O^ . P e/i . D . iT^'/t = Nr"0"Ps [*213-158 . *256-21] 

*256-22. \-:/j,eNO.:^.MlM^'fiea 
Dem. 

h . *256-203 .DI-:Hp.Pe/*.D. Nr'(il!f t M^'fi) = Nr'Ps . 
[*253-24] D.ilf^il%'/ieft (1) 

l-.(l). D\-:fi=^A.'D..MlM^'fi6a (2) 

h.(2).*250-4.Dh.Prop 

*256-221. h-.fie'NO.'D.MlM'fien [*256-202] 

*256-3. h . ilf 6 ft [*256-221 . *250-7] 

*256-31. 1- : a ! ilf . D . 2, = 2^ = M/O^ 
Dem. 

I- . *255-51-53 . D f- : Hp . D . M'Or = t% w J/'2, . 
[*205-196.*256-l] D . 2^ = min^il^'O, 

[*206-42.*201-63] =M^'Or 

[*250-42.*256-101] = 2^^ : D I- . Prop 

We shall have, for every finite v, Vr = vjii, where Vr will he defined as the 
ordinal corresponding to v, i.e. as 

ft n G"v. 

(This is a single ordinal when p is finite ; otherwise, it is the sum of a class 
of ordinals.) This subject will be considered in the next section. 



78 SERIES [part V 

*256-32. I- :. aM^^ . = ia,^e NoO :a=}=Oy./3 = a-i-i.v.a=0,.;S = 2, 
Dem. 

h . *255-65 . D h : a 6 N„0 - I'O^ . D . M'« = t'(«-i- 1) u M'(a+i) . 
[*205-196] D.a-i-i = miiiVil/'a. 

[*206-4<2.*201-63] D.a+i = i^i'a (1) 

I- . (1) . *256-31 . D h . Prop 

*256-4. l-.ls~eNO 
Dem. 

l-.*153-36. D\-:Rel,.'D.G'Rel. 

[*200-12.*25012] D . E ~ 6 n (1) 

h . (1) . *251-122 . D h : a e NO . D . a n 1, = A (2) 

h.(2).*153-34.Dl-.Prop 

*256-41. \-.N=M\jOrils^(i'h)t(i'^ [(*256-02)] 

*256-411. 1- :. aNl3 . b : a= 0, . /Sei'l, w a'M.v . 

a = 1, . y8 e a'M .v. a, Be a'M . aM^ [*256-41] 

*256-412. [■:M=A.D.N=Qrils.Ne2r [*256-41] 

*256-413. h : M=Or J, 2^ . D . JV = 0^ i Ij a 0^ J, 2, o 1^ J, 2^ . iVe 1 + 2^ 
[*256-41 . *161-211] 

*256-414. h : Q'ilf ~ el .D .N=Orils^Ml a'M 
Dem. 

h . *204-46 . *256-101 . D 

\-:'Sp.'3_lM.D.N=0r*\-M[, a'M c; 0, i 1, w (I'l,) ^ C'(M p a'J/) 
[*161-101] = 04 1, o (I'O^ u t'l,) f G'{M I a'M) w ilf I d'M 

[*i60-i] =o,ii,4^#Ca'iif (1) 

l-.(l).*256-412.Dh.Prop 

*256-42. h : a ! if . D . Nr'iV= 1 + Nr'ilf 
Dew. 
I- .*256-414 . D I- : Hp . Q'Jf ~6l . D . Nr'JV= 2^ + Nr'(ilf p a'if) 
[*181-57] =1 + 1 + Nr'(if t a'ilf ) 

[*204-46] =l + Nr'Jf (1) 

H.(l).*256-413.DI-.Prop 

*256-43. hiiVeli-t'A [*256-412-42] 



SECTION D] the series OF ORDINALS 79 

*256-41 I- :. Pe fl . D : P^Q'P less if . = . P lessiV . g ! M 

Bern. • 

h.*255l7-601.D 

h : . Hp . D : P t a'P less M.= A + Nr'P ^ (I'P < 1 + Nr'ilf (1) 

f- . *256-412-42 . D h : P = A . D . P less N (2) 

l-.*255-51. Dh:.P = A.D:PtCI'Plessiif. = .a!ikf (3) 

h.(2).(3). DI-:.P = A.D:Pta'Plessilf. = .PlessiV.a!ilf (4) 
h.*200-35.*255-51.DI-:. a'Pel.D:Pt(I'Plesslf. = .a!ilf (5) 

l-.*256-42. DI-:.Hp.a'Pel.a[!i/.D.PlessiV" (6) 

l-.(5).(6). Dh:.Hp.a'P6l.D:P^a'PlessJlf. = .a!il/.PlessiV (7) 
h . *204-46 . D h :. Hp . a ! P . Q'P ~ e 1 . D : 1 -j-Nr'P I a'P = Nr'P : 
[(1)] D:Pta'Plessilf. = .Nr'P<l-i-Nr'ilf. 

[*256-101-42] = . Nr'P < Nr'A^ . g ! ilf (8) 

h . (4) . (7) . (8) . D h . Prop 

We now make use of the above propositions to show that every well- 
ordered relation P of the type we start from is less than N, where N is to 
hold between ordinals of the- type to which Nor'P belongs. This proposition 
embodies what Burali-Forti's paradox becomes when account is taken of 
types. 

*256-5. higlJf.Pen.D. N„r;(Ps I D'P^) e D'(if ^ <'N„r'P)s 
Dem. 

V . *256-2 . *25313 . D h : Hp . D . N„r ;(Ps l D'Ps) e D'M, (1) 

h . (1) . *150-22 . D h : Hp . D . N„r"D'Ps C to'G'M, . 
[*213-141] D . Npr'P e t,'G'M, . 

[*63-53] D.CG'i/s = «'N„r'P (2) 

I- . (1) . (2) . D h . Prop 

*256-51. h : P 6 n . D . N„rJ(Ps I B'P,) smor P I Q'P [*253-463] 

*256-52. F : a ! ilf . P e n . D . P t CI'P less M p i'Nor'P [*256-5-51 . *254-182] 

*256-53. l-:Pen.D.Plessi\^p«'Nor'P 

Dem. 

h . *256-44-52 .DhrHp-alJf.D.P less iV^ «'N„r'P (1) 

|-.*256-102. DI-:Hp.ilf=A.D.P = A. 

[*256-43] D.P less i\r (2) 

h . (1) . (2) . D I- . Prop 

*256-54. h : P 6 O . D . Nr (P)'(i\r p i'Nor'P) = A 
Dem. 

V . *256-53 . D h :. Hp . D : Q ef'P . Dq . ~ {QsmoriVt ^'N^r^PJ : 
[*15211] :i:fPn Ni-'{Ni «'N„r'P) = A : 

[(*65-04)] D : Nr (P)'(iV p f 'N„r'P) = A : . D h . Prop 



80 SERIES [part V 



*256-55. 


\-:Pen. 


D. 








Nr (Py(N I «'N„r'P) = Nr (P)'(iV I i 


!'«'P) = Nr (Py[N (P, P)} = A 


Bern. 


\- . *155-12 
[*63-105] 


.DI-.PeN„r'P. 
DI-.Pefo'NoT'P. 








[*63-53] 


DI-.<'«'P = i'N„r'P 




(1) 




J-.(l)- 


DI-.Nr(P)'(iV[:«'N„r'P) 


= NT(Py(NltH'P) 


(2) 




[(*65-12)] 




= Nr (P)'{i\^(P,P)} 


(3) 




K(2).(3). 


, *256-54 . D h . Prop 






*256-56. 


\-.(NtX) 


less{iVt:<fVX)} 






Bern. 











h . *256'43-53 . D I- . (iV ^ X) less {N I (fN^v'N ^ \)} (1) 

h.*156-12. Dh.iV^^XeNor'i^^X,. 
[*63-105] DI-.iV"^X6«o'N„r'iV"t\. 

£*63-53] D h . i'f'JV l\ = f'Not'N [, X (2) 

l-.*64-16. Dh.N^Xet'ito'X^to'X). 
[(*64-01)] Dh.N^Xetoo'X (3) 

I- . (2) . (3) . D I- . «'i„„'\ = «'N„r'iV p X, (4) 

h . (1) . (4) . D I- . Prop 
When types are neglected, the above proposition appears as 

NlessN, 

which is impossible, and embodies Burali-Forti's paradox. In the form 
proved above, however, the paradox has disappeared, and we have instead 
the proposition that in higher types longer series are possible than in lower 
ones. 



*257. THE TRANSFINITE ANCESTRAL RELATION. 

Summary of *257. 

In this number, we are concerned with an extension of the notions of 
-Bjjf and R^. This extension requires two relations, R and Q. It is most 
easily explained by first defining the " transfinite posterity " of a term x with 

respect to R and Q; this class is an extension of R^'x. This class is 
generated as follows. Let us suppose, to aid the imagination, that Q is more 
or less serial in character, and that i2 is a many-one relation contained in Q. 
Then the transfinite posterity of x with respect to R and Q is generated as 
follows : Starting from x, we travel down the posterity of x with respect to R 

<— *- . . . 

(i.e. R^'x) as long as we can ; if the whole class R^'x has a limit with respect 

to Q, we begin again with this limit, which is to be included in the trans- 
finite posterity of x with respect to R and Q ; if the limit is y, we travel 

down R^'y, and include the limit of this class with respect to Q, and so on, as 
long as we still have either terms belonging to D'iJ or classes belonging to 
Q'ltg. The whole of the terms so obtainable constitute the transfinite 
posterity of a; with respect to R and Q, which we will denote* by (i2*Q)'a;. 

In order to obtain a symbolic definition of this class, let us call a class <r 

" transfinitely hereditary " when not only R"a- C o-, as in the ordinary 

hereditary class, but also if we take any existent sub-class fi oi a- n G'Q, if fi 

has a limit with respect to Q, that limit is to be a member of a: Thus a is 

to be such that the i2-successor of any member of cr belongs to a and the 

Q-limit of any existent sub-class of o- n G'Q belongs to o- (so long as these 

\j — ► 

exist). That is, B"a C a and /* C o- . g ! /tt n G'Q . D^ . It^'/it C a. Using the 

notion of the derivative of a class with respect to Q, introduced in *216, the 

condition /* C o- . g ! /n a G'Q . D^ . Itg'/* C a reduces to hq'a- C <t, in virtue of 
*216'1. Hence <t is transfinitely hereditary with respect to R and Q if 

R"<r\jhQ'(TCa. 

* This meaning for R*Q has no connection with the meaning temporarily assigned to this 
symbol in *95. 

B. &w. III. 6 



82 SERIES [part V 

We may now define the transfinite posterity of x with respect to R and Q 
as all members of G'Q which belong to every transfinitely hereditary class to 
which X belongs, i.e. we put 

{B*Qyx = C'Qn§lxea-.R"a-yjBQ'<TCcr.X-yea-} Df. 

Then the analogue of B^is ^p{ye(R*Qyx}. This relation, however, is 
less important than the analogue of R^^ limited to the posterity of x. This 
analogue, assuming Q to be transitive, will be Q ^ (R^nQYx. For this we 
introduce the two notations Qjg^ and Q (R, x), the latter being more con- 
venient when either i2 or a; is replaced by a more complicated expression. 
Thus we put 

Qi^ = Q(R,x) = Qt(R*Qyx Df. 

If Q is a well-ordered series and R = Qi, Q^ is merely the series Q 

*— *— 
beginning with x, and (R*Qyx = Q^'x = Q'x\j I'x if xeG'Q. Thus in this 

case, if a; = B'Q, Q^ = Q. But the importance of Qji^ is in cases where Q is 

not completely serial, but becomes so when limited to (i2*Q)'a;. In these 

cases, Q will, in applications, almost always be logical inclusion combined with 

diversity, or the converse of this ; i.e. it will be either 

or MN(M(lN.MJi=N), 

A. 

or the converse of one of these. In the case of o/3 (a C /3 . a =j= /3), we have 

Itg = s P (- Q'maxg) . tig =p [ (— a^min^), 
as will be proved in *258. 

In the present number, we are concerned in proving that, under certain 
circumstances, Qg^ e fi. The proof proceeds on the lines of Zermelo's second 
proof* of his theorem that if a selection exists from all the existent sub- 
classes of a given class, then the given class can be well-ordered. 

Before proceeding to treat of this subject, however, it is necessary to 
prove some elementary properties of {R^QYx. These are given in the 
propositions preceding *257'2. 

We have 
*257-ll. \-:xea. B"a w S^'tr C o- . D . (E*Q)'a; C a 

Thus in order to prove that (i2*Q)*a; is contained in a class a, we have 
to prove (1) that x belongs to or, (2) that the ii-successors of members of fl- 
are members of cr, i.e. that <7 is hereditary with respect to B, (3) that the 
derivative of a with respect to Q is contained in a, i.e. that if /* is any 
existent sub-class of o- r> G'Q which has a Q-limit, this limit is a member of <t. 

* " Neuer Beweis tax die Moglichkeit einer Wohlordnnng," Math. Annalen, lxv. p. 107 (1907). 
His first proof, which was somewhat more complicated, was published in Math. Annalen, lix. 
p. 514 (1904). 



SECTION D] the TKANSFINITE ANCESTRAL RELATION 83 

*257111. [■.{R*QyxCG'Q 
*25712. \-:xeC'Q. = .xe(B* QYx 

*257123. h-.RCQ.D. B"{R*Qyx C (E*Q)'a; 

I.e. if It<lQ,{B*Qyx is hereditary with respect to B. The hypothesis 
i2 G Q is required for most of the properties of (i2*Q)'a!. 

*257125. hiBQQ.xeC'Q.D .*R^'x C {B*Qyx 

Thus if a; e G'Q, the i2-posterity of x is contained in (i?*Q)'a;. 

*25713. t- : /i C {B^Qyx . g ! /i . D . ItgV C (E*Q)'« 

*25714. f- : iJ G Q . D . (is:*Q)'a; C ^s'^ 

Thus (B^Qyx is wholly contained in the Q-posterity oi x. 

The following propositions (*257"2 — "SG) are concerned in proving 
Qjjj.6li, with a suitable hypothesis. This hypothesis is 

Q e nVJ n trans . B e Rl'Q r. Cls -> 1 . Itg [^ CI ex'(iS:*Q)'a: e 1 -» Cls. 
We assume, to begin with, only part of this hypothesis, namely, 

Q e Rl'J" n trans . B e Rl'Q n Cls -^ 1. 
Thus to prove Qex e Ser, we only have to prove Q^^ e connex, i.e. 

y e {B^Qyx . D . {B*Qyx QQ'y, 
or, what comes to the same thing, 

{B^qyxCp'Q'^B^qyx. 

Let us pu t 0-1 = (iJ * Qyx np'Q"{B*Qyx.. 

Then any member of o-j may be called a " connected term," because it is con- 

nected by Q or Q with every other term of (i2*Q)'aj. (A connected relation 
is then a relation whose field consists entirely of connected terms.) We wish 
to prove that o-i is a transfinitely hereditary class, and therefore equal to 
{B^QYx. We do this, not directly, but by combining o-j with another class 
ffa defined as follows. Consider those members z of (B^Qyx which are such 

that their successors in Q^^ consist of B'z and its successors in Q^^, i.e. put 

r = (B^QYx n t [Qj^'z = (Q^^'B'z}. 

It will be observed that, even when Q is transitive, Q^ and (Qbx)^ are still 

useful. In this case, (Qs^)* = Qsx ^ ^^ (^'Qme . so that {Qiia,)^'B'z consists of 

B'z and its successors in Q^^. We then consider the class a^ consisting of 
those terms y whose predecessors are all members of t, i.e. we put 

<r, = (B*Qyx n^{zQy.ze (B^Qyx . 3, . q'j^'z = (Q^^'B'z}. 

Finally we put a-= ffifs a-^, i-e- 



a = {B^qyx n p'Q"{B*Qyx ng{zQy.ze {R*Qyx . D, .Q^^Jz = {Qj^J^'B'z}. 

6—2 



84 SERIES [part V 

The reason for this process is that it is easier to prove that o- is a transfinitely 
hereditary class than it is to prove this directly for a-^ ; and the result follows 
immediately for ai when it has been proved for a. 

We have then to prove R"a C a . Bq'a C a. 

The first step is to prove 

yea.D. Qj^'y = Qju'R'y « t'B'y- 
This is proved by transfinite induction, by showing that 

is a transfinitely hereditary class, whence the result, because, by hypothesis. 

The proof that Q^'y w Q^'R'y is a transfinitely hereditary class is as follows. 
If z €*Q^'R'y, R'z e*Q^'R'y. liz = y, R'z = R'y. 

\i ze Qbx'V' tlien since by the hypothesis QbJz = (.Qib^^'R'z, we have 

< - " — » 

y e {Qb^^'R'z, i.e. R'z e Q^'y. 

Hence z e (R*Qyx n {Q^'y w Q^'R'y) .Z>.R'ze Q^'y u Q*'E'y. 
We have next to prove 

^L C {R^QYx n (Q^'y Jq^'R^y) . g ! ;. . D ."UqV CQ^'y ^%'R'y. 

If a ! /t f^%'R% thenlteV CQ^'R'y. 

—* —> —* 

It fiC Q^'y -y eii, then y e maxg'/i, and Itq'/i = A. 

If /i C Q'y, we have y ep'Q"iJi,, whence w It^/t . D . ~ (yQw), whence, since 
y, by hypothesis, is a connected term, wQ^y. 

Hence in any case Itg V C Q^^'y u Q^'R'y. Hence Q^^'y w Q^'R'y is 
hereditary, and therefore contains (i2*Q)'a! ; and hence 

This shows that iJ'y is a member of a^. For by hypothesis this holds 
of all predecessors of y, and we have now shown (1) that it also holds 

of y, (2) that y is the only predecessor of R'y which does not precede y. 
This is the first step towards proving that a is transfinitely hereditary. 

It follows immediately, from what has now been proved, that ii y ea, R'y 
(if it exists) is a connected term. For by hypothesis 

{R*Q)'xCQ^'yyj*Q'y, 
whence, by what we have just proved, 

{R*Q)'xCQ'R'yyj%'R'y, 



SECTION D] the TEANSFINITE ANCESTRAL RELATION 85 

whence R'y is a connected ^rm. Hence R'y e a. Hence R"cr C a. 
It remains to prove Sq'o- C o-. 

Just as R"<T C o- was proved by proving Q'y = Q^'R'y, so Sg'o- C o- is 
proved by proving 

provided /itCo-. g; !/t .~a ! maxg*/*; 

w — > 
and this is proved by showing that Q"/i u Q^'ltq'n is a transfinitely heredi- 
tary class. 

To show that Q"ii w Q^"\iQ'fi is a transfinitely hereditary class if 

— » 
/t C o- . a ! /i . ~ g; ! maxg'/t, 

we observe that by hypothesis 



Hence R^z e {Qn^^^^fJi' \ and hence, since by hypothesis fi C Q^'ytt, 

R'zeQ^",,. 

Hence ^"{(Q*i?)'« r. Q"/*} C {Q^Ryx n Q"^. 

Also obviously R"Q^"ltQ'fi C Qj|j"ltQ'/i- 

" — > 
Hence putting p = (Q*i2)'a! n (Q'V " Q*"ltcV)> 

we have ii"jO C p. 

We have now to prove Sq'p C p, 

— > -> 

ie. a C p . a ! a . ~ a ! maxg'a . D . Itg'a C p. 

If a C Q"fi, it is obvious (since p, is composed entirely of connected terms) 
— » — » 

that seq^'a C Q"ii w Itg'/t- 

>^ — » 
On the other hand, if a ! * '^ Q*"lto'/*' ^^^"^ *" ^ Q'V> if i^' exists, does not 

affect the value of the limit of a, which is the limit of a n Q^"ltQ'fi, which is 

obviously contained in Q^"\tQ'fi. Hence Sq'/* C p. Hence fi is transfinitely 
hereditary, and we have 

/i C o- . a ! M ■ ~ a ! maxg'p, . 3 . (R^QYoo C Q"/j. w Q5|e"ltQ'p. 
At this point it is necessary to assume 

ItQ [ CI ex'iR^QYx 6 1 ^ Cls. 
This being assumed, we have, by what has just been proved, 

p, c o- . a ! /^ ■ a ! ^k'H- ■ 3 ■ (R*Qy«! c Q"p » Q*'!*^'/" ■ 

D . (ii!*Q)'« c'JltgV w Q^'ltgV 



86 SERIES [part V 

Hence Itg'/tt is a connected term. Hence 

We only require further 



/i C o- . a ! /i . a ! ltQ> . D : ^Q Itg'/t . z e (R^QYx . D, . Q^Jz = (Qb^)*'-R'^- 

Now by what we have just proved, zQltQ'fj,. = .Z6Q"fi ; and by the 
definition of a, since fiCcr, we have 

Hence we arrive at Sq'o- C o-. Since we have already proved R"<t C o-, it 
follows that <7 is hereditary, and (R^Qyas C o-, i.e. 

2/ 6 (R*Qyx :Oy:y ep^"{R^ QYx : zQj^y . D, .'q^'^ = (Qjj.)*'-B'^, 

<— <— — ^ 

i.e. Qj^ e connex : ^ e D'Qua, • 3z ■ Qs^'z = {Qm^h'R'z. 

Hence Qjjj.eSer. Hence also the immediate successor of every term z in 

I>'Qe^ is R'z, so that 

D'Q^, C D'E . (Qj,,), = R i (R*Qyx. 

To show that Q^ e il, we observe that every class contained in D'Qjj^, has 
a sequent, namely 

— > 
« C D'Qjj^ . a ! maxg'a . D . seq {Qn^ya = E'maxg'a, 

a C D'Qjja. . a ! « ■ ~ a ! maxQ'a . D . seq (QB^)'a = It^'a, 

whence a C B'Qs^ . D. . E ! seq (QaJ'a, 

which shows that Qji^eD,. 

The first derivative of Q^^, is SQ'{Q*Ryx, and its last term, if any, is 

^i'{(Q*Ryx-T>'R}, i.e. \iQ%Q^Ryx i\J)'R}. 

The hypothesis required for Qbx e£l\s the same as for Q]^ e Ser, namely, 

Q 6 Rl'J n trans . R e UVQ n Cls -^ 1 . Itg [ CI ex'(i?* Q)'a; e 1 ^ Cls. 

In order that Q^^ may not be null, we require further xeJy'R. 

The next set of propositions (*257*5 — "SG) are designed to prove that, 
subject to the above hypothesis together with x e D'R, Q^^ is the only value 
of P fulfilling the following conditions : 

(1) P is transitive. 

(2) G'P is contained in {R^Qyx. 

(3) If z is any member of D'P, R'z is its immediate successor. 

(4) If a is any existent class contained in G'P and having no maximum, 
Itg'a is its P-limit. 



SECTION D] the TRANSFINITE ANCESTRAL RELATION 87 

This proposition is essential for what may be called "transfinite inductive 
definitions," i.e. definitions (©f a series by defining the successor of every term, 
and the successor of every class having no maximum. 

The . following illustration may make this clear. Suppose iZ is a many- 
one relation of classes to individuals ; suppose we start with some class a, and 

proceed to a w I'R'a, a w I'R'a u L'E'(a w I'R'a), and so on. At the end of 
this series we put its sum, i.e. its limit with respect to the relation (C n J) ; 

let the sum be ^8. We then proceed with ^ u I'R'fi, and so on, as long as 
possible. The series ends with a sum which is not a member of D'R, if there 
is such a sum. It is evident that the series is uniquely determined by the 
above method of generation ; the above-mentioned propositions give symbolic 
expression to the process expressed in words by "and so on, as long as 
possible." 



*25701. (R*Qya! = C'Qn§{ccea.R"<TyjSQ'a-Ccj-.D,.y€a-} Df 
*257-02. QB.= Q(R,«>) = Qt(R*Qy^ Df 

*257-l. I- :. 2/ 6 (R*Qyx . = :y eC'Q-.x eo- . R"(t u Sg'a Ca.O^.yea 
[(*257-01)] 

*257101. i-::ye (R^Qyx . = :.yeG'Q :. 

xe(7 . R"(T Co-:/iCo-.a!/tn C'Q . D^ . Itg'/i Ca-'.D^.yea 
[*257-l . *216-1] 

*257102. H :: 2/ 6 (R^QYx . = :.y eO'Q:. 

^ —* —* 

xea.R"(rC<Ti iiC a .'^XjJi.n C'Q. ^glmaxg'/i. D^. seq^'/otCo-: "Ha.yecy 

[*257-101 . *2071] 
*25711. l-:a!6(7.^"o-uSQVCo-.D.(it:*Q)'a;Co- [*257-l] 

Almost all proofs of propositions concerning {R^QYx use this proposition. 
*257111. \-.(R*Qya;CC'Q [*257-l] 

*257-12. i-:xeC'Q. = .xe(R*Qyx [*257-l] 

*257-121. \-:R(-Q.ye (R^Qyx .D.R'yC (i?* Qyx 

Dem. 
h . *257-l . D 1- :. Hp . yRz .'Dixea. R"(r u Bg'a C o- . D^. «/ e o- : yRz.ze C'Q: 
[*37-l] -D'.zeO'Q:xe<T.R"aCa. Sq'o- C o- . D^ . ^r e <t : 

[*257-l] ':i:ze {R*Qyx :. D h . Prop 



88 

*257122. h 
*257123. h 
*257124. h 
*257125. h 
*257126. h 



SBBIBS 



iJ G Q . /* C (iJ*Q)'a; . D . ii' V C (R*Qyx 

RGQ.-^. R"(R*Qyx C (JB*Q)'a; 

E G Q . D . !R5„"(E*Q)'a; C (i2*Q)'a; 

i? G Q . « 6 a'Q . D . R^'x C (i2*Q)'a; 

RQQ.ooe T>'R . ~ (a;i?a;) . D . {R*Qyx ~ e u 1 [*257125] 

/i C (R*Qyx . a ! /i . D . IV/i C (R^Qya; 



[PABT V 

[*257121] 
[*257-122] 
[*257'123] 
[*257-12-124] 



*25713. h 1 

Bern, 

h . *257-101 . *10-1 . *221 . D I- :: /* C (i?*Q)'a! . D :. 

^ — » 

a!eo-.E"o-Co-:i/Ca-.a!vn C'Q . D„ . Itg'v C<r:D./iCa- (1) 

h.(l).Fact.Dh::Hp.D:. 

^ — > 

a; 6 o- . R"<r C<T:vC<T.<^lvr\ C'Q . D„ . Uq'i' Co-O./tCo-.g!/* (2) 

f-.*101.*257-lll.D 

— » 
h :. K C o- . a; ! V r> C'Q . D^ . ItgS C o- : D : Hp .(iCa.y Itg/i ."^ .yea (3) 

l-.(2).(3).DI-::Hp.i/ltg/i.D:. 

aj 6 o- . i?"o- Co-ii/Co-.alvrt CQ . D^ . It^'v C cr : 3 . y eo- (4) 
I- . (4) . *1011-21 . *257-101 . D h : Hp . 2/ Itg^ti ■ 3 • 2/ e (i2*Q)V : 3 H . Prop 

*257131. h . SQ'(i2*e)'a; C {R*Qyx [*257-13 . *216-1] 

*257132. h : « C CI ex'(i?*Q)'a; . D . Itg"* C (i2* Q)'a; [*257-13] 

*25714. V'.RQ.Q.O.{R*Qyx CQ^'x 

Dem. 
V . *90-163 . D h : Hp . D . R"Q^'x C Q:j^'x (1) 

I- . *20615 . D h : /t C Qjje'a; . z Itg/t . g ! /* . D . ^; ep'Q"n . g ! /t . /* C Q^i^'a; . 
[*40-61.*90-163] D . e Q"/x . Q"/t C V*'« ■ 

[*22-46] l^.zeQ^'x (2) 

h . (1) . (2) . *25711 . D h : Hp . « e C"Q . D . (i2*Q)'« C Q^'a; (3) 

h . *87-261-29 . *60-33 . (*216-01) . D 

I- : Hp . D . R"i- C'Q) = A . Sg'(- C'Q) = A (4) 

I- . (4) . *257-ll . D h : Hp . « ~ 6 C'Q . D . (B*Q)'a; C - C'Q . 
[*257;111] D.(i2*Q)'a! = A (5) 

h . (3) , (5) . D h . Prop 



*257-141. I- : i2 G Q . D . i2"C'Q o Sq'C'Q C G'Q [*216111 . *37-201-16] 



SECTION D] the TEANSFINITE ANCESTRAL SELATION 89 

*257142. h-.RCQ.^eO'Q.D. {R*Qya)=p {xe<7. B"cr w Sq'^ Ca.D^.yea} 
Bern. * 

l-.*257-141.Dh:Hp.D.^ta'e<7.^"<rwVo-Cff.3„.y6o-}CC"Q (1) 
h . (1) . *257-l . D f- . Prop 

*25715. h : y e (R*Qyx . z e (-B*Q)'2/ . 3 . ^ e (i?* Q)'a; 
Dem. 

I" . *257'1 . D h :. ii"o- u Sq'ct Co-.Diajeo-.D.yeo-iyeo-.D.^^eo-; 
[Syll] D:a!eo-.D.^6o- (1) 

I- . (1) . *2o7-l . D h . Prop 

*25716. h : a; e (7'Q - D'^ . D . (i2*Q)'« = t'a; 
i)em. 

h . *257-12 . D h : Hp . D . a; 6 (i2*Q)'a; (1) 

V . *37-261-29 . D I- : Hp . D . R"i'x = A (2) 

h . *205'18 . D h : Hp . ~ g ! m&XQ'i'x . D . xQx . 

[*206-42] D . I^qg'i'x = A (3) 

h . (3) . *21 6-101 . D h : Hp . D . Sq'i'x = A (4) 

h . (2) . (4) . D h : Hp . D . ^"I'a; w Sq'i'x C t'/c . 

[*257-ll] D . (E*Q)'a; C t'a; (5) 

h . (1) . (5) . D H . Prop 
We now begin the proof (completed in *257'34) that under certain cir- 
cumstances Qji^eil. We first prove that the class <t introduced in *257"2 is 
transfinitely hereditary, and this requires as a preliminary the proof that 

* < ^ . 

if yea; the class {Q^^^'y w {Qjjr^^'R'y is transfinitely hereditary. This 

preliminary is provided by *257"2"21. The hypothesis of *257'2 is not all 
used in *257'2, but is introduced because it is required in the set of pro- 
positions of which this is the first. 
*257-2. h : . Q 6 Rl' J n trans . i2 e El'Q n Cls -» 1 . 

a = {R^qyx^p^"{R^yx n § {zQ^y . D, .q'^'z = {Q^^'R'^z].0 :^ 

yea.ze {QM " (QL)*'^'^ .zeWR.:> .R'ze (Q^'y u (QL)*'-R'2/ 
Bern. 

h . *90-163 . *37-62 . *257-123 . D 



hz.RQQ.ElR'z.D-.ze {Qn^h'R'y . D .R'z e {Q^)^'R'y (1) 

h . *30-37 . D I- : E ! E'0 . ^ = 2/ . D . E'^ = ^'2/ (2) 



I- . *201-18 . *91-52 . *32-182 . D 



(- : Hp . ^ e Q^'y . D . Q^'z = {Q^)^'R'z . y e Qj^'z . 
[*13-13] ■ D.y€(Q^;)^'R'z. 

[*32182] D.R'ze(QM (3) 

I- . (1) . (2) . (3) . *71-161 . D h . Prop 



90 SEEIES [part V 

*257-21. h : Hp *2o7-2 .yea.^C (Q^^'y u ^^'R'y . a ! /. . D . 

Dem. 

V . *201-14-15 . *206134 . D 

— > — > 

I- . *205"38 • D I- : Hp . /a C Qj^'y . y e /i . D . y e maxg'/i . 

[*207-ll] 3.15!^ = A (2) 

l-.*40-55.*206-143.D 

V: fxC Q'y . w Itg/i . D . y ep'Q"/ji . w ~ e Q"p'Q"fJ, . 

[*37-l] D.~(yQ«;) (3) 

h . *257-13 . D h :. Hp (3) . Hp . D : yQw . v . wQ^y : 

[(3)] ^iwQ*^ (4) 

l-.(l).(2).(4).DI-.Prop 



*257-211. h : Hp *257-2 . y e <7 . D . (E*Q)'a; C (Q^)*'2/ u (Qj,,)^'R'y 

Dem. ^ 

l-.*257-14.DI-:Hp.^6(Q^,V2/ (1) 

l-.(l).*257-2-21-ll.DI-.Prop 



*257-22. h : Hp *257-2 .yea.D. Q^'y = {Q^)^'R'y . (Qi^^^'y = Qn^'R'y 
Dem. 

I- . *257-211 . D h : Hp . D . iiQ^^'R'y = (jR^Qyx - (^S^'y 

[Hp] ^ =Q;.'2/ (1) 

Similarly h : Hp . D . (qj^'y = S/^'2/ (2) 

h . (1) . (2) . D h . Prop 

It is to be understood that (Qjtx)^'R'y = A if ~ E ! R'y. 

*257-23. h : Hp *257-2 . D . jB"o- C o- 

Dem. 
l-.*257-22. DI-:.Hp.2/6(rnD'i?.D:0QE'y.D,.QL,'^ = (Q^'^'^ (1) 
l-.*257-22-211.Dh:Hp. 2/60- nD'i2.D.(iJ*Q)'a; = Q^/E'2/u(Q^'E'2/ (2) 
l-.(l).(2). DI-:Hp.2/6<TftD'i2.D.E'2/eo-:DI-.Prop 

The above proposition gives the first stage in the proof that a is trans- 
finitely hereditary. The second stage, similarly, requires as a preliminary 
the proof that if fi is an existent sub-class of a- having no maximum, then 

is a transfinitely hereditary class. This proof is provided by *257'24!241242. 



SECTION D] the TBANSFINITE ANCESTRAL RELATION 91 

*257-24. I- : Hp *257-2 ./iCff-al/i.^g! r^xg'fi . D .R"Qj^"p C Q]^"fi 
Bern. • 

1- . *91o2 . *201-18 . D h : Hp . 6 Qj,,'V ■ ^ -Viex'^ = (Sj*'-R'.2 ■ 
[*37-46.*13-12] D.a!(Q^*'E'^n/.. 

[*37-46] D.E'^6(Q^VV (1) 

h.*205-123. DhiHp.D./iCQs/V (2) 

h . ( 1 ) . (2) . D I- : Hp . ^ 6 Qje,"/i . D . E '^ 6 Qjj^' V : 3 I- ■ Prop 

*257-241. h : Hp *257-24. D .^"{Q^'V v. (Q^,)*"K'/.} C Q^/VC(QW*""ite'/* 
-Dem. 

H . *90-164 .DhzRQQ.D. R"(Q^h"ltQ'^, C (Q^vKv (1) 

l-.(l).*257-24.DI-.Prop 

*257-242. h : Hp *257-24 - p = Qji^"fj. u (Qa« Vlt^V . 

— » — > 

a C p . a ! a . ~ a ! maxg'a . D . It^'a C p 
Dem. 

l-.*206-15. DI-:Hp.a!/inp'Q""«-wUQa.D.a!/i-Q'w (1) 

f- . *201-521 .Dh:Hp./iCo-.D./i- Q'w C Q^'w (2) 

l-.(l).(2). Dh:Hp(l).D.a!/.nQ;i.'w (3) 

l-.*205-123.Dh:Hp.D./tCQ"/t (4) 

l-.(3).(4). Dh:Hp(l).D.W6(2^'V (5) 

l-.*206-24. DI-:Hp.^CQ"a.oCQ"/i.D. Va = V/i, (6) 

h . *20615 . D I- : Hp . a ! a n (Qj^V'ltiv . D . h^'a C (Q^)*"!^^ (7) 
l-.(5).(6).(7).Dh.Prop 

*257 243. I- : Hp *257-24 . D.(i2*Q)'a; = Qj^''/^ u p'OL' V [*40-53.*205-123] 

*257-25. h : Hp *257-24 . D . (E*Q)'« = Qjj^'V u (Qjte)*"^^'^ 

I- . *257-242 . D h:HpO. V{Qie."/*«(Qiex)*""itQV}CQ^'Vv^(4.)*"H'eV (1) 
I- . (1) . *257-241 . D h . Prop 

*257-251. I- : Hp *257-24 . D . (Qjio. Vlt^V ^P^bx"h- 
Dem. 
V . *257-25-243 . D I- : Hp . D . Q^",i u (Qje^vI^gV = Qisx'V "P'S/V ■ 
[*200-53.*24-481] D . (0^)*'%^ = p'S."^ = 31-. Prop 



92 SERIES [PART V 

*257-252. h : Hp *257-24 . g Ip'Stx'V ■ 3 ■ Qi^'V =i5'3Bx"ltQV • 3 ! IteV 
J5em. 

I- . *257-251 . *37-29 . D h : Hp . D . g ! ItgV ^ (^) 

[*200-53,*40-62] D . iJ'Qjjx"^ V C (ii:*Q)'a; - (Qij.)*"lte V 

[*257-251] C{R*Qyx-p'Qs^"ti 

[Hp.*10-57.*257-243] C QgJ'ii (2) 

h . *201-51 . *40-67 . D I- : Hp . D . Q^' V C p'S."ltQ> (3) 

h . (1) . (2) . (3) . D h . Prop 
In order to complete the proof that o- is a hereditary class, we have to 
introduce the additional hypothesis 

Itg 1^ CI ex'(iJ*Q)'« 6 1 ^ Cls. 
With the help of this hypothesis, the last stage of the proof is provided by 
the following proposition. 

*257-26. h : Hp*257-2 . Itg \ CI ex\R*Qyx e 1 -> Cls . D . Sq'o- C a- 

Dem. 
h . *257-251-252 .Dhi.Hp./iCo-.gl^.a! ItgV • ^ ■ 

(R*Qyx ='Q^'ltQ V V. (Q^*'ltQ V • ^'ItaV = Qb."m- ■■ 
[Hp] D : ItgV ep'^"(R*Qyx : yQnMl^ ■ ^v ■ Vb/2/ = ^M^h'^'y ■ 
[Hp] D : ItgV 6 o- :. D h . Prop 

*257-261. h : Hp *257-26 . D . (i?*Q)'« = o- [*257-ll-23-26] 

*257-27. I- : Q e Rl' J n trans . J? e Rl'Q n Cls -» 1 . 

Itg \- CI ex'(-R*Q)'a; e 1 -> Cls . D . 

Q^,Ser.Qj^ = (R\Q^)\;{R*Qyx 
Dem. 

h . *257-261 . D 

I- : Hp . D . (i2*Q)'a; Cp'^"{R*Qyx n § {zQ^^y . D, . q"^,'^ = (Q^^'-R'^} (1) 

f- . (1) . D h :: Hp . D :. Qjjj, 6 connex :. ^ e D'Qjj^ . Dg : ^Qb^-w . =^ . ^fi | (Qj{a;)*w :. 

[*5-32.*4-71.*257-121] 

D :. Qj{^ e connex :. zQ^f^w . =j,„ . ^ e D'Qj,^ . ^jB | Q^w . w e C'Qj^ :■ 
[*36-13.*257-121] D :. Q^o, e connex . Qb^ = (R \ Q*) t iR*Qyx " 3 I" ■ Prop 

We have thus proved that Q^j. is a series. No additional hypothesis is 
required to prove that it is well-ordered, as we shall now show. 

-» ♦- 

*257-28. h : Hp *257-27 . fi C (R*Qyx .^I/m. maxgV = A . a Ip'Qbx"/^ ■ ^ ■ 

P'Qb."/^ = (4.)*"KV • Qii/V = p'QrAi^ [*257-251-27] 



SECTION D] the TEANSFINITE ANCESTRAL RELATION 93 

*257-281. h : Hp *257-28 . E ! ItgV . D . 

V^bJ'h- = (W*'ltg> . QjL' V ='«i!»'lte'/^ [*267-28] 

*257-29. h : Hp *257-27 . x e D'jB . D . CQ^^ = (R*Qyx . B'Qj^ = x 
Dem. 

\- . *257-27-126 . *202-55 . D 1- : Hp . D . G'Qjg^ = (R*Qyx (1) 

l-.*257-14. Dh:Hp.D.(E*Q)'a!-i'a;CQjto'a; (2) 

h . (1) . (2) . D h . Prop 

*257-291. h : Hp *257-27 . « ~ e B'B .D.Qjt^ = A [*2.57-16 . *200-35] 

*257-3. I- : Hp *257-27 . D . B'Qj^ = B'B n (R^QYx 
Dem. 

h . *257-27 . D h :. Hp . y e (E*Q)'a; . D : g ! Q'y . = . g ! Q^'E'y . 
[*257-141] =.E!^'2/:.Dl-.Prop 

-> «— 

*257-31. h : Hp *257-27 . /j. C (R*Qyx . g ! /* . ~ g ! maxg'/i . g ! p'Q^/ V- ^ ■ 

seq(Qijx)V = ltQV [*257-28] 

*257-32. h : Hp *257-27 . /* C (i?*Q)'a; . g ! maxe'/t . g ! ij'Vi^'V ■ ^ ■ 

seq (Qjja:)> = -R'max (Qj^)'/* 
i)em. 

l-.*257-3.Dh:Hp.D./tCD'i2. 

— > —* ^ 

[*257-27.Transp] D . Q*'max {Qs^YfJi = Q'E'max (Qb^)V : D I- . Prop 

*257-33. h : Hp *257-27 . ytt C (ii*Q)'a; . g ! /* . g ! p'Q^' V ■ 3 ■ E ! seq (Qjj,)'/^ 
[*257-31-32] 

The above proposition together with *257"27 shows that Q^^ is well- 
ordered, in virtue of *250'123. 

*257-34. h : Hp *257-27 .O.Q^eD. 
Dem. 
h.*257-291. DI-:Hp.a;~eD'i2.D.Qjt„6fi (1) 

|-.*257-29.*20614.Dh:Hp.a;eD'i2.D.seqp'A = a; . (2) 

|-.(2).*257-33.D 

h :. Hp . a; 6 D'ii; . D : /^ C (R*Qyx . g Ij^'Okt"/* • 3^ • E ! seq {QsJ'fi : 
[*257-29.*206-131] D : g Ip'QbJ'(im n C'Qj^) . D^ . E ! seq (Qjj^)> : 
[*250123.*257-27] D : Q^ e Xi (3) 

I- . (1) . (3) . D h . Prop 



94 SERIES [part V 

*257-35. h : Hp *237-27 .D.Rl (i2*Q)'a; = (Q^), . R I (R*Qyx e 1 -♦ 1 
Bern. 

y . *257-32 . 3 h :. Hp . D : y e D'Q^. • 3 ■ seq (Q^^yi-'y = R'y (1) 
I- . (1) . *206-43 . *204-7 . D h . Prop 

*257-36. I- : Hp *257-27 . x e B'R . D . 

G'Qa. = {R*Q)'x . Q'Q^ = {R^yx - I'x . 

B'Q^ = X Tb'Qj^ = {R*Qyx - T>'R [*257-29-3] 
The following propositions are concerned in showing that a relation P 
which satisfies the hypothesis of *257'5 is identical with Q^, thus showing 
that this hypothesis is suflScient to determine P. 

*257-5. h : Hp *257-27 . P e trans . G'P C (i2* Qyx .P-P'=Rt(R* Qyx . 
Itp [ CI ex'{R0Qyx = Itg p CI eK'{R*Qyx . D . P G J . C'P = {R*Qyx 
The above hypothesis is not all necessary for the present proposition, 
but it is necessary for the series of propositions of which this is the first. 
Dem. 
h . *37-41 . D h :. Hp . D : D'(P - P^ = R"{R*Qyx n (iJ*Q)'a; 
[*257-36] ={R*QyxnT)'R (1) 

h . *32-14 . D h : Hp . D .ltp'{(P*Q)'a; f^ D'R} = l?e'{(E*Q)'a; r^ D'R} 
[*257-36] =(P*Q)'a;-D'E (2) 

h . (1) . (2) . D h : Hp . D . (R*Qyx C G'P . 
[Hp] •2.(R*Qyx = G'P (3) 

I- . (3) . D h : Hp . D :xeT>'P . D . a;P - P= (-R'a;) • 

[*34-5.Transp] D.~(a;Pa;) (4) 

I- . (3) . (4) . D h . Prop 

*257-51. h : Hp *257-5 .O.G'P = %'x 
Dem. 

\- . *257-123 . *90-16 , D 1- : Hp . D . P'^P^i^'a; C P^'x (1) 

h . *9013 . D 1- : Hp . D . ltQ"Cl ex'P^'a; = ltp"Cl ex'%'x . 

[*90-163.*40-61] D.ltQ"Clex'Pi^'a;CPi,f'a; (2) 

h.(l).(2). D\-:B.^.D.(R*QyxCP^'x (3) 

l-.(3).*257-5.Dl-.Prop 

In order to prove P = Q^ we first prove Peil. The proof proceeds as 
for Qjjj,, but in some points it is easier. It is merely outlined below, as it 
closely resembles the proof for Qg^. 

*257-52. h:Hp*257-5. 

a = G'Pn p'*P"C'P n p {zPy . D^ . P'z = P*'^'^) .D.R"aCa 



SECTION D] the TRANSFINITE ANCESTRAL RELATION 95 

Dem. 
h . *34-5 . Transp . *201 18*D h :. P^ = E ^ (-B*Q)'a; . y ep'*P"G'P . D : 

zF (B'y) . D . ~ (yPz) : zP^y ."^.zP (R'y) : 
[Hp] :>:zP{B'y).= .zP^y (1) 

As in *257-2-21, using Itp |'Clex'(i?*Q)'a; = ltQ I^Cl ex\R*Q)% we prove 
^:B.p.yea-nI>'R.p = P^'y^P^'B'y.:i.R"pCp.SQ'pCp. 

^D.(R*Qyx = f^'yyj%'R'y (2) 
I- . (1) . (2) . D h : Hp . 2/6<r n D'i? . D .P'2/ = P5,f'^'2/ (3) 

h . (1) . (3) . D h : Hp . 2/ e a n D'E .D .R'yecrzDh . Prop 

*257-521. h : Hp *257-52 .yLtCo-.a!/i.~a! maxpV . D . 

(P* Q)'a; = P"p, u P*"ltp> 
[Proof as in *257*25, by similar stages] 

*257-53. h :. Hp *257-5 . D : P e Ser : ^ e D'P . D^ . ^'z=*P^'R'z 
[Proofasin*257-27] 

*257-54. h:Hp*257-5.D.Pen [Proof as in *257-34] 

*257-55. h : Hp *257-5 . o- = ^ (P'^^ = Qj^'y) . D . R"tT C a- 
i)em. 

h . *257-53 . D h : Hp . 2/ e C/'P . D . P'P'j' = G'P - %'R'y 
[*257-53] =C'P-'P'y 

[*257-53] =P'yyji'y (1) 

l-.(l). DI-:Hp.2/e<7.D.P'.fl'2/="4.'yv.i'y 

[*257-22] ='QR^'R'y : 3 H . Prop 

*257-551. h : Hp *257-55 . D . Sq'o- C o- 
Dem. 

I- . *257-63 . D 

— » 
f-:Hp./*Co-.a!/i.^ = UqV . D . P'« = {(R^QYx n /*} u P"/* 

[Hp] =_{CB*Q)'a'«/*}wQ«.'V 

[*257'27] ='QRa,'z : 3 f- ■ Prop 

*257-56. h : Hp *257-5 .D.P = Qb^ 
Bern. 

h . *257-51-54 . DI-:Hp.D.P'a! = A. 

[*257-36] D.P'a;="Ste'« ' (1) 

I- . (1) . *257-55-551 .D\- •..B.p.D lyeCP .Oy.P'y ='QiJy :. D h . Prop 

This proves that the conditions in the hjrpothesis of *257"5 are sufficient 
to determine P. 



*258. ZERMELO'S THEOREM. 

Summary of *258. 

In this number, we shall first show the applicability of the propositions 
of *257 to the case where the Q of that number is replaced by logical 
inclusion combined with diversity, i.e. by any one of the four relations : 

a^(aC/3.a + ^), Sy8(;SCa.a + /3), 

MN(M(LN.M^N), MN(NQM .M=^N). 
If we put Q=aj8(aC/3.a + /3), 

and if « is any class of classes, then s'k is the maximum of k with respect to 
Q if s'ksk, and the sequent of k with respect to Q if s'xf^eK (*258'111); 
similarly p'k is the minimum of k if p'x e k and the precedent of k ifp'ic^e k 
(*258'101*111). Hence every class of classes has a unique maximum or 
a unique sequent with respect to Q, and every class of classes has a unique 
minimum or a unique precedent (*258"12) ; we have, moreover, 

\tQ = s\ (- a'maxg) . tie =p P (- a'miiiQ) (*25813-131). 

Hence Itg, tlge 1 — » 01s (*258'14), and Q and Q therefore satisfy the most 

exacting part of the hypothesis of *257"27. Also Q and Q are Dedekindian 
relations (*258"14). (They are not series, because they are not connected.) 

An exactly similar argument applies to MM^MQN .M^ N). Hence if 
Q is any one of the above four relations, and if iJ is a many-one contained in 
Q, it follows from *257"34 that Q with its field limited to the transfinite 
posterity of any term is a well-ordered series. If we take Q = a^{aC^.a^^), 
and take any initial term a, our series proceeds to continually larger classes, 
proceeding to the limit by taking the logical sum, i.e. if k is any existent 
sub-class of the posterity of a, s'« = limaxg'/t = limax (Qjjn)'* (*258'21*22), 
where Qoa bas the meaning defined in *257. This process stops with 
s'\D'R r\ (R^QYx] if D'J? n (iJ*Q)'a! has no maximum ; otherwise, it stops 
with the iZ-successor of this maximum, which is maxg'fO'iJn (JB*Q)'a;}. 
If, on the other hand, we take Q to be the converse of the above, we proceed 
to continually smaller classes, and the limit of any set of classes k having no 
last term is p'k. In this case, if, starting from a, every existent sub-class of 
a belongs to D'iJ, the process of diminution cannot stop short of A. This is 



SECTION D] ZEEMELO'S THEOREM 97 

the process applied in Zermeb/s theorem. We have there a class /i, assumed 
to be not a unit class, aqiil a selective relation >Si for existent sub-classes of 
^, i.e. a relation 8 for which S e e^,' CI ex' fi. Then our relation B is the 
relation of a to a — I'S'a, i.e. the relation of an existent sub-class of /j, to the 
class resulting from taking away its /S-representative. Thus Qs^ is a 
well-ordered series, which starts from /i and ends with A. Omitting the 
final A, 8 selects a representative from every member of the field of Qbil, 
and the series of these representatives, i.e. S'Qr^, is similar to Q^^ with the 
final A omitted. Moreover every member of fi occurs among these repre- 
sentatives, for, if SB be any member of /i, let k be the class of those members 
of G'Qsi,. of which a; is a member. (There are such classes, because /jl e G'Qbx 
and X e /JL.) Then xep'x, and by what was said earlier, p'K is a member of 
G'Qb^. Hence, by the definition of «, p'k ex, and therefore p'K = mAXQ'K. 
But no class smaller than p'k can belong to k, and therefore p'K — i'8'p'k is 
not a member of «, and therefore x is not a member oip'K — i'8'p'k. Hence 
x = 8'p'k, and therefore x occurs among the representatives of members of 
G'Qsii, which was to be proved. (The above is an abbreviated rendering of 
the symbolic proof given below in *258"301.) Hence the field of 8'>Qb^ is /*, 
and therefore there is a well-ordered series having /i for its field, provided 
64 '01 ex'fi is not null (*258"32). This is Zermelo's theorem. 

The converse of Zermelo's theorem has been already proved (*250'51). 
Hence the assumption that a selection can be made from all the existent 
sub-classes of fi is equivalent to the assumption that /jl can be well-ordered 
or is a unit class, i.e. 

*258-36. h:/jLe G"£l u 1 . = . g ! e^'Cl ex V 

Hence also, by *88'33, the multiplicative axiom is equivalent to the 
assumption that all classes except unit classes can be well-ordered, i.e. 

*258-37. V : Mult ax . = . 0"n u 1 = 01s 

Hence also, in virtue of *255'73, the multiplicative axiom implies that of 
any two unequal existent cardinals one must be the greater, i.e. 

#258-39. l-::Multax. D:./i, i/eNoC. D :/i^v. v.;u.>i' 



#258'1. \-:.Q = &^{aCp .a^^).'^:s'KeK.'^ . s'k= max^'/c 

Bern. 
\- .*205"101 . D h :: Hp . D :.7maX(jK .= :76«:a6/«;.Da.~(7Ca.74a): 
[Transp] = : 7 e «r : a e k . a =|= 7 . D. . ~ (7 C a) (1) 

H . (1) . *101 . D H :: Hp . s*« e K . D :. 

7 maxQ« .= :76«:a6/«;.a=}=7.Da.~(7Ca): s'k 4= 7 . D . ~ (7 C s'k) : 
[*40'13] =:7eK:aeK.a=|=7'3a.~(7Ca):s'/(: = 7: 

[Transp.*40"13] = : 7 e « . s'/c = 7 : 
[Hp] = : «'k = 7 :: D f- . Prop 

B. & W. III. 7 



98 SERIES [part V 

*258101. h : Hp *258-l . p'/c e « . D . p'/e = mine'/c [Proof as in *258-l] 

*258-ll. I- : Hp *2581 . s'/c ~ e /e . D . seq^'/c = s'k 
Dem. 

h.*40-53. DI-:Hp.D.;)'Q"« = $(a6«:.Da-aC7.a + 7) 
[Hp.*40151.*10-29] =9(s'«C7) (1) 

I- . 3|^0-1 . *22-42-46 .'^V.s'K=p'f} (s'k C 7) (2) 

I- . (2) . *258-101 . D h : Hp . D . s'« = ming'i^ (s'k C 7) 
[(1)] = seqg'/c : D h . Prop 

*258111. h : Hp *258-l . jsV ~ e « . D . prec^'/c = ja'/c [Proof as in *258-ll] 

*25812. h :. Hp *258-l . D : E ! max^'/t . v . E ! seqe'« : 

E ! ming'/K . v . E ! precg'/c [*258-1101-ll-lll] 

*25813. h : Hp *258-l . D . Itg = s C (- a'maxg) 
Dem. 

h . *258-l . Transp . D h : Hp . ~ g ! max^'K . D . s'« ~ e « . 

[*258-l 1] D . Uq'k = s'k : D h . Prop 

*258131. I- : Hp *2581 . D . tig = j3 1^ (- aiming) [Proof as in *25813] 

*25814. h:Hp*258-l.D.Q,Q6Ded.lte,tlQ6l^Cls [*258-12-13131] 

*258-2. \- : Hp *258-l . R e Rl'Q n Cls ^ 1 . D . Q^ e n 

Dem. 

\- . *258-14 . D h : Hp . D . Hp *257-27 (1) 

I- . (1) . *257-34 . D h . Prop 

*258-201. \-:Q = a$(j3Ca.aJpl3).Rem'Qr^C\s-^1.0.Qsaen 
[Proof as in *268-2] 

*258-202. [■:Q = M(MQN.M=^N).Bem'QnCls-^l.D.Qji^6n 
*258-203. \-:Q = MM{N(lM.M^N).Bem'QnC\a^l.D.Qsxea' 

*258-21. I- : Hp *258-2 . « C (i? * QYa . D . «'« = limaxQ*« 
Dem. 
h . *258-13 . D h : Hp . ~ a ! maxg'/c . D . s'k = Itg'K (1) 

h . *258-2 . D I- :. Hp . a ! maxg'K . D : (37) :7eK:aeK.D..aC7: 
[*40-151] D:s'k6k: 

[*268-l] D : s'k = maxg'K (2) 

I- . (1) . (2) . D I- . Prop 

*258-211. I- : Hp*258-201 . k C (B*Q)'a .D.p'K = limax^'K 



SECTION D] ZERMELO's THEOREM 99 

*258-22. h : Hp *258-2 . a e D'i? . « C (i2*Q)'a . g ! « . D . s'/e = limax (Q^)'* 
Dem. • 

I- . *258-21 . D h : Hp . s'k ~ 6 « . D . s'k = Itg'/c . 

[*257-13] D.s'KeJR^QYa. 

[*210-233] D .. s'k = limax (Q^)'«: :0b. Prop 

*258-221. h : Hp*258-201 .aeB'B . /cC (i2*Q)'a . D .p'K= (i?*Q)'«: 

*258-23. f- : Hp *258-2 . a e B'R . D . Q^j, e Ded . s'(E*Q)'a = B'Qm^ 
[*258-2-22 . *250-23 . *205-121] 

*258-231. h : Hp *258-201 . a e D'R . D . Q^ e Ded . p'{R*Qya = B'Qz^ 

*258-24. h:Hp*258-2.D. 

(B^Qya = 0(aea . R"a C o- . s"Clex'o- C o- - D„ . ;S e o-) 
i)em. 
h.*2581-13.*257-l.D 

h : Hp . 3 . (i?*Q)'a €/§(«€ o- . R"a C a- . s"Cl ex'o- C o- . D. . /3 6<r) (1) 

h.*257-123.DI-:Hp.D.^"(E*Q)'aC(i2*Q)'a (2) 

l-.*258-22. Dh:Hp./tC(^*Q)'a.a!/i.D.sVe(-R*Q)'a (3) 

h.*267-12. Dh:Hp.D.a6(i?*Q)'a (4) 
h . (2) . (3) . (4) . D 

f- :. Hp : a 6 o- . R^a C tr . s"Cl ex'o- Co-.D,.j8e<r:D.;86 (iJ*Q)'a; (5) 

h . (1) . (5) . D h . Prop 

*258-241. h : Hp *258-201 . D . 

(i2*Q)'a = ;§ (a e o- . ^"o- C o- . ^"01 ex'o- C <r . D„ . /S 6 0-) 

*258-242. h : Hp *258-202 . D . 

(i?*Q)'Z = 7(X 6 o- . E"o- C o- . s"Cl ex'o- C o- . D, . Fe o-) 

*258-243. V : Hp*258-203 . D . 

(i2*Q)'Z = f (Z 6 o- . E"o- C o- . p"Cl ex'o- C o- . D„ . Fe o-) 

*2583. I- : Q = aj§(jg C a . a + /8) . /Sfee^'Cl ex'/i . 

i2 = a;8 (a e CI ex'/* . /3 = a - I'/S'a) . D . Qjj^ e fl . iSJQjj^smor Q^^ ^ (- I'A) 
Bern. 
V . *80-14 . D I- : Hp . D . iJ G Q . ii! e Cls ^ 1 . D'i? = 01 ex'/t . O'i? = Cl'/t (1) 
l-.(l).*258-201.DI-:Hp.D.Qi^efl (2) 

H.*257-35. Dh:Hp.D.i2^C"Qjj^el->l. 

[(l).Hp] D.fifrC'Qj2^6l->l (3) 

h . *257-14 . D I- : Hp . D . O'Qm,. C Ol'/i (4) 

l-.*80-14. DI-:Hp.D.a'/S = Clex'/t (5) 

h.(3).(4).(5). DI-:Hp.D.-Sf;QB^smorQjj^P(-i'A) (6) 

1- . (2) . (6) . D I- . Prop 

7—2 



100 SERIES [part V 

*258-301. h : Hp *258-3 . ic e /i . K = G'Qr^ n e '« . D . a; = S'p'ic 
Dem. 

|-.*257-36. DI-iHp.D./ieO'Qjjj,. 

[Hp] D.^lK (1) 

h . (1) . *258-241 . D h : Hp . D . p'« 6 (R^QYfi . 

[*257-36] :i.p'KeC'Qs^ (2) 

l-.*40-l. Dh:Hp.D.a!ey«: (3) 

h.(2).(3). Dh:Hp.D.p'«:e«. 

[*258-l 01 ] 3 . 2j'«: = maxQ'« (4) 

h.(4). Dh:Hp.D.(p'A;-t'<Sfy/«;)~6«. 

[*257-121.Hp] 0.xr^€(p'K-i'Sy>c) (5) 

l-.(3).(5). DI-:Hp.D.a!6i'Sy«: Dh.Prop 

*258-31. h:Hp*258-3./i~6l.D.(7'^;(3B^=/i 
i)em. 

h . *80-14 . D h : Hp . 3 . Q'/Sf = 01 ex'/* . 

[*150-36.*257-14] D . S'Qr^ = /SJQjj^ p (- t'A) . G'Qm^ p (- I'A) C Q'/Sf . 
[*150-22] D . O'^SJQjj^ = 8"G'Qs^ t (- t'^) • 

[*202-54.*257-125] D . O'/SJQjj^ = S"(G'Qm^ - I'A) (1) 

h . *8321 . D F- : Hp . D . S"G'Qs^ C fi (2) 

h . *258-241-301 .DI-:Hp.a;eyu,.D.a!6 S"{(E*Q)V - t'A} . 
[*257-36] D . « 6 S"(G'Qb^- I'A) (3) 

h . (2) . (3) . D h : Hp . D . S"(G'Qb^ - I'A) = /^ (4) 

I- . (1) . (4) . D h . Prop 

*258-32. h:iCi~6l.a!e4'ClexV.D./*6C'"Il [*258-3-31] 
This is Zermelo's theorem. 

*258-321. h : Hp *258-3 . /SQjj^a . D . 5f';8 ~ e a 

h . *250-242 . D h :. Hp . D : a = (Qs^y^ ■ v . (Qz^y^Qe^a : 
[*257-35.Hp] D : a C ;8 - i'<Sf'/8 :. D h . Prop 

*258-33. h : Hp *258-3 . /* ~ e 1 . P = /SJQjs^ . D . /S= minp f' CI ex'/t 
Dem. 

h.*80-14. Dl-:Hp.aCjit.g!a.D.S'a£a (1) 

h.*258-321. Dh :Hp(l).a;6a.D.~(a;8)./3Qij^a.a! = fi('^. 

[*150-4.Hp] D.~(a;Pfi"a) (2) 

h . (1) . (2) . *2051 . D h : Hp (1) . D . 8'ammp a . 
[*258-3] D.S'a= miup'a : D h . Prop 



SECTION D] ZEKMELO'S THEOREM 101 

*258-34. l-:./:i~6l.D: 

iSf e e^'Cl ex^ . = . (gP) . P 6 n . O'P = /^ . fif= minp f CI ex V 
[*260-5 . *258-33] 
*258-35. \- : fi e G"a. = . fi<^ el.^ I et^'Cl ex' fi [*200-12.*250-51.*258-32] 
*258-36. l-:/i6(7"I2ul. = .a!eA'Clex'M [*258-35 . *60-37 . *83-901] 

*258-37. h : Mult ax . = , Cil u 1 = Cls [*258-36 . *88-33] 

*258-38. h : . Mult ax . D : Nc'a < Nc'/3 . v . Nc'a = Nc'^S . v . Nc'a > Nc'/8 
[*255-73 . *258-37 . *ll7-54-55] 

*258-39. h:: Multax.D:./i,i/6N„C.D:/t<i;. v./*>i' [*258-38] 



*259. INDUCTIVELY DEFINED CORRELATIONS. : ., 

Summary of *259. 

In the theory of well-ordered relations, we often have ' occasion to define 
a relation (vyhich is generally of the nature of a correlation) by thfe following 
process : Given a relation S, let W'8 be a relation (generally a couple) which 
is a function of S. Let us put 

At^'8=Sk>W'S. 

Then, starting from A, we form the series 

A, Ajy'A, Ajy'A^r'A, etc., 

each of which contains all its predecessors. We proceed to the limit by, 

< . 

taking the sum of all these relations, i.e. s'( J. jj?-)5|f'A ; we then proceed to 

^ < . 

A^r's'{Aj^)^'A., and so on, as long as possible. The sum of all the relations 

so obtained is a function of W, and is often important. 

As an example, we may consider the correlation of two well-ordered 
series P, Q, which is dealt with in *259'2 — "25 below. In this case, we put 

W=xf{X = seqp'D'T J, seqg'a'T}. 

Hence Tf'A = i^'A = 5'P4,5'Q = lp4, Ig, 

ATy'AT^'A=lpilQK)2pi2Q, 
and so on. 

Proceeding in this fashion, we can continue until one at least of the 
two series P, Q is exhausted. We thus pbtain a new proof that, of any two 
well-ordered series, one must be similar to a section of the other. 

For convenience of notation, let us put temporarily 

A = §T{S(LT.S^T) Dft. 

We then have A eRl'J^n trans. ^.^^eRl'^. n Cls-»1, which is part of the 
hypothesis of *257'27 and following propositions. The rest of this hypothesis 
follows by analogy from *2.58'14. We now put 

W^ = s'(Aj^*AyA Df. 

Then W^ correlates the whole of P with part or the whole of Q, or vice 
versa. This is proved in *259"25, below. 



SECTION D] inductively DEFINED CORRELATIONS 103 

For other values of W, we get other results, often of a useful kind ; for 
example we shall have cession to use the methods of this number in *273, 
which deals with series similar to the series of rationals. 

The present number gives, first, some elementary properties of {A^fr^AyA 
and Wj^ for a general relation W, concerning which we only assume that 
W'S is never contained in S, i.e. TFn(G) = A (except in *259121-13, where 
we also assume Tf e 1 -* 01s). We then proceed to deal specially with the 
case where 

F = Zy |X = seqp'D'f J, seqg'a'T) 
as explained above. 

*259-01. A = ST(SGT.S^T) Dft [*259] 
*25902. AT^ = ST(T=SyjW'S) Dft [*259] 
*259-03. W^ = s'{Aw*AyA Df 

In the following propositions, which result from those of *258, it is 
essential to have Aj^QA. For this we require that W'S, when it exists, 
shall not be contained in S. It will be observed that, according to the above 
definition, 

A^^SfiSCT). 
Hence instead of using " G " as a relation, which is notationally awkward, we 
shall use A^. Thus the condition we wish to impose upon W is that we are 
never to have (W'S)A^S. This is insured by 

which accordingly appears as hypothesis in the following propositions. 

*2591. h : .4 6 Rl'J n trans . It^ e 1 -> Cls : 

Wf\A^-^ A . D . Aj^elLVA a Cls^ 1 . ^ {Ajy, A) efi 
Bern. 

As in *25814, h . It^ e 1 ^ Cls (1) 

|-.*20ri8. DI-:.Hp.D:ilfF»Sf.D.~(ilfG5f) (2) 

h . (2) . (*259-02) .Df-:.Hp.D:/Sf4pK^-^-'SGr.S=|=2'. 
[(*259-01)] :i.SAT (3) 

H . (1) . (3) . *258'202 . 3 h . Prop 
In the following proposition, the notation A (Aw, A) is that defined in 
*25702, adopted because A^ cannot conveniently be used as a suffix. 
*25911. \-:ElW'A.WnA^ = A.O. 

W^ = B'Cnv'A {Aw, A) . s"CV(Aw*AyA C (^^*^)'A 
Dem. 
H . *258-242 . *259-l . 3 f : Hp . \ C (^ b,*^)'A .D.s'Xe {Aw* Ay A (1) 
H.(l). Oh:Bp.D.W^e{Aj^*AyA (2) 

l-.*41-13. D\- :B.-p. Te{Aw*AyA-i'W^.D.TAW^ (8) 

I- . (1) . (2) . (3) . 3 h . Prop 



104 SERIES [part V 

*259111. h:.WnA^ = A.8,Te{A^r*AyA.:>:SCT.v.T(-8 
[*259-l . *257-36] 

*25912. \-'.Se-D'Ajr. = .ElW'S [(*259-02)] 

*259121. h : If 6 1 -* Cls . 3 . DM ^= a'Tf [*25912] 

*259122. h: W n A^= A.^W^y.-K'^iAjy^AyAnT {'^(xTy)} .■D.oo{W's'\)y 
Dem. 



V . *259-ll . 


DI-:Hp.D.s'\e(A^*^)'A. 


(1) 


[Hp] 


D.s'XeX 


(2) 


h.(l).(2).*257-3.Dh:Hp.D.s'\eD'^^. 




[*259-12] 


D . E ! W's'X 


(3) 


l-.(3). 


DI-:Hp.D.(s'\)^(^^'s'\). 




[*257-121] 


D . A ^'s'X e {Ajr^Ayk - X . 




[Hp] 


D.x{Aj^'s'\)y 


(4) 


h.(2).(4). 


3 h : Hp . D . ~ {« (s'.X) y}.x{A j^'s'X) y . 




[(*259-02)] 


D.«(TF's'\)2/:DI-.Prop 




i913. V-.WnA:)^^. 


A. . F6 1 -> Cls . D . F^ = i'Tf"(^,p*^)'A 




Dem. 






h . *259-122 


. 3 h : Hp . D . TT^ G s'F"(^^*4yA 


(1) 


1- . *257-123 


. D 1- : Hp . D . s'TF"(^^*^)'A G F^ 


(2) 



I- . (1) . (2) . D h . Prop 

*25914. I- :. Fn^5,j = A : )ge(^^*^)'A a 1 ->Cls a d'W . D^. 

Tf' /Se 1 ->Cls . a'/Sr A a'F'/Sf= A : D . F^ 6 1 -♦ Cls 
Bern. 

V . *71-24 . (*259-02) . D h :. Hp . D : 

.Sf 6 (A jr*AyA A 1 -> 01s . D . Aj^'S e (^ ^*4)'A a 1 ^ Cls (1) 
h . *259-lll . D I- :. Hp . 5f, T eiAj^^AyA .D : SCT .v . TCS (2) 

I- . (2) . D h ■.I[^.XC(Aw*AyA.a!{s'X)z.y(s'X)z.-^.('3^T).TeX.xTz^yTz (3) 
f- . (3) . D h : Hp . X C (Aj^^AyA a 1 -* 01s . x (s'X) z . y (s'X) z . 'i^ .sil=y (4) 
h . (4) . D h : Hp . A, C (^ ^*^)'A a 1 ^ 01s . 3 . s'\ e 1 -» 01s (5) 

t- . (1) . (5) . *258-242 . D h : Hp . D . {Aj^^Ayk CI -> Cls . 
[*259-ll] D.F^6l-»Cls:DI-.Prop 

*259141. I-:. FA^^ = A:>Sfe(43.*^)'AACl8-*lAa'F.Ds■ 
F'>Sf601s^l .D'S'AD'F'/Sf=A:D. F^eOls^l; 
[Proof as in *25914] ; 



SECTION D] inductively DEFINED CORRELATIONS 105 

*259-15. i-:.WnA^=A:Se(A^*AyAr\l^lna'W.Ds. '• • 

F'/Sf e 1 -> 1 . D'/?*n D'W'S=A.a'Sna'W'S = A:^.W^el-*l 
[*259-14-141] 

The following proposition is a lemma for *273-23. 

*25916. h:.W f^A^ = A:T6(A^^*AyAna'W.Pl'D'T-T>Q.DJ,. 

PtiA^'T) = (A^'T)iQ:D: 

Pl'D'Wj,= WJQ:Te(Af^*AyA.'2j.:PtB'T=r'Q 
Dem. 

h . *259-lll . D h :. Hp . X. C (J.^*^)'A . D : 

a; (P p D's'X) y .= . (gT) .Te\.x{P^ DT) y (1) 

h . (1) . D h :. Hp . \ C (^^*J.)'A : 7 e\ . Dy. P ^ D'T= T''Q:D: 

x{P\,'D's'X)y. = .{'3^T).TeX.x(T'>Q)y. 

[*259-lll] = . (a>Sf, T).8,Te\.x {S\ Q\T)y • 

[*150-1] =.a7{(s'X);<2}y (2) 

1- . (2) . *258-242 . D h : Hp .Te^Ay^^Ayk . D . P ^ D'2'=T5Q (3) 

h . (3) . *259-ll . D h . Prop 

The two following propositions are lemmas for *273'22'212. 

*259-17. V■..Wf^A^ = A■.S6{AJfr*AyAr^a'W.^:is^ 

a'/S n a'F'>S = A : D . a 1^ (^^*^)'A e 1 -♦ 1 
Dem. 

V . *250-242 . *267-35 . *269-l . D ; . 

h :. Hp . 8, Te{Aw*AyA . >S+ T. D : Aw'8Q.T.v . Aj^'T G S : 

[(*259-02)] ■^■.a'W'SCd'T.M.a'W'TQa'Si 

[Hp] D : a'yS + a'T :. 3 1- . Prop 

*259171. hi.WhA:i^ = A:8e{Aw*AyAna.'W.'^s- 

D'5f n D'F'<S = A : 3 . D f (J.y*2iyA e 1 -> 1 

[Proof as in *259-l7] 

*259-2. h! TF = lf{X = seqp'D'rj,seqQ'a'r}.D.F^6l-*l.]irn^5le=A 

Dem. 
H.*72182.Df-:.Hp.D:T6a'Tf .D. TF'Tel-*! (1) 

l-.*206-2. DI-:.Hp.D:yea'F.D.D'2'nD'TF'T=A.a'2'na'F'r=A (2) 
h.(2).*55-134.DI-:Hp.?'ea'F.D.~(F'rGr) (3) 

1- . (1) . (2) . (8) . *259-15 . 3 h . Prop 



106 SERIES [part V 

*259-21. h : Hp *259-2 .Q'QJ.D. WJQ GP.D'W^CG'P. Q' W^.C G'Q 
Dem. 

h.*206133.DI-:Hp.r€a'F.D.(F'r);Q = A (1) 

h . *206-21 . D h : Hp (1) . D . seqg'a'T^ e Q^d'T . 

[*37-461] D . ( W'T) \Q\T=k (2) 

l-.*206-18. Dh:Hp(l).D.D'^;^CC"P (3) 

h . (3) . *41-43 . *258-242 . D f- : Hp . D . D' TT^ C C'P (4) 

Similarly h : Hp . D . Q' Tf^ C G'Q (5) 

h . (4) . *206-132 . D I- : Hp (1) . Te{Aj^^Ayk . D . seqp'D'T6^'P"D'T. 
[*4016] D . seqp'D'T ep'P"2'"Q'seqe'a'T . 

[*40-6r] D . (2'"'Q'seqQ'a'r) t I'seqp'DT G P (6) 

I- .(1).(2) . (6). D h : Hp(l). T6(^^*^)'A. fJQ GP.D.(2^'T);QGP (7) 
h . *259-lll . D h :: \ C (^^*4)'A . « {(s'X)5Q} 2/ . D :. 

[*ll-62.*10-23] D :. T 6 X . Dr . TJQ G P : D . xPy (8) 

I- . (8) . Comm . D h :. \C(^^*^)'A : TeX.Dr-^'QGP : D.(s'X);QGP (9) 
I- . (7) . (9) . *258-242 . D f- :. Hp . D : Te (A^*^yA .D.T'QGP: 
[*259-ll] D:F^;QGP (10) 

h . (10) . (4) . (5) . D h . Prop 

*259-211. h : Hp*259-2 . P^G J. D . WJP<LQ [Proof as in *259-21] 

*259-22. \- : Hp *259-2 . P e connex . D , D"(^^*^)'A C sect'P 
Bern. 

I- . *211-22 . D h : Hp . r 6 Q' F . B'Te sect'P . D . D' j! ^T e sect'P (1) 
I- . *211-63 . D h : D"X C sect'P . D . D's'X e sect'P (2) 

h . (1) . (2) . *258-242 . D I- . Prop 

*259-221. H : Hp*259-2 . Qeconnex . D . a'-'(il^*^)'A C sect'Q 

*259-222. h : Hp*259 2 . PeSer . E ! B'P .Q^QJ. Te^A^^Ayk . D . 

T>Q e C'Ps [*259-21-22 . *213161] 

*259-223. h : Hp *259-2 . Q e Ser . E ! 5'Q . P^ 6 / . T e (^ ^*.4 )'A . D . 

T'PeG'Qs 

*259-23. h : Hp *259-2 . P, Q e Sern Q'P .Te(Af^*AyA.D . 

(gJf.iV) . Me C'P, .NeG'Q, . TeMeiSm N [*259-2-21-222-223] 



SECTION D] inductively DEFINED CORRELATIONS 107 

*259-24. h :. Hp *259-2 . ^, Q e 11 . D : D' F^ = O'P . v . Q' Tf^ = C'Q 

Dem. • 

l-.*206-18.DI-:Hp.P = A.D.F^ = A (1) 

h.*206-18.Dh:Hp.Q = A.D.F^ = A (2) 

f-.(l).(2).DI-:.Hp:P = A.v.Q = A:D:D'F^ = C"P.v.a'F^ = 0'Q (3) 
1- . *259-ll . *257-36 . D h : Hp . g ! P . a ! Q . D . Tf^ ~ e D'^l^ . 
[*259-12] 3 . ~ (E ! seqp'D' F^ . E ! seq^'Q' F^) (4) 

I- . (4) . *252-l . *259-22-221 . 3 

h :. Hp . a ! P . a ! Q . D : D'F^ = C'P . V . Q'F^ = a'Q (5) 

h.(3).(5).Df-.Prop 

*259-25. h :. Hp *259-24 . D : (g/S) . /3 e sect'Q . Wj, eP imor (Q ^ ;8) . v . 

(aa) . a 6 sect'P . F^ e (P ^ a) iSof Q [*259-23-24] 

The above affords a new proof of *254'37, which asserts that if P and Q 
are well-ordered series, one must be similar to a section of .the other. In 
virtue of *259"25 (which has been proved without using the propositions of 
*254), F^ is the correlator which correlates the whole of one. series with 
part or the whole of the other. 

It will be observed that the relations (J.jp*J.)'A are the class of corre- 
lators of sections of P with sectiops of Q, provided P, Q e li — I'A ; i:e, 

f-:Hp*2o9-2.P,Qen-t'A.D. 

{Ayfi^Ayk=f[{'^M,N) . MeC'F, . Ne G'Q, . Te Jlf s"mor N}. 



SECTION E. 

FINITE AND INFINITE SERIES AND ORDINALS. 

Summary of Section E. 

In the present section we shall be concerned first with the distinction of 
finite and infinite as applied to series and ordinals. We shall then establish 
the distinguishing properties of finite ordinals, and shall deal with the 
smallest of infinite ordinals, namely m, the ordinal number of a progression. 
Finally we shall briefly consider certain special ordinals, and the series of 
cardinals applicable to well-ordered infinite series, namely the series of 
" Alephs," as they are called after Cantor's usage. 

In dealing with the finite and the infinite as applied to series, we have 
coi^stant need of the relation (Pi)po, where P is the generating relation of 
the series. We have 

X (Pi)po y . = .P{xh-y)6 Cls induct — I'A, 

i.e. " x{P^^y" holds when, and only when, there is a finite number of 
intermediaries between x and y. When P is finite, we have 

■P = (-Pi)po> 

but we may have this when P is not finite. The infinite series for which 
this holds are progressions and their converses (which we will call regres- 
sions), and series consisting of a regression followed by a progression, of which 
an instance is afforded by the negative and positive finite integers in order 
of magnitude. 



*260. ON FINITE INTEEVALS IN A SERIES. 

Swmmafy of *260. 

In the- present number we are concerned with the relation which holds 
between x and y when the interval P{x\—y) is an inductive class other than 
A, or when the interval P(x\-\y) is an inductive class of at least two terms. 
This relation holds if x and y have any relation of the class fin'P (defined in 
*121). We will call this relation Pj^. Thus we put 

Pf„ = s'fin'P Df. 

Then aoPf^y holds when xP„y, where v is an inductive cardinal other 
than (*260"1). This relation will take us from x to any later term which 
can be reached without passing to the limit. But if in the interval P{x—\y) 
there is any term which has no immediate predecessor, i.e. any member of 
G'P—d'Pi, then we shall not have xPf^y. Thus P,n confines us to terms 
which are at a finite distance from our starting-point. We shall find that if 
P 6 fl,, a necessary condition for the finitude of P is P = Pj^. This is not 
a sufficient condition, since it does not exclude progressions, but these are the 
only infinite series it admits, and these are excluded by the assumption 

ElB'P. 

Although Pfn is not in general serial when P or Ppo is serial, it becomes 
serial when confined to the posterity or the ancestry or the family of any 
term with respect to itself (*260"32"4). When a series P is well-ordered, the 
whole series can be divided into constituent series, each of which is the 
family of any one of its members with respect to Pj^ (except when P has 
a last term which has no immediate predecessor, in which case this last term 
must be omitted). (Of. *264.) Each of these series (except the last, possibly) 
is a progression, and the last is either finite or a progression. Hence every 
infinite well-ordered series consists of a series of progressions followed by 
a finite tail (which may be null); hence the cardinal of the field of an infinite 
well-ordered series is a multiple of Xo. These results will be proved later ; 
for the present we are concerned with the proof that the family of any term 
with respect to Pj„ is a series of which the generating relation is Pf^ with 
its field confined to that family. 



110 SERIES [part V 

In the present number we are chiefly concerned with the relations of 
Pfa to P,. We have 

*260-27. h : Ppo 6 Ser . D . P,„ = (P,\, 

This proposition will be used very frequently throughout this section. 
Without any hypothesis we have 

*26012. i-.p,„ePp„ 

We have also 

*26015. l-.P,„ = (Ppo),„ 

Hence whatever properties of Pf„ result from the hypothesis that P is 
a series will result from the weaker hypothesis that Ppo is a series. 

If Ppo is a series, Pfn is contained in diversity and is transitive (*260'202), 
but not in general connected. 

In comparing Pj^ and (Pi)po, we constantly need the proposition 

*260-22. h : Ppo 6 Ser . D . {P,\ = A . P, e 1 -* 1 . (P^po G J 

From *260'3 to the end of the number, we are concerned with the result 
of limiting the field of Pj„ to the ancestry, posterity or family of some 
member of its field. We have 

*260-33. l-:Ppo6Ser.a;6D'Pi.Pi = E.D. 

An D ('''«' ^ K'^) = i%'^) 1 -Bpo = {&'*) 1 i2}po = {R r (Uo'^)}po 

*260-34. h : Hp *260-33 . D . {P^ ^( 'a; w p't^'a;)], = (%'x) ^R = E [%o'«! 



*26001. Pf„ = i'fin'P Df 

*2601. t- : xPf^ y. = . (jsy) . z/ e NC induct - I'O . mF„y 
[*121-121.(*260-01)] 

*26011. h : xPf^ y . = . P (a; M y) 6 Cls induct - - 1 
Dem. 

(-.*260-l.*121-ll.D 

VixPf^y. s . (ai/) . 1/ e NC induct - t'O . P (a; m j/) e v +o 1 ■ 
[*120-472] = . (a/Lt) . yti e NC induct - I'O - I'l . P (a; m y) e ^ . 
[*120-2] = . P (a; M y) € Cls induct -0-1:31-. Prop 

*26012. l-.P,„CPpo 
Dem. 

h .*121-321 .*117-511 . D h: i/eNC induct- t'O .:i.P,Q.P^^ (1) 
I- . (1) . *2601 . D h . Prop 



SECTION E] on finite INTERVALS IN A SERIES 111 

*26013. [- : a;Pj„ y .D .P{xh-y), P(x-\y)eCls induct - I'A 

Dem. • 

h .*260-12 ,*121-2r22 . D I- : Hp . D .P(a!i-2/),P(«-iy)e- t'A (1) 

H . *91-54 . (*121011-012-013) . D 
h.P{a!y-y)CP(wh^y).P(x-iy)CP(xi-ty). 

[*120-481.*26011] D h : Hp . D . P (a!t- 2/), P (a; -1 2/) e Cls induct (2) 

h . (1) . (2) . D h . Prop 

*260131. I- :. Ppo G J . D : xPi^ y . = . P(a)\-y)eC\s induct - I'A . 

= .P{x—\y)e Cls induct — t'A 
i)em. 
I- . *121-22 . D h : P (« f- 2/) e 01s induct - t'A . D . a;Pp„2/ ■ (1) 

[*121-242.*91-o4] ':>.P(xt-\y) = P(a;i-y)vi'y . 

[*120-251] D.P (a; My) 6 Cls induct (2) 

1- ■ (1) . *12r242 . DI-:Hp.Hp(l).D.a;,2/6P(a;My).«4=2/. 
[*52-41] D.P(a;i-i2/)~eOul (3) 

h . (2) . (3) . *260-ll . D 1- : Hp . Hp (1) . D . xP^y (4) 

Similarly 1- : Hp. P (a;—)?/) e Cls induct . D.a;Pfn 3/ (5) 

h . (4) . (5) . *260-13 . D h . Prop 

*26014. l-:P6(Cls->l)w(l^Cls).PpoGJ".D.P(„ = Pp„ 

Dem. 

h . *121-52 . D I- : Hp . D . s'finid«P = P* . 

[(*260-01)] O.Ptr, = P^-Po 

[*12l-302] =P^-^I[C'P 

[*91-541] =Ppo:3H.Prop 

*26015. l-.P,„ = (Ppo)f„ [*2601.*121-254] 

*26016. l-.(P),„ = Pfa [*260-l . *121-26] 

*26017. h : Ppo e Ser .xP^^y .Zi .P{x^y) = G'[P^^ lP{x^y)]. 

X = 5'{Ppo lP{x^y)].y = B'Cn^'{P,o D P (^ m y)} 
Dem. 
I- . *121-242 . D h : Hp . 3 . a;, y e P (a; M y) . « =^ y . (1) 

[*52-41] D.P(a!M2/)~el. 

[*202-55] D . a'{Pp„ pP (a; m y)} = P (a; m y) (2) 

l-,*91-542. 31-:. Hp. D :06P(a!M2/) .^4=^. 3 .a;{PpoPP(a!i-i2/)}^:: 
[(l).*205-35] D : a; = min {Pp„ 1^ P (a; h y),}'P (« m y) : 

[(2).*205-12] D : a; = 5'{Pp, t P («! HH j/)} (3) 

Similarly h : Hp . 3 . y = i?'Cn v'{Pp„ ^ P (« m y)} (4) 

h . (2) . (3) . (4) . 3 h . Prop 



112 SERIES [part V 

The folIowiDg propositions are concerned in proving that if Ppo e Ser, 
Pfn = (Pi)po and P„ = (Pi)„. Note that '■ x (Pi)poy " means that we can get 
from X to yhj a. finite number of steps from one term to the next, so that 
the series contains no limit-points between x and y. The relation "x (Pj), y " 
means that v — c 1 intermediate terms 

^l> ^2, ■^S) ••• ^p—el 

can be found, each of which has the relation Pi to its neighbour, and such 
that xPiZ and z,_^iPiy. Thus we have to prove that, provided Ppo is a series, 
this occurs when, and only when, the number of terms in the interval 
P {xv-^y) is V +c !• 

«260'2. I- : Ppo e connex . xP^y . yP^z .1i .P {x\-\z) = P{x)-\y)^j P {y\-\z) 

I)em. 
h . *20ri4-15 . Dl-:Hp.D.P(a;i-iy)CP(«M^).P(yi-i5)CP(a;i-i^) (1) 
h . *202-13-103 . D h :. Hp . xP^w . D : wP^y . v . yP:j^w (2) 

h . (2) . *121-103 . D 

h :. H^ . w e P (xv-i z) . D : xP^w . wP^y . v . yP^w . wP^z z 
[*121-103] D : w e P (« M 2/) u P (y m 0) (3) 

1- . (1) . (3) . D h . Prop 

*260'201. h : Ppo e connex . D . Pjn e trans 

Dem. 
I- . *260-12 . D h : xPj^y . yPt^z . D . xP^y . yP^z (1) 

1:.(1).*260-2.D 

f- : Hp . xPf^y . yPfj^z ,0 ,P{xy-tz) = P(xi-iy)\j P(yt-fz), (2) 

[*260-ll.*120-7l] D.P(«M^)eCls induct (3) 

h . *60-32-37l .DhiaeOwl./SCa.D.^SeOul: 

[Transp] DI-:;S~60wl.;SCa.D.a~60wl (4) 

h . (2) . *260-ll . D 

h : Hp . a;P,„2/ ■ yPfn^ ■ 3 . P(«i-ijr)~eO w 1 . P(a;i-i2/) C P(a!M^:) . 
[(4)] D.P(a;M^)~eOul (5) 

h . (3) . (5) . *260-ll . D h : Hp . xPt^^y . yPf^z . D . xPi^z : D f- . Prop 

*260-202. h : Ppo 6 Ser . D . P,„ e El'J" n trans 
i)em. 

I- . *260-12 . D h : Ppo G J . D . P,„ G J (1) 

I- . (1) . *260-201 . D f- . Prop 

We shall not have in general Ppo e Ser . 3 . P,„ e Ser, because P,„ is in 
general not connected. Pf„ only relates two terms which are at a finite 
distance from each other, and hence divides Ppo into a number of mutually 
exclusive parts. We shall only have P,„ e Ser when every interval in the 
series is finite. 



SECTION E] on finite INTERVALS IN A SERIES 113 

*260-21. I- : Ppo e Ser . xP^y .yP^z .:> . P {x^z) = P {x\-iy) yj i'z 
Dem. • 

V . *121-304 .D h -.Rtp .0 . P (y^z) = I'y w I'z (1) 

H.*121-242.DI-:Hp.D.2/eP(«M2/) (2) 

I- . *260-2 . D h : Hp . D . P (a; m 0) = P (« m y) w P (yt-tz) 
[(l)-(2)] = P (« M y) w t'^ : 3 h . Prop 

*260-22. h : Pp„ e Ser . D . (P,\ = P, . p, e 1 -* 1 . (P,\, Q J 
Dem, 

h.*121-254. DKP, = (P,„X (1) 

l-.(l).*204-7.DH:Hp.D.Piel->l (2) 

h . *121-305 . D I- : Hp . D . P, G P . 
[*91-59] D.(POp„CP,„. 

[*204-l] D.(P,),„eJ- (3) 

1- . (1) . (2) . (3) . *121-31 . D h . Prop 
*260-23. h : Pp„ 6 Ser . 1/ 6 NO induct . D . (P0„ e 1 -* 1 
[*121-342 , *260-22] 

*260-24. h : Pp„ 6 Ser . v e NC induct . x (Pj). y ■ x (Pi),+„. ^ . D . yP,z 
Dem. 

h . *121-35 . *260-22 . D h : Hp . D . a; {(PO. | PJ ^ . 
[*341] D . (gw) . X {P^\w .wP^z. 

[*260-23.Hp] D.yPi^iDh.Prop 

*260-25. 1- : Ppo e Ser . P = Pj . xB^y .D .P(xt-ty) = R{x)-iy) 

Dem. 
h . *260-24 . D H : Hp . z/ 6 NC induct . xR,y. xR^+.^z . P (xMy) = R (x^y) . 3 . 

yRz .P{xh-iy) = R(xh^y). 
[*260-21] :>.P{xh-iz) = R(x>-ty)yji'z 

[*260-22.*12I •371-304] =R(xi-tz) (1) 

h • (1) . D I- :. Hp . V e NC induct : xR^y . Dj, . P (a; i-i y) = P (a; m ?/) : D : 

xR^+„jZ . Da . P (« M ^) = P (a; M £r) (2) 
h . *121-301-22-242 . D f- : Hp . xR,y .D .P(xi->y) = i'x = R(xi-ty) (3) 

h . (2) . (3) . Induct . D 

I- :. Hp .Dive NC induct . xR^y . D . P (a; i-i y) = P (a; m y) : 
[*121-12] D : fi' e finid'P . a;% . D . P (a; M y) = P (« M y) : 
[*121-52.*260-22] D : xR^y . D . P (a; m y) = P (a; m 3/) :. D h . Prop 

In the above proposition, "Induct" refers to *12013. The '^<j}^" of 
*120'13 is replaced by 

xRty.Oy.P(x\r-\y) = R(xMy). 

R. & W. III. 8 



114 SERIES [part V 

Thus (2), in the above proof, is (when v is replaced by |) 

f e NO induct . <^^ . D . ^ (f +„ 1), 
and (3) is ^0. 

Hence, by *120"13, we have 

a e NO induct . D . </>«, 
i.e. V 6 NO induct . D : xH^y . Dj, . JB (a; m y), 

which is the inference drawn in the above proof. 

Wherever " Induct " is given as a reference, it indicates a process such as 
the above, making use of *120"13 or *120'11. 

*260-251. h : F^ e Ser . D . (P^po G P,„ 
Bern,. 
|-.*260-25.DI-:Hp.i2 = Pi.a!Epoy.D.P(a!tHy) = i?;(a;i-i2/). (1) 
[*12r45.*260-22] D . P (« m y) e Cls induct (2) 

1- . *121-242 . (1) . *260-22 . D h : Hp(l) . D . *, yeP{x^y) .x^y, 
[*52-41] D.P(a;M2/)~eOul (3) 

|-.(2).(3).DI-:Hp.a;(Pi)poy.3.P(«i-i2/)6Clsinduct-0-l. 
[*260-ll] D . xPt^y Oh. Prop 

*260-26. I- : . Ppo e Ser . i? = Pi . xR^y .2:xP^y. = . xB^y 

Bern. 

h . *260-25 . D h :. Hp . D : P (iCM?/) = P (xt-iy) : 

[*121-11] D : xP^y . = . xR,y :. D h . Prop 

*260-261. I- : Ppo 6 Ser . J/ e NO induct - t'O . xP^y . xP^+^^z . D . yP^z 

Dem. 
|-.*121-ll.Dh :Hp.D.Nc'P(«My) = j/+„l.Nc'P(a!i-i^) = z;+„2. (1) 
[*1 20-32] 3-2/ + ^ (2) 

h . (1) . *120-428 . D h : Hp . D . Nc'P(«t-i^) > Nc'P(a!hHy) . 
[*ll7-222.Transp] D . ~ {P (« m 0) C P (a; m 2/)} . 

[*121-103.*201 •14-15] D.~(0P5^y). (3) 

[*202-103] D . yP^^z . 

[*202-l71] D.P(a;i-i^) = P(a;M2/)wP(2/-i^). 

[*120-41.(1).(3)] D . P (2/ -H 5) 6 1 . 

[*121-242.(2)] D.P(2/i-.ir)62. 

[*121-11] D.2/Pi^:Df-.Prop 

« 

*260-27. f- : Ppo e Ser . D . P,^ = (POp„ 

Dem. 
V . *260-261 . D h : Hp . 1/ e NO induct - t'O . xP,y . xP,+^^z . x (Pi)po2/ . D . 

yPiZ.x{P,)j„y. 
[*9r511] 3.«(Pi)p„^ (1) 



SECTION E] on finite INTERVALS IN A SERIES 115 

I- . (1) . D h :. Hp . i; 6 NO induct - I'O : xPyy . Dj, . a; (P,)po 2/ : 3 : 

xP,+^,z.:),.x{P,)^z (2) 
I- . *91-502i . D h : xP^y . D . a; {P^\„y (3) 

h . (2) . (3) . *120-47 . D h :. Hp . D : I. e NC induct - I'O .X-P.d (Pi)po = 
[*260-l] D:P,„C(P,)po ' (4) 

h . (4) . *260-251 . D h . Prop 

*260-28. h : Pp„ 6 Ser . i; 6 NC induct - t'O . D . P„ = (P,)v= (Pfn)- 
-Dem. 

I- . *260-26 . 3 1- :. Hp . D : a; (P^poy . xP^y . = , « (P^po^ ■ oo {P,\y (1) 
h . *2601 . 3 I- : Hp . xP,y . 3 . xP^^y . 

[*260-27] :>-a>(Pdvoy (2) 

t- . *121-321 . D h : Hp . ^ (P,\y .0.x {P,)^y (3) 

h.(l).(2).(3).Dh:.Hp.D:a;P,y. = .a;(P0„2/ (4) 

l-.*121-254. DI-.(PO. = KA)po}.- 

[*26a-27] D h : Hp . D . (PO. = (P,n). (5) 

h . (4) . (5) . 3 h . Prop 

The above proposition does not hold in general when k = 0, for if P is a 
compact series, Pi = A, so that (Pi)o = A, but Po = / \G'P. 

*260-29. I- : Ppo 6 Ser . xP^^y . 3 . P (a; i-h 2/) = Pj (a; m y) = P,„ (« i-i y) 
Bern. 

h . *260-27-25 . 3 h : Hp . 3 . P (a; M 2/) = Pi (a; M y) 
[*121-253.*260-27] = P,„ (a; m y) : 3 1- . Prop 

The following propositions are mainly concerned with the result of 
confining the field of Pjn to the posterity of a single term. 

*260-3. I- : Pp„ e Ser . 3 . D'P,„ = D'Pj . a'P,„ = Q'Pi . C'Pf„ = O'Pi 
[*260-27.*91-504] 

*260-31. h:Pp„eSer.a;eD'Pi.3. 

G'{Pt. D(t'« ^%»\ = tP^'x = I'a; u %,'x 
Dem. 



h . *260'27 . 3 I- : Hp . 3 . fc'a; w P^^'x = I'x u (Pi)p/a; 

[*96-14] ^ =^'«= (1) 

l-.*260-3. 3h:Hp.3.a!P,n'a:. 

[*36-13] ^3 . {w) . X {Pto I {I'x yj%^'x)} y (2) 

h . *3613 . 3 h : 2/ e Kn'« . 3 . a; {P^ ^ (*'« " ^fn'^?)} 2/ . 

[*io-24] 3 ■ (a^) ■ ^.{Pfn D (*'* « K^ )} y (3) 

h . (2) . (3) . 3 h : Hp . 3 . 1'a,' w P,„'arC C'lP^ ^(I'a; « K'*)} • 

[*37-41] 3 . 1'x w P,„'a; = a'{P,„ p (I'a; w P^'n'*)} (4) 

I- . (1) . (4) . 3 I- . Prpp 

8—2 



116 SERIES [part V 

*260 32. l-:PpoeSer.D. 
Dem. 

V . *26o-i2 . D h . Pf„ t (!'*• u Kn'«^) e Ppo D (t'^; " K'*) (i) 

h . *260-3 . *200-35 . 3 

1- : Hp . « ~ e D'Pi . D . Pf„ t (t'« v. Kn'«) = A = Pp„ D {t'co u K'*') (2) 

h . *201-521 . *260-27 . D 



h : Hp . a; 6 D'P, . D . Pf„ t (i'«? u Pfn'a;) = (POp„ D {P.)*«> ■ 
[*202-14.*260-22]D . Pf„ ^(I'a; u P,„'a;) e connex . 
[*260 202] 3 . Pfn D (t'« « K'*) e Ser . (3) 

[(1).*260-31.*204-41] D . P,„ l{l'x w ^/^) = Pp„ l{i'x u Pj^^'a;) (4) 

f- . (2) . (3) . (4) . D t- . Prop 

*260-33. h : Ppo e Ser . a; e D'Pj . Pj = i? . 3 . 

Dem.. 

h . *260-27-31 . D f- : Hp . D . P,„ CC^'^^ " An'«) = -Kpo D^*'*: 
[*96-16.*91-602] = (E*'a!)1 Epo (1) 

[*96-13] ={&21i2}p„ (2) 

[*96-2.*260-22] = {R rCRpo'«')}po (3) 

h.(l).(2).(3).DI-.Prop 

*260-34. 1- : Hp *260-33 . D . {P,„ I (I'x w Pin'«)}i = (-B*'^)1 i? = i^ rXD'a; 

jDem. 

h . *2'60-33 . *121-254 . D 

h : Hp . D . {P,„ DCi'a; u K'*)}! = K^*''^)! ii}i = {R rK>^> (1) 
f- . (1) . *121-31 . *260-22 . D h . Prop 

The following propositions are concerned with the result of confining the 
field of P(n to a single family. 

*260-4. h : Ppo e Ser . D . Pf„ C Kn'« e Ser . 

C"(P,„ DK'^) = K'^ = (K)*'^ . P,n'^ ~ e 1 
jDem. 

1- . *260-27 . *97-l7 . D h : Hp . D . P,„ D K'« =(Pi)po D(K)*'a; ■ 

[*202-15.*260-22] D . P,^t -Pfn'a' e connex . 

[*260-202.*204-42] D . P,„ t-Pfn'« e Ser (1) 



SECTION E] on finite INTERVALS *IN A SERIES 117 

h.*97-18.DI-^a'(P,„tK'«') = Kn'«' ^ (2) 

I- . (2) . *260-202 . *200-l 2 . D h : Hp . D . Ka''^ ~ e > (3) 

h . *260-27 . *97-l 7 . D h : Hp . D . P,^'cc = (Pj*'* (*) 
h.(l),.(2).(3).(4).DI-.Prop 

*260-41. l-:Ppo6Ser.i2 = P,.D. 



De 



m. 

I- . *260-27 . *97-l7 . D h : Hp . D . P,„ p P,„'a; = i?p„ I %'a! ^ (1) 
h . *97-13 . D f- : Hp . y 6 Pji^'a; . yR^„ z.D.ze B^'^R^'x « Epo"i2*'« ■ 
[*92-31 1 .*260-22] D . ^ e fl^'a; u P^'a; . 

[*9713.*36-13] 3 . y (Ppo W^;) ^ (2) 

I- . *35-21-441 . D h . Pp„ pp'jie'a; G (P*'a;)1 R^ (3) 

h . (2) . (3) . D h : Hp . D , Ppo ^P*'^ = (P*'^)1 Pp„ (4) 

Similarly h : Hp . D . Pp„ ^ %'x = R^ [%'w (5) 
I- . (1) . (4) . (5) . D h . Prop 

*260-42. h : Hp *260-41 . D . P,„ ^K'^ = (^*'*1 i2)po = (R \-%'^)vo 
Pern. 

h . *92-32 . *260-22 . 3 h : Hp . D . P^'P^i^'a; C R^'x . 
[*96-lll] D . (%'xy^ Pp„ = {{R£x)^ RU (1) 

Similarly h : Hp . D . Pp„ \-^^'x = {P pSle'^'lpo (2) 

h . (1) . (2) . *260-41 . D h . Prop 



*260-43. l-:PpoeSer.D. 



J)em. 



{Pf. tK^r = Px D K'^' = (K.'^) 1 Pi = Pi r (Kn'a;) 



|-.*260-42.*l 21 -254.3 

h : Hp . P =P, . D . {P,„ tK„^, = {(J*'«') 1 -Rli 

[*121-31.*260-22] =^^'x)^R 

[*97-17.*260-27] = (Kn''^) 1 A ~ (1) 

Similarly h : Hp . 3 . {P,„ ^ K'^']. = -Pi TAn'* (2) 

K (1) . (2) . *35-l 1 . D h : Hp . D . {Pf„ fp,„'^}, =P, [:*?,/« (3) 

H.(l).(2).(3).Dh.Prop 

Observe that the two series Pfn ^Pf„'x and Pfj, pPfn'y are either identical 
or have no common terms in their fields. This results immediately from 

*97"16, since the fields of the two series are (Pi)^'x and {Pi)^'y. 



*261. FINITE AND INFINITE SERIES. 

Summary of *261. 

In this number we define finite and infinite series, and we show that, 
where well-ordered series are concerned, there is only one kind of finitude, 
i.e. there is not the distinction, which exists in cardinals, between "in- 
ductive" and "non-reflexive." We also give various equivalent forms of 
the distinction between finite and infinite series, and some of the simpler 
properties of each. The propositions of this number are numerous and 
important. 

We define an infinite series as one whose field is a reflexive class, and a 
finite series as one which is not infinite. Thus we put 

Ser infin = Ser n C"Cls refl Df, 

n infin = n n 0"Cls refl Df, 

Ser fin = Ser — Ser infin Df, 

n fin = n - O infin Df. 

We also put, to begin with, 

XI induct = XI n a"Cls ind net Df, 

but in the course of this number we prove 

*261-42. t- . X2 fin = X2 induct 

so that the symbol " XI induct " is not required after the present 
number. 

After some preliminary propositions, we proceed (*261-2ff.) to various 
criteria of finitude and infinity. We have 

*261-25. h:.PeSer.D: 

G'P 6 Cls induct - t'A . = . P = P,„ . E ! B'P . E ! 5'P 

The condition P = Pf^ insures that every interval is finite, but this still 
leaves it possible for our series to be a progression, or its con-verse, or the 
converse of a progression followed by a progression {i.e. the type of the nega- 
tive and positive finite integers in order of magnitude). The third of these 



SECTION E] finite AND INFINITE SERIES 119 

possibilities is excluded by either E ! B'P or E ! B'P ; the second is excluded 

by E ! B'P, and the first by E ! B'P. We have 

*261-212. h :. Pefl . 3 : a'P, = a'P . = . P = (POpo . = ■ P = Pf„ 

" Q'Pi = CE'P " means that every term except the first has an immediate 
predecessor. We have 

*261-26. h : P e Ser . a C C'P . a ! o . a e Cls induct . D . E ! minp'a . E ! maxp'a 

and 

*261-27. h :. P e Ser : a C O'P . a ! a . Da . E ! minp'a . E ! maxp'a : D . 

P = P,„. a'P 6 Cls induct 
whence we obtain 

*261-28. hxPeSer.D:. 

a C C'P . a ! a , Da ■ E ! minp'a . E ! maxp'a : = .0'Pe Cls induct 

I.e. a series whose field is inductive is one in which every existent sub- 
class of the field has both a minimum and a maximum. 

From the above, together with an inductive proof that every inductive 
class which is not a unit class is the field of some series, we obtain 

*261-29. h. Cls induct = 

1 u G"P{P 6 Ser : o C O'P . a ! a . Da . E ! minp'a . E ! maxp'a} 
= 1 w 0"(n ft Cnv"0) 

The above proposition is interesting as giving an alternative method of 
treating inductive classes. Instead of the definitions adopted in *120, we 
might have taken the above proposition as the definition of inductive classes, 
putting 

NO induct = Nc"Cls induct Df. 

We should thus wholly avoid the use of mathematical induction in de- 
finitions; hence if such avoidance were in any way desirable, it could be 
secured by dealing with series before introducing the distinction of finite 
and infinite, and then defining inductive classes as the fields of series which 
are well-ordered backwards as well as forwards. The inductive properties of 
such classes would then be deduced from *261"27, together with *260'27, in 
virtue of which we have 

P e fl n Cnv"f2 .D.P = (P^po. 
whence, by *91"62, 

I- ::P efi r. Cnv"i2 . D :. ooPy . = : Pi"^ C/* . P^'xe/ju. D^.yefi. 

In virtue of this proposition, if 7 is the field of a well-ordered series P 
whose converse is well-ordered, then any property which is inherited with 
respect to Pj belongs to all the successors of x (where xey) if it belongs to 
the immediate successor of x. Hence mathematical induction follows. 



120 SERIES [part V 

From the above we obtain at once 

*261-31. h :. P 6 Ser . D : C'P 6 Cls induct . = . P, P « fl 

I.e. series whose fields are inductive are the same as inductive well- 
ordered series, and are also the same as well-ordered series whose converses 
are well-ordered. Hence also we obtain 

*261-33. hrP.Qefl.QGP.D.QeOinduct 

I.e. a descending well-ordered series of terms chosen out of a well-ordered 
series must be finite. This proposition, which is due to Cantor, has been used 
by him in many proofs. 

We have 

*261-35. h :. P 6 fl . D : C'P 6 Cls induct - t'A . = . Q'P^ = d'P . E ! B'P 

In *253"51 and following propositions we have already had the hypothesis 

CI 'Pi = CE'P . E ! B'P, which now turns out to be equivalent to the hypothesis 
that our series is finite and not null. Thus we have 

*261-36. h i.Pefl . D : G'P e Cls induct - I'A . = . Nr'P=t= l-j-Nr'P 

*261"4 and following propositions are concerned in proving that a well- 
ordered series which is not inductive always contains progressions, and in 
deducing consequences from this. We have 

*261-4. h : P 6 n - O induct . D . {{P;)^'B'P] ^ P^ e Prog 

The above proposition is very important, for many reasons. One of its 
most important consequences is that, if P is a well-ordered series which is 
not inductive, its field contains an Ko, and is therefore a reflexive class 
(*261"401). Hence a class which can be well-ordered is either inductive or 
reflexive (*261'43), and a well-ordered series is either inductive or infinite 
according to the definitions given above (*261*4!l). Hence where well- 
ordered series are concerned, the two ways of defining finite and infinite 
(namely those in *120 and *124) give equivalent results. This cannot (so 
far as is known) be proved for classes in general without assuming the multi- 
plicative axiom. 

From the above-mentioned propositions it follows that a well-ordered 
series is one in which Pi limited to the posterity of B'P with respect to Pi 
is a progression in the sense of *122 (*261'44), and that any class contained 
in a well-ordered series is either inductive or reflexive (*261"46). 

The number ends with some propositions in ordinal arithmetic. We 
prove that P'^ is well-ordered if P is well-ordered and Q is a finite well-ordered 
series (*261'62) ; that if i? is a finite well-ordered series, and P is less than Q 
(in the sense of *264), then P^ is less than Q^ ; and that a finite well-ordered 
series is less than an infinite one (*261'65). 



SECTION E] finite AND INFINITE SERIES 121 

*261-01. Sermfin = SernG"Clsrefl Df 

*26102. n infin = H n C'Cls refl Df 

*26103. Serfia=Ser-Sermfia Df 

*26r04. Iifin = n-nmfin Df 

*261-05. n induct = 12 n C"Cls induct Df 

*2611. hiPeSer infin. = .P 6 Ser.C'PeCls refl [(*261-01)] 

*26111. h: Pen infin. s.PeO.C'PeClsrefl [(*261-02)] 

*26112. h : Pe Ser fin . = . P e Ser - Ser infin . = . PeSer . C"P~eClarefl 
[(*261-03)] 

*26113. h : Peflfin . = . PeO- n infin . = . Peft . 0'P~eClsrefl 
[(*261-04)] 

*26114. h : P e fi induct .= .Peil.C'P e.Cls induct [(*26105)] 

*261-15. I- : P e Ser infin . P smor Q . D . Q e Ser infin 
Bern. 

l-.*261-l.DI-:Hp.D.P6Ser.O*PeClsrefl.PsmorQ. 
[*204-21.*151-18] D . Q 6 Ser . O'P 6 Cls refl . G'P sm G'Q . 
[*124-18] D.QeSer.C'QeClsrefl. 

[*2611] D . Q 6 Ser infin Oh. Prop 

*261-151. h : P 6 Ser infin . D . Nr'P C Ser infin [*261-15] 

*261-152. h : P 6 Ser infin . = . N„r'P C Ser infin . = . g ! N„r'P n Ser infin 
[*261-151.*155-12] 

*261-153. J- : P 6 Ser infin . = . (gQ) . P smor Q . Q e Ser infin 
[*261-15 . *1.5113] 

*261-16. h : P 6 n infin . P smor Q . D . Q e fi infin 

[Proof as in *261-15, using *261-11 . *251-111 . *151-18 . *12418] 

*261-161. f- iPeXiinfin. 3. Nr'P Cfl infin [*26ri6] 

*261-162. h : P 6 fl infin . = . Nor'P C il infin . = . g ! N„r'P n Ser infin 
[*261-161 . *15512] 

*261163. h:P6fiinfin.= .(aQ).PsmorQ.Q6liinfin [*261-16 . *151-13] 

*261-17. HiPeSerfin.PsmorQ.D.QeSerfin [*261-15 . Transp] 

*261-171. hcPeSerfin.D.Nr'PCSerfin [*261-17] 

*261172. I- : P e Ser fin . = . N^r'P C Ser fin . h . g ! N„r'P n Ser fin 
[*261-171.*155-12] 

*26li73. h : P 6 Ser fin . H . (gQ) . P smor Q . Q e Ser fin [*26117 . *15ri3] 



122 SERIES [part V 



18. hrPenfin.PsmorQ.D.Qeflfin [*261-16 . Transp] 

181. hiPellfin.D.Nr'PCOfin [*261-18] 

182. I- : P e n fin . = . N„r'P C £1 fin . = . g ! Nof'P n Q fin 
[*261-181.*155-12] 

183. h:P6nfin.= .'(aQ).PsmorQ.Q€fifin [*261-18 .*loll3] 

19. h : P e 12 induct . P smor Q . D . Q e H induct 
[Proof as in *261-16, using *120-214 instead of *124-18] 

191. h : Pen induct. D.Nr'P CXI induct [*261-19] 

192. hzPeil induct . = . N„r'P C Q, induct . = . g ! Nor'P n O, induct 
[*261191.*155-12] 

193. hzPeO, induct . = . (aQ) . P smor Q.QeD, induct 
[*26ri9.*15]-13] 

2. l-:Ppo6Connex.(5'P)P,„(jB'P).D.O'P6Clsinduct 
Dem. 

h .*202-181 . 3 h : Hp . D . G'P = P(B'P\r-iB'P) . 
[*260-ll.Hp] :>.C'Pe Cls induct Oh. Prop 

*261-21. h : P e connex . P = P,„ . E ! B'P . E ! B'P .D.C'Pe Cls induct 
Dem. 

h . *202-103 . *93-101 . D h : Hp . D . (B'P) P (B'P) . 

[Hp] ■^.(B'P)P,^(B'P)- 

[*261 -2] D.C'Pe Cls induct Oh. Prop 

*261211. h : P 6 Ser . D . ^rip'{Pa! - (Kv*} C d'P - Q'P, 
Dem. 

h.*91-511.*121 -305.3 

h :. Hp . D : 2/ e P'a; n (Pi)po'« . yP^z .D.zeP'xn (P\^'x : 

<- 



*261 
*261 
«261 

*261 
«261 

*261 
*261 

*261 

«261 



[Transp] DizeP'x- (P,)^^'x .yP^z .:> .y e-P'x^J - (P,)^'x (1) 

h.*91-5O2.Dh:.0 6P'a!-(P,)po'«. '^\zeP'x-%'x\ 

[*201-63] D : Hp . 3 . aiP^^ (2) 

h . *201-63 . D h : Hp . xF'z . yP^z . D . ~ (yPx) .y^x. 

[*202-103] D , xPy (3) 

h.(2).(3).Dh:Hp.^6P'a;- (P^'x .yP,z ."D .ye^'x . 

[(1)] D.yeP'x-(%j^'x. 



[*201-63] D.yeP'zn{P'x-(P,)^'x}. 

[*20514] D.^~emm/{Kc-(Kwa;} (4) 

h . (4) . Transp . D 

h : Hp . 6 minp'{P'a! - (P,)^,'x} . D . ~ (gy) . yP,z : D h . Prop 



SECTION e] finite AND INFINITE SERIES 123 

*261-212. l-:.P6n.D:a'Pi = a'P. = .P = (P,)po. = .P = Pfn 
Bern. • 

I- . *121-305 . D h : Hp . D . (P^po G P (1) 

H . (1) . D h : Hp . P + (POpo . D . {'3X, y) . xPy . ~ KP0po2/l ■ 

[*32-18] D.(a«;).a!P'a;-(Kv«'- 

[*250-121] D . (a«) . E ! minp'lP'^ - (Ku'*} ■ 

[*261-211] O.'^ia'P-a'P^ (2) 

I- . (2) . Transp . D I- : Hp . Q'P = Q'P, . D . P = (P^po (3) 

h . *91-504 . 3 h : P = (P,\^ . D . Q'P = a'P. (4) 

h . (3) . (4) . D h :. Hp . D : Q'P. = Q'P . = . P = (P^po . 
[*260-27] =.P = Pt^:.D\-. Prop 



*261-22. h : P 6 Ser . C"P e Cls ind uct . D . P = P,„ . D'P = D'P, . Q'P = Q'P, 
Dem. 

h . *260-12 . *201-18 . D h : Hp . D . P,„ G P (1) 
I- . *121-242 . D h : Hp . xPy .':i.x,yeP{x\-\y).x^y. 

[*52-41] D.P(a;M2/)~60ul (2) 

- I-.*120-481. DI-:Hp.D.P(a!M2/)6Clsmduct (3) 

I- . (2) . (3) . *260-l 1 . D h : Hp . a;P2/ . D . xPf^y (4) 

l-.(l).(4). Df-:Hp.D.P = P,„. (5) 

[*260-3] D . D'P = D'P, . Q'P = G'P, (6) 

I- . (5) . (6) . D h . Prop 

*261-23. h : PeSer . D'Pi = D'P.~ E ! 5'P . g ! P. D . C'PeCls refl 
Dem. 



^ <- 



h.*91-52. DI-.P,"(PO*'a; = (P,)po'a' " (1) 

h.*91-54.*260-22.D 



h : Hp . a; e (7'P . D . (Pi V« = ''« »^ (-POpo'^ ■ a^ ~ e (-Pi)po'« (2) 

l-.*93-ll. Dh:Hp.D.D'Pi = C"P. (3) 

[*9018] D.(KVa'CD'P, (4) 

I- . *260-22 . D I- : Hp . D . Pi e 1 -> 1 (5) 

h . (1) . (2) . (4) . (5) . *r3-21 . *91-74 . D 
<- — <- 



h : . Hp . D : a; e G'P . D . (P,)*'^ sm (POpo'a^ ■ (Pi)po'* C (P, V«' ■ 

^ a!(PO*'a!-(POi^o'«'- 

[*124-16] 3 . (P'O^'x 6 Cls refl (6) 

I- . (6) . (3) . (4) . D I- : Hp . D . a ! Cls refl n Cl'CP . 
[*124141] " D.C'Pe Cls refl : D h - Prop 



124 SERIES [PABT V 

*261-24. Y-iFeSer.G'Pe Cls induct - I'A . D . E ! £'P . E ! fi'P 

Bern. 

h . *261-22 . D h : Hp . D . D'P = D'P^ . 

[*261-23.Transp] D . E ! 5'P (1) 

h.(l)^. DI-:Hp.D.E!5'P (2) 

h . (1) . (2) . D h . Prop 

*261-25. h i.PeSer . D : C'P e Cls induct - t'A . = .P = P,„.E!£'P. ElB'P 

[*261-22-24-21] 

When P = P,„ . E ! 5'P . ~ E ! B'P, P is a progression ; 

when P = P,„ . E ! 5'P . ~ E ! £'P, Pisa regression 

(i.e. the converse of a progression) ; and when 

P = P,„ . ~ E ! 5'P . ~ E ! B'P, 

P is the sum of a regression and a progression. These propositions will be 
proved in the next number. 

*261-26. 1- : P € Ser . a C C'P . a ! a .« 6 Cls induct . D . E ! minp'a . E ! maxp'a 
Dem. 
i- . *205-17 .Dt-:Hp.«el,D.EI minp'a . E ! maxp'a (1) 

h . *202-55 .Dh:Hp.a~6l.D. a=0'(P I a) . 
[*261-24] D . E ! B'{P ^ a) . E ! B'Cnv'(P ^ a) . 

[*205-42] D . E ! minp'a . E ! maxp'a (2) 

h . (1) . (2) . D h . Prop 

*261-27. h :. P 6 Ser : a C C'P . a ! a . 3^ . E ! minp'a . E ! maxp'a : D . 

P = P,„.C"P 6 Cls induct 
Dem. 

t-.*250121 .DhiHp.D.Pefi. 

[*250'21] D.D'P = D'P,. 

[*260-3] D.D'P = D'P,„ (1) 

I- . (1) . D I- : Hp . ajP,„2/ . 2/ 6 D'P . 3 . 2/ e D'P,„ . ^P,„y . 

[*260-201] D.ye P,„"P,n'a; . 

[*260-12.*201-18] O.ye P"%n'x . 

[*205-lll] D.2/ + maxp'P,/a! (2) 

I- . (1) . (2) . Transp . D h : Hp . a? e D'P . D . maxp^P,„'ar = B'P (3) 

h . *260121-13 . D h : Hp . a ! P . D . E ! .B'P . 

[(3)] 0.(B'P)Pt„{B'P)- 



SKCTION E] finite AND INFINITE SERIES 125 

[*260-l 1] D . P (B'P M B'P) 6 Cls induct . 

[*202-181] * D.CP 6 Cls induct (4) 

f- . *120-212 . D h : P = A . D . C'P 6 Cls induct (5) 

H.(4).(5). Df-iHp.D.CPeClsinduct. (6) 

[*261-22] 3.P = Pf„ (7) 
1- . (6) . (7) . 3 I- . Prop 

#261-28. hiiPeSer.D:. 

a C a'P . a ! a . Da . E ! minp'o . E ! maxp'a •. = .G'P€ Cls induct 
[*261-26-27] 

*261-281. I- : 7 e Cls induct - 1 . D . 7 e 0"Ser 

Dem. 

y . *204,-24 . D h . A e 0"Ser (1) 

f-.*52-22. Dl-.Aut'icel (2) 

I- . *52-22 . D h : a; = 2/ . D . t'a; u I'y e 1 (3) 

I- . *204-25 . :>\-:x^y.O .I'x^Ji'ye C'Ser (4) 

l-.(3).(4). D\-.i'xyJi'yelwC"Sev. 

[*52-l] Dhzyel.D.yyJi'yelyj C'Ser (5) 

f-.*51-2. 0[-:yeG"Ser.yey.D.y\Ji'yeG"Ser (6) 

h . *204-61 . *161-14 . D h : 7 e 0"Ser .^ly. y^ey .0 .yyj I'ye C"Ser (7) 

l-.(6).(7). Dl-:760"Ser.a!7.D.7ut'2/60"Ser (8) 

h.(2).(5).(8). Dh:7elwO"Ser.D.7ui'y6luC"Ser (9) 

h . (1) . (9) . *120-26 . D h : 7 e Cls induct . D . 7 e 1 w C"Ser : D h . Prop 

*261-29. I- . Cls induct = 

1 w G"P {P 6 Ser : a C C'P . a ! a . D, . E ! min^'a . E ! maxp'a} 
= 1 u C"(n n Cnv"n) 
Dem. 
h . *261-281 . D h :. 7 6Cls induct- 1 . D : (gP) : PeSer .y=G'P : 
[*261-28] 

D : (aP) : P e Ser : a C C'P . a ! a . D. . E ! miup'a . E ! maxp'a : 7 = G'P : 

[*37-6] D : 7 e G"P{P e Ser : a C (7'P. a ! « . D„ . E ! minp'a . E ! maxp'a} (1) 
h.*261-28.Df-:.PeSer: 

a C G'P . a ! a ■ 3a - E ! minp'a . E ! maxp'a : 7 = O'P : 3 . 7 e Cls induct : . 
[*37-6] 3 h : 7 e C"P (P e Ser : a C O'P . a ! « . D„ . E ! minp'a . E ! maxp'a) . D . 

7 6 Cls induct (2) 

V . *120-213 . D h . 1 C Cls induct (3) 
h.(l).(2).(3).D^ 

V . Cls induct = 0"P {P € Ser : a C (7'P . a ! « ■ 3. ■ E ! minp'a . E ! maxp'a} 
[*250-121] = G"(n n Cnv"n) .31-. Prop 



126 SEBiES [part V 

The following four propositions are immediate consequences of the 
propositions already proved. 

*26r3. l-::PeSer.D:. 

G'P e Cls induct . = : P e fl : a C O'P . g ! a . D„ . E ! maxp'a 
[*261-28 . *250121] 

*261-31. h :. P e Ser. DiO'Pe Cls induct. = .P,Pefl [*261-3 . *250121] 

*261-32. h . Ser n 0"Cls induct = n induct = n n Onv"n [*261'31-14] 

On account of this proposition, we do not introduce the notation " Ser 

induct " for " Ser n (7"Cls induct," because a series whose field is inductive 
is a well-ordered series, and therefore the notation " fl induct " gives all that 
is wanted. 

*261-33. \-iP,Qea.QQ.P.D.Qea induct 
Dem. 

I- . *204-2 . D h : Hp . D . Q e Ser . Q G P . 

[*2o0-14] D.QeSernBord. 

[*2.50-12] D.Qen. 

[*261-32] D . Q 6 n induct : 3 h . Prop 

This proposition (which is due to Cantor) is of great importance in the 
theory of well-ordered series. It shows that, however great a well-ordered 
series may be, any descending well-ordered series contained in it must be 
finite. (A descending series in a given series is a series contained in the 
converse of the given series.) 

*26134. 1- : P e fl . Q'Pi = Q'P . E ! B'P .D.G'Pe Cls induct 
Dem. 

h . *250-23 . *214-12 . D h :. Hp . a C G'P . D : E ! maxp'a . v . E ! seqp'a (1) 
h . *206-181 . D h : Hp . a C C"P . a ! a . E ! seqp'a . D . seqp'a e a'P, . 
[*204-7] D . E ! P/seqp'a . 

[*206-451] D . E ! maxp'a (2) 

h . (1) . (2) . D h :. Hp . 3 : aC O'P . a ! a . D, . E ! maxp'a : 
[*261-3] D:C'Pe Cls induct :. D h . Prop 

*26135. h :. P e fl . D : O'P e Cls induct -i'A. = . Q'P, = Q'P .El B'P 
[*261-22-24-34] 

Observe that "a'Pi = Q'P . E ! £'P" occurs as hypothesis in *253-5l 
and some succeeding propositions. Thus this hypothesis is equivalent to the 
hypothesis that the field of P is an inductive existent class. It follows that 



SECTION E] finite AND INFINITE SERIES 127 

if P is an inductive well-ordered series, Nr'Ps = Nr'P, whereas if P is a 
well-ordered series which fc not inductive, Nr'Ps = Nr'P 4- 1 ; also that 

*261-36. 1- :. P 6 ft . D : C'P 6 Cls induct - t'A . = . Nr'P + 1 + Nr'P 
[*253-573 . *261-35] 

*261-37. h :. P e O . D : O'P e Cls induct . = . 1 -f- Nr'P = Nr'P -j- 1 
[*253-574 . *261-35 . *161-2-201] 

*261-38. h :. P 6 n . 3 : O'P e Cls induct - t'A . D . Nr'P, = Nr'P : 

G'P ~ 6 Cls induct - t'A . D . Nr'P, = Nr'P + 1 
[*253-56 . *261-35] 

*261-4. hiPeil-D, induct . D . {^,)^'B'P} 1 P, e Prog 

Dem. 
t-.*204-7. DI-:Hp.ii = Pi.D.Pel-»l (1) 

h . *120-212 . D h :. Hp . 3 : a ! P : 
[*250-13] D : E ! 5'P : 

[*250-21] D:ii = P,.D.5'PeD'i2 (2) 

l-.*260-22. DI-:Hp.ii = Pi.D.EpoG/ (3) 

h . *93103 . *202-52 . D 

\-:Peil.B = P,.^l%'B'P-D'P.:>.B'P6%'B'P. 
[*93-101.*91-54] D . (5'P) iipo (B'P) . 

[*260-27] D.(B'P)Pt,{B'P). 

[*261-2] 3 . G'P e Cls induct (4) 

1- . (4) . Transp . D h : Hp . jB = P^ . D . R^'B'P C D'P . 
[*250-21] D . B^'B'P C D'R (5) 

h . (1) , (2) . (3) . (5) . D I- : Hp . i2 = Pi . D . 

Rel->l.B'Pe B'R . ~ {{B'P) R^ (B'P)} . ^^'B'P C D'P . 
[*122-52] D . (^^'B'P) 1 i? e Prog : D h . Prop 

*261-401. h : P e II - n induct . D . g ! «„ « Cl'O'P . G'P e Cls ret! 
Dem. 

I- . *261-4 . *123-1 . D H : Hp . D . J)'{{P^'B'P} 1 Px e N„ (1) 

l-.*121-305. Dh:Hp.D.D'{(PxV5'P}1PjC0'P " (2) 

|-.(1).(2). Dh:Hp.D.a!N„nCl'0'P. (3) 

[*12415] D.C'PeClsrefl (4) • 

h . (3) . (4) . 3 h . Prop 



128 SERIES [part V 

*261-41. f- . fl - n induct = « infin [*261-401 .*261-11 .*124-271] 

*261-42. h . fl fin = fl induct [*261-41 . Transp . *124-27l] 

We shall henceforth use " n fin " in preference to " XI induct." 

*261-43. h . G"n C Cls induct w Cls refl [*261-4pll4] 

*261-431. hiPefl-i'A.D. 

{(Ah'B'P} 1 A = P, WB^ = A t (I'B'P u %,'B'P) 

= (i'B'Pyj%'B'P)'{P, 
Dem. 

I- . *25013-21 . D f- : Hp . D . £'P 6 D'P, . (1) 

[*260-31] D . I'B'P yj%^'B'P = (P^'B'P (2) 

i- . (1) . *260-27 . D h : Hp . D .%„'B'P = (P^'B'P - 

[*260-34] O.P,\- K'B'P = mh'B'P} 1 P, (3) 

[(2)] _ ={i'B'P^%^'B'P)^P, (4) 

h . (3) . (4) . *35-l 1 . D I- : Hp . D . {(P^)^'B'P\ ^P, = P^t {i'B'P « K'^'-P) (5) 
h . (3) . (4) . (5) . D h . Prop 

*261-44. I- :. P e n . D : P, f Pf/^'-P e Prog . = . P e II infin 
Dem. 

h . *123-1 . 3 1- : Pefi . P^ f PJn'^'^ e Prog . D . g ! K„ n Cl'C'P - 

[*124-15] D.CPeClsrefl. 

[*261-1] D. Pell infin (1) 

h . *261-4-431-41 . D h : P e fl infin .':).P,\- PJn'-B'P e Prog (2) 

1- . (1) . (2) . D h . Prop 

*261-45. I-. O infin = nnP{P,pP,„'B'Pe Prog} [*261-44] 

*261-46. y-.Pen.D. GVG'P C Cls induct v^ Cls red 
Bern. 

h . *250-141 . *202-55 . D 

h:Hp.aCO'P.a~el.D.Ptaefl.a=G'(PDa). 
[*261-43] D . a 6 Cls induct u Cls refl (1) 

I- . *120-213 . D h : a e 1 . D . a 6 Cls induct (2) 

I- . (1) . (2) . D h . Prop 

*261-47. h :. P e n . a C O'P . D : o 6 Cls induct . s . a~ e Cls refl 
[*261-46 . *124-271] ^ 

*26r6. h :. Pen . C'P C fi . Nc'C'P = v . Dp. n'Pefi : 

v 6 Nc induct - t'O — t'l : D : 

Dem. 
h . *204-272 . D I- : Nc'D'Q = 1 . Q e Ser . D . Q e 2, . 
[*56-112] D.0'Qe2 (1) 



SECTION E] finite AND INFINITE SERIES 129 

h . (1) . Transp . D h : Q e fl . O'Q C fi . Nc'G'Q = v +„ 1 . 

• veNC induct- i'0-fc'l.D.D'Q~6l (2) 
h . *261-24 . D I- : Hp(2) . D . E ! £'Q . 
[(2).*204-461] 3 . Q = Q D D'Q 4» 5'Q - 

[*l72-32] D . n'Q smor n'(Q t D'Q) x B'Q (3) 

h . *110-63 . D H : Hp (2) . D . Nc'D'Q +„ 1 = ,; +„ 1 . 

[*120-311] 3 . Nc'D'Q = 1/ (4) 

H . (4) . D h :. Hp (2) : P 6 XI . C'P C fi . Nc'O'P = .. . Dp . H'P e fi : D . 

n'(QDD'Q)en. 

[(3).*251-55] D . n'Q e n (5) 

h . (5) . Exp . D 

h :. Hp . D : QeO . O'QC n . Nc'(7'Q = «; +„ 1 . D . n'Qefi :. D h . Prop 

*26r61. l-:Penfiii.C7'PCn.D.n'Pen 

Dem. 
h . *26 1 -6 . D h : : ^i; . = „ : P 6 XI . O'P C n . Nc'O'P = v . Dp . H 'P e XI : . D : . 
1/ 6 Nc induct - I'O - t'l . D : ^v . D . ^ (y +e 1) (1) 

h . *200-12 . D I- .~(aP) . PeXi . O'PCXl . Nc'(7'P= 1 . 
[*10-53] Dh:Hp(l).D.^l (2) 

h . *172-13 . *250-4 . D h : Hp(l). D. (/)0 (3) 

h . *l72-23 .*251-55 . D h :. F+Z. F, Ze O . D : H'CFJ, ^, n'(-^4, F) e XI : 
[*55-54.*204-13] D iPeSer . 0'P= t'Fw i'^. D . O'PeXl (4) 

l-.(4).*54-101.DI-:Hp(l).D.^2 (5) 

I- . (2) . (3) . (5) . Dh:.Hp(l).D:^0:./6t'0ui'l.^K.D.<|)(i'+„l) (6) 
h . (1) . (6) . D h :. Hp (1) . D : J/ e NC induct .<}>v.O^ .^(v+^l):<f>0 : 

[*120-13] D : a e NC induct . D^ . (/>« (7) 

h . (7) .*13-191 . D h iPeXl . O'PCXl . Nc'C'PeNC induct . Dp. H'PeXl : 
[*261-14-42] D h : P eX2 fin . G'P C X2 . Dp . U'P e XI : D h . Prpp 

*26162. h:PeXi.QeXlfin.D.P«eXl 
Bern. 

I-.*251-51. DhiHp.giP.D.PiJQeXi (1) 

l-.*165-26. Dh-.Rp.O.C'Pi'QCO. (2) 

■J 

h . (1) . *165-25 . *261-18 . D I- : Hp . g ! P . D . P jt^ JQ e Xl fin (3) 

h.(l).(2).(3).*261-61.Dh:Hp.a!P.D.n'P4,;Q6Xl. 

[*176-181-182] D.P«6ft (4) 

l-.*l76-151.*250-4. DI-:P = A.D.pe6Xl (5) 

h . (4) . (5) . D h . Prop 

H. &W. III. 9 



130 SERIES [part V 

*261-63. ^zElB'R.PQQ.xeG'Qn p'Q"C'P . D . 

(I'a;) t G'B e C'Q^ np'^"G'P^ 
Bern. 

V . *11612 . D I- : Hp . D . {I'x) t G'B e (G'Q f G'R)a'G'R . 

[*176-14] D . (I'x) t G'R e G'Q^ (1) 

h . *11612 . *9311 . D h :. Hp . S e {G'P t G'R^G'R . T = (t'a;) t O'i? . 3 : 

(fif'5'E) Q (T'5'E) : ~ (ay) . yR (B'R) : 

[*10-53] D : (S'B'R) Q (T'B'R) : yR (B'R) .y^B'R.Oy. S'y = T'y : 

[*l76-19.(l)]D:^f((2«)r (2) 

h . (2) . *17616 . D h :. Hp . 3 : <S e G'P^ . D . /S (Q«) {(t'«) t G'R] (3) 

I- . (1) . (3) . D h . Prop 

*261-64. I- : E e n fin - t'A . Pless Q . D . P« less Q« 

Bern. 
h .*254-55 . D I- : Hp . D . (aF) . P' smor P . P' G Q . a ! G'Qnp'Q'"G'P' . 
[*261-63.*25013] D . (aP') ■ P' smor P . P' G Q . a ! 0'Q« n p'^"C'{PY - 
[*l76-35-22] D . {'^M) . If smor P« . M G Q^ . a ! G'Q'' n p^^"G'M . 

[*254-55.*261'62] D . P* less Q« : D h . Prop 

*261-65. l-:PeOinfin.QeI2fin.D.QlessP 
Bern. 

h . *261-ll-14-42 . D h : Hp . D . G'P e Cls refl . G'Q e Cls induct . 
[*124-26] D . Nc'C'P > Nc'C'Q . 

[*255-75] D . Q less P : D h . Prop 



*262. FINITE ORDINALS. 

Summary of *262. 

Finite ordinals are defined as the ordinals of finite well-ordered series ; 
infinite ordinals are defined as the ordinals of infinite well-ordered series. 
In virtue of *261"42, finite ordinals are those whose members have fields 
which are inductive, and are also those whose members have fields which are 
not reflexive. Finite ordinals have the formal properties which cardinals have 
but which relation-numbers and ordinals in general do not have, i.e. their 
sums and products are commutative, and the distributive law holds in the 
form 

;a >C (v + ot) = (a* X v) -i- (/i X isr), 

as well as in the form 

(v + ot) X /i = (j' X /".) -i- (si- X /i), 

which was proved generally in *184'35. 

The distinguishing properties of finite ordinals are most readily 
established by means of their correspondence with inductive cardinals. In 
general, two well-ordered series whose fields have the same cardinal need 
not be ordinally similar, but when the cardinal of the fields is inductive, 
the two series must be ordinally similar. Hence the ordinal of a finite well- 
ordered series is determined by the cardinal of the field of the series. We 
put generally 

^l^ = D,r^C"^l Df. 

The result is that, if fi is an inductive cardinal, fi,f is the ordinal of all those 
series whose fields have /t members. Thus there is a one-one correspondence 
of inductive cardinals and finite ordinals ; and in virtue of this correspondence, 
the formal properties of finite ordinals can be deduced from those of inductive 
cardinals. 

It will be observed that, according to the definitions already given, 

h.O^ = ilnC"Aby *250-43, 

h . 2, = n n C"2 by *250-44. 

9—2 



132 SKEiEs [part V 

Hence the notations 0^, 2^ are particular cases of the general notation fir. 
But in virtue of *200-12, we have, by the definition of fir, 

h . 1^ = A, 
so that 1, does not take its place in the series of finite ordinals. 
Our definitions in this number are 

NO fin = Nor"Il fin Df, 
NO infin = Nor"n infin Df, 
/A,. = flnO"/i Df. 
It will be observed that for the sake of convenience we have defined NO fin 
and NO infin so as to exclude A. The definition of /i, is chiefly useful when 
fi is an inductive cardinal. 

The number begins with various elementary propositions, partly embody- 
ing the definitions, partly concerned with fi^. We have 

*262-12. l-:Pe/t,.= .P6«.a'Pe/i 

*262-18. t-:/ieNC.a!/v.D./t= G"iir 

This proposition does not require that (ir should be a relation-number. 
If /i is a reflexive cardinal, /i, is not a relation-number unless it is null, 
because series of many different relation-numbers can be made with a given 
cardinal number of terms. When /i is a cardinal, "g ! fi" means that classes 
having /ti terms can be well-ordered. 

^26219. h :. ytt, 1/ e NO . g ! /*, . D : /* = i; . = . /t^ = v. 

Thus the relation of fi to jir is one-one so long as /t is the cardinal number 
of a class which can be well-ordered. 

We next prove that if /t is an inductive cardinal other than A or 1, /^^ is 
a finite ordinal, and that every finite ordinal is of the form /t, for an appro- 
priate fi. We have 

*262'21. I- : ytt e NC induct - I'A - t'l . D . g ! /^^ 

*262-24. h : /t 6 NO induct - t'A - I'l . D . /i^ e NO fin 

We prove this by means of an inductive proof that two series are similar 
if their fields are inductive and similar. 

*262-26. V-.ae NO fin . s . (g/t) . /* e N„C induct - I'l. a = fi^ 

Hence we easily obtain the properties of finite ordinals from those of the 
corresponding cardinals. Assuming that fi, v are inductive cardinals other 
than 1, we have 

. *262-33. Ilr + Vr = (ji +0 V)r 

*262-35. /i, -}- 1 = (/i -He 1),, if At + 0, 

*262-43. flr'kVr=(jJiXe v)r 



SECTION E] finite ORDINALS 133 

*262-53. fj^ exp^ Vr = ifi'^, if v + 0, 

*262'7. /i "> V . = . fifi> Vr 

Hence if a, /S, 7 are finite ordinals, 

*262-6. a+^ = i3+tt 

*262-61. aX(S = /3xa 

*262-62. ax(;8-i-7) = (o>Cy8) + (aX7) 

*262-63. (a x 0) exp,. 7 = (o exp^ 7) X (/3 exp^ 7) 

Thus the arithmetic of finite ordinals obeys the same formal laws as the 
arithmetic of inductive cardinals. 



*26201. NO fin = N„r"fl fin Df 

*26202. NO infin = N„r"r2 infin Df 

*26203. fj.r = nnG"/i Df 

*262-l. l-:a6N0fin.s.(aP).Penfin.a = N„r'P [(*262-01)] 

*26211. I- : a 6 NO infin . = . (gP) . P e fl infin . a = Nor'P [(*26202)] 

*262111. h :. a e NO fin . = : a 6 N„0 :«=t=i + «.v.a=0^: 

= : a 6 NO :a=fi-i-a.v.a = Or 
Dem. 

I- . *262-l . D 

l-:.aeN0fin. = :a6N„0:(aP).P6nfin.«=Nr'P: 

[*261-36] = : aeN„0 : (gP) : Nr'P + 1 + Nr'P . v . P = A : «=Nr'P : 

[(*255-03)] =:a6N„O:a4=l + a.v.a=0r: (1) 

[*180-4.*155-5]= : oeNO : a+ i-j-a . V . a = Or (2) 

H . (1) . (2) . D I- . Prop 

*262112. h : a £ NO infin . = . aeNoO - I'O, . 1 + a = a 
[*262-lll , Transp . *261-13] 

*26212. \-:Pe,ir. = .Pen.C'Pe/j. [(*26203)] 

*26213. H : Nr'PeNOfin . = .Pefl fin . =. PeQ . C'PeCls induct 

Dem. 
1- . *262-l . D 1-: Nr'PeNO fin . = . (gQ) . Q e fl fin . Nr'P = N„r'Q . 
[*152-35.*155-13] = . (gQ) . Q e li fin . P smor Q . 

[*261-183] =.Penfin. (I) 

[*261-4214] =. Pe fit. O'P 6 Cls induct (2) 

h . (1) . (2) . D f- . Prop 



134 SERIES [part V 

*26214. h : Nr'P e NO infin . s . P e fl infin .= . F eCl . G'P eC\s refl 
[Proofa8m*262-13] 

*26215. h : . a 6 NoO . 3 : a e NO fin . = . G"a e NO induct 
Dem. 

1- . *262-13 . *120-21 . D 

|-:N„r'P6NOfin. = .Pefl.N„c'(7'P6NCinduct " (1) 

I- . (1) . *251-1 . D 

h :. N„r'P e NO . D : Nor'P e NO fin . = . N„c'(7'P e NO induct . 

[*152-7] = . C'Nor'P e NO induct (2) 

l-.(2).*155-2.Dh.Prop 

*26216. l-:.aeN„O.D: 

a 6 NO infin . = . G"a ~ e NC induct . = . C"a e NO refl 
[Proof as in *262-15] 

*26217. h:Pef2.D.P6(Nc'C"P),. 

Dem. 

I- . *100-3 . D h . C'P 6 Nc'C'P (1) 

h . (1) . *26212 . D h . Prop 

*26218. l-:/i6NC.a!/ir.D./i=(7"/ir 

Z)em. 

|-.*262-12. Dh.C'firCfi (1) 

l-.*262-12. 'D\-:ae/j..Pe/j.r.D.a,G'P6fi (2) 

I- . (2) . *100-5. D I- : Hp . a e /i . P 6/ir . D . asm (7'P . 

[*73-l] D.(aS).;S'6l^l.a = D'/S.C"P = a'/S. 

[*151-1.*1 50-23] D . (a/S) . iSfJPsmorP . C'SiP^a. 

[*251-111.*262-12] D . (g^f) . /SJPe O . G'S'P^a . 

[*262-12.Hp] D.(afif)./Sf;Pe/i^.C">Sf;P=a. 

[*37-6] D.aeG"fir (3) 

I- . (3) . *10-23 . D h : Hp . D . /t C 0' ^ (4) 

h . (1) . (4) . D I- . Prop 

*26219. \-:. ii,ve NO . g ! /t,. . D : /* = y . = . /i^ = I'r 
Dem. 

l-.*262-12.Dh:/4 = i'.D./ir=i/^ (1) 

h . *262-18 . D H : Hp . /i, = i/^ . D .fi=G"vr 
[*262-18] = 1^ (2) 

h . (1) . (2) . D I- . Prop 



SECTION E] FINITE ORDINALS 135 

*262-2. h . Cls induct - 1 = G"(n n Onv"12) 
Bern. • 

h . *261-29 . D h . Cls induct - 1 = C"(f2 n Cnv"Xl) - 1 
[*200-12] = (7"(fi n Cnv"n) . D h . Prop 

*262-21. h : /i 6 NO induct - I'A - I'l . D . g ! /^^ 
Bern. 
h .*120-2 .*100-43 . D I- : Hp . D .(ga) . a e /* .- a e Cls induct . a~el . 
[*262-2] D.(ao,P).a6/i.Pen.O'P = a. 

[*262-12] D.a!/*,:DI-.Prop 

*262-211. h : a 6 Cls induct - 1 . D . g ! (Nc'a), n f„„'a 

-Dem. 
h . *262-21 . *108-12 . 3 h : Hp . D . g ! (N„c'aX . a e N„c'ffl . 
[*26212] D . (gP) . P e (Nc'o), . G'P e N„c'a . a e N„c'a . 

[*63-13] D . (gP) . P e (No'a), . O'P e t'a . 

[*64-24.*35-9] D , (gP) . P e (Nc'aX . P e <'(o | a) . 

[*64-ll] D . a ! (Nc'a), n «„o'a : D h . Prop 

*262-212. h:/i=|=0./i4=l.P6(/*+elV.D.Pta'P6yiv 

i)em. 
h.*262-12. DhiHp.D.C'Pe/i+el.Pefl. (1) 

[*110-4] D./4 6NC-t'A (2) 

I- . *93103 . *250-13 . D I- : Hp . D . O'P = I'B'P w a'P . B'P ~ e Q'P . 
[*1 10-63] D . Nc'C'P = Nc'a'P +„ 1 . 

[(1).(2)] D./i+el = Nc'a'P+„l. 

[*120-311.(1)] D.yii = Nc'a'P.P6n. 

[*202-55.*250-141] D . /i = Nc'C"(P p Q'P) . P ^ Q'P e fi . 

[*26212.*100-3.(2)] O.Pl Q'P e /i^ : D h . Prop 

*262-213. \-:.fi^0.fij=l:P,Qefir- Dp.q. PsmorQ : D : 

P, Q 6 (yi* +0 l)r . 3p,Q . P smor Q 
Bern. 

|-.*262'21212.*120-124.D 

l-:Hp.P.(26(/i+„l),.D.PCa'P,Qta'<26/^.P,Q6ll-t'A. 

[*ll-l.Hp] D . P t a'P smor Q la'Q . P, Qe fl - t'A . 

[*25017] D . P stnor Q : D I- . Prop 



136 SERIES [part V 

*262-22. h : ;x 6 NO induct .P,Qefir.D.P smor Q 

Bern. 
I- . *153-101 . *262-12 . 3 h : P, Qe Or . D . P smor Q (1) 

h.*200-12. D\-.lr = A. 

[*10-53] D h : P, Q e Ir . D . P smor Q (2) 

l-.*153-202. Dt-:P, Q6 2,. .D.PsmorQ (3) 

H . (2) . (3) . *2-02 . D\-:.fj. = 0.v.fi = l: 

P,Q6iXr.Dp,Q.P smor Q : D : P, Q e (ytt +e !> ■ 3p.Q . P smor Q (4) 
I- . (4) . *262-213 . D 

l-:.P, Qe/i,. Dp,Q.P smor Q: D : P,Q e {/j, +^1\ . Dp,q . P smor Q (5) 

h . (5) . (1) . Induct . D h . Prop 

*262-23. I-:.P, Q6nfin.D:C"PsraC"Q.= .PsmorQ 
Dem. 
l-.*262-17-13.D 

h : Hp . C'P sm G'Q.O.P,Qe (Nc'G'P\ . Nc'C'P e NC induct . 
[*262-22] D.P smor Q (1) 

h.(l).*15118.DI-.Prop 

The above is the fundamental proposition in the theory of finite ordinals, 
since it enables us to reduce relations among finite ordinals to relations among 
the corresponding cardinals. 

*262-24. h : /i 6 NC induct - I'A - t'l . D . /t^ e NO fin 
Dem. 

h . *262-21 . D h : Hp . D . a ! /i, (1) 

l-.*262-22. Dh-.Tl-p.Pe/ir.D.firCm'P (2) 

h . *26212 .*151-18 .Oh:Pe/j,r.O. Nr'P C/t, (3) 

l-.(2).(3). Di-:Hp.P6/i,.D./Xr = Nr'P (4) 

(-.(1).(4). DhiHp.D./^eNR-i'A (5) 

t- . *262-12 . D I- : Hp . P 6 /v . D . O'P 6 Cls induct . 

[*262-13.(4).(5)] D . /i, e NO fin (6) 

h . (1) . (6) . D h . Prop 

*262-241. h :. /i e NC induct . P e £2 . D : nt, = Nr'P . = . /i = Nc'CP 
Dem. 

f- . *100-3 . D I- : Hp . /I = Nc'C'P .D.G'Pe/i. 

[*26212] D.Pefir. 

[*152-45.*262-24] D . /^^ = Nr'P (1) 

h . *152-3 . *262-18 . D h : Hp . yti^ = Nr'P . D . /i = C'Nr'P . 
[*152-7] D./ii = Nc'0'P (2) 

I- . (1) . (2) . 3 h . Prop 



SECTION E] FINITE ORDINALS 137 

*262-25. I- : (g^*) . ;* e NC induct - I'l - t'A . a = ^, . = . a e NO fin 

Bern. • 

I- . *262-l-13 . D 

I- : aeNOfin . D . (gP) .Peafin . a = Nr'P . Nc'O'Pe NO induct . 
[*262-241] D . (gP) . P 6 n fin . a = Nr'P . (Nc'CP)^ = Nr'P . 

Nc'O'P e NO induct . 
[*13-172] D . (gP) . a = (Nc'O'P), . Nc'O'P e NO induct . 

[*20012.*2621.*15513] D.(a/i,).^6N0induct-t'l-i'A.a=A'>- W 
I- .*264-24 . D h : (g/i) . /^eNC induct- I'l - I'A . a = /t, . D . aeNOfin (2) 
h . (1) . (2) . D h . Prop 

*262-26. h : a 6 NO fin . = . (g^) . /^ e N„0 induct - t'l . a = /x, 
[*262-25 . *103-13-34] 

*262-27. h:o,/36NOfin.D.a + ;8eNOfin 

Dem. 
I-.*180-21. DhzHp.Pea.Qe^.D.P + Qea + yS (1) 

h.*251-24. Dh:Hp.D.a + /36NO (2) 

h . *180-111 . D h : Hp (1) . D . Nc'C"(P + Q) = Nc'(C"P + O'Q) 
[*110-3] = Nc'O'P +e Nc'O'Q (3) 

I- . *262-13 . 3 h : Hp (1) . D . Nc'O'P, Nc'O'Q e NO induct . 
[*120-45] D. Nc'O'P +oNc'0'Qe NO induct (4) 

I- . (1) . (2) . *155-26 . *251-122 . D 

h:Hp(l).D.P+,Qen.a + /3 = N„r'(P + Q) (5) 

l-.(3).(4). Df-:Hp(l).D.0'(P + Q)6Clsinduct (6) 

h . (5) . (6) . *262-l . *261-42 . D h : Hp (1) . D . a + /3 e NO fin (7) 

I- . *262-l . *15o-13 . D h : Hp . D . g ! a . g ! (S (8) 

h . (7) . (8) . D I- . Prop 

*262-271. H:a,y8eN0fin.D.aXi86N0fin 

[Proof as in *262'27, using *1S4-12 . *166-12 . *251-55 . *120-5] 

*262-272. h : a, y8 e NO fin . D . a exp, ;S e NO fin 

[Proof as in *262-27, using *186-1 . *176-14 . *261-62 . *120-52] 

*262-31. I- : /i, j; 6 NO . D . /Ay-i- j/y C (/i +e vX 

Dem. 
h.*180-2.D 

\-'..Ze(ir + Vr. = : (aP, Q) . /tr = N„r'P . v^ = N„r'Q . Z sraor (P + Q) : (1) 
[*180-111.*151-18] D : (gP, Q) . /*, = N„r'P . i/, = Nor'Q . O'Z sm (G'P + O'Q) : 
[*155-12] D : (gP, QyPefjLr.Qevr.O'Zsm {C'P + O'Q) : 

[*26212] D : (gP, Q).G'P efi.C'Qev . G'Z sm (G'P + G'Q) : 

[*1 10-21] "D iRp.D. G'Z € ft, +^v (2) 



138 SERIES [PART V 

h.(l).*262-12.*155-12.D 

[*251-25.*180-11-12.(*180-01)] D.^eO (3) 

f- . (2) . (3) . *262-12 . 3 h :. Hp . D iZefir + Vr . D . -Z'eC/i +„ i/), :. 3 h . Prop 

*262-32. h : fi, i; eNC induct .Pefir-Qevr-D-P + Qe/ir + Vr 
Bern. 

I- . *200-12 . *262-12 . D h : Hp . D . /i, z; 6 - I'l - t'A . 

[*262-24] D.^^.z^^eNO. 

[*180-21] D.P + Qe/i^ + i/^iDI-.Prop 

*262-33. \-:/i,ve'NG induct -t'l .D . fir + Vr = (fi+a v)r 

Bern. 
\- . *262-12 .:> b i. /ji= A .V . v = A zD : iJLr = A .V . vr = A : 
[*180-4] :>:fir + Vr = A (1) 

h .*110"4. DI-:./i=A.v.j' = A:D./i+oi' = A. 

[*262-12] D.{fJL+,v)r = A (2) 

|-.*262-32.Dh:Hp.P6/i,.Qei/^.D.P + Q6/t,-i-i',. (3) 

[*180-42.*152-45] D . /^i, + v^ = Nr'(P + Q) (4) 

h . (3) . *262-31 . D h : Hp (3) . D . P + Q 6 ((li +0 i')r . 
[*120-45.*262-24] .P + Qe(fi+^v)r.{ti+e v)r e NR . 

[*152-45] D.(/.+„j;), = Nr'(P + Q) (5) 

h . (4) . (5) . *10-23 . *262-21 . D l- : Hp . g ! yct . g ! v . D . /^r + I'r = (/* +c i')r (6) 
h . (1) . (2) . (6) . D h . Prop 

The above proposition still holds (as we shall now prove) when one of 
fi and V is equal to 1, but not both. When both are equal to 1, fj.r + Vr = A, 
while (ji +e v)r = 2r. 

*262-34. l-:/[i6NC-i'0.D./i, + iC(/t+el),. 

Bern. 
I- .*181-2 . 3 h i.^e/i^-i-i . = : (gP,*) ./i^ = Nor'P. Zsmor(P4»«) (1) 

h . *181-6 . *152-7 . D h : a ! P . D . Nc'C"(P 4> «) = Nc'G'P +„ 1 (2) 

l-.(l).(2).D 

h :. Hp . D : Ze/^, + 1 . D . (gP) . /^, = N„r'P . 'Nc'G'Z= Nc'<7'P+„ 1 . 
[*262-24112] D . (gP) . ,ir = N„r'P . Nc'(7'^= fi+.l . 

[*100-3] D.O'^e/i+ol (3) 

h . (1) . *262-12 . *15512 . 3 h : ^e/it^ + l . D . (gP) . Pe O . /i, = N„r'P . 
[*2511-132] D./i^+ieNO. 

[*251-i22] O.ZeD, (4) 

I- . (3) . (4) . *262-12 .D\-:.E.p.0:Zefi,.+ i.D.Z6(u,+,l\:.0\-. Prop 



SECTION E] FINITE ORDINALS 139 

*262-341. h : ,16'NCmdnct . P e ^Lr .^ . P hxe fir+i 
Dem. • 

h . *200-12 . *26212 . D h : Hp . D . ^ e - I'l - t'A . 

[*262-24] D.^^eNO. 

[*181-21] D.P4*a;e/i,.-i-l:DI-.Prop 

*26235. h : ;ii 6 NO induct - t'O - I'l . 3 . /i, + 1 = (/t* +c l)r 

Dem. 
l-.*262-12. DI-:/^ = A.D./i, = A. 

[*181-4] D. /*, + ! = A (1) 

h.*110-4. DI-:/* = A.D./^+„l = A. 

[*262-12] 3. (/.+„!), = A (2) 

h.*262-341. Dh:Hp.P6/i,.D.P4»«6Ai,+ i. (3) 

[*181-42.*152-45] D . /i,+ 1 = Nr'(P -f* a;) (4) 

f- . (3).*262-34 . D h : Hp . Pe/i,. D . P-i+a'eC/ii+ol)^ . 
[*1 20-45 .*262-24] D . P 4* a; 6</i +„ l)r ■ (/^ +c l)r ^ NR . 

[*152-45] D.(/*+„lV = Nr'(P4*«;) . (5) 

I- . (4) . (5) . D h : Hp . a ! /x^ . D . /ir + 1 = ((li +„ 1), : 

[*262-21] DK:Hp.a!/..D.;c*, + i = (M+„l). (6) 

h . (1) . (2) . (6) . D h . Prop 

*262-36. h : /i 6 NO induct - I'O - t'l . 3 . 1 + A^r = (1 +« /^V 

[Proof as in *262-35, by means of analogues of *262-34-341] 

*262-41. \-\(j.,ve NC . D . /i^ X Vr C (/* x^ v\ 

[Proof as in *262-31, using *1841-5 .*113-21] 

*26242. Vifx.ve NC induct . P e fir . Q e Vr - . P x Q e firk v,. 
[Proof as in *262-32, using *184-12J 

*262-43. \-:fi,ve NO induct - t'l . 3 . /*r X Vr = (/* ^c ")»• 

[Proof as in *262-33, using *184-11 . *113-204 . *184-15 . *120-5] 

*262-51. i-zfie NC . v e NC induct . D . /t^ exp, v^ C (/i*"), 
Dem. 

h.*186-5. DI-:/x,,i/^eN„R.j/4=0.i2e/i,.exp^z/,.D.O'i2e(a"/irr'''' (1) 

H . *186"11 . D h : i2 e /Xy exp, v^ . D . g ! /ti^ . g ! i/^ (2) 

t- . (1) . (2) .*262-18 . D H : Hp . i/ + . i? e /x^ exp,. j;, . D . CiJe/x" (3) 
f-.*262-12. Dh./t^Cfl. 

[(2).*251-1.*186-11] Dhiiie/irexp^iV.D./x^eNO (4) 

H.*262-24. Dh:Hp.i;+l'.i'4=A.D.j/^eN0fin (5) 

h.(2).(4).(5).*261-62.pi-:Hp.i'=t=l.i2e/irexp^i'^.D.iJ:6fl (6) 

h.(2).*200-12.DI-:i2e/irexp^i/^.b.i'=j=l (7) 
[■ . (3) .(6) .(7) . 0\- -..Up .D : R e /jLrex^rVr .0 . Re n .O'Re fi" . 
[*262-12] D-P6(/i,'')r:.3l-.Prop 



140 SERIES [PABT V 

*262-52. h : /i, j; e NO induct .Pefir-Qevr.'^.(P exp Q) e {fi^ exp, i^^) 
Dem. 
V . *200-12 . *26212 . D h : Hp . D . ^, j; e - I'l - t'A . 
[*262-24] D./Lir,i/,6N0. 

[*186-13.*152-45] D . (P exp Q) e {nr exp^ j/^) : D H . Prop 



*262-53. h : /i, 1/ 6 NO induct - t'l . y =1= . D . /*^ exp^ Vr = (/i'')r 




i)em. 




h . *26212 . *18611 . Dh:.nt = A.v.z/ = A:D./i^ exp^ j/^ = A 


(1) 


h .*116-204 . *26212 .DI-:.//, = A.v.i/ = A:D. (A<.-')r = A 


(2) 


1- . *262-52 . D h : Hp . P e /i^ . Q e i/r . 3 ■ (i' exp Q) e (Mr expr I'r) ■ 


(3) 


[*186-13.*152-45] 3 . Nr'(P exp Q) = /^,. exp, v. 


(4) 


h . (3) . *262-51 . D h : Hp (8) . D . (P exp Q) e {fi^ 


(5) 


K . (5) . *120-52 . D 1- : Hp (3) . 3 . M" 6 NO induct 


(6) 


l-.(5). DI-:Hp(3).D.a!(/.-)r. 




[*200-12.*262-12] ^-A'-' + l 


(7) 


1- . (6) . (7) . *262-24 . D f- : Hp . D . (/.■'), e NO 


(8) 


1- . (5) . (8) . *152-45 . 3 h : Hp (3) . D . Nr'(P exp Q) = (m-), . 




[(4)] D.^irexprVr = (jJ-'')r 


(9) 


1- . (9) . *262'21 . DI-:Hp.a!yit.a!i/.D. jit, exp, v, = (/it"). 


(10) 


1- . (1) . (2) . (10) .Oh. Prop 





We are now in a position to establish the commutative property of 
addition and multiplication of finite ordinals. This is effected by means 
of *262-33 and *262-43. 

*262-6. l-:a,/36NOfin.D.a + /S = /3 + a 

Dem. 
h . *262-26 . D h : Hp . D . (g/i, v).fi,pe'NG induct -.t'l . a = yti, . ^9 = y, . 
[*13'12] D . (a/i, p).ij,,ve NO induct — t'l . a-i-/3 = /it,-}- 1/, . a = /*, . ^8 = v, . 
[*262-33] D . (gyit, v)./i,ve NO induct - I'l . a -j- /8 = (/i -*-„ i/), . o = //,, . y8 = k, . 
[*110'51] D -(a/t, i*) . /i, V 6 NO induct — I'l . a + ^ =^ (v +^ /i)r . a = /jl,. ■ ^ = Vr . 
[*262-33] D . (a/i, v).fi,ve NO induct - i*l . a + /3 = j/, 4- /^r ■« = /*«•• ;S = v, . 
[*13-22] D.a + /8 = /3-j-a:Dl-.Prop 

*262-61. h:a,;8eNOfin.D.ax/3 = /3xa 

[Proof as in *262-6, using *262-43 and *113-27] 

*262-62. l-:a,^,76N0fin.D.ax(;S + 7) = (a>C/8) + (aX7) 
Dem. 
h .*262-27-61 . D h : Hp . D . ax(/8 + 7) = (/8-i-7)xa 
[*184-35] =(;Sxa)-j-(7>:a) 

[*262-61] =(a><;8)-i-(a>C7):Dt-.Prop 



SECTION E] finite ORDINALS 141 

*262-63. I- : a, /3, 7 6 NO fin . D . (a X (8) exp,. 7 = (a exp, 7) X (/3 exp, 7) 

Dem. • 

1- . *262-26 . D 

f- : Hp . D . (g/i, v, •sr) . /i, v, bt e NC induct — t'l . a = /*, . /3 = i/^ . 7 = bt, (1) 

h . *262-43 . D 

I- : /i, V, TO- 6 NC induct — I'l . D . (/i^ >C Vr) exp^ in-, = (/i x„ 1/)^ exp^ tn-r • (2) 

h . *113-602 .Dh:Ai = 0.i' = O.D./iX„i;+l (3) 

l-.*ll7-631.Dh:/4,i;6NC-t'0-t'l.D./iX„r=|=l (4) 

h.(3).(4). Dh:Hp(2).D./i,Xei/ + l (5) 

h.*120-5. Dl-:Hp(2).D.yttx„x/eN0induct (6) 

H . (5) . (6) . *262-53 . D h : Hp (2) . w 4= 0^ . D . (/t x„ v)^ exp, to,. = {(/i x„ 1/)=^}, 
[*116-55] =(yit'^x,i/-), (7) 

h . *ll7-652 . D h : Hp (7) . /i 4= 0, . D . /*'' > /i Xe TO . 

[*117-631] D.M'^ + 1 (8) 

V . *116-311 . 3 h : Hp (7) . /i = 0, . D . <=j= 1 (9) 

l-.(8).(9). Dh:Hp(7).D./i'^+l (10) 

Similarly 1- : Hp (7) . D . i/='4= 1 (11) 

h . (10) . (11) . *120-52 . *262-43 . D 1- : Hp (7) . D . (/a-^ x„ v'^^ = (m^^), X {y^)r 
[*262'53] = (/ir exp, to,) X (I'r exp^ to,) (12) 

K(2).(7).(12).D 

h : Hp (7) . D . (/i, X Vr) exp, to, = (/*, exp, to,) x (v, exp, to,) (13) 

l-.(l).(13).*262a9.D 

1- : Hp.74=0, .D.(aXy8)exp,7 = (aexp,7)x(/3exp,7) (14) 

I- . *186-2 . *184-16 . D 

h:Hp.7 = 0,.D.(aX/8)exp,7 = 0,.(aexp,7)x(/3exp,7) = 0, (15) 

h . (14) . (15) . D h . Prop 

*262-64. h:aeNOfin.D.a + l = l + a 
Dem. 

V . *262-35-36-26 . *110-51 . D h : Hp . a =(= 0, . D . «+ 1 = 1 + a (1) 

h.*161-2-201. Dh:a = 0,.D.a-i-l = 0,.i + o = 0, (2) 

h . (1) . (2) . D 1- . Prop 

*262-65. h:a,/3eNOfin.yS + 0,.D.ax(/3-i-l) = (aXyS) + a 

Dem. 

|-.*262-61.Dh:Hp.D.aX(/8-i-i) = (/3 + l)xa 

[*184-41] =(^X«) + a 

[*262-61] = (a X /3) -i- a : D h . Prop 

*262-66. h:a,/8eNOtin.;8 + 0,.D.ax(l-i-iS) = a+(«X/8) 
[Proofasin*262-65] 



1 42 SERIES [part V 

*262-7. \-i.iJ,,ve NO induct - t'l . D : /i > i; . s . /i^ > v, 
Dem. 
h . *262-21 . *11712 . D h : Hp . /t > v . D . a ! /i, . a ! v, . 

[*26218] D . yic = 0"flr ■ " = G"Vr . (1) 

[*255-76.*262-24] D./ji^>Vr (2) 

I- . *1 20-441 . D h : Hp . ~ (/i > J/) . D . /i < i; (3) 

h . (1) . 3 h : Hp . /(i < v . D . /i, < r, (4) 

h . *262-21 . D h : Hp . /i = sm"// . D . (gP) . /x = N„c'C"P . /j, = sm"i/ . 
i:*103-4] 3 . (gP) . fj, = Noc'a'P . v = Nc'O'P . 

[*262-241] 3 . (gP) . /*, = N„r'P . i/^ = Nr'P - 

[*1554] "D , /ir = smor"i/ (5) 

h . (4) . (5) . *117-104 . 3 h : Hp . /i < 1/ . D . /ir < I'r (6) 

I- . (3) . (6) . *255-483 . 3 h : Hp . ~ (/i > i^) . 3 . ~ (,it^ > Vr) (7) 
h . (2) , (7) . 3 I- . Prop 

*262-71. h : a 6 NO fin - t'O^ . 3 . (g^S) . /S e NO fin - I'O, u t'l . a = ^ + i 

h . *262-ll . *261-24 .3l-:Hp,3.g!ar. a'(B \ Cnv) (1) 

I- . (1) . *204-483 . (*18104) . 3 h . Prop 

*262-8. l-:o,/8eN0.7eN0fin.a</i.3.aexp,7<;8exp,7 [*261-64] 

*262-81. h :a,^€ NoO . 7 e NO fin . a exp^ 7 = /8 exp, 7 . 3 . a = smor"jS 
Bern. 

h . *262-8 . Transp . *25542 . 3 1- : Hp . 3 . ~ (a < yS) . ~ (« > ^) . 
[*255112] 3.« = smor"/3:3l-.Prop 

*262-82. h : a e NO fin . ;8 e NO infin . 3 . a < /3 [*261-65] 

*262-83. 1- : aeN„0 -t'O^ . /8,7 6N0 fin . y8< 7 . 3 . aexp,^ < aexp^7 
Dem. 

l-.*255-33.3l-:.Hp.3:(giir).weNO-t'0^.7 = /3-|-CT.v./3=t=0^.7=;S + i (1) 

l-.*254-51.3h:QGP.3.~(PlessQ) (2) 

I- . (2) . *255-l . 3 h : 7 = /S + t!!- . 3 . ~ (7 < isr) (3) 

h . (3) . *262-82 . Transp . 3 h : Hp . 7 = ;8 + or . 3 . w e NO fin (4) 

h.*186-14.3l-:Hp(4).t3- + 0,./84=0^.3.aexp,7 = (aexp^;S)><(aexp^tir) (5) 
I- . *262-7l-272 . 3 1- : Hp(5) . 3 .(gS) . SeNR-i'O, w t'l . aexp^;8 = S + i . 

[(5).(4).*256-573] 3 . a exp^ 7 > a exp^ /3 (6) 

h . *255-51 . 3 1- : Hp (4) . ^4=0^ . j8 = 0^ . 3 . aexp^ 7 > aexp^/8 (7) 
h . *1 86-22 . 3 h : Hp . /3 + 0, . 7 = )S 4- 1 . 3 . a exp, 7 = (a exp, /8) >^ ^8 . 

[*262-7l.*255-673] 3 . a exp^7 > o exp, /3 (8) 

h . (1) . (6) . (7) . (8) . 3 h : Hp . 3 . a exp^ 7 > a exp^ /3 : 3 h . Prop 

*262-84. h:PeXl-t'A.Q,P€nfin.QlessP.3.P«lessP^ [*262-83] 



*263. PROGRESSIONS. 

Summary of *263. 

If iJ is a progression in the sense defined in *122, i.e. a one-one relation 
whose field is the posterity of its first term, then R^ is a serial relation, and 
the series generated by R^^ is of the type which Cantor calls o), i.e. the 
smallest of infinite series. It is easy to prove that all progressions are 
ordinally similar, and that, if all inductive cardinals exist, the series of 
inductive cardinals in order of magnitude is of the type a. Thus a is an 
ordinal number, which is not null if the axiom of infinity holds. 

Most of the properties of m are easily deduced from the cofresponding 
properties of "Prog," which have been proved in *122. The definition is 

« = P{(aiE).i2eProg.P = i2po} Df 

The axiom of infinity implies that " less to greater " with its field con- 
fined to inductive cardinals is a member of m, or, what comes to the same 
thing but is easier to prove, that {(NO induct) 1(+gl)}po is a member of m 
(*263'12). Thus the axiom of infinity for the type of x implies the existence 
of CD in the type f^'x (*263'132) ; and generally the existence of to in any 
type of relations is equivalent to the existence of No in the type of their 
fields (*263-131), because No = D"© = C"(o (*263-101). 

By using the fact that in a progression R (in the sense of *122) all the 
terms are values of v^, where every inductive cardinal occurs as a value of v 
(which was proved in *122), we easily deduce that if there are progressions 
they are the series that are ordinally similar to the series of inductive 
cardinals (*263*161). Hence both "Prog" and w are relation-numbers 
(*263'162'19). Moreover, by *122"21'23, a> consists of well-ordered series 
(*263'11). Hence w is an ordinal number (*263"2). 

We next prove that progressions are infinite series (*263'23), and that 
a series contained in a progression is finite if it has a maximum (*263"27), 
and is a progression if it has no maximum (*263"26). It follows that, 
assuming the existence of progressions or the axiom of infinity, to is the 
smallest ordinal which is greater than all the finite ordinals (*263*31"32). 
Connected with this is the fact that the predecessors of any term in a 
progression are an inductive class (*263"412). 



144 SERIES [part V 

*263"44"48 give various formulae for a>, any one of which might be taken 
as the definition. We have 

*263-44. h . ffl = n - t'A n P (a'P, = O'P . ~ E ! B'P) 

I.e. progressions are existent well-ordered series in which every term 
except the first has an immediate predecessor, and there is no last term. 

*263-46. h.« = XlnP(E!£'P,.~E!S'P) 

I.e. progressions are well-ordered series in which there is only one term 
having an immediate successor but no immediate predecessor, and there is 
no last term. 

*263-47. h . o) = a n P {« C C'P . D, : n 6 Cls induct . = . g ! O'P n p'P"a} 

I.e. a progression is a well-ordered series in which any sub-class a stops 
short of some point of the series if a is inductive, but not otherwise. This 
proposition will be useful in the next section. 

*263-49. h.n fin w« = nnP(a'P, = a'P) = 11 nP(P = P,„) 

I.e. finite well-ordered series and progressions together are those well- 
ordered series in which every term except the first has an immediate pre- 
decessor, and are also those in which every interval is an inductive class. 

From *261'45 it follows that, if P is an infinite well-ordered series, P 
confined to the terms at a finite distance from B'P is a progression, i.e. 

*263-5. h : P e n infin . 3 . P t {I'B'P u Pf„'£'P) e « 

Hence it follows at once that an infinite ordinal is at least as great as «u, 
and therefore infinite ordinals other than m are greater than m, i.e. 

*263-54. I- : a 6 NO infin - I'w . D . a > «u 

The remaining propositions of this number are occupied in proving to X 2^=0) 
(*263'63) and « X a = « if « is finite and not zero (*26366). It is not the 
case that 2, X «» = a> or a X w = o>. 

Cantor has varied his definitions of multiplication as regards the order of 
the factors. Originally, he adopted the same rule as we have adopted, but 
in later works he inverted the rule, so that what we call 2, x<u he calls 
ft) >C 2y, and vice versa. Thus with his definitions in his later works, 2^ X <» = tu 
but (oX2r^o). We have reverted to his earlier practice, for various reasons, 
but chiefly in order to have Nr'n'(P J, Q) = Nr'P X Nr'Q (cf. *172). Which- 
ever rule we adopt, there are some inconveniences, so that the question as to 
which is chosen is not of great importance. 



SECTION e] progressions 145 

*263-01. a)=P{(ai?).i2^Prog.P=Epo} Df 

*263 02. iV = ;* {/li 6 NC induct . i/ = (/*+„ 1) n «.'/*} Dft [*263] 

The above temporary definition of N is the same as that in *123. 
*2631. f-:Pea).= .(aE).E6Prog.P = i2po [(*26301)] 
*263101. h . N„ = D"a) = 0"« [*1231 . *122141 . *91-504] 

«263-ll. h . m C O 
Dem. 

I- . *122-28-141 .*2631 .DhiPew.aCO'P.a'a.D.E! minp'a (1) 
H . (1) . *250125 . D h . Prop 

*26312. h : Infin a,s..O.N^„ea [*123-25 . *2631] 

*26313. h : a ! N„ («) . = . a ! a. A <"'« 

Dem. 

h . *263-101 . (*65-02) . D 

h : a ! N„(a:) . = . (aP) .Pew. G'P etH'x . 

[*64-57.*63-5] = . (aP) .Pea.Pe P"x : D h . Prop 

*263131. h : a ! (>*o). . = . a ! « '^ <»'« [Proof as in *263-13] 

*263-132. h : Infin ax (a;) . = . a ! <» " *^'« • 
i)em. 
h . *125-23 . *26313 . D h : Infin ax (as) . = . a ! « « «"'«"«. 
[(*64-011014)] = . a ! 0) n «^'a; : 3 I" • Prop 

This proposition asserts that, if the number of individuals of the same 
type as os is not an inductive number, then there is a progression whose 
terms are of the type of fx. This progression will be that of the inductive 
cardinals which are applicable to classes whose terms are of the same type 
as iv. 

*26314. \-:Re Prog . P = iipo . D . P = P,„ = i2,„ . i2 = P^ 

Dem. 

h . *121-254 . 3 1- : Hp . D . Pi = i2i . 

[*121-31.*1221-16] D.Pi = E. (1) 

[Hp] D.(POpo = P. 

[*260-27.*26811] O.Pt^ = P. (2) 

[*26015.Hp] D.Rt^ = P (3) 

h . (1) . (2) . (3) . D h . Prop 

R. & W. III. 10 



146 SERIES [part V 

*263141. I- : P 6 » . D . P, e Prog . P = (PO,n = (P>)po 
Dem. 
V . *263-l . D h : Hp . D . (gii) . R e Prog .P = B^. 
[*263-14] D . (aJ2) . R e Prog . P, = i? . P = iJf^ . P = iJpo . 

[*13-195] D . P, 6 Prog . P = (POf„ = (POpo OK Prop 

The above proposition shows that every interval P(a;My) in a progres- 
sion is an inductive class. 

*263142. y-.R.Se Prog .R^ = S^o-^ ■ R = S 
Dem. 

h . *263-14 . D h : Hp . D . E = (8^\ 

[*26314] =S:D\-.PTop 

*263143. h : P, Q 6 to . Px = Qi . D . P = Q 

Dem. 
l-.*2631.Dh:Hp.D.(ai?,<S).i2,/SeProg.P=iJp„.Q = /Sfp„.P, = Qi. 
[*26314]D.(ai2,S).i2,/S6Prog.P = Epo.Q = Sp„.i? = Pi./S=Qi.P, = Q,. 
[*13-17] D.(aE,/S).i2,S6Prog.P = i?p„.Q = 5fp„.iJ = fif. 
[*13-17] D.P = Q:DI-.Prop 

*263-15. \- : ReFiog . S = ^i! {v e'NCindMct . 0}= (v +^l)ii\ .'D . S e Rsiadi N 

Dem. 
h.*123-3. DI-:Hp.D.Sel->l.D'S = D'P.a'>S = NCinduct (1) 
h . *123-21 . D I- . NC induct = G'JSf (2) 

h . *110-56-643 . D h : Hp . (/i +0 l)N(v +„ 1) . D . i; +„ 1 = /i +,, 2 (3) 

h . (3) . D h :. Hp . D : 

a!{S''N)y. = . (a/u.) . /i 6 NO induct . a; = (/i +„ !)« . y = (yit +„ 2)u . 
[*121-332131] = . (a/i) . ^ e NO induct . (B'R) R^ x . (jS'A) {R^ \R)y. 

[*122-341.*121-342] =.xRy (4) 

I- . (1) . (2) . (4) . D h . Prop 

*263151. h : ii 6 Prog . D . iJ smor iV [*263-15] 

*263152. ViRe Prog . Q smor R.'^.Qe Prog [*123-32] 

*263-16. l-:i2 6Prog.D.Prog = Nr'i2 = Nr'iV [*263-151-152] 

*263161. l-:a!Prog.D.Prog = Nr'iV [*263-16] 

*263162. l-.ProgeNR [*263-161 . *154-242] 



SECTION e] progressions 147 

*26317. h : P 6 ft) . D . « = Nr'P = Nt'N^„ 
Dem. • 

H.*263-l. DK:Hp.D.(ai?).i2eProg.P = iJpo. 

[*263-151] D.(aii:).Esmoriyr.p = iJp^. 

[*151-56] D.PsmoriyTp,. (1) 

[*152-321] D.lSTr'P = Nr'iVp„ (2) 

V . *151-59 . D h : P e o) . Q smor P.O.Qi smor Pj . 

[*263-141152] D.Q,6Prog (3) 

H . *150-83 .-^h-.Peco.SeQ slnor P . D . (QOpo = S'(Pi\o 

[*263-141] = S'P 

[*151-11] = Q (4) 

h . (3) . (4) . *263-l . 3 h : P 6 o) . Q smor P.D.Qem (5) 

h.(l). Dl-:P,Qeo).D.PsmorQ (6) 

h . (5) . (6) . D h : P e « . D . o> = Nr'P (7) 

h . (7) . (2) . 3 h . Prop 

*26318. l-:a!a).D.w = Nr'iVp„ [*263-l7] 

*26319. h . o) e NR [*263-18 . *154-242] 

*263-2. h. 0)6 NO |:*263-1911.*256-54] 

*263-22. I- : P e ft) . D . Q'P C D'P . ~ E ! £'P . E ! P'P 

[*122-141 . *2631 . *122-11] 

*263-23. h . ft) C II infin 

Dem. 
h . *261-35 . Transp . *263-ll-22 . D h : P e o) . D . O'P ~ e Cls induct - I'A (1) 
H . *263-22 . D h : P € ft) . D . a ! C7'P (2) 

h . (1) . (2) . D h : P € ft) . D . C'P ~ 6 Cls induct . 
[*261-14-41.*263-11] D . P e fl infin : D h . Prop 

*263-24. h : a ! ft) . D . ft) e NO infin [*262-14 . *263-17-23] 

*263-26. h : P e ft) . a ! a n C'P . ~ E ! maxp'a . D . Pp a e o) 

h.*263-l.*205123.D 

h : Hp . D . (gi?) . i2 e Prog .P = R^.^laf^C'R.af^G'RC i2po"« . 

[*122-442-45] D . (gi?) . R e Prog . P = Ppo . P ^ a -:- (P ^ a)= e Prog . 

[*263-l] D.P^aeft):Dh.Prop 

*263-27. h : P e ft) . E ! maxp'a . D . P^ aefl fin 



Bern. 



I- . *122-43 . *2631 . D i- : Hp . D . a n O'P e Cls induct . 
[*37-41.*120-481] D . (7'(P D a) € Cls induct (1) 

h . *26311 . *260-141 . D h : Hp . D . P p a e n (2) 

h . (1) . (2) . *261-14-42 . D h . Prop 

10—2 



148 SERIES [part V 

*263-28. l-:P6a).D.SeinRl'PC town fin [*204-421 . *263-26-27] 

*263 29. hiPew.QeOfin.D.QlessP [*261-65 . *263-23] 

*263-3. h : P e o) . D . less'P = 11 fin 
Dem. 

h.*254-l.*263-l7.D 

h-.Peto.Q less P . 3 . g ! Nr'Q n Rl'P . Q ~ e « . Q e fi . 

[*26317] D . (gP) . i? e Nr'Q n Rl'P . P ~ e » . 

[*263-28] D . (gP) . P e Nr'Q r. fl fin . 

[*261-183] D . Q 6 n fin (1) 

h . (1) . *263-29 . D h . Prop 

*263-31. h:. a!Q).D:a<(B.B.aeNOfin 

Dem. 
h . *25517 . *263-l7 . D h :. P e eo . D : Nr'Q < « . = . Q less P . 
[*263-3] s . Q e fl fin . 

[*262-13] s.Nr'QeNOfin: 

[*152-4] D : a 6 NR . a < ft) . s . a e NR . a e NO fin : 

[*255-12.*262-l.*152-4] D : a < to . = . a e NO fin :. D h . Prop 

*263-32. l-:.Infinax.D:a<a). = .a6N0fin [*263-3112] 

*263-33. h:a<a).D.aeNOfin 
Dem. 

h.*2551.*155-13.DI-:Hp.D.a!ft) (1) 

I- . (1) . *263'31 . D h . Prop 

*263-34. h . 1 + ft) = ft) 
Dem. 

h . *262-112 . *263-24 .DI-:Hp.a!o).D.l+ft) = o) (1) 

l-.*181-4. Dh:o)=A.D.l + ft)=A (2) 

h . (1) . (2) . D h . Prop 

*263-35. l-:aeNOfin.D.a + ft) = ft) 

Dem. 
y . *180-61 . *263-18 .3l-:a!ft).D.0, + ft) = a) (1) 

V . *180-4 . Dl-:o) = A.D.O, + o) = A (2) 

h . (1) . (2) . Oh.0r + m = m (3) 

I- . *181-57 .*263-34 .DI-.2, + o)=i-i-ft) 

[*263-34] =0) (4) 

1- . *262-36 . D h : /i 6 NC induct - I'O - t'l . D . (/* +o 1), + o) = /v + 1 + <« 
[*263-34.*181 -68] = /V + o) (5) 

h.(5). D h :/i6NOmduct-t'0-i'l./i, + w = o).3 . (/t+el), + ffl= ft) (6) 
h . (4) . (6) . Induct . D h : /i e NC induct -I'O- I'l .D .fir + (o = (o (7) 

h . (3) . (7) . D I- : /i e NO induct - t'l . D . /tt, 4- w = o) : 

[*262-26] DI-:a6N0fin.D.a + a) = ft):Dl-. Prop 



SECTION E] progressions 149 

*263-4. I- : P e o) . D . B'P, C O fin . Nr"D'Ps = NO fin 
Bern. 

h . *254-182 . D f- : Hp . D . T>'P, C l^s'P - 

[*263-3] D . D'Ps C fi fin (1) 

h.*263-31. Df-:.Hp.D:a<Nr'P. = .a6N0fin: 

[*256-ll] D : a 6 Nr"D'Ps . = . a e NO fin (2) 

f- . (1) . (2) . D h . Prop 

*263-401. h: Pern, a 6 sect'P - I'A - I'G'P . D . E ! maxi.'a 

Dem. 

h . *250-65 .Dh:Hp.3.PCa~e Nr'P . 

[*26317] D.P^a~6fi). 

[*263-26.Transp] D . E ! maxp'o : D h . Prop 

*263-402. h : P 6 « . D . sect'P - t'A - I'G'P = P^"C'P 

Dem. 
h . *205-131-22 . *263-401 . D 

h : Hp . a 6 sect'P - t'A - I'G'P . D . a u P"a = P'maxp'a w I'rnaxp'a . 
[*2111.*91-54] D . a = P^f'maxp'a . 

[*205-lll] D.a6'P^"C'P (1) 

1- . *211-313 . D h . ^^"C'P C sect'P (2) 

h.*90-12. Dh.P5|,"0'PC-t'A (3) 

h . *205-197 . 3 h : Hp . a; 6 O'P . D . E ! maxp'P;^'* . 
[*263-22] D . 'P^'w 4= G'P (4) 

h . (2) . (3) . (4) . D I- : Hp . D . ^^"C'P C sect'P - t'A - I'C'P (5) 

h . (1) . (5) . D h . Prop 

*263-41. y:Pea}.D.P,l B'P, = P t' P*' P 

2)e»re, 

h.*213-ll-141-151.D 

h :. Hp . D : Q (Ps P D'Ps)P . = ■ (ga.iS) . a, /36sect'P-t'A-i'(7'P.aC/3.«+)8. 

Q = Pta.i? = Pp/3. 
[*263-402] 

= .(aa;,2/).a.,2/eC''P.P*'ajCP*'y.P*'«J + P*'i/.Q=PCP*'flJ.P=PW^ 
[*200-391] 

= . (a;., y) . ^, 2^ 6 C"P . P*'a; C P*'y . a? + 2/ • Q = -P D Ale'^; . -B = -P Wy . 
[*204-32.*90-12] 

= .('^a,,y).ooP^y.x^y.'p^'xC%'y.Q = Pt%'a,.R = PtP^'y. 



150 SERIES [PABTV 

[*201-18] = . (ga;, y).xPy.Q = P[, %'x .E = P ^ P*'y . 
[*1501] =.Q{P f'P^'P) R'..Oi-. Prop 

*263-411. I- : P e « . D . C'D'P, ='p^"a'P w t'A 
Dem. 

I-.*213141.*263-402.D 

h : Hp . D . C'D'P, = 0"PI"'P^"C'P 
[*93103] = (7"P ^ "P*"a'P u I'CP tP^'B'P 

[*201-521.*202-55] ='p^"a'P u t'C'P^ P^'B'P 
[*201-521.*200-35] = P^"(1'P u t'A : D h . Prop 

*263-412. h : P 6 ft) . D . P'a;, P^^'a; e Cls induct 
i)em. 

f- . *205197 . D f- : Hp . a; 6 O'P . D . E ! max^'P^^'a; . 
[*263-27.*202-55.*120-213] D . P^j^'* e Cls induct . (1) 

[*120-481] D.P'ice Cls induct (2) 

l-.(l).(2).Dh.Prop 

*263-42. h : P 6 ft) . D . sgm'P = A J, (C'P) 
Dem. 
h.*212-21.*211-12.D 

l-:.Hp.D:a(sgm'P)/3. = .a = P"a./3 = P";8.«C;8.« + /3 (1) 

h.(l).*211-l.*205-123.D 

h : Hp . a (sgm'P) /3 . D . a, ;8 e sect'P . ~ E ! maxp'a . ~E ! maxp'/8 . 
[*263-401] D.a,^ei'Awi'a'P (2) 

l-.(l).(2). DI-:Hp.a{sgm'P)|S.D.a=A.jS=C'P (3) 

l-.*37-29. DI-:a = A.D.a = P"a (4) 

t- . *263-22 . D H : Hp . y8 = C'P . D . /3 = P"0 (5) 

h.(l).(4).(5). DH:Hp.a = A.|S = C"P.D.a(sgm'P)/8 (6) 

h . (3) . (6) . D I- . Prop 

*263-43. H:Peft).D.a'Pi = a'P 

Dem. 

h . *263141 . D h : Hp . D . Q'P = a'(POpo 

[*91-504] = a'P, : D I- . Prop 



SECTION E] PROGRESSIONS 151 

*263431. l-:Pefi-t'A.a'Pi = a'P.~E!5'P.D.P6a. 
Bern. 

V . *261-35 . Transp . D h ; Hp . D . P e fl infin . 

[*26r44] D . P, [%^'B'P e Prog . 

[*261-212] D.PifP'B'PeProg. 

[*202-524] D. Pie Prog (1) 

l-.*261-212. Dl-:Hp.D.P = (POpo (2) 

h . (1) . (2) . *263-l . D h . Prop 

*263-44. !-.(» = ft -t'AnP(a'Pi = a'P.~E!£'P) [*263-43-22-431] 

*263-45. h.w = n-t'Ar.P(P = Pf„.~E!5'P) [*261-212 . *263-44] 

*263-46. h.o, = ftnP(E!B'P,.~E!5'P) 

Dem. 
h . *121-305 . *93-101 . D 

h : P 6 ft . ~ E ! £'P . Q'Pi =1= d'P .3.3! Q'P - d'Pi . Q'P = D'P - I'B'P . 
[*250-21] D.a!D'P,-a'P,-t'S'P. 

[*93101] D . a ! B'Pi - t'5'P (1) 

h . *121-305 . *25021 . D h : P e ft - I'A . D . £'P e B'P, (2) 

1-.(1).(2). DI-:P6ft.~E!£'P.a'Pi4=a'P.D.^'P,~6l. 

[*53-3] D . ~ E ! 5'P, (3) 

I- . (3) . Transp . D I- : P e ft . E ! B'P, . ~ E ! B'P . D . Q'P, = O'P . 

[*263-44] D.Pew (4) 

h . *250-21 . *263-44 . D h : P e <u . D . £'P, = B'P . 

[*250-13] 3 . E ! B'P, (5) 

h . (5) . *263-44 . DH:Pe«.D.E!5'P,.~E!B'P (6) 

f- . (4) . (6) . D h . Prop 

*263-47. h . « = ft n P {a C O'P . D. : a 6 Cls induct . = . g ! (7'P r> ^'P"a| 

Dem. 
\- . *254-52 . D I- : Pe « . o C O'P . a ! O'P np'P"a . D . (P ^ a)lessP - 
[*263-3] D.Ppaeftfin. 

[*261-42-14] D . 0'(P t a) e Cls induct . 

[*202-55.*120-213] D . a e Cls induct (1) 

h . *261-26 . D I- -.Peto.aCG'P . aeCls induct . g ! a . D . E Imaxp'a. 
[*263-22] D . a ! P'maxp'a . 

[*205-65.*40-69] D . 3 ! C"Pn^'P"a (2) 



152 SEEiES [part V 

I- . (1) . (2) . *40-2 . D 

H :. Peo) .aC C'P . D : a e Cls induct . = . g I C'Pnp^"a (3) 

l-.*40-2.*120'212.D 

h :: Pen :. aC C'P . D, : aeCls induct . = .g ! C'Pnp'P'"a :. 3 .g ! P (4) 

h . (4) . *200-51 . D h : Hp (4) . D . C'P~ e Cls induct (5) 

h . *250-16 . D 

h : Hp (4) . g ! Q'P - Q'P, . D . P'miDp'(a'P - a'P^) e Cls induct . 

[*261-26] D . E ! maxp'?minp'(a'P - Q'P,) . 

[*205-252] D.minp'(a'P-a'Pi)ea'Pi (6) 

t- . (6) . Transp . 3 h : Hp (4) . D . Q'P, = Q'P (7) 

h . (5) . (7) • *261-34 . D h : Hp (4) . D . ~ E ! 5'P (8) 

h . (4) . (7) . (8) . DI-:Hp(4).D.P6<» (9) 

1- . (3) . (9) . D h . Prop 

*263-48. I- . 0. = n A P {a C C'P . D. : a~e Clsrefl . s . g ! C'P nj3'P"a} 
[*263-47 . *261-47] 

*263-49. h.Xlfinu 6) = nnP(a'Pi = a'P) = n ft P(P = P,„) 
Dem. 

I- . *261'22 . *263-44 .Dh:P6nfinwa).D. a'P, = a'P (1) 

l-.*261-34.*263-44.DI-:P6fl.a'Pi = a'P.D.Peflfinw(B (2) 
h . (1) . (2) . D h . n fin u ft, = fl ft P (Q'Px = Q'P) 

[*26r212] =fiftP(P = P,J.Dh.Prop 

*263-491. l-:PeOfinwo,.D.P = (P^po ■ Pa = (PiV 

h . *263-49 . *261-212 . D I- : Hp . D . P = (P,)p„ . (1) 

[*91-602.*121103] D .P(an-iy) = P^(xiHy) . 

[*121-11] D.P„ = (P,)„ (2) 

I- . (1) . (2) . D 1- . Prop 

*263-5. I- : P 6 n infin . D . P ^ (t'5'P w P,n'-B'P) e co 

Bern. 
h . *261-45 . 3 h : Hp . D . P, T P,/5'P e Prog (1) 

h . *260-33-27 . D h : Hp . D . (P, ['p,„'B'P)^ = P,„ ^ (I'^'P « P,n'-B'P) 
[*260-32] =PUi'fi'PwP,„'5'P) (2) 

t- . (1) . (2) . *263-l . D h . Prop 



SECTION E] PROGRESSIONS 153 

*263-51. hiPeflinfin.D. 

I'B'P uKn'-B'P e D'(Pe n /) . I'B'P yj^tu'B'P e a'sgm'P 
Bern. 

f- . *263-5-22 . D h : Hp . D . ~ E ! ma.xp'(i'B'P ^%^'B'P) (1) 

F- . *26011 . D h : Hp . 2/ 6 a'P - P,/5'P . a; e'Pt^.'B'P . D . 

P (5'P hH 2/) ~ 6 Cls induct . P (5'P m a;) e Cls induct . 

[*120-49] 3 . Nc'P {B'P m ?/) > Nc'P {B'P w x) . 

[*117-222.Transp]D.~(yP«) (2) : 

I- . (2) . Transp . D h : Hp . D . P"K'£'P C ^P w K'-B'P (3) 

I- . (3) . *93-101 . D h : Hp . 3 . P"{i'B'P C %^'B'P) C I'B'P u %^'B'P (4) 

H.(l).(4).*211-41.Dh:Hp.D.i'jB'PwP,n'^'^eD'(PeA/). (5) 

[*212152] D . i'£'P yj%^'B'P e Q'sgm'P (6) 

f- . (5) . (6) . D h . Prop 

*263-52. hiPeXlinfin-w.D. (ga;) . « e Q'P . P,n'5'P u l'5'P = P'a; 
i)em. 
I- .*263'49 . Transp . D I- : Hp . D . g ! a'P-a'P, . 
[*260-27] D.^ia'P -%^'B'P . 

[*250-121] 3 . E ! niinp'(a'-P - Pfn'-B'-P) ■ 

[*263-51.*206-25.*211-726] D . (ga;) . a; e Q'P . P^'^'P w I'B'P^'P'x: 

D h . Prop 

*263-53. h : P 6 n infin - <» . D . Nr'P > tu 
Dem. 

h . *253-13 . *263'52 . D h : Hp . D , P t; (Pi^'B'P u t'£'P) e D'Ps . 
[*263o] D . a ! w n D'Ps . 

[*255-17.*26318] D . Nr'P > « : 3 h . Prop 

The above proposition shows that « is the smallest of infinite ordinals. 
The same fact is otherwise expressed by the following proposition. 

*263'54. 1- : a e NO infin - I'w . D . a > m [*263-53] 

*263-55. h:Pea).D.P!6w-i-i.s'P6&) + i 

DeTn. 

h.*253-511.*263-44.DI-:Hp.D.Psec»-i-i (1) 

h . *2.52-372 . *263-44 . D h : Hp . D . s'P e to + 1 (2) 

h . (1) . (2) . D h . Prop 



154 SERIES [part V 

The following propositions are lemmas for proving w X 2^ = eo (*26363). 

*263-6. i- :: P eSer . x^y . M = P X (x iy) .0 :. RM,S .= : 

(gw) . u 6 C'P .R = oi;lu.8=ylu.v. (gw, v) . uPjV . R~y ],u.S = a!lv 
Bern. 
h . *1 66- 1 1 1 . D I- : . Hp . mPi; . E = a; 4, M : 5f = a; J, i; . V . /S = 2/ 4, ?) : D . 

RM{ylu).{yiu)M8. ' 

[*201-63.*204-55] D.'^(RM,S) (1) 

Similarly \-:.Rp.uPv .R^y iu.S = y iv.D .'^{RM^S) (2) 

h.*166-lll.D 

I- : Hp . uPw . wPv .R = ylu.S==xlv.D. RM(x lw).{x\,w) MS . 
[*201-63.*204-55] D.~(i2M„Sf) (3) 

h . (1) . (2) . (3) . Transp . *166111 . 3 
h:.Hp. RMiS.O:{'^u).R = xlu.S = yiu.ueG'P.v. 

{'^u,v).uPTV.R = y ^u.S=xlv (4) 

\-.*166in.D\-:'H.i>.R = xlu.S=yiu.RM(xiv).O.SM(xiv) (5) 
h . *166111 .'^\-:.Rp.R = xiu.S = ylu. RM(y lv).D:u = v.v. uPv : 
[*166-111] D:ylv = S.v.SM(yiv) (6) 

h.*166-lll.D 

\-:'S-p.R = yiu.S = xiv. uP^v . RM {ylw).O.SM(yiw) (I) 

\- . *166-111 .D\-:.ll^.R = yiu.8 = xiv. uP^v . RM{x J, w) . 3 : 

xiw = 8.^.8M{x\,w) (8) 
h . (5) . (6) . (7) . (8) . D I- : . Hp : « e C'P . i2 = a; J, M . /S = 2/ 4, M . V - 

uP,v.R = yiu.8 = xlv:0.RM^S (9) 
h . (4) . (9) . 3 I- . Prop 

*263-61. I- : P 6 Ser . a; + 2/ . Jlf = P X (« 4, 2/) . 3 . d'M, = y yCPux 4,"a'P, 
[*263-6] 

*263-62. h:P6(».a!=t=2/.3.Px(a;4,2/)6o> 
Dem. 
h . *263-61-43 . D h : Hp . D . Q'jP x (a; j y)], = 2/ ^"O'P u x l"a'P 
[*166111] =a'{Px{xiy)} (1) 

|-.*251-55. DI-:Hp.D.Px(a!4,2/)6ii (2) 

h.*166-14. DI-:Hp.D.Px(a;4,2/)6-t'A (3) 

\- . *166-16 . *263-22 . D h : Hp . D . S'Cnv'{P x (a: 4, y)} = A (4) 

h . (1) . (2) . (3) . (4) . *263-44 .Dh:Hp.D.Px(a;4,2/)6a):DI-. Prop 

«26363. \-.(oX2r = co 
Dem. 

I- . *263'62-l7 . DI-:P6«.Qe2,.D.Nr'(PxQ) = ffl (1) 

h . *184-13 . *26317 .DI-:P6«.Qe2,.D. Nr'(P x Q) = « X 2, (2) 

|-.(1).(2). Df-:a!w.a!2^.D.«x2,= w (3) 

\- . *184-11 . DI-:<a = A.D.«BX2^ = A (4) 



SECTION E] PEOGEESSIONS 155 

I- . *123-14 . *263101 .Df-:a!ft).D.a!2. 

[*262-21] • D.a!2, (5) 

l-.(3).(4).(5).Dh.Prop 
The following propositions are lemmas for proving *263'66. 

*263-64. y :P,QeSer .aieCF .zQ,w . M = P X Q .D .(z ia!)M,{w la:) 
Bern. 
l-.*166-lll.D[-:Hp.D.(0 4,a!)M(«;4,a;) (1) 

h . *166-111 . D I- :. Hp . (a: I «) JIf (m J, y) . D : xPy .w .x = y. zQu : 
[*204'7l] D : xPy .v.x = y.u = w.v.x = y. wQy : 

[*166-111] ■^■.{wlx)M{u\,y).y.{iu\,x)^(u\,y) (2) 

h.(2).*204-55.DI-:Hp(2).D.~ {(M42/)ilf(wJ,a')} (3) 

h . (1) . (3) . *201-63 . D h . Prop 

*263-641. V \ P,qe^ev . z = B'Q.w = B'Q .xP^y . M=^ P X Q .O . 

{z\,x)M,{wiy) 
Dem. 

h . *166-111 . 3 h : Hp . D . {zix)M{w i y) (1) 

h . *166111 . D h :. Hp . (^ j «) Jf (m J, ?;) . D : a;P?; : 

[*204-7l] D:v = y.w.yPv (2) 

h.(2).*166-lll.D 

I- :. Hp . {z lx)M(u^v) . D :ulv = w ^y .v .(w ly)M(u^v): 

[*204-55] D : ~ {(m 4, ») M{w J, y)} (3) 

h . (1) . (3) . *201-63 . D h . Prop 

*263-642. h : P, Q 6 Ser . ilf = P X Q . D . (C'P x Q'QO C Q'ilf, [*263-64] 

*263-643. h : P, QeSer.E ! 5'Q . E ! 5'Q.ilf =Px Q.D.(5'Q)4,"a'P,Ca'ifi 
[*263-64] 

*263-65. l-iPew.Qeflfin-i'A.D.PxQeo) 
Dem. 

h.*25r55.Dh:Hp.D.PxQ6n (1) 

h.*166-14.DH:Hp.D.PxQe-i'A (2) 

h . *263-642;643 . *261-24 . D 

I- : Hp . D . (C'P X a'QO w (B'Q) i"a'P, C a'(Pi X Q), . 
[*263-49] D . (C'P X Q'Q) w (5'Q) l"a'P C a'(P x QX . 

[*166-12-16] D . a'(P X Q) - ^'(P X Q) C a'(P x Q\ . 

[*93-101 .*201-63] D . a'(P x Q) = a'(P x Q), (3) 

I- . *166-16 . *263-22 . D h : Hp . 3 . fi'Cnv'(P x Q) = A (4) 

I- . (1) , (2) . (3) . (4) . *263-44 . D J- . Prop 

*263-66. l-:a6NOfin-t'0,.D.ft)Xa = «» [*263'65] 
The proof proceeds as in *263'63. 



*264. DEEIVATIVES OP WELL-ORDERED SERIES. 

Summary of *264. 

The principal purpose of the present number is to show that every 
infinite well-ordered series is the sum of a series of progressions followed 
by a finite tag, which may be null. For this purpose, we proceed as follows : 
If X is any member of G'P, it must belong to the family, with respect 

to P„ of some member of G'P-d'P^, unless x = B'P and £'P~ea'Pi. 

Assuming that we have either ~E!B'P or S'PeCE'Pi, and assuming 
further that P is an infinite well-ordered series other than a progression, 
it follows that every member of G'P belongs to the family, with respect 
to Pj, of some member of CV'P, because, by *216-611, C'V'P = D'P, - Q'Pi 
in the circumstances contemplated (*264'15). Now P limited to any one 

family with respect to Pj is a progression, unless that family includes B'P ; 

and if it includes B'P, it is finite. Hence our proposition follows. 

An important consequence of the above proposition is that every cardinal 
which is not inductive and is applicable to classes that can be well-ordered is 
a multiple of «„ (*264-48). 

For the purposes of this number we need a notation for the series of 
series each of which consists of the family of some member of C'VP. We 
therefore put 

Pp, = P D ; {P^h'^^'P Dft [*264]. 
Here " pr '' is intended to suggest " progression." When P e fl infin — ca, 
Ppr is the series of progressions (possibly ending in a finite tag) whose 
sum is P (or P ^ D'P, in one case). Before using this definition, some 
preliminary considerations are necessary. V'P is the series of limit-points 
of P, including B'P. In order that V'P may exist, there must be at 
least one limit-point besides B'P. Now the limit-points of a series are 
G'P-a'P^,i.e. the limit-points other than B'P are Q'P-Q'Pi (*216-21). 
Hence when B'P exists and Q'P — CE 'Pi exists, V'P exists. Hence by 
*263'49, 

*26413. h:.PeIl.D:a!V'P. = .Pen infin - m 



SECTION E] derivatives OF WELL-OEDEEED SEEIES 157 

I.e. a well-ordered series whose derivative exists is one which is infinite 
and not a progression. W» have similarly 

*26414. h : P e n infin - « . D . G'V'P = G'P - d'P^ 
and 

*26412. h:P6n.D.a'V'P = a'P-a'Pi 

We next proceed (*264"2 — •261) to study the posterity of a term x 

with respect to Pj, i.e. the series P^{P^^'x. We show that if this series 

has a last term, it is finite (*264-21), and ends with B'P (*264-2), while 
if not, and if xeCP^, i.e. if x has either an immediate successor or an 
immediate predecessor, the series is a progression (*264*22). Hence we have 

*26423. l-:.Pe0.a;eC"V'Pn(7'P,.D: 

E ! maxp'(PiVa! . = .x = B'Cnw'V'P . E ! B'P 

Moreover, if xeC'Pi, the ancestry of x with respect to Pi must end with 
a member of the derivative of P, i.e. 



*264-233. h : P e n infin -a.xe G'P^ . D . minp'(Pi)*'a; e G'V'P 

We thus arrive at the result that if P has a last term, so has V'P 
(*264'24), and if x is any member of the derivative except the last, the 

series P^{P^^'x is a progression (*264'25), while if x is the last term of 

the derivative, and the series P has a last term, then P ^ {P^^'x is finite 
(*264'252). Moreover the supposition that P ends with a member of the 
derivative is equivalent to the supposition that P ends with a term which 
has no immediate predecessor (*264'26). 

We now proceed (*264'3 — "403) to consider the relation Pp, defined 
above. If we take any term y in a well-ordered series, there is some term 
X belonging to G'P — Q'Pi such that the family of y with respect to Pj 
is the posterity of x. This results from *264*283 above. Thus we may 
. divide the field of P into mutually exclusive stretches, each of which is the 
posterity of some member of O'P — Q'Pi with respect to Pj. The series of 
series thus obtained is Pj,j. There is an exceptional case, when the series 
ends in a term having no immediate predecessor, for then the posterity of 
this term with respect to Pi is null, and therefore Ppj omits this term. 
Otherwise, we shall have S'Ppr = P; i.6. we have 

*264-39. 1- : P e O infin - © . ~ {B'P e G'V'P) . D . 2'Pp, = P 

*264-391. \-:Pen.B'Pe G'V'P . D . t'P^, = P C D'P 

Moreover we have 

*264-36. f- : P € n . D . Ppr smor V'P . Pp, e Eel'' excl 



158 SERIES [part V 

from what was proved earlier we know that, assuming Peil, we have 
D'PpjCw (*264-401); if P has no last term, CPp^Ca; if P is infinite and 

has a last term, B'Pp^ is finite, and if the last term of P belongs to C'V^P^ 

£'Ppr = A. Hence, using *251"63, which assures us that, in virtue of 
*264-36 above, if C'Pp, C m, S'Pp, is a multiple of to, we find (*264'44) that 
every well-ordered series has an ordinal number of the form (a >C w) 4- ^, 
where a and /3 may be any ordinals, including 0^ and 1 (putting 1 X a = a to 
avoid exceptional cases). The above account omits the exceptional cases, 
which require special treatment and render the proof long; but in the end 
the above simple result is obtained. 

Since a multiple of H^ is not increased by the addition of an inductive 
cardinal, it follows (*264'44) that the cardinal number of the field of an 
infinite well-ordered series is always a multiple of tia (*264'47). Hence 
if all classes can be well-ordered, all cardinals which are not inductive are 
multiples of No. In virtue of Zermelo's theorem, the same result follows if 
the multiplicative axiom is true. 



*26401. Pp, = Pt5(A)*5V'P Dft[*264] 

*26411. H:.Pen.D:a!sgm'P. = .Peninfin 

Dem. 
l-.*263-51. DhiPeninfin.D.glsgm'P (1) 

1- . *212-152 . *211-41 . D h : P e n . a ! sgm'P . D . a ! sect'P- t'A - Q'maxp . 
[*261-28.Transp] D . P e fl infin (2) 

I- . (1) . (2) . 3 h . Prop 

*26412. h : P e n . D . a' V 'P = Q'P - a'P, 
Devi. 

I-.*216-61. DI-:Hp.a!P.D.a'V'P = a'P-a'Pi (1) 

h . *216-612 . *33-241 . D h : P = A , D . Q'V'P = A . Q'P - Q'Pi = A (2) 
I- . (1) . (2) . D h . Prop 

*26413. l-:.Pen.D:a!V'P. = .Pefi infin - o, 
Dem. 

h . *26412 . D h :. Hp . D : a ! V'P . = . g ! a'P-a'P^ . 
[*263-49] = . P e n infin - o) : D h . Prop 

*264-14. h:Peflinfin-<».D.O'V'P = a'P-a'P, [*264-13 .*216-611] 

*26415. I- :. P 6 n infin - « : ~ E ! B'P . v . B'P e a'P^ : D . G'V'P ='b'P^ 

Dem. 
V . *264-14 . *93103 . D h : Hp . ~ E ! 5'P.D.C"V'P=C"P-a'Pi.C"P=D'P. 
[*93-101.*2.50-21]- D.G'S/'P='b'P, (1) 



SECTION e] DEEIVATIVES OP WELL-ORDERED SERIES 159 

l-.*93101. DhiB'PeQ'Pi.D.C'P-a'PiCD'P (2) 

H . (2) . *264-14 . D f- : Hp .%P e Q'A . D . C'V'P = D'P - a'P^ 
[*93-101.*250-21] =B'P, (3) 

I- . (1) . (3) . D I- . Prop 



*264-2. h : P e n . E; ! msiXp'(P^%'x . D . maxp'(P,)*'a; = B'P 
Bern. 



h . *206-42-46 . 3 h : Hp . D . seqp'(PO*'« = P/maxp'(P,)*'a; . 
[*90-16] ^.^qp'(P^'a;C(P^'x. 

[*206-2] D . seqp'(Pi)*'a! = A . 

[*250126] D . maxp'(P,)*'a! = £'P : D h . Prop 

*264-21. h : P 6 fl . E ! maxp'(P,)*'« . D . 



P t (Pi)*'« 6 12 fin . P (a; M 5'P) e Cls induct 
Dem. 



f- . *20035 . D h : Hp . (Pi)*'* = I'a; . D . P r (P,)^'x = A (1 ) 

« < 

I- . *260-27 . D h : Hp . (Pi)*'a; 4= I'a; . D . a!P,„ maxp'(P,)j,f'a; . 

<- 



[*260-ll] D.P{a;i-imaxp'(P,Va;} 6 Cls induct. (2) 

[*205-2] D.G'P^ C^O^'a; e Cls induct (3) 

I- . (1) . (2) . (3) . *264-2 . D h . Prop 



*264-22. h : P e n . ~ E ! maxp'(Pi)5,e'« ■ « « C'P, . D . P ^ (Pi)*'a; e a 
Dem. 

I- . *260-32-34-27 . D h : Hp . D . {P p (PO*'«}i = {CPO*'*} 1 A ■ (1) 

[*122-52] D.{Pt(PVf}xeProg _ (2) 

h . (1) . *260-33 . D I- : Hp . D . [{P D (AVa;! Jp„ = P ^ (PO*'* (3) 

|-.(2).(3).*2631.DI-.Prop 

*264-221. h : P e fl . a; ( V'P) 2/ . D . P (« - 2/) ~ 6 Cls induct 
Dem. 
I- . *207-34 . *216-6 . D h : Hp . D . xF'y . y = Itp'P'y . 
[*207-25] D . xF'y . y = ltp'(P'a; n P'y) . 

[*20713] D . xP'y . ~ E ! maxp'(P'a; n 'p'y) . 

[*261-26] D . P'a; n P'y ~ e Cls induct : D h . Prop 

*264-222. h : P e fi . P'« e Cls induct . D . a; ~ e D'V'P [*264-221 . Transp] 



160 SERIES [PAKT V 

*264-223. h : P e n . P (« - 2/) ~ 6 Cls induct . D . g ! CI'V'P a P (« -i 2/) 
Dem. 
V . *261-3 . D h : Hp . D . (ga) .aCP(«-2/).a!a.~E! maxp'a . 
[*250-122] D.(aa).aCP(a;-y).a!a.E!ltp'a. 

[*206-213] D.(aa).aCP(a!-2/).a!a. VaeP(a;-i2/). 

[*206-181] D . a ! dtp A Q'P n P (« -) 2/) . 

[*216-602] D . a ! a'V'P A P "(« -H y) : D I- . Prop 

*264-224. h : P e n . a; = 5'Cnv'V'P . E ! P'P . D . P'a; e Cls induct 
Dem. 
\- . *264-223 . Transp . D h : Hp . D . P (« - B'P) e Cls induct : D h . Prop 



*264-225. h :. P 6 n . a; e O'Pi . D : E ! maxp'(P,)*'a; . = . (Pi)*'* e Cls induct 

[*264-21-22] 

*264-23. \-:.Peil.X€G'V'PnG'F,.:i: 

E ! maxp'CPO*'* ■ ^ ■ a; = i5'Cn v'V'P . E ! B'P 
Dem. 



I- . *264-2 . D I- : Hp . E ! maxp'(Pi)i^'a; . D . E ! B'P (1) 

h . *264-21-222 . D h : Hp (1) . D . a; ~ e D'V'P . 

[*93-103] D . a; = 5'Cnv'V'P (2) 

I- . *264-224 . 3 I- : Hp . a; = 5'Cnv'V'P . E ! B'P .D.P'aie Cls induct . 
[*120-481-251] D . (Pi)*'a; e Cls induct . 

[*90-12.Hp.*261-26] 3 . E ! maxp'(P,)*'a; (3) 

1- . (1) . (2) . (3) . D h . Prop 

*264-231. h-.Pen.xeC'V'P- C'P^ . D . a; = 5'Cn v'V'P = B'P 

Dem, 

h .*2o0-21 . D h : Hp . D .a;~6D'P . 

[*93-103] D.x = B'P. (1) 

[*216-6] D.aj~6D'V'P. 

[*93-103] D.ai= B'Cny'V'P (2) 

h . (1) . (2) . 3 h . Prop 

*264-232. l-:.Pe0.a;6a'V'P.D: 

(P^'x 6 Cls induct . = .x = B'Cnv'V'P .El B'P 

This proposition differs from *264'23 by not assuming that x e G'Py. 

If B'P has no immediate predecessor, B'PeC'V'P-C'Pi, so that B'P 
satisfies the hypothesis of *264-232, but not that of *264-23. 



SECTION E] derivatives OF WELL-ORDERED SERIES 161 

Dem. 

l-.*90-13. Dl-:Hp.(*Pi%'a; = A.D.a;~eO'P,. 

[*264-231] D.a; = B'Cnv'V'P.E!B'P (1) 

I- . *120-212 . D h : Hp (1) . D . {P^'x e Cls induct (2) 

h . *264-225 . D 

h :. Hp . a ! (Pj)^'ai . D : {Pi)^'x e Cls induct . = . E ! maxp*(Pi)*'« ■ 

[*264-23] =.a; = £'Cnv'V'P.E!5'P (3) 

h . (1) . (2) . (3) . D h . Prop 

*264-233. I- : P e fi infin - to . « e G'P, . D . mmp'(Pi)^'ai e G'V'P 
Dem. 

l-.*250-121. D h : Hp . D . E ! minp'(Pi V« (1) 

h . *90-l72 . D h : Hp . y (PO* « . zP^y . D . ^ e (Pi)*'* n P'y . 

[*205-14] D . y + mini.'(Pi)*'a; (2) 

> 

I- . (2) . Transp . D h : Hp . y = miiip'(Pi)*'« . D . y ~ e (I 'Pi . 

[*26414J D.yeC'V'P (3) 

I- . (1) . (3) . 3 I- . Prop 

*264-24. h : P 6 n infin . E ! 5'P . D . E ! 5'Cnv'V'P 
Dem. 

I- . *26412 . D h : Hp . 5'P ~ 6 O'Pi .D.B'Pe G'V'P . 

[*216-6] D . B'P = S'Cnv' V'P (1) 

y . *264-233 . *263-22 . D I- : Hp . B'P e G'P^ . 3 . minp'(Pi)*'JB'P e C'V'P (2) 

h . *205-55 . D I- : Hp (2) . « = minp'(P,)*'5'P . D . 5'P = maxp'(PiVa; . 

[*264-23.(2)] D . a; = B'Cnv' V'P (3) 

h . (1) . (3) . D I- . Prop 



*264-25. hzPen.xe D'V'P . D . P p (Pi)*'a; e co 
Dem. 

h . *264-232 . *250-21 . D h : Hp . D . (Pi)*'a!~ e Cls induct . x e D'Pi . 

[*264-225] D . ~ E ! maxp'(P,)*'a; . x e D'Pi . 

[*264-22] O.Pt [Pih'x e o) : D h- . Prop 

*264-251. h : P e X2 . ~ E ! £'P . a; e 0' V'P . D . P ^ (Pi)*'a; e o) 

Dem. 

h . *250-21 . D h : Hp . 3 . a; 6 D'Pi . 

[*264-23.Hp] D . ~ E ! maxp'(PiVa! . x e B'P, . 

[*264-22] 'ii.Pt (Pi)*'* e o) : D 1- . Prop 

E. ifcW. III. 11 



162 



SERIES 



[part V 



*264-252. \-:Pen.ElB'P.x = B'Cnv'V'P . D . P p (P,)*'« e £1 fin 
Dem. 



f- . *264-23 . D f- : Hp . a; 6 C'P^ . 3 . E ! ma.xp'(Pi)^'x . 
[*264-21] D.Pl (Pi)*'« e ft fin 



(1) 
(2) 



(1) 
(2) 
(3) 



I- . *90-14 . 3 h : «~ 6 C'Pr .O.Pl {P,)^'x = A 
1- . (1) . (2) . D f- . Prop 

*264 26. h :. P e ft . D : J5'P 6 C'V'P . = .ElB'P. P'P~ e Q'P, 
Dem. 

I-.*14-21. D h-.B'P 6 G'V'P.D. El B'P 
h . *264-12 . D I- : Hp . B'P e G'V'P . D . B'P ~ e Q'Pi 
h . *26412 . D h : Hp . £'P~ e Q'Pj . D . 5'P e G'V'P 
1- . (1) . (2) . (3) . D f- . Prop 

*264-261. h :. P e ft . D : ~ (B'P e C V'P) . = .C'P = C'P, 

Dem. 
h . *264-26 . D h :: Hp . D :. ~ (B'P e G'V'P) 
[*202-52] 
[*250-21] 
[*121-322] 

*264-3. I- : QP^,R . = .(^x,;/) .x(V'P)y .Q = Pl (P^'x . E = P t {P^'y 

[(*264-01)] 
*264-31. I- :. P e Ser . D : QPj^B . s . 

(•^x,y).x,yeG'P-a'P,.xPy.Q = Pt(P^'x.R = Pltp7)'^'y 

[*207-35.*264-3.*216-6] 

*26432. h . O'Pp, = P t "C^O*"^'' V'P [*150-22 . (*264-01)] 
*264-321. h : P e Ser . a; e C" V'P . D . (P^'x ~ e 1 

I- . *216-611 . D I- : Hp . D . a; e (7'P - Q'Pi 



~ E ! 5'P . V . £'P e a'P, : 

£'PCa'P,: 

(7'PCC'Pi: 

C'P = C"Pi::DI-.Prop 

<- 



h.*90'14. Dl-:a;~6C"Pi.D.(PiVa;=A 

I- . *1 21-305 . D h : Hp . a; 6 D'Pi . D . a ! (Pi)*'* -I'x. 

[*90-12] D.(Pi)*'a!~el 

l-.(l).(2).(3).DI-.Prop 

*264-33. h : P e Ser . D . G"G'P^, = CPO*"C" V'P 
[*264-321 . *202-55 . *264-32] 



(1) 
(2) 

(3) 



SECTION E] derivatives OF WELL-ORDERED SERIES 163 

*264-34. \-:P6n.x,yeC'P.Pl(P^'x = Pl(P^'y.D.x = y 
Dem. * 



h . *264-321 . *202-55 . D I- : Hp . D . (Pi)*'* = (Pi)*'2/ (1) 

I- . (1) . *9012 . D h : Hp . a; 6 C'P^ . D . a; (Pi)*y . y {P;)^x . 
[*260-22.*91-541] ■^.x = y (2) 

h.*250-21. Dh:Hp.a;~6a'Pi.D.a! = 5'P (3) 

h . (1) . *9012-14 . D h : Hp . a;~ 6 C'P, . D . y^ e O'P, . 

[*250-21] D.y^B'P (4) 

h.(3).(4). DI-:Hp.a!~6C'Pi.D.a; = 2/ (5) 

h . (2) . (5) . D h . Prop 

*264-341. h : P 6 Ser . a?, 2/ 6 G'V'P . x (P;)^y . D . a; = y 

Dem. 

h . *216-611 . 3 h : Hp . D . y ~ 6 Q'Pj . 

[*91-504] D.~{a;(Pi)p„2/}. 

[*91-54] D . a; = 2/ : D I- . Prop 

*264-35. 1- : P e Ser . a;, 2/ e G' V'P . g ! (Pa)*'a; n (Pi)*'y .O.x = y 

Dem. 

h . *96-302 . 3 I- :. Hp . D : a; (Pi)*2/ . v . y (Pi)** : 

[*264-341] D:x = y:.D\-. Prop 

*264-36. h : P e £2 . D . Ppr smor V'P . Pp, e Rel'^ excl [*264-34-35] 

The following propositions .lead up to *264'39'391. 
*26437. h : P 6 11 infin - « . D . s'C'P^^ = P^ 

Dem. 
h . *264-32 . D h :. Hp . D : a;(s'a'Pp,) y. = . (ga) . aeG^'P.x, y e (P^)^'a.xPy . 
[*260-32-27] = . (act) . a e C'VP^^, y e (Pi)*'a .joPf^y . 

[*264'233-35] = . (ga) . a = minp'(Pi)*'« = mmp'{Pi)^'y . xPf^y . 

[*13-195] = . mmp'(P,)^x = minp'(Pi)*'2/ ■ *Pfn2/ (1) 

h . *260-27 . 3 h : Hp . a;P,„2/ . D . (Pi)*'*C (Pi)*'^/ . _ 
[*205-5] 3 . minp'(Pi)*'a; = minp'(Pi)*'y (2) 

h . (1 ) . (2) . D 1- : . Hp . 3 : a; (i'CPp,) y. = . xPf^y : . D h . Prop 

*264-371. h : P e Ser . a (V'P) 6.3. (Pi)*'a C P'6 
Dem. 

l-.*216-6. 3H:Hp.3.a6P'6 (1) 

h . *204-71 . 3 1- : Hp . a;P6 . xP,y . ~ (2/P6) .D.y = b. 
[*33-14] 3.6ea'Pi (2) 

I- . (2) . Transp . *216-611 . 3 h :. Hp . 3 : arP6 . a;Piy .O.yPb (3) 

h . (1) . (3) . *90112 , 3 I- :. Hp . 3 : ct(Pi)*a! . 3 . a;P6 :. 3 h . Prop 

11—2 



164 SERIES [PART V 

*264-372. 1- : P 6 Ser . D . PJPp, G P-P,„ 

Dem. 
l-.*264-3-321.*20255.D 

h :. Hp . D : a; (PJPp,) y. = . (ga, 6) . a (V'P) b.xe (P^'a . y e (PO^'b . (1) 
[*264-37l] D . xPy (2) 

h . *264-35 . D h : Hp . a (V'P) 6 . a; 6 (PiVa . 2/ e (P,)*'6 . 3 . 2/ ~ e (A)*'a ■ 
[*90-l'/] D.2/~6(P,Va;. 

[*260-27] D.~(«P,„2/) (3) 

I- . (1) . (2) . (3) . D h : Hp . D . F'P^, G P-P^ : D h . Prop 

*264-373. h : P e 11 . ~ (B'P e C'V'P) . D . P-P,„ G PJPp, 

Bern. 
h.*264-261-233.*263-49.D 



h : Hp .!c{P^Pt^)y. D . mmp\P,)^'x. vamp\P,)^'y e G'V'P (1) 

> > 

1- . *96-301 . 3 h : . Hp . mmp'{Pi)^'ai = miup'(P^)^'y .D:x (Pi)* y .v.y {P^^x : 

[*260-27] :>:x = y.v. xPj^y . v . yPi^x (2) 

h . (2) . Transp . D h : Hp (1 ) . D . mmp'{P.,)^'x 4= xa.mp\P;)^'y (3) 

h . (1) . *264-371 . D h : Hp . minp'(P,)*'2/ P minp'(Pi)*'a; . D . yPx (4) 

h . (4) . Transp . D h : Hp (1) . D . ~ {minp'(P,)*'2/ P minp'(PiVa;j (5) 

h . (3) . (5) . D h : Hp (1) . D . minp'{P^'x P mmp'(P^ _ (6) 

h . (1) . (6) . 3 h : Hp (1) . D . (ga, 6) . a (V'P) 6 . a; e (PO*'a . y e{P,)^'b . 

[*264-3-321 .*202-55] D . » (PJPp,) y : D h . Prop 

*264'38. h : P 6 n . ~ (£'P e G'V'P) . D . F>P^, = P-Pfn [*264-372-373] 

*264-381. h : P 6 fl . 5'P e (7' V'P . D . PJPp, = P t D'P -^P,„ 
Bern. 

H . *264-33 . D h : Hp . D . s'G"G'P^^ C C"P, . 
[*264-26.*42-2] D . 5'P ~ e O'PJPp, . 

[*264-372] D.P;Pp,GP^D'PiP,„ (1) 

h . *250-21 . D I- : Hp .x{PlIi'P-Pt^)y . '^.x,ye G'P, . 
[*264-233.*263-49] D . minp'(Pi)5u'a;, minp'(P,y2/ e G'V'P (2) 

Thence as in the proof of *264"373, 

l-:Hp.a;(P^D'P-P,„)2/.D.a;(P;Pp,)2/ (3) 

I- . (1) . (3) . D h . Prop 



SECTION E] derivatives OF WELL-ORDERED SERIES 165 

*264-39. h : P e fl infin -^ . ~ (5'P e G'V'P) . D . S'Pp^ = P 
[*264-37-38 . *26012 . *1621] 

*264-391. hzPea.B'Pe G'V'P . D . 2'Pp, = P^ D'P 
Dem. 

h.*264-13.Dh:Hp.D.P60infin-a) (1) 

h . *260-27 . D h : Hp . D . Pfa = P,„ C C'P, 

[*264-26] =Pt^lI)'P (2) 

I- . (1) . (2) . *264-37 . *260-12 . D h : Hp . D . s'C'P^, = P,„ . P,„ G P ^ D'P (3) 
F . (3) . *264-381 . D h . Prop 

*264-4. h : P 6 n . ~ E ! 5'P . D . C'P^^ C w [*264-251 -32] 

*264-401. h : P e n . D . D'Pp, C « 
Bern. 

h . *151-5 . *264-34 . D I- : Hp . D . D'Pp, = P t"(PO^"D' V'P (1) 
h . (1) . *264-25 . D I- . Prop 

*264-402. I- : P e fl infin . E ! B'P . D . 5'Pp, e il fin 

Dem. 

h . *264-24 . D h : Hp . D . E ! P'Cnv' V'P . 

[*151-5.*264-34] D.5'Pp, = Pp(P,y£'Cnv'V'P . 

[*264-252] D . B'P^, e II fin : D h . Prop 

*264-403. \-:P€a.B'Pe G'V'P . D . B'Pj,, = A 
Dem. 

h . *264-26-231 . D I- : Hp . D . B'P ~ e C'P^ . S'P = 5'Cnv' V'P . 
[*9014] D . CPO^'-B'Cnv' V'P = A . 

[*151-5.*264-34] D . P'Pp, = A : 3 h . Prop 

The following propositions deal with the various different cases that arise. 
Their net result is expressed in *264'44. 

*264-41. h:P6Xlinfin-«.~E!5'P.D.Nr'P = Nr'V'Pxw 
Dem,. 

I- . *264-36-4 . D h : Hp . D. . Pp, e Rel"" excl n Nr'V'P . G'Pj,, C m . 
[*251-63] 3 . S'Ppr 6 Nr' V'P x « . 

[*264-39] D . P 6 Nr ' V'P X w : D I- . Prop 



166 SERIES [part V 

*264-42. h : P € n . 5'P ~ 6 G'V'P . V'P e 2, . D . Nr'P = « + Nr'^'Pp, 
Dem. 

h . *264-36 . D 1- : Hp . D . Pp, = (£'Pp,) i (-B'Pp,) . 
[*] 62-3.*264-3913] -:>.P = B'P^.^B'P^^ . 
[*264-36-401] D . Nr'P = « + jS'Pp, : D h . Prop 

*264-421. h: Pen. 5'P6C"V'P.V'Pe2r.D.Nr'P= 0)4-1 
Dem. 

h . *264-36 . 3 : Hp . D . Pp, = (B'P^,) i {B'P^) . 
[*162-3.*264-39113] "^.Pl D'P = B'P^.^B'P^, 
[*264-403.*160-21] = B'P^, . 

[*264-401] D.PtD'Peo). 

[*204-461 J D . P 6 ft) + 1 : D h . Prop 

*264-422. h : P 6 fl infia - w . 5'P ~ e C" V'P . V'P ~ e 2, . D . 

Nr'P = {Nr'(V'P)t (D'V'P) x «} -i-Nr'P'Pp, 
Dem. 

V . *264-36 . *204-272 . D h : Hp . D . D'Pp, ~ e 1 . 
[*204-461.*264-24-36] 3 ■ Ppr = Ppr D D'Pp, -f> 5'Pp, . 

[*162-43.*264-39] D . P = S'(Pp, t D'Pp,) 4^fi'Ppr (1) 

h . *264-36-401 . *251-63 . D 

I- : Hp . D . Nr'S'(Pp, t D'^pr) = Nr'(V'P) C (D'V'P) x o, (2) 

h . (1) . (2) . *264-36 . D h . Prop 

*264-423. h : P 6 fl . 5'P 6 C" V'P . y 'P ~ e 2^ . D . 

Nr'P = {Nr'(V'P)t: (D'V'P) x ft)} + 1 
i)em. 

As in *264-422, 

h : Hp . D . Pp, = Pp, t D'Pp, -t» 5'Pp, . 

[*162-43.*264-391] D . P^ D'P = 2'(Pp,p D'Pp,)4i5'Pp, 

[*264-403] =S'(Pp,t;D'Pp,) (i) 

(- . *264-36-401 . *251-63 . D 

f : Hp . D . Nr'2'(Pp, [. D'Pp,) = Nr'(V'P) ^ (D' V'P) x lo (2) 

I- . *204-461 . 3 H : Hp . 3 . Nr'P = Nr'(P ^ D'P) + 1 (3) 

l-.a).(2).(3).DI-.Prop 



SECTION E] derivatives OF WELL-ORDERED SERIES 167 

*264-429. lxa = a Df 

This definition is merely intended to enable us to include 1 with ordinals 
in general formulae. 

*264-44. l-:P6n.D.(aa,^).a6N0wt'l.^eN0finui'I.Nr'P=(ax») + ;8 
Dem. 

V . *1 60-22 . *166-13 . D I- : P e fi fin . D . Nr'P = (0^ >(oy) + Nr'P (1) 

h.*160-21. Dh:P=a>.D.Nr'P = (i>(»)4-0, (2) 

F.*264-41.*160-21.D 

I- : Pefl infin - » . ~ E ! 5'P . D . (ga) . aeNO . Nr'P = (a x to) 4- 0, (3) 

h . *264-42-402 . D 

f-:P6ni5'P~6C"V'P.V'Pe2,.D.(a/3)./8eNOfin.Nr'P=(i>(«)-i-/8 (4) 
l-.*264-421. Dh:P6n.5'P6C"V'P.V'P62^.D.Nr'P = (l>C<») + i (5) 
t- . *264-422-402 . D |-:Pe flinfin - « . 5'P~eO'V'P . V'P~e2^. 3. 

(aa,^).a6NO.;3 6NOfin.Nr'P = (aXffl)-i-/3 (6) 
l-.*264-423.DI-:P6n.5'P6C'V'P.V'P~6 2^.D. 

(a«).a6N0.Nr'P = (ax«)-i-l (7) 
1- . (1) . (2) . (3) . (4) . (5) . (6) . (7) . D h . Prop 

The following propositions apply the above results to the cardinal number 
of the field of a well-ordered series, 

*264-45. I- : P 6 n . V'P 6 2^ . D . Nc'(7'P = N„ 
Dem. 

I- . *26442-402 . *180-7l . *152-7 . D 

I- : Hp . 5'P ~ 6 G'V'P . D . (g/*) .fie VIC induct . Nc'C'P = C'o) +„ fi . 
[*263101.*1 23-41] D . Nc'C'P = N„ (1) 

f- . *264-421 . *181-62 . 3 h : Hp . 5'P e C" V'P . D . Nc'C'P = G"co +^ 1 
[*263-101.*123-4] =N„ (2) 

f- . (1) . (2) . D h . Prop 

*264-451. h : P 6 n infin - « . ~ E ! 5'P . D . Nc'CP = Nc'C'V'P x„ N„ 
Dem. 

h . si5264-41 . *184-5 . D h : Hp . D . Nc'O'P = Nc'C V'P x„ 0"« 
[*263-101] = Ne'C" V'P x^ K„ : D I- . Prop 

*264-452. h : P e n infin - « . V 'P ~ e 2^ . 5'P ~ e C" V'P . D . 

Nc'a'P = Nc'D'V'PXeNo 
Dem. 

h . *264-422 . *184-5 . *180-71 . D 

h : Hp . D . (a/.) . /* 6 NO induct . Nc'O'P = (Nc'D'V'P x„ N„) -!-„ /^ (1) 



168 SERIES [part V 

h . *123-43 . *117-62 . D h : Hp . /* e NO induct . D . /^ < Nc'D'V'P Xe N„ . 
[«117-561] D . (Nc'D'V'P x„ N„)+e/i < (Nc'D' V'P x, N„) +, (Nc'D'V'P x, N„) 
[*123-421.*113-43] < Nc'D'V'P x^ K„ (2) 

h . (1) . (2) . *ll7-6-25 . D t- : Hp . 3 . Nc'CP = Nc'D'V'P Xe N„ : D h . Prop 

*264-453. h:P6nmfin-«.E!B'P.V'P~6 2,. D.Nc'0'P = Nc'D'V'P XeN„ 
Dem. 
As in *264-452, 

h . *264-423 . D h : Hp . £'P e C" V'P . D . Nc'CP = Nc'D'V'P Xe K (1) 
h . (1) . *264-452 . D H . Prop 

*264-46. h : P e n infin - 6, . D . Nc'O'P = Nc'O'V'P x„ No 

Dem. 
F . *123-421 . *264-45 . D h : Hp . V'P e 2^ . D . Nc'CP = Nc'C'V'P x^ii„ (1) 
h . *264-453 . D 

h : Hp . E ! B'P . V'P ~ e 2^ . Nc'C" V'P = /i +„ 1 . D . Nc'O'P = /i, Xe K„ 
[*123-421.*113-43] = (fi x„ N„) +« (/i x^ N„) (2) 

h.*ll7-571-6.D 

h:Hp.D./tiX„N„<(/i+„l)x„N„.(/i+„l)x„N„<(/xx„N„)+,(/iX„N„) (3) 
h . (2) . (3) . D h : Hp . D . Nc'O'P = (/^ +„ 1) x„ No 

[Hp] =Nc'C'V'Px„N„ (4) 

f- . (1) . (4) . *264-451 . D h . Prop 

*264-47. h:Peninfin.D.(aM).y^6NC-i'O.Nc'0'P = /iX<,No [*264-46] 
t-48. h : « 6 0"n - Cls induct . 3 . Nc'a e D'( x„ N„) [*264-47] 



*265. THE SERIES OF ALEPHS. 

Summary of *265. 

In the present number, we shall confine ourselves to the most elementary 
properties of the ordinals and cardinals considered. The most important 
propositions to be proved are the existence-theorems. These all depend 
upon the axiom of infinity; moreover, as the numbers concerned grow 
greater, the existence-theorems require continually higher types. 

In virtue of the definition in *262, (Ko),. is the class of well-ordered series 
whose fields have No terms. This is not an ordinal number, but the logical 
sum of a certain class of ordinal numbers, namely of Nr"(No)r. 

a>i is the smallest ordinal whose field has more than No terms. We do 
not, however, take this as the definition of eoi : we define eoi as the class of 
relations P siich that the relations less than P (in the sense of *254) are 
those well-ordered series which are finite or have No terms in their fields, i.e. 

ft), = P {l^s'P = (No), u n fin} Df. 

By *2.54'401 it follows immediately that if Peoji, P is a well-ordered 
series and lUi is its ordinal number (*265"11). Hence lOi is an ordinal number 
(*265'12), though we need the axiom of infinity to show that o)i exists. 

Assuming the axiom of infinity, the existence-theorem for (o^ is derived 
from the series of ordinals which are finite or belong to series of No terms. 
For notational convenience, we temporarily define this series as N; thus 
N= «) I {NO fin u Nr"(NoW Dft [*265]. 
It is also convenient temporarily to write M for '' <• " : thus 

Jf=< Dft[*265]. 

It is easy to prove that if No exists, N is an ojj (*265'25). Hence we 
obtain the existence-theorem for ai in either of the forms: 

*265-27. h : a ! No n i'a . D . a ! ft), n «"'<„„'o 
*265-28. V : Infin ax (a?) . D . g ! o»i « t^H^'x 

It is easy to prove that oii is greater than the ordinal number of any 
series of No terms (*265'3), and that if coi exists, 

iif'ft), = NO fin u Nr"(No)r (*265-35), 

i.e. the ordinals less than o)i are those that apply to series of No terms or of 
a finite number of terms. 



170 SERIES [part V 

We define Ni as 0"(o^, i.e. as the class of those classes which can be 
arranged in a series whose ordinal number is Wi. It follows from *152'71 
that Ni so defined is a cardinal number (*265'33), and that if N,, exists, 
Ni > No (*265-34). 

In a precisely analogous fashion we can put 

ft), = P {less'P = (Ni)r « (No)r ^ O fin} Df. 

K = G"m, Df. 

Theorems similar to those mentioned above can be proved for a^ and Kj 
by similar methods. We can proceed to to^ and N„, where v is any ordinal 
number. But our methods of proving existence-theorems fail if v is not 
finite, since at each stage the existence-theorem is proved in a higher type 
and we know of no meaning that can be assigned to types whose order 
is not finite. 

It is easy to prove that the sum of two ordinals which are less than a)i is 
less than m^. Much of the accepted theory of (No), and ft>i depends upon the 
proposition that the limit of any progression of ordinals less than Wi is less 
than o)i, so that in the series N, every progression has a limit within the series. 
This proposition — or at any rate the current proof of it — depends upon the 
multiplicative axiom. The proof, in outline, is as follows : 

It is easy to prove that an ordinal which has No predecessors must be 
a member of Nr"(No)r, i.e. must be, in Cantor's language, an ordinal of the 
second class. Now consider any progression P contained in N, i.e. consider 
a series Hj, ffj, . . . a,,... of increasing ordinals of the second class. The interval 
between any two consecutive terms of this series is either finite or has No 
terms. Hence N"G'P, i.e. the class of ordinals preceding the limit of our 
series, is the sum of No classes each of which is finite or has No terms. It is 
then argued that, because No XoNo = No, the whole class N"G'P must consist 
of No terms. This conclusion, however, except in special cases, requires the 
multiplicative axiom, since it depends upon *113"32, i.e. 

h ;. Mult ax . 3 : /t, J/ e NO . KevnCl excVfi .D .s'xe fix^v. 

It follows that, unless for those who regard the multiplicative axiom as 
certain, it cannot be regarded as proved that Mi is not the limit of a pro- 
gression of smaller ordinals. With this, much of the recognized theory of 
ordinals of the second class becomes doubtful. For example, Cantor pro- 
ceeds to define a host of ordinals of the second class as the limits of given 
series of such ordinals. It is probable that, in regard to all the ordinals which 
he has defined in this way, a proof that they belong to the second class can 
be found, by actually arranging the finite integers in a series of the specified 
type. But the mere fact that they are limits of progressions of nunjbers of 



SECTION E] the series OF ALEPHS 171 

the second class does not, of itself, suffice to prove that they are of the second 
class. • 

As another example we may mention the very interesting work of 
Hausdorff*, much of which is based upon the proposition that a term which 
IS the limit of an toj chosen out of a given series cannot be the limit of an 
ft) chosen out of the same series. This proposition is a consequence of the 
proposition that o)i is not the limit of a progression of smaller ordinals, and 
must therefore be regarded as doubtful. Hausdorff constructs by means of 
it many remarkable series, for example, compact series in which no pro- 
gression or regression has a limit. The existence of such series appears, 
however, to be open to question, unless the multiplicative axiom is assumed. 

It is not improbable that a proof, independent of the multiplicative axiom, 
can be found for the proposition that eo^ is not the limit of a progression ; but 
until such a proof is forthcoming, the proposition cannot be regarded as 
certain. 



*26501. ft)i = P {less'P = (N„)^ u fi fin} Df 

*26502. {<x = (?"ft>i Df 

*265-03. ft)2 = P {l^s'P = (Ni)r ^ (No)r ^ fi fin} Df 
*26504. N2 = 0"«B, Df 

etc. 
*26505. M=< Dft[*265] 

This definition is revived from *256. 
*265-06. iV^=JlfnNOfinwNr"(K„)^} Dft [*265] 

The existence-theorem for (Oi is derived from N, since, if Xo exists, Neoii. 
*2651. 1- : . P 6 ft), . = : Q less P . =q . Q e £1 . C'Q e Cls induct u N„ 
[(*26501)] 

*26511. h:Peo),.D.ft)i = Nr'P.P6ll 

Dem. 
h . *265-l . D h : Hp . D . A less P - 

[*2541] :^.P6a (1) 

h . *2o4-401 . (1) . (*265-01) . D h : Hp . Q e ft), . D . Q smor P (2) 

h . *254-401 . (1) . (*265-01) . D h : Hp . Q smor P.:). less'Q = (NoV w fi fin . 
[(*265-01)] D.^eft), (3) 

I- . (1) . (2) . (3) . D h . Prop 

* Vntersuchungen ilber Ordmtngstypen. Berichte der mathematisch-pbysischen Klasse der 
Eoniglich Sachsisehen Gesellschaft der Wissenachaften zu Leipzig, Feb. 1906 and f eb. 1907. 



172 SERIES [part V 

*26512. h . «, 6 NO [*265-ll . *256-54] 

*26513. l-:aeNOinfin.D.Jlf^Jl/'aea 
Dem. 
V . *256-202 . D h : P e n infin . D . Nr'JIf t (M'Nr'P) = Nr'(P p Q'P) 
[*262112] =Nr'P (1) 

I- . (1) . *262-ll . D h . Prop 

*265-2. l-.a'i\r=NOfin-i'0,wNr"(K„V = i^'0, [*255-51] 

*265-21. I- : a ! K„ . a 6 NO fin u Nr"(N„X ■ ^ ■ 

Jf t 'M'a less iV" . aJf (Nr'i^) . a C \^'N 
Dem. 

V . *253-13 . *265-2 . D h : Hp . a e NO fin u Nr"(No), .O.M^'M'ae T>'F, . 

[*254-182] -^.MlM'a less N (1) 

I- . (1) . *265-13 . Df-:Hp.«eNr"(N„),.D.aJlf(Nr'iV) (2) 

h . (2) . *263-31-101 . D h : Hp . aeNO fin . D . aMa, . ioM(Nt'N) . 

[*256-l] D . ailf (Nr'iV") (3) 

l-.(2).(3). Dh:Hp.aeNOfinuNr"(N„X.D.aM(Nr'i^. (4) 

[*255-17] D.aCl^'iV (5) 

t- . (1) . (4) . (5) . D h . Prop 

*265-22. l-:a!N„.D.nfinu(N„),C1^8'JV" [*265-21] 

*265-23. h:P6D'i\^s.3.(aa).a6N0finuNr"(N„),.P=JlfpM'a.Nr'P = a 
[*265-2 . *25313 . *26513 . *262-7 . *120-429] 

*26524. h:P6D'i\rs.D.P6f2finu(N„), [*265-23] 

*265-25. l-:a!N„.D.iVe«i 
Dem. 

V . *254-4112 . D h : PlessiV. D . (gQ) . Q e D'N, . P smor Q . 
[*265-24.*261-18.*151-18] D .P ellfin w(N„X (1) 

I- . (1) . *265-22 . D I- : Hp . D . \^s'N=n fin u (N„), . 
[*265-l] D . iV 6 0)1 : D h . Prop 

*265-26. 1- : a 6 No . D . N„r;(less ^ C^'d'a) e m^ . N„r;(less t G"Cl'a) = N 

Dem. 
h.*254-431.*150'37.D 

I- . N„r;(less t G"Cl'a) = (N„r;iess) p N„r"(0 n C"Cl'oi) (1) 

I- . *123-16 . D h : a 6 No . D . N„r"(n n 0"Cl'a) C NO fin u N„r"(No)^ (2) 



SECTION E] the series OF ALEPHS 173 

I- . *12314 . *262-18-21 . D h : a e «„ ■ /* e NO induct - t'l . 3 . g ! /it, n C"CVa : 
[*262-25] D*:a6N„.i;eNOf3n.D.a!i/nC"Cl'a. 

[*152-45] ■^.pe Nor"C"Cl'a (3) 

l-.*152-7.Dh:Pe(N„X.a6N„.D,aeO"N„r'P. 

[*60-34.] D . Nr'P e N„r"0"Cl'a (4) 

I- . (3) . (4) . D h : a 6 No . D . NO 6n u Nr"(N„), C N„r"(G"Cl'a n fl) (5) 

I- . (2) . (5) . D h : o 6 N„ . D . NO fin w Nr"(No), = N„r"(0"01'a n O) (6) 

I- . (1) . (6) . (*255-01 . *265-05-06) . D h : o e N„ . D . N„r;(less ^ a"Cl'a) = iV . 
[*265-25] D . N„r ;(less l G"Cl'a) e ajj : D I- . Prop 

*265-27. h : a ! No n i'a . D . a ! a, n «"%o'a 
i)em. 

l-.*64-55.DI-:/S6«'a.G'PC/3.D.P6C« (1) 

l-.(l). 0\-:/3et'a.0.G"C['^Ct^'a. 

[*1 55-12.*63-5] D . Nor"C'"01'y8 C t't^'a . 

[*64-57] 3.N„r;(less^a"Cl'j8)e«"'i„o'a (2) 

h . (2) . *265-26 . 3 h . Prop 

*265-28. h : Infin ax («) . D . g ! wi a f^'t^'x 

Dem. 

V . *123-37 . D h : Hp . D . a ! No « «'«"« . 
[*265-27] D . a I «i n t'^'t^H^'x . 

[*64'312] D . a ! Ml rt ^"'^'''a! : D h . Prop 

Propositions concerning N^ and m^, and generally N„ and «»„, where v is 
an inductive cardinal, are proved precisely as the above propositions are 
proved. There is not, however, so far as we know, any proof of the existence 
of Alephs and Omegas with infinite suflfixes, owing to the fact that the type 
increases with each successive existence-theorem, and that infinite types 
appear to be meaningless. 

*265-3. h : a 6 Nr"(No), .li.a^ciwi [*265-22-25] 

*265-31. l-:a!No-3-Ni>Ko 

Dem. 

h . *26o-25 . D h : Hp . D . C'iVe Ni (1) 

l-.*265-2. Dh.NOfin-t'O^CG'iV (2) 

I- . *262-19-21 . *12327 . D h : Hp . D . NO fin - 1% e N„ (3) 

|-.(2).(3). Dh:Hp.D.Nc'(7'iV>N„ (4) 

h . (1) . (4) . D f- . Prop 



174 SERIES [part V 

*265-32. t-:a!No.D.N„ + Ni.N„nNi = A 
Bern. 

l-.*265-3.Dt-:P6a.O'P6N„.D.P~6ft),. 

[(*265-02)] D.C"P~eN, (1) 

I- . (1) . *26218 . (*265'02) .Df-.N„nN, = A.DI-. Prop 

*265-33. h . Nj 6 NC [*152-71 . *265-12] 

*265-34. h : a ! No . D . ^e, > No [*265-31-32-33 . *255-74] 

*265-35. h : a ! 0,1 . D . M'w, = NO fin u Nr"(No), 
Dem. 

V . *265-3 . *263-31 . D h : Hp . D . NO fiu u Nr"(No)r C M'<o, (1) 

V . *265-ll . D h : P e wi . Nr'Q e M'co^ . D . Q less P . 

[*265-l] D.Nr'Q6N0finwNr"(N„), (2) 

h . (1) . (2) . D h . Prop 

*265-351. h : P e Ml . = . a ! tB, . Nr"D'Ps = NO fin u Nr"(No)r 
Dem. 
l-.*256-ll.*265-35.D 

h : a ! 0,1 . Nr"D'Ps = NO fin w Nr"(N„), . = . a '■ «i • M'^'P = M'w, . 
[*2561.*204-34] s .Pea,,: D h . Prop 

*265-352. l-:P6o,i.D.Nr"D'Ps = if'ftJi [*265-35-351] 

*265-36. \-:a,0e Nr"(N„V . D . a + /8 e Nr"(N„V 
Dem. 

I- . *1 80-71 . D h : Hp . D . C"(a + y3) = C"a +„ 0"/3 

[*262-i2] =^^o+o^«o 

[*123-421] = No . 

[*262-12] ^.a + ^e Nr"(i*o)r : ^ I" ■ Prop 

*265-361. h . «, /3 e NO fin w Nr"(No)r . 3 . a + /3 e NO fin u Nr"(N„)r 
[Proof as in *265-36, using *120-45 and *123-41] 

*265-4. h : P 6 0,1 . a C (7'P . P^"a e 01s induct w N„ . D . a lp'P''a 
Dem. 

h . *2651 . D h : Hp . D . (P t P*"a) less P . 

[*254-51] O.P^"a=^G'F. 

[*202-504] D . a lp'P''a Oh. Prop 

*265-401. h : P 6 0,1 . a C a'P . P"a e Cls induct w N„ . 3 . a ! p'P"a 
Dem. 

h . *205131 . D h : Hp . D . P^"a = P"a \j maxp'a . 
[*205-3.*120-251.*123-4] D . P^"a e Cls induct w No . 
[*26o4] D.a!p'P"a:3f-.Prop 



SECTION E] the series OF ALEPHS 175 

*265-41. h : P 6 ffli . D ^"G'P C K„ w Cls induct . ^^"G'P C N„ u Cls induct 
Dem. • 

h . *254-l 82 . D h : . Hp . D : a; 6 a'P . D . (P D Ip'x) less P . 

[*265-l] D. ^a; 6 No u Cls induct (1) 

H . (1) . *120-251 . *123-4 . D h :.Hp . 3 : a; e C'P . D . "p^'x e N„ w Cls induct (2) 

1- . (1) . (2) . D 1- . Prop 

*265-42. I- : P e wi . D . Q'P C D'P 
Z)em. 

h . *265-4-41 . 3 h : Hp . a; e d'P . D . g ! fP"i'x . 
[*5301-31] D . a; 6 D'P : D h . Prop 

*265-43. h : P 6 <ui . ar e (7'P . D . P p P,„'a; e « . E ! Itp'P'^'a; 
Pem. 

1- . *264-2 . *265-42 . D h : Hp . D . ~ E ! maxp'Pf^'a; . (1) 

[*264-22] D . P t K'« e <o (2) 

h . (2) . *265-41 , *123-421 . D h : Hp . D . P"Kn'« e ^<o • 
[*265-401] D . a ! ^'P"^„'a; . 

[(1).*250123] D . E ! Itp'Kn'* (3) 

h . (2) . (3) . D h . Prop 

*265 431. V-.Pea^.QdP.xeG'Q.^'x C%^'x . D . g ! p'P"G'Q 
Dem. 

V . *265-43 . D h : Hp . D . C'Q C P'ltp'Pj/a; Oh. Prop 

*265-44. |-:P6«,.a;eC"P.D.PCP*'a;eo)i 

Pern. 
l-.*253-13.Dl-:Hp.D.D'(PtPj|e'a;)s=P{(ay).a;P*y.P = PtP(a;H-2/)j (1) 
h . *254-101 . D h : Hp . xP^y . D . Nr'P ^ P (« ^- y) ^ Nr'P I P'y . 
[*265-352] D . Nr'P lP(xt-y)e M'w, (2) 

h . *265-352 . D h : Hp . D . Nr'P [JP'x e M'to, (3) 

h . (3) . *265-361-35 . D 
1- : Hp .aeM'fOi . D . Nr'P ^P'a! + a e M'co, . 
[*265-351] D . (ay) . Nr'P lP'x + a = Nr'P I P'y . 

[*253-47-ll] D . (ay) . a'P*?/ • Nr'P ^ P'a; -i- a = Nr'P I P'y . 



176 SERIES [part V 

[*204-45] D . (ay) . xP^y . Nr'P p P'a; + o = Nr'P p P'a; + Nr'P ^ P (a; i- y) . 

[*2o5-564] D . (32/) . ^Pj^y . a = Nr'P ^ (a; i- y) • 

[(1)] D . a 6 Nr"D'(P t P*'«)» (4) 

1- . (2) . (4) . D h : Hp . D . Nr"D'(P IP^'x), = ^'«, . 

[*265-35-351] "^.P^ P^'x e <oi : D h . Prop 

*265-441. h : P e Ser . e, i? 6 « n Rl'P . ii G Q . 3 . 

P"a'i2 = P"G'Q . Q"G'B = C'Q 
Dem. 

h . *263-27 . Transp . 3 I- : Hp . D . ~ E ! ib&xq'G'B . 

[*205123] D . C'ii C Q"G'R . (1) 

[*37-2] D . P"G'i2 C P"Q"G'R 

[*37'15-2] CP"G'Q (2) 

I- • *263-47 . Transp . D h : Hp . D . p'Q"G'R = A . 

[(1).*202-51] O.G'Q= Q"G'R . (3) 

[*201-5.Hp] D.P"G'QCP"G'R. 

[(2)] D . P"G'i2 = P"G'Q (4) 

I- . (3) . (4) . D h . Prop 

*265-45. h :.Pea>i.QGP:«eC'Q.Da;.a!Q'«-i'fn'«:Qe«. 

S = S ^ {« 6 (7'Q . y = mmQ'(V« -X'«)} ■R = S [%'B'Q : D . 

R^em.R^CQ. P"C'R^ = P"0'Q 
Dem. 

f-.*32181.DI-:Hp.D.SGQ. (1) 

[*91-59.*201-18] D.B^GQ (2) 

h . *26311 . D 1- :. Up . D : «6 O'Q . D^ . E ! S'a; : 

[*71-571] D-.SeCls-^l.G'QCD'S: 

[(1)] D:5f6Cls-*l.a'/SfCD'/8f: 

[*122-51.*96-2l] D : P 6 Prog : 

[*2631] D : i?p„ 6 « (3) 

h . (2) . (3) . *265-441 . D h : Hp . D . P"G'R = P"G'Q (4) 

I- . (2) . (3) . (4) . D h . Prop 

*265-451. h :. Hp *265-45 . 3 : a; e C'P . D . P (« f- Ei'a;) e N„ 
Bern. 

h . *265-45 . *26314 . D h :. Hp . D : a; e O'iJ . D . Ri'x=8'x . 
[Hp] 3 . Ri'tB e P'a; - Pf^'x . 

[*260-131] D.P(a!t-i?i'a;)~eClsinduct (1) 

H . *265-41 . D h :. Hp . D : a; e O'P . D . P (a; i- ^'a;) e N„ w Cls induct (2) 

I- , (1) . (2) . D h . Prop 



SECTION E] the series OF ALEPHS 177 

*265-452. h : Hp *265-45 . w ! -P (» ►- -Ri'*) « -P (2/ >- -Bi'y) . D ■ « = 2/ 
Dem. 

V . *201-18 . D h : . Hp . D : a;P (^/y) . yP (P,'a;) : 

[*14-21] D:a),yeG'R.xP (E,'y) . yP (A'a;) : 

[*204-41.*265-45] 3 : «J?p„ (B^'y) . yR^„ (R^'x) : 

[*204'7l] 3 : a; = y . V . osR^oy •.y = x.v . yR^x : 

[*4-41] 3 : a; = y . V . a;22poy . yR^x : 

[*204-13.*265-45] D : « = y : . D h . Prop 

*265-453. h : Hp *265-45 . k = a {(ga;) . a; e O'i? . a = P (a; ^- ^x'a;)} . D . 

« e No n 01 excl'No ■ «'« = P"G'P « P*"G'R [*265-451-452] 

*265-454. h :. Hp *265-453 : « e No n 01 excl'No . 3« . «'« e «« : ^ ■ 

P"C'R n P^"G'R 6 N„ [*265-453] 

*265-46. l-:.Peffli.Qewn Rl'P zxeO'Q .D^.'^lQ'x- P,„'a; : 

K 6 No rt 01 excl'No . 3. . s'«: 6 No : 3 ■ P"Cf'Q e N„ 
[*265-41-454 . *123-421] 

*265-461. I- : Hp *265-46 . D . a ! p'P"0'Q [*265-46-401] 

*265-47. h : . P e eoi . Q e « n R1«P : « e N„ n 01 excl'No . D« . s'/e e N„ : 3 . 

a!^'P"a'Q [*265-461-431] 

*26548. l-:.«6Non 01 excl'No. D^.s'/eeNoOiPeoji.QewnRl'P. 3. E!ltp'Q 
[*265-47.*250123] 

*265481. h : Mult ax . 3 . Hp *265-48 [*118-32 . *123-52] 

*265-49. l-:.Multax.3:P6a),.Q6«nRl'P.3.E! VQ [*265-48-481] 

This proposition shows that, assuming the multiplicative axiom, any 
progression of ordinals of the second class {i.e. consisting of series having No 
terms) has a limit in the second class, because Necoi. 

*265-5. 1- :Peft>i . Qew . C'QC a'P . ~E ! maxp'G'Q . 

R = ^{xeG'Q.y = mmQ'{P'x a q"'«)} .S = R[ B^'B'Q . 3 . 

8^ew.S,o(^P. P"G'S,, = P"C'Q 
Bern. 

h.*20511. 3t-:Hp.3.EGP.PeQ. (1) 

[*20ri8] 3.^poGP./SpoeQ (2) 

I- . *205-197 . 3 h : Hp . a; e (7'Q . Q^'x C P^'x . 3 . a; = maxp'^^^'a; (3) 

h . *263-412 . *261-26 . 3 h : Hp . a; e O'Q . 3 . E ! maxp'Q'a; (4) 

E &W III. 12 



178 SERIES [part V 

I- . (3) . (4) . *205193 . D h : Hp . a; 6 O'Q . %'x C P^'a; . D . E ! inaxp'(7'Q (5) 
h . (5) . Transp . D h :. Hp . D : a;6 O'Q . D . g ! Q^'x-P^'x . 

[*91-542.*202103] 3 . g ^^'x n*P'x . 

[*250121] D . E ! E'a; (6) 

h . (1) . (6) . *122-51 . D h : Hp . D . (Sf 6 Prog . 

[*263-l] 3 . -Sfpo e o) (7) 

h . (2) . (7) . *265-441 . 3 h : Hp . D . P"C'S^^ = P"G'Q (8) 

h . (2) . (7) . (8) . D h . Prop 

*265-51. I- : Hp *265-48 . P e Wi . a e N„ n QVC'P . ~ E ! maxp'a . D . E ! Itp'a 
Dem. 

h . *265-5 . D h : Hp . D . (a,S) .Sewn Rl'P . P"C'S= a (1) 

h . (1) . *265-48 . D h . Prop 

The following propositions follow easily. 
*265-52. h:.Hp*265-48.P6Wi.D: 

o n CP e N„ w Cls induct . = . g ! (7'P n^'P"(« n (7'P) [*265-51-41] 

*265-53. t- :: Hp *265-48 . 3 :. P e «i . = : 

P e Xi : a ft G'P e No w Cls induct . =„ . g ! C'P n p'P^ '(« ft O'P) 

*265-54. I- : P 6 «i . D . a'V'P C ltp"0"(6j ft Rl'P) [*265'5] 

/.e. every limit-point in an w, is the limit of a progression, which is what 
(following Hausdorff) may be conveniently called an w-limit. 

*265-56. I- : P 6 «, . D . a' V'P = ltp"(7"(« ft Rl'P) [*265-54 . *216-602] 

This proposition does not, like *265'48, assert that every progression in 
P has a limit, and therefore it does not require the hypothesis of *26548. 



SECTION F. 

COMPACT SERIES, RATIONAL SERIES, AND CONTINUOUS SERIES. 

Summary of Section F. 

A compact series is one in which there is a term between any two, 
i.e. in which P G P", where P is the generating relation. We may call 
any relation P compact when P QP'; then a transitive compact relation 
will be one for which P = P". Hence a serial relation P is compact when- 
ever P = P^. Compact series in general have certain properties, some of 
which have been already proved ; but the majority of the interesting pro- 
positions in this subject come from adding some other condition besides 
compactness. Thus series having Dedekindian continuity, which have many 
important properties, are such as are compact and Dedekindian. Bational 
series (i.e. such as are ordinally similar to the series of all rational numbers, 
positive and negative, or, what is equivalent, to the series of rational proper 
fractions) are defined as such as are compact, without beginning or end, and 
consisting of ti^ terms. Such series, also, have many important properties. 
A continuous series (in Cantor's sense) is a Dedekindian series containing 
a rational series in such a way that there are terms of the rational series 
between any two terms of the given series. This species of compact series 
also has many important properties. It consists of all series ordinally similar 
to the series of real numbers including and oo . 



12—2 



*270. COMPACT SERIES. 

Summary of *270. 

The propositions of the present number are mostly either obvious or 
repetitions of previously proved propositions. The latter are repeated here 
for convenience of reference. 

We put comp = P (P G P^) Df, 

so that the class of compact series is Ser n comp. We have 

*27011. h :. P 6 comp . = : xPy . D^,^ . g ! P'« n P'y 

*270-34. h : P 6 trans n comp . 3 . s'P = sgm'P 

The proposition s'P* = sgm'Pj^ , which was proved in *212, is a particular 
case of the above. 

*270-41. J- : P 6 Ser n comp . D . Nr'P C Ser n comp 

I.e. a series which is similar to a compact series is a compact series. 

*270-56. 1- : P 6 Ser . Q 6 fi . ~ E ! 5'P . ~ E ! B'Q . D . P« e Ser n comp 

This proposition gives us a means of manufacturing compact series of 
various types, such as toexp^w, wexprCOi, etc. 



*27001. comp = P(PGP'') Df 

Here " comp " is an abbreviation for " compact." " Compact " series are 
the same as the series which Cantor calls " tiberall dicht." 



*2701. t- 

*27011. h 

*27012. V 

*27013. h 

*27014. h 



P 6 comp . = . P C P= [(*270-01)] 

. P 6 comp . = : xPy . 3^,,^ . g ! P'x n P'y [*270-l] 

Pecomp. = .Pecomp [*270-ll] 

P e trans r. comp . = .P = P^ [*2701 . *201-1] 



Pe Ser n comp . = . Pe Rl'/n connex . P=P^ . = . Pe Ser . P= P» 
[*270-13] 

*27015. h:PeSerncomp. = .PeSer.Pi = A [*201-66 . *270-14] 



SECTION F] compact SERIES 181 

*270-2. h : P e comp . D . ~ g ! m&Xp'P'x [*205-25 . *270-l] 

*270-201. h : P e comp . D . ~ g ! mmp'a'P . ~ g ! maxp'D'P 
Dem. 

H.*3r-25. DI-.imnp'a'P = P"D'P-(P2)"D'P (1) 
I- . (1) . *2701 . D h : Hp . D . minp'a'P = A (2) 

Similarly h : Hp . D . maxp'D'P = A (3) 

h . (2) . (3) . 3 h . Prop 

— > ^ — > 

*270-202. h : P 6 comp . D . ~ g ! minp'P"a . ~ a ! maxp'P"a 

[Proof as in *270-201] 
*270-203. V'.Pe comp . 3 . ~ g ! seqp't'a; [*206-42 . *270-l] 

*270-204. I- : P e Ser ft comp . E ! seqp'a . D . ~ E ! maxp'a 
[*206-451 . *270-15] 

*270-205. h : P e Ser n comp . D . Itp = seqp [*207-l . *270-204] 

*270-21. I- : P 6 Rl'/ n comp . a; e O'P . 3 . « Itp (P'x) [*207-31 . *270-l] 

*270-211. V\Pe Rl'Jn comp . D . D'ltp = G'P [*270-21] 

Thus if a relation is compact and contained in diversity, every member 
of its field is a limit-point. 

*270-212. h : P 6 connex . D'ltp = O'P . D . P e comp 
Dem. 

I- . *207-34 . D h : Hp . D . C7'P C - a\P- P') . 

[*33-251] D . a'(P- P«) = A . 

[*270-l] D . P 6 comp : D h . Prop 

*270-22. h : . Pe Rl'/n connex . D : Pe comp . = . D'ltp = C'P . = . Q'P C D'ltp 
[*270-211-212 . *207-18] 

*270-23. h-.Pe comp - I'A . D . P ~ e Bord 
Dem. 

I- . *270-201 . D h : Hp . D . (ga) . a C G'P . g ! a . ~ g ! minp'a . 
[*250101] D.P~ 6 Bord Oh. Prop 

*270-24. h : P 6 Ser A comp - t'A . D . G'P ~ e Cls induct 

Bern. 

I- .*270-23 .DI-:Hp.D.P~6n. 

[*261-31] D . O'P ~ e Cls induct Oh. Prop 

*270-3. h : P e Ser n comp . 3 . sect'P - D'Pe = P^"G'P 
[*211-351.*270-15] 



182 SERIES [part V 

*270-31. f-:P6 trans ncomp.D.D'Pe = D'(Pen/) [*211-51 . *27014] 

*270-32. h : P 6 trans n comp .'^.'P'ice B'{Pe n I) [*211-452 . *270-l] 

*270-321. H :'P"C'P C D'(Pe nI).-^.Pe comp [*211-451 . *2701] 

*270322. H :. P e trans . D : P"G'P C D'(Pe n /) . = . P e comp 
[*270-32-321] 

*270-33. h : . P e Ser . D : P 6 comp . = . Q'maxp n d'seqp = A 
[*211-551 . *270-14] 

*270-34. h : P e trans n comp . D . s'P = sgm'P [*270-31 . (*212-0102)] 

*270-35. h :. P 6 trans n connex n comp .0:Pe Ded . = . Q'maxp = - Q'seqp 
[*214-4.*27018] 

*270-351. F- : . P e Ser . D : P 6 comp a Ded . = . Q'maxp = - Q'seqp 
[*214-41.*270-14] 
A series which is compact and Dedekindian is one which has Dedekindian 
continuity. Thus the above proposition states that a series which has Dede- 
kindian continuity is a series such that every class has either a maximum or 
a sequent, but not both. 

*270-352. I- : Pe Ser n comp r\ Ded . a e sect'P. D . limaxp'a = liminp'(0'P - a) 
[*214-42] 

*270-36. h : P 6 Rl' / n comp . D . Sp'G'P = a'P.V'P = P 
[*216-2 . *270-211 . (*21605)] 

*270-4. h : P e comp . D . Nr'P C comp 
Dem. 
h.*201-2. D\-:SePi5IdiQ.D.(8>Qf = 8>Q\P = 8>Q (1) 

h .(1) .*2701 . D h : Pecomp . SfePsmor Q . D . fifJQG>Sf;Q». 

[*150-31] D.S'S'QQS'S'QK 

[*151-252] D . Q G Q" : 3 h . Prop 

*270-401. h'.Pe comp . = . Nor'P C comp [*270-4 . *155-12] 

*270-41. h : Pe Ser n comp. D. Nr'P C Ser n comp [*270-4 . *204-22] 

*270-411. I- : P 6 Ser n comp . = . Nor'P C Ser n comp [*270-41 . *15512] 

«— — > 

*270-42. \-:Pe comp . D . P ^ P^'tv, P I P^'x e comp 

Dem. 

\- . *270-ll . D h : Hp . 2/, 2^ 6 P^'m . yPz . D . (gw) . yPw . wPz . 

[*90-16] '^.(:sw).weP^'x.yPw.wPz (1) 

h . (1) . *270-ll . 3 h : Hp . D . P^P^'x e comp (2) 

— > 
Similarly h : Hp . D . P ^ P^'x e comp (3) 

f- . (2) . (3) . D h . Prop 



SECTION F] compact SERIES 183 

*270-5. h : P, Q e Ser A comp . C'P rt f7'Q = A . ~ (E ! 5'P . E ! B'Q) . D . 

• P4^QeSerncomp 

Dem. 

H . *1 60-51 . D h : Hp . D . (P4.Q)= = P» u Q** c; D'P t O'Q c; C'P f Q'Q 
[*93103.Hp] =P2c;Q»c;0'PtC"Q (1) 

h . (1) . *270-l . 3 I- : Hp . D . P^iQ G (P4.Q)'' (2) 

h.(2).*204.-5.DI-.Prop 

*270-51. h : P 6 Ser n comp . G'P C Ser r. comp . P e Rel" excl . D . 

S'P 6 Ser n comp 
Dem. 

h.*204-52.DI-:Hp.D.S'PeSer (1) 

, l-.*1621.D 

h . (S'P)' = (s'C'Pf a (PJP)'' u (s'O'P) I (P;P) c; (^JP) | (s'G'P) (2) 

I- . *2r0-l . D h : Hp . x(s'G'P) y.D. (gQ) .QeC'P. xQFy . 

[*41-13] ^.x{s'G'Pyy (3) 

I- . *270-l . D h : Hp . a; (PJP) y . D . a; (PiP^ y . 

[*163-12.*201-2] D . a; (PJP)^ y W 

h . (2) . (3) . (4) . *1621 . D h : Hp . D . S'P G (S'P)» (5) 

h . (1) . (5) . D h . Prop 
The hypothesis of *270'51 is in excess of what is required for the 
conclusion, which only requires, in place of Pecomp, that there should he 
no two consecutive relations in G'P of which the first has a last term while 
the second has a first term. This is proved in the following proposition. 

*270-52. I- : P 6 Ser A ReP excl . G'P C Ser n comp . 

B"P^"{G'P n Cnv"a'5) = A . D . S'P 6 Sern comp 
Dem. 

V . *2701 . *163-12 . D I- : Hp . D . s'G'P G {s'C'Pf (1) 

h . *201-63 . D h : Hp . D . F>P = PJP, c; PJP" (2) 

H . *93103 . D h :. Hp . QP^R . D : D'Q = O'Q . v . d'R = G'R (3) 

h . (3) . D h :. Hp . a; (PJPi) y . D : 

(aQ, R):xeI>'Q.yeG'R.v.x60'Q. yed'R : QP^R : 
[*3313-131-17] 

D : (ad-B,^) -.xQz.zeG'Q.yeC'R.v.xeG'Q.zeG'R. zRy : QPiiJ : 
[*i50-52.*201-63] D : x {{s'G'P) \ (PJP)} y . v . a; {(PJP) | (s'CP)} y : 
[*162-1] ■:i:x{t'Pyy (4) 

h .*1 63-12 .*201-2. D h : Hp . D . PJP^ = (P'P)'' (5) 

f- . (2) . (5) . *162-1 . D h : Hp . D . P;P G (2'P)" (6) 

I- . (1) . (6) . *162-1 . D h : Hp . D . S'P G {VPf (7) 

h . (4) . (7) . *204-52 . D h . Prop 



184 SERIES [PART V 

*270'521. I- :. P 6 Ser n ReP exel . O'P C Ser n comp : 
C'P n Cnv"a'B = A . V . C'P n a'5 = A : D . S'P 6 Ser A comp [*270-52] 

*270-53. l-iPeSer. Qe Ser n comp. ~(E!5'Q.E!S'Q).D.PxQe Ser n comp 
Dem. 
I-.*1661. :>V.PxQ=X'Qy>P (1) 

|-.*165-21.DI-.Q_4,;P6RePexcl. (2) 

h .*165-25.*204-21. D h : Hp . g ! P. D . QJ, JPeSer (3) 

h . *165-26 . *270-4 . 3 h : Hp . D . (7'Q J, JP C Ser n comp (4) 

h . *151-5 . *165-26 . D h : Hp . ~ E ! 5'Q . D . C'Q J, ;P o a'5 = A (5) 

I- .*151-5 .*165-26 . D h : Hp.~E!5'Q. D .G'Q_4,;PnCnv"a'J5=A (6) 

I- . (1) . (2) . (3) . (4) . (5) . (6) . *270-521 . D * 
h : Hp . a ! P . D . P X Q e Ser n comp (7) 

H . *16613 .DI-:P = A.D.PxQ6Sern comp (8) 

h . (7) . (8) . 3 h . Prop 

*270-54 h : Pe Ser n comp . ~ E ! B'P .xr^eG'P . D . P-f+aJeSer n comp 
Dem. 

h.*204-51. DI-:Hp.D.P-f*a!eSer (1) 

h.*16ri. Dh:Hp.D.(P4>a!)'' = P''wD'Pti'a; 
[*93-103] =P^Ki C'P t I'x (2) 

h . (2) . *2701 . D h : Hp . D . P+>a; G (P +>«)'' (3) 

h . (1) . (3) . 3 h , Prop 

*270-541. h : P e Ser n comp . ~ E ! B'P . a; ~ e C'P . D . a; «f P e Ser a comp 
[Proof as in *270-54] 

*270-55. h : P 6 n . C'P C Ser . ~ E ! 5'P . O'P n Cnv"<l'B = A . D . 

II'P e Ser a comp 
Dem. 
h.*251-3.DI-:Hp,D.n'P6Ser (1) 

h . *250-21 . *93-103 . D 

1- : Hp . Q e C'P . ilf 6 Pa'C'P . D . (ga;) . {M'P^'Q) (P^'Q) x (2) 

h . *200-43 . D 

Vi'K^{^).{M'K'Q){h'Q)iio.L=M\{-i'P,'Q)Kix^{P,'Q).-D.M{li'P)L (3) 
I- . *200-43 . D 

I- : Hp(3) . Ne F^'G'P. (M'Q) Q(N'Q) . MfP'Q = N\-'P'Q . D . i(n'P) A^ (4) 
|-.(2).(3).(4).D _^ 

h : Hp . M,NeF^'C'P .QeC'P. (M'Q) Q (N'Q) . M['p'Q = N['P'Q . D . 

(^L).M{U'P)L.LiU'P)N (5) 
h . (5) . *200-43 . D I- : Hp . 3 . H'P G (U'Py (6) 

I- . (1) . (6) . D h. Prop 



SECTION f] compact SERIES 185 

*270-66. h:P6Ser.QeXl.~E!5'P.~E!£'Q.D.P«6Serrtcomp 

Bern. • 

I-.*176-151. Dh:P = A.D.P«eSerncomp (1) 

h.*176-181-182. Df-.P«smorn'PJ,;Q (2) 

h . *165-25 . *251121 . D h : Hp . g ! P . D .'p J, JQ e « (3) 

h . *165-26 . *204-21 . Dh : Hp. 3. CPj^JQCSer (4) 

h.*165-25.*151-5. D h : Hp.g !P. D .~E!P'Cnv'P J,;Q (5) 

h.*165-26.*151-5. D f- : Hp. D . O'P J, JQ n C!nv"a'P = A (6) 

H . (3) . (4) . (5) . (6) . *270-55 . 3 h : Hp . g ! P . D . H'P J, JQ e Ser a comp . 
[(2).*270-41] 3 . P« € Ser n comp (7) 

h.(l).(7).Dh.-Prop 

By means of the above proposition, compact series can be manufactured by 
taking series of such types as a exp^ to, to exp^ Wj, a>i exprO>, etc. Any power 
o exp, ^ consists of compact series, if y8 is an ordinal having no immediate 
predecessor, and a is any serial number having no immediate predecessor 
(i.e. not formed by adding i to a serial number). 



*271. MEDIAN CLASSES IN SERIES. 

Summary of *27l. 

We shall call a class, a a " median " class in P if a C G'P and there is a 
member of a between any two terms of which one has the relation P to the 
other. When this is the case, we have 

xPy . y„^y . (g^) .zed. xPz . zPy, 
i.e. PGPI'alP. 

Thus P cannot contain any median class unless P is compact. Conversely, 

if P is compact, G'P is a median class. Hence relations containing median 

classes are the same as compact relations. Median classes are important in 

dealing with rational and continuous series : the rationals are a median class 

in the series of real numbers, and the series which Cantor calls continuous 

are characterized by the fact that, in addition to being Dedekindian, they 

contain a median class which forms a series of the same type as the rationals. 

— » _ _ 

If P is a compact series, the class P"Q.'P is a median class in the series s'P 

(*271'31). This fact is used in proving that the series of segments of a 

rational series is a continuous series. 

Our definition is 

med = aP(aCG'P.PeP|^a|P) Df. 
— » 
Thus med'P will be the median classes of P, and " Pe Q'med " means that 

there are median classes of P. We have Q'med = comp (*271"18); also 

*27115. h:amedP.D.P,Ppa6comp 

*271-16. I- : (a n G'P) med P . = . (a n D'P) med P . = . (a n Q'P) med P . 

= .(anD'Pna'P)medP 
If P is a series, and a C G'P, a is a median class when, and only when, its 
derivative is d'P, i.e. 

*271-2. hz.PeSer.aCO'P. D:amedP. = .a'P = V« 
An important proposition is 

*271-39. h:P,Q6SerADed.amedP./3medQ . (P I a)smor (Q t^).:^. 

PsraorQ 

I.e. if P and Q are Dedekindian series, and a, ^ are median classes of P 

and Q respectively, then if P ^ a and Q\,^ are similar, so are P and Q. This 



SECTION F] median CLASSES IN SERIES 187 

* 
proposition is proved by showing that P is similar to the series of segments 
of P ^ «, the correlator beiug Itp with its converse domain limited (*27l'37). 
Another important proposition is 

*271-4. h:SePsmofQ.iSmedQ.D.(S'"/8)medP 

I.e. a correlator of P with Q correlates median classes with median 



The above two propositions are used in *275"3'31, which prove that two 
series which are continuous (in Cantor's sense) are similar, and that a series 
similar to a continuous series is continuous. 



*27101. med = aP(aCC"P.PCPfa|P) Df 
*2711. Vi.oLmediP . = '.olCC'P .P(lP\a\P: = \ 

aCG'P: xPy . D„,j, . g ! a n P'x r. P'y [(*27l-0l)] 
*27111. f-:amedP. = .amedP [*27ll] 

*27113. l-:amedP.;8C0'P.D.(aui8)medP [*27l-l] 

*271-14. h : a med P . D . C7'P t a med (P t «) 
Dem. 

h.*271-l.D 

h :. a med P . D : «, 3/ e a . xPy . "^x.y • (a^s') • ^ e a . aiPz . zPy . 
[*35-102] '^^^y.{'^z).zea.x{P^a)z.z{P^a.)yi 

[*35-102.*27l-l] D : C'P ^ a med (P t a) :. D h . Prop 

*27115. l-:amedP.D.P,PPaecomp 
Dem. 

h.*27ri. Dh:Hp.D.PGP». 

[*270-l] :i.Pe comp (1) 

h . (1) .*271-14 . D h : Hp . D . P ^ oecomp (2) 

h . (1) . (2) . D h . Prop 

*27116. I- : (o n C'P) med P . = . (a n D'P) med P . = . (a n O'P) med P - 

= .(anD'Pna'P)medP 
Dem. 
h.*27l-l.*33-15.D 

I- :. (o n G'P) med P . = : a!P2/ . D^.y . a ! a rt D'P n P'x n P'y : 

i*27l-l] =:(anD'P)medP (1) 

h . *2711 . *33151 . D h : (a n C'P) medP . = . (a n Q'P) med P (2) 

I- . *271-1 . *33-15'151 . D 

h : . (a n C'P) med P . = : a;P?/ . D«, ^ . g ! a n D *P ft Q'P n P 'a; n P'y : 

[*27l-l] =:(aftD'Pfta'P)medP (3) 

h . (1) . (2) . (3) . D h . Prop 



188 SERIES [PART V 

*27117. I- : P e comp . D . G'P, D'P,<['P e i^d'P 
Dem. 

h . *35-452 . *270-l . D h : Pecomp . D . PQ.P\G.'P \ P . 

[*271-1] D . Q'P 6 med'P . (1) 

[*271-13] O.C'Pe i^d'P . (2) 

[*27116] D . D'P e i^d'P (3) 

f- . (1) . (2) . (3) . D h . Prop 

*27118. y . Q'med = comp [*2711517] 

*27r2. h:.PeSer.aC(7'P.D:amedP. = .a'P = Sp'o [*216-13.*271-1] 

*271-3. hiPe m'J n trans . a med P . D . P"a med (s'P) 

Dew*, 
h . *271-15 . *270-34 . 3 h : Hp . 3 . s'P = sgm'P . 

[*212-11] D.s'P = ^9{/3,76D'(PeA7).a!7-/8} (1) 

h . (1) . *211-12 . D h : Hp . /3 (s'P) 7 . D . g ! 7 - /3 . P"7 = 7 . P"/3 = /3 . 
[*37-l] '2.(^a;,y).xey-^.a!Py.yey. 

[*27l"l] D . (ga;, y,z).x6y-^ . xPz . zPy .zea.yey. 

[*20112] D . (gas, y,z).xey-^. xPz . zPi/ . e a . y e 7 . ~ (yP^) - 

[*32-18] D , (a^) . ^ea . a ! P'z-^. 'g^ly-P'z. 

[(l),*270-322] D.(a0).^€a.yS(s'P)(P'«).(P'^)(s'P)7 (2) 

h.(2).*27l-1.3l-.Prop 

*271-31. f- : P 6 Rl' J rt trans n comp . 3 . P"a'P med (s'P) [*271-3-l7] 

The following propositions lead up to the proposition 
*271-37. h : P e Ser n Ded . a med P . D . Itp r C"s'(P ta)eP sSor {s'(P ta)] 

whence, if a is a median class of P, P is similar to the series of segments of 
P^a. This proposition is used in proving that every continuous series is 
similar to the series of segments of a rational series. 

*271-32. h : P 6 Ser . P = P C a . iS 6 D'Pe . E I Itp'/S . D . ^=R"fi=a n'P'W^ 

Dem. 

— » -♦ 

V . *205-9 . 3 I- : Hp . a rt G'P ~ e 1 . 3 . maxjj'/3 = maxp (o n /3) 

[*37-413.*21111] = maxp'/3 

[*207-13] = A (1) 

h . (1) . *200-35 . D I- : Hp . 3 . r^xjj'jg = A . 

[*211-42-12] 3 . /8 = P"j8 (2) 

h . *207 -231 . 3 h : Hp . 3 . P"^ = P'ltj.'/3 . 

[*37-413] 3.P"(8 = aAP'lt/^ (3) 

h . (2) . (3) . 3 h . Prop 



SECTION F] MEDIAN CLASSES IN SERIES 189 

*271-321. h : Pe Ser . E = P^ a . D . Itpl^D'iJeel-^l 
D&n. • 

f- . *271-32 . D h : Hp . /8, 7 e D'Ee . 1 V/8 = ltp'7 . D . /3 = 7 : D I- . Prop 

*271322. h : P e Ser . iJ = P p a . D . ItpJs'iZ C P 

Dem. 
f- . *212-23 . D h :. Hp . D : x(\tph'R) y. = . 

(a/S,7).A7eD'Pe./8C7./8 + 7.a; = lV/3.y = ltp'7- 
[*207 •231] D . (3/8, 7) . |S, 7 e D'Ee . ^ C 7 . ^8 + 7 . P'a; = P";8 . "P'y = P"7 . 
r*37-2.*2ri-321] D . P'« C P'2/ . a; + y . 
[*204-33] li.xPyi.'^V. Prop 

*271-33. I- : P e trans . a med P .':i.P'x = P'\ol n P'a;) 
i)em. 

V . *201501 . D F : Hp . D . P"'p'xZP'x . 

[*37-2] D.P"(anP'a;)CP'a! (1) 

I- . *27l-l , D H :. Hp . D : yPx . D . (g^) . yPz .zea. zPx . 

[*37-l] D.ye P"(a n P'a;) (2) 

h . (1) . (2) . 3 h . Prop 

*271 331. h : Hp *27l-33 . R = Pla .'D .an'P'x = R'\a a P'x) 
Dem. 

h . *271-33 . 3 h : Hp . 3 . o n P'aj = a n P"(o n P'a;) 

[*37-413] = R"(a n P'a;) : D h . Prop 

*271-332. h : P e Ser . a med P.xeC'P .O.x = ltp'(a n P'x) 
Dem. 

I- . *271-331 . D h : Hp . D . a n P'a; C P"(a n P'x) . 

[*205-123] D . maxp'(a n P'a;) = A (1) 

h.(l).*271-33.D 

h : Hp . D . a; e (7'P . P'a; = P"(a n P'a;) . ~ E ! maxp'(a r« P'a;) . 

[*207-521] 0.x = ltp'(« n P'a;) : 3 I- . Prop 

*271-34. h : P e Ser . a med P . 3 . P = ltp;s'(P I a) 
Dem. 

h . *271-331 . *211-11 .3l-:Hp.i2 = Pta.3.anP'a;e B'R, (1) 

I- . *204-33 . 3 h : Hp . xPy .D .an P'xCa n P'y (2) 

h . *271-332 . 3 h : Hp . xPy . 3 . a; = ltp'(a n P'a;) . y = ltp'(a n P'y) . (3) 

[*204-I ] 3 . « n P'a; + a n P'y (4) 



190 SERIES [PAKT V 

h . (1) . (2) . (4) . *212-23 . D 

l-:.Hp.J? = P^a.D:a;Py.D.(anP'a;)(s'iJ:)(ar.P'y) (5) 

H . (3). (5) . D I- :. Hp . D : aiPy . D . a; {ltp'^'(P p a)} y (6) 

h . (6) . *27l-322 . D I- . Prop 

*27135. h : a med P . D . D'(P D a)^ C - Q'maxp 

Dem. 
I- . *37-413 . *211-11 . D 

h :. iS 6 D'(P ^ a)e . D : (gp) . ^ = a n P"(p n a) : (1) 

[*37-l] D:(ap):a;6^.D».(ay).yepno.a!Py (2) 

h . (2) . *271-1 . 3 

h :. Hp . ;S 6 D'(P t a)c . D 1(3/3) : a; 6/3 . Da,.(ay,«).a;P5.^6a.5^P2/.ye/3na. 
[(1)] D«.(a^).a;P^.^€^. 

[*371] D^.xeP"^ (3) 

h . (3) . *205-123 . D I- : Hp . /3 e D'(P ^ a)e . D . maxp'yS = A : D h . Prop 

*27r36. h:P6Ded.ainedP.D.D'(Ppo)eCa'ltp [*27l-35 .*214-101] 

*271-37. 1- : P 6 Ser n Ded . a med P . D . Itp p C's'(P D «) e -P smof {s'(P t «)} 
[*271-321-3436 . *151-22] 

*271-38. l-:P6SerADed.amedP.D.Psmor{s'(P^a)} [*27l-37] 

*271-39. h :P,QeSernDed.amedP./3medQ.(P^a)smor(Q^/3).D. 

P smor Q 
Bern. 

h . *212-72 . D h : Hp . D . {s'(P ^ a)J smor {s'(P ^ ^)} (1) 

V . *271-38 . D h : Hp . D . P smor {s'(P D «)} • Q smor {s'(Q t ^)] (2) 
h.(l).(2). DH.Prop 

This proposition is used in proving that all continuous series are similar, 
by means of the fact that such series contain rational series as media;ns, and 
that all rational series are similar. 

*271-4. ViSeP s15m Q . ^ med Q . D . (,Sf"/S) med P 
Dem. 

h.*35-354.*74-14.Dh:Hp.D.Q|'^|S=Q|/S|'/S"/3. 

[*150-1] •^.S-'{Q\^) = {8'Q)\S"$. 

[*151-11] -^■{SKQ\mW'Q) = {P\S"p)\P (1) 

l-.*72-6. DI-:Hp.D.((2|'jS)|,Si/S = QI^/8. 

[*150-1] ':>-mQimi.Sm = 8\Q\^\Q\8 (2) 

h.(2).*27l-l. DH:Hp.D.5|Q|<SG{fi';(Q|'^)j|(5(;Q). 
[*15M1.(1)] D . P G (P I' 5"/3) I P . 

[*271-1] D . (S"/S) med P : D I- . Prop 



*272. SIMILARITY OF POSITION. 

Swmmary of *272. 

If P, Q are two serial relations, and T is a correlator which correlates 
some terms of C'P with some terms of C'Q, we say that two terms x and y, 
of which OB belongs to C'P and y to C'Q, have similar positions with respect 
to T if y comes after the correlates of all members of D'T which x comes 
after, and y comes before the correlates of all members of D'T which a; comes 
before. This notion is useful for inductive definitions of correlations. If we 
start by correlating any two terms Xi, y^, and take another term x^ coming 
(say) after x^, a term y^ having similarity of position with respect to x^ ^ y^ 
must come after y^. Suppose now we take x^ between x^ and x^. Then 
a term ys having similarity of position with respect to Xi ^ y^Ki x^ ^ y^ must 
come between yi and 2/2 ; and so on. A correlation T constructed in this way 

will be such that T''QQ.P . hp G Q. If the whole of C'P and C'Q can be 
obtained by prolonging the construction long enough, T will at last become 
a correlator of P and Q. This is the principle of Cantor's proof that any two 
rational series are similar. 

As a rule, when the notion of similarity of position is useful, the relation 
T will be one-one, but this is not assumed in the definition. We write 
" xTp(iy'' for " x and y have similar positions in P and Q respectively with 
respect to T" or, as we may express it more shortly, " the P-position of x is 
T-similar to the Q-position of y." The definition is 

Tpq = ^{xeC'P.ye C'Q . D'T n P'x C T''Q'y . D'T r^'x C T''Q'y . 

D'Tni'xCT'y] Df. 

This definition states that the predecessors of x which have T-correlates are 
to be correlated with predecessors of y, the successors of x which have 
T-correlates are to be correlated with successors of y, and if x itself has 
a T-correlate, y is to be a T-correlate of x. 

When T is a many-one relation, the definition becomes somewhat simpler. 
We then have 

*27213. H :: Te Cls -» 1 . D :. xTpqy . = : 

xeC'P.yeC'Q zzeB'TnP'x . D, . T'zQy :zeD'Tr\ P'x . 0, . yQPz : 

xeD'T.O.y^T'x 



192 SERIES [part V 

We have 

*27216. I- . (D'r)1 TpQ G T 

That is, a term which has a correlate cannot have similarity of position with 
any term except one with which it is correlated. A member of C'P n T>'T 
will have similarity of position with its correlate (assuming yeCls— »1) if 

P I B'TQT'>Q . T"C'P C G'Q (*2r2-18). 

Under ordinary circumstances, a term which is not a member of T>'T 
cannot have similarity of position with any member of Q.'T (*272'2). When 
T is many-one and its domain is contained in C'P, and P and Q are series, 
and X has no T-correlate, we have (*272'21) 

xTpQy .= \xeG'P.yeC'QiZ6 DT r> P'x . =^ . T'zQy, 

i.e. in this case, x and y have similar positions if the predecessors of x which 
have correlates are the terms whose correlates precede y. In this case, if 
xeG'P, we have (*272-212) 

Tp^'x = C'QnP (D'T n "P'x = f'Q'y't =G'Qf^^ (D'T n Ip'x = T"Q^'y). 

We next investigate the condition for G'P = T>'TpQ, i.e. the condition 
required in order that every member of G'P may have similarity of position 
with some member of G'Q. A suflBcient condition is 

P, Q 6 Ser . Q e comp . Te Cls-*1 . D'T e Cls induct . P l D'T CT'Q .'g^lQ. 

T"G'PC'D'Qna'Q 
as is proved in *272'34. 

We next consider the reversibility of Tpg, i.e. the condition that the 

converse of TpQ should be (T)qp. A sufficient condition is 

P, Q e Ser . Te 1->1 . DTC G'P . G'TCG'Q (*272-42). 

Finally, we have two propositions on the addition of another couple x^yto 
T. With the above-mentioned hypothesis of *272-42, if xTp^y and T'Q G P, 
putting W=T^x\, y, we shall have PIT>'W=W>Q (*272-51), so that the 
hypothesis we had for T still holds for W. 

The propositions of this number are in the nature of lemmas for 
Cantor's proof that any two rational series are similar, which is given 
in *273. 



*27201. TpQ = ^[xeG'P.ye G'Q . DT n P'x C T"Q'y . 

D'T n Ip'x C f'Q'y . D'T n I'x C T'y} Df 
*2721. 1- : xTp^y . = .xeG'P .yeG'Q .B'T n P'x C T"'Q'y . 

D'T A P'x C T'^'y . D'T n i'x CT'y [(*27201)] 



SECTION F] similarity OF POSITION 193 

*27211. hzxeCP."^. 

n p'P'CD'T n I'x) 
Dem. 

h . *2721 . D h : Hp . D . 

%Q'a! = G'Q n § {zeB'Tn'P'x . D, . sT\ Qy :ze'D'Tn*P'x . D, . zT\ Qy : 

z 6 D'T n I'x . D^ . ^Ty} 

[*40-51-53] = C'Q n j3'Q'"^"(D'r n P'a;) n p'Q"'y"(D'r n P'a;) 

n p^"(D'T nL'x):D\-. Prop 

*272111. h : « 6 (7'P . D . 

[*272-ll . *4018] 

•*27212. h :: xTpQy . = :.xeC'P .yeC'Qi.ze B'T. -D.-.zPx .:i .zT\Qy: 

zPx.D.zT\Qy:z = x.6.zTy [*2r2-l] 

*272-13. h :: Te Cls->1 . D :. xTpQy . = :xeG'P .y eC'Qi 

zeB'Tn'p'x . D, . T'zQy : ^eDTn P'« . D^ . yQT'z -.xeD'T .:> .y=T'x 
[*272-12.*7l-701] 

*272-131. 1- : 2'e Cls-*1 .xeC'P.D. 

%q'x = G'Q n p'^"T"P'x u Q"T"'P'x u P'iD'T n t'<r)} 
[*272-lll.*71-613] 

*27214. \-:x6C'P- B'T . D . 

[*272-lll.*40-18] 

*272141. I- : a; 6 C'P - D'!r . D . 

IVe'a; =0'Qn§ (D'T n P'a; C 2'"Q'2/ . I>'T n*P'x C ^'^y) 
[*272-l] 

*272-15. \-:Te 01s-»l . a; e O'P - DT . D . 

Tpq'x = O'Q n ^'Q""2'"P'a; n p'Q"T"'p'x 
[*272-131 . *4018] 

*272-16. \-.(D'T)'\TpqQT 
Bern. 

1- . *272-12 . D h : a; e D'T . xTp^y .'D.xTy.Db. Prop 

R. &W. III. 13 



194 SERIES [part V 

*272161. h : TeCls^l .PID'TQTIQ . D . (DT)-] TpQ = G'P^T\C'Q 
Bern. 

I-.*150-41. -^V:n^.zeT>'T.zPx.xTy.-:i.T'zQy (1) 

l-.*150-41. -^V:ze'D'T.xPz.xTy.:i.yQT'z (2) 

I- . (1) . (2) . *272-13 . D h : Hp . «ry .« e C'P . y 6 O'Q . D . xTp^y (3) 
h . (3) . *272-16 . D h . Prop 

*27217. I- : Te Cls->1 . P I D'TQ. T'Q . D'T C G'P . G'T CG'Q.D. 

T=(D'T)^TpQ [*272161] 
The hypothesis of *272"17 is satisfied in all the important uses of TpQ. 

*272171. I- : Hp *272-l7 .a;eD'T. D .%q'x= I'T'x [*27217] 

*27218. \-:Te Cls^l . P ^ B'TdTiQ . T"G'P C G'Q . x e G'P n DT. D . 

Tp,i'x=T'x 
Bern. 

h . *150-41 . D h :. Hp . D : ^ 6 D'T n Ip'x . D, . {T'z) Q (T'x) (1) 

h . *150-41 . D t- :. Hp . D : ^ 6 B'T n P'a; . D, . (T'x) Q (T'z) (2) 

h . *37-61 . D h : Hp . D . !"« 6 CQ (3) 

I- . (1) . (2) . (3) . *272-13 . D 1- : Hp . D . xTpg (T'x) (4) 

h . *272-13 . D h : Hp . xTp^y . 3 . y = T'a; (5) 

1- . (4) . (5) . D t- . Prop 

*272-2. h:TeCls-*l.D'rCC'P.Peconnex.QG/.a;~eD'r.D. 

Dem. 

h . *272-13 . D I- : Hp . xTpQy . z e B'T n P'x .li .T'z^y (1) 

l-.*272-13 .':> V :B.^ . xTpQy .z eT>'T (y P'x .:> .T'z^ y (2) 

1- . (1) . (2) . D h : Hp . xTpQy . ^ e DT. D . T'^ + y : D f- . Prop 

*272-201. h : TeCls-^l . D'T C G'P . P e connex . g ! DTpg- DT. D . 

a'TCG'Q 
Bern. 

V . *202-104 . D h :. Hp .zeTf'T .xTpQy . a;~eD'r. D : ^P« . v . a;P^ : 

[*272-13] D : T'^Qy . v . yQ (?'^) : 

[*33-132] D : r^ e C'Q : . D h . Prop 

*272-21. V :: Te Cls^l . DTC C'P . P, Q e Ser . «~ eD'^ . D :. 

xTpQy . = :xeG'P.yeG'Q:ze'D'T nP'x . =, . T'zQy 
Bern. 

\- .*2l2-2 .Dh :.JIy, . z eB'T .xTpQy .D : xJf=z .yJf=T'2 : 

[*204-3.*272-201] D:xPz . = .<^(zPx):yQ(T'z). = .n^{(T"z)Qy} (1) 



SECTION F] similarity OF POSITION 195 

h . (1) . *27213 . D I- ::. Hp . D :: xTp^y . = :. 

xeC'P .yeC'Q -..zeBV. ^,:zPx.O. T'zQy \r^{zPx) . D . ^(^'0) Qy (2) 
F . (2) . D 1- : : Hp . D :. xTpgy .= :xeC'P .yeC'Q:ze D'T. zPx . =, . T'zQy : : 

D h . Prop 
*272-211. h :: Hp *2r2-21 . D :. xTpQy . = : 

xeC'P.yeG'Q-.ze B'T n P'x .=,.yQ (T'z) [Proof as in *272-21] 
*272-212. h : Hp *272-21 .xeG'P.D. 

Tpq'x = G'Qr^'^ (DT n 'P'x = r""5y) = G'Qn§ (B'T n p"'* = T'^'y) 
[*272-21-211] 

*272-22. h : r e Cls-> 1 . P, Q 6 trans . aiTpg y . ^, w e D'T . a; e P (0 - w) . D . 

i)em. 

I- . *272-13 . D 1- : Hp . 3 . T'zQy . yQT'w : D f- . Prop 

*272-221. h : r 6 Cls-* 1 . P, Q 6 trans . a ! D'Tpq nP(z-w).D. (T'z) Q (T'w) 

[*272-22] 
*272-23. h : . r 6 01s -* 1 , P, Q e trans : 

z(Pl I>'T)w. D^,» . a ! D'TpQ nP(z-w):D.PlI>'T(lT'Q 
Dem. 

h . *272-221 . D f- :. Hp . D : ^(P t D'T)w. D . (T'z)Q(T'w) . 
[*150-41] D . z (T'>Q) w :. D I- . Prop 

*272-24. \-:J)'TnG'P = A.:).TpQ=G'P'fG'Q [*2721] 
*272-3. hiTeCls-^l.-SGr.D.TpQGSfpQ 
Dem. 

h . *272-13 . D h :. Hp . xTpQy . D : ^ e DT. ^Pa; . D . T'zQy : 

[*72-9] D : 5 e D'(Sf . ^Pa; . D . S'zQy (1) 

Similarly h :. Hp . aiTpQ^ . D : ^ e D'-Sf . ^P^ . D . 2/Q,S'^ (2) 

\-.^272-lS.'D\-:.B.p.xTpQy.D:zeD'T.z = x.O.T'2 = y: 

[*72-9] D:0eD';Sf.^ = a;.D.,S'^ = y (3) 

h . (1) . (2) . (3) . *272-13 . D h : Hp . xTpQy . D xSpqy : D h . Prop 

The following propositions lead up to *272'34. 

*272-31. h : P, Q 6 Ser . T e 01s -* 1 . a; ~ 6 B'T . z = maxp'(D'r nP'x) . 

w = minp'CDT n P'a;) .PIB'TQT'Q.D .%q'x = Q (T'z - T'w) 
Dem. 

h . *205-21 . D h : Hp . ueD'T r^P'x- I'z .li.uPz. 

' [*160-41.Hp] O.T'uQT'z (1) 

13—2 



196 SERIES [part V 

h . (1). D I- : B.^.yeQ{T'z-T'w).ue'D'Tn'P'x.:i . T'uQy (2) 

Similarly h : Hp . y e Q (f'^ - T'lv) .ueJ)'Tn%x .0 .yQT'u (3) 

1- . (2) . (3) . *272-13 . D I- : Hp . 2/ 6 Q (r'2 - T'w) . D . xTpQy (4) 

I-.*272-22. D}-:'S.^.O.*Tpq'xCQ{T'z-I"w) (5) 
I- . (4) . (5) . D 1- . Prop 

*272-32. I- : P, Q e Ser . T e Cls-> 1 . D'T C P'a; . 

Dem. 

\- . *272-13 . D h :: Hp . D :. xTpQy . = : m e D'T . D„ . T'ttQ;/ ' (1) 

h . *205-21 . D h : Hp . 16 e DT -t'z.:>. uPz . 
[*150-41.Hp] D . T'uQPz (2) 

I- . (2) . D h : . Hp . 2/ e Q'T'z . D : m e DT . D„ . T'uQy : 

[(1)] 3:*ypgy (3) 

f-.(l). D\-:Rp.xTpQy.D.T'zQy (4) 

h . (3) . (4) . D I- . Prop 

*272-321. \-'.P,QeSer.TeCls-^l.'D'TCP'x. 

PlD'T(lT'Q.w = minp'B'T . 3 .%q'x =Q'T'z 
[Proof as in *272-32] 

*272-33. 1- : P.QeSer . Qecomp . TeCIs^l . D'TeCls induct . 

P t DTG yJQ . D . (P"D'rn P"D'T) - DTC DTp^ 
Dem. 

I- . *261-26 . D h : Hp . a ! B'T nP'x.D.El maxp'(D'Tn^'x) (1) 

I- . *261-26 . D h : Hp . a ! B'T n P'a; . D . E ! mmp'(D'T n P'a;) (2) 

I- . *20511111 . D 

I- : Hp . a; ~ e DT . ^ = iaa.Xp'(D'T n P'a;) . w = mmp'(D'T n P'a;) . D . ^Pw . 

[*150-41] D . T'zQT'w . 

[*270-ll] D . a ! Q (7"^ - ?'«;) . 

[*272-31] 3.a!Vp<2'<« (3) 

h.(l).(2).(3).D _^ ^ 

h : Hp . a;~6 D'T. a ! DTn P'a; . a ! T>'Tn*P'x . D . a;6D'2'pQ : 3 l" ■ Prop 



SECTION f] similarity OP POSITION 197 

*272-331. h : Hp *272-33 . a ! Q . T"G'P C D'Q . D . G'F n ^'P"D'r C DTpg 

Dem. 
h . *261-26 . D h : Hp . a ! DT n C"P . D . E ! maxp'D'T (1) 

I- . *272-32 . D h : Hp . a; ep^"D'T . z = maxp'DT . D . %^'x =^'T'z . 
[*33-4] D.aiKe'* (2) 

f- . (1) . (2) . D I- : Hp. aje^'P'^DT. a ! DTn C'P . D . aseD'^pQ (3) 

t- . *35-85 . *272-24 . D I- : Hp . DT n O'P = A . D . (7'P C D'Tpg (4) 

h . (3) . (4) . D h . Prop 

*272-332. h : Hp *272-33 . a ! Q ■ r"C"P C Q'Q . D . O'P n ^'P"D'r C B'Tpq 
[Proof as in *272-331] 

*272-34. h : Hp *272-33 . a ! Q • T"C'P C D'Q n Q'Q . D . G'P = D'Tp<j 
[*272-33-3:31-332-18 . *202-505] 

The following propositions are lemmas for *272'42. 

*272-4. h:P,QeSeT.Tel->l.I)'TCG'P.a'TCG'Q. 

a; ~6 D'T . xTpQy . D . y(T)Qpic 
Dem. 

h . *272-21 . D h :. Hp . D : a; eC'P . y 6 C'Q -.zeD'TnP'a; . =^ . T'zQy : 

[*72-243] ■D-.xeG'P.yeG'Q: (T'w) Pa; . =» . w e aT . wQy : 

[*272-21] D:2/(?')epa!:.Dl-.Prop 

*272-41. h : P, Q 6 Ser . r e 1 ^ 1 . DT C C'P . Q'T C G'Q . 

a3eD'r.ajrpQ2/.D.2/(?)cpa; 
i)em. 

h . *272-13 . D h :: Hp . D :. a; 6 C'P . 2/ = T'a: : 

zeD'Tn P'x . 0, . T'^Qy : ^ e DT n P'a: . 3^ . yQ{T'z) :. 
[*204-3] D :. a; 6 G'P . 2/ = T'a; : ^ 6 D'T n P'a; . D^ . T'zQy : 

^ 6 D'y - t'a; - P'a; . D^ . 7"^ =f y . ~ {(2"^) Qy} : . 
[Transp] Dz.xeC'P .y=I"x:.Z6l>'T- I'x . D^ : ^Pa; . = . (7"^)Qy :. 
[*204-l] Dz.xeC'P. y=T'x :.zeT)'T. X-zPoo. = . {T'z) Qy :. 
[*72-243] D:.xeG'P.y = T'x:. (T'w) Px.=^.we a'T . wQy :. 
[*71-362] D:.yeG'Q.x = T'y:. (T'w) Px.=^.we a'T . wQy :. 
[*14-21.*33-43] :):.yeG'Q.x=T'y:.we a'T . D,„ : (T'w) Pa; . = . wQy :. 
[*204-3] D z.yeC'Q .x=T'y :. w eOTn Q'y . D,„ . ^'wPa; : 

w 6 a'T n Q'2/ . D^ . xP(T'w) :. 
[*27213] D:.y(r)Qpa;:01-.Prop 



Bern. 



198 SERIES [part V 

*272-42. \-:F,QeSeT.T 6 1-^1. T>'TCC'P.a'TCG'Q.O.(T)Qp = TpQ 
Bern. 

V . *272-4-41 . D I- : Hp . D . TpQ G {T)qp (1) 

K(l)|. Dh:Hp.D.Cnv'(r)gpGrpQ (2) 

t- . (1) . (2) . D I- . Prop 

*272-43. 1- : P, Q 6 Ser n comp - t'A .Tel^l. D'TC D'P n Q'P . 
QT C D'Q n a'Q .P t 'D'T= T>Q . D'TeCls induct . D . 
D'rpQ = C"P.a'TpQ = C"Q 

h . *272-34 . D F : Hp . D . DTpQ = O'P (1) 

I- .*i5o-36 . D h . 2';Q=r;Qp a'y. r;p= ?;pc dt (2) 

l-.(2). \-:n^.D.PtJ)'T=T>Qia'T. 

[*151-25] :i.Qia'T=hPl'D'T 

[(2)] =hp (3) 

F . *1 20-214 . D h : Hp . D . Q'T e Cls induct (4) 

f- . (3) . (4) . *272-34 . D h : Hp . D . C'Q = D'(?)qp 

[*272-42] = a'TpQ (5) 

I- . (1) . (5) . D h - Prop 

*272-5. \-:P,QeSer.Te Cls ->1 . DTC C'P . ^Tpgy .TiQCP.D. 

(Tva;ly)iQ(lP 
Dem. 
V . *150-75 . D 

I- : Hp . D . (To a; 4, 2/)5Q = T5Q » T"Q'y f I'x kj I'a; t r"Q'2/ (1) 

h . *272-212 . D h : Hp . a; ~ 6 B'T . D . T"^?/ C P'x . T"Q'y C P'a; (2) 

h.(l).(2). DI-:Hp.a;~6D'T.D.(rc;a;J,2/);QGP (3) 

h.*272-16. DI-:a;6D'r.D.7'wa;4,2/=r (4) 
I- . (3) . (4) . D h . Prop 

*272-51. h : P, Q 6 Sep . Te 1-»1 . DTC C'P . OTC C'Q . 

xTpQy.PlT>'T=T'Q. W=Tvj x^y ."^ .PID'W^ W'Q 
Dem. 

h.*272-5. DI-:Hp.D.F;QGP (1) 

h . *272-42 . D h : Hp . D . y(T)Qpx (2) 

h . *150-36 . *151-26 . D h : Hp . D . hp = Ql QT (3) 

h . (2) . (3) . *272-5 . D I- : Hp. D . FJPG Q (4) 

I- . (1) ; (4) . *150-36 . D h : Hp . D . W'Q G P ^ D'Tf . FS (P D D'F) G Q . 
[*151-26] D.PtD'TF=TF;Q:Df-.Prop 



*273. RATIONAL SERIES. 

Summary of *273. 

A " rational series " is a series ordinally similar to the series of all positive 
and negative rational numbers in order of magnitude, or, what is equi- 
valent, a series ordinally similar to the series of all rational proper fractions 
(0 excluded). This characteristic of rational series is not, however, the most 
convenient for purposes of definition. Following Cantor, we define a rational 
series as one which is compact, has no beginning or end, and has No terms in 
its field. Thus the field of a rational series can be arranged in a progression, 
and this is the source of the special properties which distinguish rational 
series from other compact series. 

Kational proper fractions can be arranged in a progression in many ways, 
for example the following : If two fractions (in «heir lowest terms) have the 
same denominator, put the one with the smaller numerator first ; if they have 
different denominators, put the one with the smaller denominator first. We 
thus obtain the series 

11213123^1 

^' 'S' "S' 4' 4' 5' T> 6' 6' F'"'' 

This series is a progression, and contains all rational proper fractions. 

Conversely, the natural numbers can be arranged in a rational series. 
Take, e.g., the following arrangement: Express the numbers in the dyadic 
scale, so that every number is of the form 

2 2''(/i6«;), 

where « is a finite class of integers. The relation of the number to k is 
one-one. Arrange the various k's by the principle of first differences, i.e. 
form the series M^i ^ (Cls induct - t'A), where M is the relation " less than " 
among finite integers. The resulting series is a rational series ; thus the 
integers are arranged in a rational series by virtue of their correlation with 
the classes k. This arrangement places all the odd numbers before all the 
even numbers, all numbers of the form '^v+2 before all numbers of the 
form 4sv, and so on. If two numbers are expressed in the dyadic scale, 
their relative position in the series is determined by the first digit (starting 
from the right) which is not the same in the two numbers : the one in which 
this digit is 1 precedes the one in which it is 0. 



200 SERIES [part V 

The two chief propositions in regard to rational series are (1) that any 
two rational series are ordinally similar, (2) that if i? is a progression, its 
finite existent sub-classes arranged by the principle of first differences form 
a rational series. The second of these propositions is proved by showing 
(a) that the finite existent sub-classes arranged by first differences form 
a compact series, (b) that the finite existent sub-classes arranged by last 
differences form a progression. By this means, given any progression, we 
can specify a relation which arranges its terms in a rational series. For if T 
is a correlator of our progression R with the progression 

iJie I (Cls induct - I'A), 

then T'Ra. t (Cls induct - I'A) 

is a rational series whose field is C'R. Hence rational series exist in any 
type in which progressions exist. 

The arrangement of the finite sub-classes of a progression, with the 
resultant existence-theorem for rational series, will be dealt with in the 
following number. In the present number, we shall be concerned with the 
proof that any two rational series are ordinally similar. 

The proof of the similarity of any two rational series is due to Cantor. 
It is long and rather complicated ; in outline, it is as follows. 

Let P, Q be two rational series, and R, S two progressions whose fields 
are C'P and C'Q respectively. Construct a series of correlations of parts of P 
with parts of Q on the following plan : Begin with A, and if T is any correla- 
tion, let the next be 

T\j seq^'D'T 4 mins'^Ps'seq/DT. 

Then the sum of all the correlations generated from A by this law of 
succession will be a correlation of P with Q. It will be seen that, if 
we put 

W = xf{X = sec^s'B'Tl mms'TpQ'seqs'^'T], 

the relation which is to be shown to be a correlator of P and Q is W^, in the 
sense defined in *259. Thus we have to prove 

W^el-^l.a'W^ = G'Q.P = WJQ. 

Wji e 1 — » 1 results immediately from *259"15. 

P D D'TF^ = F^5Q results immediately from *259-16 and *272-51. 

Thus it remains to prove D'F^ = C'P . Q'Tf^ = C'Q. 

D'W^ = C'P is easily proved. ~By induction, if T is one of the series 
of partial correlators, D'T e Cls induct, and therefore E ! seqjj'D'T, by *263"47, 

and by *272-34, C'P = I)'Tpq; hence g ! Tpe'seq^'DT, and therefore, by 

*250"121, E \mixis'TpQ'seqji''D'T. Hence T has a successor, which correlates 



SECTION F] RATIONAL SERIES 201 

seqjj'DT with mins'TpQ'seqjj'DT. Hence the successor, in R, of every 
member of C'R which has a correlate, has a correlate ; hence by induction 
every member of C'R (i.e. of G'P) has a correlate. Hence D' W^ = C'P. 

The proof of Q'TT^ = C'Q is more difficult. As before, let T be one of the 
series of partial correlators. We have to prove that there is a correlator which 
has seqs'CET in its converse domain ; when this is proved, the result follows 
by induction. To prove this, put 

a; = mms'TpQ'se(is'<I'T. 

X exists, in virtue of *272-43. Also since D' W^ = G'P, it follows from *259-13 
that there is a partial correlator U such that 

X = seqjj'D'ZJ. 

We then have to prove seq^'OT = mius'Upq'x. 

Put y = seqs'aT. Then S'y C a'T. Hence, by *272-2, 'S'y n Upt^'x = A. 

Thus it xUpgy, it follows that y = mins' Upq'x. To prove xUpqy, observe that 

TQ.U. UpQ dTpQ.Pl T)'U= U'Q. 

We have ueD'U.O. »-(uTpQy), by *272-2. Hence, by the definition of Tpq, 
we have, if m e D' U, 

(a^) . z e Tt'T .zPu.r^ (T'zQy) . v . (g^) .zeD'T. uPz . ~ (yQT'z). 

In the first case, we have ('^z).ze'D'T.zPu.r^{zPx), because xTpqy 
Hence, since x^z because a; ~ g T)'T, 

('^z).ze'D'T.zPu.xPz. 
Similarly, in the second case, 

('3^z).ze'D'T.uPz.zPx. 
The second case is incompatible with xPu, and the first with uPx. Hence 

xPu . D . (a^) .zeD'T.xPz . zPu : uPx . D . (g;^) .zeD'T . uPz . zPx. 
But, since xTpqy, xPz .'^.yQ (T'z) ."^ .yQ{ U'z), because TGU, and since 
PtT>'U=U''Q, zPu.O.(U'z)Q( U'u). 

Hence xPu . D . yQ (U'u), and similarly uPx.'D .(U'u)Qy. Hehce xUpgy. 

Hence y = miug' Upq'x, and therefore y belongs to the converse domain of the 
next correlator after U. Hence every term of C'Q belongs to the converse 
domain of some correlator, and therefore to d'W^. Hence W^ correlates P 
and Q, and P and Q are ordinally similar. 



202 SERIES [part V 

*27301. 7; = SerncoinpnC"N„nP(D'P = a'P) Df 

Following Cantor, we use t? for the class of rational ISeries. 

*273-02. RspqT =T\j seq^'D'T J, mins'Tpg'seqjj'D'r Dft [*273] 

*273-03. (ii;Sf)pQ = (B^Q VA Dft [*273] 

*273-04. T^pQ = s'{RS)pQ Dft [*273] 

^BSPQ ■will l>6 shown to be a correlator of P with Q when P and Q are 
rational series, and R and >S are progressions whose fields are O'F and C'Q 
respectively. 

*2731. l-:P67;.H.PeSerncomp.C"P6Ko.D'P = a'P [(*273-01)] 

*27311. h :. Pel? . = : PeSer n comp . D'P= Q'P: (gE) .Rew.C'P = G'R 
[*2731 . *263101] 

*273-2. h :F = Xr{Z = seqB'D'T4,min/VpQ'seq^'D'T} . D. 

[*257-125 . *258-242 . (*273-02-03-04 . *259-02-03)] 

Here the temporary definitions of *259 are revived. 

The second of the above inclusions might be changed into an equality, 
but it is not necessary for our purposes to prove this. 

*273-21. h : Hp *273-2 . D . D' F^ C C'iJ . Q' W^ C C'8 
Bern. 

V . *259-13 . D h : Hp . D . D' Tf ^ = s'D" W"{A ^*.4)'A (1) 

V . *206-18 . D i- : Hp . XeD'Tf . 3 . D'Z C G'R (2) 
l-.(l).(2).Dt-:Hp.D.D'F^CC"J? (3) 
Similarly 1- : Hp . D . Q' If ^ C C'S (4) 
h . (3) . (4) . D I- . Prop 

*273-211. h : Hp *273-2 .T ed'W ."? .B'T r^T>'W'T = ^ [*206-2] 

*273-212. h : Hp *273-2 . D . F^ e Cls -> 1 . D f (4 ,^*J.)'A e 1 -> 1 
[*273-211 . *259-141-17l] 

*273-22. h : Hp *273-2 . C"P = O'P . P e connex .Q(LJ.-:i. 

Bern. 
V . *273-211-212-21 . *206-2 . (*25903) . D 
|-:.Hp.D:2'e(il^*^)'Ar.a'F.D.reCls-*l.D'rCC"P.seqB'D'r~6D'T. 

[*272-2] D . min/rps'seqB'DT ~ e QT (1) 

H.(l).Dh:.Hp.D:re(^^*4)'Ana'F.Dr.a'rna'F'r=A: 
[*259-14-l7] D:F^6l->Cls.ap(4,^*^)6l-*l (2) 

I- . (2) . *273-212 . D h . Prop 



SECTION F] RATIONAL SERIES 203 

*273 23. h : Hp*273-2 . P, Q e Ser . G'P = G'R . C'Q = G'S . Te {A ^*4)'A . D . 

p^D'T=r;Q 
Bern. 

h .*272-51 .*273-21 . D h : Hp. TeO'll^. D .Pf D'I^'r=(I„/r);Q (1) 
h.(l).*25916.DI-.Prop 

*273-24. V'.Te {RS)pq . D . D'f, Q'T e Cls induct 
Bern. 

h . *120-251 . D 

I- :. Hp . 3 : T e D'^ ^ . DT e Cls induct . D . D'^j^Te Cls induct : 
[*90-112] ::>ik{Ayfr)^T.O.D'Te Cls induct : 

[*273-2.(*273-03)] D : Te{RS)pQ . D . D'T e Cls induct (1) 

Similarly h :. Hp . D : T e (ii>S')pQ . D . QT e Cls induct (2) 

h . (1) . (2) . D h . Prop 

*273-25. \-:P,Qev.G'P = 0'E. C'Q = G'S . Te(RS)pQ . D . 

B'TpQ = G'P . a'TpQ = C'Q 
Dem. 

h . *273-l . D 

I- : Hp . D . P, Q 6 Ser n comp . G'P = B'P = Q'P . C'Q = D'Q = Q'Q (1) 
h . *273-l . *263-44 . D h : Hp . D . a ! P . a ! Q (2) 

I- . (1) . (2) . *273-22-23-24 . *272-4.3 . D h . Prop 

*273-26. \-:.P,Q€v.R.86(o. G'P = G'R . O'Q = G'8.D: 

Te{R8)pQ . D . E ! seq^'DT . E ! ininsTpe'seq^'DT 
i)em. 

I- . *273-21 . *263-47 . *273-24 . D 1- : Hp . Te(RS)pQ . D . a ! O'i? np'R"D'T. 
[*250-122] D . E ! seq^'D'T (1) 

h . (1) . *273-25 . D h : Hp . D . a ! Trg'seq^'D'T . 

[*250-121.*2721] D . E ! mms'TpQ'seqjt''D'T (2) 

h . (1) . (2) . D h . Prop 

*273-27. h : Hp *273-2 . Hp *273-26 . D . (RS)pq CG'W . {RS)pq C DM ^ 
[*273-26] 

*273-271. I- : Hp *273-26 . T e (RS)pq . D . seqjj'DT e B'T^spq 
Dem. 

h .*273-2 . D h : Hp . Hp*273-2 . D . T€{RS)pQf\'D'ATr.'^- ATr'Te(RS)pQ (1) 
h . *273-2 . D 

h : Hp . Hp *273-2 . :re (i2/S)pQ . E ! I ^'T . D . seq^'DT e D'A ^'T (2) 

h.(l).(2).*273-27.D 

h : Hp . Hp *273-2 . D . ^ p^T e (ii<S)pQ . seq^'D'T e B'A ^'T . 
[*273-2.(*27304)] D . seq^'D'^ e B'Trspq : 3 I" ■ Prop 



204 



SERIES [part V 



*273-272. h : Hp*273-26 . D . I)"(R8)pq = R"C'R 
Dem. 

h . *206-401 . D h : Hp . T6(RS)i.q ■ D'r= E'a; .a;eG'R.D.x = seq/DT. 

[*204-71.*250-21] D .'D'Rspq'T = R'R,'x (1) 

h.*25013. Dh:B.p.D.'D'A='R'B'R (2) 
h . (1) . (2) . *90-131 . D h :. Hp . D : T(Rspq)^A . 3 . J)'Te'R"0'R : 

[(*273-03)] 0:I>"(R8)pqC'R"G'R (3) 

h . (1) . (*27303) . D 

h : . Hp . D : a; 6 O'E . iJ'a; e D"(E,S)pq . D . !r' A'a; e I>"(RS)pq (4) 

l-.(2). DF:Hp.D.E'£'iJ6D"(Ji^)pQ (5) 

J- . (4) . (5) . *90-112 . D h :. Hp . D : a; 6 (R^'B'R .D.'r'x€ J)"{RS)pq (6) 

t- .*263-43 . *250-21 . D h : Hp . D . <7'i2 = C'Ej . B'R = B'R, (7) 
h . (6) . (7) . *263141 . *122-1141 . D 

hz.K^.O-.xeC'R.D.R'cce I>"(RS)pq (8) 
h . (3) . (8) . D F - Prop 

*273-28. I- : Hp *272-26 . D . Tjispq e 1 -* 1 . D'T^spq = G'P.P= Tsspq'Q 
Dem. 

h.*273-2-22.DI-:Hp.D.rRSPQ6l->l (1) 

I- . *273-272 . 3 h : Hp . D . D'Tbspq = s'R"G'R 

[*26322] = G'R (2) 

I- . *273-2-23 . D h : Hp . D . P t D'^bspq = TnspQ'Q • 

[(2)] ■^.P = Tj^p^->q (3) 

I- . (1) . (2) . (3) . D I- . Prop 

In order to prove TpspQ e P smor Q, it only remains to prove 

a'T^PQ = G'Q. 

*273-3. I- :. Hp*273-2 . T, f76(4^*4)'A . D : D'TCD'ZJ. = .TGU 
Dem. 

h.*33-263. libzTGU.O.D'TCB'U (1) 

h.*259-lll. DI-:.Hp.D:yGi7".v.fyGT (2) 

h.*33-263. D\-:U<1T.T>'TCI>'U.D.I>'T='D'U (3) 

h . (3). *273-212 . D I- : Hp . UCT-B'TC I>'U. D.T=U (4) 
h.(2).(4). Db-.R^.B'TCD'U.O.TQU (5) 

h . (1) . (5) .31-. Prop 



SECTION F] BATIONAL SERIES 205 

*273-31. I- : Hp *273-26 . T e (RS)pq .yeC'S- d'T .~S'yCa'T.O. 

(aa-'.tT) . X = mins'TpQ'y . Ue(RS)pQ . x = seqs'D'U 
Dem. 

V . *273-25 . *250121 . D I- : Hp . D . (ga;) . x = mius'TpQ'y (1) 

I- . *273-272 . D h : Hp . a; = min^'TpQ'y . D . (g CT) . CTe {RS)pq . B'U^'l'x . 

[*206-401] D.(^U).Ue(RS)pQ.x=8e(is''0'U (2) 

I- . (1) . (2) . D h . Prop 

*273-32. I- : Hp *273-31 . x = mins' TpQ'y . U e (RS)pq . x = seqjR'D' CT . D . 

xUpQy.TGU 
Bern. 

h . *205-14 . D h :. Hp . uRx . D : ~ (uTpgy) : 

[*272-13] •^■.(•^z):z6l)'T:zPu.'^(T'zQy).v.uPz.'^(yQT'z) (1) 

I- . *272-2-42 . DI-:Hp.D.a;~6D'y. - (2) 

[*273-272] D.B'Tc'R'x (3) 

h.*273-272. DI-:Hp.D.^'« = D'J7" (4) 

l-.(3).(4).*273-3.Dh:Hp.D.rGU- (5) 

h.(l).*272-13.D 

t- :. Hp . uRx . D : (ga) -.zeB'T-.zPu .r^(zPx) . v . uPz .'^{xPz) (6) 

1- . *204-l . D 1- :. Hp . D : uPx . zPu . D . zPx : xPu . uPz . D . xPz (7) 

I- . (6) . (7) . (4) . D h :. Hp . M e D'CT. D : uPw . D . (-g^z) . z e D'T. uPz .~{xPz) : 

xPu . D . (a^) . ^ 6 J)'T . ^Pm . ~(0Pa;) : 

[(2)] : mP« . D . (a«^) . z e B'T . mP^ . zPx : 

a;PM.D.(a^).0 6D'r.^PM.aiP^ (8) 
h . *272-13 . *273-23 . D 

h:Hp.MeD'f7'.^6D'r.MP0.^Pa;.D.(&'M)Q(^'^:).(r'^)Qy. 
[(5)] :>-(U'u)Qy (9) 

Similarly I- : Hp.MeD'^J.^reDT.^PM .ajPi? . D . yQC^'w) (10) 

K (8). (9). (10). 3 

h :. Hp. MeD'f7. D : wP« . D . (^'m) Qy.xPu.'D. yQ{U'u) (11) 

h . (11) .*272-13 . 3 h : Hp . D . xUpQy (12) 

h. (5). (12). 31-. Prop 



206 SERIES [PABT V 

*273-33. 1- : Hp *273-32 . 3 . y = mins' Upq'x . x {Rspq^ U) y 
Bern. 

h . *273-32 . D h : Hp . D .'s'y C a'U. 

[*272-2-42] D .'S'y nUpQ'x = A (1) 

h . (1) . *273-32 . *205-14 . D h : Hp . D . y = mins' Uj.q'x (2) 

h . (2) . (*273-02) . D 1- : Hp . D . a; (Rspq'U) y.Dh. Prop 

*273-34. h : Hp *273-31 . D . y e (J'Thspq 
Bern. 

h . *273-31-33 . D I- : Hp . 3 . (a ?7) . [/■« {RS)pq . y e d'RsQP V ■ 
[*90-16.(*273-03)] D .('s^W) .W €(RS)pQ.yea'W . 

[(*273-04)] D.ye a'T^spQ : 3 I" ■ Prop 

*273-35. h : Hp *273-26 . D . a'TsspQ = G'Q 
Bern. 

t- . *273-34 .:>\-:R'p.yeC'S.S'yC a'Tj^sPQ -O.ye a'TssPQ (1) 
h . (1) . *250-34 . D h . Prop 

*273-36. h : Hp *273-26 . D . T^spq e P smof Q [*273-28-35] 

*273-4. \-:P,Qer,.D.P smor Q 
Bern. 
h . *273-ll . D I- : Hp . D . (gi?, S).R,8 eto .C'P = G'R . C'Q = C'S . 
[*273-36] D . (gJS, /S) . T^pg e P smor Q : D t- . Prop 

*273-41. hzPev.PsmorQ.D.Qev 

Bern. 

h . *270-41 . D h : Hp . D . Q e Ser n comp (1) 

h . *15118 . *123-321 . D t- : Hp . D . O'Q f {<„ (2) 

h . *151-5 . D h : Hp . D . D'Q = a'Q (3) 

I- . (1) . (2) . (3) . *273-l . D h . Prop 

*273-42. l-:Pe7?.D.»7 = Nr'P [*273-4-41] 
*273-43. f- . t; 6 NR [*273-42 . *266-54] 

The following propositions are easy to prove: 

h : Q 6 Ser ft C"ii„ .Pe^j.D.QxPe^?, 
whence h : a e NR n Cl'Ser . G"a = N„.D.aX77 = i7; 

and 

I- : P 6 t; . Q 6 Ser n C"N„ .«€ C'P . D . a; i ;Q e Nr'Q n Rl'(e X P) . Q X P 6 77, 
whence, from the fact that all tj's are similar, 

\-:PevQeSerrx C^'N, . D . a ! Nr'Q « Rl'P. 
Thus an rj contains series of all the order-types composed of J4o terms. 



*274. ON SERIES OF FINITE SUB-CLASSES OF A SERIES. 

Summary of *274. 

In the present number, we shall be concerned with the construction of 
a rational series consisting of the finite existent sub-classes of a progression. 
When the finite sub-classes of a progression (excluding A) are arranged by 
the principle of first diiferences, the result is a rational series. When they 
are arranged by the principle of last differences, the result is a progression. 
These two propositions, with the consequent existence-theorems, are to be 
proved in the present number. 

We define "P, " as P„i with its field limited to finite existent classes. 
(For the definition of P^i, see *170"01.) In the present number, we shall be 
chiefly concerned with P, when Pew, but it has interesting properties in 
many other cases. 

Our definition is 

Pr, = Pel D (Cls induct - I'A) Df. 

We shall be concerned in this number not only with P,, but also with 

P,„ ^ (Cls induct - I'A). This is Cnv'(P),. Thus if we put P=Q, the 
hypothesis that P e Q as used in studying Pj,. ^ (Cls induct — I'A) is 

equivalent to the hypothesis that Q e fi as used in studying Gnv'Q,, 

i.e. Q,. Thus the study of P^j and Pi,, with their fields limited to inductive 
existent classes may be replaced by the study of P, in the two cases where 

(1) P 6 fl, (2) P 6 fl. The second case is the simpler, and is considered first. 
We have first, however, a collection of propositions which only assume that 
P is a series. 

Since an inductive existent class in a series must have a maximum and 
a minimum, we have 

*27412. l-::P6Ser.D:.aP,/8. = : 

-> -> 

a, jS 6 CI induct'C'P - I'A : (30) . ^ e a - ^8 . a « P'« = /3 n P'a 

We have 
*27417. I- : O'P ~ 6 1 . D . O'P, = CI induct'G'P - i' A 



208 SERIES [part V 

Whenever P is a series, P, is a series (*2'74"18). If P has a last term, the 
class consisting of this last term only is the last term of P, ; if P has no last 
term, P, has no last term (*274'191). If G'P is an inductive existent class, 
the first term of P, is G'P (*274-194); if not, P, has no first term (*274195). 
Hence if P has no last term, P, has no first or last term, and we have 
D'P, = Q.'Pn (*274'196). Thus of the characteristics used in defining i), 

we have P, eSer whenever PeSer, and D'P, = C['P, whenever ~E!5'P. 

We next prove 

*274-22. h : P e n . D . P, e fi 

which, in virtue of what was said above, is equivalent to 
P 6 n . D . Pi„ t (Cls induct - t'A) e fi, 
that is : The principle of last differences applied to the inductive existent 
sub-classes of any well-ordered series gives a well-ordered series. 

To prove *274'22, since we already know that P, is a series, we only have 
to prove that every existent sub-class of CP,, has a maximum with respect 
to P,. This is proved as follows. 

Let K be any existent sub- class of CI induct'O'P — I'A. Consider the 
minima of all the members of k : these minima all exist, because k is 
composed of inductive classes. Then in virtue of the nature of the principle 
of first differences, members of « which have a later minimum come later 
than those that have an earlier minimum. Hence if we consider minp"«, 
the classes whose minimum is the maximum of miu/'K (which exists, because 

P e O) are later than any other members of k. Put 

Xi = maxp'minp"/(; . Ki = k n minp'a^. 

Thus «! consists of those members of k which have the largest minimum, 
and members of Kj come later than any other members of k. Similarly the 
latest members of «■] will be those that have the greatest second term. 
That is, if we take away the (common) first term from each member of k^, 
and if Xi is the resulting class of classes, we have to apply to \i precisely 
the same process as we have already applied to k. Thus we are led 
to put 

sOi = maxp'minp"« . Ki = k a minp'aji .\ = (— i'x^"ki, 
x^ = maxp'minp'% . K2 = \^r\ minp'aJa . X^ = (- i'x^"k2, 
and so on. The series oci, x^, ... is an ascending series in P, and is therefore 
finite, by *261*33. It therefore has a last term, say «„. Then the class 
I'iCi w I'x^ w . . . u I'x, is a member of k, and is easily shown to be its 
maximum. Hence every existent sub-class k of O'P, has a maximum, and 

therefore P, e O. 



SECTION F] on SEBIES OF FINITE SUB-CLASSES OF A SERIES 209 

In order to symbolize the above process, we put 

Pm'it = niaxpmmp"/e Dft, 

Tp'K = (- i'P„,'k)"{ic n mnp'Pm'ic) - I'A Dft, 

Mp'K = P„"(Tp)#'K Dft. 

w < 

Thea Pm'« is what we called »!, Tp'x is what we called Xj, {Tp)^'k is 

the class k, \, Xj, ... Xn, and Mp'k is the class x^, x^, x^, ... x,. Thus what 
we have to prove is 

Mp'K = max {Py)'K, 

which is proved in *274215. 

We prove next 

*274-25. I- : P 6 o) . 3 . P, e o) 

For this purpose we use *26344, namely 

« = n - I'A rt P (a'Pi = d'P . ~E ! B'P)- 

Thus it only remains to prove 

D'(P,X = D'P,.~E!5'P,. 

~ E ! J5'P, follows from *274-195, and D'(P,)i = D'P, is proved without any 
difficulty ; hence our proposition follows. 

From *274'2517, by substituting P for P, we obtain 
*274-26. h : P e ft) . D . Pie t (01s induct - I'A) e a, . 

C"P,c t (Cls induct - I'A) = CI induct'O'P - t'A 
whence it follows immediately that 

*274-27. h : a 6 No . D . CI induct'a e «„ ■ CI induct'a - I'A e «„ 
I.e. a class of ii„ terms contains No inductive sub-classes. 
We now have to prove 

*274-33. \-:Peco.D.P„6v 

In virtue of *274-17-27, we have O'P, e ti„ ; and by *274-18, P, e Ser. 
Thus it only remains to prove P, e comp . D'P, = Q'P,. The second of 
these results immediately from *274196. As for P, e comp, if oiP,^, 

a w yS 6 Cls induct, and therefore g ! p'P"(a w /3) ; but if a; e p'P"{a w j8), we 
have aP, (/3 w I'a;) . (/8 u t'x) P,^ ; hence P, G P,''. This completes the proof 
that P, e J?. 

The proposition holds not only if Pew, but if P is any series which has 
no last term and whose field has Nj terms (*274"32). 

Finally, we deal with the existence of rj (*274'4 — •46). If P e »>, P is 
similar to Pio t (Cls induct - t'A), by *274-26; and if T is a correlator of 

B. & W. III. 14 



210 SEBIES [part V 

these two, T'P,, is an 77 whose field is C'P (*274-4). Hence the existence 
of 7] in any type is equivalent to the existence of a in that type (*274"41). 
Hence we have merely to apply previous propositions on the existence of w. 



*27401. P, = Pelt (01s induct -I'A) Df 

*27402. P„'«:=maxp'minp"« Dft [*274] 

*27403. Tp'K = (- i'P^'k)"(k n imnp'P^'«) - t'A 

Dft [*274] 

*27404. Mp'K = P^"{TpyK Dft[*274] 

*2741. V : aP,/3 . = . a, ^S e CI induct'C^'P - I'A . g ! a - ;8 - P"{^ - a) 
[*170-1 . (*27401)] 

*27411. 1- : P 6 Ser . a 6 CI induct'O'P - t'A . D . E ! minp'a . E ! maxp'o 
[*261-26] 

*274-lll. h : P e Ser . ~ E ! 5'P . a e CI induct'C'P . D . g !p'P"a 

V . *274-ll . D h : Hp . a ! a . D . maxp'a e D'P . 

[*205-65] D.a!yP"a (1) 

l-.(l).*40-2.DI-.Prop 

*27412. I- : : P e Ser . D :. ap,^ . = : 

a, j8 e CI induct'O'P - t'A : (a^) . e a - ^ . o n P'^ = |8 ft P'0 
D&m. 

I-.*170-2.D 

I- :. a, /3 e CI induct'a'P-i'A:(a0). a e a- /3. a nP'ir = /3 ft P'if:D.aP,/3 (1) 

h . *274-ll . D h : Hp . aP,yS . D . E ! minp' {a - jS - P"(iS - a)} . 

[*170-23.*205-192] "^ .{'^z).ar^P'z = ^r^P'z (2) 

h . (1) . (2) . D h . Prop 

*27413. l-.P,et(Cls induct- t'A) = Cnv'(P), [*1 70-101 . (*274-01)] 

*27414. h :: P 6 Ser . 3 :. a {Pi„ t (Cls induct - t'A)})8 . = : 

a, /3 e CI induct'C'P - t'A : (g^) .ze^-OL.ar\P'z = ^r^P'z 
[*27412-13] 

*27415. h : a, /3 e CI induct'C'P -t'A.;8Ca./3=|=a.D. aP^^ 
[*170-16 . *274-l] 

*274151. I- : a e CI induct'CP - 1 . a; e a . D . aP,(t'a;) [*274-15] 



SECTION F] on series OF FINITE SUB-CLASSES OF A SERIES 211 

*27416. l-:a[!P,. = .0'P~eOwl 
Dem. 

l-.*274-l. Dhia'.P^.D.gia'P (1) 

V . *274151 . D h : O'P ~ e w 1 . D . a I P, (2) 

h .*60-38 . D h : O'Pel . D . ~ (ga, /8) . a, ^8 e Cl'O'P- I'A . g ! a-/3. 
[*274-l] D.P, = A (3) 

h . (1) . (2) . (3) . D h . Prop 

*27417. h : (7'P ~ e 1 . D . C'P, = CI induct'O'P - I'A 

Dem. 

h . *274-151 . D h . 01 induct'O'P - I'A - 1 C D'P, (1) 

h . *2r4151 . D h : ȣ C'P . G'P + t'a; . D . I'a; e Q'P, (2) 

h . (2) . D h : Hp . D . GVG'P n 1 C Q'P, (3) 

f-.(l).(3). Dh:Hp.D.Clinduct'C"P-t'ACC?'P, (4) 

h . (4) . *2741 . D h , Prop 

*274171. ViP^Q.J.xPy.-:i. (I'x) P„ (I'y) [*274-l] 

*274-18. h:PeSer.D.P,6Ser 

Dem. 
h.*201-14.D 
I- :. Hp. 06 a -/8. w 6/3 -7. a nP'^ = i8r.P'0.j8nP'w = 7 nP'w.D: 

zPw .0 .zea — y .an P'z = 70 P'0 (1) 
h . *201-14 . D 1- :. Hp (1) . D : wPz . D . w e a - 7 . a n P'w = 7 n P'w (2) 
I- . (1) . (2) . *202-103 . *27412 . D h : Hp . aP,^ . /3P,7 . D . aP,7 (3) 

I- . *27411 . D 

h : Hp . a,^6 Clinduct'C'P- t'A . a + 18 . D . (g^) . = minp'{(o-^)w (i8-«)} . 
[*20514] D . (a^) . 6 (a - /8) u (/3 - a) . a n P'0 = iS A P'a . 

[*274-12] D.a(P,c;P,)/3 (4) 

f-.(3).(4).*l70-l7.Df-.Prop 

*27419. h : P 6 connex . P" C J . 3 . 5'P, = i"£'P 

Dem. 
H . *274-151 . 3 h . CI induct'CP - 1 C D'P, (1) 

h.*274l7l. DI-:Hp.D.i"D'PCD'P, (2) 

h . (1) . (2) . *274-17 . D h : Hp . D . 5'P, C t^'P'P (3) 

h . *202-524 . D 

|-:Hp.a;e£'P./3eCl'a'P-l'A..'C~6i8.D.a;6P"(/3-l'a!) (4) 

14—2 



212 SERIES [part V 

l-.(4).D 

I- : Hp . a; e'S'P . D . ~ (3^8) . ;8 e CI induct'CP - t'A . g ! i'x-^-P"(^-l'ai) . 
[*274-l] D.t'a;~6D'P, (5) 

h . (5) . *27417 . D h : Hp . D . t"'B'P CB'P„ (6) 

h . (3) . (6) . D H . Prop 

*274191. \-:.Pe connex . P» G J" . D : E ! 5'P . D . 5'P, = I'B'P : 

~ E ! £'P . D . P'P, = A [*27419] 
*274192. I- :. P 6 connex . P" G / . D : E ! £'P . = . E ! B'P^ [*274-191] 

*274193. h . B'P,, = I'C'P n (Cls induct - I'A - 1) 
Bern. 

h.*274-151. D I- :0'Pe Cls induct -I'A-l.D.O'Pe^'P, (1) 

h . *274-16l7 . D h : C'P ~ € (Cls induct - I'A - 1) . D . C'P ~ e C'P, (2) 

1-.*27415. D hiaeClinduct'O'P-t'A. a; 6 C'P- a. D. (aw t'a!)P,o (3) 

l-.(3). Dh.Clinduct'C'P-t'A-t'a'PCa'P, (4) 

h . (4) . Transp . *2741 . D h . "b'P^ C (CI induct'C'P - I'A) n I'C'P (5) 

l-.(5).*27416. D t- .£'P,C (Cls induct -i' A -l)nt'0'P (6) 

h . (1) . (2) . (6) . D h . Prop 

*274194. \-:C'Pe Cls induct - t'A - 1 . D . B'P„ = G'P [*274193] 

*274195. I- : O'P ~ 6 Cls induct . D . S'P, = A [*274193] 

*274196. h : P 6 Ser . ~ E ! 5'P . D . D'P, = Q'P, 

'^^'^- l-.*274-192. DI-:Hp.D.£'P, = A (1) 

I- . *274-195-16 . *261-24 . D h : Hp . D .'B'P„ = A (2) 

h . (1) . (2) . D I- . Prop 

The following propositions give the proof of P e 11 . D . P, e O (*274'22). 

*274-2. h : P 6 fi . « C CP, . g ! k . D . E ! P^'k . PJk e minp"« 
[*274-l-ll . *250-121 . (*274-02)] 

*274-201. \-:^e Tp'k . = . (ga) .aex. miup'a = P^'k .^ = cl- l'P,„^k . g ! j8 
[(*27403)] 

*274-202. h : E ! P„'« . D . E ! Tp'«: [(*27408) . *14-21] 

*274203. h :. Hp *274'2 . D : Tp'/e = A . = . k n minp'P„'« = i'i'P^'k 

Dem. 
h . *274-2-202 . D 

1- : : Hp . D : . Tp'k=A . s : ~ (ga, 0).aeK . niinp'a = Pm'ic . /3 = a - I'Pm'ic ■ g ! /8 : 

<— 
[*13'191] = : a e « n minp'Pm'/c . Da . a - I'Pm'ic = A : 

«— 
[*274-2] = : a e « n minp'Pm'* .=«■«= t'Pm'ic :: D H . Prop 



SKCTION f] on series OF FINITE SUB-CLASSES OF A SERIES 213 

*274-204. h : « C G'P, . « (Tp)*\ . D . \ C (7'P, 
i)em. • 

y . *120-481 . *274-201 . D h : « C Cls induct . E ! Tp'k.:^.Tp'k C Cls induct (1) 

I- . *274-201 . D h : « C Cl'G'P . E ! fp'« . D . fp'* C Gl'C'P - t'A (2) 

h . (1) . (2) . *274-16 . D h : « C C'P, . E ! Tp'« . 3 . Tp'/e C CP, (3) 

h . (3) . Induct . D f- . Prop 

*274-205. h : P 6 Ser . E ! Pr^'Tp'X . D . (P™'\,) P (P,„'Tp'X) 
Dem. 

h . *274-201 . *205-21 . D I- : Hp . /3 e Tp'X . D . /3 C P'P^'X (1) 

h . *205-ll . (*274-02) . D h : Hp . 3 . Pm'Tp'X e s'Tp'X (2) 

h . (1) . (2) . D h . Prop 

*274-206. h : Hp *274-205 . k {Tp)^X . D . (P™'«) P (P^'fp'X) 
Dem. 

h . *14-21 . (*27402) . D 1- : E ! PJT/X . D . E ! P^'X, (1) 

1- . (1) . Induct . D I- : Hp . D . E ! P^'k (2) 

h . (2) . *274-205 . Induct . D h . Prop 

*274-207. h : P e fl . «: (rp)*X . P^'X = maxp'i/p'/c . D . 

^El Pm'Tp'X .Tp'\ = A 
Bern, 

h . *274-205 . Transp . D h : Hp . 3 . ~ E ! P^'h'^ ■ 

[*274-204-2.Transp] D . Tp'A, = A : D h . Prop 

*274208. h:.Pen.KC(7'P,.a!K.D: 

A 6 (^V)*'* : (gA,) . « (rp)*5t . \ n minp'P^'X = I'l'P™' \ . Tp'X = A 

Dem. 

f- . *250-121 . 3 h : Hp . D . E ! maxp'Jlfp'/e (1) 

h . (1) . *274-207-203-204 . D h . Prop 
*274-21. h : /3 e Tp'/c . D . jS w t'Pm'« e « [*274-201] 

*274-211. h : « (Tp)*\ . ^S e \ . D . |8 w P™"T'p (« i- A.) e « 

Dej». 
h . *274-21 . D h :. Hp : /3 6 \ . 3^ . /3 u Pm"Tp (/«: i- \) e k : D : 

7eTp'\.DY.7uP^"rp(«i-r/\)e« (1) 
h . *274-21 . (1) . Induct . D h . Prop 



\ 
\ 



214 SERIES [part V 

*274-212. h : PeXl . /tC C'P, . g ! k . D . Jl/p'* e/e 

Z)em. 
f-.*274-208-211.D 

I- : Hp . D . (gX.). «(Tp)^\ . ?p'X = A . l'P„'\ e \ . I'Pm^X w P„"yp(Ki-\) e k . 
[*121-103] D . (gX) . P^"rp (« M \) 6 « . PJ'Tp (« m \) = P„,"(Tp)^'k : 

D h . Prop 

*274-213. h : P e Ser . « C C'P, . a e « . « (7'p)5|,\ . P'P»'\ n ilfp'/e C a . D . 

a-(P'Pm'\r.ilfp'«;)6\ 
-Dem. 

h .*274-201 . D h : Hp . K = \. D . a-(P'P^'\ n Jlfp'«) = a . 

[*13-12] D.a-(P'P^'\nifp'«)6/<: (1) 

h . *274-206 . D 

h :. Hp : ^ e « . P'P™'\ n ilfp'* C ^8 . D^ . /3 - (P'Pr^'X n Mp'k) e X : D : 

/3 e « . P'Pr^'Tp'X n il/p'/c C /8 . D . P'P^'X n ifp'« C /3 . P,„'X e /9 . 

{/8 - (P'P»'X n ilfp'K)} 6 X . P^'X 6 {^ - (P'PJX n il/p'«)} . 

[*274-201] D . {/3 - (P'P„'X « il/p'/e) - t'P„,'\} e Tp'X . 

[*274-206] D.{/3-(P'P™'rp'XnJ/p'«)}6rp'X (2) 

h . (1) . (2) . Induct . D h . Prop 

*274-214. h-.Pen.KC C'P„ . a e /c - I'i/p'* . 3 . aP,(il/p'«) 

Dem. 
h . *274-212 . D f- :. Hp . D : Mp'k e Cls induct : (1) 

[*17016] D : Mp'k C a . D . aP,(il/p'«) (2) 

I- .*274-ll . (1). D I- : Hp . a ! Mp'k- a. "^ . E I minp'(ifp'«; - a) . 
[*205-14.(*274-04)] D . (gX) . « (rp)^,^ X . P^'X ~ e a . P'P^'X n Mp'k C a . 
[*274-213] D . (gX) . « (Tp)^X . P^'X ~ e a . a - (P'P^'^ n Jlfp'/e) e X . 

'P'Prr.'XnMp'KCa. 
[*274-201] D . (aX, z) . K {Tp)^ X.z = minp'fa - {P'Pr^'X a ifp'«)} . 

zP (Pm'X) . P'P™' X f^ Mp'k C a . 
[*3118] D . (30) . e a - Mp'k . Ifp'/e r> P'« C a . 

[*17011] D . aP,{Mp'K) (3) 

h . (2) . (3) . D I- . Prop 

*274-215. h : P € 12 . « C C'P, . g ! « . D . Mp'k = max (P,)'« [*274-212-214] 



SECTION F] on series OF FINITE SUB-CLASSES OF A SERIES 215 

*274-22. h:Pen.D.P,ef2 
Dem. • 

h . *274-215 . D h : Hp . D . E !! max (P,)"C1 ex'CP, . 

[*250125] D . P, e n : D h . Prop 

The following propositions constitute the proof of 

Pe<o.D.P,6«» (*274-25). 

*274-221. h : P 6 Ser . P'raax^'a e Cls induct . a e 01 induct'G'P -I'A-i 'B'P . 

/8 = (a - I'maxp'a) u P'maxp'a . D , aP,;S 
Dem. 

h.*205-55. Dh:Hp.£'P6«.D.a!a-i'maxp'a (1) 

h.*202oll. Dh:Hp.5'P~ea.D.5'P6P'maxp'a (2) 

h . *93101 . D h : Hp . ~ E ! B'P . D . g ! P'maxp'o (3) 

l-.(l).(2).(3).DI-:Hp.D.a!^ (4) 

I- . *120-481-71 . D h : Hp . D . /3 e Cls induct (5) 

b . *205-21 . *200-361 . D h : Hp . D . /3 n P'maxp'a = a n P'maxp'a (6) 
t- . (4) . (5) . (6) . D h : Hp . D . a, /S 6 Clinduct'C'P - I'A . maxp'a 6 « - /3 . 

« r\ P'maxp'a = ffrt P'maxp'a . 
[*27412] D . aP,/9 : D I- . Prop 

*274-222. h : Hp *274-221 . aP,7 . maxp'a ey.D. ^P^y 

Dem. 

— > -♦ 

|- . *27412 . D |- : Hp . D . (g^;) . ^ e a - 7 . « =f maxp'a . a n P'2: = 71-1 P'^r . 

[*201-14.*205-21.Hp] D . (g^) .Z6ff-y.^nP'z = ynP'z. 
[*274-12] D . /SP,7 Oh. Prop 

*274-223. h : Hp *274-221 . aP,7 . maxp'a ~ e 7 . 7 4= /3 . 3 . /SP,7 

Dem. 

— > — > 

h . *274-12 . D h :. Hp . D : (g^) .zea — y- I'maxp'a . a n P'z = 70 P'^: . v . 

-♦ — > 

a n P'maxp'a = 7 a P'maxp'a (1) 

1-.*201-14.*205-21.D 

— » —* 

\- :. Hp : (a«) . 5 6 a — 7 — I'maxp'a . a r> P'2: = 70 P'^ : D . y3P,7 (2) 

-♦ — » 

h . *205'21 . D h : Hp . a n P'maxp'a = 70 P'maxp'a . D . 

-♦ 
a — I'maxp'a = 70 P'maxp'a (3) 

h . *202101 . D h : Hp . D . 7 C P'maxp'a u P'maxp'a (4) 

-> — > 

h . (3) . (4) . 3 t- :. Hp . a n P'maxp'a = 7 n P'maxp'a . D : 7 C /8 : 

[*17016.(*274-01)] D:7=|=/3.D./3P,7 (5) 

h.(l).(2).(5).Dh.Prop 



216 SERIES [part V 

*274-224. h : Hp*274-221 . aP^r^ . y34= 7 . D . /3P,7 [*274-222-223] 

*274-23. h:Hp*274-221.D.«(P,)i/3 [*274-221-224.*204-72] 

*274-25. f-:Peft).D.P,6« 
DeTn,. 

V . *274-22-16 . D h : Hp . D . P, 6 n - t'A (1) 

l-.*274-191-l7.D 

h : Hp . a 6 D'P, . D . a e CI induct'C'P - t'A - I'B'P (2) 

h.*263-412.*274-ll .D 

h : Hp . a 6 CI induct'a'P - I'A . D . P'maxp'a e Cls induct (3) 

h . (2) . (3) . *274-23 . D I- : Hp . a e D'P, . D . « e D'(P,X (4) 

h . (1) . (4) . *274-195 . *121-323 . D 

t- : Hp . D . P, 6 n - I'A . D'P, = D'(P,). . ~ E ! £'P, . 

[*263-44] D . P, 6 » : D h . Prop 

*274-26. h : P 6 oj . D . Pie t (Cls iuduct - t'A) e « . 

C'Pie p (Cls induct - I'A) = CI induct'CP - I'A 
Dem. 

I- . *274-13 . D h : Q = P . D . Pie p(Cls induct - I'A) = Q, (1) 

l-.*274-25.DI-:P6M.Q = P.D.Q,6ft) (2) 

I- . *27417 .Dt-:Pe«.Q = P.D. G'Q, = CI induct' C"P - I'A (3) 
h . (1) . (2) . (3) . 3 t- . Prop 

*274-27. 1- : a 6 {<„ . D . CI induct'a e N„ . CI induct'a - I'A e «„ 
Dem. 

h . *263-101 . D h : Hp . D . (gP) .Peo).a=G'P. 
[*274-26] D . (gif ) . if e « . CI induct'a - t'A = CM . 

[*263101] D. CI induct'a- t'A e No . (1) 

[*123-4J D. CI induct'a 6 No (2) 

h . (1) . (2) . 3 h . Prop 

The following propositions constitute the proof of 
Pem.D.P^er) (*274-33). 

*274-3. h : P e Ser . aP,y8 . x e p'P"(a u ^) . D . oP,(/S u I'ai) . (/3 w t'a;) P„^ 
Dem. 

|-.*200-53. DI-:Hp.«6a.D.;SnP'2r = (^wt'a;)ftP'5 (1) 

l-.*200-5. DI-:Hp.^^ea-/3.D.^;ea-(/3wt'a;) (2) 

h . (1) . (2) . *274-12 . 3 h : Hp . D . aP,(/8 u I'a;) (3) 

h . *200-5 . *170-16 . 3 h : Hp . D . (/3 u t'a;) P,/3 (4) 
h . (3) . (4) . D I- . Prop 



SECTION f] on series OF FINITE SUB-CLASSES OF A SERIES 217 



*274-31. h : P e Ser . ~ E ! 5'P . D . P 6 Ser A comp 




Dem. ♦ 




I- . *274-l . *120-71 .31-: aP,/3 . D . a u ^ e 01s induct - t'A 


(1) 


h .(1).*27411 . D 1- : Hp . aP,^ . D .E ! maxp'(au^) . 




[*93103] D . a ! P'maxp'(a w /3) . 




[*205-67] D.a!^'P"(au^). 




[*274-3] D . aP,^/3 


(2) 


h . (2) . *27418 . D h . Prop 




*274-32. 1- : P e Ser n G"N„ . ~ E ! 5'P . D . P, e 77 




Dem. 




h . *274-31 . 3 h : Hp . D . P, 6 Ser n comp 


(1) 


h . *274-196 . D 1- : Hp . D . D'P, = Q'P, 


(2) 


1- . *274-27-17 . D h : Hp . D . C'P, e K„ 


(3) 



h . (1) . (2) . (3) . *273-l . D h . Prop 
*274-33. l-:Pe<o.D.P, e»; [*274-32 . *263-10111-22] 
This is the principal proposition of the present number. 

*274-34. l-iaeNo. D.a!9?n C'(C1 induct'a - t'A) 

Dem. 
h . *263-101 . D h : Hp . D . (gP) . P e « . C"P = a . 

[*274-33l7] D . (gilf ) .Mev G'M= 01 induct'a - t'A : D l- . Prop 

The following propositions are concerned with the existence-theorem 
for -0. They all follow from *274-33. 

*274-4. h-.Pew. T=^i'P mm {P^^ I (Cls induct -I'A)} . D . T'P„evnG'C'P 
Bern. 
h . *274-26-l7 . Dh:Hp.D.a'r=a'P, (1) 

I- . (1) . *151-11-131 . D h : Hp . D . ^TJP, smor P, . G'T''P„ = G'P . 
[*274-33.*273-41] D . TJP, e v . C'T'>P„ = C'P : 3 h . Prop 

*274-41. I- : a ! 0) n «'P . = . a ! 7; n «'P 
Bern. 

h . *274-4 . D I- : Qeffl n i'P . 3 . (<^R).Rev. G'R = O'Q . 
[*64-24] 3 . a ! 7? n i'P (1) 

I- . *273-ll . 3 h : i2 e jy n <'P . 3 . (aQ) .Qea .G'Q = G'R. 
[*64-24] 3 . a ! <" « «'-P (2) 

h . (1) . (2) . 3 h . Prop 

*274-42. h:aeN„.3.a!'?"C''o [*274-4-26 . *263-l7 . *250-6 . *263-101] 

*274-43. h . No = G"ri [*273-l . *274-42] 

*274-44. h : a ! No " fa . = ■ a ! ^7 -^ *oo'a [*263-131 . *274-41] 

*274-45. h:a!^{o(«). = ■a!'?'^«"'« [*263-13 . *274-41] 

*274-46. h : Infin ax («) . = . a ! 1? " «^'a; [*263-132 . *274-41] 



*275. CONTINUOUS SERIES. 

Summary of *275. 

The definition of continuity to be given in this number is due to Cantor. 
A different and not equivalent definition was given by Dedekind: series 
which are continuous in Cantor's sense are also continuous in Dedekind's 
sense, but not vice versa. Cantor's definition has the advantage (among 
others) that two series which are continuous in his sense are ordinally 
similar, which is not necessarily the case with series that are continuous in 
Dedekind's sense. Dedekind's definition of '"continuous series" is, in our 
language, " series which are compact and Dedekindian." Cantor's definition 
(after a certain amount of simplification) is " series which are Dedekindian 
and contain an N„ as a median class." In the case of the real numbers, the 
rationals are a median class of this sort. 

An equivalent definition to the above is that a continuous series is a 
Dedekindian series whose converse domain is the derivative of a contained 
rational series (*275'13). 

Following Cantor, we shall use for the class of continuous series. 

In what follows, we prove first that the series of segments of a rational 
series is a continuous series, i.e. 

*275-21. l-:P67;.D.s'P6^ 

_ — > 
The contained No is P"G'P. The proposition follows at once from 

*271'31. On its importance, see remarks on *275'21 below. 

From this proposition, it follows that if tj exists in any type, d exists in 
the next type (275"22), whence the existence of in suflficiently high types 
follows from the axiom of infinity (*27525). 

To prove that any two continuous series are similar, we use *27l"39. By 
the definition, if P and Q are continuous, they contain respectively two 
median classes a and /3, such that P^a and Q p /8 are rational series. Hence 
by *273-4, PpasmorQp/3, and therefore PsmorQ, by *27l-39. Also 
obviously PeO .P smor Q .D .Qed. Hence 

*275-32. \-:Pe0.O.0 = 'Nr'P 

and 
*275-33. h.^eNR 



SECTION F] continuous SERIES 219 

*27501. e=SeTnJ)ednmed"iio Df 

*275-l. hrPe^.s.PeSernDed.alNoAi^d'P 
[(*275-01)] 

*27511. [■:.Pee. = :PeSeinBed:(-s.a).aeiio.Bp'a = a'P.aCG'P 
[*275-l . *27l-2] 

*27512. f-;:Pe6'. = :.P6SernDed:.(aa):a€No: 

ajPy . D«,„ . a ! a rt P (a; - y) : a C C'P [*275-l . *271-1] 

*27513. i- :.P e e .= :P eSeT nI)ed:(<aR). RQP .Rev .8p'G'R = a'P 

Bern. 
I- . *273-l . *27l-2 . D 

h:P€Sern'Ded.RQP.R6v.Sp'G'R = a'P.:>.G'Reiio-G'Re^d'P. 

[*2751] D.Pee (1) 

f-.*27ri6.D|-:amedP.)8 = anD'Pna'P.D./3medP. (2) 

[*271-15] D.P C^ecomp (3) 

h.*123-17.Dh:Hp(2).PeSer.a€N„nCl'C'P.D./36N„rtCl'(7'P (4) 

h.*27l-l. DI-:/SmedP.D.P"/3 = D'P.P"yS = a'P (5) 

I- . (5) . *37-41 . (2) . Dh:Hp(2).D.D'(PD/S) = /8.a'(PCj8) = ;8 (6) 

h . (3) . (4) . (6) . *273-l . D I- : Hp(4) . D . P p /SeT, (7) 

I- . (2) . *27l-2 . D h : Hp (4) . D . Sp'(7'(P t /3) = Q'P (8) 

H . (7) . (8) . *275-l . D h : P 6 5 . 3 . (g^S) . P p /3 e i? . Sp'C'CP l^) = Q'P (9) 
h . (1) . (9) . D h . Prop 

*27514. \-.d = Gn\"e 
Bern. 

I- . *214-14 . *271-11 . D 1- : P e Ser ft Ded . a 6 N„ n i^d'P . = . 

PeSernDed.aeNo^med'P (1) 

h.(l).*275-l.Dh.Prop 

*275-2. h : P e J? . D . s'P 6 Ser A Ded . 'P"G'P e N„ . 'P"G'P e mwl's'P 
Bern. 

h.*214-33. Dh:Hp.D.s'PeSernDed (1) 

h . *204-35 . D I- : Hp . D . P"G'P sm G'P . 

[*2731.*123-321] :i .'P"G'P e H, (2) 

h . *271-31 . *2731 . D h : Hp . D .'P"G'P e i^d's'P (3) 

h . (1) . (2) . (3) . D h . Prop 



220 SERIES [part V 

*275-21. h:Pe7?.D.s'Pe5 [*275-21] 

This proposition is of great importance, particularly in the theory of real 
numbers. We shall define the real numbers as segments of the series of 
rational numbers, in order to be sure of their existence. Thus if P is the 
series of rational numbers, s'P> which may be taken to be the series of real 
numbers, is continuous. If P is the series of rational proper fractions, 
excluding 0, s'P is the series of real proper fractions together with and 1 : 
this series is continuous in virtue of the above proposition. 

The above proposition is also useful as enabling us to deduce the existence 
of 8 from that of i;, and thence from that of No, and thence from the axiom 
of infinity. A rise of type is, however, required for the existence-theorems, 
which are given in the following propositions. 



*275-22. 


i-iaS'/oCa.^-a!^"^"'" 


Dem. 






h . *64-o5 . D h : a ! 9? A i«,'a . D . (aP) .Pev-G'PC *„'« . 




[*63-37l] D.(aP).Pe9?.(7'P6«'a. 




[*275-21] D.i's^Q).Qed.C'QCt'a. 




[*64-57] D . a ! n «"'« : D 1- . Prop 


*275-23. 


1- : a ! K„ n i'a . D . a ! ^ n «"'a [*274-44 . *275-22] 


*275-24. 


h : a ! No («) . D . a ! n i^'a; [*275-23 . *64-31-312 . (*65-02)] 


*275-25. 


h : Infin a,x(iv) .':> .-^Id nt^'x 


Dem. 






1- . *123-37 . D h : Hp . D . a ! No («='«) • 




[*275-24] D . a ! ^ " «""<"« • 




[*64-312] D . a ! ^ " ^'« : 3 f- • Prop 


*275-3. 


hiP.Qe^.D.PsmorQ 



Dem. 
h . *27513 . D I- :. Hp . D : P, Q 6 Ser o Ded : 

('^R,S).B,S6v.R<^P-8QQ.C'R€^d'P.C'Semed'Q: 
[*204-41] D:P,Q6SernDed:(aa,i8).amedP./8medQ.P^a,QCi8e7?: 
[*273-4] D:P,Q6SernDed:(aa,/3).amedP.i8medQ.(P^a)smor(Qp;S): 
[*27l-39] D : P smor Q :. D h . Prop 

*275-31. l-:Pe5.PsmorQ.D.Qe^ 

Dem. 

-» — » 

h . *27l-4 . D h : Psmor Q . a ! No -^ med'P . D . a '■ N„ n med'Q (1) 

h . *204-21 . *214-6 . D 1- : P 6 Ser n Ded . P smor Q . D . Q e Ser a Ded (2) 
h . (1) . (2) . *2751 . D h . Prop 

*275-32. h:P6^.D.^ = Nr'P [*275-3-31] 

*275-33. h . 6 NR [*275-32 . *256-54] 



*276. ON SEEIES OF INFINITE SUB-CLASSES OF A SERIES. 

Summary o/" *276. 

The subject of the present number bears the same relation to ^ as that of 
*274 bears to t). We shall consider, in the present number, the arrangement 
of all the infinite sub-classes of a series (together with A) by the principle of 
first differences, i.e. the relation 

■Pel D(-Cls induct wi'A), 

where P is the given series. This relation we will call Pg. It consists 
of Pel with its field limited to terms not belonging to C'P^ (*276-12). It 

will (under a certain hypothesis) contain a part similar to P„ namely P^i 
with its field limited to complements of finite sub-classes of G'P. Hence 
if Peco, Pf will contain an 17, whose field is composed of the complements 
of members of CP, (*276'2). The field of this 17 will be a median class of P«. 
We shall find, also, that Pg e Ser, if Pe ft (*27614), and Pg e Ded, if Pe O infin 
(*276-4). Hence 

*27641. \-:Peo>.O.Pge0 

Also, since Pew. D. 01 'O'P 62"°, and since C"P,eN„, we shall have C"P9e2'*' 
(*276"42). This result is important, since it gives the proposition 

*276-43. h.G"0 = 2^ 

The proof that Pg is Dedekindian if P is an infinite well-ordered series 
is somewhat complicated. We proceed by proving that every sub-class of 
C'Pg has a lower limit or a minimum. In this proof, we observe first of all 
that 

C'P = B'Pg.A = B'Pe (*276-121). 
Hence G'P is the lower limit of the null-class, and A is the minimum of t'A ; 
also if K is any existent sub-class of G'Pg, other than I'A, we have 

liimn (P«)'« = limin (P«)'(/f - I'A). 
Hence if we can prove 

K C G'Pg .a!K.A~e«:.D^.E! limin (Pg)'* (A), 

we shall have CI ex'G'Pg C Q'limin (Pg), 



222 SERIES [part V 

whence, by *214"12'14, we shall have Pg e Ded. Thus we have to prove (A), 
ie. « C D'Pfl . a ! « . D^ . E » limin (Pe)'K, which is *276-39. To prove this 
proposition, consider mmp'(s'K—p'K). This exists unless wel; it is the 
first term which belongs to some members of k but not to others. Those 
members of k to which it belongs precede (in the order Pj) those to which it 

does not belong. Let us call those to which it belongs Tp'k, so that 

^ <— 

Tp = « \ {\ = « n e'minj'(s'K —p'k)}. 

Put also Pm'« = minp'(s'K — p'k) Dft, 

so that we may put Tp'k = k n e-'Pm'it Dft. 

Then if we put A=k\{\Ck.X^k), Tp and A fulfil the hypotheses of *258, 
and we have 

A{Tp,K)en,. 

The series A {Tp, «) proceeds to smaller and smaller sub-classes of k, of which 
any one, say X, consists of terms which come earlier (in the order Pj) than 
any other sub-class of ic not belonging to \. By *258'231, the s&T\esA{Tp, k) 
has an end, namely 

p'(Tp*AyK. 

If this is not null, it must consist of a single term, which will be the minimum 
of K (*276'33). But if it is null, we proceed as follows. Put 

Pti'K = s'^ {(a\) .\6{Tp^AYK.y = p'-Kn P'Pm'M Dft. 
Then Pa'/c will be the lower limit of k. 

In the first place, we easily prove that, since p'{Tp^A)'K = A, if 

\6(?p*4y«-t'A, 

Pm'^ and Tp'X both exist (*276'341). Hence every member of k has 

predecessors in k, and k has no minimum. In the second place, we show 

that 

\{A(Tp.K)}fi.^llM.D. (P«,'\) P (Prn'fi) (*276-34-342), 

-> -♦ 

and that aeX.D.p'Xn P'P^'T^ = an P'P,n'\ (*276-353). 

Hence we find that 

\{A(Tp,K)]/j,.aefi.D. p'\ n P'Pr„.'X =p'ii n P'P^'X = « r. P'PJX . 

D . p'X n'P'Pr^'X C p'fi n'P'PJ/i . 

{p'lJL n'P'PJ/j,) n'P'P^'X = p'Xr^'p'PJ^, 
whence it follows that 

\ 6 {Tp^Ayic - fc'A . D . p'\ n P'P^'X = Pti'« n P'PJX, 

whence, by what was stated above, 

X e (Tp^AYk .aeX.D.an P'P„'X = Pa'ic n P'Pm'^^ (*276-354). 



SECTION F] on series OF INFINITE SUB-CLASSES OF A SERIES 223 

Again, if ae/t, the product of all the members of {Tp^A)'k to which a 
belongs is a member of (Sp^AYk to which a belongs, but if we call this 

product X, Pm'X<^ea. (because, if P^'Xea, aeTp'X, which is contrary to the 
definition of A,). Hence we have 

a 6 « . D . (Pti'«) P« a (*276-36). 
It only remains to prove 

(Pti'«) P»^ ■ 3 ■ (aa) . a e « . oPg/S (*276-3r). 
By the hypothesis, and the definition of P^k, we have 

(a^, X) . \ e {Tp^AYk .zep'Xn P'P„,'\ - j8 . Pt,'« nP'2 = 0n P'z. 
Since this involves E ! Pm'K it involves X =^ A, hence, by what was stated 

above, it involves 

-> -» -» 

(gir, X, a) . X 6 (Tp*Ayie .aeX.zean P'P^'X - /3 . Pti'« nP'2=^n P'z. 

-* — > 

Hence we obtain ^nP'zC P^'k n P'P^'K 

and Pa'KnP'P^'X = anP'P^'\, 

— » 
whence yS n P'^^ C a. 

Hence, by *17011, we have aPg/S. 

This completes the proof of P^'k = tl (Pa)'« (*276"38). Hence, combining 
the two cases, we find that « has a minimum if g lp'{Tp*AyK, and a lower 
limit if ~a ! p\Tp^AyK. Hence E ! limin (P«)*«, in either case (*276-39). 

This completes the proof of Pg e Ded if P e O infin. 



*27601. Pe = Pel D(-Cls induct ut'A) Df 

*27602. J. = S/3(;SCa.i8 + a) Dft [*276] 

*276-03. P„'X = minp'(s'X-p'X) Dft [*276] 

*276-04. Tp = X/i {/i = X n e'P^'X} Dft [*276] 

*276-05. Pt/« = s'f^{(aX).X6(rp*i!)'«;-i'A.7=^'XnP'P^'X} Dft[*276] 

*276-l. h : aP«y8 . = . «, /3 e (Cl'O'P- Cls induct) w I'A . g ! o - /3 - P"(/S - a) 
[*1701 . (*276'01)] 

*27611. H : : P e fi . D : . aP« (8 . = : «, /3 e (Cl'C'P - Cls induct) u I'A : 

(a«) .zea-^.anP'2 = ^nP'z [*251-35 . (*276-01)] 
*27612. h : C'P ~ e 1 . 3 . P, = Pel D (- C'P,) [*274-l7 . *2761 . *170- 1] 
*276121. I- : C'P ~ e Cls induct . D . 

B'Pt = A . B'Pe = C'P . C'Pg = (Cl'C'P - Cls induct) w t'A 
[*l70-31-32-38.(*27601)] 



224 SERIES [part V 

*276122. h : a'P ~ 6 u 1 . 3 . 0'P, w G'Pe = Cl'O'P [*276-121 . *27417] 
*276123. h : a'P ~ 6 Cls induct . s . g ! P^ [*2761-121] 

*27613. h : O'P ~ e w 1 . D . Nc'C'P, +„ Nc'CP^ = 2^^'^'^ 

[*2r6-122.*116-72] 
*27614. hiPefl.D.PgeSer [*251-36 . (*276-01)] 
*276-2. h : P 6 0) . D . (O'P -)"(C1 induct'O'P- I'A) e X„ n i^d'Pg 



I- . *24.-492 . D I- . (0'P-)"(C1 induct'a'P- I'A) sm (01 induct'O'P - t'A) (1) 

f- . (1) . *274-27 . D h : Hp . D . (G'P -)"(C1 induct'C'P - I'A) e N„ (2) 

l-.*200-361.*263-47.D 

-> -> <- 

h : Hp . aP(,/3 . ^ 6 « - y3 . a r. P'^: = (8 n P'^ . 7 = ;8 u P'« . D . 

7 n P'^ = /8 n P'^ = n P'^: . ^: 6 a — 7 . 7~ 6 Cls induct . 

[*276-ll.*l70-16] 3 . aP«7 . 7P«;S (3) 

1- . *263-47 . D h : Hp (3) . D . O'P - 7 6 Cls induct (4) 
h . (3) . (4) . *276-ll . D 

h : Hp . aPfl^ . D . (37) .C'P-ye Cls induct . aPe7 . yPgl3 (5) 
h.*120-7l.Transp.D 

I- : Hp . a e CI induct'CP -I'A.D. (G'P - a) ~ e Cls induct (6) 

h . (6) . *276121 . D h : Hp . D . (G'P -)"(C1 induct'C'P - I'A) C G'Pg (7) 
h . (2) . (5) . *271-1 . (7) . D h . Prop 

The following propositions constitute the proof of 
P 6 12 infin . D . Pg e Ded (*276-4). 

*276-3. 1- : . E ! P^'X, . D : a e Tp'X . s . a e X . P^'X e a : P»'X, = minp'(s'A, - p'X) 
[(*27603-04)] 

*276-301. I- : Peli . \ C Cl'O'P- t'A .X^eOul.D.E! P^'X, . E ! Tp'X 
Bern. 

l-.*40-12-13. Dl-:.^'\ = s'\.D:a,/8e\.Da,p.« = /S (1) 

h . (1) . Transp . *40-23 . D I- : Hp . D . g !s'«-p'« . 
[*250-121] D . E ! minp'(s'K - p'«) : D h . Prop 

*276-302. h : E ! P^'X, . D . P^'X 6 p'fp'X - 1)'\ [*276-3] 

*276303. \-.Tp(-A. (Tp)^ G A 
Dem. 

h . *276-3 . D h : /iTpX . D . /i C X (1) 

l-.*276-302.DI-:/irp\.D./t+X (2) 

h . (1) . (2) . *201-18 . 3 h . Prop 



SECTION F] on series OF INFINITE SUB-CLASSES OF A SERIES 225 

*276'304. \-:^l{A {Tp, k)}\.D . /iC\.p'XCp'/jL. ij,^\. p'X^p'ii 
[*276-302-303] * 

*276-305. V.A(Tp,K)eD, [*2o8-201 . *276-303] 

*276-31. l-:P6n.a!\.\C CVO'P - t'A . \ ~ e D'Tp . 3 . 

X, 6 1 . s'\ = p'\ = I'X [*276-301 . Transp] 

*276-32. l-:.Pen.\~60«-<l.\CD'P8.D: 

-» -» 

P^'A, 6 p'Tp'X - p'A, : a 6 \ . D„ . a r. P'P^'A, = p'X n P'Pm'X 

Dem. 

V . *276-301 . D h : Hp . D . E ! Tp'\ . E ! P^'X . (1) 

[*276-302] D . P™'\ e^'Tp'X - p'\ (2) 

h . (1) . *276-3 ; D h : Hp . D . P'F,„'X n s'X, = P'P^'X n p'X (3) 

h . (2) . (3) . D I- . Prop 

*276-321. I- : Hp *276-32 . a e Tp'\ . ^ e \ - Tp'X . . aPg^ 
Dem. 
h . *276-3-32 . D h : Hp . D . P^'X e a - /3 . a n P'Pm''>^ = /8 n P'P^'X . 
[*27611] D . 0P9/3 Oh. Prop 

*276-322. h : Hp *276-32 . /j, e (Tp^Ayx .ae/i.^eX- /j-.D . aPg^ 

Dem. 
h .*40"23.Dh :. pC(Tp*Ay\ ."s^l p : fj-ep .aeii.^6\- /i. D^,a,p ■ aP«/8 : D : 

aep'p. fieX-p'p.':ia,fi.aPg0 (1) 
h . (1) . *276-321 . *258-241 . D h . Prop 

*276-33. h : Hp *276-32 . a ! p'(Tp*Ayx . D . 't'p'(rp*^)'\, = min {PeYX 

Dem. 
h . *276-31 . *258-231 . D h : Hp . D .p'{Tp*Ayx e 1 (1) 

I- . (1) . *276-322 . Dh:liip.a6\-p'{Tp*Ay\.:>.{i,'p'(Tp^A.y\]Pea (2) 
F- . (1) . (2) . D h . Prop 

*276-331. I- : Hp *276-32 . g ! p'(Tp*Ay\ . D . E ! min (Ps)'^ [*276-33] 

*276-34. h : Hp *276-32 . fj,Tp\ . /^ e B'Tp . D . (P„'X) P (P^V) 

h . *276-3 . 3 h : Hp . 3 . PJ\ = minp'(s'X - p'\) (1) 

h . *276-3-304 . 3 h : Hp . D . P^'/* e (s'X - p'X) (2) 

1- . *276-302 . 3 h : Hp . 3 . P^'Xep'/j. . P^'fi ~ ep'p, . 
[*1312] 3.P^'X + P^'/i (3) 

h . (I) . (2) . (3) . 3 I- . Prop 

R. & W. III. 15 



226 SERIES [part V 

*276-341. f-:.Hp*276-32.p'(rp*4)'X = A.D: 

Pr„."iTp*Ayx C P"P„,"(Tp*Ay\ . P^'\Tp*Ay\ ~ e Cls induct : 

/i e {Tp*Ayx - t'A . D^ . E ! Tp',x . E ! P^V 
Dem. 

V . *258-231 . *2r6-301 . D 

h :. Hp . D : /i 6 (rp*^)'X - iy(Tp*Ayx . D . E ! Tp'yu, . E ! PJfA, : 
[*276-34.Hp] D:f,€iTp*AyK.E\P„,',i.D.(PJfi)P(P„,'Tp'fi) (1) 
I- . (1) . *261-26 . Transp . D I- . Prop 

*276-34:2. h : Hp *276-341 . \ {A (Tp, «)}/.. E ! P^V ■ ^ • (^m'^) P (PmV) 

Dem. 
h . *276-3 . D 

I- :: Hp : p C (rp*4)'K . g ! p . g ! jj'/a : D :. P^'pV e s'pV - iJ'p'/o :• 
[*40-lll ] D :. (aa) .aep'p. Pm'p'p e « : (a«) -aep'p. Pm'p'P ~ e a :. 

[*40'1.*11"26] D:.\e/3.DA:(aa).a6X.PTO'p'/3 6a: (ga) .ae\.P„'p'p<^ea:. 
[*40-l-ll] D:.\6/j.DA.P^ype(s'X-y\) (1) 

h . (1) . *276-302 . D h :. Hp (1) . D : 

Tp'X ep.Xep.'D^. PJX e p'Tp'X . PJp'p ~ 6 p'Tp'X : 

[*13-12] D:Tp'\ep.Xep.D;,.P„'X + P„yp (2) 

h . (1) . (2) . *276-3 . D h : Hp (1) . Tp'X ep.Xep.D. (P^'X) P {Pm'p'p) (3) 
t- . (3) . *276-34 . *258-241 . D I- . Prop 

*276-35. l-:.Pen.«;CD'Pfl.a!/«:. p'{Tp^A)'K = A . D : 

X e {Tp*AyK - I'A . D . P^'X e^'^p'X n P'P^'Tp'X 
Dem. 

h . *276-341 . D h : Hp . X e {Tp^AyK - t'A . D . E ! Tp'X . 

[*276-302-34] D . P^'X e jj'Tp'X n P'PJTp'X : D I- . Prop 

*276-351. I- : Hp *276-35 . D . P^"(rp*^)'/«; C Pa'/c 
Dem. 

l-.*276-3. DI-.~E!P»'A (1) 

h . *276-35 . (*276-05) . D h : Hp . X e (Tp*^)'« - t'A . D . P^'X e Pt,'« (2) 
h . (1) . (2) . D I- . Prop 
*276-352. h : Hp *276-35 . D . P^'k ~ e Cls induct [*276-35r341] 
*276-353. f- : Hp *276-35 . X e (2V*^)'k .X{A{Tp,k)]/i . ae jj, .:i . 

p'X n 'P'PJX = p'fi r^'P'P^'X = an P'P^'X 
Dem. 
l-.*276-304. DhiHp.D.aeX (1) 

h . *276-35-31 . Transp . D h : Hp . D . E ! P^'X . X~ eO u 1 (2) 

I- . (1) . (2) . *276-32 . D h : Hp . D . _p'X n P'P^'X = an P'P^'X (3) 



SECTION F] on series OF INFINITE SUB-CLASSES OF A SERIES 227 

h . (3) . D 1- :. Hp . D^yS 6/i .':>p.ar^ P'P,„'X = /8 n P'P^'X (4) 

1- . (4) . D h : Hp . D . a n IP'PJX = p'fi n 'P'P,^'X (5) 

h . (3) . (5) . D h . Prop 

*276-354. I- : Hp *276-35 . \ e (Tp*AyK .aeX.D. 

Pa'K n'P'PJX = p'X nlP'P^'X = an P'P^'X. 
Bern. 

h . *276-353 . D 1- : Hp . g ! ^ . \ {^ (Tp, «)) /i . D . 

-> — > 

[*22-47] 0.(p'fin'P'Pj,jL)nP'P^'XCp'\n'P'PJ\ (1) 

1- . *276-353 . 3 h : Hp . ytt {4 (rp , «)} X . D . ;, V '^ ^'i'mV = i)'\ n P'P^V 
[*276-342] Cij'XnP'P^'X, (2) 

h . (1) . (2) . *276-305 . D 

h :Hp . ^L6(Tp*AyK - I'A . D .(p'^nP'PJ^) n P'PJXCp'X nP'PJX (3) 
h . (3) . *276-32 . (*27605) . D h . Prop 

*276-355. h : Hp *276-35 . a e «: . D . (gX) . X e (Tp*AyK . a e X . P^'X ~ e a 
Bern. 

f- .*40-l . D h :. Hp . D : (gX) . X6(2V*4)'« . a~ eX : 
[*276-305] D : (gX) : X e (Tp* J.)'* . a ~ e X : /i {^ (?>, «)} X . D^ . a e /i (1) 

h . *40-l .D\-:./ji{A (Tp, «)} X . D^ . aeyii : X =2)'J. (Tp, k)'X : D . aeX (2) 
h . (1) . (2) . Transp . D 

1- : Hp . D . (gX, /t) . /*, X e (Tp* J.)'/e . X = ?pV . a e /x . a ~ e X . 
[*276-3] D . (a/t) . /I e (Tp*AyK .ae/i. P^'fJ. ~ e a : D h . Prop 

*276-36. h : Hp *276-35 . a e « . D . (Pu'k) PgO 

l-.*276-351-355-354.D 

I- : Hp . D . (gX) . X e (Tp*^)'* . P^'XePu'/c - a . P^'KnP'P^'X = a nP'P^'X . 
[*276-352] D . (Pa'«) P^a : D h . Prop 

*276-361. h : Hp *276-35 . D . « C P^'Pti'/e [*276-36] 

*276-37. h : Hp *276-35 . (Pa'x) Pg^ ■ 3 ■ (ga) . a e k . oPeyg 
Bern. 

h . *276-l 1 . D h : Hp . D . (g^) . ^: e Pa'x - /S . Pu'/c nP'z = ^ nP'z . 

[(*27605)] D . (gi?, X) . X e (Tp*^)'k .zep'Xn P'P,n'\ - ^ . 

Pa'fcr^P'z^^n'P'z. 
[*276-354] D . (gz, X, o) . X e (rp*^!)'^ .ae\.zea-0. 

'P'z CP'PJX . a n P'P^'X = /8 n P'P^'X . 

[Fact.*276-304] .{'^z,a).a€K .zea-0 .^ nP'zCa. 
[*170-11] 3 . (ga) . a e « . oPg^ Oh. Prop 

15—2 



228 SERIES [part V 

*276-38. l-iPeO./cCD'Pe.a!*:. f(Tp^A )'« = A . D . P^'k = tl {Pt)'K 
[*276-361-37] 

*276-381. h : P 6 n . K C D'P, . a ! « . ^'(Tp*^)'* = A . D . E ! tl (Pfl)'« 
[*276-38] 

*276-39. h : P 6 fl . K C D'Pj . g ! « . D . E ! limin {P^Yk. [*2r6-331-381] 

In the following proposition, the only reason why P has to be infinite is 
in order that Pg may exist ; for " Ded " was so defined as to exclude A. 

*276-4. h : P e O infin . D . P^ e Ded 

Dem. 
h.*276-121.*207-3.*20518.DI-:Hp.D.liminp'A = O'P.liminp't'A = A (1) 
1- . *206-7 . D h : Hp . « C G'Pe . A e k . « + t'A . D . 

prec {PgYK = prec (Pe)'(K - I'A) (2) 
I- . *205-192 . D h : Hp (2) . D . nim {P^Yk = min (Pe)'(K - I'A) (3) 

h . (2) . (3) . D h : Hp (2) . D . li^n {P;)'k. = lii^n {PeY{ic - t'A) . 
[*276-39] D . E ! limin (P,)'k; (4) 

h . (1) . (4) . *276-39 . D H :. Hp . 3 : K C G'P, . D, . E ! limin (Pe)'* : 
[*214-12-14] D : Pfl e Ded :. D h . Prop 

*276-41. l-:P6ft).D.P«60 [*276-2-414 . *2751] 

*276-42. I- : P 6 o) . D . CP^ e 2No 
i)em. 

I- . *27613 . *274-27 . 3 h : Hp . D . Nc'CP^ +„ N„ = 2Ko (1) 

I- . *276-2 . D h : Hp . D . (g^) . Nc'0'P« = /^ +„ N„ . 

[*123-421] D . Nc'O'Pe +„ X„ = Nc'O'Pg (2) 

I- . (1) . (2) . D h . Prop 

*276-43. I- . G"d = 2»<» 

Bern. 

h . *276-42-41 .Dl-:a!a).D.a! C'd n 2K« . 

[*100-42.*27533.*152-71] D . G"0 = 2^0 (1) 

h.*276-ll.*263-101. :>\-:co = A.D.e = A (2) 

h . *263-101 . *116-204 . D h : « = A . D . 2K» = A (3) 

|-.(2).(3). DI-:qj = A.D.O"0 = 2«» (4) 

I- . (1) . (4) . D h . Prop 

The propositions proved in the present number are capable of being to 
some extent generalized. Also we can prove 

i- . 6 = ((0 exp,. m) + 1. 



SECTION F] on series OF INFINITE SUB-CLASSES OF A SERIES 229 

For this purpose, we prove first that if P, Q are well-ordered series, P^ is 

Dedekindian (except that«lf ~ E ! B'P, P^ has no last term) ; i.e. we prove 

P.Qen . D : XC C«PO . a !\. Da . E ! limin(P«)'\. 

For this purpose, assuming X C O'P^ ■ a ! ^, put 

Qr„.'X = ming'^ (s'X'y ~ e u 1), 

Tp'X = X n # {M'Qm'X = mmp's'X'Qrn'X], 

A — Xp,(ij,CX . fi^ X), 

(PQYX = s'N {(a/x) . /. e {Tp*Ayx . N=(p'^i) IQ'Qm'i^}. 

We can then show, by steps closely analogous to those in the proof of P^eDed, 
that we have 

a ! p'{Tp*Ayx . D . ^LY{Tp*Ayx = min (P«)'X, 

~a lp'(Tp*Ayx . D . (PQyx = prec (P«)'X,, 

whence, in either ease, E ! limin (P«)'A,. 

Hence we have 

|-:P,Q6n.E!£'P.D.P«6Ded, 

f-:P,QeIl.~E!£'P.^~6C"P«.D.P«-f*^6Ded. 

We have therefore h . (a> exp^ w) -j- 1 C Ded. 

We now have to prove 

Q 6 (to exp^ «) -j- 1 . 3 . a ! No " med'Q. 

For this purpose, it will be sufficient to prove 

P 6 o) . D . a '. No '>med'(PO. 

The No in question will be the class of those members of G'(P^) in which, 
from a certain point onward, the correlate of every member of C'P is B'P. 
We have 
ilf (PO N. = :M, Ne(C'P t G'PVG'P : 

(aa;) .xeC'P. MlP'x = NfP'x . (M'x) P (JV'a;). 
Now consider the relation 

L = M [P^'x ^Jyi P,'x c; (l'B'P) t P'P/a;, 

where (ilf'P,'«)P2/. 

Then M (P^) L . L (P^) K Also L has B'P for the correlate of every term 
after Pi'a;. Hence it is determined by the correlates of the terms up to and 

including P^'x. Thus, putting z = Pi'x, we have to consider the class of 
relations 

fji = x {(a^) . ^ 6 a'P . z 6 1 ^ cis . a'x = p*'^ . D'Z c cp}. 



230 SERIES [part V 

If X 6 /i, Z vy (I'B'P) '[ P'maxp'Q'Z is a member of G'P^. We have there- 
fore only to show that /a e No. 

To show that /tteNo, we observe that if X e /j,, D'X and d'X are both 
inductive classes; hence each has a maximum. Let X and X' be two 
members of fi, and let us put 

X = maxp'D'Z . x' = maxp'D'Z' . y = maxp'Q'Z . y' = maxp'a'X'. 

If x = fip and y = vp, put « +p2/ = (yu. +e i')p. Then put X before Z' if 
{x 4-p y) P («' +p y'), or it x+py = x +p y' . yPy". But if a; +p y = «' +p y' and 
y= y', i.e. if x = x' .y = ]f , take the immediate predecessors of m, y, x', y' in 
D'Z, Q'Z, D'Z', Q'Z' respectively, and apply the same tests to them, and 
so on, until we come to a difference. In this way, we obtain an arrangement 
by last differences (in a slightly extended sense), and this arrangement is 
easily shown to be an m. Hence /a e No. Hence the class 

" = 7 {(aZ) . Z 6 yii . 7 = Z vy {I'B'P) f V'maxp'Q'Z} 

is an No, and we have already shown that it is a median class of G'P^, 
Hence 

t-:P6a).D.a!N„n vaeA\PP). 

The same class will be a median class of P^ -^ Z, if Z~ e G'P^. Hence 

I- : Pe <B . Z'^eC'P'' . D . g ! No n ^A'{P''-^Z). 

Hence, by what was proved earlier, 

V -.Peto.Zr^eG'P^ .:> .{P'-^Z)ee, 

i.e. h . (o) exp, w)-\-\ = 6. 



PART VI. 

QUANTITY. 



SUMMARY OF PART VI. 

The purpose of this Part is to explain the kinds of applications of 
numbers which may be called measurement. For this purpose, we have 
first to consider generalizations of number. The numbers dealt with hitherto 
have been only integers (cardinal or ordinal) ; accordingly, in Section A, we 
consider positive and negative integers, ratios, and real numbers. (Complex 
numbers are dealt with later, under geometry, because they do not form 
a one-dimensional series.) 

In Section B, we deal with what may be called " kinds " of quantity :. 
thus e.g. masses, spatial distances, velocities, each form one kind of quantity. 
We consider each kind of quantity as what may be called a " vector-family," 
i.e. a class of one-one relations all having the same converse domain, and all 
having their domain contained in their converse domain. In such a case as 
spatial distances, the applicability of this view is obvious ; in such a case 
as masses, the view becomes applicable by considering e.g. one gramme 
as + one gramme, i.e. as the relation of a mass m to a mass m' when m 
exceeds m' by one gramme. What is commonly called simply one gramme 
will then be the mass which has the relation + one gramme to the zero 
of mass. The reasons for. treating quantities as vectors will be explained in 
Section B. Various different kinds of vector-families will be considered, the 
object being to obtain families whose members are capable of measurement 
either by means of ratios or by means of real numbers. 

Section C is concerned with measurement, i.e. with the discovery of 
ratios, or of the relations expressed by real numbers, between the members 
of a vector-family. A family of vectors is measurable if it contains 
a member T (the unit) such that any other member S has to y a relation 
which is either a ratio or a real number. It will be shown that certain 
sorts of vector-families are in this sense measurable, and that measurement 
so defined has the mathematical properties which we expect it to possess. 

Section D deals with cyclic families of vectors, such as angles or elliptic 
straight lines. The theory of measurement as applied to such families 
presents peculiar features, owing to the fact that any number of complete 
revolutions may be added to a vector without altering it. Thus there is not 
a single ratio of two vectors, but many ratios, of which we select one as the 
principal ratio. 



SECTION A. 

GENERALIZATION OF NUMBER. 

Swmmary of Section A. 

In this section, we ftrst define the series of positive and negative 
integers. If /i is a cardinal, the corresponding positive and negative 
integers are the relations +o/i and -„/*, or rather (+|.At) t (NC induct — I'A) 
and (— ,. jj) ^ (NC induct — I'A). (It will be observed that a positive integer 
must not be confounded with the corresponding signless integer, for while 
the former is a relation, the latter is a class of classes.) We next proceed to 
numerically-defined powers of relations, i.e. to R", where v is an inductive 
cardinal. We have already defined R^ and R^, but for the definition of ratio 
it is important to define R" generally. If ii e 1 — > 1 . iJpp G J, we shall have 
R'' = R,, and if ReSev, we shall have {Riy = Ry. But these equations do 
not hold in general, and in particular if RCI and v=^0, R'' = R but i?„ = A. 
After a number devoted to relative primes, we proceed to the definition 
of signless ratios, thence to the multiplication and addition of signless ratios, 
thence to negative ratios, and thence to the generalized addition and 
multiplication which includes negative ratios. (In the case of ratios, signless 
ratios are identical with positive ratios. This is possible because signless 
ratios, unlike signless integers, are already relations.) We then proceed 
to the definition of real numbers, positive and negative, and to the addition 
and multiplication of real numbers. At each stage, we prove the com- 
mutative, associative, and distributive laws, and whatever else may seem 
necessary, for the particular kind of addition and multiplication in question. 

Great difiiculties are caused, in this section, by the existence-theorems 
and the question of types. These difficulties disappear if the axiom of 
infinity is assumed, but it seems improper to make the theory of (say) 2/3 
depend upon the assumption that the number of objects in the universe 
is not finite. We have, accordingly, taken pains not to make this 
assumption, except where, as in the theor}' of real numbers, it is really 
essential, and not merely convenient. When the axiom of infinity is 
required, it is always explicitly stated in the hypothesis, so that our 
propositions, as enunciated, are true even if the axiom of infinity is false. 



*300. POSITIVE AND NEGATIVE INTEGERS, AND NUMEEICAL 

RELATIONS. 

Summary of *300. 

In this number, we introduce three definitions. We first define " TJ" as 
meaning the relation which holds between fi+aV and /t whenever /j. and v 
are existent inductive cardinals of the same type, and v + O, and /i+c" exists 
in this type. Thus U is the relation " greater than " confined to existent 
inductive cardinals of the same type. The definition is : 

*300-01. C/'=(+„l)poC(NCinduct-t'A) Df 

Then if fj, is an inductive cardinal which exists in the type in question, 

Uy, and Ufj, are the corresponding positive and negative integers, where " Uf^" 
has the meaning defined in *121. It will be observed that OU^/i, so that 
f/j» exists, when fi exists in the type in question. We prove (*300'15) that 
Z7 is a series, and (*300'14) that its field consists of all existent inductive 
cardinals of the type in question, its domain consists of all its field except 0, 
and its converse domain of all its field except the greatest (if any). If the 
axiom of infinity holds, G'U consists of all inductive cardinals. 

It will be observed that IT arranges the inductive cardinals in descending 
order of magnitude. The reason for choosing this order instead of the 
converse is that U is less required in its serial use than as leading to the 
functional relations U^. As explained at the end of Part I, Section D, there 
is a broad difference between functional and serial relations, and this 
produces, where one relation (or its derivatives) is to have both uses, a 
certain conflict of convenience as to the sense in which the relation is to be 
taken. Considered as arranging the integers in a series, U would naturally 
be defined so as to arrange them in ascending order of magnitude, as was 
done with "N" in *123. But considered as functional relations, it is more 
convenient and more natural to take (say) +,, 1 as the relation to start with, 
and —el as its converse. Thus we want /lUiV when /j, = v+e'i-, i.e. we want 
U^'v = v+cl; and this requires the definition of U given above. 

We prove in this number (*300'23) that Z7is well-ordered, and (*300-21-22) 
is either finite or a progression. We also prove (*300'17'18) that, if /j, is any 



236 QUANTixr [part vi 

typically indefinite inductive cardinal, /i and /i+el will belong to C'U if U 
is taken in a suflficiently high type. 

Our other two definitions in this number define two classes of relations 
which are of vital importance in the theory of ratio. We define numerical 
relations, which are called " Rel num," as one-one relations whose powers are 
all contained in diversity, i.e. we put 

*30002. Rel num = (1 -> 1) n ^ (Pot'R C BA'J) Df 

We thus have (*300-3) 

1- : i? 6 Rel num . = . Rel^l . R^^QJ. 

It will be remembered that the hypothesis i26(Cls->l)u(l— >Cls). J?poG J 
played a great part in *121, and in all later work which depended upon *121. 

When both R and R fulfil this hypothesis, we have R e Rel num, and 
vice versa. We prove (*300'44) that if (t is an inductive cardinal not zero, 

and P is a series, then P^ is a numerical relation, and so is P„. If P is an 

endless well-ordered series, finid'P {i.e. the class of relations P„) is what 

(in Section B) we shall call a vector-family : P^ is the vector which carries 
a term a steps along the series. 

In order to be able to deal with zero, we have to consider the application 
of ratios, not only to such relations as are numerical in the above sense, 
but also to relations contained in identity, because a relation contained 
in identity may be regarded as a zero vector, so that {e.g.) if P is a 
series, / f" G'F will have a zero ratio to P, if a- is an inductive cardinal 
other than 0. 

We therefore introduce a class " Rel uura id " consisting of numerical 
relations together with such as are contained in identity; these maybe called 
numerical or identical relations. They may be defined as one-one relations 
whose powers, other than R„, are contained in diversity, because, if i? G /, 
there are no powers other than Rg. Thus we put 

*30003. Rel num id = (1 -> 1) n E (Potid'i? - I'Ro C Rl'J) Df 

and we then prove 
*300-33. I- . Rel num id = Rl'J w Rel num 

For the application of ratio, it is important to know under what circum- 
stances there exists a numerical relation R such that R^ is not null. We 
prove (*300'45) that if a- is an inductive cardinal, and P is a series of 

o--|-ol terms, then {B'P) P^{B'P). Now we also prove (*300-44) that if 
P is a series, and R = Pi, P„ = Ra and R is a numerical relation. Hence 
it follows, by *262-211, that if o- =)= and a is a class of o- +„ 1 terms, there is 



SECTION A] POSITIVE AND NEGATIVE INTEGERS 237 

a numerical relation R whose field is of the same type as a and for which R„ 
exists. Remembering *39©'14 (quoted above), this proposition is : 

*300-46. hzaea'U-i'O.D. 

(gP, iJ) . P 6 ((X +e l)r . -K = Pi . -R 6 Kel num . t'G'B = t^'a- . (B'B) R„ (B'R) 
We have conversely (*300-47) 
I- : £ eRel num . g ! iJ^ . D . o-eNCind . g ! (o- +„ 1) a t'G'B . o- n t'G'Bea'U, 
where " NO ind " has the meaning defined in *126, i.e. "treNCind" means 
that o- is a typicallj' indefinite cardinal. 

The number ends by propositions proving (*300'52) that U^ is a 
numerical relation, that (*300'57) 

±l{Ui),n(U^)^.:i.^XaVeG'U.^X^v = T}X^fi, 
and analogous theorems. 

*30001. U = (+0 l)po t (N C induct - I'A) Df 

*30002. Rel num = (1 -♦ 1) n ^ (Pot'i? C El'J) Df 

*300-03. Rel num id = (1 -» 1) n ^ (Potid'P - l'B„ C Rl'J) Df 
*3001. h :/t£/'i'.s./i(+ol)p„i;./i,i/eNC induct -t'A [(*300-01)] 
*30011. [■:./iUv. = : 

II, V e NC induct - t'A : (gX.) . X, e NC induct — t'O . jit = v +o^ : 
= : fi,ve NO induct — t'A : (g;\) .\=fO./*=v+eX.: 
s : /i, 1/ 6 NC induct - t'A : (a\) . \ e NO - c'O . /j. = v +a\ 
[*300-l . *120-42-428-462-452 . *110-4] 

«300'12. i-:fjLUv. = .fi,ve NO induct — t'A . i/ < /t . 

= . fi,ve NC induct . v < /* . 

= • fie NO induct . i/ < /* 
[*300-ll . *ll7-3 . *120-42 . *ll7-26 . *110-6 . *117-15 . *120-48] 
*30013. V.UQ.J [*30012 . *117-42] 

*300-14. \-.G'U= NC induct - t'A . 'D'U= NC induct - t'A - t'O . 

Q' CT = NO induct n D (g ! i; +e 1) = «> (i' +b 1 e NO induct - t'A) . 

B'U=Q 
[*30012 . *117-511 . *120122 . *101-241 . *120-429-422] 

*30015. h - fTe Ser [*300-13 . *1 20-441] 

*30016. h : a e Cls induct . D . N„c'a eC'U r, t^'a . N„c'a eG'{Ul t^'a) 



Bern. 



h . *120-21 . D h : Hp . D . N„c'« e NC induct (1) 

h . *103-13 . D h . Noc'a 4= A (2) 

I- . *103-11 . 3 h . N„c'a e fa (3) 
h . (1) . (2) . (3) . *300-14 . D h . Prop 



238 QUANTITY [PAET VI 

*30017. h : /i e NC ind . D . (ga) . fint'aeC'U . fieG'iUl f'a) 
Dem. 

V . *126-1 . D I- : Hp . D . (ga) . a e Cls induct . /j, = Nc'a . g ! /t . 

[*103-34] D . (ga) . a e Cls induct .fint'a= N„c'a (1) 

I- . (1) .*300-16 . D I- : Hp . D . (aa) .finfaeCU. (2) 

[*65-13] D.(aa)./z6C"fr./iC«'a. 

[*63-5] D.(ao)./xeC"?7./t6«"a (3) 

h . (2) . (3) . D h . Prop 

*30018. I- : /A 6 NC ind . D . 

(a<r) . 2^ 6 (7'( (7 1 «"<7) . i/J. +^1) n t'a 6 C'U . ^ e a'(U l t^a) 
[*126-13-15 . *300-l7-14] 

*300181. h: ^e'NC'md. fin t'aeG'U.D. 

2" n f'a. eG'U.(fi+^l)n f'a e C U . fj. n f'a e a'U 
[*1 26-23 . *300-14] 

*300-2. h : Infin aK.D.U=N'j„ 

Here N has the meaning defined in *263'02. 
Bern. 

t- . *300-1 . *1251 . D h :. Hp . D : /iZZi; . s . ju,. 1/ 6 NC induct . fi (+„ l)po v . 
[*120-l.*91-574] = . K+o 1)* . ;tt (+„ l)p„ V . 

[*96-13] =-/*{(+cl)r(+^'0]po^- 

[(*263-02.*l 20-01)] = . vN^ /i :. D h . Prop 

*300-21. h : Infin ax . D . CTe o [*300-2 . *263-12] 

*300-22. I- : ~ Infin ax . D . C/" 6 fl induct 
Dem. 

\- . *125-16-24 . Transp . D h : Hp . D . C" ?7 e Cls induct (1) 

I- . (1) . *300-15 . *261-32 . D h . Prop 

*300-23. h.Ueil [*300-21-22] 

*300-231. \-:/iU^v. = .fi,ve^C induct - I'A.fi^v+^l . 
= .fie NC induct — I'A . fi = v+cl. 
= . /* 6 NC induct - t'A - I'O .v = fi—„l . 
= .ve NC induct — t'A . i/ = /t — ^ 1 
Dem. 
l-.*300-15-12.*201-63.D 



[*120-429] = 
[*ll7-25] = 



fi,ve NC induct — t'A . i/ < yu. : ~ (gX) . i; < \ . \ < /x : 
/*, 1/ e NC induct - I'A .v <./jLiv+el'^/ji. /j.'^v+al : 
/i, 1/ e NC induct — I'A . /A = 1/ +(. 1 (1) 



I- . (1) . *120-422-424-423 . D h . Prop 



SECTION a] positive AND NEGATIVE INTEGERS 239 

*300-232. f- : /t 6 NO induct . D . 

U,. = (+0 /t) t (^C induct - t'A) . U„ = (-„ /*) I (NO induct - t'A) 

For the definition of f/)., see *121'02. 

i)em. 
I- . *121-302 . *300-15 .Dl-:pC/'„o-. = .aea'l7.p = (r. 
[*300-14.*110-6] = . p, o- € NO induct - t'A . p = o- +„ (1) 

h . *260-22-28 . *121-332 . D 

f- : ^/^ = (+c IJ) t (NC induct - I'A) . D . f7^+„i = (+„ /*) p (NO induct - I'A) | fT-j 
[*300-231] = (+„ /i) t (NC induct - t'A) | (+„ 1) I (NO induct - t'A) 

[*120-45-452] ={+e(/x+ol)}D(NC induct -t'A) (2) 

h. (1). (2). Induct. Dh. Prop 

*300-24. h : /i e NC induct . D . D' J/^ = U^'ij, = NO induct n i) (v > ;u,) 
[*300-232 . *lir-31 . *120-45] 

*300-25. h : /i 6 NC induct . D . 

B'U^ = U'fj. = NC induct n t> {v < /j,) = U (0 i- fi) 
[*300-232-24-12] 

*300-26. V:fi6G'U. = .,j,U^0. = .±lU^l (G'U) [*300-23214 . *110-6] 

Here the fi in " U^" is of higher type than the /m in " /leCU," because 
the interval f7 (0 i— i /i) is composed of members eiach of which is of the same 
type as ft,. 

*300-3. l-:i2eRelnum. = .E6l->1.2?poGJ. = .J26l->l.Pot'i2CRl'J 
[(*300-02)] 

*300-31. I- : ii 6 Eel num iA. = .Rel^l. Potid'jR - i'iJ„ C Rl'J 
[(*300-03)] 

*300311. V\RQ.I. = .Ra = R. = .B = I\C'R 
Bern. 

h . *20113-18 . Dh:.iJ;G/.D:a;6 0'i?.D.i?j|5'a;niij^'a! = i'a; (1) 

|-.(l).*12111.Dh:i2G7. D ./pC'i2Gi2„. 

[*121-3] D.i?„ = /pO'i2. 

[*72-92] D.Ro = R = I[C'R (2) 

l-.*121-3. D[-:Ro=B.':>.R<lI (3) 

h . (2) . (3) . D h . Prop 

*300-312. h : i? G / . D . Potid'i? = I'R = t'i2„ [*300-311 . *50-72 . Induct] 

*300-313. h:Re Rel num id . D . i2* - E, G J [*300-31 . *91-55] 



240 QUANTITY [part VI 

*300-32. l-:i?6Relnumid. D.i?o = /rC"i2 
Dem. 

V . *91-35 . D H . / 1^ C'iS; 6 Potid'i? - Rl ex'J" (1) 

f- . (1) . *300-31 . 3 h . Prop 

*300-321. h : E 6 Rel num id.iJ=t=i2„.D.i2GJ".a!i? [*300-31] 

*300-322. h : ii G J^. D . i?p„ n i?„ = A 
Dem. 

h.*121-3.Dh:a;i?p„3/.a;4=2/.D.~(a;i2„2/) (1) 

l-.*50-24.DF:.Hp.D:~(a;Ba;): (2) 

[*91-57] D : a;i2p„ x.:3.x (B^„ \R)x. 

[*121-103.(2)] D . E (« i-i «) 4= i'« . 

[*121-11] D.~(a;ii„«) (3) 

1- . (1) . (3) . D I- . Prop 

*300-323. h : i? 6 Rel num id . E =t= i?„ . D . iilpo G J 
Dem. 

h . *300-321-322 . D I- : Hp . D . Epo n E„= A . 

[*300-32] D . iJpo n / p C"i? = A : D h . Prop 

*300-324. h :. E 6 Rel num id . 3 : E G / . v . E e Rel num 

Dem. 
h . *300-311-323 . D h :. Hp . D : ie G / . V . iJpo G J (1) 

I- . *300-32 . D h : ii 6 Rel num id . i?p„ G J . D . Potid'iJ-i'i?„=Pot'i2 (2) 

I- . (2) . *300-31 . D h : i? e Rel num id . Ep„ G J" . D . Pot'E C Rl' J (3) 

I- . (1) . (3) . *300-3 . D 1- . Prop 

*300-325. [■■.RQI.D.Rel&el num id 
Dem. 

I- . *300-312 . D h : Hp . D . Potid'i? - I'R, = A (1) 

t- . (1) . *300-31 . D h . Prop 

*300-326. I- : E e Rel num .D.ReRel num id 
Dem. 

i-.*121-3.*300-3. DI-:Hp.D.iS;„~6Pot'ii (1) 

1- . *121-302 . *300-3 . 3 I- : Hp . D . i2„ = / 1' C'i? (2) 

I- . (1) . (2) . *91-35 . DI-:Hp.D.Potid'i2-i'i?o = Pot'iJ (3) 

I- . (3) . *300-3-31 . D h . Prop 

*300-33. h . Rel num id = Rl'/ u Rel num [*300-324-325-326] 

*300-34. 1- . A 6 Rel num [*300-3 . *72-l] 

*3004. h . Rel num = Onv"Rel num [*300-3 . *91-522] 

*300-41. I- . Rel num id = Cnv"Rel num id [*300-31 . *91-521] 



SECTION a] positive AND NEGATIVE INTEGERS 241 

*300-42. I- : jK e Eel num . D . Pot'R C Eel num 
Dem. 

I- . *91-6 . *92-102 . D 

I- : E 6 Rel num . P e Fot'B . D . P e 1 -» 1 . Pot'P C Bl'J . 

[*300-3] D . P e Rel num Oh. Prop 

«300'43. I- : E 6 Rel num id . D . Potid'J? C Eel num id 
Dem. 

I- . *300-325-312 . D h : i? G / . D . Potid'iJ C Eel num id (1) 

h.*300-325. Dt-./pO'EeEelnumid <2) 

h . (2) . *300'42-326 . 3 h : iJ e Eel num . D . Potid'fl C Eel num id (3) 
h . (1) . (3) . *300-33 . D h . Prop 

*300-44. l-:.PeSer.o-eNCind.D: 

P„, P„ 6 Eel num id : o- 4= . D . P, = (Pi), . P„, P„ e Eel num 
Dem. 

h . *121-302 . *300-325 . D I- : Hp . o- = . D . P„, P, e Eel num id (1) 

h.*260-28. Dl-:Hp.o- + O.D.P, = (P,)a (2) 

h . *300-3 . *260-22 . D h :. Hp . D : P^ e Eel num : 

[*121-5.*300-42] D : <r 4= . D . (PiV e Eel num . 

[(2).*300-4] D . Pa, P„ e Eel num (3) 

h . (1) . (2) . (3) . D h . Prop 

*300-45. h : o- 6 NC ind . P e (o- +c l)r ■ 3 • (B'P) P, (B'P) 

For the definition of (a- +„ 1),, see *262-03. 

Dem. 

h.*26212.Dh:Hp-D.P6a.a'P€o-+el. 

[*202181.*261-24] D . (B'P) P, (B'P) Oh. Prop 

*300-46. \-:<7€a'U-L'0.D. 

(gP, P) . P e (o- +0 l)r . -B = Pi . -R 6 Eel num . t'G'B = «„'<r . (B'B) B, (B'B) 

Dem. 
h . *30014 . D h : Hp . D . (ga) . a e Cls induct . t'a = t^'a . a e o- +o 1 . 
[*262-211] D . (gP) . P e (a +„ 1), . t'C'P = t,'a- - 

[*300-45] D . (gP) . P e (o- +„ 1 V . f'O'P = t,'a- . (B'P) P„ (B'P) . 

[*300-44.*261-22] D . (gP, i?) . P e (o- +e l)r ■ P = Pi - P e Eel num . 

t'G'B = </o- . (P'P) B^ (B'B) : D h . Prop 

B. & W. Ill, 16 



242 QUANTITY [part VI 

*300-47. hiiJeRelnum. giiZ^.D. 

o-eNCind . a ! (a+„ 1) r^ t'G'R . a n t'G'Rea'U 
Dem. 

I- . *121-11 . D 1- : Hp . D . (ga;, y) . -B (« m y) e o- +c 1 ■ 
[*121-46] D . o- +„ 1 6 NO ind . a ! (o- +c 1) n t'C'R . 

[*120-422.*300-14] D . o- e NO ind . g ! (o- +c 1) " t'G'B . 

ant'G'Rea'U-.Oh.Prop 
*300-48. h:i2G/.7/=t=O.D.i?„ = A 
Dem. 
\- . *300-312-311 . *91-55 . D h : E G 7 . D . J?^ = / f C"i2 (1) 

l-.(l).*121-103.DI-:i?G/.D.i?(a;M2/)=C"i?nt'a;ni'2/ (2) 

h . (2) . *121-11 . D 1- :. i? G 7 . D : xR,y . = .G'Rn I'x ni'yev+^l. 
[*ll7-222] D.r+el^Nc'i'a;. 

[*117-54.*120-124] D.i/+„1 = 1. 

[*110-641.*120-311] D . 1/ = (3) 

h . (3) . Transp . D h . Prop 

*300-481. h : i? 6 Rel num id . i/ + . D . (i?„), = A . {R,\ G R, 

Bern. 

h . *300-32-48 . D I- : Hp . D . (R,\ = A (1) 

h . *300-43-32 . D f- : Hp . D . (i?,)„ = / 1^ G'R, . 

[*121-322.*300-32] D . (R,\ G E„ (2) 

h . (1) . (2) . D h . Prop 

*300-49. h : B 6 Rel num . A ~ e Pot'E . D . O'i? ~ e Cls induct 

JDem. 
I- . *121-5 . D h :. Hp . D : i/eNCinduct . D . g ! i2„ . 
[*12111] D . a ! (z/ +,, 1) n Cl'Ci? :. D I- . Prop 

*300-491. h : (gi?) . E e Rel num . A ~ e Pot'i? . D . In fin ax [*300-49] 
*300-5. I-. CTieRelnum [*300a5-44] 

*300-51. h.Uoe Rel num id [*300-15-44] 

*300-511. \-.U^ = i TJX [*300-21-22 . *263-491] 

*300-52. l-:/ieNCind-i'O.D. f/^eRelnum [*300-15-44] 
*300-53. f-.(Xel)tC"C/eRelnumid [*300-325 .*113-621] 

*300-54. I- : Infin ax . /,t e D't/" - I'l . 3 . (x„ /a) |; D' i7e Rel num 
Bern. 
h.*120-51. DI-:Hp.D.(x„/i)t:D'f;'6l-»l (1) 

V . *126-61 . *113-621 . D F :. Hp . D : (0 {(x,/i) ^ D'C7} a- . D . ^ > o- : 
[*ll7-47-42] D:((x„/*)pD'fr}p„GJ (2) 

h . (1) . (2) . *300-3 . D h . Prop 



SECTION a] positive AND NEGATIVE INTEGERS 243 

*300-55. l-:a!i?pnE,.D.a!(/3+„l)nf'C"i2.p = o- [*12ril . *120-31] 
*300-551. h : a ! J2p n E^ . = . a ! Bp . p = o- [*300-55] 

*300-552. h : i? e Rel num . 3 . (i?|), G J?fx„„ 
Dem. 
H.*121-36. Dh:Hp.^,i;6NOincl-i'0.D.(i?f), = i?j>,,, (1) 

I- .*300-481 .DI-:Hp.f = 0.i' + O.D. (i?f). = A (2) 

h . *300-32-311 . *113-602 .Dh:Hp.^ = 0.i/ = O.D. (J?f), = iJf x.. (3) 
h.*300-481.*113-602. D h :Hp. f 4=0 . i; = . D . (J2f),ei?fKov (4) 
|-.*300-47. DI-:Hp.~(^,i/6NCind).D.(Ej), = A (5) 

I- . (1) . (2) . (3) . (4) . (5) . D h . Prop 

*300-56. h : i? 6 Rel num . g ! {R(\ n {R^ ■ 3 . 

^x^v = vx<,IJ'-{^Xov)r^t'C'Rea'U 
Dem. 

\- . *300-552 . D I- : Hp . D . a ! J?f>,„, A i?,^.,. (1) 

I- . (1) . *300-55 . D I- . Prop 
*300-57. l-:a!(C7-j)„n(f7,V.D.^x„i;6C'i7.^x„i; = ^x„/x 

h . *300-5-511-56-552 . D I- : Hp . D . ^ x„ z/ = i? x„ ;u, . a ! ?7jx.. (1) 

I- . (1) . *300-26 . D h . Prop 

By *30056, we have, with the above hypothesis, (^ x^ v) n t'G' Ued'U. 
But here the 17 in Q.'U is of higher type than the U in (^ x,. k) n t'G'U or in 
the hypothesis. In the type of the U in the hypothesis, we have ^x^veC'U, 
not necessarily ^x^ved'U. 

*300-571. V 1.^,7, e'D'U.:i:^\{Ui\f^{Ur;)^. = .^x^veG'U.^x^v = -nx^fi. 

Dem. 
|-.*300-26.DI-:|x„i'6C"f7.fx„i/ = ,,Xe/i.D.(^x,i'){t7-fx..Af7-,x„40 (1) 
I- .*121-36 . D h : Hp . Hp(l) ./. + . Z/ + O.D. f7jx.. = (^a- i^,x.^ = (f7,V (2) 
I- . *300-32 . D h : Hp . Hp (1) . i; = . D . ( [7j), = / p 0' f7f . 
[*300-26] D.0{([7jV}0 (3) 

Similarly h : Hp. Hp(l) ./4 = 0.D .0|(f7,)^} (4) 

H .*113-602 . D h : Hp . Hp (1) . 1/ = . D . /A = (5) 

l-.(l).(2).(3).(4).(5).Dh:Hp.Hp(l).D.a!(f/^).n(C7,V (6) 

f- . (6) . *300-57 . D I- . Prop 

*300-572. V;.^^V>'U.:>:<^\{Ui)y. = ,^x^veQ'V I *300-571 ^ 

16—2 



*301. NUMERICALLY DEFINED POWERS OF RELATIONS. 

Summary of *301. 

In this number, we have to exhibit the powers of a relation R, i.e. the 
various members of Potid'J?, as of the form R", where o- is an inductive 
cardinal. We have already had R^ = R\R and R^ = R^\ R. What we need 
is a definition which shall give 

Now R" is a function of R and o- ; thus we have to exhibit R" in the form 
S'ff, where 8 will be a function of R. That is, we have to define the relation 
)S as a relation of R" to cr, and S must be such that, if it holds between 
R" and a, it holds between R"'^''^ and 0-+0 1. Thus we may take /S as a sum 
of couples, such that if one couple is R" ^ tr, the next is (i?" \R) I (a- +„ 1), 
i.e. such that, if one couple is Q ^ cr, the next is (Q\R) j, (o- +(. l)i Now 

(Q I i?) 4, (c7 +„ 1) = (I i?) II K i)'(Q U). 

Hence, since we want to have R'' = I\' G'R, our class of couples is 

M[M{(\R)U-,l)}^{{irG'R) iO}]. 
Calling this class num (R), we may therefore put 

i?-'={s'num(i?)}'o- Df. 
If we put (I R) II (-(. l) = Rp, the above definitions are 

num (R) = (R^'iil r G'R) i 0} Dft, 
R''={s'mim{R)Ya- Df. 

But the g,bove definition of Rp requires some modification before it can be 
considered quite correct. With the above definition, we have 

RAQio-) = (Q\R)i(a+,l) (1). 

Now since num (R) is defined by means of (Rp)^, and since the definition 

of JSjif contains the hjrpothesis R"/j, C //., it follows that, if num (R) is to be 
significant, the relation — ^ 1 which appears in the definition of Rp . must 
be homogeneous, so that, in (1), a- and tr+^l must be of the same type. 
Hence a; though typically ambiguous, cannot be typically indefinite; 



SECTION a] numerically DEFINED POWERS OF RELATIONS 245 

therefore, if the axiom of infinity is not true, we shall sooner or later arrive 
at ff = A as we travel up tlH inductive cardinals. In that case, we shall have 

iJ"-'! I (a -e 1) 6 num (R) . (R"-'^ \R)iAe num (R), 

(R'-'^R \R)iAe num (R), etc. 

Now if (for example) i? is a cyclic relation, such as that of an angle of 
a polygon to the next angle to the left, we shall not have 
^T-a = ij<r-ci|j2 or R''-'^\R = R''-''^\R\R. 

Hence s'num (R) will fail to be one-many, and i2°' will fail to exist. Hence 
it becomes desirable to restrict a- to cardinals which exist in some assigned 

type, i.e. to replace -„ 1 by (— ^ 1) ^ (NO induct — t'A), i.e. by C/i. 

Thus we now put Rp = (| R) \\ U^ Dft. 

But even this definition is not quite complete, because the type of U is 
not assigned. It makes some difference how the type of U is assigned, for 
if we take as the type of C'U a, type lower than that of i'N„c'<'i2, we may 
find that our numbers become A before we have ceased to obtain fresh 
powers of R. 

For example, suppose the total number of individuals were four, and that 

these were a, x, y, z. Let us write x \,{a,y,...)kv sc ]^ a<ax \,yK) Then 

consider the relation R = x ]^ {a,y) v) a ^ y v» y ]^ {x,z). Then 

R^ = x ],{x,y,z)^Jai{x,z)v^y \, {a, y), 

R' = x\, {a,y,x,z)yjal {a,y)vty \^ {!B,y,z\ 

R* = a)l(y,x,z,a)K/al(y,x,z)vyl {a,y,x,z), 

R' = x l(a,x,y,z)Kia I (a,x,y,z)K/y I (a,x,y,z). 

After this, R^ = R^\R = R' [R''^ etc. But up to R^, each power of R is 
difierent from all its predecessors. If we take t'G'U=t''N„cH'G'R, G'U 
consists only of the numbers 0, 1, 2, 3, 4, and is thus inadequate to deal with 
R\ Hence the type in which we take U must be a suflBciently high type, 
which must increase with the type of iJ. Hence we take G'Uin the type of 
t'Noc'i'iJ, i.e. in the type of t^'R. This is secured by writing U p i^'R in 
place of 17 in the definition of Rp. Hence the final definitions for R" are : 

*301-01. Rp = (\R)\\(U^lf'R) Dft [*301] 

*30102. num (R) = (^VK^ T (^'^) i (0 '^ *"-K)} ^^^ [*301] 

*301-03. R' = {s'nxxm{R)Ya Df 

The two temporary definitions 5it301'01'02 are only to extend to the 
present number. 



246 QUANTITY [part VI 

With the above definitions we have 

*30116. \-:,j,eG'Unf'R. = .E\B^ 

*301-2. \-.R'' = I[G'R.R' = R 

*301-21. h : 1/ s a' C n f 2? . D . iJ-'+'i = R-\R 

*301-23. h:fj,+^veC'UnP"R.:^. R^+"' = R/'\R'' = R^\Rr 

*301-26. l-:P6Potid'i2. = .(a<7).P=i2'' 

I.e. the powers of R are the various relations R". This proposition might 
have been not universally true if we had taken ?7 in a lower type. 

*301-3. ViR(lI.<TeG'Unt"R.:i.R''==R = R, = I[G'R 

It is largely for the sake of this proposition that we require powers 
of relations in dealing with ratio, rather than finid'iJ. For we have 
-R G / . <r =^ . D . Ra = A, so that R„ does not give what is wanted if 
RGI. On the other hand (*301'41), if EeRelnum, we have R" = R„ 
iiaeG'U r\P'R. Thus as applied to numerical relations, R^ may always 
replace R". 

We have, whatever R may be, 
*301-504. }r:fi,veG'Un f'G'R .v^O.D. (Ri^y = iJ^x-" 

The importance of this number will appear in connection with ratios. 



*301-01. Rp = (\R)\\{U,l. f'R) Dft [*301] 



*301-02. num (R) = (Rp)^'{{I T G'R) J, (0 n t"R)} Dft [*301] 

*30103. iJ''= {s'num (i2))'o- Df 

*301-1. \-:<rea'(Ul V'R) .:> .Rp'{Q i C7) = {Q\R) \, \{a +„ 1) n t^'R] 
[*55-61 . (*.301-01)] 

*301101. V:<Tea\Vl 1?'R) . = . o- e Q' T/n V>'R . = . o- e Q' f/" . <r C t^'R 
[*63-5] 

*30ri02. V:ae(l'{Ult"R).= . 

(gX) . \ e 01s induct . g ! - X . i? e «o'X, . o- = Noc'X 
[*30014.*1 03-11] 

*301103. V:<yea\Ul P'R) . = . 

(gX.) . X 6 CIs induct .g!-X.i?6X.(7 = Njc'X 
[*301-102.*r3-71-72] 

*301104. \-:aea'(Ul t^'R) . = . (o- +„ 1) n t^'R e NC induct - t'A 
[*301-101.*300-14] 

*301105. f- : o- 6 a'( P C P'R) . = . (aX) . X e CIs induct . i? e X . o- +„ 1 = Noc'X 
[*301104] 



SECTION a] numerically DEFINED POWERS OF RELATIONS 247 

*301106. \-:(7ea'(Ulf'R). = . (gX) . \ e Cls induct . Reto'X. a-+^l = N„c'\ 
[*301-104] • 

*301107. h : o- 6 a'( C/"^ f'R) . = . o- e NC ind . iJ e s'{a +, 1) 
[*30M06 . *126-1] 

*30111. h'.aea'iUl 1^'R) . = . E ! Rj,'(Q J, a-) [*3011] 

*30ri2. \- : M€nnm(:R) .-D .{'3,P,a).P eVotid'R.cT eO'U ni^'R .M = P I a- 
[*95-22] 

*301-13. \- :Pl06mim(R). D.P = I\-G'R 

Bern. 

h . *90-31 . (*301-02) . D 

h.: P I fienum (R) - 1'{(7 \- C'R) | 0} . D . 

iPi^.){(R,)^\Rp}{(I[G'R)iO}. 

[*30-33.*301-l] D . (P i /i) {RphiR i 1) . 

[*95-22] D.fiU^l. 

[*300-24] D./ii + O (1) 

h . (1) . Transp . D h . Prop 

*301-14. h:P i ^,Qi fi6niim{R).D.P = Q 

Dem. 
V . *120-124 . *90-31 . D 
h:{Si(/.+„l)}(E^)*{(/ra'i?)iO}.D. 

{S i (;. +„ 1)1 {R, i (E^)*} {(/ r G'R) I 01 (1) 

h . (1) . (*301-02) . *3()1-12 . *300-14 . D 

h : ,Sf 4, (/i +„ 1 ) 6 num (22) . D . S 4, (/. +c 1) e -R/'num (22) . a ! /i +„ 1 . 
[*301-1] 

D. (aP,i').p; 1/61^111(2?). ,sfi(/* +ei)=(P|i2)i(i'+„i). a! /*+ci- 

[*55-202.*120-311] 

D.(aP).PJ,/*6num(22).S4,(M+el) = (-P|-B)i(/^+cl) (2) 

h . (2) . D h :. P 4, /i, Q i M e num (22) . Dp, Q . P = Q : D : 

;Sf4,(|,x+el),ri0i*+ol)enum(22).Ds,i..;S=2' (3) 

I- . (3) . *3011213 . Induct . D h . Prop 

*301141. I- . a's'num (22) = C'Un 1^'R 
Dem. 

h . *3011 . D 

h : o- e Q' U" n f '22 . a- e a's'num (22) . D . (<r +,. 1) e a's'num (22) (1) 

H . (1) . *300-14 . Induct . 3 h . Prop 



248 QUANTITY [part VI 

*30115. H . s'num (E) e 1 ^ Cls 
Dem. 

\- .*301-14> .Oh : M,N€num(R) .'3^ia'M na'N .D . M = N (1) 
h . (1) . *72-32 . D h . Prop 

*301-16. b:fieO'Ur^f'B. = .^lR^ [*301141-15 . (*301-03)] 

*30r2. i-.Ro^I^G'R.R'^R [*301-13-16-1 .(*30103)] 

*301-201. h : V e C'U n f'R .:y .{R' I v) e num (E) 
Dem. 

\- . *30116 . (*301-03) . D I- : Hp . D . i?" {s'num (R)} v . 
[*4.1-11] D • (gJlf ) . If 6 num (iJ) . R^Mv . 
[*301-12] D . {'S^M,?, a-) .Memim{R) . ilf = P i a- . R'Mv . 
[*55-13] D.(E''4,v)6num(iJ):DI-.Prop 

*301-21. \-:v€a'Un t"R . D . R'+'^ = R''\R 
Dem. 

\- . *301-1-201 . D h : Hp . D . i?''+»i J, (,; +„ 1), (R-' | i?) | (v +« 1) e num (R) . 
[*301-14] D.ii-+»i = JB''|i?:DI-.Prop 

*301-22. h : E ! iJ" . D . i?" e Potid'E [*301-2011216] 

*301-23. h : ,j,+^v e G'U n t^'R .D . RI-+"' = Ri'\R'' = R-'lRi^ 
[*301-21 . Induct] 

*301-24. h :. <7 6 NO ind : /i< o- . z^ < /i . D^, „ . E'' + E" : D . 

2) em. 

h . *120-442 . D h : Hp . /t< o- . 1/ < o- . i?'' = E" . D . /* = K (1) 
h . (1) . *73-14 . *301-15 . D 

h : Hp . D . Nc'P {(a/t) . /t < ff . P = i?"} = Nc'^ (/t < o-) (2) 

h. (2). *1 20-57. Dh. Prop 

*301-241. I- : Hp *301-24 .0 . a nt''Rea'{Ul V'R) . R'+'i^ = R''\R 
[*301-24-104-21] 

*30r242. }■ :<reC'U nf'R./j.'^ff .V < /JL.R'' = R'' .0 .R-'IR^ R''-'f'+<"'+'^ 
m. 

\- . *120-412-416 . D h : Hp . D . o- = (o- -e ya) +0 /* ■ 
[*301-23] D.R'' = R'-"'-\R^. 

[Hp.*30r21] D.E°-|i2 = P°-'=''|i2"+'=i 

[*301-23] = iJ-'-oM+c^+ci .Oh. Prop 



SECTION a] numerically DEFINED POWERS OF RELATIONS 249 

*301-25. h:(ao-).P = iJ''.D.(a[T).P|i2 = i?'- [*301-16-241-242] 

*30r26. h:P6Potid'i?. = .(a(r).P = i?-- 
Dem. 

V . *301-2&-2 . Induct .DViPe Potid'i? . D . (a<r) .P=-R' (1) 

I- . (1) . *301-22 . D h . Prop 

*30r3. \-:RCI.a6C'Uni^'B.D.R'^=R = R„ = I[C'R 
[*300-312.*301-16-26] 

*30r31. l-:i2G/.o- + O.D.i2„ = A [*300 48] 

The above proposition is the same' as *300'48, but is repeated here to 
show the relations of R„ and R". 

*301-32. l-:.i2G7.a!ii.D:a!i2,. = .o- = [*300-311 .*301-31] 

*301-4. h : P e Rel num . o- e G'U n t^'R .D.R^^R" 
Dem. 

h . *301-2 . *121-302 . D h : Hp . D . P„ = P» (1) 

h.*301-21.*121-332.D 

h :. Hp . ff 6 a' ?7 A ^'P . D : P, = P' . D . P,+,i = R-'+-^ (2) 

h . (1) . (2) . Induct . D h . Prop 

*301-41. hzR.SeRel num . g ! P'' n P" . D . /j, = z/ . a ! (/li +„ 1) n t'G'R 
[*301-4-16 . *300-55] 

*301-5. h : /tt Xe v e O'E/ ft f 'P . /i =f . V + . D . (P")" = P^x-" 
Bern. 

|-.*117-62-32. Dh:Ki).O.fi,veC'Unt'"R (1) 

h . (1) . *301-16-2 . D h : Hp . D . {Ri^y = P^x=i (2) 

1- . *301 -23 . D h : i; +„ 1 6 C ZJ n f 'P . D . (P'')-+=i = (P")- 1 P*' (3) 

h.(3).*301-23.D 

\-:(jix^i')+^/jLeG'Un t^'R . (P")" = P^X'" . D . (P'')-+'=i = pO'Xcv)+c,. (4) 

I- . (4) . *113-67l . D 

I- :. (R^y = P'^X"- . 3 : /i x,(i;+, 1)6 C'Cr r> f P . D . (iJ^y+d = ij^xc(.+ci) (5) 
h . *ll7-57r32 . D t- : /^ Xo (z/ +c 1) e G'Unt^'R . D . /^ x„ v e C'i7 n i»'P (6) 
h . (5) . (6) . D h :. /* Xe 1/ 6 C" f7n «»'P . D . (P'')-' = P''X=-' : 3 : 

/u, X, (j/ +c 1) e a'CT A f P . D . (P-y+oi = P^xc(v+ci) (7) 
h . (1) . (2) . (7) . Induct .Oh. Prop 

*301-501. \-:fi = 0.veG'Unf'R.D. (P")- = P^x^^ [*301-2-3] 

*301-502. hzfi.ve G'Uf\f'G'R .0 . fi x^veG'Unt^'R. (/t x^ i^) a i"P e C"J7 

Bern, 
y . *300-14 . *120'5 . D h : Hp . a ! (/A x„ y) A ««^ 3 . (^ Xe I-) A i='P e O'fT (1) 
h . *300-14 . D h : Hp . D . (ga, /3) . a e ^ . /8 e v . a, ^ e i'O'P . 

[*113-251] D . (ga, /3) . a X /3 e /i Xo v . a, )S 6 i'O'P . 

[*113-17.*64-61] D . (ga, |8) . a x /8 e (/* x,, i/) a «"P (2) 

h . (1) . (2) . *65-13 . D h . Prop 



250 QUANTITY [part VI 

*301-503. I- : i; 6 NO ind . J. n t'C'R e G' U^ {t"G'R) . 3 . «/ a f'R eG\Ul f'B) 

Dem. 

h . *S00-14 . D h : Hp . D . (ga) .aevn t'G'E . 

[*106-2] D . (ga;, o) . J, x"a e v n P'R (1 ) 

h . (1) . *300-14 . D h . Prop 

*301-504. \-:iJ.,veC'Un f'G'R . i/ + . D . (Bf^y = E'^xc 
[*301-5-501-502-508] 

*301-505. h :. ^ 6 D'[^. D : a ! {{+, ^)IG'UY . = .^x,veC'U 
Dem. 
h . *120-452 . D h : a ! {(+„ ^^ G'U}' . = . g ! {(+« r)D C'C/"}" ■ f ^ (7'C/. 
[*300-232] = . a ! ( ?7f )■' . f 6 C" f/" (1) 

h . (1) . *300-52 . *301-4 . D 

h :. Hp . D : a ! {(+„ ^)[; C'C/}- . = . a ! (fTf), . f e O'f^. 
[*300-572] =.f XeveCfTi.Dh.Prop 

*301-51. \-:.l'neB'U.D:±l\{+,^)lG'UYn{(+,r,)lG'U}^. = . 

h . *301-505 . *300-232 . *301-4 . D 

[*300-571] =.fx„v6 0'D'.fx„i/ = 77Xe/t:.Dh. Prop 

*301-52. i-:veT>'Un t"R . D . (x,, /a)" = x„ (/a") 

1- . *301-2 . *113-204 . *116-204-321 . D h . (Xe yti)' = Xe(/ii) (1) 

I- . *301-21 .Dh-.ve a'Un f'R.:>. (x„ /i)''+'=i= (x,, /*)" | (x^fi) (2) 

h . (2) . D h : 1/ 6 a' [/-n i^'iJ . (x„ /i)" = x„ dj,") . D . (x„ /i)-'+'i = x„ (fj.") | (x, ^) 
[*116-52-321] =x„(/i''+"i) (3) 

h . (1) . (3) . Induct .Dh. Prop 



*302. ON RELATIVE PRIMES. 

Summary of *302. 

The present number is merely preparatory for the definition and discussion 
of ratios. We want, of course, to give a definition of ratio which shall ensure 
that /m/v = (//, Xj t)/(v Xe t). Hence in defining /x/v in any given type, we 
cannot exact that fi and v themselves should exist in that type, but only 
that, if p/a is the same ratio in its lowest terms, p and o- should exist in that 
type. Hence, if we are not to assume the axiom of infinity, it is necessary to 
deal with relative primes before defining ratios. 

The theory of relative primes is concerned with typically indefinite in- 
ductive cardinals (NC ind). It will be observed that we have three different 
sorts of inductive cardinals, namely NC ind, NC induct, and C U. NC ind 
consists of typically indefinite cardinals, NC induct consists of all cardinals 
of some one type, and G'U consists of all existent cardinals of some 
one type. If the axiom of infinity holds, we have (7'[/= NC induct, and 
NC ind = sm"NC induct. But neither of these is true if the axiom of 
infinity does not hold. It will be found that, where we require typically 
definite cardinals, it is G'U or d'U ox D'U that we require, not NC induct; 
that is to say, we almost always want to exclude A, and sometimes we want 
to exclude the greatest existent cardinal of the type in question, or to 
exclude 0. Thus "NC induct" will seldom occur in what follows. The 
cases in which C'U or D'U or Q'ZJ occurs are of two sorts: (1) where we 
are proving typically definite existent-theorems, (2) where we are concerned 
with series, as e.g. in *300, where we considered the series of existent 
cardinals, or in *304 below, where we shall consider the series of ratios. 
Wherever series are concerned, we must have typical definiteness, because 
the definition of "PeSer" involves G'P, and therefore only a homogeneous 
relation can be serial. This is a particular instance of the fact that when we 
require numbers as apparent variables (as e.g. in the theory of real numbers), 
typical definiteness becomes essential. Many propositions containing the 
hypothesis "fie NC ind " (where /i is a real variable) do not allow of /* 
being turned into an apparent variable, because this requires that fi should 
be fixed in one type, and our original proposition may demand that the 



252 QUANTITY [part VI 

type in which /i is fixed should be a function of /i. For example, *30017 
states 

1- : /i 6 NO ind . D . (ga) ./ji,eC'(Ul fa). 

If we fix the type of fi, we thereby also fix the type of a, and the proposition 
becomes false unless the axiom of infinity is true. In fact, the proposition 
demands that, the greater ^ becomes, the higher must the type of a become. 
" NC ind " is not a strictly correct idea, and the primitive proposition *9"13 
does not apply without reserve to propositions in which it occurs. We have 
introduced it because it immensely simplifies the expression of many proposi- 
tions, and because it is easy to avoid the errors to which it might give rise if 
used without remembering that it is a concession to convenience. 

It will be found that, when we are not concerned with existence-theorems, 
or with numbers as apparent variables, " NC ind " is almost always the notion 
required. This applies to all cases where we are only concerned with addition, 
multiplication, subtraction and division; it applies to ratios except when 
ratios are considered as forming a series, or when we are investigating their 
existence. As regards the use of an "NCind" as an apparent variable, 
there is a distinction between " all values " and " some value." If we have 
"pe NCind," "(g/a)" will often be legitimate when "(/>)" is not. The 
reason of this is that, if we are to fix upon one typically indefinite cardinal, 
it will be possible to assign one definite type in which it exists ; e.g. there are 
at least two classes, four classes of classes, sixteen classes of classes of classes, 
and so on. But if we are making a statement about all typically indefinite 
inductive cardinals, it will not be true unless there is a type such that our 
statement holds of all inductive cardinals in this type. 

In virtue of *300"17, if we have "joe NC ind," we may replace it by "peC U" 
if we may take U in as high a type as we please, or if, on account of the rest 
of our proposition, p cannot be greater than some assigned inductive cardinal 
so long as the hypothesis of our proposition is true. 

The above remarks will enable the reader to test the uses of typically 
indefinite inductive cardinals as apparent variables, and the passage from 
propositions concerning NC ind to propositions concerning C U. 

We define p as prime to a- when both are typically indefinite cardinals and 
1 is their only common factor, i.e. we put 
*302-01. Prm = joff{|0,o-6NCind:p = |=XoT.a- = i;XoT.Df,,,T-T = l} Df 

In this definition, ^, r/, r may be taken to be typically indefinite cardinals, 
because, when jO=^XoT.o- = 7;XoT,we must have ^^p.ri^a- .r^p. t^ a; 
so that f , 7), T cannot grow indefinitely (with a given p and o-) while the 
hypothesis remains true. 

We define " (p, a-) Prm^ (fi, v) " as meaning that p is prime to cr, that t is 
not zero, and /a = p x^ t . i* = o- x^ r, i.e. pja is /x/v in its lowest terms, and t is 
the highest common factor of /* and v. The definition is : 



SECTION a] on relative PRIMES 253 

*302-02. (p, a-) Prm^ {,1, v). = . 

p ftm o- . T 6 NC ind — I'O . p, = p x^t . v = <t x^t Df 

We then put further 
*302-03. (p, a) Prm (ji, v). = . (gr) . (p, a) Prm^ (/i, v) Df 

Here again there is no ©bjection to t as an apparent variable, because t 
must be not greater than p, and v. "(./a, a) Prm (/*, v) " secures that p/o- is /i/v 
in its lowest terms. 

We also define, in this number, the lowest common multiple and the 
highest common factor. 

Our definition of "Prm" is so framed that every inductive cardinal is prime 
to 1 (*302-12), that 1 is the only number which is prime to itself (*302"13), 
and the only number which is prime to (*302"14). 

After a number of preliminary propositions, we arrive at the result that 
if p. and v are not both zero, and ^ and rj are not both zero, the existence of 
a couple p, a- which is prime both to fi, v and to f, »? is equivalent to 
/x Xo 1; = V x„ ^, i.e. 

*302-34. h : . /i, 1/, ^, 7? e NC ind . ~ (/i = V = 0) . ~ (^ = 1; = 0) . D : 

/iXei7=i'Xef. = . (ap, a) . {p, 0-) Prm {p., v) . (p, a) Prm (1^, 77) 

We have also 
«302-36. h : /i, V 6 NC ind . ~ (/i = v = 0) . = . (gp, o-) . {p, a) Prm (p,, v) 
*302-38. I- : (p, 0-) Prm (/*, v) . (f, -n) Prm (/*, v) . D . p = f . o- = 17 

I.e. there is only one way of reducing a fraction to its lowest terms. 

We prove also (*30215) that if p,, v are typically indefinite cardinals, which 
both exist in the type of X (i.e. p.^, VKeC'U), then 

(p, o) Prm {p., v). = . (p, 0-) Prm (p.^, v^). 
This enables us, when we wish, to substitute typically definite cardinals for 
the typically indefinite p, and v. 



«302-01. Prm = pff{p,o-eNCind:p = f XeT.o- = i7X<,T.Df,,,T.T = l} Df 

«302-02. (p,<r)PrmT (/*,!/). = . 

p Prm a .T€ NC ind — I'O . p, = p x^r . v = a x^t Df 
Here p,, v are to be typically indefinite in the same way as p x^ t and a x^ t. 
Thus if, in some one type, px^T and ax^r are both null, that does not justify 
us in writing (p, a) Prm, (A, A), because there are other types in which px^r 
and ff Xj T are not null. On this subject, cf. *126. 

*30203. (p, 0-) Prm (At, i'). = .(aT).(p,o-) Prm, (/*,!/) Df 

*302-04. hcf {p,, v) = (7t) {(ap, a-) . (p, a) Prm, (/*, v)] Df 

*302-05. 1cm {p., v) = (7?) {(gp, cr, t) . (p, <r) Prm, {p.,v).^ = px^a x^ t] Df 



264 QUANTITY [PAKT VI 

*3021. hz.p Prm a .= :p,a-e NO ind : p = ^ x^t . a = rj x^r . Df,,,T ■ t = 1 
[(*302-01)] 

*30211. h : p Prm c7 . s . o- Prm p [*302-l] 

*30212. h : p Prm 1 . = . p e NO ind [*302-l . *117-631-61] 

*30213. h : p Prm p. = .p = l 
JDem. 

l-.*302-12.Dt-:p = l.D.pPrmp (1) 

h . *3021 . D h :. p Prm p.D:p = lx„p.D.p = l: 
[*113-621] D:p = l (2) 

I- . (1) . (2) . 3 1- . Prop 

*30214. 1- : Prm jj,. = .fi=l 
Dem. 

|-.*302-12.Dl-:/i==l.D.0Prm/i (1) 

l-.*302-l. DI-:.OPrmyit.D:0 = OXo/i./it=l Xe/u,. D ./i=l : 
[*113-60r621] D:/i = l (2) 

h . (1) . (2) . D I- . Prop 

*30215. F :. M, 1/ 6 NC ind . /i^, VKeC'U.D: 

(p, a) Prm (/i, i/) . = . (p, o-) Prm (p,x, vx) 
Dem. 

h . *126-101 . *300-14 . D 

h :. Hp . D : p Pim a .re NO ind — I'O . /* = p x^ t . i/ = o- x^ r . = . 

p Prm o- . T 6 NC ind — I'O . /ix = p Xo t . v^ = o- x^ t (1) 

h . (1) . (*302-02-03) . D h . Prop 

«302'2. I- : fi;veC'U.<^(/j, = v = 0). k = t{('3^p,(7). /j, = px^r . i' = ff Xgr}. D . 

E ! max ( ^)'« . max ( UYk eD'U 
Bern. 

|-.*113-621.Dh:Hp.D.l6« (1) 

h . *ll7-62 . *113-602 . Transp . D 

h :. Hp . re/t. D : T^/t. v.T^i/ (2) 

h . (1) . (2) . *300-21-22 . *261-26 . *300-26 . D I- . Prop 

In the above proposition we write " max ( UYk " rather than " min ( UYk," 
because, since U arranges the natural numbers in descending order, " min ( f7 )'« " 
is the greatest number which is a member of k, and therefore it is less con- 

fusing to speak of this number as " mux (UYk." 



SECTION a] on EELATIVE PRIMES 255 

*302-21. I- : Hp*302-2 . t = max(?7)'K ./i = pXoT.i/=o-x„T.D. 

Dem. 

h . *1312 . D h : Hp . /3 = /a' Xo t' . o- = 0-' Xe t' . D . 

/* = p' Xe t' Xo T . 1/ = 0-' Xo t' Xg T . 

[*113'602,Transp.Hp] D . t' x^ t 4= . t' x^ t < t - 

[*120-511.*117-62] D . t' = 1 (1) 

h.(l).*302-l.DI-:Hp.D.pPrmo- (2) 

I- . (2) . *302-2 . (*30202) . D h . Prop 

*302-22. h:./j,,ve NC ind . ~ (^^ = i; = 0) . D : (ap, cr, t) . (p, o-) Prm, (ft, v) : 

(ap. o-)-(/'.o-)^rm(/i, i/) 
[*302-2-21 . *300-17 . (*302-03)] 

*30223. V :. /i, V eD'U.D : ("S^p, (t) : p, a eD'V . fji, x^a = V x^ p : 

f, 7? eD'^T". /it Xoi? = K X,, ^ . Df,, . f > p . i; > o- 
Dem. 

h . *300-23 . *113-27 . D 

h : Hp . « = D'O^n p {(go-) . /* x„ o- = i/ Xep} . D . E ! min (^)'« (1) 

I- . (1) . *300-12 . D 

h :.Hp. D : ('S.p>o^) : p,(reD'U./MXea-=vXgp: 

f,776D'?7./iX„iy = i/x„^.Djf.,.f>p (2) 
I- . *120-51 . D 

h : Hp . p, a- e D' cr. /i Xo ff = I' x„ p . ju, x„ 7? = v Xgf . D . p Xg 17 = o- Xe f (3) 

h . *117-571 . D h : Hp (3) . I, ■>; 6 D'C/". I > p . D . f x„ o- > p Xe <r (4) 

I- . *126-51 . D I- : Hp (4) . ff > 1? . D . p Xo o- > p Xg 7; (5) 

I- . (4) . (5) . D I- :. Hp (4) . D : cr > 1? . D . f Xe or > p Xb 77 : 
[Transp] D : ^ x^ o- = p x^ j; . D . 17 ^ o- (6) 

h . (2) . (3) . (6) . D h . Prop 

*302-24. h :./i,i/,p,creNCind-i'0.yttx„o- = z/x„p : 

/i Xg 77 = 1/ Xj 1^. ^, i;eD'f/^. Df,,.f^p.i7^o-:D.p Prm <r 
i)em. 
h . *302-l . D 

I- : p, o-6D'C/".~(pPrmo-). D . (a^, 17, t) . t 4= 1 .p = ^XoT.o- = 77XoT 
[*113-203-602.*120-511.*117-62] 

D.('S,^,V,T).lv,T€l>'U-l'l.^<p.V<<r-p = ^-X«r.a- = rjX„T (1) 
|-.*120'51 . D I- :/i, y, p, o-6D'Z7./iXoO- = i/ Xop.p = ^XoT . o- = i7 XoT. D. 

/iXo77 = Z/Xef (2) 

I- . (1) . (2) . D I- : /i, v, p, o- 6 D'f7. /i Xo o- = I' Xe p . ~ (p Prm cr) . D . 

(a^. 77) . /i Xo 77 = «/ Xo f . ^, 7/ e D'CT. |< p . 77 < o- (3) 
h . (3) . Transp . *300-l7 . D h - Prop 



256 QUANTITY [part VI 

*302-25. h : p.^eD'U. D . (ga, y8) .aeC'U. j8 < f ./> = (« x„ ?)+oy3 

Bern. 
y . *117-62 . *120-428 . D 1- : Hp . D . p < (/a +„ 1) x^ ^ (1) 

h . (1) . *300-23 . D h : Hp . D . E ! min ( Uyci {aeG'U.p< (a +c 1) x„ ^} . 
[*120-414-416] 3 . (ga) . a e C" t/". /a < (a +„ 1) x„ ^^ . p > a x^ f . 

[*lir-31.*120'4o2] D.(a«,/3).a,/3eC'C.p<(a+„l)x<,|.p = (ax„f)+e^. 
[*113-671] D.(aa,^).a,^e(7'[/-.p<(ax„?)+„?.p = (ax„f)+„;8. 

[*120-442.*ll7-561.Transp] 

D . (a«, /3) . a 6 C" tr. /S < ^ p = (a x„ ^) +„ ;8 : D h . Prop 

*302-26. 1- : Hp *302-24 . D . (p, o-) Prm (;a, j/) 

Bern. 
h . *302-25 . D 

|-:Hp.D.(aa,;8,7,S)./t = (ax„p)+„y8.i' = (7X„o-)+„S.;8<p.S<<r (1) 
h . *113-43 . D 
1- : /t = (a x„ p) +e /3 . 1/ = (7 x„ 0-) +„ S . /3 < p . S < 0- . yu. Xo o- = V x„ p . D . 

(a x„ p x„ 0-) +e (^ x„ 0-) = (7 x„ p X,, <r) +„ (S x„ p) . /S < p . 8 < o- - (2) 
[*117-31.*120-452.*113-671] 

D . a x„ p x„ o- < (7 +„ 1) x„ p x„ o- . 7 x„ p Xo <7 < (a +0 1) Xo p Xe ff - 
[*126-51] D.a<7+„1.7<a+„l. 

[*120-429-442.*ll7-25] D . a = 7 (3) 

h . (2) . (3) . *120-41 . DI-:Hp (2). D.;8XeO-=Sxe p. /8<p.S<o-. 
[Hp] D . ;8 = . S = (4) 

h . (3) . (4) . D h : Hp (2) . D . p, = a x„ p . 1/ = a Xe o- (6) 

h . (1) . (5) . *302-24 . D h . Prop 

*302'27. \-:fJ',v, p, a;^,T}e NC ind - t'O . p. x^ o- = 1/ x^ p . /i x^ 17 = i/ x^ ^ . D . 

?x„o- = i7Xop 
Dem. 

|-.*113'27.Di- :Hp.D.^XeJ'XoCr = 7; X„/4X„o- 
[Hp] =»;x„i/x„p. 

[*126-41] D.|:x„o- = 77X„p:DI-.Prop 

*302-28. 1- : Hp*302-24 . ^, 97 eNC ind - I'O ./i x^i; = 1/ x^ f . D . 

(p, <7) Prm (f, v) [*302-26-27 . *300-l7] 

*302-29. f- : Hp*302-28 . ^Prm rj .:i .^ = p .■>] = a 
Bern. 

V . *302-28-l . D 

I- :.Hp. 3:(aa).?= «x„p,')7 = aXocr:^=aXep.»; = aXeO-.D„.a=l: 
[*1 4-122] D:|:=lx„p.97 = lx„<7:.DI-. Prop 



SECTION a] on relative PRIMES 257 

*302-3. h:/i,i;,f,97eNCind-t'O.AiX„97 = i/x„^.D. 

• (ap. 0-) ■ {P> 0-) Prm {fi, v) . (p, a) Prm (|, 17) 

Dem. 

I- . *302-23-24 . D 

h :. Hp . D : (gp, o-) : p Prm a- . p,a-e NC ind — t'O . /i Xg o- = k Xo p : 

a, /SeD'D". /t Xo/3 = i'X„a. D„,p .o^/o.^S^o-: 
[*302-26-28] 3 : (gp, a) . (p, o-) Prm {/i, v) . (p; a) Prm (f, ^) :. 3 h . Prop 

*302-31. f- : {p, a) Prm (/t, i^) . /t Prm v .'2 .,j. = p .v = tT 

Bern, 
y- . *302-l . (*302-02-03) . D 

h :. Hp . D : (gr) . /jl = px^T .v = tr x^t: fi = p x^r .v = (r x^t .'^j.t=1 : 
[*1 4-122] D./i = pXol.J'=o-x„l:.DI-. Prop 

«302'32. I- : ^Prm rj . /* Prm v .^XgV='t]XgfjL.O .^ = jj,.tj = v 
Bern. 

b . *302-3-31 . D 

h : Hp . D . (gp, a-) . p Prm a-.^ = p.fi = p.r] = a-,v=tr:'Di-. Prop 

*302-33. I- : . /i, i^, f , 1; e NC ind - t'O . D : 

fj.x^r) = vx„^. = . (gp, 0-) . (p, 0-) Prm (ji, v) . (p, a-) Prm (f, 17) 
Z)em. 

h . Id . (*302-02-03) . D h : (p, <r) Prm (/*, v) . (p, o-) Prm (^, 17) . D . 

(a'''. T' ) • 1", t' 6 D'Cr. /;t = p Xg T . 1/ = O- Xg T . ^ = p Xg t'. ?/ = (7 Xg t'. 

[*113-27] D . (gr.V) . /* x„ 7? = p x„ o- x^ t x^ t' = i/ x„ ^ (1) 

I- . (1) . *302-3 . D h . Prop 

*302-34. h:./i,j',|:,i?eNOmd.~(/i = i/ = 0).~(^ = '>7 = 0).D: 

/i Xei? = 1/ Xef . = . (gp, 0-) . (p,ff)Prm(/i, z/) . (p, o-)Prm(f,5y) 
i)em. 

• h.*113-602.Dh:Hp.ytt=O.i/=l=0.D.f=O. 77=1=0 (1) 

|-.*113-602-621.D 

\- •.fi = .v=^0 .^=0 .r]=^0 .D .fi=0 XgV.v=lx^v.^ = Ox^Tl .r] = lXaTi • 
[*302-14] D.(0,l)Prm(At,i').(0,l)Prm(|,97) (2) 

h . (1) . (2) . D h : Hp . ^ = . i; =f= . D . 

("KP' <^) ■ (P> o") Pi™ (m. v) . (p, 0-) Prm (f, 17) (3) 
Similarly I- : Hp . v = . /i =[= . D . 

(a/3> °") ■ (P> <^) Prni (/*> v) . (p, a) Prm (f , 77) (4) 
h . (3) . (4) . *302-33 . D h . Prop 

*302-35. H :. /*, 1/ e NC ind . ~ (/i = i; = 0) . p Prm a.D: 

p.x„a = vx„p. = .{p,a) Prm (fi, v) [*302-3414-31] 

H.&W. III. 17 



2o8 QUANTITY [part VI 

*302-36. h : /A, 1/ 6 NO ind . ~ (/A = i; = 0) . = . (gp, a) . (p, a) Prm (fi, v) 

Bern. 
\- . *302-14 . D t- :. (p, a) Prm (/t, i/) . D : 

/3, o- 6 NC ind . ~ (jO = ff = 0) : (gr) . t e NC ind - t'O . /* = p x,. t . k = o- x^ t : 
[*120-5.*113-602] -D:fi,ve'NCind.'^(ji = v = 0) (1) 

h . (1) . *302-22 . D h . Prop 

*302-37. I- : (p, a-) Prm (/*, v) . = . 

/i, z^ 6 NC ind . ~ (/J, = . y = 0) . /3 Prm a . /mx^ct = v x^p [*302-35-36] 

*302-38. h : (p, tr) Prm (/a, i/) . (^, i?) Prm (/t, z;) . D . p = | . a = t; 

Dem. 
h . *302'37 . D h : Hp . D . p Prm o- . ^ Prm r) . fiXca- = v x^p . fjLX^r) = vXa^. 

r^(fl=O.V = 0) (1) 

I- . (1) .*302-14 .*113'602 .Dh:Hp./it = 0.D./3 = 0.^ = 0.o--l.i;=l (2) 
h . (1) . *30214 . *113-602 .DI-:Hp.j' = 0.D./3 = l.^=l.o- = 0.7; = 0(3) 
t- . *302-27 .Dh:Hp./i=t=0.j/ + O.D.px<,97 = o-Xef. 

[(1) . *302-32] 0.p=^.a- = v (4) 

h . (2) . (3) . (4) . D h . Prop 

*302-39. h : (p, a) Prm (/*, z/) . D . /i ^ p . v > o- 

Dem. 
I- . *302-23-36 .31-:. /i, z/eD'C/". D : 

(aP> o") : (p. o") Prm (/i,v) : ^.rjeB'U. fix^r) = v x^^ .D^^^ . ^"^p .rj^a: 
[*113-27] D : (ap, 0-) . (/>, cr) Prm (/*, i/) . /i ^ p . i/ ^ o- : 

[*302-38] D : {p, <r) Prm (/i, i-) . D . /i > /> . i/ > o- (1) 

h . *302-37-14 . D h : /i = . (p, o-)Prm (/i, i/) . D . i/ + . p = . o- = 1 (2) 

Similarly h : i/ =0. (|0, o-)Prm(/i, v) . D .^=|= O.p = 1 .a- = (3) 

f-.(2).(3).D t-:.(p,o-)Prm(/i,v):/i = 0.v.i' = 0:D./i>p.7/>o- (4) 
h . (1) . (4) . *302-36 . *300-l7 . D I- . Prop 

*302-4. h : /*, 1/ e NO ind . ~ (/A = i; = 0) . D . E ! hcf (/i, I/) 

Z>em. 
1- . *302-22 . 3 h : Hp . D . (gp, a, t) . (p, o-) Prm^ {p,, v) (1) 

h . *302-38 . (*302-02-03) . 3 

h : (p, 0-) Prm^ (/i, I/) . (f,77) Prm^ (/A, J/) . D . p = f . o- = i; . ^ = p Xg T . /* = I x„ or . 
[*1 26-41] D.T = CT (2) 

h . (1) . (2) . (*302-04) . D I- . Prop 

*302-41. h : ytt, 1/ 6 NC ind . ~ (yu, = J/ = 0) . D . E ! 1cm (/it, J/) 
[Proof as in *302-4] 



SECTION a] on relative PRIMES 259 

*302-42. h : /i, v eNC iud . ~ (/* = v = 0) . D . hcf (/4, v) Xolcm {/i, v) = fix^v 

Bern. • 

h . *302-4-41 . (*302-0405) . D h : Hp . D . 

(aP, 0-, t) . /i = p Xo T . V = (7 X„ T . hcf (/i, i;) = T . Icm (/i, I/) = /3 x„ o- Xo T . 

[*113-27.*116-34] D . (gp, o-, t) . /t x„ v = p x,, <7 Xj r" - 

hcf (/i, v) Xe 1cm (/*, I/) = p Xo o- Xo t' : D h . Prop 

*302'43. h : (p, a) Prm (/t, i/) . D . p Xg hcf {fi, v) = /j, . a x^ hcf (/x, v) = v 
[*302-4 . (*3020204)] 

«302'44. h : (p, <r) Prm (/*, v) . D . p x^ v = 1cm (/i, v) = a-Xgfi 
[*302-41 . (*302 0205)] 

*302-45. h : (p, a-) Prm (/i, i;) . ?, »? e NC ind . ~ (^ = 9; = 0) . /* x„ t; = i; x„ f . D . 

lcm(^,77) = px,^ = o-Xei7 
Dem. 

\- . *302-37 . D h : Hp . D . (p, 0-) Prm (^ v) (1) 

h . (1) . *302-44 . D I- . Prop 



17—2 



*303. RATIOS. 

Swmmary of *303. 

In this number, we give the definition and elementary properties of the 
ratio fijv. Most of the important applications of ratios are to numerical or 
identical relations, i.e. to relations which may, in a certain sense, be called 
vectors. Neglecting identical relations for the moment, let us consider 
numerical relations, and to fix our ideas, let us take distances on a line. 
A distance on a line is a one-one relation whose converse domain (and its 
domain too) is the whole line. If we call two such distances R and 8, we 
may say that they have the ratio fijv if, starting from some point x,' 
V repetitions of R take us to the same point y as we reach by fi repetitions 
of S, i.e. if xR^y . xS^^y. Thus R and 8 will have the ratio i^jv if g ! E" n >S^ 
In order, however, to insure that fi/v = pja if {p, a) Prra (fj,, v), it is necessary, 
in general, to substitute g ! iJ"' n /S'' for g; ! iZ" n 8^^. (In the above case 
of distances on a line, the two are equivalent.) Thus we shall say that R has 
the ratio /i/v to 8 if (gp, c) . (p, <t) Prm {fi, v).%\R'' f\ 8''. 

If we apply the above definition to identical relations, we find that, 
if RQI .80.1, R has the ratio fi/v to 8 provided '^lRf\8, i.e. provided 
a ! G'R n CS. This application is required for dealing with zero quantities 
and zero ratios. 

Thus we are led to the following definition of ratios : 

*30301. fi/v = RS{{'3_p,a).(p,a)Frm(ij„v).'3_lR''nSi'} Df 

This definition, as it stands, requires justification in two respects : (1) we 
commonly think of ratios as applying to magnitudes other than relations, 
(2) we should not commonly include as examples of ratio certain cases included 
in the above definition. These two points must now be considered. 

(1) In applying our theory to (say) the ratio of two masses, we note that 
the idea of quantity (say, of mass) in any usage depends upon a comparison 
of different quantities. The "vector quantity" R, which relates a quantity jjij 
with a quantity m^, is the relation arising from the existence of some definite 
physical process of addition by which a body of mass rrti will be transformed 
into another body of mass m^. Thus a- such steps, symbolized by R'^, 



SECTION A] RATIOS 261 

represents the addition of the mass o- (wij — mi). Similarly if S is the relation 
between M3 and M^ which*,rises from the process of addition turning a body 
of mass Ml into another body of mass M^, then Sf symbolizes the addition of 
the mass p (M^ — M^. Now g ! iJ"' o /S*" means that there are a pair of masses 
m and m', such that mli''m' and mSi'm. In other words, if we take a body A 
of mass m and transform it so as to turn it into another of mass m+tr (m^ — m^, 
we obtain a body of the same mass m as if we proceeded to transform A into 
a body of mass m + p{M^—M^. Hence (T{'rrh-rrh) = p{M^ — M^; that is 
(wis — rrh)/(M2 — Ml) = p/a. But in our symbolism the addition of m^ — mi is 
represented by the vector quantity R, and that of M^ — M^ by the vector 
quantity S; so in our symbolism It has to S the ratio of p to a. 

Thus to say that an entity possesses p, units of quantity means that, taking 
U to represent the unit vector quantity, Ui^ relates the zero of quantity — 
whatever that may mean in reference to that kind of quantity — with the 
quantity possessed by that entity. 

It can be claimed for this method of symbolizing the ideas of quantity 
(a) that it is always a possible method of procedure whatever view be taken 
of it as a representation of first principles, and (/8) that it directly represents 
the principle "No quantity of any kind without a comparison of different 
quantities of that kind." 

Furthermore analogously to our treatment of cardinal and ordinal numbers, 
we can define any definite quantity of some kind, say any definite quantity of 
mass, as being merely the class of all "bodies of equal mass" with some given 
body.* The zero mass will be the class of all bodies of zero mass ; and if there 
are no bodies with the properties that a body of zero mass should have, this 
class reduces to A in the appropriate type. 

Thus the application of our symbolism to concrete cases demands the 
existence of a determinate test of " equality of quantity " of different entities, 
and a determinate process of " addition of quantity." The formal properties 
which the process of addition must possess are discussed in the numbers 
concerned with vector families. 

(2) Having now shown that cases apparently excluded by our definition 
of ratio can be included, we have to show that no harm is done by our inclusion 
of cases which would naturally be excluded. In order that ratio may agree 
with our expectations it is necessary that the two relations R and S, whose 
ratio we are considering, should have the same converse domain. Otherwise 
we get such cases as the following : Let P, Q be two series, and suppose* 
B'P = B'Q, 5p = 6q, 11p = 9q, 13p = 25q, but that P and Q have no other 
terms in common. Then we shall have, it R = Pi.S=Qi, 

{B'P)R'5p.(B'P)S'5p, 

* For notation, cf. »121. 



262 QUANTITY [part VI 

whence it follows that R has to /S the ratio 5/4, i.e. we have R(5/4i)S. But 
we shall also have R {8/10)8 and i? (24/12) (S, i.e. R(4i/5)S and R (2/1)8. 
Thus our definition does not make different ratios incompatible. In practical 
applications, however, when R and 8 are confined to one vector-family, 
different ratios do become incompatible, as will be proved at the beginning 
of Section C. And so long as we are not concerned with the applications 
which constitute measurement, the important thing about our definition of 
ratio is that it should yield the usual arithmetical properties, in particular the 
fundamental property 

fi/v = p/a- . = .IJ,Xea-=VXeP, 

which is proved, with our definition, in *303"39. Thus any further restriction 
in the definition would constitute an unnecessary complication. 

In virtue of our definition of fi/v, fj,/v = A if yti and v are not both inductive 
cardinals, or if /n = i/ = (*303-ll-14). We have (*303-13) \-.fi/v = Gnv'{v/ix), 
i.e. the converse of a ratio is its reciprocal. If /x. = 0, and R {fi/v) 8, R must 
have a part in common with identity (which we may express by saying that 
iJ is a zero vector), and 8 may be any numerical or identical relation whose 
field has a member which has the relation R to itself (*303"15). Thus if v, a 
are inductive cardinals other than 0, 0/v = O/o-. The common value of ratios 
whose numerator is is the zero ratio, which we call Og (where "q" is intended 
to suggest "quantity"). The definition of Oj is 

*30302. Og = s'0/"NC induct Df 

In like manner, if fi and p are inductive cardinals other than 0, we have 
fi/0 = p/0. The common value of such ratios we call oo g, putting 

*303-03. oOj = sV0"NCinduct Df 

The properties of ratios require various existence-theorems, and in estab- 
lishing existence-theorems without assuming the axiom of infinity, the question 
of types requires considerable care. We have 

*303-211. h : (p, 0-) Prm (fx,, v).O.fi/v = p/a- 

so that the existence of fi/v does not depend upon fi and v, but upon p 
and 0-, where p/a is fi/v in its lowest terms. We may, therefore, in consider- 
ing existence-theorems, confine ourselves, in the first instance, to prime 
ratios. 

To prove the existence of (p/cr) [, t'R, when p Prm o-, we take two relations 
R and 8 both contained in identity. These have the ratio p/a provided their 
fields have a member in common and El R". El 8''. By *301'16, this requires 
p,ae C'{ Ul f'R). Thus we have 

*303 25. l-:.|oPrm(r.D: 

'3_l(p/a)lfR . = .p,aeG'(Ul t"R). = . p {R), a (R) e C U 



SECTION A] RATIOS 263 

But this existence-theorem, which is obtained by supposing R and S 
contained in identity, is i^t much use in practice : what we require is the 
existence of a ratio between numerical relations. For this purpose, assuming 
p'^cr and a-^0, let \ be a class of such a type that Nc'i'X "^p+^l. (Such 
a class can always be found in some type, by *300'18.) Then we have 
p^ed'U, and we can construct a series Q such that C'Q is of the same type 
as X and 'Nc'O'Q = p+^1. (This is proved in *262'211.) We can then choose 
out of Q a series P having the same beginning and end, and consisting of 
o- +0 1 terms. We then have 

(B'Q) (Q.y (B'Q) . (B'Q) {P,y (B'Q). 

Hence Pj and Qi have the ratio p/a: A similar argument applies if o- ^ p 
and p =^ 0. Thus we arrive at the proposition : 

*303-322. I- : p Prm o- . p^, o-^ e J)'Un QT/. D . a ! (p/a) I (Rel num n too'X) 

I.e. if p is prime to a and neither is 0, and p+ol, <r+o 1 both exist in the 
type of X, then there are numerical relations having the ratio pja- and having 
their fields of the same type as \. 

The case when either p or o- is requires separate treatment. If R has 
to 8 the ratio O/o", R must be partly contained in identity (*303"15) ; hence 
we have to find a hypothesis for a ! (0/<r) \ Rel num, since g; ! (O/cr)^ Rel num 
is impossible. Since O/o- = 0/1, we only require the existence of 2 in the 
appropriate type, i.e. we have 

*303-63. 1- : a ! 2a . 3 . a ! O3 P (Rel num n t^'X) 

It will be remembered that a ! 2^ is demonstrable except in the lowest 
type. 

In the above propositions, /* and v and p and a have been typically in- 
definite. Ratios of typically definite inductive cardinals are dealt with by 
means of *302'15, which gives at once 

*303'27. h : /i, v e NC ind . /i^ , Kx e 0' IT". D . /t/v = nJvx 

I.e. a ratio may, without changing its value, have its numerator and 
denominator specified as belonging to any type in which both exist. This 
enables us to take p and cr as typically definite cardinals in *303'322, thus 
obtaining the proposition 

*303-332. h:.p Prm a-.'D-.'s^l (p/o-)^ (Rel num r. t^^'p) . = . p,(TeI)'Un a'U 
The above existence- theorems are useful in proving 
a/0 = y/S. = .ax,B = l3x,y. 
We proceed as follows: We first show (*303'34) that, if p,a- are inductive 
cardinals other than 0, and p+gl, cr -f-^ 1 exist in the type of X, we can find 
numerical relations R and 8 such that '^IR' f\ 81', but ij > o- . D . ~ a ! R^. 



264 QUANTITY [part VI 

This is done by taking two series P and Q having the same beginning 
and end, and having G'P e o- +e 1 . G'Q ep+^l. Then if J? = Pi and 8 = Qi, 
we have 

{B'P) R' (B'P) . (B'P) S" (B'P) :7,><r.D.B^=A, 
whence the result. From this proposition it follows immediately that if 
p Prm 0- . f Prm 77 . 17 > o", and if p^, a-^eD'Urid'U, we can find an R and 
an 8 such that R (p/a-) fif . ~ {i2 (f /•»?) 8}. A similar argument applies if ?? < a- 
or ^ > p or f < p. Hence we find, by transposition, 

*303-341. h:px,o-x6D'f7'na'fr./3Prmo-.^Prm97.(p/o-)^*„„'^. = (?/'7)DC^-3- 

p=^.a-=Ti 

From this point on, the argument offers no difficulty. For if we have 
«//3 = 7/S . (p, a) Prm (a, /3) . (^ r,) Pnn (7, 8), 
we have, by *303-341-211, p = ^.ff = 7). Hence, by *302-32, we have 
a Xj 8 = jS Xj 7. What is approximately the converse, i.e. 

*303-23. h : /i, I/, 1^, 7? e NO ind . 

~ (/* = I' = 0) ■ ~ (f = »? = 0) . /i x„ 77 = 1/ Xe I . D . /i/v = ^/t; 

follows at once from *303'211 and *302"3. Hence, after dealing with 
special cases, we find 
*303-38. h : . a, /3, 7, S 6 NC ind : 

ax,/3Aea'C7.v.7;„8;,ea'?7":~(a = /3 = 0).~(7 = S=0):D: 
(a/;3) I «„„'X = (7/S) t «„„'\ . = . a Xe 8 = ;8 Xe 7 

It will be observed that a//3 is typically indefinite, like Nc'f. But in 
order to insure that a//3 = 7/8 however the type may be determined, it is only 
necessary to insure that this equation holds in a type in which (a/yS)^ Rel num 
exists. When we write simply " a//3 = 7/8," we shall mean that this equation 
holds however the type may be determined ; in other words, that it holds in 
a type in which (a/^S) ^ Rel num exists. (There always is such a type, if 
a, /8 6 NC ind - I'O, in virtue of *303-322 and *300-18.) Thus we have 
*303-391. h :. a, ^ 6 NC ind . ax, /8a e a' Cr. ~ (a = ^ = 0) . D : 

(a//3) D «oo'^ = (7/8) D 4o':^- . = . a//3 = 7/8 . = . a x„ S = yS x^ 7 

and, in virtue of *303'38, we have 
*303-39. l-:.a,;8,7,86NCind.~(a = ;S = 0).~(7=8 = 0). D: 

a/iS = 7/8. = .aXeS = /3Xo7 

This proposition is, of course, essential to the justification of our definition 
of ratios. 

The remaining propositions of *303 consist (1) of applications of the 
theory of ratio to powers of a given numerical relation, (2) of properties 
of Og and oo ,, (3) of a few properties of the class of ratios. This last set 
of propositions depends upon two new definitions, which must be briefly 
explained. 



SECTION A] RATIOS 265 

We have already explained that fijv is typically indefinite. Thus if we call 
It/v a " ratio," ratios are, Hke " NO ind," not strictly a class, because every 
class must be confined within some one type. Nevertheless it is convenient, 
just as in the case of NO ind, to treat ratios as if they formed a class ; and, 
with similar precautions, we can avoid the errors into which we might be led 
by treating them as a proper class. We therefore put 

*303-04. Rat = Z{(a/i,v)./i,i'6NCind.i' + 0.Z = /t/i;} Df 

("The condition v 4= is only introduced because it is usually convenient 
to exclude oo g.) It will be observed that fijv is still typically indefinite if /i 
and V are typically definite. This results from *303-27. But we often want 
typically definite ratios. We want these defined in types in which there are 
numerical relations having the ratios in question. Hence we put 

*303 05. Rat def = 1 {(g/*, v) . /i, i/ e D' CTn Q' 17. Z = {^jv) I t^'p] Df 

Here " def" stands for " definite," and /i, v are typically definite inductive 
cardinals. The desired properties of "Rat def" result from *303-322. It 
should be observed that, besides consisting of typically definite ratios, 
"Rat def" differs from "Rat" by the exclusion of Og. This is merely for 
reasons of convenience. 

The properties of " Rat " and " Rat def" follow immediately from previous 
propositions. We have 

*303 721. h : Z e Rat - t'Oj . D . (g/i) . Z I t^'/jL e Rat def 

*303-73. l-zZeRatdef.D.alZ^Relnum 

By *303-322 ; and by *303-391, 

*303-76. h :. Z, Ye Rat . Z^ t^'p e Rat def . D : Z^ t^^'p = 7^ tn'p . = .X=Y 

If the axiom of infinity holds, every member of " Rat '' except Og becomes 
a member of " Rat def" as soon as it is made typically definite. Hence 

*303-78. I- : Infin ax . D . Rat def = Rat - 1% 

The uses of " Rat " and " Rat def" differ just as the uses of " NO ind " and 
"NO induct" differ. The distinction is only important so long as the axiom 
of infinity is not assumed. 



*30301. ixlv = RS[{<3^p,a).{p,a)Vi-vii{^,v).'3_\R''hS''} Df 

In the above definition, p, <r, fi, v are typically ambiguous, but p, a- must 
(by *301"16) exist in the type of t'R, while /t, v need not do so ; /t, v cannot, 
however, be null in all types, by *30017. 

*30302. 05 = s'0/"NC induct Df 

*30303. 00, = sVO"NC induct Df 



266 QUANTITY [PABT VI 

*303-04. Rat = Z{(a/tt,i;)./i,i;€NCmd.i;4=0.Z = /i/i'} Df 

*30305. Ea,tde{=X{{'3^fi,v).iJ.,ve'D'Ur^a'U.X = {filv)lti^'fi} Df 

*3031. \-:B(fj,jv)S. = .{'^p,<T).{p,a)'Prm{^,v).±lR'nS'' [(*30:3'01)] 

*303-ll. f-:~(^,z.eNCind).D./i/i^ = A [*303-l . *302-36] 

*303-13. h . yti/i/ = Cnv'(v/fi) [*303-l . *302-ll] 

*30314. h.O/0 = A [*303-l . *302-36] 

*30315. \-:R{0/v)S. = .veNCind-i'0.'^lRnI[C'S. 

= .ve'NCmd-L'0. '3^1 G'SnfcixRx) 
Bern. 

\- . *302-14-38 . *3031 . D 

\- 1 R(0/v) S . = . V e'NGmd- I'O .±\ R' n S' . 

[*301-2] = . 1/ 6 NC ind - I'O . a ! E A / 1' C/S : D h . Prop 

*303151. [-z.R.Se'Rel num id.D:R (O/v) S. = . 

z; 6 NC ind - I'O . J? e Rl'/ , S e Rel num id . a ! C"i2 n C'/Sf 
[*303-15 . *300-324-3] 

*30316. h : R(,j./0)S .^ . fie'NCmd- I'O .■3,1 S n I[ C'R . 

= . At 6 NC ind - I'O . a ! C'iJ n ^ (xSx) [*303-15-13] 

*303161. h :. iJ, »S e Rel num id.O-.R (fijO) S. = . 

/tieNCind - I'O .ii eRelnum id . SeRl'/. g ! G'R n G'S 
|;*303151-13] 

*303-17. \-:.iJ,,ve'NCind-i'0.R,8e Rel num id . R {fijv) S . D : 

i?, S e Rl'/ . V . E, 5f e Rel num 
Dem. 

V . *3031 . *113-602 . D 

h ::Hp. D i.E^^eRelnumid : (g/s, o") . jo, cr e NC ind - t'O . g ! i?°' n fSf*" :. 

[*300-33.*301-3] 

D :. ,Sf 6 Rel num id : . i? e Rl'I : {^p) . p e NC ind - t'O . g ! ii n fifp : v : 

ii 6 Rel num : (gp, a).p,a6 NC ind - t'O . g ! iJ" n /Sp :. 
[*300-3] D :. S 6 Rel uum id :. i? e Rl'/. g ! /n,Sfpo . v . ii e Rel num . g! Jn8j„ :. 
[*300-3-33] D:.R,Se Rl'/ . v . J?, S e Rel num :: D I- . Prop 



SECTION A] RATIOS 267 

*303 18. \-:.,jL,ve D'H^t^'R . R, SeRl'I . D : 

R(/^)S. = .R{Olv)S. = .R(,i.jO)S. = .'^[C'RnC'S 
[*303-1151-16 . *301-3] 

*303181. 1- : a ! (,j,/v) . = . (gp, <t) . (p, a) Prm (jj,, v) 

Bern. 
h . *303-l . D h : a ! (^/v) . D . (gp, a) . (p, a) Prm (/m, v) (1) 

I- . *301-3.*300-325-17. D h : (p, o-) Prm (/n, i-) . D. (aa;).(a;4a!)(/i/j/)(a;4,a;) (2) 
h . (1) . (2) . D h . Prop 

In the above proposition, if ^jv is typically indefinite, so that " a ! fijv " 
only asserts existence in a suflSciently high type, p, <t may also be typically 
indefinite. But if /i/i/ is to be taken in a definite type, p and o- must be taken 
in the corresponding type, and must not be null in that type. This is proved 
later. 

*303182. h :. 0/0 = /t/i; . = : ~ (/*, i/ e NC ind) . v . /* = v = 

Here the equation 0/0 = p.jv is assumed to hold in a sufficiently high type. 
J)e/m. 

V . *303-14 . D 1- :. 0/0 = /i/i/ . D : /i/i/ = A : 

[*303181.*302-36] D : ~ (/t, i^ e NC ind - t'O) . v . /t = i/ = (1 ) 

h.(l).*.3031114.DI-.Prop 

*30319. V : R {y^jv) S. = .R (fi/v) S [*303-l . *121-26] 

*303-2. t- :. (p, a) Prm (//, i;) . D : i? (,jl/v) 5f . = . g ! iJ- A .S" 
Bern. 

h.*303-l. :y\-:'Sp.±lR-'f^8i'.D.R(fj,/v)S (1) 

h . *302 38 . *303-l . D h : Hp . E (p./v) S .D .^IR' nS" (2) 

h . (1) . (2) . D (- . Prop 

*303-21. i-:.pFima-.D:R(p/a)S. = .'3_lR''nS'' [*302-31 . *303-l] 

*303-211. h : (p, 0-) Prm (yii, v).D.p./v = p/a [*303-2-21] 

«303'22. I- : p Prm a-. fi,ve NO ind . ~ (^u = v=0). p. x^ (7= vx^p .D. fijv = pja- 
[*302 37.*303-211] 

*303-23. \-:p,,v,^,r)e NCind . ~ (/*= v= 0).~(^ = ■»? = 0) . /i x^t? = v x^ ^, D. 

W" = f/'7 [*302-3 . *303-211] 

jf(303'24. t- : yLt, v 6 NC ind . ~ (/i = v = 0) . D . (a/o, o-) . p Prm o- . njv = jo/o- 
[*303-211.*302-22] 

The following propositions give typically definite existence-theorems for 
ratios. 



268 QUANTITY [part VI 

*303-25. \-:.p'Pima.:^:'3^l(pla)l,t'R. = .p,<y€G'(Ule'R). = .p{R),<T{R)eG'U 

I.e. if p Prm a, there are relations of the same type as R and having the 
ratio pja when, and only when, the number of relations of the same type as R 
is at least as great as p and at least as great as o-. 

Dem. 
V . *303-21 . D h :. Hp . D : a ! (p/o-)^ «'J2 . D . (g/Sf, T) . E ! /Sf' . E I^o.^f, Tet'R. 
[*30116] 0.p,c-eG'Uit"R (1) 

l-.*301-16-3.DI-:.Hp.D: 

p, aeC'Ul t^'R.xeto'G'R .D.(xlx)p = (xlxy = a;iic (2) 

h . (2) . *303-21 . D 

1- :. Hp . D : p, o- 6 G'Ul f'R . xet.'G'R . D . (a; J, «) {pja) {x J, x) (3) 

I- . (1) . (3) . *63'18 . D h . Prop 

*303-251. I- : /*, 1/ e G' ?7t «"i2 . ~ (/i = I' = 0) . D . a ! {filv) I t'R 
Bern. 

V . *302-36-39 . D h : Hp . D . (gp, o-) . (p, o-) Prm (/i, i/) . /t > p . i' > o- . 
[*117-32] D . (ap, 0-) . {p, a) Prm (/t, v).p,aeG'Ul, t"R . 

[*303-211-25] 3 .,a ! (fju/v) ^ ^'E : D I- . Prop 

*303-252. 1- : yit, j; e NC ind n 0' [/"t t"G'R . ~ (/i, = i- = 0) . D . g ! (At/i/) D ^'-R 
Bern. 

h . *64-51-55 . D h : /i = Nc'a . a e t'G'R . x e t,'G'R . D . 4 x"ol e /x a t"R (1) 
h . (1) . *300-14 . D F : Hp . 3 . /i, i; e CfTt «"iJ (2) 

h.(2).*303-251.Dh.Prop 

In the above proof, jx, v are assumed to be typically indefinite. If they 
are typically definite, sm"/j, and sm"!/ must be substituted for p, and v on the 
right-hand side of (1) and (2). The hypothesis " fj.,ve'NCind n C'Ulf'G'R" 
is a convenient abbreviation for 

"fj,,ve NC ind . M rt i'O'i?, v n i'O'E e C" CTp ««(7'i? . " 
By *65-13, 
;a n t'G'R € G'U[. t^'G'R . = . /i C t'G'R .fieG'U^ f'G'R . = .^e G'Ult^G'R. 

Bat " fMeG'U^f'G'R" requires that /j, should be typically definite, whereas 
"/tteNOind" requires that p, should be typically indefinite. Hence the 
hypothesis of *303'252 is only defensible as an abbreviation, meaning 
"ya, 1^6 NCind, and if /m, v are given the suitable typical definition, they 
become members of G'Utf'G'R." 

*303-253. I- : /i, i; 6 NC ind n G' Ul f'X . ~ (^ = i; = 0) . D . g ! (p,/v) I «„„'X 
[*303-252] 

*303-254. I- : /4, V 6 NC ind . p.^, vk e G'U. ~ (^ = z/ = 0) . D . g ! (fi/v)ltoo'\ 
[*303-253 . (*65-01)] 



SECTION a] ratios 269 

«303-26. I- : /A, v e NO ind . ~ (^ = v = 0) . D . (gX.) . g ! (jjl/v) I t^'X 
[*303-254 . *300f 7] 

*303-27. \-:/j.,ve'NGind.iJLK,VKeG'U.O./jilv = iJ,KlvK [*30215 . *303-l] 

*303-3. h : p Prm o- . g ! Po^"" . D . Pp (p/o-) P" 
Dem. 

h . *30116 . *14-21 . D I- : Hp . D . p x„ o- e C'f/'n <"P (1) 

h . (1) , *301-5 . D h : Hp . p + . o- + . D . (Pp)"' = PoX""- = (P")" . 

[*303-21] :>.F''{p/a-)P- (2) 

l-.*301-2. DI-:Hp.p = O.D.P'' = /r(7'P = P''x»-.g[!/r(7'P (3) 

h . *302-14 .Dh:Hp.p = O.D.o-=l. 

[*301-2] D.P'' = P (4) 

h . (3) . (4) . D h : Hp . p = . D . a ! (P")" n (P")" . 

[*303-21] D.P''(p/o-)P°- (5) 

Similarly h : Hp. o- = . D.P'-CpMP'' (6) 

h.(2).(5).(6).DI-.Prop 

*303-31. H : p Prm o- . p + . o- + . (p x„ o-) n i'X, e Q' [7 . D . 

(aP) . P 6 Rel num n «„„'X, . Pp (p/a) P" 
JDem. 

I- . *300-46 . *301-4 . D h : Hp . D . (gP) . P e Rel num . (B'P) Pp^cr (B'P) (1) 

h . (1) . *303-3 . D h . Prop 

*303-311. l-:p^,<r;,6a'P'-l'0.p><7.D.(aP,Q).Pe(p+„l),.Q6(o-+„l),. 

P,QeC>.-Qe-P-5'P=P'Q.P'P=5'Q 

Dem. 

I- .*262-21 . D h : Hp . 3 .g! (p+ol),oC>. (1) 

h . *117-22 . D I- : Hp . P 6 (p +0 1 ), . 3 . (ga) . a C (7'P . a e o- +o 1 (2) 
h . *261-26 . *205-732 . D 
l-:Hp.P6(p+el>.«C(7'P.aeo-+„l. 

yS = (a - I'minp'a - t'maxj 'a) w i'5'P w i'J5'P. D . ^8 e o- +o 1 . 
[*250-141.*202-55] D.Pti86(o-+el)r (3) 

I- . (1) . (2) . (3) . *205-55 . D h . Prop 

*303-32. h : p Prm o-.p>o-.o-=|=0.pAe(I'f7.D. 

a ! (p/o-) C (Rel num n t^'X) n M (Ep„ G 8^) 
Bern. 

|-.*303-311.Df-:Hp.D.(aP,Q).Pe(p+,l),.Q6(o-+„lV.P,Q6C>.. 

Q(IP.B'P = B'Q.B'P = B'Q (1) 
h . *300-44-45 . *301-4 . D 

h : Hp . P e (p +0 l)r . /Sf = Pi . D . iS e Rel num . (B'P) So (B'P) (2) 



270 QUANTITY [part VI 

Similarly 

h : Hp . Q 6 (a- +„ l)r . i? = Q, . D . i? e Rel num . (B'Q) R' {B'Q) (3) 

h . (1) . (2) . (3) . *261-35-212 . D 

h : Hp . D . (gi?, 8).B,Se Rel num n «„„'\ . B^^ QS^^.'S^IR-' nS" (4) 

h . (4) . *303-21 . D h . Prop 

*303-321. h : p Prm a . p + . a + . px, o-a e Q' f/". D . g ! (p/a) I (Rel num n «oo'^.) 
[*303-3213] 

*303-322. I- : p Prm a- .pK.a-^eD'Una'U.D .'g^l (p/cr) I (Rel num n t^'\) 
[*303-321] 

*303-323. h : /i, i; e NC ind - t'O . D . (gX) . g ! (/i/i/) I (Rel num n i™'\) 
[*303-322] 

*303-324. h : /i, v 6 NC ind . fj,^,Vf, e D'Cr. ~ (/i Prm v).^ . 

a ! du./!/) I (Rel num n <oo'X) 
Dem. 
h . *302-22 . D h : Hp . D . 

(a/3, cr,T)./3Prmcr./3=t=0. cr=|=O.T=t=O.T=t=l . /i4 = jC» X„T.i'=crX<,T. g! /ix . a! Vx . 
[*303-2-21] 

D . (a/3, 0") . p Prm o- . p + . a-=|= . /i/i' = p/o- . a ! (p +0 1)a . a ! (o" +0 1)a ■ 
[*303-321] 3 . a ! (/^MD Rel num : D h . Prop 

In order to the existence of (/i/v) ^ Rel num in any given type, it is 
by no means necessary to have fj,,veD'U in the corresponding type. If 
pFrma .p,(T eH'U rid'U, (p x„T)/(a x^t) will exist, however great t may 
be, because (p Xg t)/(o- x^ t) = p/a. 

*303-33. h : a ! (P'/v) t (Rel num n t^'X) . = . 

(a/3, 0-) . (/>, <7) Prm (fj,, v) . /3a, o-x 6 D'CTn Q'tT 
Dem. 

h . *303-322-211 . D 

h : {p, a) Prm (/i, i/) . px, tx e D' C/" n Q' fT" . D . a ! (W") t (Rel num n ^„„'X) (1) 

|-.*303-1811516-211.D 

h :. a ! (/ti/") D (Rel num n <„„'X) . D : (a/>, cr) . (p, o-) Prm (;4, v) . p =t= . o- =|= . 

a ! ipla) I (Rel num n «„'\) : 

[*303-21] 3 : (ap, 0-) . (p. a) Prm (/i, i;) . p 4= . o- + : 

(a-B, S).B,Se Rel num n «„„'X . a ! -R°' <=» -S'' : 

[*301-41] D : (ap, 0-) . (p, 0-) Prm (/ct, I/) . p 4= . <7 4= . 

a ! (p +0 1) n «„'X . a ! (o- +c 1) n «„'\ (2) 
h . (1) . (2) . D h . Prop 

*303'331. h :. p Prm o- . D : a ! (p/o-) D (Rel num n to^'X) • = .pK,'y>.eI>'U nd'U 
[*303-33 . *302-31] 



SECTION A] RATIOS 271 

*303-332. h :. (0 Prm o- . D : a ! (pja) f (Rel num r. ii/p) . = . p,o eD'U nd'U 
[*303-331] • 

In this proposition, p, a are typically definite cardinals, whereas in 
*303'331 they are typically indefinite. 

*303-34. h : p.o-eNCind .p^,o-xeD'i7n a'U . 7,><T.':i. 

(giJ, S).R,Se Rel num n i„,'\ . g ! ii'' n 6'" . ~ {g ! i?" n 8^ 

Note that ~ {g ! iJi n S^} does not imply E ! iZ") or E ! SK 

Dem. 

h .*303-311 . D 1- : Hp. D .(aP,Q,ii,£f) .Pe(p+„1V. Q6(o-+„l),. 

P,Q€too'X.B'P = B'Q.B'P = B'Q.R = P,.S=Q, (1) 

As in *303-32 Dem, 

\-.{l).0h:-RTp.:i.(^P.Q,R,S).Pe{p+,l)r.Q6{.7+,l)r.S = P,.R=Q,. 

R, S e Rel num . (B'P) (R' n S") (B'P) . 
[*121-4.8.*202181.*301-4.*300-44] 

D . (aE,*S) . i?, fife Rel num n «„„'X, . g ! i?" n fifp . ~(a ! iJi) : D h . Prop 

*303-341. h : p^, o-^eB'Una'U.pPrma-. ^Frmr,. (p/<r)l t^''X.= (^lv)tt^''^- 3- 

p = ^.a = rj 
Dem. 

h . *303-34-21 . D h : p;i, o-A 6 D'CTn Q'tT . p Prm <r . f Prm 17 . i; > tr . D . 

(/'/o-)DCx + (r/'7)DCx (1) 

I- . (1) . Transp . *3021 . D h : Hp . D . «? < o- (2) 

h . (2) . *303-13 . D[-:Hp.3.f<p (3) 

h . (2) . (3) . *ll7-32 . DI-:Hp.D.^;,,(T^6a'f7' (4) 
h . *303-322 . D h : Hp . D . a ! (^/r,) ^ Rel num . 

[*303-ll-15-16] D . f 4= . 1/ 4= (5) 

I- . (2) . (4) . (5) . D h : Hp . D . ^>,, vk e D'U^a'U. 

"^'>-^'>-|^] ^-<'?-P<^ (6) 

- . (2) . (3) . (6) . D h . Prop 

*303-35. \-:l^€a'U.^PTrQ7,.(0/l)lt^'X^{^lv)tV\.0.^ = 0.v = l 

Bern. 
\- .*300-14. D I- :Hp. "2 .(•g^x.y) .x=^y .x.yet^'X. 

[*303-15] D . (ga;, 3/) . a; 4= y . (« J, «) (0/1) (« J, y) . a; J, «, a; J, y e <oo'>. ■ 

[Hp] D.(aa;,2/).a; + 2/.(«J,«)(f/9?)(a;4,y). 

[*303-16-l7.Transp] D . ^ = . (1) 

[*302-14] D . 7? = 1 (2) 

I- . (1) . (2) . D h . Prop 



272 QUANTITY [part VI 

*303-36. V :. p^.cr^ea'U .V . ^^,rj>.ea'U : p Prm a . ^ Prm t? : D : 

Dem. 
h . *30014 . *302-14 . D 

h i.px.o-^eCI'C/'.pPrm o-.~(p;^,o-xeD'?7).D:/3 = 0.ff=l.v.jO = l.(7 = 0: 
[*303-35-13] D : f Prm ,, . (p/<r) [; t^'X = (^/^) ^ «„„'\ .D.p=^.a = v (1) 

l-.(l).*303-341.DI-.Prop 

*303-37. h :. a,/3 eNCind n a'CiJt «"X) . ~(a = /3= 0) . v . 

7,S6NCindna'(I/t«"\).~(7=S = 0):D: 

(a//3) t «oo'X= (7/8) D<„„'X .D.ax,S = /3x„7 
Z)em. 

1- . *302-36 . *303-211 . D I- : a, j3 6 NO ind . ax, /9^ 6 a' iT" . ~ (a = ;S = 0) . D . 

(ap,<r).(p,<r)Prni(a,/3)./5/<7 = a/^ (1) 
t- . (1) . *303-254-181 . D h : Hp (1) . (a/^) [, t„,'\ = (7/8) l «„„'\ . D . 

(af '?)■(?,'?) Prm (7, 8) (2) 
h . (1) . (2) . *302-21-22 . *303-211 . D 
I- : Hp (2) . D . (ap, <7, ^, ^) . (p, <7) Prm («, ^) . (?, ,7) Prm (7, 8) . p, ^ 6 a' f^ . 

[*303-36] D . (ap, «7) . (p, <7) Prm (a, /3) . (p, a) Prm (7, 8) . 

[*302-34] D.aXo8 = /3Xe7 (3) 

Similarly 

l-:7,86NCind.7,,8,ea'U-.~(7=S = 0).(a/;8)t:C^- = (7/S)DC:v.3. 

aXeS = /3x„7 (4) 
1- . (3) . (4) . D h . Prop 

*303-371. I- : «, ;S, 7, 8 6 NO ind . a^, /S^, 7^, 8^ e C'C . ~ (a Prm /3 . 7 Prm 8) . 

(a//3) t ioo'X = (7/8) D «»'>. . 3 ■ a x„ 8 = ^ Xe 7 
[Proof as in *303-37] 

*303-38. h:.a,A7,SeNCiDd:ax,/3x6a'C/'.v.7x,8xea't/': 

~(a = /3 = 0).~(7 = 8 = 0):D: 
(a//8) D VX = (7/8) t ioo'X. ■ = ■ a Xe 8 = ^ X, 7 [*303-37-23] 
*303-381. l-:.a,A7,8 6NCind.ax,/8x,7x,8xe(7'[7.~(aPrm/3.7Prm8).D: 
(a//S) C *co'^. = (7/8) t «<„,'\ . = . a Xe 8 = yS x„ 7 [*303-37r23] 
*303-39. l-:.a,A7.86NCind.~(a = ^ = 0).~(7 = S = 0).D: 

a/j8 = 7/8 . = . a x„ 8 = yS Xo 7 [*303-38 . *300 18] 
*303-391. l-:.a,;SeNCind.ax,y8A6a'[/.~(a = iS = 0).D: 

(«//8)D «oo'X = (7/8) D «oo'>^ . = . a//8 = 7/8 . = . a x„ 8 = ^ x„7 
[*303-38-264-ll-14] 



SECTION A] RATIOS 273 

Thus when a/yS is used as a typically indefinite symbol, we obtain the 
same results as if we suppo^d it defined as of a type too'X, where a +o 1 and 
^+ol both exist in the type of \, i.e. Nc%'\>a.Nc%'X>/3. 

*303-392. l-:.a,/3ea'Cr.~(a = jS = 0).D: (a/^) P «n'a = (7/S) D «ii'a . = . 

a/jS = 7/S . = . a Xe S = /3 Xe 7 [*303-39r27] 

This proposition differs from *303391 by the fact that a, jS have become 
typically definite. It will be observed that even when a and /S are typically 
definite, a/fi, like ax^^, remains typically indefinite. 

*303-4. \-:.p Prm a . i? e Rel uum .:i:R^ (p/a) i2^ .= . 3 ! R,,^,^ 
[*303-3-21 . *301-4] 

*303-41. l-::/t,veNOind.~(/* = 0.i; = 0).D:. 

J2eRelnum . ^=lcm{fi,v). D : R^itijv) R, . = .g;!i2f 
Dem. 

h.*303-2.*300-44.D 

1- :. Hp ./*=f0.j/ + 0.i26 Eel num . {p, a) Prm^ (/i, v) . D : 

Rn i/J'/v) i2„ . = . a ! i?,.Xo<r f^ -Bi-X.p ■ 

[*302-37] =.a!i2^x.. (1) 

h . (1) . *302-44 . 3 h :. Hp(l) . f = 1cm (ji. v). D : R^{fj,/v)R, . = . g ! iij (2) 

h . (2) . *302-22 . D 

I- :. Hp./i=|=0. i/=t=0.i2eRelnum.f =lcm(/i, J/). D :i?j»(/ct/i/)i?,. = .a;!i2f (3) 

l-.*302-44.D 

h :.Hp./i = 0.i2eRelnum . f =lcm(/i, v). D: ^=0: 

[*303-15] 0:R^{/j,/v)Ry. = .'3_lRi (4) 

Similarly 

h :.Hp .v = .iJeRelnum. f = Icm (fi,v). D : Rh(/m/v) R, . = .g; !i2| (5) 

f- . (3) . (4) . (5) . D h . Prop 

*303-42. h : . Hp *303-41 . ^ = lcm (jj., v). D : U^, (/i/v) U,. = . 1cm (/i, v)eG'U 
[*303-41 . *300-26] 

*303"43. h :. Infin ax . D : ;i4, 1/ e NC ind . ~ (/i = 1/ = 0) . D^, „ . CT^ (fJ^/v) U, 
[*303-42 . *30014] 

*303-44. V :. Hp *303-42 . P e Ser . D : P^ (/t/v) P,, . = . g ! Pf 
[*303-41 . *300-44] 

*303-45. F : P e fi in fin . /*, 1/ 6 NC ind . ~ ((li = . !» = 0) . D . P^ (/it/j/) P„ 
[*300-44 . *303-44] 

*303-46. V:.{p,a) Prm (/i, i;) . f , t? e NC ind . E e Rel num . D : 

i?j (/i/v) i2, . = . f Xe o- = t; Xo fj . a ! -Kfxoff 
Z>em. 

f-.*303-211.3 

h :. Hp . D : Ri(/jilv)R,, . = . R{(p/<t)R^ . 

[*303-21] =.a!i2jx„.-^i2,xoP- 

[*300-55] =.|XoO- = 9? Xe/3.a[!iJfxc<r:-3l"-I'rop 

R. &W. III. 18 



274 QUANTITY [part VI 

*303-461. I- :. /i, i;, f , 97 6 NC ind . ~ (/i = 1/ = 0) . ~ (^ = 9? = 0). E 6 Rel num . D : 

Rl {/j-jv) i2, . s . ^ Xo i; = t; Xo ^ . a ! Biomit.i) 
Bern. 

V . *302-45 . D 

I- : Hp . (/J, 0-) Prm (^, t?) . D . f x^ <r = 1cm (^, i?) (1) 

h . *302-35 . D 

h : Hp . (p, a) Prm (/t, i') . f x^ o- = ly x^ p . D . (p, o") Prm (^, »?) . (2) 

[*302-34] D . ^ Xe 1/ = 77 x„ ;li (3) 

1- . *302-35-37 . D 

I- :Hp.(p,o-)Prm(/i,i/).f XoV = 77Xe/t.D.^XeO- = '>7 x^p (4) 

h . (1) . (2) . (3) . (4) . *303-42 . D h . Prop 

*303-47. h : . Hp *303-461 . A ~ e Pot'i? . D : J?f (/^/i/) i?,. = .fx,i/ = 7?x„/i 
[*303-461] 

*303-471. h :. /i, I/, f , 7? 6 NO ind . ~ (/i = 1/ = 0) . ~ (f = 17 = 0) . P e fi infin . D : 

Pf (m»P, . = ■ ^ x„i' = i7 x„/i 
[*303-47.*300-44] 

*303-48. !-:./*, K, 1,7/ eNOind.~(^ = v = 0).~(f = i7 = 0).D: 

Ul(/j,/v) f/,. = .^x„i/ = iy x„/4.1cm(f,7;)6(7'Cr 
[*3O3-461.*3O0-26] 

*303-49. h : : Infin ax . D : . /a, i/, ^, 77 e NO ind . ~ (/a = v = 0) . D : 

^fW") C;. = . ?XoV = 77X„/i 
Dem. 
h . *303-15 . D h :. /i, v, ?, 77 e NC ind . /i = . i^ 4= . D : 
CTj Oii/i;) U^. = .U^e Rl'/ . CT, e Rel num id . 
[*1 20-42] =.^=0. 

[^113-602] =.^Xci' = 77X,;ii (1) 

Similarly 

|-:./i,i',f,77eNCind./*4=0. v = O.D: Ui(im/v) U,,. = . ^ x^v = r} x^fi (2) 

h . (1) . (2) . *303-48 . D h . Prop 

*303-5. h : p, o- e NC ind - t'O . a ! (p +0 o-)a . D . 

{^P,Q).Pe(p+,l)r.Qe{a+,l\.P,QeWX. 

B'P = B'Q . B'P = B'Q . C'P n O'Q = t'£'P « I'B'P 
Bern. 

h . *110-202 . *120-417 . D 

|-:Hp.D.(aa,/3).a,/8eC^.aep+cl./Se<7-el.aft^ = A (1) 



SECTION a] ratios 275 

h . *262-2 . D ^ 

l-:Hp.a,^6«„'X,.a6p+ol./8eo--el.aft/3 = A.o-4=2.D. 

[*251-131-141] D . (gP, 8,Q).P,8,Qean t^'X .G'P = ol.C'S=^ . 

Q = B'P^S-{*B'P. G'P n C'Q = I'B'P u I'B'Q (2) 
h . *262-2 . D h : Hp . a, /3 6 «„'\ . a e p +0 1 . /8 = I'a; . « ~ e a . (7 = 2 . D . 

(aP,Q).P,QeC't.P6n.G'P=a.Q = (5'P)4,^+>5'P (3) 
I- . (1) . (2) . (3) . D I- . Prop 

*303-51. h : . p Prm o- . p 4= . o- + . g ! (p +« o-);, . D : 

(ai2, S):E,S6 Rel num n i„„'X, . i? (p/a-)S:^/'ri^p/a-. D^,, . <^B{^/t))S 

Dem. 
l-.*300-44:45.*301-4.D 
l-:Hp.P6(p+„l)j^.Qe(<7+„l),.;Sf=Pi.J? = (2i. 

P'P = 5'Q . B'P = B'Q . O'P n O'Q = t'J'P u I'B'P . D . g ! iJ' n ^S" (1) 

I- . *301-41 . D h : Hp (1) . ~ (^ = p . ,, = 0-) . D . Pi A /Sff = A (2) 

I- . (1) . (2) . *303-21 . D 

h :. Hp(l) . p : P(p/o-)i^ : ^Prmi? . ~(^ = p . ,, = 0-) . Df , .,^P(|/,,)fif: 

[*303-36] b : P (p/<r) <S : ^ Prm t; . ^/,, + p/a . D^., . ~ P (^/,,) S : 

[*302-22.*303-211] 

[*303182] D : P (p/cr) 5f : ^/i; =j= p/o- . Dj,, . ~ P (1^/,,) S (3) 

1- . (3) . *300-44 . *303-5 . D h . Prop 

*303-52. I- :. /i, 1/ e NC ind - I'O . a ! (/* +„ i')^ . D : 

(gP, S) : P, ;Sf e «„„'\ . P (,./,/) fif : ?/^ + /x/f . Dj., . ~ P (^/t?) -S 

h . *303-24 . *302-39 . D 

h : Hp . D . (g[p, cr) . p Prm cr . /^/j; = p/er . p 4= . o- =)= . g; ! p +e o- (1) 

h . (1) . *303-51 . D F . Prop 

*303-6. h : 1/ e NO ind - t'O . D . O/i; = Og [*303'1 5] 

*303-61. h:i'6NOind-t'0.D.i'/0=oOj [*303-16] 

*303-62. \-.Og = Cnv'oo, = RS{'3_lBr^I[G'S) [*303-6-61-13-15] 

«303-621. h . Og I' Rel num id = Cnv'(Rel num id ^ oo g) 

= P;S (P G / . >Sf e Rel num id . g ! G'P n G'S) [*303-6-6113-151] 

18—2 



276 QUANTITY [part VI 

*303-63. I- : a ! 2x . D . a ! Og I' (Rel num n t^'X) 
Bern. 

h . *303-15-6 , D h : fl;4= 2/ . D . /Og (a; J, 3/) : D h . Prop 

*303-631. h : a ! 2x . D . a ! (Rel num n «„o'X.) ^ oo , [*303-63-62] 

*303-65. 1- : a ! 2^ . D . Og t «oo'A, + 00 g p 4„'\ 
Dem. 
h . *303-62 . D h : a; =1= y . D . lOg (a; I y) . ~ {/oo g (a; 4, 2/)} : D I- . Prop 

*303-66. h :. a ! 2x . D : (fi/v) tt^'\ = Og. = . fi = . v eNCind- I'O 

Dem. 
l-.*303-6.DI-:y[4 = 0.i/eNCind-i'0.D./i/i' = 0g (1) 

h . *303-615 . D 

I- :/i/i; = Og. D .At/j' = ii^(E6Rl'7. ^feRel num id -a ! G'BnG'S) (2) 

h . *300'3 . D h : Hp . D . (a«, y).a;4=y.«J.2/6 Rel num n too'\ . 
[*10-24] D . a ! (Rel num id - Rl'J) n t^'X (3) 

|-.(2).(3).*303-ll-l7.D 

h ::Stp .D :. (fj,lv)lt^'X = Og.D : fi,v e'NC ind : (1 = .V . v = (4) 

l-.(2).(3).*303-16.D 

1- :. Hp . D : (fi/v) ^ i^'X, = 0, . 3 . ~ (yti =|= . z/ = 0) (5) 

h . (4) . (5) . D I- : . Hp . D : (/t/i/) ^ «„„'\ = 0, . D . /i = . i/ e NO ind - I'O (6) 
h . (1) . (6) . D h . Prop 

*303-67. h :. a ! 2x . 3 : (/a/j/) t t^o'X =<x> g. = .v = . fieNG ind-i'O 
[*303-66-62] 

*303-7. h : Z 6 Rat . s . (a/t, v) . /i, y e NO ind . v + . X = /^/i; 
[(*303-04)] 

*303-71. h : ZeRat def . = . (a/*, v) . ,jL,veI)'Un Q'U .X = (,ji./v)lt,r'/i 
[(*303-05)] 

*30372. h : Z e Rat . D . ('gji) .'3_lXl t^'fi [*303-26] 

*303-721. h : Z e Rat - t'Og , D . (a/*) . Z I ta'fJi. e Rat def 
[*300-18.*303-7-7l] 

*303-73. I- : Z e Rat def . D . a ! Z t Rel num [*303-322-324] 

*303-731. b:.p Prm <7.:^: {pja) I t^'p e Rat def . = . p, er e D'CTn Q'tT 
[*303-7l . *302-39] 



SECTION a] ratios 277 

*303-74. b :. pfrma . X=(j}la-)ltn'p-':>:t^.Xinelnnm. = .p,a-6'D'Una'U 
[*303-332] • 

*303-75. h : Z 6 Eat . a ! Z ^ («,//* n Rel num) . D . X ^ ^u'/i e Rat def 
[*303-74-7l] 

*30376. h :. X, Fe Eat . Z ^ t^^'p e Eat def . D : X ^ «u'p = F ^ t^^'p . = . X= F 
[*303-391] 

*303-77. h : . lafin ax . D : /4, v e NC ind - t'O . D . /i/i/ e Eat def 
[*30014 . *303-7l] 

*303-78. h:Infinax.D.Ratdef=Eat-t'05 [*303-7-77] 

The above two propositions assume that fi/v in the first, and "Eat" in 
the second, have been made typically definite, but they hold however the 
type may be defined. 



*304. THE SERIES OF RATIOS. 

Summary of *304. 

In this number we consider the relation of greater and less among ratios, 
and the series generated by this relation. We need two different notations, 
one for greater and less between typically indefinite ratios, the other for 
greater and less between ratios of the same type. The former is more 
useful where we are dealing merely with inequalities between specified ratios, 
but the latter is necessary when we wish to consider the series of ratios in 
order of magnitude, since a series must be composed of terms which are all 
of the same type. We put 

*30401. X <, F. = . (g/i, V, p, a-) . /i,v,p,a-6 NO ind .a-^0./j,x^a-<.vx^p. 

X = p.lv.Y=pl<T Df 

This definition is so framed as to include Og but exclude oo q. For the 
relation "less than" among rationals of given type (excluding Og), we use 
the letter H, to suggest tj (defined in *273), because, if the axiom of infinity 
holds, the series of rationals of a given type is an t). The definition is 

*30402. jS" = 1 F {Z, 7 e Rat def . X <^ F} Df 

When we wish to include Og in the series, we use the notation H' ; thus 

*30403. £r' = XF{Z, F6Ratdefwt'0g.X<^F| Df 

(It will be observed that here I'Og acquires typical definiteness through 
the fact that it must be of the same type as "Ratdef" in order to make 
" Rat def w t'Og " significant.) 

If the axiom of infinity does not hold, H and H' will be finite series : 
if v +0 1 is the greatest integer in a given type (i/ > 1), the first term of H 
is Ijv and the last is vjl (*304'281). In a higher type, we shall get a larger 
series for H, but at no stage shall we get an infinite series. If, on the other 
hand, the axiom of infinity does hold, H is & compact series (*304'3) without 
beginning or end (*304"31) and having Ko terms in its field (*304'32), 
i.e. fl" is an ?? (*304-33). In this case, ' C'iZ" = D'S" = Rat - I'O, (*304-34), 
i.e. any rational other than Og, as soon as it is made typically definite, belongs 
to C'H. 



SECTION a] the series OF RATIOS 279 

Under all circumstances, S is a series (*304-23), and H exists in the 
type tfa'^ if 3 exists vo^ the type t'\ (*304-27). In the same case, 
(?'iZ"=Ratdef (*304-28). Similar propositions hold for H'. 

G'H' consists of typically definite ratios, and if X is any ratio, there are 
types in which X belongs to C'E' (*304-62). If the axiom of infinity holds, 
every ratio is a member of G'H' in every type (*304-49). 



*30401. X <, r . = . (a/i, V, p, a-) .fi,v,p,a-e NC ind . cr =j= . /i Xo <r < v Xj p . 

X = fi/v.Y=pl<T Df 

*30402. H = X7{X,YeB,Sitdei.X<,.¥} Df 

*30403. fl' = 1 F {Z, 7 e Rat def u i% . X <r Y] Df 

*3041. h : X <.r Y. = . (a/A, V, p, a) . fi,v,p,<Te NC ind . fiXecr <.v x^p . 

X = filv. Y= pItT [(*304-01)] 

*304-ll. 1- : fijv <r p/a-. = . ir/p <r vjp. [*304-l] 

*30412. V\X<rY. = . Y<rX [*30411 . *303-13] 

*30413. h : X <^ F . D . Z, Fe Eat . F+ 0, 
Dem. 

h . *117-5 .3h:/itXoO-<z/Xjp.D.i/XojO=|=0. 

[*113-602] D.i; + 0./) + (1) 

h . (1) . *304'1 . *303-7 . D I- . Prop 

*30414. h : XEY . = . Z, F e Rat def . Z <, F [(*304-02)] 

*304-15. h : XHY. = . (g/t, v, p, a) . p,, v, p, <7 eD'fTo a'f7 . 

Z = (/i/v) I t^^'fi . F= (p/a) t tn'fi ./iX^a-KvX^p 
[*304-14-1.*303-71] 

*304-151. 1- : ZffF . = . (^M, N,p).M<rN.Ml t^'p., N I tn'fi e Rat def . 

X^Ml t^^'fi . F = iV ^ <u'/t [*304-15] 

*304-152. h:.p Prm v . p Prm o- . D : {(ji/v) I iu'/i} H {(p/a) I t^^'p] . = . 

p,/v <r p/a- .p.,v,p,<r6'D'Uf\a'U [*304-161 . *303-731] 

*30416. h : (p,/v) H (p/a) . = . (cr/p) H (v/p.) [*304-lo] 

*304161. h : XHY. = . FffZ [*304-12151] 

*304-2. h.HdJ 

Dem. 
h . *803-37 . D 

I- •.p.,v,p,a-e'D'U nd'U .{p,lv)^tn'p = {plff)^ tii'p,. D./i XcO- = i' Xgp . 
[*304-15] D . ~ {(p,/v) H (p/a)} (1) 

h . (1) . Transp . ^ H . Prop 

*304-201. h - ~ (Z <,Z) [Proof as in *304-2] 



280 QUANTITY [PAET VI 

*304-21. h. 5^ 6 trans 

Dem. 
V . *30415 . D h : XHY . 7HZ . D . 

px.'nKaX.^.X^ {ixjv) i t,,'^ . Y = (p/a) \; t,,'u . Z = (f/,?) ^ «n V (1) 
I-.*117-571.*120-51.D 
h'.fji,v,p,a;^,r]eI)'Ur\Q.'U.fj,Xe(7<vXgp.pX„r]<a-Xe^.':y. 

yu, x„ o- Xg 1? < V Xo p Xe i; < K x„ a- x„ f . 
[*126-51]D./iX„9?<i'X„^ (2) 

h . (1) . (2) . D f- . Prop 

*304-211. h : X <^ F . F <, ^ . D . Z <, ^ [Proof as in *S04-21] 

*304-22. h.He connex 

Dem. 
h . *126-33 .0\-:.fji,,v,p,aeB'U na'U .D: 

IJ,x„cr <vXeP . v./iXecr = i;Xo/J.v.ytt Xo<r>i/Xo/3 (1) 
h.(l).*304-15.DI-.Prop 

*304-221. \-:.X,Ye'Rsit.D:X<r7.v.X=7.v.Y<rX [Proof as in *304-22] 

*304-23. h.HeSer [*304-2-21-22] 

*304 24. h : /^, .. 6 D' f7 n a'f/" . 1/ =(= 1 . D . (^/v) H [nl{v -„ 1)} 
Dem. 

V . *120-414-415-416 . D h : Hp . D . i/ -„ 1 e I)'Ur^ Q'U (1) 

I- . (1) . *304-15 . D h . Prop 

*304 241. h-./jueD'U.^+^lea'U.I). (fi/l) H {{^ +„ l)/lj 
Dem. 

l-.*300-14. DhiHp.D./i, lea'C7 (1) 

h . *300-14 . *120124 . D h : Hp . D . ^u, +„ 1 e D'U" (2) 

l-.(l).(2).*304-15.DI-.Prop 

*304-25. l-:/i,i/6D'f/'na'C/'.~(/i+el = 5'f/.i/=l).D.M/i;6D'ir.i///i6a'fl 
[*304-24-241-16] 

*304-251. h : /i +„ 1 = £'[/" . D , ^/l ~ 6 T>'H 

Dem. 

|-.*300-14.D 

1- : Hp . /), o- 6 D'Cn a'C/" . D . /3 < /i . 1 < ff . 

[*117-571] D.px„l</iXeff (1) 

h . (1) . *304-15 . D 1- . Prop 

*304 26. h :. /i Prm v.D:,j,/ve D'H .= .v/fie d'H . 

= . /Lt.i/eD'D'n a'CT. ~(/i+„ 1 =£'i7. 1;= 1) 
[*302-39 . *304-25-251-15-16] 



SECTION a] the series OP RATIOS 281 

*304-261. 1- . D'5'=i' {(a/i, v) . fi, veB'Un Q'U". ~(/i+o l=B'U.v='l) . 

• X = (jjlIv) i iu V} [*304-2o-251-15] 

*304-262. 1- . a'H= X {('^,v).^,ve'D'Ur^ a'U .r^(fji+,l = B'U. v= 1) . 

X = (yjij) I t,^'ii] [*304-261-16] 
*304-27. l-:a[!ir.= .a!3 
Dem. 

H . *30014 . D 

h :. a ! 3 . D : /i= 1 . i; = 2 . D . (It, ;; 6D'Z7n a'U". ~(/i+„ 1 = 5'?7 . i/ = 1) . 

[*304-25] D . a ! ff (1) 

t- . *304-261 . D 

I- :. a ! if . D : (a/i, i;) : /i, 1/ e D'fTn a'f/" : /* +„ 1 6 a'fr. V . V + 1 : 

[*300-14] D : (a^t) . /i> 1 . a ! /* +0 2 . V . (ai') . v > 1 . a ! " +0 1 : 

[*ll7-32] D:a!3 (2) 

h . (1) . (2) . D I- . Prop 

*304-28. h : a ! 3 . 3 . G'H = X{{'3ji,v). fi.ve'D'U r^ d'U .X = {jilv)l V/*} 

= Eat def 
Dem. 

V . *30014 . D I- :. Hp, D : /i +0 1 = ^'f/" . 3 ■ /i > 1 (1) 

f-.(]).DI-:Hp.D.~(aAi,z/)./i+„l=5'C/'.i;=l.i/+el=5'f/'.^=l (2) 
h . (2) . *304-261-262 . *303-71 . D h . Prop 

*304-281. h :. a ! 3 . D : fi/v = B'H . = .,i= I .v+^l = B'U . = . v/iJ, = B'H 
[*304-28-261-262] 

*304-282. h.O^'^eC'H [*304-27-28 . *303-66] 

*304-29. h : (fju/v) H (p/a) . /* +„ p, i/ +« o- e Q' CT . D . 

(/./«.) H {(/^ +, p)/(v +e a)] . {(m +e /,)/(. +„ <r)} H {pi a) 
Dem. 

f- . *3041 . D h : Hp . D . /i x^ cr < v Xj p . 

[*126-5] D . /i Xo (z/ +e o-)< v x„ (/i +e p) . 

{fj. +c p) Xe «r < (l/ +0 ff) Xe p . (1) 

I- . (1) . *304-l . D I- . Prop 
*304-3. I- : Infin ax . D . iT e Ser n comp [*304-29-23] 

*304-31. t- : Infin ax . D . ~ E ! B'H . ~ E ! B'H ' [*304-281 . *30014] 

*304-32. 1- : Infin ax . D . G'H e K„ 
Dem. 
h . *304-15 . *303-211 . *302-22 . D 

H . Nc'(7'fi^< Nc'l {(ap, 0-) . p Prm o- . p, o- 6 D'Z7n a'Z/ . Z = p/o-j 

[*303-36] < Nc'# {(ap, ff) . p Prm o- . p, o- 6 D' CT n a' U" . if = p J, 0-} 
[*33-161] <Nc'C"t'"x,Nc'C"U' (1) 



282 QUANTITY [part VI 

h . (1) . *123-52 . *300-21 . D I- : Hp . D . Nc'C"5"< K„ (2) 

I- . *304-28 . D 

1- : Hp . D . Nc'C'S^ Nc'l {(37/) . v e B'U n a'U. X = vjl] 

[*303-36] > Nc'(D' U^a'U) 

[*300-21] > No (3) 

h . (2) . (3) . *ll7-23 . D h . Prop 

*304-33. h : Infin Ax.D.Hev [*304-3-31-32 . *273-l] 

*304-34. h : Infin ax . D . C'H = B'H = Eat - 1% [*303-78 . *304-28] 

*304-4. h : XH'T. = . X, FeKat def u t'Og . X<rT. 

- • (a*^. I', /3, 0-) . ytt, z/, /), o- 6 Q' fT" . v 4= . cr 4= . /i X,. o- < 1/ Xe jO . 

X = (fi/v) t t^^'fi . F= (p/<t) I tr^'^ [*3037 1 . (*304-03)] 
*304401. h :. Infin ax . D : Z<, F. s . X^'F [*304-4 . *303-78] 

*304-41. V.'D'H' = X{{'3^ii,v).fi,v6a'U.v^Q.'^{li,+^l = B'U.v=\). 

[Proof as in *304-261] 

*304-42. h . a'H'= X {(g^, v) . fi.v ea'U . fi^O .v^O .X = (fi/v) ^ «uV} 
*304-43. h : a ! if' . = . a ! 2 [*304-42] 

*304-44. h : a ! 2 . D . Cfi"' = X {(g/^, i;) . /i, 1/ e Q' Z7 . i; + . Z = (/i/i.)t i„V} 
[*304-41-42] 

*304-45. h : a ! 2 . D . B'H' = 0^ [*304-41-42 . *303-6] 

*304-46. l-:a!3.D.ir' = 04«f fi" [*304-45-4-271] 

*304-47. I- : Infin ax.D.H'ei+v [*304-46-33] 

*304-48. h.H'eSer 
Dem. 

h . *304-4 .Dh:a!2.~a!3.D.Zf' = 0j4 (1/1) (1) 

I- . (1) • *304-43-46-23 . D h . Prop 

*304-49. I- : Infin ax . D . C'H' = Ji'H' = Rat [*304-34-46] 

*304-5. hiXeO'fl". D.aJ-^DRelnum [*303-73 . *304-14] 

*304-51. V:Xe C'H' . D . a ! ^ T ^^1 num 
Dem. 

h . *303-63 . *304-43 . D I- : Hp . D . a ! 0, f Eel num (1) 

h . (1) . *303-73 . *304-4 . D H . Prop 

*304-52. h : Z e Eat . D . (a/i) . Z ^ <iiV e C'H' [*304'44 . *300-18] 

*304-53. f- : Z e Eat - t'O, . D . (a/i) • X I Ui'h- ^ (^'H [*304-28 . *300-18] 



*305. MULTIPLICATION OF SIMPLE RATIOS. 

Summary of *305. 

The ratios hitherto considered are called "simple" ratios in opposition 
to "generalized" ratios (introduced in *307), which include negative ratios. 
We deal with multiplication and addition first for simple ratios, and then 
for generalized ratios. In this number we are only concerned with the 
multiplication of simple ratios. 

In defining multiplication of ratios, we naturally frame our definition so 
as to secure that the product of fijv and pja shall be {fi x^ p)j{v x.^ o"). This 
is effected by the following definition (where "s" stands for "simple"): 

*30501. Xx.,Y=RS [(a/i, I/, jo, 0-) . /^, I/, p, 0- 6 NC ind . J- + . o- 4= . 

X = im/v.T=p/<t.R {(fi Xe p)/(v X, <t)} S] Df 
which gives us 

*305-142. h : /li, p e NO ind . 1/ =f ■ o' + . 3 . /t/i' Xg p/o- = (/i Xo p)/(v x^ <r) 

and 

*305144. h : a ! {fijv Xs pi a) . D . /i/v Xg p/o- = (jix^ p)l(v x^ a) 

The reason for the hypotheses in these propositions is that, if yu, is a 
cardinal which is not inductive, while p = and v, a are inductive and 
not 0, fj,/v = A and fj,/v Xsp/a- = A, but (/iXap)/(v Xacr) = Oq. 

For the applications of the multiplication of ratios, it is essential that we 
should have, if M, S, T belong to a suitable vector family, 

R (ji/v) S . 8(p/a) T.'^.Rifi/v Xg p/a-) T, 

e.g. we want two-thirds of five-sevenths of T to be (2/3 Xg 5/7) of T. It will 
be shown in Section that our definition satisfies this requirement. 

We prove in this number 
*305-3. h : X, Fe Rat . = . Z Xg Fe Rat 
*305-22. h :. Z Xg F= Oj . = : Z, Fe Rat : Z = Og . V . F= Og 

i.e. a product only vanishes when one of its factors vanishes ; 
*305-301. h : Z, F e Rat - t'Oj . = . Z Xg F e Rat - i% 



284 QUANTITY [part VI 

*305-25. \-:^L,v,p,o-e'D'Una'U.D. (fi/v x, p/a) I t^'fi e G'H 

Thus a product of two ratios which both exist in a given type exists in 
the next type, i.e. 

*305-26. f- : X, 7 e Rat . Z ^ t,,'fj., Y [. t^'fi e Rat def . D . (Z x, F) ^ t^'/i e C'H 

The formal laws offer no diflSculty. We prove the commutative law 
(*305"11) and the associative law (*305'41); we prove that Xxjl/l=Z 

(*305-51) and that Zx,Z = 1/1 (*305-52). Division results from 
*305-61. \-:.Ae'Rait-i%.A'eB:at.D:Ax,X = A'. = .X = A'x,A 

and the axiom of Archimedes is given bj^ 
*305-7. h : Z, 7 6 Rat - t'O, . D . (aa) . o e NO ind . 7<, (a/1 x, Z) 



*305'01. Z X, 7= RSK'Sji, v,p,<T).fji,v,p,a-e'NGmd.v^0.a-^0. 

X = p.lv.Y=pl<r.R{{^X,p)l{vX^<T)]^ Df 

*3051. V:R{Xx,Y)8. = . ('3,fji,v,p, o-) . /i, i/, p.o-eNCind . i;=t=0 . o- + . 

X = ixlv. Y^pja . R {(/i Xe p)l{v Xe a-)] 8 [(*305-01)] 

*30511. h . Z X, 7= 7 Xs Z [*3051] 

*30512. h : Z, 7 ~ e I'Oj w i' oo^ . Cnv'(Z x,Y) = Xx,Y [*3051 . *30313] 

*30513. t- : /*, V, p, a- 6 NO ind - I'O . p-jv = pfjv . pja- = pja . D . 

(p, Xe p)/{v x, 0-) = {p' x„ p')/(j'' Xo o-O 
Z)em. 

1- . *303-39 . D F : Hp . D . /t Xo v' = v x„ /a' . /3 Xo cr' = p' Xo o- . 

[*120-51] D . /4 Xo /3 Xo v' Xo 0-' = / Xe p' Xo v Xo o- - 

[*303-39] :>-(px. p)l{v Xo <7) = {p' Xo pO/Ci-' Xo <x') : 3 h . Prop 

*305131. V:v,p,(7e NC ind - t'O . 0/z/ = /*'/).' . pja- = p'/a' . D . 

(0 x„ p)/(p Xo a) = (/ Xo p'W Xo ff') 
Dem. 
l-.*303-66. Df-:Hp.D.yLt' = O.i''€NCind-i'0 (1) 

h . (l).*303-6 . D h : Hp. D. (0 x^p)/{vx^o-) = Og = {p: x,p')l(v'x,a-') : D h. Prop 

*305132. I- : /i, I/, p, o- e NC ind . V 4= . o- 4= . /it/j/ = p'/v . p/a = pja . D . 

{p xi p)/(i' Xo 0-) = (/^' Xo pO/C"' Xo o-') 
[*305-13-131] 



SECTION a] multiplication OF SIMPLE RATIOS 285 

*30514. h : /*=}= . /)4= . i/=j= . 0- + . 3 . /i/v Xsp/(7 = (/i ■x.^p)l{v x^ o") 

Bern. 
V . *3051-132 . D 
h : : Hp . D :. i2 {fijv x^ pja) S .-: 

(a/. "'. />'. o-') • /*'> "'. p'. 0-' 6 NO ind . iit/i' = /i'/v' . p/o- = p'/o-' . v' + . 0-' =f : 

i2{(MX„p)/(..x,«7)}>S (1) 
H . *303-181 . *302-36 . *120-512 . D 

h : Hp . i? {(^ x„ p)/(v x„a-)}S.D. fi, v,p,a-e NC ind (2) 

l-.(l).(2).DI-.Prop 

The condition /* =j= . p =j= is required in the above proposition because if, 
e.g. yu. = . p 6 NC infin, we shall have (if v,ae NC ind — I'O) fi/v = Oj . p/tr = A, 
whence fi/v x^ p/a = A, but (jx x^ p)/{v x^ cr) = Og. If we assume /i, p e NC ind, 
it is not necessary to assume /i 4= . p =^ 0. This is stated in *305"142. 

*305-141. h : . i; = . V . or = : D . /i/i/ Xg p/or = A 
Dem. 

h .*30S-67-ll .Dh :v = . fi',v' eiaCmd . fjL/v = /jl'/v .:> .v =0 (1) 
l-.(l).*305-l.Dh.Prop 
*305142. \- :/i,pe NC ind . v =}= . o- =|= 0. D . fi/v Xj p/o- = {/i x^ p)/(i; x^ a) 

[Proof asin*305-14] 
*305143. h : a ! (At/V Xg p/f) . D . /t, v, p, o- e NC ind . v =}= . o- =f 

Dem. 
h . *3051 . D h : a ! (fi/v x, p/o-) . D . ('^p,', v') . /, v' e NC ind . v' =j= . /i/v = p.'/v' . 
[*303-182-67] D./i,i'6NCind.i/=)=0 (1) 

Similarly H : a ! (a'/i' Xg p/v) . D . p, o- e NC ind . o- =f (2) 

I- . (1) . (2) . D h . Prop 
*305144. I- : a ! (fi/v x, p/a) .D./i/v x, p/a =(/i x„ p)/(v x^ a-) [*305-143-142] 

«305'15. 1- :. ~ (/Lt, v, p, o- 6 NC ind) .v.i' = 0.v.(7 = 0:D. /i/v Xg p/a = A 
[*305-143 . Transp] 

*30516. h:.fi,v,p,ae NC ind:/* = 0.v.p = 0:i'=|=0.o-4=0:D. 

p./v X, p/o- = Og [*305-142 . *303-6] 

*30517. f-.Zx,<»g = A [*305-141 . *303-67] 
*305-2. h : a !X x, 7. D.Z,Fe Rat 

Dem. 
h . *305-l . 3 

h : Hp . 3 . (a/i, v, p, 0-) . fj,,v,p,ae NC ind .i'4=0'O'=f^"-^~ /*/" ■ -^ ~ P/"" ■ 
[*303-7] 3 . Z, FeRat : 3 h . Prop 



286 QUANTITY [part VI 

*305-21. \-:Xx,7e Rat - I'O^ .:^.X,Y€ Rat - 1% 

Dem. 

h . *303-72 . *305-2 . 3 h : Hp . D . X, Fe Rat (1) 

h .*30516.Transp . D h : Hp . D . X + Og . 74=0^ (2) 

h . (1) . (2) . D I- . Prop 

*305-22. \-:.XXs7=0g. = :X, YeRat-.X = 0^.v .Y=Oq 

Dem. 
l-.*305-l-2-142.*303-66.D 
h :. X Xj F= Oj . = : (g/x, v,p,a).X= fi/v . Y= pja . fi,v,p,(Te NC ind . 

/u,Xo/3 = 0.i'Xo 0-4=0: 
[*303'66] = : (g/i, v, p, a) : X = fiju . F= p/a- . /j,,v,p,a-€ NC ind . 

K =t= . cr =)= : yu./j' = Og . V . jo/o- = Og : 
[*303-7] = : X, Fe Rat : X = 0, . V . F= Oj :. 3 h . Prop 

*305-222. h : X X, Fe Rat .O.X.Ye Rat [*305-21-22] 

The following propositions are lemmas designed to show that if X, F are 
ratios which exist in a given type, X Xg F exists in the next type. 

*305-23. h : /i e NC ind . D . (2 Xe /*) +e 1< 2''+»i [*ll7-652 . *120-429] 
*305-231. }-.(fji+^iy = fj?+^{2x,fj,)+„l [*116-34.*113-43-66] 

*305-232. h : /i 6 NC ind . D . /i'' < 2"+'' 

Dem. 
h . *116-311-321 . D 1- . O'' < 2»+«i (1) 

f-.*305-231. DI-:Hp.^^<2'^+o'.D.(/i+el)='<2''+=i+„(2x,/[i)+,l (2) 
h . (2) . *30.5-23 . D h : /i e NO ind . /a'' < 2''+«' . D . (/a +, If < 2''+»' +„ 2»+=' - 
[*113-66.*116-52] 3 ■ (fi +0 1)" < 2"+"' (3) 

h . (1) . (3) . Induct . D I- . Prop 

*305-24. h:/i, i/./j.o-eD'fZna'Z/.D. 

(/A Xojo) n ^'/i,(j/ Xgff) r\ t'/MeD'U r\ d'U 
Dem. 

V . *116-72 . D h : Hp . 3 . (2^+=' n t'/i) eC'U. 

[*305-232] :>.fi^nt'fi€a'U (1) 

h.*116-35. DhiHp.D./i'^ni'/ieD'f;' (2) 

Similarly \- iRp .D .v' r\t'fi,p^ nt'iu,,a^ nffieTfU na'U (3) 

l-.*117-57l.D 

I- :. Hp . D : /i Xj p < ^^ V . /i Xo /3 < /d'' : J' Xo o- < 1;= . V . z/ Xo c7 < 0-2 (4) 

h . (1) . (2) . (3) . (4) . D h . Prop 



SECTION a] multiplication OF SIMPLE RATIOS 287 

*305-25. ^:fi,v,p,a-e'D'Una'U.D.(filvx,pla)lt^'lj,eC'H 
Bern. • 

I- . *305'14 . D I- : Hp . D . fijv x, p/a- = {fi x^ p)/(v x^ cr) (1) 

I- . (1) . *304-28 . *305-24 . D h . Prop 

*305-26. h-.X.YeU&t.X^ tn'/i, Y^ <n'/* 6 Rat def . D . (Z x, F) t <ooV e G'H 
[*305-25 . *304-28] 

*305-27. h : Z, Fe Rat - I'Og . D . (g/i) .(X x,Y)H„'fie G'H 
[*305-26 . *303-721] 

*305-28. h : X, F 6 Rat . D . (a/i) . (X x, F) ^ t^'fi e O'fi^' [*305-27-22] 

*305-3. h : Z, F 6 Rat . = . X x, Fe Rat 
Dem. 

h . *305-142 . *303-7 . D h : X, Fe Rat . D . X x, Fe Rat (1) 

h . (1) . *305-222 . D h . Prop 

*305-301. h : X, Fe Rat - t'O, . = . X x, Fe Rat - I'Og 
[*305-14.2 . *303-7 . *305-21] 

*305-31. I- : (a/i) . X t t,,'fi, Y t t^'fi eG'H. = . (gi/) . (X x, F) ^ «„'«' e G'H 
[*305-301.*304-53] 

*305-32. h : (a/^) . X ^ «u V, F f «,//* e G'S"' . = . (31/) . (X x, F) p t^'v e G'H' 
[*305-3 . *304-52] 

*305-4. h :\, I/, 0-6 NO ind./* 4=0 •P + 0.t4=0.D. 

(\/fi Xgvjp) Xg{<r/T)=(X x^vx^a-yifi Xe/3 x„T)=X/fi, Xs{v/p Xsct/t) [*305-142] 

*305-41. l-.(Xx,F)x,^=Xx,(Fx,Z) [*305-4-2] 

*305-5. h : /* + . 3 . {\//jl) x, (1/1) = X/p. [*305-14-14215] 

*305-51. h:XeRat.D.Xx,(l/l) = X [*305-5] 

*305-52. l-:XeRat-t'Og.D.Xx,X = l/l 
Dem. 

f- . *30514 . *303-13 . D 

h : Hp . D . (a/x, v).p,,ve NC ind - I'O . X x, X = (/u, x„ i;)/(i; x^ /*) . 

[*303-23] D . X x, X = l/l : D I- . Prop 

*305-6. l-i.^eRat-t'Oj.XeRat.D:^! x,X = A'. = .X = ^'Xsl 
Bern. 

f-.*304-l-4.*305-32-222.D 

1- : Hp . D . (a/x, v, /3, cr, f , rj). p,v,a-e NC ind — I'O .p,^,7je NC ind . 

A = p./v . X = p/a . A' =^ ^It, (I) 



288 QUANTITY [part VI 

h . *305-142 . D f- :. yet, 1/, o- e NC ind - t'O . p, f 97 6 NO ind . D : 
W" Xs p/o- = f /'?■ = ■ (m Xo />)/(" Xe o'} = ^/v ■ 

[*303-38] = ./iXepXe1? = J'X„o-Xe^. 

[*303-38] =-p/<r = (.V x„ ?)/(/[* Xe 1?) 

[*305-142.*303'13] = ^/i? x„ Onv V/") (2) 

h . (1) . (2) . D h . Prop 

*305-61. h :. ^ 6 Rat - 1% .A'en^it.^D:Ax,X=A'. = .X = A'x,A 
[*305-6-222-32] 

*305-7. h : Z, 7e Rat - t'O, . D . (ga) . a e NC ind . F <, (a/1 x, X) 

Dem. 

h . *117-571 . *120-511 . *117-62 . D 

\- : /i,v,p,(re NC ind — t'O . f > j/ . D . 

[*3041] D . (p/o-) <r (^ Xe p Xe f Vi; . 

[*305-14] D . (p/<7) <, {/jilv X, (p Xe f )/l} (1) 

I- . (1) . *304-l . *120-5 . D h . Prop 

*305-71. h :. -? 6 Rat - t'Og .D :X <rY .- .X XsZ KrYx^Z 

Bern, 
h . *30.5-142 . D h : Hp . X <, F. D . 

(a^, v, p, 0-, ^, »;) . /A, v, p, 0-, f , i; 6 NC ind . J/ + . o- + . ^ =f . 17 4= . 
X = fj,/v . F= p/o- . Z= ^/r) . fiXgCr <iv Xe p . 
Z X, ^ = (/i Xe M" Xe '?) ■ F X, ^= (p Xe ^)/(o- Xe 17) . 

[*304-l.*126-51]D.Xx,2:<rFx,-? (1) 

|-.(l).Dt-:Hp.Zxj^<rFx,^.D.Zx,^x,^<^FxsZxgi. 
[*305-51-52] D . Z <^ 7 (2) 

h . (1) . (2) . D h . Prop 



*306. ADDITION OF SIMPLE RATIOS. 

Summary of *306. 

The addition of simple ratios is treated in a way analogous to that in 
which their multiplication is treated. We wish to secure that the sum of 
Xjv and fijv shall be (K+gfj.)/v, and that the sum of /i/v and p/a- shall be 
{(/i Xj o") +0 (j; Xgp)}j(v Xo a-). This is secured by the definition 

*306-01. X +g F= R^ [('3JI, v,p).iJ,,v,pe'NCmd.v=^0. 

X = filv.Y=plv.E {(/i +c p)lv} S] Df 
whence we obtain 

*30613. I- : 1/ + . D . /i/i/ +s /a/i; = (;ti +e p)/v 

*30614. h : 1/ =1= . o- =t= . D . /i/v +8 p/o- = {(/Li Xe 0-) +0 (i* Xo p)}/(v x^ a-) 

Our definition is so framed that oo g +8 txs , = A. This is on the whole 
convenient, though we could, of course, frame our definition so as to have 

COq+s'X) q='X> q. 

In applications, if R, S, T are members of a suitable vector-family, we 
want to have 

B (fi/v) T . 8 (p/o-) T.:).iR\S) iix/v +8 pla) T, 

e.g. if a vector R is 2/3 of T, and a vector S is 5/7 of T, we want the vector 
which consists of first travelling a distance R and then travelling a distance 
S to be (2/3 +8 5/7) of T. We shall show in Section C that our definition of 
addition fulfils this requirement. 

As in the case of products, the sum of two ratios is a ratio (*306'22), and 
the sum of two ratios which exist in a given type exists in the next type 
(*306"64). A ratio is unchanged by the addition of Og (*306'24), and a sum 
of two ratios is only Og if both the summands are Og (*306'2). No difficulty 
is offered by the formal laws : we prove the commutative law (*306'11), the 
associative law (*306'31), and the distributive law (*306'41). 

An important proposition is 
*306'52. h :. X <^ F. = : Z 6 Rat : (g^) . .^e Rat - t'O, . Z +« ^= F 
When the axiom of infinity is assumed, this proposition becomes 
XH'Y. = : X 6 C'H' : (^Z) .ZeC'H.X +,Z=Y. 

B. & W. III. 19 



290 QUANTITY [part VI 

We prove also the proposition upon which subtraction depends, namely 
*306-54. h :. Z, Fe Rat . D : Z +, F=Z +,Z. = . F= Z 



*306-01. X+,Y=RS [(3^1, v,p).fi,v,pe'!^Cmd.v^O. 

X^(i/v.Y=p/v.R{(fi+,p)/v]^ Df 

*3061. f-:iJ(Z+,F)<Sf. = .(a/i,i',p)./x,i/,p6NCind .7^ + 0. 

X = ij./v.Y=pIv.E {{/j. +, p)/v} S [(*306-01)] 

*306 11. h . Z +, F = F +, X [*3061 . *110-5l] 

*30612. I- : a ! (Z +, F) . D . Z, F 6 Rat [*3061 . *303-7] 

*306121. \-:fjL/v = M.'/v' . p/v = p'// . 3 . (At +c p)Ip = (/*' +o p')/"' 

Dem. 
h .*303-39 . D h : Hp . /i, i^, p, /*»' e NC ind . i/ + . k' + O . D . 

fi x^v = fi' XgV . p Xgv' = p' x^v . 
[*113-43] D.(f^+, p) X, v' = (ji! +, p') X, V . 

[*303-39] D.(f,+,p)/v = (p,'+,p')lv' (1) 

l-.*303-181.*302-36.D 

h : Hp . ~ (p,, V, p, pi, v', p' e NC ind) . D . (^ +c p)lv = A . (/ +„ p')/"' = A (2) 
I- . (1) . (2) . *303-67 . D h . Prop 

*306-13. \-:v=^O.D.p,/v+,p/v = (fj,+^ p)/v 
Bern. 

I- . *306-l . D F : Hp . D . (ytt +„ p)/v G /^/j/ +, /a/i' (1) 

l-.*306-121.D 

!-:/./«. = p.'/v' . p/v = p'jv' . X {(^' +, p')/v'} F . D . Z {(^ +, p)/v} Y (2) 

F . (2) . *306-l .D\-.p,/v+, p/v e (y(i +e /3)/i' (3) 

h . (1) . (3) . D I- . Prop 

*30614. F : I- + . o- + . D . /x/^ +, p/<7 = {(/. x„ <r) +e (v x„ p)}/(i; x„ a) 

Dem. 
h.*303-39.D 

h : Hp . /i, v, /), o- 6 NC ind . D . p,/v = (/a x^ o-)/(j/ x„ o-) . p/o- = (v x„ /3)/(i' x„ <r) . 
[*306-13] ^■f^/i'+eP/<T = {(/^x,<r)+,(vx„p)}/(vx,^) (1) 

I- . *30612 . *303-ll . D 

\-:r^(lu,,u,p,ae NC ind) . D . //,/z. +, p/a=A. {(fj.x^a-)+^(v x„p)}J(v x„a-)= A (2) 
h . (1) . (2) . D f- . Prop 

*306141. ]-:.v = 0.v.(T = 0:D.p./v+,p/a- = A [*306-12 . Transp . *303-7] 



SECTION a] addition OF SIMPLE RATIOS 291 

*30615. \-:iJi.lv+,p/<T=Qg. = .fi = p = 0.v,cr€'NGind-i'0 

Bern. • 

l-.*306-14.*303-66.Dh:/i = /j = 0.r',o-eNCind-i'0.D./t/j/+,p/<r=0, (I) 
h.*30612. D h : /i/i/ +« p/o- =Oy.D. /.,!/, (0,0- 6 NO ind (2) 

h.*306-141. DI-:/i/x/+,/3/o- = 0,.D.i'=|=0.o-=f (3) 

K (3) . *306-14 . D h : Hp (3) . D . {(/. x„ o-) +, (v x„ p)}/(v x„ cr) = 0, . 
[*303-66] O.(f,x,a)+,{vx,p)=^0.vx,a^Q. 

[*110-62.*113-602] D./i = /3 = 0.i' + 0.<r + (4) 

l-.(l).(2).(4).DI-.Prop 

*30616. \-.X+,7=RS[('^iJ.,v,p,a-).p.,v,p,ae'SCmd.vJrO.<7^0. 

X = p./v. Y= p/a- . R {{p, x„ o- +c J/ x„ p)/v x„ <t] S] 
[*3061412] 

*30617. h : /i = . 1/, /3, o- 6 NO ind . i; 4= . o- =t= . D . /i/i; +, jo/o- = p/o- 
Dem. 

h . *303-6 . D h : Hp . D . p/i/ = O/o- . 

[*306-13] 3 . p.lv +s p/a- = (0 +„ p)/ff : D 1- . Prop 

*3062. \-:X+,Y=0q. = .X = 0g.Y=:0j [*3061512] 

*306 22. l-:Z+,FeRat. = .Z, FeRat 

Bern. 
h . *30616 . *303-7 . D h : Z +, Fe Rat . s . 

(g/u, 7^, jO, cr) . p,v,p,a- e NC ind . X = /ci/i/ . F= p/cr .vx^a- ^0 . 
[*113-602]= . (a/i, i/,/3, a-).p,,v,p,<re'NCmd.X = p./v.Y=p/(T.v^O.<T^O. 
[*303-7] ■ = . Z, Fe Rat : 3 F . Prop 

*306-23 h : Z +, Fe Rat - I'O, . = . Z, Fe Rat . ~ (Z = F= 0,) 
[*306 22.*303-7.*306-2] 

*306-24. l-:ZeRat.D.Z+,Oy = Z [*306-l7-ll] 

*306-25. h : Z +s Fe Rat . = . a ! (Z +g F) . = . Z, Fe Rat 
[*306-12-22 . *303'26 . *30614] 
Here X +gY must be taken in a sufficiently high type, otherwise X +gY 
may be null when X, FeRat. 

*306-3. h . (X/fi +, v/p) +s o-/r = X/p, +, (v/p +, ct/t) 

Bern. 
\- . *306-14 .DI-:/i=|=0./34=O.T4=O.D. (X//i +s v/p) +, cr/r 
= {(^ Xo /») +0 (/* x-o v)]/{fi Xe p) +s ajr 

[*80614] = ((A, X„ p X„ t) +c (/i X,, jy X„ t) 4e (/i X„ /D X„ (7)}/(yti Xe p X„ t) 
[*1 13-43] = [{X Xe (/) Xe r)} +e {^ X^ ((l/ Xe t) +e (p X^ <T))W/{p. X^ (p X, t)} 
[*306-14] = V/. +, {(v Xe t) +e (p Xe (r)}/(p Xe t) 

[*306-14] = \//i+,(i//jo+,o-/T) (1) 

l-.(l).*30612.DI-.Prop 

19—2 



292 QUANTITY [part VI 

*306-31. \-.{X+,7)+sZ=X+,{Y+,Z) 

Bern, 
h . *306-3 . D\-:X = X/fi.Y=v/p.Z=a-jr.D. 

(X+,Y)+,Z = X+,(Y+,Z) (1) 
h . *306-25 . D h : ~ (gX, fi, v, p,a,T).X = X//j,.Y=v/p .Z=<7/t .D . 

(X+sY)+,Z=A.X+,iY+,Z) = A (2) 
h . (1) . (2) . D h . Prop 

*306-4. h . X//i Xj (v/p +s o-/t) = (\/yt4 Xg iz/p) +, (X//4 Xj (t/t) 

Bern, 
h . *306-14 . D h : X,y[4, 1/, p, 0-, T e NC ind . /i =1= . V + . o- + . D . 

X/fJ. X, (v/p +s 0-/t) = \//A X, {(j/ Xo t) +e (/3 Xo a)}/(p Xe t) 
[*305-14] = [\ Xe {(j/ Xe t) +e (p X„ 0-)}]/(/x X„ p X^ t) 
[*303-23] = [X Xe jU, Xe {(v Xe t) +e (p Xe 0-)}]/(^ X^pX^p. X^ t) 
[*113-43] = {(X Xe /i Xe V Xe t) +„ (^ Xc /^ Xc P Xc 0")}/(A* ^e jO Xg /i Xe t) 
[*30614] = (X Xe I/)/Ox Xe p) +s (X Xe <7)/(p, X, t) 

[*305-14] = (X///, X, vZ/a) +, (X//^ x^ (t/t) (1) 

h . *305-2 . *306-22 . D h : g ! X//i x^ (i///3 +, a/r) . D . \/p, v/p, a/r e Rat . 
[*303-7] :>-Hp(l) (2) 

l-.*306-12.*305-143.D 

^'■'k- {(V/* Xg Z^//3) +s (X//i Xg (t/t)} . D . X//LI, Iz/iO, (7/t 6 Rat . 

[*303-7] 3.Hp(l) (3) 

l-.(2).(3).D 

h : ~ Hp (1) . D . \/fj, Xg (v/p +, ct/t) = a = (X//i Xg v/p) +, (X/p. Xg o-/t) (4) 

I- . (1) . (4) . D h . Prop 

*306-41. l-.Xxg(F+g^) = (XxgF)+g(Xxg^ [*306-4-25 . *305-2] 

*306-51. \-.X+s (v/1 Xg X) = (v +e 1)/1 Xg X 

Bern. 
h . *306-12 . D t- :. a ! (Z +g (z^/l XgX)} . D : X v/l x, Z e Rat : 
[*305-3.*303-7] D : v e NO ind : {'S_p, o-) . /o, <7 e NC ind . cr =t= . X = p/o- (1) 

h . *305-2 . D h :. a ! ((i; +e 1)/1 x, Z} . D : (i- +„ 1)/1, Z e Rat : 
[*303-7.*126-31] D : 1/ 6 NC ind : {-^p, tr) . yo, o- e NO ind . o- =j= . Z = /o/<7 (2) 
I- . *305-142 . D I- : I/, ;o, o- e NC ind . o- =t= . D . I//1 Xg p/o- = (j/ x^ p)/cr . 
[*806-13] D . p/(T +g (v/l Xg /d/o-) = {/J +e (i/ Xe yo)]/o- 
[*113-6.71] ={(v+,l)x,p]/a- 

[*305-14] =(v +,!)/! Xsp/a (3) 

F . (1) . (2) . (3) . D h . Prop 



SECTION a] addition OF SIMPLE EATIOS 293 

*306-52. h :. X <r 7. = : X 6 Rat : (a^) . Ze Rat - I'Og .X+,Z=Y 

Bern. 
F.*306-13.*119 34.D 
\- : fi, V, p, a e NO ind .v^O .(T=^0 . X = /jl/v . Y= pja . fj.Xa& <.v x^p ■ 

^= (v Xo p) -0 (p-x^a).Z= ^liy Xoff).D.X+sZ={vXa p)/{v x„ a) 
[*303-23] = p/a 

[Hp] = Y (1) 

h.(l).*304-l-13.D 

h •.•X<r F. D : XeRat : (gZ) . ^eRat- t'O, .X+sZ=Y (2) 

h . *306-14 . D 
\-:fji.,v,p,ae'MCmd.v^0.p^0.a-^O.X = iJ./v.Z = p/a.Y=X+sZ.'D. 

F= {(/i X„ ff) +„ (v Xe /))}/(y Xe 0-) . [{(/i X„ ff) +„ (v X„ /a)} Xe f] > /i Xc {v X„ O") . 

[*304-l] D . X <^ 7 (3) 

h . (3) . *304-l . D h : Xe Rat . ^6 Rat - i% .X +,Z=Y .D .X <rY (4) 
I- . (2) . (4) . D h . Prop 

The above proposition requires that X and Y should be taken in a 
sufficiently high type, namely at least in a type in which, if X = fi/v and 
Y = p/a, where fi Prm v and p Prm a; (v x^ p) +„ 1 and (/i x^ o") +„ 1 are not 
null. Otherwise there may be no Z such that X+gZ =Y. 

*306-53. h:./i,j;6NCind.v4=0.o-=f=0.i7=t=0.D: 

/m/v +s p/(7 = p,/v +s ^/v-=- p/<^ = ^/'7 

h . *306-12 .Dh:HTp.fi/v+s pja- = njv +> ^/v . ~ (/o, o- e NO ind) . D . 

W" +« l/i? = A . /a/o- = A . (1) 

[*306-25] D . ~ {fi/v, ^/v 6 Rat} . 

[Hp.*303-7J D.~(?,97eNCind). 

[*303-ll.(l)] 3 . f/'? = p/a- (2) 

I- . *306-25 .D\-:Hp.fi/v+s p/a- = /jl/v +g ^/v . />, o" e NO ind . D . 

^,77 6 NO ind (3) 

h . (3) . *30614 . *303-39 . 3 

h : Hp (3) . D . [(fi x„ ff) +c (v x„ |o)} x^ v x^ t? = {(/i x^ rj) +„ (v x^ ^)} Xe i/ x„ o- . 
[*113-.43] 

D . (/i Xo O- Xe V Xe 1?) +„ (l/^ Xe p-X.a'n) = (H- Xc 0" Xc «' Xo '?) +o (''" X^ | X^ ff) . 
[*126-4] D . l/^ X„ (,0 Xe V) = "" Xe (^ Xe <7) . 

[*303-39]D./>/<7 = ^/»; (4) 

h . (2) . (4) . D h :. Hp . D : fi/v +sp/<r = p^/v+g^/v .:>.pl<T= ^/v (5) 

1- . *306-l . D\-:p/(7 = ^/v^- P'h +s pl<y = /*/«' +8 f/i; (6) 
I- . (5) . (6) . D h . Prop 



294 QUANTITY [part VI 

*306-54. t-:.X, FeRat.D:Z+,F = X+,^. = . Y=Z 
Dem. 

h . *306-25 . D h :. Hp . D : X+, Fe Rat : 

[*306'25] D:Z+, F=X+,^.D.Z6Rat (1) 

h . (1) . *306-53 . *803-7 . D h . Prop 

*306-55. l-:F<,X.D.~(a^).Z+,Z=F 

Dem. 
I- . *117-291 . *304-l . D h : Hp . D . ~ (Z <, F) . 

[*306-52] D.'^('^Z).Z6'Rat-i'0g.X+sZ=Y (1) 

t- . *306-24 . *304-l . D h : Hp . D.~(Z+,0,= F) (2) 

t-.*306-25. Dh:Hp.Z+,Z= F.D. ^eRat (3) 

I- . (1) . (2) . (3) . D h . Prop 

The following propositions are concerned with the existence of Z +g F in 
definite types. It will be showa that if Z, F exist in a given type, X +sY 
exists in the next type, i.e. if Z ^ t^^'/ji, and F I tn'fi exist, then (Z +g F) ^ iooV 
exists, where Z, F are rationals. 

*306-6. 1- : /I, peB'U n a'U. D . (/i, +„ /a) n t'^ e D'Una'U 

Dem. 

l-.*305-23.DI-:Hp.//,<p.D.M+„/3<2<'+"i (1) 

Similarly h : Hp ./3</i. D . /ti+o/s < S-^+'i (2) 

t-.(l).(2).*11672.DI-.Prop 

*306-61. \-:tJ.,v,p€D'Una'D .0. (fi/v +, p/v) n «„„'/* e Rat def 

Dem. 
\-.*B061Z-6.'D\-:RY).D.fi/v+,p/v=-(fi+„p)/v.(iu.+^p)nt'fi,vnt'iMeI>'Una'V. 
[*30371] D . (At/y +8 p/i') n ^„„V e Rat def : D h . Prop 

*306-62. [■:fi,v,peJ)'Una'V'.D. (fi/v +, pip) n t^'fj, e Rat def 

Dem. 

h . *303'39 . D I- : Hp . D . /i/i/ +, p//3 = jn/j; +, i^/i/ (1) 

I- . (1) . *306-61 . D I- . Prop 

*306-621. h : o- 6 NC ind . D . o-'' -„ a- +e 1 < 2"^ 
Dem. 

|-.*116301-311. Dl-.0^-<,0+ol<2» (1) 

|-.*116-321-331. DI-.P-el+„l<2i (2) 

I- . *ll7-55 . *126-5 . D h . 2^ -„ 2 +e 1 < 2= (3) 

I- . *305-231 . D h : Hp . o- > 1 . 0-2 -,, a- +e 1 < 2-' . D . 

(o- +„ 1)= -„ (tr +„ 1) +„ 1 < 2" +„ (2 Xe <r) . 
[*ll7-652.*116-52]D.(a-+,l)2-,(o-+„l)+„l<2-'+«i (4) 

1- . (1) . (2) . (3) . (4) . Induct . D F . Prop 



SECTION a] addition OF SIMPLE RATIOS 295 

*306-622. h : /* e NC ind - t'O . D . (/* -0 1)^ = ij? -^{1 x„ ii) +„ 1 
D&m. • 

[-.*305-231^^^:^. DI-:Hp.D.(/i-el?+c{2x,(/^-,l)}+„l = /.' (1) 

t- . *113-43 . *120-416 . D I- : Hp . D . {2 Xe (/i -e 1)} +e 2 = 2 Xe /i (2) 

K(l).(2). DI-:Hp.D.(/.-,l)^+„(2x«/.) = ^»+„l (3) 

h.(3).*119-32.DI-.Prop 

*306-623. h : /t, v, p 6 NC ind . v < ^ . p < /i . D . (/i Xe ytt) +e (i/ Xe pX 2''+«i 
Dem. 

h . *120-429 . D h : Hp . D . (/. x„ /i) +„ (i; Xo /d)< /.= +e (/i - 1)^ - 
[*120-429.*306-622] D . (/* x„^)+„(i; x„p) < (2 x,fi?) -, (2 x„y[i)+„2 
[*306-621.*126'51] < 2"+"! : D h . Prop 

*306624. h : /4, J/, /3, 0" 6 NO ind .i/</*.p^/i.o-^/t.D. 

(/. x„ <7) +„ (,; x„ p)< 2^+' 1 [*306-623] 

*306-63. h : /A, i/,/3, o- 6 D'f7 n Q'tT" . D . (/i/v +, p/a-)^too'fJ. e Rat def 

h . *306-62 .DhiHp. !- = /*. D. (At/v +« p/a-) p «ooV e ^.at def (1) 

I- . *306-624 . *305-24 . *303-7l . D 

h : Hp .!'</*. p^/t.o-^/it. 3. (/i/v +g p/(t) ^ fooV ^ ■'^^^ ^^^^ (2) 

Similarly 

h : Hp .i/</i./i^j0.o-^/i.D. (/ti/j/ +, p/a) ^ ioo'/i e Rat def (3) 

l-.(2).(3).D 

h : Hp . V < /i . (7 ^ yti . D . (//./k +g p/cr) ^ ^oo V ^ ^^^ '^^f (^) 

Similarly 

h : Hp . fj,> V . <T^/JL.0 . (/i/v +g p/a) f t^'/jL e Rat def (5) 

1-.(1).(4).(5).DI- :Hp. o-</t.D. (jM/v-\-gpla-) ^C/teRatdef (6) 

Similarly H : Hp . /* ^ o- . D . (/i/i; H-j p/cr) ^ ^oo'/ti e Rat def (7) 

h . (6) . (7) . D h . Prop 
The following propositions are immediate consequences of *306"63. 
*306-64. h : (/tt/v) I tii% (p/a-) I tn'/j, e Rat def . D .{fi/v +, p/o-) ^ «ooV « I^at def 
*306-65. F : Z, Fe Rat def . D . (X +, F) ^ t^'G"C'X e Rat def 
*306-66. h : Z, F 6 0'5^ . D . (Z +, F) t t^'CG'X e G'H 
*306-67. h : Z, F 6 C'lT' . D . (Z +s F) ^ C'C'CZ e C'iT' 



*307. GENEBALIZED RATIOS. 

Swmmary of *307. 

In this number we introduce negative ratios. If X is a ratio, what would 
ordinarily be called — X is X | Cnv. This may be seen as follows. Suppose we 

have RXS. We then have R (X | Cnv) 8. Now if R and S are vectors which 

carry us in the same direction, R and S are vectors which carry us in 
opposite directions, i.e. their ratio is negative. Hence calling the class of 
negative ratios "Eat„," we may put 

*30701. Eat„=|Cnv"Rat Df 

The sum of "Rat" and "Rat„" we will call "Rat^," where "g" stands 
for " generalized." Thus we put 

*307-011. Rat3 = RatuRat„ Df 

If njv <r pjcr, we have {(/tt/v) | Cnv} ( | Cnv' <,) {{pja) \ Cnv]. Hence 
we put 

*30702. <„=|Cnv;<, Df 

*307021. >„ = Cnv'<„ Df 

If X and Y are generalized ratios, we consider X less than Y if either 
X, Y are both positive and X <, 7, or X, Y are both negative and X >„ F, 
or X is negative and Y is positive or zero. Hence we put 

*307-03. <s = (>„)o«^)u(Rat„-t'0,)|Rat Df 

On the analogy of <„ and <.g, we put 

*30704. Hn = \Cnv)E Df 

*30705. Hg=:Hn^H' Df 

We prove in this number that if Z is a ratio, X | Cnv = Cnv | X, and 
Cnv'(Z I Cnv) = Z I Cnv (*307-21-22). We prove also 
*307'25. \-.G'EnC'H„ = A 

We prove that Og and oo, are their own negatives, but are not the nega- 
tives of anything else (*307-26-27-31). We prove Nr'Zr„ = Nr'fl' (*307-41) 
and Infin ax . D . ff^ei; (*307-46). None of the propositions of this number 
offer any difficulty. 



SECTION A] GENERALIZED RATIOS 297 

*30701. Rat„ = I Cnv"Eat Df 

*307011. Raty=RatuRat„* Df 

*30702. <„ = |Cnv;<, Df 

*307021. >„ = Cnv'<„ Df 

*30703. <j = (>„)c;«,)c;(Rat„-i'05)tRat Df 

*307031. >g = Cnv'<g Df 

*30704. Hn = \CiiwlH Df 

*30705. Hg = Hn^H' Df 

*3071. V:R{X\ Cnv) S . = . RXS [*7l-7] 

*307-ll. h : i? (I Cnv5Z) S. = . RXS [*307-l] 

*30712. h . X I Cnv I Cnv = X [*307-l] 

*30713. h : X I Cnv = F I Cnv . = . X = F [*307-12] 

*30714. h : F= X I Cnv . = . X = F I Cnv [*307-12] 

*30715. l-:a[!X^/«:. = .a!/c1(X|Cnv)P(Cnv"«;) [*307-l] 

*30716. !-:.«: = Cnv"«;.D:g[!Xt«-.s.a!(X|Cnv)tK [*307-15] 

*307-2. h . (fi/v) I Cnv = Cnv | (fijv) [*307-l . *30319] 

*307-21. f-:X6Ratui'oo5.D.X|Cnv = Cnv|X [*307-2 . *303-7-67] 

*307-22. l-:XeRatut'oOj.D.Cnv'(X|Cnv) = X|Cnv [*307-21] 

*307-23. \-.Gn\"0'Hn = G'H„ [*304-28 . *30313 . *307-22] 

*30724. h:fj,,v,p,a-eQ.'U.fi Prm v . p Prm o-.p^o-.<r=|=O.D. 

'3_l{p/a)-(fi/v)\Cnv 
Z>em. 

I- . *303-32 . h :. Hp . D : (gP, Q) . P, Q € Rel num . Ppo G Qpo . P (p/o-) Q : 

[*303-21] D : (gP, Q).P,Qe Rel num . Pp„ G Qp„ . g ! P' A Qp : 

[*300-3] D : (gP, Q).P,Qe Rel num . a ! P- n Qc . P- n Q'' = A : 

[*303-21] D : (gP, Q) . P (p/a) Q . ~ {P (/*/i') Q] :. D h . Prop 

*307 25. h . G'H n a'JS'n = A 

Dem. 
l-.*307-24.*303-13.D 

I- : /i, v, p, <7 eC['f/"./iPrm v . pFrma- . D . /^/v =|= (p/<r) | Cnv (1) 

b . *302-22 . *303-211 . *304-27-28 . D h : X, Fe G'H . D . 

(a/*' ^> p,o-)./i,v,p,(reQ^'U./i Prm v . p Prm o- . X = /i/v . Y=p/cr (2) 
h.(l).(2).Dh:X,Fe(7'fl".D.X+F|Cnv:Dh.Prop 



298 QUANTITY [part VI 

*307-26. h . O5 1 Cnv = Og = Cnv j Og 
Dem. 

l-.*307-2. Dh.Og|Ciiv = CnviOg (1) 

h . *303-6-15 . *.307-l . D 1- : ii (Oj I Cnv) /S . = . a ! ii n / f C'^ . 
[*33-22] = . a ! ii n / p C'>Sf . 

[*308-15] = . RQ^S (2) 

h . (1) . (2) . D I- . Prop 

*307-27. h . 00 g I Cnv = 00 , = Cnv I X , [*307-26 . *303-62] 

*307-3. l-:X6C"if.D.g[!(X|Cnv)tRelnum [*304-5 .*30716 .*300-4] 

*307-31. l-:X6Rat-i'03.D.X|Cnv=j=0,.X|Cnv=t=oOg 
[*307-3.*304-53.*303-62] 

*307-4. h : XHnY . = . (X | Cnv) H{Y\ Cnv) [*150-41 . (*307-04)] 

*307-41. h.Nr'ir„ = Nr'ir [*307-13 . (*307 04)] 

*30742. 1- : Infin ax . D . Nr'if„ = Hi-'En = v [*307-41 . *304-33] 

*307-43. h-.XeC'Hn.O.'iilXl'Relaum [*307-3] 

*307-44. h . O4, 00 5 ~ 6 C'H„ [*307-31] 

*307-45. h . 'Nr'Hg = Nr'^ + 1 + Nr'i^ [*307-25-41 . (*307-05)] 

*307-46. I- : Infin ax.D.Hgerj [*307-45 . *304-33] 

This proposition requires rj + i+rj =ri, which is easily proved. 



*308. ADDITION OF GENERALIZED RATIOS. 

Summary of *308. 

In this number we have to extend addition sp as to include negative 
ratios as addenda, and for this purpose we have to define subtraction of 
simple ratios. This is defined as follows: 

*308-01. X-,Y=RS {(a^) : Z, F, Z e Rat : ^ +8 F= X . RZ8 . v . 

Z+,X=Y.RZS] Df 
That is to say, if F <,. X, X — j F is the ratio which must be added to F to 
give X, while if X <,. Y,X-gY is the negative of the ratio which must be 
added to X to give F. Thus we have 

*30813. h:. F<rX.v.F6Rat.F=X:D.X-,F=(7^(Z+,F=X) 

*30814. l-:.X<^F.v.XeRat.F=X:D.X-,F={(jZ)(Z+,X=F)}|Cnv 

We have, of course, X-gO,= X (*308-22), Oj-,X = X|Cnv (*308-2:3), 
and X— gX=Og (*308-I2). Existence-theorems for X— jF are closely 
analogous to those -for X+^F and X x, F. Also we have 

*308-2. h : X, F 6 Rat . = . X -, F 6 Rat, 

We define the sum of two generalized ratios by means of the sums and 
differences of simple ratios, as follows : 

*30802. X +j F= (X +, F) w (X -, F I Cnv) c; 

(F-,X|Cnv)c;(X|Cnv+,F|Cnv)|Cnv Df 

Of the four relations which occur in the above definition, all but one 
must be null if neither X nor F is 0,. Thus if X and F are positive, 
X-jF|Cnv, F-sX|Cnv, and X|Cnv-f-gF|Cnv are null; if X is positive 
and F negative, X+«F, F-,X|Cnv, and X| Cnv +j F| Cnv are null; if X 
and Fare both negative, X-(-, F,X — j F| Cnv, and F— ,X | Cav are null. 

If X is Og and F is positive, 

X +. F= F-, X I Cnv . X -, F| Cnv = (X | Cnv +, Y\ Cnv) | Cnv ^ A. 
If both X and Fare Oj, all four relations are Og. 



300 QUANTITY [part VI 

Hence we find 
*308-32. h : X, Fe Rat . D . Z +ff F= Z+, F 
*308-321. h : Z 6 Rat . F6Rat„ , D . Z +3 F = X -, F| Cnv 
*308-322. h : F 6 Rat . Z 6 Rat„ . D . Z +3 F= F-. Z | Cnv 
*308-323. h : Z, Fe Rat„ . D . Z +^ F= (Z | Cnv +, F| Cnv) | Cnv 

The existence-theorems for X +gY are closely analogous to those for 
Z +8 F, and the formal laws oifer no difficulty. We have 

*308-52. hr.Z, FeRat^. D :Z+s F=Z+jZ. = . F=Z 

*308-54. 1- : Z, F 6 Rat^ . D . (gZ) . -^e Rat^ .X+gZ=Y 

*308-56. h :. Z <p F. = : Z 6 Ratj, : (g^) . Z e Rat - l% .X+gZ=Y 

*308-72. \-:{X+g Z) <g {X+gZ'). = .X6 Rat^, .Z<gZ' 



*308-01. Z -, F= RS {(gZ) : Z, F, ^e Rat : ^+8 F= Z . RZS . v . 

Z+sX = Y.RZS} Df 
*30802. Z+jF=(Z+,F)o(Z-8F|Cnv)va 

(F-,Z|Cnv)vy(Z|0nv+8F!Cnv)|Cnv Df 

*3081. h : F<^Z.D.Z-8F=ES{(a^).^6Rat.^+,F=Z.i?^S} 

Bern. 

h . *306-55 . D 1- : Hp . D . ~ (g^) . ^ +« Z = F (1) 

h . (1) . (*308-01) . D h . Prop 

*30811. h : Z <^ F. D . Z -, F=E§{(a2) . ZeRat . Z+sX= Y.RZS} 

Dem. 

h . *306-55 . D h : Hp . D . ~ (g^) .Z+,Y=X (1) 

h . (1) . (*308-01) . D h . Prop 
*30812. h : Z e Rat . Z = F . D . Z -8 F= 0, [*306-54-24] 

*30813. h:. F<,Z.v. FeRat. F=Z: D .Z-, F= (?^)(^+, F= Z) 
Bern. 

I- . *306-52-24 . D h : Hp . D . (g^) . Z+, F= Z . ZeRat (1) 

h.*306-54. Dh:Hp.Z-|-,F=Z.Z'+,F=Z.D.-Z' = -?' (2) 
t- . (1) . (2) . *308-112 . D h . Prop 
*30814. l-:.Z<,F.v.ZeRat.Z=F:D.Z-,F=f(?Z)(^+,Z=F)}|Cnv 

[Proof as in *308-13] 
*30815. I-:~(Z, FeRat).D.Z-, F=A [(*308-01)] 

*30816. I-:Z, FeRat. Y+,Z = X .D . X -,Y= Z 
Bern. 

l-.*306-55.*304-221.DI-:.Hp.D: Y<rX.v . FeRat. F = Z (1) 
|-.(l).*308-13.DI-.Prop 



SECTION a] addition OF GENEBALIZED RATIOS 301 

*30817. h:X,YeRa,t.X+sZ=Y.-^.X-sY=Z\Gtiv [*306-55.*308-14] 

*30818. I-:F<,Z.D.5'-, FeEat-t'O, 
Dem. 

h.*306-52.DH:Hp.D.(aZ).ZeRat-t'0,. Y+sZ = X (1) 
h . (1) . *308-13 . D h . Prop 

*30819. \-:X<rY.O.X-,Y€ Rat„ - t'O, 
Bern. 

h . *306-52 . D h : Hp . D . (g^) . Ze Eat - I'O^ .X+,Z=Y (1) 
h . (1) . *308-14 . D h . Prop 

*308-2. h : X, 7 6 Rat . = . Z -, Fe Rat^ [*308-12-18-19-15] 

*308-21. h : Z -, F= ( F-, Z) | Cnv = Cnv |(F-, Z) 

i)em. 
I- . *308-13-14 . D 

h:.Z<rF.v.ZeRat-i'0,.Z=F:D.Z-,F=(F-,Z)|Cnv (1) 

h . *30813-14 . *30712 . D 

h:. F<rZ.v.F6Rat-t'0j.F = Z:D.Z-, F=(F-,Z)|Cnv (2) 

h . (1) . (2) . *304-221 . D f- : Z, Fe Rat . D . Z -, Y=(Y-, X) \ Cnv (3) 

[*307-21.*308-21 =Cnv|(F-sZ) (4) 

h . (3) . (4) . *308-15 .31-. Prop 

*308-22. f-:Z6Rat.D.Z-,0j = Z [*306-24 . *308-13] 

*308-23. h:ZeRat.D.Oj-,Z = Z|Ciiv [*308-21-22] 

*308-24. h : (v/p) <r (X//i) . D . \//* -, v/p = {(K x^ p) -, (p. x„ v)}/(p. x„ p) 

Dem. 

H.*3041.DI-:Hp.D.\Xe/3>jtiXoy (1) 

h . *303-23 . *306-13 . (1) . 3 

h : Hp . 3 . {(\ Xe /)) - (ji X, v)]l(p, X, /)) +, v/p = 

[{(\ X^p)-(jM Xe I/)} +0 Ot x„ !/)]/(/* x„ p)- 

[*303-23.*119-34] = \//i (2) 

h . (1) . (2) . *30816 . D h . Prop 

*308-241. h : (X/p.) <r(v/p) . D . X/m -, v/p = [{(a* x^ v) -„ (\ x^ p)}/(/i x„p)]| Cnv 
[*308-24-21] 

*308-25. \-:\,p,v,pe'D'Una'U.vlp <r \//i . D . (V/* "s "/p) D *oo V « C''^' 
i)em. 
h . *305-24 . D 

h : Hp. D. f(\ x,p)-,(^iX,v)]nt'pL,(p. x,p)r^t'^leT>'U r^a'U (1) 
h . (1) . *308-24 . *304-28 . D h . Prop 



302 QUANTITY [part VI 

*308-251. \-:X,fi,v,pe'D'Un a'U . \/fi <r v/p.O. (X/fi -, v/p) I <»'/* e G'Hn 
[*305-24.*308-24.1] 

*308-252. [■:X,fi,v,p6J)'Una'U.D.(X/fi-, v/p) I t^'p. e G'Hg 
[*308 25-251-12] 

*308-26. h : Z, Fe Eat . Z t im'p., Y I t^.'/j. e G'H' . D . (Z -, F) t t^'p. e G'Hg 
[*308-252 . *304-28] 

*308-261. h : X, F 6 G'H' . D . (X -, F) ^ t^'G"C'X e G'Hg [*308-26] 

*308-3. l-:a!(X-,F|Cnv).D.X6Rat. FeRat^ 
[*30815.*30712] 

*308-301. h : a ! (X I Cnv +, F I Cnv) .O.X.Ye Eat„ [*30612 . *307-23-12] 

*308-31. h : a ! (X +^ F) . D . X, Fe Rat^ [*:306-12 . *308-3-301 . (*308-02)] 

*308-32. h:X, Fe Rat. D.X +3 F=X+sF 

Dem. 
h . *308-3-301 . *307-25 . (*308-02) . D 

l-:X, FeRat-i'Og.D.X+^F=X+,F (1) 

I- . *306-24 . *308-22-3301 . D 

l-:X6Rat-t'0g.F=0,.D.X+pF=X = X+,F (2) 

h . *306-24 . *308-3-301 . D H : X = Og . F= Oy . D . X +3 F= 0, = X +, F (3) 
K(2).(3).D 

hr.XeRat. Y=Oj.v. Ff Rat .X = 0,: D .X+j F=X +s F (4) 

h . (1) . (4) . D h . Prop 

*308-321. F : X e Rat . Fe Rat„ . D . X +^ F= X -, F| Cnv 
[*30612 . *308-3-301 . *307-25 . (*308-02)] 

*308-322. h : Fe Rat . X 6 Rat„ . D . X +5 F= F-^ X | Cnv 
[*306-12 . *308-3-301 . *307-25 . (*30802)] 

*308-323. h : X, Fe Rat„ .:>.X+gY= (X |.Cnv +, Y\ Cnv) i Cnv 
[*306-12 . *308-3-301 . *307-25 . (*308-02)] 

*308-33. h : X +^ F 6 Rat^ . = . X, F e Rat^ 
[*306-22 . *308-2-32-31] 

*308-4. h . X +3 F= Y+gX [*306-ll . (*308-02)] 

*308-41. I- . X +j F= (X I Cnv +g Y\ Cnv) | Cnv 

Dem. 
h . *30712 . *34-26 . (*308-02) . 3 
I- . (X i Cnv +^ F| Cnv) | Cnv = (X | Cnv +, Y\ Cnv) | Cnv u (X | Cnv -, F) | Cnv 

w (F| Cnv -,X) I Cnv o (X +, F) 
[*308-21J = (X I Cnv va Y\ Cnv) | Cnv a (F-,X | Cnv) 

c;(X-,F|Cnv)w(X+,F) 
[(*308-02)] = X +^ F . D h . Prop 



SECTION aJ addition OF GENERALIZED RATIOS 303 

*308-411. h.(X+gY)\Cn\ = X\ Cnv +j, 7 \ Onv [*308-41 . *30712] 

*308-412. h : X\Cay +gY\^nv = Z\Cn\ . = .X +gY=Z 
[*308-411 . *30713] 

*308-42. h : X, F e Rat . D . (X -, F) +j F= i: 

Dem. 
h . *30812-32 . *306-24 . D h : Hp . Z = F. D . (X -, F) 4-^ F= X (1) 

h . *308-18-32 . D 1- : Hp . F<rX . D . (X -, F) +^ F=.(X -. F) +, F 
[*308-13] =X (2) 

I- . *308-19-322 . D h : Hp . X <^ F. D . (X -, F) +s F= F-,(X -, F) | Cnv 
[*308-21] =F-,(F-,X) (3) 

h . *30813 . D F : Hp (3) . D . X +, ( F-, X) = F . 
[*308-16-18] D.X=F-,(F-,X) (4) 

l-.(3).(4). DI-:Hp.X<,F.D.(X-,F)+^F=X (5) 

I- . (1) . (2) . (5) . *304-221 . D h . Prop 

*308-43. I- : X, Fe Rat . D . (X +y F) -, F= X ' 
Dem. 

h . *308-32 . D h : Hp . D . X +y F= X +, F . 
[*308-16.*306-22] D . (X +^ F) -, F= X : D h . Prop 

*308-44. h :. X, ^eRat .D:X-sZ= Y-,Z. = . X= F 

Dem. 
l-.*308-13-1415.DI-:X=F.D.X-,Z=F-,-^ (1) 

h.*308-2. DI-:Hp.X-,Z=F-,Z.D. FeRat. 

[*308-42] D.(Y-,Z)+,Z=Y. 

[Hp] D.{X-,Z)+,Z=Y. 

[*308-42] D.X^Y (2) 

I- . (1) . (2) . D h . Prop 

*308-45. H :. X.^eRat . D : Z-,X = Z-, Y. = .X=Y 
[*308-44-21.*30ri3] 

*308-46. h : X, Fe Rat . F+ Oj . D . (X -, F) <g X 

Bern. 
h . *308-19 . D h : X <r F . D . (X -s F) e Eat„ - I'Oq . X e Rat . 
[(*307-03)] 3 . (X -, F) <g X (1) 

I- . *30812 . D h : Hp . X= F. D . X -s F= 0, . 

[*304-46.(*307-03)] 3 ■ (X -, F) <^ X (2) 

h . *30813-18 . D h : Hp . F<,X . D . (X-, F)+, F= X . X-,FeRat-i'0,. 
[*306-52] D.(X-sY)<rX. 

[(*307 -03)] 3 . (X -, F) <^ X (3) 

h . (1) . (2) . (3) . D h . Prop 



304 QUANTITY [part VI 

*308-47. hzXeB.at.ZZeRa.Xj-i'Og.li.X-.Y^X+.Z 
Bern. 

h . *306-52 . *308-46 . D h : Hp . D . (X - F)<, (Z +, Z) . 
[*304-201] D.Z-,F+Z+,^:DI-.Prop 

*308-51. h :. X e Rat^ .D:X +gY=X . = .¥=0, 

Dem. 
l-.*30833. Dh:.Hp.D:X+(,F=X.D.reRatj (1) 

l-.*308 32. Dh:XeRat.F=Oj.D.X+jF=X+»F 
[*306-24] = X (2) 

I- . *308-322 . D h : X 6 Rat„ . F= 0, . D . X 4-^ F= F-« X | Cnv 
[*308 23.*307-12] . =X (3) 

l-.(2).(3). DI-i.Hp.D:F=Os.D.X+^F=X (4) 

h . *308-32 . D h : X, Fe Rat . X +^ F= X . D . X +, F= X . 
[*306-24-54] 3 . F= 0, (5) 

l-.*308-321.DI-:X6Rat.F6Rat„.X+3F=X.D.X-,F|Cav = X. 
[*308-22-45] D . F| Cnv = O4 . 

[*307-2] D.F=0, (6) 

h . *308-322 . D 1- : X 6 Rat„ . Fe Rat . X +s F= X . D . F-, X I Cnv = X 
[*308-23.*307-12] = Og -, X | Cuv . 

[*308-44] 3.F=0j (7) 

h . *308323 . *307-14 . D 

l-:X, F6Rat„.X+^F=X.D.X|Cnv+,F|Cnv = X|Cnv. 
[(5).*307-26] D.F=03 (8) 

h . (1) . (5) . (6) . (7) . (8) . D h :. Hp . D : X +^ F= X . D . F= Oj (9) 

I- . (4) . (9) . D h . Prop 

*308-52. h :. X, FeRat^. D :X+g Y=X+gZ. = .Y=Z 

Bern. 
h . *308 321-47 . D h : X, F 6 Rat . F+ O5 . X +3 F= X +^ -^ . D . Z ~ e Rat„ (1) 
|-.*308-ol. D\-:XeUsitg.Y=Oj.X+gY=X+gZ.D.Z=Oj (2) 

t- . (1) . (2) . *308-33 . D h : X, F e Rat . X +^ F= X +^ Z . D . ^ e Rat (3) 

I- . (3) . *308-32 . DI-:X,F6Rat.X+jF=X+jZ.D.X+,F=X+,Z. 
[*306-54] :i.Y=Z (4) 

h . (4) . *308-323 . *307-13 . D h : X, Fe Rat„ .X +gY=X +gZ .0 .Y=Z (5) 
I- . *308-321-32-47 . D 
h:XeRat. FeRat„. X+y F=X+^^. D .^^eRat- I'O, (6) 

f-.(2)5-5.Transp.D 

hiXeRat. FeRat„-i'0g.X+jF=X+^^.D.Z=t=05 (7) 



SECTION a] addition OF GENERALIZED RATIOS 305 

h . (6) . (7) . *308-33 . D 

h : X eRat . YeUatn- I'Og.M +gY= X +gZ .'^ . Z eU&tn (8) 

I- . (8) . *308-321 . D h : Hp(8) . 3 . X-, F| Cnv = Z-,^| Cnv . 
[*308-45.*307-13] 3 . F= Z (9) 

h . (9) . *308'411 . *307-13 . 3 

h:X6Rat„.F6Rat.Z+j,F=Z+j^.D.F=Z (10) 

h . (4) . (5) . (9) . (10) . 3 h : Hp . Z +j F= Z +3 Z . D . F= ^ (11) 

h . (11) . (*30802) . D I- . Prop 

*308-53. F : Z, Fe Ratj . D . Z +3 (F+^ Z I Cnv) = F 

Bern. 
h . *308-321 . *307-12 . D h : Z, Fe Rat . D . X +g{Y+gX \Cny) =X+g(Y-,X) 
[*308-4-42] = F (1) 

h . *308-32 . D 

h :Z6Rat„. F6Rat.D.Z+^(F+j,Z|Cnv) = Z+j,(F+,Z|0nv) 
[*:308-4-321.*306-22] =(F+,Z| Gnv)-,Z|Cnv 

[*308-43-32] =F (2) 

h . *308-323 . *307-12 . D 
h : Z 6 Rat . Fe Rat„ . D . Z +g(Y+gX\ Cnv) = Z+.,(F| Cav +,Z) | Cnv 



[*308-321.*306-22] 


= Z-,(F|Cnv+,Z) 




[*308-17.*307-12] 


= F 


(3) 


h . ( 1) . D h : Z, Fe Rat„ . D . Z 1 Cnv +y (F 1 Cnv +j Z 1 Cnv 1 Cnv) = F| Cnv . 


[*308-411] 


D.ZiCnv+p(F+3Z|Cnv)|Cnv=F|Cnv. 




[*308-412] 


D.Z+^(F+yZ|Cnv)=F 


(4) 


K(l).(2).(3).(4). 


D I- . Prop 




*308-54. h : Z, Fe ] 


Ratg . 3 . (aZ) . ZeUatg .X+gZ=Y [*308-53-33] 


1 


*308-55. h :.Z, F.^eRat^. D :X+gZ= Y. = .X= F+^Z| Cnv 




Bern. 






1- . *30S-53-52-4 . D h 


:Hp.Z+3^=F.D.F+jZ|Cnv = Z 


(1) 


h . *308o3-4 . D h 


:Hp.F+^^|Cnv = Z.D.Z+j^=F 


(2) 



h . (1) . (2) . D h . Prop 

*308-56. h :. Z <^ F . = : Z e Rat^ : (g-?) . -^ e Rat - t'Oj . Z +j ^= F 

Bern. 
h . *306-52 . *308-32 . D 

h:.Z<^F. = :ZeRat:(a^).ZeRat-t'Og.Z+jZ=F: (1) 

[*306-52-25] D : Fe Rat : (-^Z) .ZeRat-i%.X +gZ=Y (2) 

H.(2)?lCn^4^.3 
A., Y 

h:.Z>„F.D:Z6Rat„:(aZ).ZeRat-l'0j.FlCnv+p-? = Z|Cnv: 

[*308-55-412] :i\X^ Rat„ : (g^ . Z e Rat - t'O, .X-VgZ^Y (3) 

K. & w. III. 20 



306 QUANTITY [part VI 

\- . *308-82-53 . *806-23 . D h : X e Rat„ . Fe Rat . D . 

Y+gX\Guv.e-Ra,t-i%.X+g(Y+gX\Gnv)=Y (4) 
1-.(1).(2).(3).(4).(*307-03).D 

\-:.X<gY.D:XeB.a.tg:('^Z).Z6Ra.t-i%.X+gZ=Y (5) 

I- . *35-103 . (*307-03) . D h : X e Eat„ - I'O, . Fe Eat . D . X <y F (6) 

h . *308-55-412 . D 

h:X,Y6'Ratn.Ze'B,a.t-i%.X+gZ=Y.:3.X\Cnv=Y\Gnv+sZ. 
[*30(J-52] D . X >„ F (7) 

h . (6) . (7) . D 1- :. XeRat„: (a-^) . ZeRat- I'Og .X+gZ= Y::^.X<gY {8) 
I- . (1) . (8) . D h :.XeRaty : (a-?) . ^eRat - i%.X+gZ= F: D . X <3 F (9) 
h . (5) . (9) . 3 h . Prop 

*308-561. I- :. X <„ F. = : Fe Rat^ : (gZ) . ^e Rat - I'Og .X+gZ=Y 
[*308-56-33] 

*308-57. h : X <y F. = . X e Rat^ . F+^^X j Cnv e Rat - I'O, . 

= .Ye Redg . F +^ X | Cnv e Rat - I'O, 
Bern. 

V . *308-55-56-4. . D 

1- :. X <^ F. = : XeRat^ : (a^) . ZeB,at-i% . Z= F+jX| Cnv (1) 

l-.*308-55-561-4.D 

\-:.X<gY.= :Y6 Rat^ : (g^ . ^ e Rat - t'O, . -? = F +j X | Cnv (2) 

h . (1) . (2) . D h . Prop 

*308-6. h : X, F^eRat . D . (X +gY)+gZ= X +giY+gZ) 
[*308-32 . *306-22-31] 

*308-601. \-:X,Y,Ze'Ra.tn.O.(X+gY)+gZ = X+g{Y+gZ) 

Dem. 
h.*308-323.*307-12 

h : Hp . D . (X +j F) +3 Z= (X 1 Cnv +, F| Cnv) I Cnv +^ (Z I Cnv) I Cnv 
[*308-411] = {(X I Cnv +, Y\ Cnv) +g Z \ Cnv} | Cnv 

[*308-6.*306-22] = {X | Cnv +^ ( F | Cnv +g Z \ Cnv)} | Cnv 

[*308-411] =X+^(F|Cnv+j^|Cnv)|Cnv 

[*308-323] = X +^ ( F +3 Z) : D I- . Prop 

*308-602. \-:\fi, v, p,a,re NC ind . /it, p, t ~ e I'O . D . 

(\//t +, vjp) -, aJT = (\//A -g o-/t) +g vjp 
Bern. 

h . *308-24 . D h : Hp . o-/t <rX//i . D . 

(X//t +g I/V/j) -, 0-/t = {(\ X„ p X„ t) 4e (,a X„ 1/ Xo t) -0 (/i Xe p X^ 0-)}/(/i X„ /J X^ t) . 
(\//t -« 0-/t) +g l//p = {(\ Xe jO X„t) -„ (/i Xe p Xe a) +„ (m Xel/ X„t)}/(ji, X^p XJt) (1) 



SECTION a] addition OF GENERALIZED RATIOS 307 

h . *308-241 . D h : Hp . X/fi +, v/p <r a/r.D. (X/fi +, v/p) -, a/r 

= [{(/* Xo P Xc 0-) - C^ Xo P Xc t) - 0^ Xo " Xc t)}/(m Xo P Xo t)] I Cnv . 

(V/l, -, 0-/t) +3 v/p = [{(jJL X„ t) -, (\ Xe 0-)}/(;lt Xe t)] | Cnv +j v//) 

[*308-322'21] 

= [{(/i Xe p Xo 0-) - (X X„ /> X„ t) - (m X„ V X„ T)j/(/t Xe /O Xe t)] | CdV (2) 

h . *308-24.-24.1 . D h : Hp . \//i <r (t/t . a/r <, \//i +, i//p . D . 

( V/* +« "/P) -s 0-/t = {(>■ Xc p Xe t) +e (/* Xe V X^ t) -e (/i Xe p X^ (t)}/(jj, X^ p X^ t) . 

(X/fi -, a/r) -{-gv/p = [{{p. Xe ff) -e (\ Xe T)}/{p. x^ t)] | Cdv +g v/p 

[*308-322-21] = {(X x„ p x„ t) +« (p- x, v x^ t) -„ (/i Xe p Xe a)}/(p, x„ p x^ r) (3) 

l-.*308-16-12.D 

I- : Hp . X/yii ^a/r ."^ . {X/p, +« v/p) — s a/r = v/p = (X.//t — g o-/t) +j v/p (4) 

l-.*308-12-53-l7.D 

h : Hp . X/p. +e v/p = a/T .0. (X/p, +, v/p) -« ff/r = Og = (.\//i -, ct/t) +g v/p (5) 

h . (1) . (2) . (3) . (4) . (5) . D h . Prop 

*308-61. I- : Z, F, Ze Rat . D . (Z +y F) -, Z= (Z -, -^) +j F 
[*308-602-32] 

*308-62. h : Z, FeRat . ZeRatn . D . (Z+j F) +jZ=Z+j(F+j^ 

Z)em. 
h . *308-33-321 . D h : Hp . D . (Z +^ F) +g Z= (Z +J, F) -, ^ I Cnv 
[*308-4] = {Y+g X) -,Z\ Cnv 

[*308-61] =(Y-,Z\Onv)+gX 

[*308-4] = Z +/F-, Z I Cnv) 

[*308-321] = Z +s (F+ff ^) : D I- . Prop 

*308-621. h:Z, FeRat™. -?eRat.D.(Z+jF)+^-^=Z+3(F+^Z) 

Dem. 
h.*308-62.D 

I- : Hp . 3 . (Z| Cnv+^ Y\ Cnv)4-j-^|Cnv = Z| Cnv+^(Fi Cnv+^^l Cnv) . 
[*308-411] D . (Z 4-j F) I Cnv +5 -^ i Cnv = Z | Cnv +g(Y+g Z) | Cnv 
[*308-411] ={X+g(Y+gZ)}\ Cnv . 

[*308-412] D . (X+gY)+gZ=X+g(Y+gZ): D h . Prop 

*308-63. \-.(X+gY)+gZ=X+g{Y+gZ) 

Bern. 
h . *308-6-601-62-621 . D 

h : Z, F ZeRat^. D .(Z+(, Y)+gZ=X +giY+gZ) (1) 

h . *308-31-33 . D 

I- : ~(Z, F^eRatj) . D . (Z 4-^ Y)+gZ=k.X +g(Y+gZ) = A (2) 

I- . (1) . (2) . D h . Prop 

20--2 



308 QUANTITY [part VI 

*308-71. h : Z e Rat^ .Z <gZ' .:> .{X +gZ) <g {X +g Z') 
Bern. 
Y . *308-57 . D h : Hp . D . Z' +y ^1 Cnv e Rat - 1'% . 
[*308-56] D.(X+gZ)<g {(X +^ Z) +^ (Z' +g Z \ Cnv)} . 

[*308-63-53] -^.(X-^-gZXg (X +gZ'):^\-. Prop 

*30872. \-:(X+g Z) <g {X+gZ'). = .Xe Rat^ .Z<gZ' 

Bern. 
h . *308-33 .:)[-:(^X+gZ)<g (X +, Z').D. X, Z, Z' e Rat^ (1) 

h.*308-57.D 

1- : (X+,^)<,(X +,-?')■ 3 ■ {(^+.-^')+.(^ + -^)iCnv} eRat-t'Og. 
[*308-411-63-53] D . (^' +p 2' | Cnv) e Rat - i'% (2) 

h . (1) . (2) .*308-57 . D h : (Z +, Z)<, (X +, ^') . D . ^<,Z' (3) 

f- . (1) . (3) . *308-7l . D I- . Prop 

*308-8. h : Z, F 6 Rat, . X I U.'ix, Y i «„ V e G'H„ . D . (Z +, F) ^ «ooV e C'i?, 
[*308-32-321 •322-323 . *306-()4 . *308-26] 

*308-81. h : Z, F e C'H^ . D . (Z +, F) ^ WC'G'X e G'H„ [*308-8] 



*309. MULTIPLICATION OF GENERALIZED RATIOS. 

Summary of *309. 

The subject of this number is simpler than that of *308, because it 
requires nothing analogous to the consideration of subtraction. The product 
of two generalized ratios is defined as follows : 

*30901. X x„ Y= (Z X, 7) w (Z I Cnv x^ Y\ Cnv) 

^J{Xx,Y\ Cnv) I Cnv w (X \ Cnv x, F) | Cnv Df 

As in *308, three of the four products concerned in this definition will 
be null in any given case (unless X = Oq or Y=Og). Hence 

*30914. h:X,F6Rat.D.Xx,F=XxsF 

*309141. 1- : Z e Rat . Fe Eat„ . D . Z x^ F= (Z x, F| Cnv) | Cnv 

*309 142. h : Fe Rat . Z 6 Rat„ . 3 . Z x, F= (Z j Cnv x, F) | Cnv 

*309143. h : X, Fe Rat„ . D . Z x^ F= Z | Cnv x^ F | Cnv 

The propositions of this number are merely generalizations of those of 
*305. The proofs of the formal laws are straightforward, but the pi:oof of the 
distributive law (*309'37) is long, because of ^ihe multiplicity of different 
cases. 



*309-01. Z X, F= (Z Xs F) c; (Z I Cnv x, Y\ Cnv) 

w (Z X, F| Cnv) I Cnv c; (Z | Cnv x, F) ] Cnv Df 

*3091. h . Z X, F= (Z X, F) w (Z I Cn V X, F I Cnv) 

va (Z Xg F I Cnv) | Cnv o (Z | Cnv x , F) | Cn v [(*309-01)] 

*309-101. I- : Z e Rat - 1% . D . Z | Cnv x, F= A [*305-2 . *307-25] 

*309102. 1- : Z e Rat„ - l% . D . Z x, F= A [*305-2 . *307-25] 

*30911. t- : a ! Z Xj F . D . Z, Fe Rat, [*S05-2 . *3()91] 

*309-12. h.XXgY=:YXgX [*305-ll . *3091] 

*309121. h . Z X, F= Z I Cnv x^ Y\ Cnv 

= (Xx„Y\ Cnv) I Cnv = (Z | Cnv x^ F) | Cnv [*3091 . *307-12] 



310 QUANTITY [part VI 

*309-122. 1- . X X, Fj Cnv = Z I Cnv x„ F= (Z X, F) I Cnv 
[*309-121 , *307-12] 

*30913. l-:X,FeB,at-i'0,.D.XXi,F=Xx,F [*309-l •101-12] 

*309131. \- :. X =0g. Y eUsit- I'Og .w . Y=Og . X eB-At - L%:D . 

Xx,F=Zx,F=Oj 
Dem. 

h.*309101.D 

I- :X = 0, . FeRat- I'Oj .D.Xx„Y=(Xx, F) t;(Z| Cnv x, F)| Cnv . 

[*307-26.*305-22] D . X x, F= Z x, F= 0, (1) 

h . (1) . *309-12 . D h : F= 0, . X 6 Rat - 1% . D . Z x, F= Z x^ F= 0, (2) 

I- . (1) . (2) . D f- . Prop 

*309133. h : Z = Og . F= 0, . D . Z X, F=Z X, F=05 
[*309a . *307-26 . *305-22] 

*30914. l-:Z,F6Rat.D.Zx,F=ZxgF [*30913131133] 

*309141. h : Z 6 Rat . Fe Rat„ . D . Z x, F = (Z x, F| Cnv) | Cnv 
[*309-121-14] 

*309142. h : Fe Rat . Z 6 Rat„ . D . Z x^ F= (Z | Cnv x, F) | Cnv 
[*309-141'12] 

*309143. h : Z, Fe Rat„ . D . Z x, F= Z | Cnv x, F| Cnv [*309-14-121] 

*30915. h : X, Fe Raty . = .XXgYe Rat, 
Bern. 
h . *305-3 . *309-14143 . 3 

f-:.Z, FeRat.v.Z, FeRat^O.Zx^FeRat ' (1) 

h . *305-3 . *309-141142 . D 

l-:.ZeRat.FeRat„.v.ZeRat„. FeRatO.ZXgFeRat™ ' (2) 
h.(l).(2). DI-:Z,F6Rat„.D.ZXyFeRat„ (3) 

h . *303-72 . (*307-01011) .■^h:Xx,Ye Rat^ .D.'^lXx.Y (4) 

h . (4) . *30911 . DI-:Zx^FeRat^.D.Z,F6Rat, (5) 

1- . (3) . (5) . D f- . Prop 

*30916. l-.(Zx^F)x,^ = Zx,(Fx,Z) [*305-41 . *309-l] 

*30917. h : Z, F ~ 6 t'O, w t'c» , . D . Z x^ F= Cnv'(Z x, F) 

Dem. 
h . *3091 . 3 h . Z X, F= (Z X, F) w (Z I Cnv X, F I Cnv) 

va (Z X, F| Cnv) | Cnv w (Z | Cnv x, F).| Cnv (1) 



SECTION a] multiplication OF GENERALIZED RATIOS 311 

H.*305-12.Dh:Hp.D.Xx,F=Cnv'(Zx,F) (2) 

h . *307-22 . D I- : X 6 Rat . i:^ Z I Onv = Cnv'(X | Cnv) (3) 

l-.(3). Dh:Z6Eat.X = Z|Cnv.D.i"|Cnv = (i|Cnv)|Cnv 

[*307-12] =Z 

[*307-14] =Cnv'(Z|Cnv) (4) 

h . (3) . (4) . D I- : Xe Rat^ . 3 . Z | Cnv = Cnv'(Z ] Cnv) (5) 

l-.(2).(5).DI-:Hp.Z,F6Rat,.D. 

Z|Cnvx,F|Cnv=Cnv'(Z|Cnvx,r|Cnv). 

Z X, F| Cnv = Cnv'(Z x, Y\ Cnv), . 

Z|CnvXsF=Cnv'(Z|Cavx,F) (6) 

h . (1) . (2) . (6) . *309-l . D h : Hp . Z, Fe Rat, .li.Xx^Y^' Cnv'(Z x, F) (7) 

l-.*303-13-7. DI-:Z, FeRatj-i'Og.H.Z, FeRatj-t'Oj (8) 

h . (8) . *309-ll . D 

h : ~(Z, FeRat, u t'oo ,) . 3 .Z x, F= A. Cnv'(Z x„ F) = A (9) 

h . (7) . (9) . D I- . Prop 

*309-21. h :. Z, FeRatg :Z = 03 . v . F=Og : = .Z x, F=05 

Bern. 
h . *309-14-141 . *305-22 . *307-26 . D h : Z e Rat, . F= 0, . D . Z x, F= Oj (1) 
h.*309-15. Dh:ZXjF=Og.D.Z, FeRaty (2) 

I- . (2) . *30914141-142143 . *307-26 . 3 
l-:.Zx, F=Og.D:Zx,F=0,.V.Z|Cnvx, F|Cnv = Oy. 

V . Z X, F| Cnv = Og . V . Z I Cnv x, F= 0, : 
[*305-22.*307-26] D : Z = 0, . v . F= O5 (3) 

l-.(l).(2).(3).DI-.Prop 

*309-22. I-:Z, FeRat^-t'Oj.s.ZXjFeRatj-i'Og [*309-21 . Transp] 

*309-23. h : Z 6 Rat, - I'O, . D . Z x, Z = 1/1 
Bern. 

I-.*30913. Dh:Z6Rat-i'05.D.Zx,Z = Zx,Z 

[*305-52] = 1/1 (1) 

h . *309121 . *307-22 . 3 h : Fe Rat - 1% . Z = F| Cnv . D . Z x, Z = F x, F 

[(1)] =1/1 (2) 

h . (1) . (2) . D h . Prop 

*309-24. h :ZeRat,. D.Z X, 1/1 =Z 
Bern. 
l-.*309-14. DF:ZeRat.D.Zx,l/l = Zxsl/l 

[*305-51] ■■■"■^ =Z (1) 

h . (1) . *309-142 . D f- : Z e Rat„ . D . Z x, 1/1 = (Z | Cnv) [Cnv ' 
[*307-12] =Z ' (2) 

I- . (1) . (2) . D h . Prop 



312 QUANTITY [part VI 

*309-25. \-:.X,AeB,atg.Aj=Og.:i:XXgA = A'. = .X = A'XgA 
Dem. 

h . *30&-23-24-16 . 3 h :Hp. D.Z = Z x^il x,I (1) 

l-.(l). Dh:'S.^.XXgA = A'.D.X = A'XgA (2) 

h.(l)'^.*30915.DI-:Hp.D.^' = ^'x,^ x.A (3) 

h.(3). D\-:Rp.X^A'XgA.D.Xx,A = A' (4) 

f- . (2) . (4) . 3 I- . Prop 

*309-251. \-:.X,A'€'Ra.t„.A^Og.D:Xx,A = A'. = .X = A'XgA 
[*309-25-15] 

*309-26. I- : X, FeKat^, .X^Og.O. (g^ . ZeRat^ .Xx„Z=Y 
Bern. 

l-.*309-25.Dt-:Hp.^=Fx,Z.D.^x,Z=F (1) 

l-.(l).*309-l 512. Dh. Prop 

*309-31. I- : Z, FeEat . ^eRat, . D . (Z+, F) x«-? = (Z x„^+,(Fx^Z) 

Dem. 
h.*308-32.*30914.D 
h:Hp.^6Rat.D.(Z+,F)x,Z=(Z+sF)x,Z. 

XXgZ=XxsZ.YXgZ=Yx,Z. 
[*306-41] D.(X+,Y)x,Z=(Xx„Z)+,(7x„Z) (1) 

h . *309122 . D 

h : Hp . TT 6 Rat . ^ = F I Cnv . D . (Z +„ F) X, ^= {(Z +, F) Xg Tf } I Cav 
[(1)] ={(Zx,F)+,(Fx,F)}|Ciiv 

[*308-411.*309-122] ={X XgZ)+g(YXgZ) (2) 

I- . (1) . (2) . D h . Prop 

*309-311. h : Z, Fe Rat„ . ^e Rat, .0 .(X +gY)XgZ=(X x„Z) +» (F x^ Z) 
Dem. 
I- . *308-41 . *309-122 . D 

h : Hp . 3 . (Z+„ F) x,Z= {(X \ Cnv+, Y\ Cnv) x,^} | Cnv 
[*309-31] = {(Z| Cnv x,Z)+,(F| Cnv x,Z)} \ Cnv 

[*309-122.*308-41] ={X x,Z)+g(Yx,Z):D\- . Prop 

*309-32. h : (v/p) <r ( X//*) . ff/r e Rat . D . 

(\//t -, I^/p) X„ 0-/t = {((\ Xo p) -e (/i X„ k)) Xe 0-}/(^ X„ p X^ t) 

Dem. 

V . *308-24 . D h : Hp . D . V/* -, j//(0 = ((\ Xe p) -„ (fi x, v))/fi x,p (1) 
h . (1) . *30914 . *305-142 . D h . Prop 



SECTION a] multiplication OF GENERALIZED RATIOS 313 

*309-33. h : X//t, vjp, aJT e Rat . D . 

^\//i -, vjp) Xj (ff/r) = (\/^ X, ff/r) -g (i^/p X, ff/r) 
Dem. 

h . *309-14 . D h : Hp . D . Xjp. Xg a/r = X//a x« o-/t . v/p x^ c/t = v/p Xj (t/t . 
[*305-142] D . V^ X, a/T = (\ x„ <7)/(/. Xe t) . v/p x„ cr/r = (v x, a)/(p x, t) (1) 
h . (1) . *308-24 . D 
h : Hp . (v/p) <^ (\//t) . D . (X,//i X, «7/t) -, {v/p X, a/r) = 

{(\ Xe 0-) X„ (p Xe t) -c (/* Xe t) X^ (v X„ <T)]/{p. X„ p X^ T=) 

[*303-38] = {(\ Xc a-x^p)- (ji x^ v x^ a)\/{ii x^ p x,, t) 

[*309-32] ={X/p.-,v/p)x,a/T (2) 

h . (2) . D F : Hp . (X//i) <^(/;/p) . D . 

(i//p X(, <7/t) -s {X/fi x„ o-/t) = (v/p -, \//i) X, (t/t . 
[*308-21.*309-122] D .(\/^ x,a/r)-,iv/p x,a/r) = {X/p.-,v/p) x^a/r (3) 
H . *30812 . *309-21 . D 
I- : Hp . \//i = i//p . D . (X//it -, i//p) Xj o-/t = Og . 

{X/p. X J a/r) -,, (i//p Xj <7/t) = 0, (4) 

f- . (2) . (3) . (4) . D I- . Prop 

*309-34. V : X, F.ZeRat . D . (Z-, F) x^Z=(Z x„^)-,(Fx^^) 
[*309-33] 

*309-35. h : X.^eRat . FeRatn. D . (Z+^ F) x„Z={X XgZ)+„(YXgZ) 

Dem. 
h .*308-321 . D h : Hp . D . Z+„ F=X-, F| Cnv . 

{Xx„Z)+„{Yx„Z) = {Xx,Z)-,{Y\Cx,yx,Z) (1) 
h . (1) . *309-34 . D h . Prop 

*309-36. h:Z,ZeRat„.F6Rat.D.(Z+jF)x,Z=(Zxj^+^(Fx,Z) 

Dem. 
h.*308-41.*309-121.D 

I- : Hp . 3 . Z +, F= (Z I Cnv +, Y\ Cnv) | Cnv . Z Xj^=Z |Cnv x,Z| Cnv . 

Fx,Z=F|Cnvx,Z|Ciiv. 
[*309-122] D . (Z +, F) X, ^ = (Z I Cnv +, 7 \ Cnv) x, Z | Cn v . 

(Zx,Z)+,(Fx^Z) = (Z|Cnvx,^|Cnv)+,(F|Cnvx,Z|Cnv) (1) 
I- . (1) . *309-35 . D h . Prop 

*309-361. h : Z e Rat^ . Ye Rat„ . ^ e Rat . D . 

(Z+, F) x,-?=(Z x,^)+„(Fx,Z) [*309-311-36] 



314 QUANTITY [part VI 

*309-362. [■:X,Z6Ra.t,.YeRatn.:^.{X+,Y)x,Z = {Xx„Z)+^(VXgZ) 

Dem. 
h . *309122 . *308-41 . D 
h . (Z +, 7) X, ^ = {(X +, 7) X, ^ I Cnv} I Cnv . 

(Zx,Z)+,(7x,^) = {(Zx,^|Cnv)+,(7x,^|Cnv)}|Cnv (1) 

h.*309-361.D 
h : Hp . ^6Eat„ . D . (Z+j 7) x^-^j Cnv 

= (Zx,^jCnv)+,(7x„Z|Cnv) (2) 

h.(l).(2).DI-:Hp.^eRat„.D.(Z+,7)x,^ = (Zx,^)+,(7x,2) (3) 
h . (3) . *309-361 . D f- . Prop 

*309-363. h : Z, Y.ZeB.aig.'D .(X+g 7) XgZ= (Z XgZ)+g(YxgZ) 

Dem. 
h.*309-35-12.*308-4.D 

\-:Y,Z6'Rat.Xe'R&tr,.D.(X+gY)XgZ = {XXgZ)+g(YxgZ) (1) 

h . *309-36 . D 

\-:YeB&t.X,ZeUa.t„.-^.(X+gY)XgZ = {XxgZ)+g(YxgZ) (2) 

l-.(l).(2).D 

h:Z6Rat„. Yenab-ZeBatg.-^ .{X+gY)XgZ={X XgZ)+g(YxgZ) (3) 
h . (3) . *309-31 . D 

h : Ze Rat(, . 76Rat . -^eRat^. D . (X +g 7) XgZ = {X XgZ) +g(YxgZ) (4) 
h . (4) . *309-362 . D f- . Prop 

*309-37. V.{X+gY)XgZ = (Z x^ ^) +s ( 7 Xg Z) 
[*309-363-ll-15 .*308-31-33] 

*309-41. V\.AeRaX-i%.':>:{A XgX)<gY. = .X <g(YxgA) 

Dem. 
I- . *308-56 .OV:.(AxgX)<gY. = : 

AxgXeEaitg:{:^Z).Ze'Ra,t-i'Og.(AXgX)+gZ=Y (1) 
I- . (1) . *309-15 . D h :: Hp . D :. (4 XgX) <gY. = : 

Xe'Ra,tg:('^Z).ZeB,a,t-i'Og.(AxgX)+gZ=Y: 
[*309-25-37-23-24] D : Z eEatj : (gZ) . ^eRat- t'O, . Z +j(^Xg^)= YxgA : 
[*305-31.*30913] D:Xe Rat^ : (gZ') . Z' e Rat - (,% . X +gZ'=YxgA: 
[*308-56] D:X<g(YXgA) (2) 

Similarly H :. Hp . D : Z <g{YxgA) .D.{A XgX) <g 7 (3) 

1- . (2) . (3) . D I- . Prop 



SECTION A] MULTIPLICATION OF GENERALIZED RATIOS 315 

*309-42. [--..A 6Rat„ - I'O^. 0:{A XgX)<gY . = . (TxgA) <gX 
Dem. 
h . *307-4 . *309-122 . D 

h :. Hp . D : (^ XgX) <gY. = .{Y\ Cnv) <g(A | Cnv XgX) . 
[*309-41.*307-22] = . (7 1 Cnv x^ ^ ] Cnv) <g X . 

[*309-121] =.{YxgA)<gX:.0\-.Frop 

*309-5. h : Z, Fe Ratj . X t <n V. Y I t^^'t^ e C'Hg .D.(XxgY)t t^'f^ e C'Hg 
[*309-14-141-142-143 . *305-26] 

*309-51. h : Z, F 6 C'Hg . 3 . (Z x ^ F) p t^'CC'X e C'Hg [*309-5] 



*310. THE SERIES OF REAL NUMBERS. 

Summary of *310. 

Real numbers, as opposed to ratios, are required primarily in order to 
obtain a Dedekindian series, so as to secure limits to sets of rationals having 
no rational limit. If rationals and irrationals are to form one series, it is 
necessary to give some definition of " rationals " other than " ratios," since 
the series of ratios (assuming the axiom of infinity) is not Dedekindian, and 
is not part of any arithmetically definable Dedekindian series. But in virtue 
of the propositions of jj(212, the series of segments of the series of ratios, 
i.e. the series s'-ff, is Dedekindian, and this series contains a series, namely 

H'^H, which is ordinally similar to H. Thus the properties which we desire 

real numbers to have will result if we identify them* with segments of H, 

-* 
and give the name " rational real numbers " to segments of the form H'X, 

i.e. to segments which have ratios as limits. Thus H'X is the rational real 
number corresponding to the ratio X, and a real number in general is of the 
form H"X, where \ is a class of ratios. H"X will be irrational when X, has 
no limit in H. 

Since real numbers involve classes of ratios, the ratios concerned must be 
of some one type, and cannot be typically indefinite. Thus, as might be 
expected, hardly any of the properties of real numbers can be proved without 
assuming the axiom of infinity. In the present number, however, we shall 
be mainly concerned with just those few simple properties which are inde- 
pendent of the axiom of infinity. 

The series ^'H, by which real numbers are to be defined, has both a 
beginning and an end, namely A and D'H (which = G'H if the axiom of 
infinity holds). D'H will be infinity among real numbers. It is not con- 
venient to include it in the series of real numbers as defined, just as it was 
not convenient to include oo 5 in the series H or H'. Again A is not 
naturally to be taken as the zero of real numbers, which should rather be 
taken as being I'Oq. Thus we are led to the two following definitions, in 
which is the series of positive real numbers other than zero and infinity, 

* On this definition of real numbers, cf. Principles of Mathematics, Chap, xxxiii. 



SECTION A] THE SERIES OF REAL NUMBERS 317 

while @' is the series of zero and the positive real numbers other than 
infinity : • 

*31001. e = (s'ff) D (- i'A - t'D'S") Df 
*310011. @' = t'Oy*f @ Df 

These notations are framed on the analogy of H and H', the letter @ 
being chosen to suggest 6, the relation-number of the continuum. Although ^ 
we do not have Nr'@=^, we have Nr's'jH'=^, and therefore (*310-15) 
1 + Nr'@ + i = ^, and Nr'©' + 1 = ^ (assuming the axiom of infinity). Thus 
the relation-number of @ is simply that of a with the ends cut off. 

We put further, on the analogy of ff„, Hg, 
*31002. @„ = (s'ir„)D(-t'A-t'D'£r„) Df 
*310021. ©'„ = t'Oj«f ®„ Df 

*31003. @g=®„4:@' Df 

Thus @„ is the series of negative real numbers, @'„ the series of zero and 
the negative real numbers, @j the series of negative and positive real numbers 
(infinity always excluded). The class of positive real numbers is C®, 
of negative real numbers G'%n, of all real numbers (excluding infinity) 
C"@ u t't'Og.u 0'%n- If v is a positive, real number, | Cnv"i/ is the corre- 
sponding negative real number (*310"16). The properties of 0, @„, @j in 
respect of limits, continuity, etc., result from the properties of 6 as proved in 
*275, and from the properties of series of segments as proved in *212. 

Instead of taking the series of segments as constituting the real numbers, 
it is possible to take the series of their relational sums, i.e. s>@. This 
depends on the fact that s'@smor@ (*310"33). The chief advantage of 
s5@ is that it is of the same type as the series of ratios. We shall show in 
*314 how to construct the arithmetic of real numbers defined as the relational 
sums of segments ; until then, we shall regard real numbers as segments of 
the series of ratios. 



*31001. @ = (s'fl)D(-i'A-t'D'ir) Df 

*310011. ©' = t'O4«f0 Df 

*31002. @„=(s'^„)C(-i'A-i'D'ir„) Df 

*310021. ®'„ = t'05«f @„ Df 

*31003. @p=@„4^©' Df 

*3101. F . ©, ©', ®n, ®'n, ®g 6 Sor [*304-23 . *307-41-2o . *204-5 . *212-31] 

*31011. V : fj,®v . = .fj,,ve B'He - I'A - I'D'H .^Cv./jl^v. 
s . /i, i; e D'iTe . a I/n-a! D'H - v .'g^l v — fi . 
= . /i, 1/ 6 D's'if n Q's'-ff . M C 1/ . /i + I' 
[*212-23-132 . *211-61 . (*310-01)] 



318 QUANTITY [part VI 

*310111. h : fi@nv . = .fi,ve D'(ir„)e - I'A - I'H'Hn .fiCv.fi^v. 
= .li,ve D'(Hn)e . a ! /i . a ! D'^„ - 1/ . a ! i» - /x . 
= .fjL,ve J)'<s'H„ n a'^'Hr, .(iCv-n^v [(*310-02)] 

*310112. h :. II® gv . = : /it©„i/ . v . /i@i/ . v . 

/i e (?'@„ . i; e I'l'O, w 0'@ . v . /x = I'Oj . k e C"@ [(*31003)] 

*310-113. h :. /t®'!/ . = ifi=i% .v€C'@.v. ijBv [(*310-011)] 

*310114. 1- :. fiWnV . = :iM= 1% . v e (7'®„ . v . /t@„z/ [(*310-021)] 

*31012. h - 0'@ = D's'5" n a's'H = D'^e - t'A - I'B'H . 

G'@n= Ji'^'Hn n a'^'Hn = D'(ir„)e - t'A - t'D'if^ [*212-132] 

*310121. F . C"@ C CI ex'D'ff . (7'0„ C 01 ex'D'^„ [*310-12] 

*310 122. t-:a!3. = .a!@- = -!i[!@'- = -a!@»- = -a!0'»- = -a!®ff 
[*212-14 . *16M3 . *304-27.] 

*310123. h : a ! 3 . D . G'& = l'i% u G'@ . C"©'„ = i'i% u G'e„ . 

C'®g = 0'@„ w I't'O, w C"@ [*310-122 . *161-14] 

*310 13. \-.G'@n G'@n = A . s'G'@ a «'(?'©„ = A 

Bern. 
h . *31011-111 . D h : /i 6 C"@ . 1/ 6 C"@„ . D . /* C D'if . i/ C D'ir„ . a V ■ H ! " • 
[*307-25] D./i+i'./t<M' = A:Dh.Prop 

*310131. l-.t'0y^eO'@ua'@„ [*304-282] 

*31014. h . @n smor © [*212-72 . *307-41] 

*310-15. h : Infin ax . D . ©' -f> O'^', ©'„ -t> C'fi^„, G'H^ *f @g -f* O'S'e 61 
[*304-33 . *310-14 . *275-21] 

*310151. h : Infin ax . D . ©', ©'„ e Ser n comp n semi Ded 
[*310-15 . *275-l . *27l-18 . *214-74] 

*31016. l-:z/6a'©. = .|Cnv"i;eC"©„ [*310-12 . (*30704)] 

*31017. I- . I Cnv"| Cnv"!/ = v [*307-12] 

*31018. f- : /i = I Cn v"v . = .v=\ Cnv' > [*31017] 

*31019. \-:/j. = v. = .\Gnv"/jL = \Cuv"v [*310-17] 

*310-31. \-:/jLeC'®yj G'®„ . D . a ! (s'ti) t I^el num [*304-5 . *310121] 



SECTION a] the series OF REAL NUMBERS 319 

*310-32. h :. ij,,veC'@g.D : s'fi = s'v . = . fi = v 

Bern. • 

f-.*310-31.*303-62.D 
I- : /A 6 G'© u C"®„ . V = t'Og . D . a ! (s'/ii) D Rel num . ~ g ! (s'l;) ^ Rel num . 

D.s'/t + s'i/ (1) 

f- . *31012-31 . *307-25 . D I- : ^ e C"@ . i/ e C'@„ . D . sV 4= s'v (2) 

h . *31011 . D h :. /t@j; . D : a ! K - yii : 
[*310-121] D : (3(0, a): p/a-ev:^/r)efi . Df,, . ^/i? 4= /s/o" : 
[*303-52] D : (gp. a, R,8): p/<Tev .R {pi a) S : ^/t; e /. . Df „ . ~ {i? (^/i,) S] : 
[*41-11] D:a!s'i--^s'/i (3) 

h . (3) . *310-1 . D h : /I, i/€ C"@ . /i + r . D . sV + s'i* (4) 

Similarly h: fi,ve G'@n ./j.^v.'D. s'fi 4= s'v (5) 

h . (1) . (2) . (4) . (5) . D f- :. Hp . D : m4= i; . D . s'/i + «'" (6) 

I- . (6) . Transp . D h . Prop 

*310-33. l-.s5@smor® .s5@„smor@„.s;@5smor©g [*310-32] 



*311. ADDITION OF CONCORDANT REAL NUMBERS. 

Summary of *311. 

We define a set of real numbers as concordant when all are positive 
or zero, or all are negative or zero, i.e. when all belong to G'®' or all belong 
to G'&n. Given two concordant real numbers /a and v, we define the sum of 
fi and V as the class of sums, in the sense of *308, of a member of fi and a 
member of v, i.e. as 

W{('^M,N).Me(i.Nev. W=M+gN], 

i.e. as s'fi+a"v, in virtue of *40"7. It is easy to prove that, assuming the 

axiom of infinity, the sum so defined has the properties we require of a sum. 
We denote the sum so defined by " /x+pv." In order to insure that /M+pV 
shall be A unless /i, v are concordant real numbers, we put 

*311-02. fi-\-pV = X {concord (/*, j/) . Z e s'n +g"v] Df 

Thus if a, V are concordant real numbers, fi+pv — s'iu,+g"v (*311*11); if 

not, fi+pv —A (*311'1). A definition of addition which applies to real 
numbers of opposite sign will be given in *312. 

The comnmtative and associative laws for +p (*311*12'121) follow at 
once from the corresponding laws for +g. Assuming the axiom of infinity, 
we prove without much difficulty that the sum of two positive real numbers 
is a positive real number (*311'27), and the sum of two negative real 
numbers is a negative real number (*311'42). In these proofs, when propo- 
sitions of previous numbers involving " Rat " are used, " Rat " is replaced by 
G'H' and " Rat - I'O, " by G'H. This is legitimate in virtue of *304-49-34. 
In *311'511 we prove (assuming the axiom of infinity) that if ^ is a positive 
real number, and Y is any positive ratio, however small, there are members 
X oi ^ such that Y +gX is not a member of ^, i.e. given any positive real 
number, there are rationals differing from it by less than any assigned positive 
rational. This proposition is useful, and is used in proving that if ^, rj are 
positive rationals, each is less than |^+j,i; (*311'52). The converse of this 
proposition, i.e. the proposition that, if fi@v, there is a positive real number 



SECTION a] addition OF CONCOEDANT REAL NUMBERS 321 

\ such that v = fi+pX, is proved in *311"621'64, after a considerable amount 
of work. Thus we have • 

*311-65. h :: Infin ax . D :. fi®v .= :/i,ve G'@ : (gX) . X e C'@ .v = fi+p\ 

We have, of course, a corresponding proposition for ©„ (*311"66). From 
*311'65 we deduce without difficulty that if fi is less than v (/i, v being 
positive real numbers), then \ +^ /* is less than X+pV (\ being a positive 
real number), i.e. 

*311-73. h : Infin ax . \ e C© . ii®v . D . (\ +p /i) @ (\ +p v) 

whence (with the corresponding proposition for @„) we deduce 

*311'75. I- :• Infin ax . concord (\, /x) . D ; X 4-p /* = \ +p i/ . = . /i = i* 
which secures the uniqueness of subtraction. 



*3ir01. concord (//,, V,... ). = :/u, J/,... eO'®'.v. /A, V,... 6 (?'©'„ Df 

*31102. /[4 +pi' = X {concord (/i, I/). Z€s'/i+j"i/} Df 

*3111. I- : ~ concord (/A, J/) . D . /t +J, V = A [(*31102)] 

*311'11. 1- : concord (/*, i^) . D . 

fi+pv=s'n+g"v= W[{'^M,N).Mefi.Nev.W=M^-gN] 

[(*31102)] 

*31112. V.ii+pV=v+piM [*311-1-11 . *308-4] 

*311-121. h . (\ +p /i) +p V = \ +p (/i +p v) [*311-1-11 . *308-63] 

«311'13. h : concord (/i, v) . = . concord (| Cnv'Vi | Cnv"i;) 
[*310-16 . (*311-01)] 

*31114. h : concord (/*, | Cnv"i;) . = . concord (| Cnv'V, v) [*311-13 .*310-l7] 

*31115. I- : concord (/i, I Cnv"i/) . D . ~ concord (/i, I/) [*310-13-16] 

*311-2. h : Infin a.x.^CC'E . X eC'H .O.X +g"H"^ = H"X +/'^ '^ ^'^ 

Bern. 
I- . *308-72 . *304-34-401 . 3 h :. Hp . D : Fe Z +g"H"^ . = . 

i'3^Z,Z').Z'e^.ZeG'H.Y=X+gZ.{X+gZ)HiX+gZ'). 
[*37-6] =.('^Z,Y').ZeC'H.Y=X+gZ.Y'6X +g"^ - YHY' . 
[*306-52] = . YeH"X +/'? ■ ^HY :. 3 h . Prop 

*311-21. h : Infin a^x. ^CG'H .^l^ .X eC'H.D . H^'X C H"X +g"^ 
Dem. 

h . *306-52 . *304-401 . D f- :. Hp . D : Fe f . D . Z£r(Z +p F) : 
[*40-5r61] D:XeH"X+g"^ (1) 

h . (1) . *304-23 . D h . Prop 

E. & W. III. 21 



322 QUANTITY [part VI 

*311-22. I- : Infinax . f C C'i? . g l^.XeC'H. D . 

H"X +/'! = H^'X w X +g"H"^ 
Bern. 

[■.*304:-23.:>\-.H"X+g"^={H"X+g"^nH^'X)^(H"X+g"^n^'X) (1) 
h.(l).*3ir2-21.DI-.Prop 

*311-23. h : Infin a.x .^eC'@ .X eC'H .D . H"X +p"f = H^'X u X +g"H"^ 
[*311-22.*310-12] 

*311-24. V :. Infin ax . f e C'@ . Ye G'H.D: 

{^Z).ZHY. YeZ+g"^: Yes'^+g"H'Y 
Bern. 

l-.*304-31.Df-:Hp.D.(aTf). We^. WHY. 
[*306-62] 3 .(a^, W) . We^.ZHY. Y=Z+gW:D\-. Prop 

*311-25. h : Infin ax . f 17 e C"@ . D . f C ^+j,i; . j; C ^ +p t; 
Dem. 

I-.*310-12. ■^h-.B.p.Yev^^.H'YCT,. 
[*311-24] O.Yes'^+g"v ' (1) 

f-.(l).*311-ll.Dh:Hp.D.7;Cf+pi7 (2) 

l-.(2).*31112.Dl-:Hp.D.^C?+j,77 (3) 

h . (2) . (3) . D h . Prop 

*311-26. h : Infin ax . f , 7; e C"@ . D . H"(^ +pv) = ^+pV 
Dem. 
I- .*311-23 . D h :. Hp. D : FeT? . D.H"(^+g Y) = H^'Yyj(H"^)+g Y: 

[*3iril.*310-12] D:H"(^+pv) = -S*"v'^(^+pV) 

[*311-25.*310-12] = ? +p9y :. D h . Prop 

*311-27. h : Infin ax . ^, 77 e (7'@ . D . f +p 17 e (7'© 
Bern. 
h . *311-25 . *310-12 . D h : Hp . D . a ! f +p 9? . 

[*311-26.*310-12] D . ^ +p ly e 0'@ u t'D'if (1) 

|-.*310-12.*211-703.D 

h : Hp . D . (gi/, N).M,Ne B'H . if ep'^"^ . i\r ep'H"f) . 
[*308-32-72.*306-23] D . (,'iM,N) , Jf +3 JVep'^"(^+j,i?) n D'if (2) 

I- . (2) . *200'5 . D f- : Hp . D . ^+pr,=^'D'H (3) 

h . (1) . (3) . D F . Prop 

The axiom of infinity is essential to the truth of the above proposition, for 
if it fails we have E ! B'H . B'H ~ e f +p »;, while fieC® .1^ .B'H efi. 



SECTION a] addition OF CONCORDANT REAL NCMBEKS 323 

*311-31. h . I Cnv"(/i +p v) = (I Ciiy"fi) +p (| Cnv"i;) 
Dem. • 

I- . *311-13-1 . D 

h : ~ concord (^, j/) . D . | Ciiv"(/i, +pv) = A.(\ Cnv"/*) +p (I Cnv"v) = A (1) 
I- . *31113-11 . D h : concord (/*, i/) . D . | Cnv' V +pv) = \ Cnv"s'/i +g"v 

[*308-411] =s'{\Gny",i) +g"Q Cnv"v) ' (2) 

h . (1) . (2) . D h . Prop 

*311-32. l-.|Cnv"(/i+ylOnv"j/) = (|Cnv"/i)+pi; [*311-31 .*310-17] 

*311-33l. [■.fi+pV = \Cnv"{(\Cav"/j.)+p(\Cnv"v)} [*3ir31 .*31018] 

*311'41. I- : Infin ax. ii,v e 0'©„ .'^./iC/ji.+pV.vC/jL+pV 
Dem. 
I- . *311-25 . *310-16 . D h : Hp . D . I Cnv"/* C (| Cav"/i) +p (| Cnv"*/) . 
[*311-33.*310-17] D./iC/i+pi; (1) 

Similarly 1- : Hp .0 .vC/jb+pV (2) 

h.(l).(2).Dh.Prop 

*311-42. h : Infin ax.pb.ve 0'@„ .O-fi+pve C"@„ 
Dem. 

I- . *311-27 . *310-16 . D h : Hp . D . (I Cnv'V) +p (| Cnv'S) e C'@ . 
[*311'33.*310-16] O./i+pve a'@„ Oh. Prop 

*311-43. h : /t 6 G'@g . D . /* +, t'Og = /* 
i)e»i. 

h.*31111.DI-:Hp.D./i+pi'0,= #{(ailf) . Jfe/*. W=M+gOg} 
[*308-51] =/*:Dh.Prop 

*311-44. h : Infin ax . concord (jj,,v).'^ ./jL+pve G'&g [*311-27-42-43] 

«311'45. f- :. Infin ax . concord (/t, v) : /t=|= t'Og . v . i; = t'Og zD .fiCfi+pV 
[*311-25-41-43] 

*311-51. 1- : Infin ax . f e D'^e -I'A.Ye G'H . Y+g"^ C f . D . ^ = G'^ = D'H 

Dem. 
h.*38-13.Dh:Hp.Z6?.D. F+jZef. 

[*306-52] D.Fe| (1) 

F.*306-51.D 

|-:Hp.I.6NCind.Ze^F+g(I;/lx.Z)e^D.F+^{(l/+ol)/lx,Z}€^ (2) 
I- . (1) . (2) . Induct . D h : Hp . j; eNCind . Ze^ D . F+g(i//l x.Z) e^ (3) 
|-.*305-7.*306-52.D 

i- zR^.X e^ .ZeC'H .:> .{'^v) .V eiaCmd. ZH {Y+g{v/l XsX)} (4) 

h . (3) . (4) . D h : Hp . -? e (7'^ . D . Z e f : D h . Prop 

21—2 



324 QUANTITY [PART VI 

*311-511. l-:Infinax.feC"©.F6C'ir.D.(aZ).Z6f.F+jZ~e^ 
[*311-51 . Transp] 

*311-52. h : Infin ax . f , ,7 e C'0 . D . f @ (f +p v) 
Bern. 
l-.*311-511.DI-:.Hp.D:FeO'fi'.D.(aZ).Ze^Z+3F~e^: 
[*311-11] D : (gX, Y).X+, Fe (f +pv)-^ ■ 

[*310-ll.*311-27] D : ^@ (f +p 1?) :. D I- . Prop 

*311'53. h : Infin ax . f, t? e a'@„ . D . f ©„ (|^ +p v) [*311-52-33] 

*3ir56. I-:. Infin ax. f 6 C"@g.D:f =^+^7?. = . »? = i'0, [*3111-43-52-53] 

*31157. h :: Infin ax.D:.^=f+j,i7.= :?=A.v.f6 G'%g . r, = I'O, 
[*311-o6-l] 

*3ir58. h:Infinax./i6C"@.D./i = ^'> [*304-3 . *270-31] 

*311-6. V iluf^na.x . ii@v . X,Y ev - II . XHY . M € iJL .-;> . M +g{Y ->X)ev 
Bern. 

h . *310-11 . D I- : Hp . D . MHX . 

[*308-42-72] D.{M+g{Y-sX)}HY (1) 

h.(l).*311-58.Dh.Prop 

*311-61. h : Infin ax . fi,@v . 

\ = L[(^X,Y).X,Yev-fi.XHY.L=Y-sX}.D. 
s'fji,+g"XCp [*311-6] 

*311-62. I- : Infin ax . ,j,(&p .Xev-fji.D. (gF) .Yev-/JL. XHY 
Dem. 

h . *311-58 . D h : Hp . D . Z 6 H"v - H"/j, : D h . Prop 

*311-621. h : Hp*311-61 . D . \e C"@ 

Bern, 
h . *311-62 . D h : Hp . D . a ! \ (1) 

I- . *308-46 . D I- : Hp . D . \ C H"v (2) 

h.*311-62. Dy:RY>.X,Yev-fi.XHY.:).{'^Z).Z€v-fi. YHZ . 
[*308-42-72] ■^.{'3^Z).Zev-ii.{Y-,X)H{Z-,X) (3) 

l-.(3).*37-l.DI-:Hp.D.\C^"\ (4) 

h.*308-56-42-72.D 

I- : Hp . Z, Fei; -/i . XHY. LH{Y-, X).-::i.XH{X +gL) . (Z +gL) HY . 
[*310-ll.*308-43] -^.LeX (5) 

I- . (5) . *37-l . D h : Hp . D . H"\ C \ (6) 

I- . (1) . (2) . (4) . (6) . D F : Hp . D . \ 6 D'^e - t'A - t'D'^ . 
[*310-12] D . \ 6 C"@ : D h . Prop 



SECTION a] addition OF CONCORDANT REAL NUMBERS S25 

*311-63. h :Inhna.x .V eC® . X ev . N € G'H .-^ .('a^L) . LHN .X -hgLev 

Bern. • 

h . *311-58 . D h : Hp . D . (gF) . Fe i; . XHY (1) 

I- . *308-42 . D h : Hp. Yev.XHY.Z= Y-,X.ZHN.:>.ZHN.X+gZ6v (2) 
F . *308-42-72 . D 

i-:RTp.Yev.XEY.Z=Y-,X.NH^Z.LHN.'^.LHN.X+gL€v (3) 
h.(3).*311-58.D 

f-:Hp. Yev.XHY.Z=Y-,X.NH^Z.-^.{'aL).LHN.X+gLev (4) 
h . (1) . (2) . (4) . D h . Prop 

*311-631. h : Infin ax . fi@v . iVe /* . D . 

(^M,X,Y).Mefi.X,Y€v-fji.XHY.N=M+g(Y-,X) 
Dem. 

h . *311-58 . *308-72 . 3 

Vz'S.^.Xev-ii.LHN.X+gLev.Y=X^-gL.M=N-gL.:i. 

Me^L.X,Yev-/i.XHY.N=M+g(Y-,X) (1) 

h . (1) . *311-63 . D h . Prop 

*311-632. I- : Infin ax . fi®v . JVe v - /i . D . 

('aM,W).Me^L.M+gW,N+gW€v-iM.{M+gW)H(N+gW) 
Dem. 

I- . *306-52 . *311-63-58 . D h : Hp . D . (g Tf ) . F e (7'5 . iV +j F e v - /li (1) 

h.*311-511.Dh:Hp. WeG'H.0.('3^M).Mefji.M+gW'^efi (2) 

h.*311-58. Oi-zKp.Me/M.Nev-fi.WeC'H.D.MHN'.WeC'H. 

[*308-72] D . (M+g W) H(N+g W) (3) 

I- . (3) . *311-58 . D h : Hp (3) . iV+j F e j; . 3 . Jlf+j TT 6 1/ (4) 

J- . (2) . (4) . D 

l-:Hp. W6C'H.F+gWev-^L.0.('3^M).M€fi.M+gW€v-fi (5) 

h . (1) . (3) . (5) . D h . Prop 

*311-633. h : Infin ax . fi@v .Nev.'^. 

('S.M.X, Y).M6^L.X,Y€v-fi. XHY. N=M+g(Y-,X) 
Dem. 
V . *308-61-4-63 . D 

\-:R^.MHN.X = M+gW.Y=N+gW.:y.N=M+g{Y-,X) (1) 

h . *311-632 . *308-72 . D h : Hp . J\r~ e/i . D . (gM, W,X,Y). 

Mefi.X=M+gW.Y^N+gW.XHY.MHN.X,Yev-^i, (2) 
h.(l).(2).Dl-:Hp.iV~e/t.D. 

(aJf,Z,F).Jlfe^.Z,Fei;-/i.Z5'F.i\r=Jlf+3(F-,Z) (3) 
|-.(3).*311-631.Dh.Prop 



326 QUANTITY [part VI 

*31164. l-:Hp*311-61.D.i/ = /i+j,\ 

Dem. 

V . *311-633 . D . I. C sV +g"\ (1) 

I- . (1) . *311-621-61 . D h : Hp . D . \ 6 C© . j; = sV H-/'\ . 

[*31111] D.i;=/i+p\:Dh.Prop 

*311-65. h : : Infin ax . D : . ii@v . = :fi,v6 G'& : (gX) . \ e C'@ .v = fi+pX 
[*311-52-64] 

*3H-66. I- : : Infin ax . D : . yu,©™!/ . = : /t, v e C'©„ : (gX) . \ e G'@n .v = ii+p\ 

Dem. 
\- . *310-11111 . D h : /i©„v . = . (I Cnv'V) © (| Onv"i/) (1) 

H.(l).*311-65.Dh::Hp.D:. 

/i@nv . = : I Cnv' V 6 C'@ : (gX) . X e C© . | Cnv"v = | Cnv"/i +j, X : 
[*311-32.*31016-19] = : /i 6 (7'0„ : (gX) . X e (?'©„ .v = /j,+p\::D}- . Prop 

*311-73. I- : Infin ax . X e C© . fi@v . D . (X +j, /*) © (X +p v) 
Dem. 

I- . *31 1-65 . D h : Hp . D . (g/j) . /a e C© . v = /x +3, /o . 
[*311-121] D.(gp).pea'@.X+pi; = (X+p/i)+^p (1) 

h .*3H-27 . D h : Hp . 3 . X +j, /i,, X +p v e (7'© (2) 

I- . (1) . (2) . *311-65 . D h . Prop 

*311-731, h : Infin ax . X e 0'©„ . /i©„i; . D . (X +p /*) ©„ (X -^-p v) [*311-73] 

*311-74. I- :. Infin ax : X,/* e (7'© . v . X, /* e (7'©„ :D:X+j,yu, = X+pZ/. = ./i = v 

h.*3ir271. DI-:X,ytie(?'©.X+3,/t = X+pj'.D,i/6a'© (1) 

I- . *311-73 . Transp . D 1- : Hp (1) . D . ~ (/i©v) . ~ (j;@yu,) (2) 

h.(l).(2).*310-l. Dh:Hp(l).D./* = i/ (3) 

Similarly I" :X,/ieC"©„.X+pyii = X+pV . D ./i = v (4) 

h . (3) . (4) . D h . Prop 

*311'75. I- :. Infin ax . concord (X, )it) . D : X +p /i = X 4-^ i; . = . yu, = v 
[*311-74-43] 



*312. ALGEBRAIC ADDITION OF EEAL NUMBERS. 

Summary o/" *3]2. 

In this number we extend the definition of addition so as to apply to real 
numbers of opposite sign. As in *308, this requires a previous definition of 
subtraction. We define subtraction as follows : If there is a \ such that 
v+p\ = fi, then /i — J, V is \ ; if there is a X such that /i +p X = v, then /i—pvis 
I Cnv"\, i.e. the negative of X,; in any other case, /i— pV = A. The formal 
definition is : 

*31201. ii-pV = X{{'g\)i\iJL,veG'®gi 

V +p\ = fi . X e\ .V . fi +pX = v . X 6\ Cnv"X} Df 

Hence assuming the axiom of infinity we have 

v{@\J ©„)./* . 3 ■ M -J, I- = (J^) {v+pX = ii) (*31218), 
/i (@ w @„) i; . D . /t -p i; = (7\) (ji +p I Cnv"\ = v) (*312-181), 
\6G'eg.D.\-p\ = i'05 (*312191). 

The algebraic sum of /* and v is defined as /i+pv if /t and v are of the 
same sign, and as yn — p | Cnv"j; if /* and v are of opposite signs ; i.e. we put 

*31202. ii+aV = {ii.+pv)yj{fi-p\C'D.v"v) Df 

This definition is justified because either /i+pv or fi—p\Cnw"v must 
always be A. Thus we have 

*31232. I- : concord (ji,v).'D . fi+aV = /i+pV 

^12'33. f- : ~ concord (/*, z/) . D . /* +„ i* = /* -p | Onv"i> 

The propositions proved are analogous to those of previous numbers, and 
offer no difficulty. 



*31201. fj.-pV = X{(<3X):X,ti,veC'®g: 

p+p'K = /i.X eX.v ./i+pX = v.X e\ Cnv"\} Df 

*31202. /i+aV = (jji+pv)\j(jjL-p\Cnv"v) Df 



328 QUANTITY [part VI 

*3121. }-:.Xe/i-pV.= : fi,v6G'®g:(^X) :XeC'@g: 

v+p\ = /jL.Xe\.v.ii+p\ = v.Xe\Cnv"X [(*311-01)] 

*312-11. h : ^ concord (fi,v).0.fi-pv = A [*311-l-27-42-43] 

*31212. h : Infin ax . v®fi . D . 

fi-pV = X {(a\) . \ 6 G'@ .v+pX = iJ,.Xe\} = (?\) {v+p\ = /i) 
Dem. 
I- . *3111-65 . Dh:Hp.D.~(a\)./i+j,X. = v (1) 

h.(l).*312-l. 0\-:Bp.D.fi-pV = X{(<3\).\eG'&.v+p\ = fi.Xe\} (2) 
I- . (2) . *311-74 . D h . Prop 
*31213. h : Infin ax . fi@v . D . 

fi-pv = X {{'s\) .XeC'® . ii+p\ = v . X e\Gny"\] 
= I Cqv"(7\) (fji,+p\ = v) [Proof as in *312-12] 

*31214. f- : Infin ax . i;@„/t . D . 

fi-pV = X {(gX) . \ 6 0'@n .v+pX=fi.XeX} 

= (;\) (v +p \ = /*) [Proof as in *312-12] 

*31215. I- : Infin ax . fjL®„v . D . 

fi-pv = l {(a\) . X 6 G'@n ./i+pX = v.Xe\ Cnv"\} 
= I Cnv"(7\) {fi+pX = v) [Proof as in *312-12] 

*31216. V-.fie G'@g . D . fl-p i'Og = fi [*312-1 . *311-43] 

*31217. h : /i e G'@g . 3 . 1'O, -p /^ = | Cnv'V [*312-1 . *311-43] 

*312-18. \-:IntiTia.x.v(&K)@n)fj,.D.fi-pV = (iX)(v+pX = fi) [*31212-14] 

*312181. h : Infin ax . /i (@ c; 0„) i; - D . /i -p j^ = | Onv"(jX) (yu, +p X = v) 

= (iX)(fi+p\Gnv"X==v) [*312-13-15] 

«312'19. h : Infin ax . concord (X, /j,) . . (X +p /i) —pX = fi 
[*31218 . *311-65-6G-43] 

*312191. h : Infin ax . X e G'@g . 3 . X -j, X = t'O, [*311-52-53-43] 
*312-2. I- . I Cnv"(^ -pv) = \ Cnv'V ~p \ Cm"v 

Dem. 
h . *312-1 . *31016 . D 
H :. X e I Cnv'V -p I Cnv"i; .= :/*, v e G'@g : 

(gX) : X 6 O'0j : | Cnv"j; +p X = | Cnv'V .XeX.v. 

I Cnv'V +pX = \ Cnv"i/ . Z e I Cnv"X : 
[*311-32] = :iJL,ve G'@g : (gX) : X e 0'®g : 

V +p I Cnv"X =/t.ZeX.v./i+p| Cnv"X =v.Xe\ Cnv"X : 
[*312-1.*310-16] =:Xe\ Cnv"(ji -pv):.Oh. Prop 

*312-201. i-.fi-p\ Cnv"!/ = j Cnv"(| Cnv'V -p v) [*312-2] 



SECTION a] algebraic ADDITION OF REAL NUMBERS 329 

*31221. h . I Cav"(i; -p,j,) = fi-pv 

Bern. • 

h . *312-1 . D h :: X 6 I Cnv"(i; -,, /*) . = :. (gF) :. /a, i; e G'@g :. 

(a\):\6C"@g:/i+p\ = i'. Fe\.Z= F|Cnv. V. 

v+pX = ii. Fe|Cnv"X.Z= F|Cnv:. 
[*31016] = \.ii.,ve G'®g :. (gX) : \ e G'®g :/ji,+p\ = v .X e\ Cnv"\ . v . 

V +p'K = /I. X eX:, 
[*312-1] =:.Z6/i-yv::Dh.Prop 

*312-211. y.fi-p\ Cnv'S = v-p\ Cnv'V [*3i2-201-21] 

*312-22. I- : Infin ax . v (6 o @„) /t . D . /* -, i; e G'@ 
JDem. 
h . *311-65 . *312-12 . D h : Hp . v@ii .D . /M-pV eC® (1) 

h . *311-66 . *312-15 . D h : Hp . /i@nv . D . | Cnv"(ji -p v) e G'@„ . 
[*310-16] ^.fji-pveG'@ (2) 

h . (1) . (2) . D h . Prop 

*312-23. h: Infin ax. /i(@w@„)i».D./i-pi;e(7'@„ [*312-21-22 . *31016] 

*312-3. [■.fi+aV = {fJL+pv)^j{/i-p\Cnv"v) [(*312-02)] 

*312-31. h : ~ (/*, 1/ e G'@g) . 3 . ^ +,. v = A [*312-3-ll . *311-1] 

*312-32. h : concord (/i, j;) . D . /i +a v = At 4p v [*312-311 .*31115] 

*312-33. h : ~ concord (fi,v).D. fi+aV=fi-p\ Cnv"j; [*312-3 . *311-1] 

*312-34. h : Infin ax . ^, v e G'@g . D . /* +„ !» e (7'©j 
[*312-32-33-22-23 . *311-44] 

*312-41. i-.fl+aV = V+afJ- 

Dem. 
h . *312-32 . *311-12 . D h : concord (/*, r) . D . /* +„ v = j» +<, /* (1) 

I- . *312-33-21 . D h : ~ concord (/*, v) . D . /* +„ v = | Cnv"(| Cnv'"i; -p fi) 

[*312-201] =v-p\Cn\"fL 

[*312-33] =v+aH- (2) 

h . (1) . (2) . D h . Prop 

«312'42. I- : Infin ax . concord (\, /*, v) . D . (\ +p fi) —p (\ +pv) = fi—pv 
Bern. 

I- . *31127-42-43 . D I- :. Hp . D : concord (\ +j, fi, X +p v, X, fi, v) : 
[*311'75] D •.X-\-pp = ii. = .(X+pp)+pV = ii+pV. 

[*311-12-121] =.{X+pv)+pp = ii+pV (1) 

Similarly I- :.Hp . 3 :/*+pP = X. = .(/*+pi')+pjO = \+yi' (2) 

h . (1) . (2) . *312-1 . D h . Prop 



330 QUANTITY [part VI 

*312-43. I- : Infin ax . concord (\, /i, v) . v (@ c; 0„) /i . D . 

(\ +pfi)—pV = X +p {fi -p v) 
Dem. 

V . *311-65-66 . D h : Hp . D . (g^o) . p e G'@g .fi=^v+pp. 
[*312-12-13-19] D . (a/j) . p e C'&g .(\+pfi)-pV = \+pp. /ji,-pV = p::>[- . Prop 

«312-44. h : Infin ax . concord (\, /i, v) . /i (@ va ®„) k . D . 

(X, +J, /t) -p V = \ -p (v -p jx) 
Dem. 

V . *311-65-66 . D h : Hp . D . (gp) . p e 0'@g .v=fi+pp. 
[*312-42-19] D . (ap) . peC'@g . {X+pfj,)-pV = X-pp. p = v-pfi: D h . Prop 

ii^312'45. h : Infin ax . concord (X, fi) . . (\ +p p.) —p fi = X +p {p, —p p) 

Dem. 
h . *31219 . *311-43 . D h : Hp . D . /* -^ /i = i% . 
[*311-43] '^.X+p(ji-pp)-=X 

[*31 219] ={X+pp)-pp.:'^V . Prop 

«312 451. I- : Infin ax . concord (\, p,,v).0 . 

(X +p p)-pv = (X +0, p) +« I Onv"j^ = X. +«()". +a I Cnv"!;) 
Dem. 

V . *312'43 . D h : Hp . i; (@ c* @„) /i . D . (X +p/i) -J, V = \ +i, (/t -jp I') 
[*312-33] =\+p(/t+„|Cnv"j/) 
[*312-32-12-14] =\+„(/^+„!Cnv"i;) (1) 

V . *312-44 . 3 h : Hp . /It (@ w @„) v ."^ .{X+pp)-pV = X-p{y-p p) 
[*312-21] =X-p\Cn^"{tL-pv) 
[*312-33-12-14] =X.+a(/i-j,i') 
[*312-33] = \ +„(/.+„ I Cnv"i^) (2) 
h . *312-45 . D I- : Hp . /i = K . D . (\ +p /It) -p J/ = \ +p (/I, -^ I/) 

[*312-33-32] = X +e (/*+<, I Cnv"j;) (3) 

h . (1) . (2) . (3) . *312'32 . *311-43 . D h . Prop 

«312'46. h : Infin ax . concord (\, /it) . 3 . (\ +„ /t) +„ v = \ +» (/tt +a v) 
Dem. 

V . *312-32 . *311-65-66-43 . D I- : Hp . concord (\, /it, v) . D . 

0^-\-alj)+aV = {X->rpp)+pV.X+a{p-+aV) = X-¥p(,p.+pV) (1) 

h . *31 2-451 . 3 

h : Hp . concord (\, p., \ Cnv"j;) . 3 . (\ +„ /a) +» k = X +» (/t +<, v) (2) 

h.*312-31.3l-:i;~eC"@j.3.(\+a/t)+„i' = A.\+„(/t+<,i') = A (3) 

f- . (1) . (2) . (3) . *311-121 .31". Prop 



SECTION A] ALGEBRAIC ADDITION OF REAL NUMBERS 331 

*312461. h : Infin ax . concord (ytt, v) . . (X +„ /J-) +av = \ +a (/* +a v) 
Dem. • 

V . *312-46 . D h : Hp . D . (i; +a /i) +a \ = V +„ (/* +« \) (1) 

1- . (1) . *312-41 . D h . Prop 

*312'47. h : Infin ax . concord (\, v) . 3 . (X, +„ /i) +0 1* = X +» (/* +« J') 
i)em. 
h . *312-461 . D h : Hp . D . (/i +„ X) +„ V = /i +« (X +„ i^) . 

[*312-41] D. (X +„/*)+„ J/ = /i+<,(X+„i') 

[*312-41] =(X+a »')+«/* 

[*312-46] = X H-a (i^ +„ 11) 

[*312-41] =X+<,(M+„i;):Dh.Prop 

*312-48. I- : Infin ax . D . (X +„ /*) +„ i; = X +„ (^ +„ v) 

Bern. 
I-.*812-31.D 

f-:~{X,/i,j/6O'0g}.D.(X+„/i)+«i'=A.X+„(/i+„j») = A (1) 

V . *31012 . D h :. X, /i, V e (?'@g . 3 : concord (X, /i) . v . concord (X, v) : 
[*312-46-47] :>:(X+aiJ.)+aV = \+a(H-+av) (2) 

h . (1) . (2) . D h . Prop 

*312-51. l-:XeC"@g.D.X+„t'05 = X [*312-32 . *311-43] 

*312-52. l-:Infinax.X6(7'@3.D.X+„|Onv"X=i'0j 
Dem. 

h . *312-33 . D h : Hp . D . X +„ I Cnv"X = X, -p X 
[*312-191] = I'Og : D h . Prop 

*312-53. H :. Infin ax . X, /t, j; e C'@g .D :'\+afi = v .= .\ = v+a\ Cnv"/* 
[*312-48-51-52] 

*312-54. h : Infin ax . X, /i e (7'©j . D . (go-) . o- e C"®g . X +„ o- = /i 
Dem. 

I- . *312-48-51-52 . D h : Hp . 3 . X +„ (| Cnv"X +<, yu.) = /* (1) 

I- . *312-34 . D 1- : Hp . 3 . 1 Cnv"X +„ /i e G'®g (2) 

h . (1) . (2) . D 1- . Prop 

*312'55. h :■ Infin ax . X, /i, v e G'®g .D:X+a/i = X+ai'.s./i = i» 
Z)em. 

I- . *312-41-63 . D f- :. Hp . D : X+a/t = X+oJ' . = . A' = (X+a v)+a I Cnv"X . 
[*312-41-48] = . yll = v +a (X +a I Cnv"X) . 

[*312'51-52] =./i = v:.Dh.Prop 



332 QUANTITY [part VI 

«312-56. I- :. Infia ax , concord (\, /*) . 3 : A,@j/i . = . (go-) . a e C'@ . \ +„ o- = /i 

Dem. 
I-.*311-65.*312-32.D 

I- : . Hp . \, /i e Ce . D : \@g/t . = . (g^) . o- e G'@ . \ +„ o- = /* (1) 

h.*3ir66.*310-16.D 

I- :. Hp . \, /t e a'e„ . D : \@g/i . = . (go-) . o- e C"@ . /i +J, I Cav"o- = \ . 
[*312o3-32] = . (ao-) . o- e Ce . \ +„ o- = /* (2) 

h.*312-51. DI-:.Hp.\ = i'Og.D:\@g/i.=.(ao-).a-e(7'0.\+aO- = /i (3) 
h , *312-53ol . D h :. Hp . /4 = t'Og . D : \®g/i . = .(aff).o- e (7'@ .\ +„ o" = /i (4) 
h . (1) . (2) . (3) . (4) . D I- . Prop 

*312'57. I- :. Infin ax . \, /* e G'®g . ~ concord (\, /*) ■ 3 : 

\0g/i . = . (ao-) . ff 6 0'@ . \ +a o- = /i 
Dem. 
h.*312-48-51-52. 3 h :\6C"®„. /ieO'® . D ./t = \+a(|Ciiv"A,+a/t) (1) 
I- . *312-32 . *311-27 . D h : Hp (1) . D . (| Cnv"\+o/i) e C'@ (2) 

t-.(l).(2). 3h:\eO'e„./i6(7'©.D.(a<7).<reC"6.\+„ff = /i, (3) 

h . *312-32 . *311-27 . *31013 . D 

h:XeC'@.yit6C'®„.D.~(aff).o-eC'@.\+aff = ^ (4) 

h.(3).(4).D 
h : . Hp . D : \ e a'@„ .fi6G'@. = . (go-) . o- e 0'© . \ +« ff = /t :. D h . Prop 

*312-58. h : . Infin ax . \, /a e C'@g . 3 : 

\ej/t . = . (a<7) . ff 6 0'@ . \ +a ff = /* [*312-56o7] 



*313. MULTIPLICATION OF REAL NUMBERS. 

Summary of *313. 

Multiplication of real numbers is simpler than addition, because it is not 
necessary to distinguish between factors of the same sign and factors of 
opposite sigtis. Thus we put 

*313-01. fjiXaV = X{fi,veG'@g.Xes'iJ.Xg"v} Df 

Thus if /J,, V are real numbers, their product is the class of products (in 
the sense of *309) of members of /i and members of v ; otherwise their product 
is A. The propositions of this number are analogous to those of previous 
numbers, and the proofs are as a rule analogous to those of *311, except in 
the case of the distributive law (*313'55). 



*31301. ^LXaV-=X{^Ji,veG'®g.Xes'^■>^g"v} Df 

Proofs in this number are mostly analogous to those for addition, and are 

therefore often omitted. 

*31311. l-:~(/i,i;6C7'©j).D./iX„i; = A 

*31312. V:u.,ve G'®g .:i ./iXaV = s'u, Xg"v 

*313-21. \-:fi,ve G'& w I'l'Og . D . /i x^ v = s'/i x,"v 

*313-22. \-:/ji,ve 0'@„ w I'l'Og . 3 . /i x,, /; = s'(\ Cnv"/i,) x,"(| Cnv"i;) 

*313-23. hzfjie (7'@„ . i; e C"@ . D . /* x„ i; = I Cnv"s'(| Cnv'V) x,"v 

*313-24. \-:fjieG'@.ve (7'@„ . D . /i x„ k = | Cnv"s'(/t x,)"| Cdv"i» 

*313-25. h.fjiXaV = \ Cnv"(| Cnv' V x„ i/) = | Onv' > x„ | Cnv"i» 

*313-26. I- . /t x„ I Cnv"i; = | Cnv"/t x„ v = | Cnv"(fi x^ v) 

*313 31. h : Infin ax . ^ e O'0 . Z e G'H .:i.X x/'f C H"X x/'^ 

*313-32. V : Infin a.s.^eG'@.Xe G'H . D . Z x/'f = H"X x /'f 

*313-33. h : Infin ax . f e 0'© . Z e G'H . 3 . Z x/'f e G'@ 

*313-34. [- : Infin ax . f e (7'0„ . Z e G'Hn ■ 3 • Z x/'f e (7'® 



334 QUANTITY [part VI 

*313-35. h : Infin ax . f e (7'@ . X e C'Hn -O.X x/'f e G'@n 

*313-351. F : Infin ax . ? e C"©„ . X e G'H . D . Z x/'f e C"®„ 

*313-36. h:^e C'@g . D . Oj x/'f = t'Oj 

*313-37. h-.Xe C'Hg ."^.X x/'t'O, = i% 

*313-38. I- : Infin ax . f e G'®g . X e C'Hg . D . Z x/'f e C'@g 

*313-41. h : Infin ax . concord (fi, v) . /x =|= I'Oj . v =|= I'O, . D . /i x^ i» e (7'@ 

*313'42. h : Infin ax . ~ concord (/m, v). n,ve 0'@g .D . /juXave (7'@„ 

*313-43. t-:./i=f'Og.v .v = i%:/ji.,veG'@g: :> . /iXaV = L'Oq 

*313-44. h : Infin ax . yit, v e G'®g .D.fiXave G'@g 

3l^313'45. \- . (lXaV = VXa/l 

«313'46. 1- : Infin ax . D . (\ x,, /i) x^ v = X x,, (^ x^ j;) 

The following propositions are concerned with the proof of the distributive 
law. 

«313'51. f- : Infin ax . concord (X, /*, v) . D . (j> x^ \) +„ (v x^ ft) = 

^[(gX, Y, Z, Z').XeX. Ye /j, . Z.Z' €v.M=(Z XgX)+g(Z' Xg 7)] 

[*313-12 . *312-32 . *311-11 . *313-41] 

*313-511. h I Infin ax .\,fj.eG'@. Z, Z' € fi. ZHZ' .X e\.0 .Z XgZ' XgX eX 
Bern. 

h . *304-l-401 . *305-14 . D h : Hp . D . (2 x^ X) if (2^' Xg X) . 

[*309-41] :i.{ZxgZ'XgX)HX. 

[*311-68] D.ZXgl'XjXeXiDh.Prop 

*313-52. I- : Infin ax . concord (\, /t, i;) . D . (v x^ \) +» {v Xaiji) = v Xa (A, +„ ti) 

Bern. 
l-.*313-61-511.Dl-:Hp.D. 
{vXa\)+a(vXaM-) = ^[i^^,Y,Z).Xe\.Yefi.Zev.M={ZxgX)+g(ZxgY)] 

[*309-37] =M[('3_X,Y,Z).Xe\.Ye/j-.Zev.M=Zxg(X+gY)] 

[*31312.*3i2-32.*311-ll] = v x„(\ +„ /i) : D h . Prop 

3K313 53. h : Infin ax . concord (\, /i) . ~ concord (X,, v).ve G'®g . D . 

(v XaX) +a(v Xa iJl) = V XaiX+a/i) 

Bern. 
I- . *313-25 . D I- . (\ +a m) x„ 1/ = I Cnv"{(\ +afi) x„, \ Cnv"i/} (1) 

h . *313'52 . D h : Hp . 3. (\+„ fi) x^ \ Cnv"i;=(\ x„ | Cnv"i;) +^{^ x„ | Cnv^'i;) 
[*313-26.*311-31] =Cnv"{(\x„i;)+„(/iX„i')} (2) 

t- . (1) . (2) . D h . Prop 



SECTION a] multiplication OF KEAL NUMBERS 335 

3N313'54:. I- : Infin ax . concord (\, v) . ~ concord (\, /i) • /ms G'@g . D . 

Bern. 
h . *312'33'34 . D h :. Hp . \ +„ /tt = p . D : concord (\, p) .v . concord (/i, p) (1) 
h . *313-52 . 3 h : Hp (1) . concord (\, p) . D . 

(pXav) +a (I Cnv"/i x„ I/) = (p +o I Cnv"/i) X.. " 
[*312-53] = X, x» v . 

[*312-53.*313-26] D . p x» i/ = (X x„ i^) +„ (/* x„ v) (2) 

Similarly h : Hp (1) . concord (/i, p) . D . p x^ v = (X Xo v) +„ (/* Xg, v) (3) 

h . (1) . (2) . (3) . D I- . Prop 

«313'55. t- : Infin ax . D . (v Xa \) +» (v x,, /*) = v x^ (\ +» /*) 
[*313-52-53-54-ll .*312-31] 



*314. EEAL NUMBERS AS RELATIONS. 

Summary of *314. 

In this number we take up the definition of real numbers suggested in 
*310, namely s"G'®g instead of G'@g. The series of real numbers is now 
s>®g instead of ®g. Everything in this number depends upon 

*310-32. I- ■../ji.,veG'@g.'D:s'/j. = s'v. = .fJi = v 

In consequence of this proposition, s [ G'®g is a correlation of the two 
sorts of real numbers, and the properties of the relational sort can be 
immediately deduced from the propositions of previous numbers. We define 
addition and multiplication of relational real numbers so as to secure that, if 
/jL, V are real numbers of our previous sort, the arithmetical sum of s'fi and 
s'v is s'(/A +o v) and their product is s'{fi x^ v). This is effected by putting 

*31401. X+rY=RS[{'3_ti,v).X = s'^.Y=s'v.R{s\^+av)]8] Df 

with a similar definition for X Xr T. The zero of real numbers is now 
Og instead of I'Og, and the negative of a real number X is X | Cnv. The 
fundamental propositions are 

*31413. \-:/ji.,v€ G'@g . D . s V +^ s'v = s'(/j, +« v) 

*31414. \-:fi,ve G'@g . D . s V x^ s'v = s'(/jl x^ v) 

in virtue of which the arithmetical properties of relational real numbers 
follow at once from those of real numbers as segments. 

Relational real numbers are useful in applying measurement by means of 
real numbers to vector-families, since it is convenient to have real numbers 
of the same type as ratios. 

For some purposes, a somewhat different definition of real numbers as 
relations is more convenient. Instead of deriving our relations from ®g, we 
may derive them from ^'Hg, i.e. we may consider the relations s"G'%'Hg 
instead of the relations s"G'@g. In virtue of *217-43, {%'Hg) I (- I'A-i'G'Hg) 
is ordinally similar to ®g; hence the requisite properties of s"G'%'Hg follow 
at once. 



SECTION a] real NUMBERS AS RELATIONS 337 

*31401. X+rY=R8 [(aAM/) .X = s'fi.Y=s'v.R {s'(,i +« v)} S] Df 

*314-02. XXrY = R^[('a,fi,v).X = s'tJ..Y=s'v.R{s'(fix^v)}S] Df 

*31403. J' = (^„)e I (C'Hn -) r {^'(Sn)e - t'A - i'G'H„} 

Kj (G'H„) i (1%) a (G'H„ u) 1^ (D'ffe - t'A - I'G'H) Df 

*31404. M+„N=RS[(^,v).M=s'^'fi.N=s'^'v.R{s'J"(^i+av)}S] Df 

*31405. Mx,N=RS[{'^,v).M=s'^'fjL.N=s'^'v.R{s'^'(fiXav)}S] Df 

*3141. h:a!Z+,F.D.X,Fes"(7'eg [*312-31 . (*314-01)] 

*31411. t-:.Iafinax.D:a[!X+,F.=.X,Fes"0'@3 [*314-1 . *312-34] 

*31412. h :. Infin ax . D : g ! Z x, F. = . X, Fe s"(7'e, 

*31413. \- : fiyve C'@g . D . s'/i +, s'v = s'(/i +„ v) 
i)em. 
h . *314-1 . (*314-01) .31-:^ {S'/m +r s'v} S. = . 

(a/3, a).p,tTe G'@g . s'/i = s'p . s'v = s'a.R [s'{p +a <t)\ 8 (1) 
h . (1) . *310-32 . D h - Prop 

*31414. \-:iJi.,v6 G'@g .O.s'/i x, s'v = s'(ji Xa v) 

*314-2. h:i2 6s"(Cf'@j-t't'0,).D.a[!ii^Relnum [*310-31] 

*314-21. h :. Infin ax . D : i2, /Sf e s"Cf'ej . = . i2 +, >Sf 6 s"G'®g . 

= .i2x,S€5"0'@3 

Dem. 

V . *314-13-14 . *312-34 . *313-44 . 3 

I- : Hp . iJ, /Sf e s"G'®g .D .R+rS,R XrSe s"G'@g (1) 

I- . (1) . *31411-12 . D I- . Prop 

*314-22. }-:Res"C'@g.'2.R+rO, = R.RXrOg = 0, 
Bern. 

f-.*314-13-14.D 

V: jjue G'@g .O.s'/jL +, 0, = s'{fi +» I'Og) . s'/i x, 0, = s'((jl x„ I'O,) . 

[*312-51.*313-43] D . s'/i +^ Oj = s> . s'/* Xr 0« = Og : D I- . Prop 

*314-23. h : Infin ax . i2 e s"C'®g . D . i? +, E f Cnv = 0, 

h . *31413 . D I- : /i e C"0g .D.s'/i +r s'\ Gnv"fi = s'(ijl +„ | Cnv"/*) . 
[*43-421] D . s> +, (s V) I Cnv = s'(/t +„ | Cnv"/*) 

[*312-52] =Og:Df-.Prop 

E. <Siw. Ill, 32 



338 QUANTITY [part VI 

*314-24. \-.B+rS = 8+rR [*312-41 . (*314-01)] 

*31425. \-.RXrS = SxrR [*313-45 . (*314-02)] 

*314-26. I- : Infin ax . D . (E +, S) +,.T = R +, (8 +, T) 
Dem. 
h .*314-13 . D h : Hp. /D,o-,T6 a'®g. i? = s'p . ;Sf = s'<7 . r = s'T . D . 

[*312-48] =5'{p+»(o-+aT)} 

[*314-13] =R+r{S+rT) (1) 

h . *314-11-21 . D 

h : ~ (a/3, <7, t) . /9, cr, T 6 0'@j . i2 = s'p ■ ^ = s'o- . r = s't . D . 

{R+rS)+rT=k.R-\-r{S-<rrT)^k (2) 

I- . (1) . (2) . D h . Prop 

*314-27. f- : Infin ax . D . (i2 x, ^) x^ 2'= E x^ (zSf x^ T) 
[*31414 . *313-46 . *314-12-21] 

*314-28. I- : Infin ax . D . (i? x^ (S) +^ {R XrT) = R x, (S +, T) 

Dem. 
h . *3141 3-14 . D h : Hp . p, (T T e (7'©3 . E = s'|0 . -Sf = s'o- . r = s't . D . 

{R Xr S) +r (R Xr T) = s'(p X^ a) +,. s'{p X„ t) 

[*314-21-13] = s'{{p x„ a) +a(p x^ t)} 

[*313-55] =^s'{px^(cr+^r)} 

[*314-21-14] = s'p X,. s'{a +„ t) 

[*314-13] = s'p x^ (s'a- +r s't) 

[Hp] =RXriS-^rT) (1) 

l-.*314-211112.D 

h . ~ (ap, 0-, t) . p, 0-, T 6 (7'@3 . B = s'p ■ -S = s'o- . y = s't . D . 

(i?x^>Sf)+^(i2x,.r) = A.i2x,(/S+^r) = A (2) 

I- . (1) . (2) . D h . Prop 

*314-4. h : Infin ax . D . c/" e {{s'Hg) ^ (- t'A - I'G'Hg)] sSor 0, 

[*217-43 . *304-31-282-23 . *307-41-44-46-25 . (*31001-011-0203)] 

*314'41. f- . s [^ (G's'Hg) 6 1 -> 1 [The proof is analogous to that of *310-32] 

*314 42. h : Infin ax . D . s^ '@g smor ©^ [*314-4'41] 

*314'5. h :. Infin ax . D : 

a ! Jf+^iV. = . a ! il/ X, iV . = . M,N€s"(D'<;'Hg - t'A) 
[*312-34 . *313-44 . *314-42 . (*314-0405)] 

*314-51. h : Infin a,x.fi,ve G'@g . 3 . 

[*314-42 . (*314-04-05)] 

The properties oi M+aN and M x„N result from this proposition exactly 
as those of X+rY and X x^ F result from *31413-14. 



SECTION B. 

VECTOR-FAMILIES. 

Summary of Section B. 

The present Section is concerned with the theory of magnitude, so far 
as this can be developed without measurement. Measurement — i.e. the 
application of ratios and real numbers to magnitudes — will be dealt with in 
Section C ; for the present, we shall confine ourselves to those properties of 
magnitude which are presupposed in measurement. But throughout this 
Section, measurement is the goal : the hypotheses introduced and the 
propositions proved will be such as are relevant to the possibility of 
measurement. 

We conceive a magnitude as a vector, i.e. as ap operation, i.e. as a 
descriptive function in the sense of *30. Thus for example, we shall so 
define our terms that 1 gramme would not be a magnitude, but the difference 
between 2 grammes and 1 gramme would be a magnitude, i.e. the relation 
" + 1 gramme " would be a magnitude. On the other hand a centimetre 
and a second will both be magnitudes according to our definition, because 
distances in space and time are vectors. It will be remembered that we 
defined ratios as relations between relations ; hence if ratios are to hold 
between magnitudes, magnitudes must be taken as relations. 

We demand of a vector (1) that it shall be a one-one relation, (2) that it 
shall be capable of indefinite repetition, i.e. that if the vector takes us from 
a to h, there shall always be a point c such that the vector takes us from 
h to c. If R is the vector, the point to which it takes us from a is R'a ; 
thus the above requisite is expressed by " E ! R'a . Do ■ E ! R'R'a," i.e. by 
" D'R C G.'R." It will be observed that the points which are starting-points 
of the vector form the class d'R, i.e. the class of possible arguments to R 
considered as a descriptive function, while the points which are the end- 
points of the vector form the class D'R, i.e. the class of values of R considered 
as a descriptive function. Since D'R C d'R, we have d'R = G'R ; thus the 
field of the vector consists of all points from which the vector can start. By 

22—2 



340 QUANTITY [part VI 

assuming I>'R C d'R, we exclude magnitudes of kinds which have a definite 
maximum, unless they are circular, like the angles at a point or the distances 
on an elliptic straight line; but, except when they are circular, such 
magnitudes are of little importance. 

According to what has just been said, if i? is a vector whose field is a, we 
have 

R€l-*l.a'R = a.J)'BCa. 

A relation which fulfils this hypothesis is called a " correspondence '' of a, 
because it makes a part of a correspond with a. The class of correspondences 
of a we denote by" cr'a," which is the cardinal correlative of "cror'P," defined 
in *208. Thus we put 

cr'a = (1 -♦ 1) n a'a n D"Cl'a Df. 

We proceed next to define a " vector-family of a." This we define as an 
existent sub-class of cr'a such that, if R and S are any two members of it, 
i? I *S = ^ I jB. We define a class of relations as " Abelian " when the relative 
product of any two members of the class is commutative, i.e. we put 

Ahel = ii(R,8eH:.Dji,s-Ii\S=8\R) Df 

Thus a vector-family of a is an existent Abelian sub-class of cr'a, i.e. writing 
" fm'a " for " vector-family of a," we put 

fm'a = AbelnClex'cr'a Df. 

The class of vector- families is then defined as everything which is a vector- 
family of some a, i.e. we put 

FM = s'D'{m Df. 

Thus a vector-family is an existent Abelian class of one-one relations 
which all have the same converse domain, and all have their domains 
contained in this common converse domain. If k is a vector-family, the 

common converse domain is i'Q."k, which is identical with s'CE"*, and will 
be called the " field " of the family. Thus we have 

l-ZKeFM . = .H:e Abel . « C 1 -> 1 . a"« e 1 . s'D"« C s'a'U. 

A vector-family may be regarded as a kind of magnitude, In order to 
render measurement possible, we require various hypotheses as to the nature 
of the family. Measurement within a given family ic is obtained by limiting 
the fields of ratios to k, i.e. by considering X^. k where X is a ratio, or ^ ^ « 
where Z is a, relational real number of the kind defined in *314. In order 
to make measurement possible, we wish k to be such that, if X is a ratio, 
X P K shall be one-one ; again, if R, 8, T are members of k, and R has the 
ratio X to 8, while 8 has the ratio Y to T, we wish R to have the ratio 
X Xg F to T, i.e. we wish to have 

ZpK|Fp«e(Zx,r)t«; 



SECTION b] vector-families 341 

again, if R has the ratio X to T, and 8 has the ratio Y to T, we wish R \ 8 
(which represents the " sum of R and 8) to have the ratio Z" + s F to T, 
i.e. we wish to have 

(Z ^ «'T) 1(7^ «'T) G (Z+, 7) P «'y. 

The above and other similar properties will be proved, with suitable 
hypotheses, in Section C ; for the present, we shall proceed with the theory 
of vector-families without explicit regard to measurement. 

The first and most important hypothesis as to a family which we consider 
is the hypothesis that it is " connected," i.e. that there is at least one member 
of its field from which we can reach any member of the field by a vector 
belonging to the family or by the converse of a vector belonging to the 
family. Such a member of the field of k we shall call a " connected point " 
of K ; the class of such points will be denoted by " conx'/t " ; the definition is 

-> <— 

conx'/c = s'<1"k r\ 6, (s'x'a w s'k'u = s'Q."k) Df. 

It will be observed that s'k'u are the points to which there is a vector from 

<— 
a, while s'k'u are the points from which there is a vector to a. The definition 

states that these two classes together make up the whole field of the family. 

We define a connected family as one which has at least one connected point, 

i.e. we put 

FM conx = ^ilf n ;J (a ! conx'/t) Df. 

The properties of connected families are many and important. Among these 
may be mentioned the following: If « is a connected family, the logical 
product of any two different members of « is null, i.e. if P, Q e k . P =^ Q, then 
Pf\Q= A, or, what comes to the same thing, if P, Q e k, and if we ever have 
P'x = Q'x, then P = Q; if P e k, all the powers of P are either members of k 
or the converses of members ; if P, Q e k, then P | Q is either a member of 
K or the converse of a member. A connected family may not form a group, 
i.e. we do not necessarily have 

P,Qe K. Dp,Q.P| QsK, 

but we shall show at a later stage (*354) that a group can be derived from a 
connected family « by merely adding to it the converses of those members of 
K (if any) whose domains are equal to their converse domains. The result 
of this addition is to give us a connected family which is a group. 

Another important property of a connected family k is that / f" s'<I"k is 
always a member of it. / \ s'Q."k is the zero vector. In a connected family, 
every vector except / [ s'(1"k is contained in diversity. For many purposes, 
the class of vectors excluding I [ s'Q."k is important. We therefore put 

K3 = «-R1'/ Df. 



342 QUANTITY [part VI 

la the study of a vector-family k, an important derived class of relations 
is the class of all relations of the form B | S, where R,SeK. The operation 
-K I S consists of an /Sf-step forward, followed by an i2-step backward ; that is 
to say, if R'S'a exists, it is obtained by moving a distance 8 forward from a 
to S'a, and then a distance R backward from S'a to R'S'a. The class of 
such relations as i2 1 >S, where R,SeK, we call «.; i.e. we put 

K, = s'(Cnv"K)\"K Df. 

The class k, will have different properties according to the nature of k. We 
may distinguish three cases : 

(1) The field of k may have a first term, i.e. there may be a member of 
s'(I"k which is not a member of s'D"«g. This case is illustrated, e.g. by a 
family of distances from left to right on the portion of a given line not lying 
to the left of a given point. This given point will then belong to s'Q."k, 
since there are vectors which start from it, but it will not belong to s'D"«g, 
since there are no vectors which end at it except the zero vector. A 
connected point a which belongs to s'(J"k but not to s'D"/cg is called the 
"initial" point, and a family which has an initial point is called an "initial" 
family. A family cannot have more than one initial point. Thus we put 

init'/e = t'(conx'« — s*D"/«g) Df, 

FM init = FMn Q'init Df. 

(2) It may happen that, even if k is not an initial family, none of the 
converses of members of «g are members of «. (If k is an initial family, this 
must happen.) This case is illustrated by the case of all distances towards 
the right on a straight line. It is also illustrated by the family of vectors of 
the form (+g X) ^ G'H, where X e G'H'. In this case, as in (1), it is possible, 
by adding suitable hypotheses, to secure that s'k^ shall be a series. This case 
divides into two, which are illustrated by the above two instances : it may 
happen, as in our first instance, that the domain of a vector is always equal 
to its converse domain, i.e. T>"k = (1"k ; or it may happen, as in our second 
instance, that the domain is only part of the converse domain. (The domain 
of (+s X) ^ G'H consists of all ratios greater than X.) 

(3) It may happen that «g contains pairs of vectors which are each 
other's converses. In this case, it is obvious that s'k^ cannot be serial, since 

R,ReK^.'^,R\R = I\ s'0."k .R\R(£. (s*«g)S so that (s'K^f is not contained 
in diversity (except in the trivial case k = I'A). 

In considering «„ we do not at first explicitly introduce any of the above 
possibilities, but it is necessary to bear them in mind in order to realize the 



SECTION b] vector-families 343 

purpose of the propositions proved concerning k^. If Z is a member of k^, 

and L = B,\B, where R,8eK, then if a is a connected point, and L'a exists, 
it follows that there is a member T of « w Cnv"* such that L'a = T'a. It is 
easy to deduce from this that L=T, hence Lex ^ Onv"*. The same holds 

if L'a exists. Hence if E ! L'a . v . E ! L'a, i.e. if a e G'L, L is a member of 
K u Cnv"K. Thus if a belongs to the field of every member of «„ we shall 
have K^ = K\J Cnv"K. We say that a family " has connexity " (not to be 
confounded with "being connected") if g ! conx'/fn^'C'/fj ; thus we put 

^Jf connex = FM n « (g ! conx'*: n p'C'K,) Df, 
and by what has just been said we have 

h : K e ^Jlf connex . D . Kj = k w Cnv"«. 
We also have h : /c e JWconnex . D . s'kq e connex 

and h :. « 6 FM conx .D : k e FM connex . = . s'kq e connex. 

It is these propositions that justify the notation "^if connex." 

It is obvious that we shall have g; ! p'C'k^ if D"k = Q"/e, unless k = I'K. 

Some illustrations will serve to make clearer the nature of the hypothesis 
g Ip'C'K,. This hypothesis states that there is at least one term a in the 
field of K such that, if R, 8 are any two members of k, we can either take an 
B-step forward from a, followed by an iS-step backward, or we can take an 
iS-step forward followed by an i2-step backward. Suppose, for example, that 
our family consists of all vectors of the form (+o fi) ^ NC induct, where 
/i 6 NC induct. Then if R is the operation of adding /*, and (Sis the operation 

of adding v, R\8 is the operation of adding v—gfi if v^ fi, and is the 

operation of subtracting fi—„v if fi> v. In the former case R\S e k, while 

in the latter case S\ReK. In the former case, if or is any inductive cardinal, 

(R I /Sf) V = y — /* +e ■!!'■ ; in the latter case, (8 1 R)'iir = /* — „ v H-^ •sr. Thus in 

either case 'meO'(R\8). Thus the family in question has connexity, and 
K, = K^J Cnv"K. i 

But now consider the family consisting of all vectors of the form 
(Xo^it) ^(NC induct — t'O), where /ieNC induct — I'O. This is an initial 
family, its initial point being 1. But it does not have connexity. If 

R = (x„ fi) I (NC induct - I'O) and 8 = (x„ v) I (NC induct - I'A), R\8is the 
operation of multiplying by v and dividing by fi, with its field confined to 
inductive cardinals other than 0. If v is prime to fi, this relation has only 
multiples of fi in its converse domain and only multiples of v in its domain. 
Hence its field consists of multiples of /i together with multiples of v. Thus 
no member of k. except I[s'(I"k, i.e. (Xo 1)^ (NC induct -t'O), has the 
whole of s'(1"k for its field, and there is no number which belongs to the 



344 QUANTITY [part VI 

field of every member of «.. The above family may be usefully borne in 
mind in considering k^, since it affords good illustrations of most of the 
general theorems concerning k^. 

If K is any family except I'A, any finite number of members of k, have an 
existent relative product, and their converse domains have an existent 
logical product. If « is a connected family, any two members L, M of k^ 
whose logical product exists, i.e. for which (gy) . L'y = M'y, are identical, and 
if x,y are any two members of s'Q"«, there is just one member of k^ such 
that x = L'y. 1{ MeK, and P is a power of M, there is some member L of 
K, such that PdL. But P is not in general itself a member of k^. For the 
application of ratio, the member of «. which contains P is important. We 
call it the "representative" of P. The general definition of a representative is 

rep/P = «'(«, ft G'P) Df 

In a connected family, Ktnd'P cannot have more than one member; hence 
if there is any member of k^ which contains P, that member is rep„'P, and if 
there is no member of k. which contains P, rep,;'P = A. 

If P I Q is any member of k,, (where P,Qe «), we shall have 

re^,'(P\Qy = P^\Q>; 

and it L, M 6 K„ we shall have 

rep/(X I M)o = rep^ '(Z" | Mo) = rep<'{(rep/i'') | (rep/Jf'')}. 

These two formulae are the most useful in determining representatives. 

In order to apply the above theory to the measurement of vectors, it is 
necessary to distinguish between open and cyclic families. An open family 
is one in which, if M e k^ — K1'/, M^^ G /, i.e. one in which no number of 
repetitions of a non-zero member of «i will bring us back to our starting- 
point. If this condition fails, as in the case of angles, or distances on the 
elliptic straight line, the problem of measurement is more complicated, since, 
if ^ is a measure of an angle, so is 2j/7r -1- Q for any integral v. The case of 
cyclic families will be considered in Section D ; for the present, we proceed 
to consider open families, and we shall still be concerned almost exclusively 
with open families in Section C. It should be observed that in cyclic 
families, as we shall define them, members of «g return into themselves, 
whereas in open families, not merely no member of k^, but no member of 
a:, — Rl*7, returns into itself. In most of the families that naturally occur, 
it happens either that no member of k^ — Rl'J returns into itself, or that 
there are members of /cg which do so. But there is no logical necessity in 
this, as the following instance shows: Consider the family consisting of 
positive and negative integral multipliers other than — 1, with their fields 
confined to positive and negative integers other than — 1. Then 1 is a 



SECTION b] vegtok-families 345 

connected point of this family, in fact the initial point. Multiplication by 
— 1 is a member of k„ since^t can be obtained by multiplying by any integer 
;* and then dividing by — /i. Also the square of multiplication by - 1 is 
contained in identity, and is the zero vector of our family. Hence there is a 
member of k^ — Rl'/ whose square is contained in identity, although no power 
of any member of Kg is contained in identity. 

In order to avoid brackets, we put 

'^id = i''i)d Df, 
i-e. «,9 = K. - Rl'/. 

Then the definition of open families is 

FM&^ = FMf^1c (s'Pot"/«:,9 C Rl'J) Df. 

Hence V:.KeFMa,^.= : ks FM:M e k^q. Dj^.M^^dJ. 

It will be observed that if n is an open family, K^ is contained in Rel num id 

(cf *300), and w.g C Rel num. Hence if M e /e.g, M-' = M, (c£ *121), and the 

propositions on intervals in *121 become available. Also if M e K^g, and 

a e s'(I"k, we have 

^ — > w -» 

M ^ M^'a e Prog . M^^ t J^*'a e «• 

The chief use of these facts is to show that the existence of open families 
implies the axiom of infinity and the existence of No. Hence as applied to 
open families, the theory of ratio undergoes the very great simplification 
which results from the axiom of infinity. 

If* is open and connected, and L, Me k^, and a is any inductive cardinal 
other than 0, we shall have L = M if L' = M'' or rep^'i' = rep, 'if" or 
a ! i"^ n M". If p, r are also inductive cardinals other than 0, we shall 
have rep/Z'' = rep/M" if ipx.r = jif<rx.T^ or jf rep/i'"<°'' = rep/ilf'>^'^ or if 
g ! XpX"'' n Jf'><«^ We have in fact 

rep^t'Z'' = rep^'M" . = .'glLi' nM" 

= .a!X'"<°''n ilfo-x«'- 
and rep/Mp = rep/it/" . = .¥>'= M' . = .p- a. 

On applying the definition of ratio (*303"01), we see from the above 
propositions that, with the above hypothesis, 

M {pi a) N . = .^\M''nNi'. = . rep/Jlf' = rep/i\r'', 

while if R, T are members of «, 

Ripla) 8.^.^ = 8". 

Further, we have, in virtue of the above propositions, 

a ! i' n Jfp . a ! Z" o Jf*' . D . yu, Xo o- = 1/ Xo jO, 

whence X, Fe G'H' .^ ! Z ^ K^g <S F^ «;<3 . D . X = F. 



346 QUANTITY [part VI 

These propositions, together with 

belong to Section C. They are mentioned here as showing why the 
propositions of this Section are useful in connection with measurement. 

We next proceed to consider serial families, which are those in which 
i'/cg is an existent serial relation. For this purpose we require the definition 
of "FM connex" already given, and also the definition of "transitive" 
families. We define a as a " transitive point " of k if 

— » — > 

i.e. if any point which can be reached from a in two non-null steps can also 
be reached in one non-null step. We define a family as transitive when it 
has at least one transitive point. If KeFMconx, the hypothesis that k is 
transitive is equivalent to the hypothesis that «g forms a group, and implies 
that K forms a group. We define a serial family as one which is transitive 
and has connexity, i.e. we put 

FM sr = FM trs n FM connex Df. 
Then if « e FM sr, s'xg is a .serial relation, so that the points of the field of k 
are arranged in a series by means of relations of distance. 

When a family is serial, the vectors also can be arranged in a series, by 
means of a relation which may be regarded as that of greater and less. After 
a short number on initial families (explained above), we proceed to the 
consideration of greater and less (&.s it may be called) among vectors. We 
may call a point y " earlier " than a point z when there is a non-null vector 
which goes from y to z, i.e. when z (s'«g) y. If ilf, iV e «, , we then say that N 
is " less " than M if the i\r-step from some point x takes us to an earlier 
point than the Jf-step. Writing V^ for " greater than " among members of 
K,, our definition is 

V, = MN {M, NeKr. (a«) ■ {M'x) (s'/cg) {N'x)] Df 
For the same relation confined to members of «, we use the notation U^ ; 
thus 

U^=V^l^ Df 
If « e FM conx, we have 

U, = P^{P,QeK'.('^T) .T e K^.P = T\Q}; 
this is generally the most serviceable formula for U^. 

If « is a serial family, U^ and F^ are series ; and if k is an initial family, 
JJ^ is similar to s'wg. 

The last number in this Section is concerned with the axiom of 
Archimedes and with the existence of sub-multiples of vectors. The axiom 
of Archimedes will be expressed by saying that if a is any member of the 



SECTION b] vector-families 347 

field of K, and R is any vector, then R^'a, for a suflBciently great finite v, will 
be later than any assigned member of the field of k. In other words, putting 
P= Cnv's'/fg, we wish to have 

« 6 O'P . D^ . (gi;) . j; e NO ind - t'O . xP (R"a), 
or, what comes to the same thing, 

P"R^'a = G'P. 

This will hold if « is a serial family and P is semi-Dedekindian (cf. *214). 
If, further, P is compact (i.e. P^ = P), then all finite sub-multiples of a given 
vector exist, i.e. 

>Sf e « . V e NC ind - I'O . D . (gZ) .L en:. 8 = 1". 

It will be observed that, according to our definition of ratio, if S = L'' and 
S^A,L has to 8 the ratio 1/v, so that L is the vth sub-multiple of 8. 

Instead of treating vector-families by the method we have adopted, we 
might have started from a double descriptive function, which we may denote 
hy x + y, and concerning which we should make various hypotheses. By the 
general notation of *38, we obtain various relations of the form +y or a; -|- . 
These relations may replace the k employed in our method. For convenience 
of notation, we may put 

'+'y = + y Df. 

+'x = x+ Df. 

Then if -|- has suitable properties, and 7 is a suitable class, -I- "7 will be a 
vector family. 

Let us assume that x + y exists when, and only when, x and y both 
belong to the class 7, and that when x and y both belong to the class 7, « + y 
also belongs to this class. Then ii x + y exists, so does x + y + y; hence 
D'+ y C (J'+ y. Further, by our assumptions, if x,yey, x+y exists, and 
therefore x e OE'-l- y. Hence yey. D . (I'+ y = y. Hence if 7 exists, 

D"+"7 6 1 . s'D"+"7 C s'a"+"7. 

If we now assume x + y = x + z. Dx,y,z • y = z, 
-♦ 
then +"'Y C 1 — > 1. Hence we now have 

-* 
+"<y e CI ex'cr'7. 

In order to obtain the Abelian property, we require 

(x + y) + z = (x + z) + y, 

which holds if -I- obeys the permutative and associative laws. Thus in this 

case, 

— » 

+"7 e fm'7. 



348 QUANTITY [part VI 

-> 

In order that +"7 may be a connected family, we require 

(:S^ct):.zey.Oi : (gy) : a = z + y.v.z=a + y. 

A sufficient, though not a necessary, condition for this is that there should be 
a zero, i.e. 

(ga) ■.Z6<y.'2z.z = a + z. 

In this case, + a is the zero vector, and if a is not the sum of two terms other 
than itself, a is the initial point of the family. 

-> 
The condition that if x, y are members of 7 so is « + y secures that +"7 

is a group. Families which are groups we denote by " FM grp." 

Thus collecting what has been said, we find that 

— » 

+"7 6 FM conx grp 

if + fulfils the following conditions : 

(1) x-\-y exists when, and only when, ie,y e<^; 

(2) a;,2/67.Da,,j,.a; + 2/e7; 

(3) x + y = x + z.'^x,y,z-y = z; 

(4) x-\-y = y + x; 

(5) {x + y) + z = x-ir{y + z); 

(6) (ga) '.ze'f.'^z.z = a + z. 

From (3) and (4) it follows that the a of (6) is unique, i.e. there cannot be 
more than one zero. 

In order to insure that our family shall have connexity, we require 
further 

(7) x,y erf .':>x,y-{'5.'^)- z erf '. X ■{- z = y .w .y + z = x; 

(8) in order that our family may be an initial family we require that 

x-\-y shall only be zero when x and y are zero. 

With this further condition, our family becomes serial. 

The above is only a sketch of one of the simplest ways of generating 
families by means of double descriptive functions. Other ways are possible, 
and by greater complication greater generality can be obtained. 

There are some advantages in the above manner of treatment. First, it 
is possible to take our magnitudes as being the x and y which appear in 
" x-\- y',' instead of having to take them as the vectors + y or « +. Secondly, 
our vector-family derives unity from the fact of being generated by the 
single operation +. Thirdly, the method is more in agreement with current 
conceptions of quantity than the method we have adopted. The choice 



SECTION B] vector FAMILIES 349 

between the two methods is a matter of taste ; but it would seem that the 
method we have adopted is Capable of somewhat greater generality than the 
other, and that it requires less new technical apparatus than the other. We 
have not elsewhere had occasion to treat of double descriptive functions 
which only exist when their arguments belong to assigned classes, though 
it is to be observed that our definitions of various kinds of addition and 
multiplication might quite easily have been so framed as to give this result. 
For instance, we might have put 

fi+oV = (?i!r) {(ga, /3) . /t = N„c'a . v = N„c'/3 . w = Nc'(a + jS)} Df. 

In that case, E ! (/i +o v) would have implied fi,ve NqC, whereas with our 
definition it is only a ! (/* +o v) that implies /*, v e NpO. The general treatment 
of double functions which only exist in certain cases would require a 
considerable logical apparatus not required elsewhere in our work, and this 
is, for us, a reason against adopting the method of treating vector-families 
which derives them, as in the above sketch, from a single function x + y. 



*330. ELEMENTARY PROPERTIES OF VECTOR-FAMILIES. 

Summary of *330. 

In this number, we begin by defining the class of " correspondences " of 
«. A " correspondence " of a is a one-one relation B which makes every 
member of a correspond to an n, i.e. which is such that, if a; e a, R'x always 
exists and is a member of a. Thus, for example, if fi is an inductive number, 
+0 fJ-, with its field limited to inductiye numbers, is a correspondence of the 
class of inductive numbers, provided the axiom of infinity holds. (Otherwise, 
(+0 m) t ^^ induct is not one-one.) The definition of correspondences of 
a is 

*33001. cr'a = (1 -» 1) n Q'a n I)"Cl'a Df 

I.e. a correspondence of a is a one-one relation whose converse domain is 
a and whose domain is contained in a. The definition should be compared 
with the definition of " cror'P " in *208. 

It will be seen that ii Re cr'a and xea, R'x exists and is an a, and 
therefore R'R'x exists and is an a, and so on. Hence all the powers of 
R exist (*330'23). Similarly if R, S, T, ... are any finite number of corre- 
spondences oi a, R\8\T\ ... exists. This is proved for two and three factors 
in *330-21-22. 

We define a " vector-family of a" as an existent Abelian class of 
correspondences of a, where an Abelian class of relations is defined as one 
such that the relative product of any two of its members is commutative. 
Thus we put 

*33002. Ahel = 1i{R,SeK.'^B,s-R\S = S\R) Df 
*33003. fm'a = Abel n 01 ex'cr'a Df 

*33004. FM = s'I)'tm Df 

It will be remembered that Potid'P and (for certain kinds of relations) 
finid'P are Abelian classes of relations (*91"34 and *121-352). If P e 1 -» 1, 
Potid'P will be a vector-family of C'P, and if further Pp^ Q J, fiuid'P will be 
the same vector-family. 

One other definition belongs to this number, namely 

*33005. /c. = s'(Cnv"«)|"/<: Df 

This definition has been sufficiently discussed in the summary of the 
present Section. 



SECTION B] elementary PROPERTIES OF VECTOR-FAMILIES 351 

After some preliminary propositions on CI ex'cr'a (*330'1 — "32) and on 
K. (*330'4! — -iS), we proceed^o such properties of families as do not require 
any further hypothesis as to the nature of the family concerned. These 
properties are mainly such as assert the existence of relative products, and 
of logical products of converse domains, or such as assert commutativeness of 
the relative product under certain circumstances. The earlier propositions 
deal with members of k, the later propositions mainly with members of «.. 
The most useful propositions are : 

*330-54. V : KeFM . Q,Re ic.^\B'x .-:i .^\R'Q'x 
*330-56. V I KeFM .Q,Re K .'&\R'a.:i .R'Q'a = Q'B'a 
*330-61. \-'.KeFM-i'i'k.L,MeK,.'^. 

a ! a'Z n d'M.'^ ! D'i n (I'M . a ! Q'i n D'if . g ! Vt'L n D'M 
*330'611. h zKeFM-i'i'A . L,MeK..D .±1 L\M 
*330-624. hzKe FM- I'l'k . Z e «. . D . A ~ e Pot'i 
*330-63. \-:K.eFM.L,MeK,.^\L'x.^\ L'M'x . D . L'M'ai = M'L'x 
*330-642. h-.KeFM- I'l'k .L,MeK,.0. (ga;) . E ! i'a; . E ! L'M'x 
*330-71. hzKeFM.P.QeK.pe'NGmd-i'O.El Pi"x . D . E ! (P | Qy'a; 
*330-72. h-.Ke FM - I'l'k . £, Jf e /Ci . p, o- e NO induct . 3 . g ! Q'Ze n a'M' 
*330-73. h :K€FM.P,Qeic . p 6NCind.E!(P 1 QY'x.D .(P\Qy"x = Pi"Q'"x 



*33001. cr'a = (1 -> 1) o G'a n D"Cl'a Df 

*330-02. Ahe\ = ii(R,SeK.Ds.s-li\S = S\R) Df 

*330-03. fm'a = Abel n CI ex'cr'a Df 

*33004. FM = s'D'fm Df 

*33Q-05. K, = s'(Cnv"«) I "k Df 

)} 

*3301. l-:«6Clex'cr'a.s.«Cl->l.a"« = t'a.D"/eCCl'a [(*330-01)] 
*33011. I- :. (ga) • /r e CI ex'cr'a . = : « C 1 -> 1 : (ga) . Q"*; = I'a . s'J)"k C a 

r*3301] 
*33012. h : « 6 CI ex'cr'a . D . s'a'U = a [*330-l . *53-02] 
*33013. h : /c 6 CI ex'cr'a . D . D"k C Cl's'a"* . s'J)"k C s'a"K [*3301-12] 
*330131. H : (ga) . « e CI ex'cr'a . = . k C 1 -> 1 . a"« e 1 . s'D"« C s'Q"/^ 

[*330-ll-12] 
*33014. h :«£ CI ex'cr'a. 3. D"/eCNc'a [*330-l] 



352 QUANTiTy [part VI 

*33015. h.Clex'cr'A = i'i'A [*330-l] 

*330151. h ; a ! a . /c e 01 ex'cr'a . D . A ~ e « [*33014] 

*33016. l-:.(aa).«;6Clex'cr'a:« + i'A:D.A~e« [*330-15-151] 

*33017. l-:a!a./«:eClex'cr'a.D.D"/cCClex's'a"A; [*33013-151] 

*33018. h :.(aa).«:6Clex'cr'a:«; + t'A:D.D"«CClexVa"K [*33015ir] 

*33019. h . i\I \a)eQ\ ex'cr'a [*3301] 

*330'2. 1- : « 6 01 ex'cr'a . i? e « . g ! T>'M n s'a"K . D . a ! E | Jf 

Dem. 

h . *330-l-12 . D h : Hp . D . a ! D'M n a'B : D h . Prop 

*330-21. h : « 6 01 ex'cr'a . « 4= t'A . ii, S e « . D . a ! -K | (S 

Dem. 

h . *33018 . D t- : Hp . D . a ! B'-Sf n s'a"K (1) 

I- . (1) . *330-2 . D h . Prop 
*330-22. h : «e01ex'cr'a . K^f^ I'A. B,8,Te k .0 .jil R\S\T 
Bern. 

h . *330-21-18 . D h : Hp . D . a ! ^'(S \ T) n s'<1"k (1) 

1- . (1) . *330-2 . D h . Prop 

*330-23. h : « 6 01 ex'cr'a . K^i'k.Re k .":> . k^^e Potid'JB 
Dem. 
h.*330-16.DI-:Hp.D.a!/rO'-B (1) 

V . *33018 . D h : Hp . P 6 Potid'i? . a ! -P • ^ ■ a ' D'P n s'(1"k . 
[*330-2] D.a!-R|P (2) 

I- . (1) . (2) . Induct . D I- . Prop 

«330-3. I- : « 6 01 ex'cr'a . I[ ae k .D . kCs'k\"k 

Dem. 

I- . *3301 .Dh:.Hp.D:i?e«.D.JB = i2|/|^a:.Dh. Prop 

*330-31. h:«601ex'cr'a.i2e/c.D.E|ii = /fs'a"« [*3301] 

*330-32. h : . « 6 01 ex'cr'a . R,Se k.O :R\8= I[ s'a"ic . = . R = 8 
Dem. 

1- . *330-31 .D[-:.Rp.O:R = S.O.R\S = I[ s'a"K (1) 

l-.*330-l. DI-:Hp. . R\R\8 = (D'R)^8 (2) 

f- . (2) . D t- :. Hp . D : E I fi'= / r s'd"* . D . -B = (D'R) 1 /S . 
[*72-92] D.R = 8ia'R. 

[*330a] D.E = /S (3) 

h . (1) . (3) . D h . Prop 



SECTION b] elementary PROPERTIES OF VECTOR-FAMILIES 353 

*330-4. \-:MeK,. = .(^B,8).B,S6K.M = R\S [(*330-05)] 

*330-41. h.Gnv"K,= K, [*330-4] 

*330-42. h : K e CI ex'cr'a ./fae/e.D.KW Cnv"* C k, 
Bern. 

h . *330-l . *60-5-51 .Dh:Hp.E6«.D.i2 = (I|'a)|i2./|'a6 Cnv"« (1) 
I- . (1) . *330-4-41 . D h . Prop 

*330-43. hiKeCl ex'cr'a . D . / 1' s'Q"* e k, [*330-31-4] 

*330-5. l-:.KeAbeU=:i?,S6«:.Da,s.iJ:|-S = ,S|iJ; [(*330-02)] 

*330-51. h:/«;efm'a.s.«eAbelnClex'cr'a [(*330-03)] 

*33052. h : « e FM . = . (ga) . k e Abel r. 01 ex'cr'a . 

= .Ke Abel . K C 1 -* 1 . a"K 6 1 . s'D"k C s'a"« 
[*330-51-131 . (*330'04)] 

*330 53. h : k eFM.Q.Rex. E ! R'Q'no.O .ElQ'x.ElR'x 
Dem. 

H . *330-5 . D I- : Hp . D . E ! Q'R'x (1) 

l-.(l).*30-5.DI-.Prop 

*330-54. h : K e FM. Q, Re K. El R'x.:>. El R'Q'x 
Dem. 

\- . *330-31-52 . D h : Hp . D . ^'a; = R'Q'Q'x (1) 

H . (1) . *330-53 . D H . Prop 

*330-541. h : « 6 i^ilf . Q, i? e /«: . D . Q"l>'R C D'i2 [*330-54] 

*330-542. \-:iceFM.ReK.O. B'R e sect's'* [*330-541 . *21] -1] 

*330-55. h : KeFM-i'i'k . Q, iJe« . D . g ! D'Q r. D'E . g ! Q"D'i2 

Dem. 

h . *330-54 , D h :. Hp . 3 -.xeJ^'R . D . Q'xeTi'R : 

[*33-43] D : a ! D'iJ . 3 . g ! D'Q r. D'iJ (1) 

h . (1) . *33016 . D h : Hp . D . 3 ! D'Q n D'i2 (2) 

H . *3301116 . D h : Hp . D . D'iJ C a'Q . a ! D'E . 

[*37-43] D . a ! Q"D'i2 (3) 

h . (2) . (3) . D h . Prop 

*330-551. V : Hp *330-55 . D . g ! Q | J2 [*330-55 . *37-32] 

B. & w. III. 23 



354 QUANTITY [PABT VI 

*330-56. \- : K € FM. Q, Bex. El R'a."^. R'Q'a = Q'R'a 
Dem. 

V . *330-oll . D h : Hp . D . Q'R'R'a = R'Q'R'a . 
[*72-24] D . Q'a = R'Q'R'a . 

[*330-31 -54] D . B'Q'a = Q'R'a : D h . Prop 

*330-561. \-:KeFM.Q,ReK.O.R\Q[T>'R = Q\R [*330'56] 

*330-562. \-:KeFM.Q,ReK.'^.R>QCQ [*330-561] 

*330-563. f- : « 6 FM . ^ e k . \ C « . D . R>s'\ G s'\ [*330-562] 

*330-57. h : K 6 Abel .R,Seie .ve'NC induct . D . ^>'|^"'= (RIS)". R\S'= S-'\R 
Bern. 

t-.*30r2. :i\-.R'>\S' = {R\S)'>.R\8<> = S<'\R (1) 

I- . *330-5 .*301-21 . D I- : Hp . i? I -Sf" = -S" I J? . D . i? I -Sf+'i = 8"+''^ \ R (2) 

l-.(l). (2). Induct. Dh:Hp.D.i2|>Sf>' = S'''|ii • (3) 

I- . (3) . *301-21 . D I- : Hp . D . iJ-'+'i 1 8"+'^ = R''\8''\R\8 (4) 

I- . (4) . *301-21 .D\-:Rp.R''\S-' = (R\8y.D. ^"+"1 1 8"+--^ = (E | /SO'+'i (5) 

h . (1) . (5) . Induct . D h : Hp . D . E" I ^f" = (i? I -Sf)" (6) 
I- . (3) . (6) . D I- . Prop 

*330-6. \-:KeFM-i'i'A.LeK,.0.'3^lL 
Dem. 
\- . *330-16-4 . D h : Hp . D . (aQ, R).Q,R€K.<3ilR.L = R\Q . 
[*330-54] D . (aQ, iJ, «) . Q, E e « . E ! E'Q'a; . Z = E | Q . 

[*34-41] D . a ! Z : D h . Prop 

*33061. \-:KeFM-i'i'A.L,MeK,.:i. 

a ! a'Z r. a'iif. a i b'l n a'Jif . a ! ci'z n d'j/. a ! D'x n D'ii/ 

i)em. 
h . *330-55-4 . D 

l-:Hp.D.(aQ,-R,'S,r).Q,i?,-Sf,2'e«.i = E|Q.lf=r|S.a!D'i2'>D'2'- 
[*330-54] 

D . (aQ,ii,*Sf, r,a;) . Q, E, S, Te/c . i = ^ I Q . ilf = r| /S. E ! E'Q'a; . E ! T'8'a; . 
[*34-41] D . (a«) . E I i'a; . E ! Jf 'a; . 

[*33-43] D . a ! ci'i ft a'iif a) 

l-.(l).*330-41.DF-.Prop 



SECTION B] elementary PROPERTIES OF VECTOR-FAMILIES 355 

*330-611. \-:KeFM- I'l'k . Z. M e «. . D . g ! i | # [*330-61 . *34-3] 

«) 
*330-612. I- : « e FM- t't'A . Z, ilf, JVe «, . D . g ! Q'Z n G'ilf n Q'iV 

Z)em. 
f- . *330-22-4 . 3 
l-:Hp.D.(aP,Q,i2,;Sf,T,F).P,Q,i2,5f,r,F6«. 

L = P\Q.M=R\S.N=T\W.^\P\R\T. 
[*330-53] D . (gP, Q, R, S, T, W, x) .P,Q,R,S,T,WeK. 

L = P\Q.M=R\S.B'=T\W.'KlP'a!. El R'x. El T'a;. 
[*330-54] D . (a*) .ElL'x.ElM'x.ElN'x-.Oh. Prop 

*330-613. h : « 6 Pilf - I't'A . i, M, iVe «, . D . g ! i | Jl/| iV 

h . *330-22-4 . D 

H : Hp. D . (aP, Q,i2,fif, T, Tf.a;) . P.Q,R,S, T,W€k. 

L = P\Q.M=R\8.N=T\W. El P'R'T'a;. 
[*330-54] D . (gP, Q, iJ, S,ie).P,Q,R,SeK . 

L = P\Q.M=R\S.ElP'R'(N'x). 
[*330-54] D . (gP, Q) . P, Q e « . Z = P I Q . E ! P'{M'N'x) . 
[*330o4] D . (a*) . E ! L'M'N'x : D h - Prop 

*330 62. h : K 6 Pilf .Ze*. .)Sfe«.D.(Sf|ZGZ|(S 
Dem. 
h.*330-561.DI-:Hp.P,Qe/e.Z = P|Q.D.iSf|PGP|5f. ' . 

[*330-5] D.<SfjP|QGP|Q|S:DI-.Prop 

*330 621. Vi.KeFM- I'l'A . Z e «. . C'P C s'Q"* . g ! P : 

<Se«.Ds-'S|-PCP|/Sf:D.a!P|Z 
Dem. 

h . *330-ll .Dh:. Hp.Q,i26/«;.Z = S|i2.D: • 

a;P^ . D . (gw, ^r) . uRx . zQy . xPy . 

[*34-l] 3 . g ! i2 I P I Q . 

[*330-5] D . a ! P I E I Q . 

[*330-561] D . g ! P I Q I 22 . 

[Hp] D . a ! i* I i :■ 3 i- • Prop 

23—2 



356 QUANTITY [part VI 

*330-622. h : Hp *330-621 . D . g ! i | P 

Dem. 
h . *33011 . *72-59 .D\-:R^.Q,BeK.L = Q\R.D.PQQ\P\Q. 
[*72-59] D . P I Q G Q I P . 

[*330-621] D . a ! Q I P I i? . 

[*330-5] D . a ! Q I P I P . 

[Hp] D . a ! Z I P : D h . Prop 

*330-623. \-:KeFM.SeK.LeK,.Me Pot'i . D . >Sf | if G ilf | <Sf 

Bern. 

h . *34-34 .Dh:Hp.fi'|ilfGJIf|<S.D./Sf|il/|iGif|<S|X. 
[*330-62] D.S\M\L<1M\L\S (1) 

I- . (1) . *330-62 . Induct . D I- . Prop 

*330-624. h : /c 6 Pif - I't'A . Z, 6 «. . D . A ~ 6 Pot'i 

Pem. 
h . *330-6 . D h : Hp . D . a ! i (1) 

h . *330-622-623 . D h : Hp . Jf e Pot'Z . a ! ^- 3 ■ H ! -M"! -^ (2) 

I- , (1) . (2) . Induct . D h :. Hp . D : ilf 6 Pot'i . Dj^ . a '■ itf :■ 3 H ■ Prop 

*330-625. \-:KeFM.L,MeK,.Qe7ot'{L\M).SeK.O.S\Q(lQ\S 
Bern. 

h.*330-62.DI-:Hp.D./Sf|i|ifGZ|J/|/S (1) 

I- .>*34.-34 . D 

' h:Hp.PePot'(Z|lf)./S|PGP|/S.D./Sf|P|P|il/GP|/S'|X|il/ 
[(1)] QE\L\M\8 (2) 

I- . (1) . (2) . Induct . D I- . Prop 

*330-626. h : a: e Pi/ - I'l'A . i, ilf e «. . 3 . A ~ e Pot'(i | M) 
Bern. 

|-.*330-611. Dh:Hp.D.a!P|if (1) 

i- . *330-621-625 . D F : Hp . Q e Pot'(i \M).^\Q.:i .^\Q\L (2) 

I- . *330-625 . D I- : Hp . Q e Pot'(Z I ilf ) . >S 6 « . 3 . (S I Q I Z G Q I /Sf I X 
[*330-62] Q.Q\L\8 (3) 

h . (2) . (3) . *330-621 . D 1- : Hp . Q e Pot'(Z \M).^\Q.:i .^\Q\L\M (4) 
F-. (1). (4). Induct. Dh. Prop 



SECTION B] elementary PROPERTIES OF VECTOR-FAMILIES 357 

*330-627. \-:ic€FM-i'L'A.L,MeK,.PeFot'M.-2.<3,lP\L.'3_lL\P 
Bern. 

h.*330-611. DI-:Hp.D.a!Jlf|i.a[!i|M (1) 

h . *330-623 . Dh:Hp.Se«.D./Sf|P|iGP|S;|Z. 

[*330-62] D.8\P\L<IP\L\8 (2) 

l-.(2),*330-622. Dl-:Hp.a!P|Z.D.a!Jf|P|i; (3) 

f- . (1) . (3) . Induct . D I- : Hp , D . a ! P I i (4) 

h . (2) . *330-621 . D h : Hp . a I Z I P . D , a ! Z I P I if (5) 

h . (1) . (5) . Induct . D h : Hp . D . a ! ^ I P (6) 

H . (4) . (6) . D h . Prop 

*330-63. h : « e Pif . i, if e K, . E ! Z'a; . E ! L'M'ai . D . L'M'x = Jf 'i'a; 
-Dem. 

h . *330-5 6 . D h : Hp . Q, P, ,?, r e « . X = Q I JS . M = ^ I r . D . 

[*330-5] =8'Q'T'R'x 

[*330'56.Hp] = S'T'Q'R'x : D h . Prop 

*330-64. Vi.KeFM.L.MeK,.'^: 

E ! i'a; . E ! L'M'x . = . E ! if '« . E ! il/'i'a; [*330-63] 

*330-641. h :. « ePJf . Z, ilf 6K. . E ! Z'a; . E ! il/'a; . D : 

E ! L'M'x . = . E ! if' Z'a; . = . L'M'x = ilf 'Z'a; [*330-63-64] 

*330-642. h : K e PM - t't'A . Z, ilf 6 «e . D . (a«) . E ! Z'a; . E ! Z'Jf' a; 

Dem. 
h . *330-21 . D 

l-:Hp.D.(aP,Q,P,'S,a;).P,Q,P,/Sf6«.Z = P|Q.lf=P|iSf.E!P'P'a;. 

[*330-53-54] D . (aP, Q, P, >Sf, «) . P, Q, P, /Sf 6 « . Z = P I Q . if = P I /Sf . 

E ! P'Q'x . E ! P'Q'B'S'x : D h . Prop 

*330'643. I- : «: e Pif . P e « . Z e «. . E ! Z'a; . 3 . P'Z'a; = L'P'x [*330-56-5] 

*330-65. \-:KeFM.Q,R,S,TeK. B'Q'x = T'8'x . D . T'Q'x = P'^S'a; 
2)em. 

f- . *72'24 . D h : Hp . 3 . Q'a; = P'T'/Sf'a; 

[*330-56] = T'R'8'x . 

[*72-24] D . r'Q'a; = P'/Sf'a; : D h - Prop 



358 QUANTITY [part VI 

*330-66. \-:.iceFM.Q,R,S,TeK. 'El R'Q'x . E ! T'S'cc . D : 

R'Q'x = T'S'o! . = . T'Q'x = R'S'iE 
Bern. 

h . *330-56 . D h : Hp . T'Q'cc = R'S'x . D . T'R'Q'a; = R'R'S'x 

[*72-241] = 8'x . 

[*72-241] D . R'Q'x = T'S'a: (!) 

h . (1) . *330-65 . D h . Prop 

*330-7. h : « e I'ilf . P, Q e « . /3 € NO ind - t'O . E ! Q'(P | Qy-'^'P'x . D . 

Q'(PjQ)p-ci'P'a, = (P|Q)p'a! 
Dem. 
l-.*330-56.*301-2.D 

I- : Hp . E ! Q'(P I Qy'P'ic . D . Q'(P | Qf'P'a) = (P | Qy'x (1) 

h.*330-56.*301-21.3 

h :. Hp : E ! QX-P I Qy-'^'P'i« ■ 3a= ■ Q'(P \ Q)"-' ^'P'a; = (P | Qy'a; : 3 : 

E ! Q'(P 1 QyP'* . 3 . Q'(P I QyP'x = (P I Qy'Q'P'x 

[*330-56.*301-21] = (P | Q)''+ "a; (2) 

h . (1) . (2) . Induct . D h . Prop 

*330-71. h : /cePJkf . P, QeK . /> eNC ind - t'O . E ! P"'*. D . E ! (JP\Qy'x 

Dem. 

V . *330-54 . D I- : Hp . E ! P"x . D . E ! (P | Qy'x (1) 

I- . *301'21 . D h :. Hp : E ! Pf'x . D», . E ! {P\Qy'x : D : 

E ! Po+oi'a; . D . E ! (P | Q)'>'P'a! . 

[*330-52] D . E ! Q'(P | QyP'x . 

[*330-7] D.E!(P|Q)''+«i'a; (2) 

h . (1) . (2) . Induct . 3 h . Prop 

*330-711. }-:iceFM.Q€ s'Pot"« . D . Q'Q = s'Q"* 
Dem. 

I- . *330-62 .DhiHp.Pe/e.D. Q'P = s'a"« (1) 

h . *37-322 . D 

I- : Hp . P e « . Q e Pot'P . Q'Q = s'a"ic . D . a'(Q | P) = s'a"/«: (2) 

h . (1) . (2) . Induct . D h . Prop 



SECTION B] elementary PROPERTIES OF VECTOR-FAMILIES 359 

*330-72. \-:KeFM- I'l'A . Z, if e «. . p, <t e NO iaduct . D . a ! (I'L" n Q'if " 
Dem. * 

t-.*330-7ll-23.D 

h : Hp . P, i2e« . D . (ga) . E iR^'a-R^'aea'P' ■ 
[*330-52] 3 . (ga) . E ! P'"B"a (1) 

h . *330-57 . D h : Hp (1) . « = P''R"a . 3 . E ! P'-'a; . E ! R-^'x (2) 

h . (2) . *330-7l . D 

I- : Hp(2) . Q, SeK . Z = P| Q . ilf = ^| Sf. D . E iZP'a;. E ! Jf-'a; . 
[*33-43] D.xe a'L" n Q'ilf »• (3) 

f- . (1) . (3) . D I- . Prop 

We have " NO induct " in the above proposition, not " NC ind," because 
it is necessary to have E ! Z*" . E ! iW", and by *301'16 this may fail if either 
p or o" is null in the type of L and M. The existence of a family does not 
imply the axiom of infinity, since the family may be cyclic. 

*330-73. \- : K € FM . P,Q e K . p eNCind .El iP\ Qy-'a: . D . 

(P\Q)'"x = Pi''Qp'a! 
Dem. 

I- . *330-56 . D I- : Hp . E ! P'y . D . Q'P'y = P'Q'y (1) 

h . (1) . D 1- : Hp . Q'Po-'i'a; = Po-'^'Q'x . E ! P'"y . D . Q'P'"y = P'Q'Po-'^'y 

[Hp] =P'Pi-'i'Q'y 

[*301-23] =P'"Q'y (2) 

I- . (1 ) . (2) . Induct . D h : Hp . E ! PCy . D , Q'Pi"y = Pi-'Q'y (3) 

1- . *301-23 . D f- : Hp . (P I Q)p'« = Pp'Qo'x . E ! (P | Qy+'^'x . D . 

(P I Qy+'^'x=P'Q'P'"Q"x 

[(3)] =P'P'"Q'Q'"a; 

[*301'23] =P/>+.i'Qp+.Ka, (4) 

1- . (4) . Induct . D h . Prop 



*331. CONNECTED FAMILIES. 

Summary of *331. 

A " connected point " of a family k is a point of the field of k from which 
every member of the field can be reached by a member of k or the converse 
of a member. That is, if a is a connected point, we are to have 

X e s'(1"k . Da, . (gi?) . Re K .a;{R\j B)a 

as well as a e s'G."k. This amounts to saying that every member of s'Q."k 

is of the form R'a or R'a, where ReK. The definition is 

*331-01. conx'« = s'a"K n S, (s'«'a u s'x'a = s'a"«) Df 

Here we include the factor s'Q."k in the definition, in order to exclude 
the case when k = t'A. If s'G"k were not included, we should have 
conx'i'A = V, whereas with the above definition conx'i'A = A. 

In the case of any other family, the factor s'Q."k makes no difference, 

since if s'Q"« exists, 

— » <— 

s'K'a u s'k'u = s'(I"k . D . a 6 G's'k, 

and if K is a family, G's'k = s'Q."k. But in the case of t'A, the factor 
s'(1"k insures that no connected point exists, thus securing, conversely, that 
a family which has a connected point is not t'A- This is convenient, since 
the case of t'A, which is trivial, would often otherwise have to be explicitly 
excluded. 

The definition would be more analogous to the definition of a connected 

relation in *202 if we put 

-> <- 

conx'/c = s'Q"k a a (s'/cg'a w s'/tg'a w I'a = s'G"k) Df. 

But this definition fails to give us the information that there is a member 
of K which relates a to itself, whereas our definition does give this informa- 
tion, and hence leads to the proof that / f" s'(1"k e k, i.e. that there is a zero 
vector. 

We say that a family " is connected " when it has at least one connected 
point, i.e. we put 

*331-02. FM conx = .fif r> « (g ! conx'«) Df 



SECTION B] connected FAMILIES 361 

When all points of the field are connected points, the family " has con- 
nexity " (cf. *334-27), profided k 4= t'A. For the present, we only assume 
that at least one of the points of the field is a connected point. To 
take an illustration: the family whose members are of the form 
(Xo 1^) D (NO induct — I'O), where fi e NO induct — t'O, has only one con- 
nected point, namely 1. If we had taken positive and negative integers, 
both as multipliers and as constituting the field, we should have had two 
connected points, namely 1 and — 1. 

Almost all our future propositions on vector-families will be confined to 
connected families. In the present number, we prove first that in a connected 
family k, the vector which relates a connected point to itself also relates any 
other member of the field to itself (*331'2), whence it follows that I \ s'0."k 
is a member of k (*331'22), and that every other member of k is wholly 
contained in diversity (*331-23), and that k u Cnv"« C k. (*331"24). We 
next prove that the product of two members of k is a member of k or of 
Onv"K (*331'33). We then proceed to consider «., and prove at once the 
two fundamental properties of «, in a connected family, namely (1) that 
between any two members of s'Q."k there is a relation which is a member 
of «, (*331"4), and (2) that two members of k^ whose logical product exists 
are identical (*331*42). From these two propositions it follows that there is 
just one member of k^ whicb relates any two members of s'(I"k (*381'43). 
Finally we prove that any power of a member of « is a member of /c u Cnv"* 
(*331'54), and that any power of a member of k^ is contained in some member 
of K, (*331-56). 

Stated symbolically, the above propositions are as follows : 

«331-2. h :. « e FM . a e conx'/e . x e s'<1"k .ReK.D: R'a = a. = . R'a; = x 

*331-22. l-:«:6^Jfconx.D./fs'a"«;6« 

*331-23. h : «; e i?'Mconx . D . « C Rl'/ w Rl'J 

*331-24. h : « 6 FM conx . D . « w Cnv"/c C k^ 

*331-33. V'.Ke ^if conx . D . s'k |"« C « u Cnv"/t 

*331'4. V\Ke FM conx .x,ye s'G."k . D . (gZ) .Lsk^.x^ L'y 

*331-42. h :. K 6 JWconx .i, ilfe/e..D:a!inM. = .i = Jlf 

*331-43. Voce FM conx .x,ye s'(I"k .'2.M(MeK,. xMy) e 1 

*331-54. V-.KeFM conx . P e « . D . Pot'P C k u Cnv"«: 

*331-56. V-.KeFM conx. LeK,.Me'PQt'L.:>.{'^N).NeK,.MQ.N 



*331-01. conx'« = s'a"/ena(syaws'«'a = s'a"«) Df 
*33102. FM conx = Pjtf n ;S (g ! conx'«) Df 



362 QUANTITY [part VI 

*3311. \-:aeconx'K. = .aes'a"K.s'K'ayJs'K'a = s'a"K [(*331-01)] 

*331-11. h:.aeconx'«.= :aes'(I"«::a;es'a"K.Da,.(aK).-B6/(;.a;(iJt(iJ)a 
[*331-1] 

*33112. h : a ! coax'* . D . k + t'A [*3311] 

— » «— 
^33113. h :. K e CI ex'cr'a . D : a e conx'/c . = .«=!= I'A . s'k'u w i'«'a = s'Q."k 

Dem. 

H . *53-24 . D h : Hp . « =)= I'A . s'k'u \j s'x'a = s'QL"* . D . g ! s'«'a w sVa . 
[*330-13] D.a6s'a"« (1) 

h . (1) . *331-112 . D h . Prop 

i|e331'131. h :: /ce CI ex'cr'a. D :. aeconx'«. = : K^l'A.:a>es'(l"/c. Da,. 

(aE).ii;e«:.a!(i2t;E)a [*331-13] 

-* 
*33114. h:.\ = K\J Cnv"« . D : a e conx'/e . = . a e s'a"« . s'\'a = s'G."k 

[*3311] 

«331'2. \- :. K 6 FM . a e coax'K . X e s'G."tc. B € K . D •.B'a = a. = .R'a; = x 
Dem. 
h.*331-ll. DI-:Hp.D.(a<Sf).5feK.L(/SwS)a (1) 

l-.*330-5. :)i-:B.^.SeK.x = S'a.R'a = a.D.E'x = S'R'a 

[Hp] =iS'a 

[Hp] ^ =^ (2) 

I- . *330-56 . D h : Hp . (Sf e «. a; = /S'a . iJ'a = a . D . -R'a; = 5'JS'a 
[Hp] =S'a 

[Hp] =«; (3) 

h . (1) . (2) . (3) . D h :. Hp . D : E'a = a . D . i2'a; = a; (4) 

Similarly 1- :. Hp. D :i2'a; = a;. D .i2'a = a (5) 

h . (4) . (5) . D h . Prop 

*331-21. h:.Ke FM . a e conx'« .ReK."^: R'a = a . = . / f s'CI"« = iJ 

Dem. 

|-.*331-2. Dh:Hp.E'a = a.D./rs'a"« = E (1) 

h.*33M. Dl-:Hp./|^s'a"« = i2.D.i2'a = a (2) 

h . (1) . (2) . D 1- . Prop 

*331-22. \-:KeFM conx . D . J I' s'Q"* e « 
Dem. 

h . *331-11 . D h : Hp . a 6 conx'* . D . (ai?) .ReK.R'a = a (1) 
I- . (1) . «331-21 . D h . Prop 



SECTION B] connected FAMILIES 363 

*331 23. \-:KeFM conx . D . k C Rl'/ w Rl'J 
Dem. 

h . *331-221 .Df-:Hp.iJeK.a!jBn/.D.iJG/:DK. Prop 

*331'24. l-:«;e^Jfconx.D.«wCav"«C«:. [*330-42 . *331-22] 

*331-25. V'.KeFM conx - 1 . 3 . g ! « n Rl'J" [*331-22-23] 

*331-26. h : « 6 ^JW conx - 1 . D . s'«, s'ki ~ e k. 
Dem. 

V . *331-22-25 . D f- : Hp . D . (ga, R,8,x).R,SeK. aRa .aSoB.a^x. 
[*71-172.*41-11] D.s'/«;~6l->l. (1) 

[*331-24] D.s'/«:.~6l-*l (2) 

h . (1) . (2) . *330-52 . D h . Prop 

*331-31. h : . /c e ^ilf . a e coux'k . as e s'(I"ic . P e k . iV e k. . D : 

P'a = N'a. = .P'os = N'x 
Dem. 

h.*331-ll.*330-4.D 

\-:B.-p. 0.('a.Q,Ii,S).Q,R,S€K.iD(QKiQ)a.N = R\8 (1) 

I- . *330-5 . D 
y:-Kp.Q,R,Seic.a!=Q'a.N=R\S.P'a = N'a.D.P'x = Q'R'S'a 

[*330-56] =R'Q'S'a 

[*330-5] ^R'S'Q'a 

[Hp] =-ZV'a' (2) 

l-.*330-56.3 

\-:-a-p.Q,R,SeK.!v = Q'a.N'=R\8.P'a = N'a.:i.P'x = Q^R'8'a 

[*330-5] =1'^'?"" 

[*330-56.Hp] ^R'S'Q'a 

[Hp] =-ZV'«' (3) 

h.(l).(2).(3).DI-:Hp.P'a = i\r'a.D.P'a!=iV'a; (4) 

Similarly h : Hp . P'a; = iV'a; . D . P'a = iV'a (5) 

h . (4) . (5) . D h . Prop 
*331-32. K :. K 6 .fif conx .Pe/e.iVe/e. .D:a!Pni\/". = .P = JV 

Dem. 
I- . *331-31 . D h :: Hp . a e conx'« . D :. », y e s'a"/«; . D : 

P'a!='N'x. = .P'a = N'a. = .P'y = N'y (1) 
I- . (1) . (*331-02) . D h : . Hp . 3 : a;, 2/ e s'Q"* . P'a; = N'a!.-^. P'y = iV'i/ : 
[*33-45.*72-94] D:a!PnJV.D.P = JV (2) 

h . *33ri2 . *33016 . D I- :. Hp . D : P = iV . D . a ! P n J\^ (3) 

H . (2) . (3) . D h . Prop 



364 QUANTITY [part VI 

*331-321. h :. « e ^ilf conx .P,QeA:.D:a!PnQ. = .P = Q [*331-32-24] 

*331-33. \-:KeFM conx ."H.s'k \"k C k w Cnv"* 

n 

Dem. 
I- . *33111 . D F- :. Hp . D : (ga) :P,Qe>c. Dp,Q . (gii!) . (P'Q'a) (R^JR)a (1) 
h . *330-5 . D 

I- : Hp . P, Q, JB 6 «. P'Q'a = E'a . <S e «. y = /Sf'a . 3 . P'Q'y = /S"P'Q'a 
[Hp] = /Sf'JJ'a 

[*330-5.Hp] = B'y (2) 

h.*330-56.D 

h : Hp . P, Q, JB 6 « . P'Q'a = R'a.86ic.y = S'a.D . P'Q'y = >S'P'Q'a 
[Hp] = ^'iJ'a 

[*330-56.Hp] =R'y (3) 

h . (2) . (3) . *33M1 .D\-:Kp.P,Q,BeK. P'Q'a = R'a .D.P\Q = R (4) 
Similarly ^ -.Kp .P,Q,ReK.P'Q'a=R'a.D .P\Q = R (5) 

|-.(1).(4).(5).D 
l-:.Hp.P,Qe«.D:(aiS:):i26«::P|Q = E.v.P|Q = E:.Dl-.Prop 

*331*4. f : « e PJfconx . a;, 2/ e s'(I"/«: . D . (gZ) .LeK,.x = L'y 

Dem. 
V.^^^l^\l.^^V'.B.p.■:i.{'^a,R,8).R,8eK.x{R^aR)a.y{8\JS)a (1) 
h.*330-56.Dh:Hp.E,5(e/«:.« = P'a.2/ = /S'a.D.a; = ^'E'2/. 
[*330-4] D.(ai).i;e«:..a! = i'2/ (2) 

I- . *331 •24-33 . 3 

V:'&p.R,8eK.x = R'a.y = S'a.-:i.R\SeK,.x={R\S)'y (3) 

h.*331 •24-33. D 

V:B.-p.R,8eK.x = R'a.y = 8'a.-^.R\8eK,.x = (R\8)'y (4) 

h . *330-4 . D 

t-:Hp.P,^6«.a; = E'a.2/ = ^'a.D.^|/SeA;,.a! = (E|iS')'^ (5) 

I- . (1) . (2) . (3) . (4) , (5) . D h . Prop 

*331-41. V-.KeFM conx . D . s'/«r. = (s'Q"*) t (s'a"*) [*331-4] 

*331^42. h :. K 6 Pif conx .L,M e k,."^ -.^X L f\M . = . L = M 
Dem. 
V . *330^6 . *331-12 .Df-:Hp.Z = ilf.D.a!inif (1) 

I- . *331-4 . D 
^ :Rp . L'x = M'x .'El L'y .0 .{"^N) . N e K,. N'x = y .El L'y . 



SECTION B] connected FAMILIES 365 

[*330-63] D . (giV) .Neic,.N'x = y .L'y = N'L'x 

[Hp] . =N'M'x 

[*330-63] =M'N'x 

[*1312] ■^.L'y = M'y (2) 

Similarly V:B.^.L'x = M'x.^\M'y ."^ .L'y = M'y (3) 

h . (2) . (3) . *7l-35 .Dh:Hp.a!Z<Sil/.D.i; = Jlf (4) 

h . (1) . (4) . D I- . Prop 

*331-43. h : « 6 FM conx .x,ye s'Q."k .2.M(MeK,. xMy) e 1 
Dem. 

l-.*33r4. DI-:Hp.3.(aJl/).(Jl/e«,.a;M2/) (1) 

1- . *331-42 . D h : Hp . i, if 6 «. . xMy . xLy .D.L = M (2) 

h . (1) . (2) . D h . Prop 

*33144. \-:.K6FMconK.P,QeK.D:'3^lPnQ.= .P = Q [*331-42-24] 

*33r45. \-:. KeFM conx. L,M,N 6 K,.':): 

•3_lL\MnN. = .L\M = N[a'(L\M) 
Dem. 

H. *330611. DhiHp.Z I if = iV'['a'(i|Jf).D. a !Z| If niV^ (1) 

l-.*330-63. D \- :B.^. L'M'x = N'x. El L'M'y.XeK,.y = X'x.:>. 

L'M'y = L'X'M'x . E ! L'M'x . E ! L'X'M'x .ElX'x. 

[*330-63] D. L'M'y = X'L'M'x. El X'x. 

[Hp] O.L'M'y = X'N'x.ElX'x. 

[*330-63] D . i'Jlf'2^ = N'X'x 

[Hp] =i\^'2/ (2) 

I- . (2) . *331-4 . D H : Hp . L'M'x = N'x . ye a'(L | if ) . D . L'M'y = N'y (3) 

h . (1) . (3) . D h . Prop 

*331-46. \-:.'B.p*B31-4<5.D:M\L = N^a'(M\L). = .L\M=N\-a'(L\M) 

Dem. 
h . *330-642-63 .D\- ■.Rf.L\M=N\- a'(L \M).D. (ga;) . M'L'x = N'x . 
[*33r45] ':>.M\L = N[a'{M\L) (1) 

Similarly \-:HTp.M\L = N[a'{M\L).D .L\M=N[a'{L\M) (2) 

f- . (1) . (2) . 3 h . Prop 

*33r47. \-:KeFMconx.L,M6K,.D.('s^N).NeK,.L\MQ.N.M\LQN 
[*331-46-45-4] 

*331-48. h : K e FM . i e «i . g ! conx'w n O'i . D . i e k w Cnv"« 

h . *330-41 . D h :. Hp . a e conx'« n C'i . D : i, i e «. : E 1 i'a . v . E ! X'a : 
[*331-11] D : X, Z e «. : (g-B) : E e /c u Ciiv"/«: : i'a = iJ'a . v . Z'a = jR'a : 
[*331-24-42] D : (aii) : i2 e « w Cnv"« :i: = iJ.v.i = iE:.3h. Prop 



366 QUANTITY [part VI 

*331-5. \-:k6 FM conx . P ex . Le k,. . L\P,P\L€k, 
Bern, 
y . *331-33 . D 

h:'Rp.Q,Reic.L = Q\R.D.('^8).SeK^jCnv"K.L\P=Q\S (1) 

h.*330-4.Dh:Hp(l).^f6«.i|P = Q|/S'.D.i|P€K. (2) 

f- . *34-2 . D 

\-:B.^{l).SeCm"K.L\P = Q\8.D.i'^T).TeK.L\P = Cnv'(T\Q). 
[*331-33] D.Z|Pe«wCnv"«. 

[*33r24] D.L\Peic, (3) 

h- . (1) . (2) . (3) . *330-41 . D h . Prop 

*331-51. h : K e FM conx . P e « . D . Pot'P C «. [*331-5 . Induct] 

*33r52. \-:KeFMcons..P,QeK.LeK,.O.P\L\QeK, [*331-5] 

*331-53. \-:k6 FM conx . P, Q e k . /a, o- e NO induct . . P" \ Q' e k, 
[*331-5 . Induct . *331-51 . *330-43] 

*331-54. h-.KeFM conx , P e «. D . Pot'P C « o Cnv"A: 
Dem. 
h . *330-711 . D h : Hp . a e conx'« . Q e Pot'P . D . E ! Q'a . 
[*:331-11] D . (gT) . 2' e « w Cnv"« . Q'a = 2"a . 

[*331-51-42-24] D . Q e « w Cnv"« : D I- . Prop 

eie331'55. I- : « e PM conx .P,Qeic,.pe NC induct . D . 

(P I Q)" G P" I Q" . P" I Q" 6 «. [*330-73 . *331-53] 

*331-56. h : « ePJf conx .Leic, .MeVot'L . D . ('^N) .NeK^.MGN 
[*331-55 . *330-4] 



*332. ON THE REPRESENTATIVE OF A RELATION IN A FAMILY. 

Summary of *332. 

We saw at the end of the last number (*331"56) that any power of a 
member of k. is contained in a member of «,. When a relation is contained 
in a member of «„ we call this member the " representative " of the relation 
in the family. For purposes connected with the application of ratio, the 
" representative " is an important function of a relation, especially when the 
relation concerned is a power of a member of «,. By the definition of ratio 
(*.3b301), we shall have L (p/a) Jlf if g ! i'' n Jlf " and p Prm o-. Now if i' 
and M'' each have a representative, then they must have the same representative 
if g ! X"^ n M" (by *331'42). Hence we are enabled to substitute an equality 
for '^l L' r\ Ml" in dealing with ratios of members of «,. The elementary 
properties of representatives are dealt with in the present number. 

We denote the representative of P in the family k by " rep^'P." In order 
to insure E ! rep,'P under all circumstances, we do not define rep.'P as the 
only member of «t which contains P, but as the logical sum of the class of 
members of «. which contain P, i.e. we put 

*33201. rep/P = sV' '^ G'P) Df 

*— 
In a connected family, if P is not null, «i r\ d'P cannot have more than 

one member (*332"21), and therefore the representative of P, if it is not null, 

must be a member of «, (*332"22). If P is a member of k^, it is its own 

representative (*332-241). 

We prove in this number that, if P, Q, R,... have existent representatives, 
the representative of their relative product (unless this product is null) is 
the representative of the relative product of their representatives (*332-37). 
Among other important propositions in this number are the following : 

*332-32. I- : K e FM conx . Z, Jf e «. . 3 . rep/(Z | M) = rep/(Jlf | L) 

*332-51. y-.KeFM conx . P, Q e « . D . rep/(P | Q) = Q | P 

*33253. h-.Ke FM conx . P, Q e /e . p e NC induct . D . rep/(P | Q)" = P" | Q" 

*332-61. \- : Ke FM conx . i e k. . D . rep^c^'Potid'^ C «. 



368 QUANTITY [part VI 

*332-8. h-.KsFM conx . i, ilf e «, . ^ e NO ind . D . 

rep/(Z|M)^ = rep/(if|ifO 

*332-81. h : « 6 ^Jf conx . v, a- e NO ind - t'O . Z e «, . D . 

rep^'i''^^""' = rep^'(rep^'X-)'' 



*33201. rep/P = s'(«,nC'P) Df 

*332-l. h . rep/P = s'(«. r,*C'P) = ^ {(gi) .Lek^.PQL. xLy] 
[(*332-01)] 

*33211. h : a ! rep/P .':i.PQ. rep/P [*332-l] 

*33212. h : a ! rep/P . D . g ! («, n^'P) [*3821] 

*33213. h . rep/A = s'/e. [*332-l] 

*33214. h : P e Q . D . rep/Q G rep/P [*3321] 

*33215. h . rep/P = Cnv'rep/P 

h . *330-41 . D h . «, n G'P = Onv"(«. n^'P) (1) 

I- . (1) . *332-l . D I- . Prop 

*33216. h : « = I'A . 3 . rep/P = A [*3321] 

*332-2. Vi.KeFM- I'l'k . D : g ! («, n Q.'P) . = . g ! rep/P 
i)em. 

h . *380-6 . D h : Hp . a ! («. n V'P) . D . g ! (k, r^'o.'P) - I'A . 
[*332-l] D . a ! rep/P (1) 

h . (1) . *33212 . D h . Prop 

*332-21. h : K e PM conx . g ! P . D . («. n G'P) e u 1 

h . *331-42 .Dh:Hp.L,il/6«,.PGi.PGif.D.i = ilf:Dl-. Prop 

*332-22. hz.Ke FM conx . g ! P . D : rep/P ek^.v. rep/P = A 
jDem. 

h . *332'21-12 . D h : Hp . a ! rep/P . D . («, n^'P) e 1 . 

[*3321] D . rep/P e «. : D h . Prop 

<— 
*33223. I- :. keFM conx .±IP.D: rep/P e «. . = • g ! («. n G'P) 

h . *332-22-2 . D h : Hp . rep/P ~ e «. . D . («. n^'P) = A (1) 
I- . *330-6 . D 1- : Hp . rep/P e «, . D . a ! rep/P . 

[*332-2] D.a!(«.AG''P) (2) 

h . (1) . (2) . D h . Prop 



SECTION B] on the REPRESENTATIVE OF A RELATION IN A FAMILY 369 

*332-231. hi./ceiWconx-l .D:rep/PeKi. = .a!P.a!(«;.n G'P) 
Dem. 

f- . *331-26 . D t- :. Hp . D : rep/P e k, . D . rep/P =t= s'lc, . 
[*332-13] D . P + A (1) 

I- . (1) . *332-23 . D I- . Prop 

*332-232. h :. « 6 ^ilf conx - 1 . D : rep/P e «;. . = . g ! P . g ! rep/P 
[*332-231-2] 

*332'24. VuKeFM conx . g ! P . 3 : X e («, n G 'P) . = . g ! rep/P . rep^'P = L 

Dem. 
I- . *332-21-l . DI-:.Hp.D:i6/<:.n^'P.D.rep/P = Z (1) 

l-.*332-2. Dh:.Hp.D:i6/«;,ne'P.D.a!rep/P (2) 

V . *332-22 . D h :. Hp . D : a ! rep/P . D . rep/P e k. : 
[*13-12] D:a'.rep/P.rep/P=L.D.Z6K. (3) 

h . (3). *33211 . D 1- :. Hp . D : a ! rep/P . rep«'P = i . D . L 6(«r. n G'P) (4) 
h . (1) . (2) . (4) . D h . Prop 

*332-241. h-.iceFM conx . P e «. . D . P = rep/P 
Dem. 

f- . *332-24 .DI-:.Hp.a[!P.D:P6«:.nG'P. = .a! rep/P . rep/P = P : 

[Hp] D:rep/P = P (1) 

l-.*330-6. Df-:Hp.~a:!P.D./e = t'A. 

[*332-13] D.rep/P = A (2) 

I- . (1) . (2) . D 1- . Prop 
*332-242. hzKe FM conx . g ! P . g ! rep/P . D . rep/P = rep/rep/P 

i)em. 

I- . *332-22 . D I- : Hp . D . rep/P e «, (1) 

t- . (1) . *332-241 . 3 h . Prop 

*332-243. h-.KeFM conx . g ! P . P G 7 f s'a"« . D . rep/P = 71' s'tt"* 
[*332-24 . *330-43] 

*332-244. h :. K 6 Pilf conx - 1 . D : 

a ! P . P G 7 1^ s'a"K . = . rep/P = 7 ^ s'a"« 
7)em. 
h . *331-26 . *330-43 . D h :. Hp . D : »'«, + 7 p s'a"« : 

[*33213] D:rep/P = 7r5'a"«.D.a!P (1) 

|-.*33211. Dl-:.Hp.D:rep/P = 7rs'a"«.D.PG7rs'a"« (2) 

h . (1) . (2) . *332-243 . D f- . Prop 

H. &w. III. 24 



370 QUANTITY [part VI 

*332-25. \-:k6FM conx . g ! P . g ! rep/Q .PdQ.D. rep/P = rep/Q 

Dem. 

h . *332-ll . 3 h : Hp . 3 . P e rep/Q (1) 

b . *332-22 . D 1- : Hp . D . rep/Q e k, (2) 

h . (1) . (2) . *332-24 . D 1- . Prop 

*332-26. f- : « 6 PMconx . g ! P n Q . g ! rep/P . g ! rep/Q . D . 

rep/P = rep/Q = rep/(P n Q) [*332-26] 

*332-27. h : K e Pilf coax . a ! P . a ! rep/Q . g ! Q n rep/P . D . rep/P = rep/Q 

Dem. 

h . *332-ll . 3 h : Hp . D . Q G rep/Q . 

[Hp] 3 . a ! rep/P n rep/Q (1) 

h . *332-22 . D h : Hp . D , rep/P, rep/Q e k, (2) 

h . (1) . (2) . *331-42 . 3 h . Prop 

*332-31. h : /c e PM conx . Z, M e k. . 3 . rep/(i \M)eK, 
[*330-611 . *331 •47-12 . *332-23] 

*332-32. h : « e FM conx .L,MeK,.'D. rep/(Z | ilf ) = rep/( if | L) 
[*330-611 . *331-47-12 . *332-24] 

«332-33. I- : /t e Pif conx . rep/P, rep/Q e «. . g ! P | Q . 3 . rep/(P | Q) 

= rep/{(rep/P) | (rep/Q)} = rep/{(rep/P) | Q} = rep/{P | rep/Qj 
Dem. 
h . *330-6 . *331-12 . 3 h : Hp . 3 . a ! rep/P . g ! rep/Q . 
[*332-ll] 3.PGrep«'P.QGrep/Q. (1) 

[Hp] 3.a!P|rep/Q (2) 

l-.*330-6.*332-31..(l).3 

h : Hp . 3 . P I rep/Q G rep/P | rep/Q . g ! rep/{rep/P | rep/Q} . 
[(2).*332-25] 

3 . rep/(P I rep/Q) = rep/{rep/P | rep/Q} . g ! rep/(P | rep.'Q) (3) 
Similarly h : Hp . 3 . rep,'{(rep/P) | Q} = rep/{(rep/P) | (rep/Q)} (4) 

l-.(l).3 l-:Hp.3.P|QGP|rep/Q. 

[Hp.(3).*332-25] 3 . rep/(P | Q) = rep/(P | rep/Q) (5) 

h . (3) . (4) . (5) . 3 h . Prop 
*332-34. l-:Hp*332-33.3.rep,'(P|Q)e/«;. [*332-3r33] 

*332-35. \-:KeFMcoQx..L,M,NeK,.D. 

rep/(i \M\N) = rep/{Z | rep/(if | iV)} = rep/[{rep/(Z | M)} | iV] 
[*330-613.*332-31-33] 
*332-36. h :Hp*332-35. 3. rep/(i; I if I iV) 6 «. [*332-35-31] 



SECTION B] on the REPRESENTATIVE OF A RELATION IN A FAMILY 371 

*332-37. i-ZKeFM oonx . rep/P, rep/Q, rep/i? eK,.'3^\P\Q\R.O. 
rep^^P I Q I -R) = rep/{rep/P | rep/Q | rep/i2} 
= rep/{rep«'P j rep/i? | rep«'Q} 
= rep/{rep,'Q | rep,'i? | rep/P} 
Dem. 

y . *332-38 . 3 

I- : Hp . D . rep/(P | Q | i?) = rep/[rep/P | rep/(Q | R)] 
[*332-33] =rep/{rep,'P|rep/(rep/Q|rep/iJ)} (1) 

[*332-35] =rep<'{rep/P|rep/Q|rep,'i?} (2) 

h . (1) . *332-32 . D 

I- : Hp . D . rep/(P | Q \ R) = rep/{rep/P | rep/(rep/P | rep/Q)} 
[*332-35] =rep/{rep/P|rep/i2|rep/Q} (3) 

> . (1) . *332-33-32 . D 

h : Hp . D . rep/(P| Q\R) = rep/[{rep/(rep/Q | rep/iJ)} | rep/P] 
[*332-35] =rep/{rep/Q|rep/P|rep/P} (4) 

I- . (2) . (3) . (4) . D h . Prop 

*332-41. l-i./eeP^conx.Z.JIf.iVeK. .D: 

rep/(Z I M) = rep/(i | i\^) . = . Jlf = iV 
J5em. 

I- . *34-34 . D 1- : Hp . rep/(i: | M) = rep/(Z | iV) . D . 

L I rep/(i I Jlf ) = i I rep/(L | N) . 

[*332-35] D . rep/(i | i | if ) = rep/(i | £ | iV) . 

[*330-31] D . rep.'Jf = rep.'iV . 

[*332-241] O.M = N::>\-. Prop 

*332-411. hz.KeFM conx .L,M,N€k..D: rep/(ilf | i) = rep/(i\^ | £) . s . M = iV 
[*332-32-41] 

*332-42. \-:KeFM conx . i, M e «. . D . CQv'rep/(Z | M) = rep/(2 [ M) 
[*332-32-15] 

*332-43. h:.KeFMconx.L,M,NeH:,.0: 

N = rep/(X \M). = .L = rep/(iV \M). = .L^ rep/(Jtf | JV) . 

= .M= rep/(iV I i) . = . il/ = rep/(i | N) 
Dem. 
h . *332-35 . *330'41 . D 

h : Hp . iV = rep/CZ | i/) . D . rep.'(i | if j M) = rep/(iV | M) . 

[*330-31] D . rep/Z = rep/(iV | M) . 

[*332-241] D.L = rep/(N'\M). (1) 

[*332-32.*330-41] D.L = rep/(^ 1 iV) (2) 

H . (1) . *330-41 . D f- : Hp . i = rep/(iV| M) . D . iV= rep/(Z/ 1 if ) (3) 

h.(l).(2).(3).Dl-.Prop 

24—2 



372 QUANTITY [part VI 

*332-44. \-:.KeFMconx.L,M,N€K,.D:iep,'{L\M) = N. = .L\MQN 
[*330-6 . *332-24-31] 

*332-45. h :. Hp *332-44 . D : rep/(X \M) = N. = . rep/(i \M\M) = I ^s'a"K 
Dem. 

V . *332-35 . D h : . Hp . D : rep/(Z | J/) = iV" . D . rep/(Z \M\N) = rep/(iV | N) 
[*332-24.*330-31] =I\s'G."k (1) 

I- . *332-35 . 3 I- :. Hp . D : rep.'(i; \M\ N) = I\s'a"K . D . 

rep/[{rep/(i: | if)} j i/'] = / T s'a"/c . 
[*332-31-43] D.rep/(Z|if) = rep/iV 

[*332-241] =iV (2) 

h . (1) . (2) . D t- . Prop 

*332-46. h :. « e FM conx . L,M e k,.D : L\M dl .= . L = M 
Dem. 

h . *330-43-611 . *332-243 . D 

l-:Hp.Z|MG7.D. rep/(i | il/) = / T s'<^"« ■ 
[*332-43.*330-43] D . Z = rep/^ 

[*332-241.*330-41] =i^ (1) 

|-.*rri91.Dh:Hp.Z = if.D.i!JlfG/ (2) 

F- . (1) . (2) . D h . Prop 

*332-51. h : /c 6 J?Wconx . P, Q e « . D . rep/(P \Q) = Q\P 
Dem. 

I- . *331-24 . *332-32 . D h : Hp . D . rep/(P | Q) = rep/(Q | P) 
[*332-241] =Q|P:DI-.Prop 

*33252. h: ice FM conx. P,Q,R,8eK.0.vep/{P\Q\R\S) = Q\S\P\R 
Bern. 

h . *330-613 . *331'12-124 . D h : Hp . D . g ! (P | Q) | (P | ;S) . 

[*332-33-51] D.rep/(P|Q|P|^) = rep/(Q|P|,SiP) (1) 

h . *330-561-611 .Df-:Hp.D.Q|P|S|PGQ|,S|P|P.a[!Q|P|^|P (2) 

l-.*331-52. D(-:Hp.D.Q|is|P|P6/e, (3) 

h , (1) . (2) . (3) . *332-24 . D h . Prop 



SECTION B] on the REPRESENTATIVE OF A RELATION IN A FAMILY 373 

*332-53. h-.KsFM conx . P, Q e « . p e NC induct . D . rep/(P | Q>' = P" | Q" 
Bern. 

H.*330-624.Dh:Hp.D.a[!(P|Q>' (1) 

H.*330-73. Df-:Hp.D.(P|Q)pGP''|Q'' (2) 

h.*331-53. Dh:Hp.D.P''|Q''6«. (3) 
h . (1) . (2) . (3) . *332-24 . D f- . Prop 

*332-61. \-:KeFM conx . i e /e, . D . rep/'Potid'i C «. 
Dem. 

h . *332-243 . *330-43 . D f- : Hp . D . rep/(/ f C'L) e «, (1) 

h . *332-31 . D h : Hp . Jf 6 Pot'Z . rep/M e k. . D . rep/fiS | rep/if } e «. (2) 
h . *330-624 . D h : Hp . Jlf e Pot'i . D . g ! Z | Jlf (3) 

f- . (2) . (3) . *332-83 . D h : Hp (2) . D . rep/(i | if ) e «. (4) 

h . (1) . (4) . Induct . D I- . Prop 

*332-62. h:«6J^ilfconx.A~6Pot'P.a!rep/P.D. 

rep/'Pot'P C rep/'Pot'rep.'P 
Dem. 

h . *332-242 . D h : Hp . D . rep^'P = rep/rep/P (1 ) 

|-.*332-22. Dh:Hp.D.rep/P6«. (2) 

h . (2) . *332-61 . D 

I- : Hp . Q 6 Pot'P . rep/Q e rep/'Pot'rep/P . D . rep/Q e k. (3) 

h . *91-36 . D h : Hp . Q 6 Pot'P . D . g ! P | Q (4) 

I- . (2) . (3) . (4) . *332-33 . D f- : Hp (3) . D . rep/(P | Q) = rep/{rep/P | rep/Qj . 

[Hp.*91 -36] D . rep/(P | Q) e rep/'Pot'rep/P (5) 

h . (1) . (5) . Induct .31-. Prop 

*332-63. h : Hp *332-62 . D . rep/'Pot'P C k, 

Bern. 

h . *332-22 . D f- : Hp . D . rep/P e «. (1) 

I- . (1) . *332-62-61 . D f- . Prop 
*332-64. V-.KeFM conx . rep/'Pot'P C k. . 3 . rep/'Pot'P C rep/'Pot'rep/P 

h.*331-26.*33213.DI-:Hp.«~el .D.A~6Pot'P (1) 

V . *330-6 . *331-12 . D h : Hp . 3 . A ~ e rep/'Pot'P (2) 

K . (1) . (2) . *332-62 . D h : Hp . «~e 1 . D . rep/'Pot'P C rep."Pot'rep/P (3) 
h . *330-43 . *331-22 . D h : Hp . k e 1 . D . «. = l'{I [ s'tt"*) = k (4) 



374 QUANTITY [PABT VI 

l-.(2).(4). *33212. DI-:Hp (4). D.PG/ 1' s'a"«:. (5) 

[*332-243-13.(4)] D . rep/P = / f s'Q"* (6) 
h . (5) .*301-3 . D h : Hp(4) . D . Pot'P = i'P . 

[(6).*332-241] D . rep/'Pot'P = I'rep/rep/P (7) 
h . (3) . (7) . D h . Prop 



1-65. h : A ~ e Pot'P . g ! rep/P . 3 . Pot'P C s'Rl"Pot'rep/P 
Dem. 
h.*332-ll.Dh:Hp.D.PGrep/P (1) 

h . (1) . D h : Hp . Q 6 Pot'P . R e Pot'rep/P .QGP.D.Q|PGi?| rep/P (2) 
h . (1) . (2) . Induct . D h . Prop 

*332-66. h : a ! rep/P . i? e Pot'rep/P . D . (gQ) . Q e Pot'P .QQR 
[Proof as in *332-65] 

*332-67. \-:KeFM coax . A ~ e Pot'P . g ! rep/P . D . 

rep/'Pot'rep/P = rep«"Pot'P 
Dem. 
h . *332-242 . D h : Hp . D . rep/rep/P = rep«'P (1) 

l-.*332-66. DI-:.Hp.D:i?ePot'rep/P.D.a!P|P (2) 

h . *332-22 . D h : Hp . D . rep/P e k, (3) 

h . (3) . *332'61 . D h :. Hp . D : i? 6 Pot'rep/P . D . rep/JB e /r. (4) 

h.(2).(3).(4).*332-33.D 

h :. Hp . D : P 6 Pot'rep/P . D . rep/(rep/P | rep/P) = rep/(P | rep,'P) (5) 
h . *332-33 . D h : Hp . P 6 Pot'rep/P . Q e Pot'P . rep/P = rep/Q . 3 . 

rep/CQ I P) = rep«'(rep/P | rep/P) 
[(5)] =rep/(P|rep/P) (6) 

h . (6) . D F- : Hp . P e Pot'rep/P . rep/P e rep/'Pot'P . D . 

rep/(P|rep/P)6rep/'Pot'P (7) 
f- . (1) . (7) . Induct . D h : Hp . 3 . rep/'Pot'rep/P C rep/'Pot'P (8) 

h . (8) . *332-62 . D h . Prop 

*332-71. \- : Ke FM conx .L,MeK,.'^. 

rep/'Pot'(i I M) = rep/'Pot'rep/(i | M) 
Dem. 

h.*330-626. DI-:Hp.D.A~6Pot'(Z|il/) (1) 

h ■ *332-31 . *330-6 . D h : Hp . 3 . g ! rep/(Z | M) (2) 

h . (1) . (2) . *332-67 . D I- . Prop 



SECTION B] on the REPEESENTATIVE OF A RELATION IN A FAMILY 375 

*332-72. h : Hp *332-7l . D . rep,"Pot'(X | M) C «, [*332-31-61-7l] 

*332-73. \-:KeFM conx .L.Meic,.'^. Pot'(Z | M) C s'Rl"Pot'rep/(X | M) 
[*332-65-31 . *330-626] 

*332-74. \-:KeFM conx .L,MeK,.Pe Fot'M . 3 . 

rep/(Z I P) = rep/(P | L) = rep/(Z | rep/P) 
Dem. 

h . *330-627 . *332-61-33 . D 

h : Hp . D . rep/(i | P) = rep/{i | rep/P} (1) 

[*332-61-32] =rep/{rep/P|Z} 

[*330-627.*332-61-33] = rep/(P | L) (2) 

h . (1) . (2) . D h . Prop 

*332-75. h : Hp *332-74 . D . g ! rep/(i ! P) [*332-74-61-31 . *330-6] 

*332-8. I- :« ei^iW conx. X,Jfe«:..|eNCind . D. 

rep/(X I My = rep/(Z« | ilfO 
Dem. 

I- . *332-243 . D 

I- : Hp . I = . D . rep/(Z | ilf )f = / p s'a"« = rep/(i^ 1 Mi) (1) 

h . *301-21 . *332-83 . *330-626 . 3 

h : Hp . rep/(Z I Jf )« = rep/(Z* | M^) • 3 • 

rep/(i i Jf )«+•! = rep/{i^ \Mi\L\M} 
[*332-37] =rep/{Z«|rep/(il/^|X)|Jlf} 

[*332-32-33] = rep/ji* | rep/(i | Jl/^ | -3^ } 

[*332-37] =rBp,'{Zf+ci|^^+"'} (2) 

H . (1) . (2) . Induct . D I- . Prop 

*332-81. h : « e FM conx . v, o- e NO ind - t'O . X e k. . D . 

rep/Z-"*"" = rep«'(rep,'Z'')' 
Dem. 

h . *301-23 . 3 h : Hp , rep/X'"^"'' = rep/Crep/i")' . D . 
rep/i'"'°'°"+°i' = rep/(Z'"'"'' | L") 
[*332-33] = rep/iCrep/Z-)"" | rep/i-} 

[*301-23] =rep/(rep/Z'')''+«i (1) 

h . (1) . Induct . D h . Prop 

*332-82. hi/cePJfconx.i/eNCind-l'O.i.Jfe/e. .3 . 

rep/(X I My = rep/{rep/(i | i/)}" 
Dem. 
h . *332-33 . 3 h : Hp . rep/(i | My = {rep/(Z \ M)}" . 3 . 

rep/(Z I My+'' = rep/[{rep/(Z I ilf )}" 1 rep/(Z | M)] 
[*301-23] =rep/{rep/(Z|ilf)}''+'i (1) 

h . (1) . *113-621 . *301-2 . Induct . 3 h . Prop 



*333. OPEN FAMILIES. 

Summary of *333. 

An " open " family is defined as one such that, if L is any member of Ki 
which is not contained in identity, then every power of L is contained in 
diversity, i.e. L^ G J. We shall often have occasion, both in this number 
and later, to consider the class k^ — Rl'/, and in later numbers we shall often 
have occasion to consider the class k — Rl'/. We therefore put 

*33301. k9 = a:-R1'/ Df 

*333-011. «.a = («,)a Df 

Thus K^g consists of all members of k^ which are not contained in identity, 
i.e. (if «: is a connected family) all members of k^ except I \ s'(1"k. The 
definition of an "open" family is 

*33302. ^ifap = ^ifn;i{s'Pot"«,gCRl'J} Df 

From the point of view of the application of ratio, the hypothesis that 
a family is open is very important. To begin with, it insures (*333"18) 
that K,g consists of "numerical" relations (cf. *300), so that if Xe^^g, we 
have Pot'i = fin'i (*333-15), and in virtue of *300'491, the existence of 
open families implies the axiom of infinity (*333"19). 

Again, in an open connected family, if L, M are two different members 

of Ki, all the powers oi L\M are contained in diversity, and therefore the 
representatives of these powers are members of Kjg ; that is, we have 

*333-22. V-.Ke FM ap conx .L,M sk^.L^M ."^ . rep/Tot'(Z | M) C /c^g 

It follows from this proposition that, with the above hypothesis, if o- is 

any inductive cardinal other than 0, L' \ M" is not contained in identity, and 
therefore L" ^ M" and rep^'X" =j= rep^'ilf". Hence by transposition we obtain 
the two propositions : 

*333-41. I- :. /c 6 FM ap conx . i, il/ e k. . o- e NO ind - I'O . D : 

rep/Z" = rep^^'ilf" .=..L = M 



SECTION B] OPEN FAMILIES 377 

*333-42. h :. Hp *333-41 .D : L' = M" . = . L = M 

Hence we obtain • 

*333-43. J- :. Hp*333-41 .::> z'g.l L' n M' . = . L = M 

This proposition shows that in an open connected family, no two 
members of «. have the ratio 1/1 unless they are identical. Again it 
follows from *333-41 that if L"^''' and M'^''^ have the same representative, 
then Z*" and M" have the same representative, and vice versa, i.e. 

*333-44. \-:.K6FMa.p conx . L,MeK, . p, o-, t e NC ind - t'O . 3': 

rep^'Z'"^'"' = rep^'if'"''"" . = . rep^'Z'" = rep^'Jl/" 

Hence we obtain two propositions which are vital for the application of 
ratio, namely : 

*333-47. h :. « 6 ^ilif ap conx . Z, JWe k. . p, o- e NO ind - t'O . D : 

rep^'Zp = rep^'M' . s . g ! Z*" n ij/" 

*333-48. h :. « e ZMap conx . L, M e k,. p, <t,t e^G ind - I'O . D : 

On comparing this last proposition with the definition of ratio (*303'01), 
it will be seen that, whether p is prime to o- or not, Z has to M the ratio a/p 
when, and only when, g ! Z*" A Jf "•, i.e. (by *333'47) when, and only when, 
rep^'Z*" = rep^'ilf". 

From *333-47 it follows also that, if Jlf e K,g, M^ and M' will not have the 
same representative unless p = a (*333"51), i.e. 

*333'51. I- :. « e FM aTp conx . if e /c^g . p, cr e NC ind . D : 

reTp^'Mf = rep^'ilf' . = .p=a 

From this it follows that no member of /e^g has any other ratio to itself 
than 1/1. Again, by *333-47-48'51, we have 

«333-53. 1- : K eiW ap conx. Z, if e «,9. a iZ'nilf''. a iZ'n if". D. 

fiXea-=vXgp 

Hence if Z and M have the two ratios p/a, p,/v, we have p/a = p,/v ; that 
is, no two members of /c^g have more than one ratio. 

The applications of ratio indicated in this summary will not be made till 
the following Section ; they are here mentioned in order to show the utility 
of the propositions of the present number. 



*33301. «:g=K-Rl'/ Df 

*333-011. /c.g = (Og Df 

*333 02. FMap = FMf\1c {s'Pot"«:.9 C Rl'J} Df 

*33303. FM ap conx = FM ap n FM conx Df 



378 QUANTITY [part VI 

*3331. h -.Me K^s. = .{^P,Q) . P,Qe K . M^P\Q .±1 M nj . 

= .MeK,.^\MhJ _ [(*333-01-011)] 

*333-101. \-'..KeFMa.^.= :KeFM:M6K,y.PeVot'M.0M.P'P(i.J: 

= :/ceFM:MeK^S.-^M-M^^(lJ [(*333-02)] 

*33311. \-:KeFMa.Tp.LeK^Q.':>.LQ.J.L^(lJ.Lr,L = A.L^L.'3^lL 
[*3331101] 

*333-12. \-:iceFMa.p conx . g ! rep/P . g ! P n J . D . 

rep/P 6 K,9 . (rep,'P)po G J 
Dem. 

V . *332H . D h : Hp . D . a ! rep/P n J". 

[*332-22.*333-l] D . rep.'P e /c^g (1) 

h . (1) . *333-101 . 3 h . Prop 

*333-13. I- : KePilf apconx . g !rep/P . g ! P n J". D . Ppo G/ 

Dem. 
I-.*33211. DI-:Hp.D.PGiep/P (1) 

h . (1) . *332-22 . D h : Hp . D . a ! (rep/P) -S J . P,„ G (rep/P)p„ . rep/P e «, . 
[*333-l] 3 . Ppo G (rep,'P)po . rep/P e /..g . 

[*333-101] D . Ppo G / : D h . Prop 

*333-14. h : « ePilf ap conx .L.Mex,. Lj=M .D .(L\ M)^<IJ 
Bern. 

F.*330'626. DI-:Hp.D.A~ePot'(£|ilf) (1) 

h . *332-31 . *330-6 . 3 h : Hp . D . g ! rep.'(i | M) (2) 

h . *332-46 . Transp • D h : Hp . D . g ! (i | if ) n / (3) 

h . (1) . (2) . (3) . *33313 . D h . Prop 

*333-15. h : « e Pi(/ ap . i e K^g . D . Pot'i = fin'X = finid'i - I'Zo 
[*121-501 . *33311-101] 

*33316. h iKeFM&^conx.LyMeic.L'^M.Ii. 

Pot'CI I ilf) = fiii'(i I M) = finid'(Z I M) - i'{L \ M\ 
[*121-501.*33314] 

*33317. h : K ePilf ap conx . g ! rep/P . g ! P n ./". D . 

Pot'P = fin'P = finid'P - t'P„ [*121-501 . *333-13] 

*33318. h : « e Pilf ap . D . «:.9 C Rel nam [*333-101 . *300-3] 

*33319. hi/cePifap-t'i'A.D.Infinax [*333'18 . *330-624 . *300-491] 

*333-2. h:a!Pilfapconx.D.Infinax [*333-19 . *331-12] 



SECTION B] OPEN FAMILIES 379 

*33321. I- : « 6 JPJl/ap conx . Z e «.g . D . rep,".Pot'X C /c,g 
Bern. « 

h . *332-61 . D I- : Hp . D . rep/'Pot'i C «. (1) 

I- . *333101 . *330-624 .Dh:.Hp.D:A~e Pot'Z . Pot'i C Rl'J" : 
[*332-ll.(l)] Drilferep/'Pot'i.D.alilfnJ' (2) 

H . (1) . (2) . *333-l . D h . Prop 

*333-22. h : « e ^M ap conx . Z, if e «. . Z 4= iW" . 3 . rep/'Pot'(i | M) C /c^g 
-Dem. 

h . *332-n . D f- : Hp . D . rep/'Pot'(Z | M) = rep."Pot'rep/(Z '\ M) (1) 

h . *332-46-ll-232-31 . D h : Hp. D . rep/(Z | M) e k^ (2) 

h . (1) . (2) . *333-21 .31-. Prop 

«333-23. H : « 6 FJl/ ap conx . A ~ e Pot'P . g ! rep,'P .'3^\P hj ."H . 

rep/'Pot'PC«,g 
i)em. 

l-.*332-62. bl-:Hp.D.rep/'Pot'PCrep,"Pot'rep/P (1) 

h . *332-ll-22 . *333-l . D I- : Hp . D . rep/P e K.g (2) 

I- . (1) . (2) . *333-21 . D t- . Prop 

*333-24. h : K 6 Pif conx . A~6 Pot'P . g ! rep.'P. i- e NO ind . g ! i; n i^'i? . D . 

rep.'P' = rep/(rep/P)- 
Dem. 

I- . *301-2 . *332-243 . D I- : Hp . D . rep««P» = I\ s'Q."k = rep/(rep,'P)» (1) 

h . *332-63 . *330-6 . *301 •16-22 . D 

I- : Hp . D . rep/P", rep/P e k, . g ! P'+^i . (2) 

[*301-21.*332-33] D . rep/P'+^i = rep/{(rep/P'') | rep/P} (3) 

I- . (2) . (3) . D h : Hp . rep/P' = rep/Crep/P)- . D . 

rep/P'+^i = rep/{rep«'(rep/P)'' | rep^'P} . 

rep/( rep/P)", rep«'P e k. (4) 

h.(2).*330-624.*301-21.3F:Hp.D.a! (rep/P)- 1 rep/P (5) 

h . (4) . (5) . *332-33 . D I- : Hp (4) . D . rep/P""!-"! = rep/{(rep/P)" | rep/P} 
[*301-21] =rep/(rep/P)"'+"i (6) 

h . (1) . (6) . Induct . D h . Prop 

A hypothesis equivalent to i/ e NO ind . g ! v r> P'R is v eG'U^ f'R. It 
is sometimes convenient to substitute this for the other. 

*333-25. h : « e FM conx . Z, if e /e^ . w e NC ind . g ! x; n f'L . D . 

rep/(Z I My = rep/{rep/(Z | M)]" 
Dem. 

V . *330-626 . *331-12 . D h : Hp . D . A ~ e Pot'(Z | M) (1) 
h . *332-31 . *330-6 . D h : Hp . 3 . g ! r6p/(Z | M) (2) 

V . (1) .(2) . *333-24 . D t- . Prop 



380 QUANTITY [part VI 

*333-32. h :« ei^ilf conx. i, ilf 6 «..p,o-ea'(C/'Cf'Z).D. a iZ" I Jlf' 
Dem. 

V . *330-61 . *301-2 . D h : Hp . D . a ! i» j Jlf» (1) 

h.*330-623. Dh:.B.^. :>:SeH:.Ds-S\I^\M''(Ll>\M'\S: (2) 

[*330-622] D:a!i>|ilf''.D.a! i>+'^i | M' (3) 

h . (2) . *330-621 . DI-:.Hp.D:a!i''|ilf''.3.a!^r-3/''+'=' (4) 

h . (1) . (3) . (4) . Induct . D h . Prop 

*333-33. h-.Ke FM conx .L,Meic,.,Te a\Ult"L) . D . 

repJ{L' I Jlf") = rep/(Z | if)' 
i)em. 

I- . *333-32 . *332-243 . D 

h : Hp . D . rep/(Z« | i/») = / 1' s'a"K = rep/(Z j if")» (1) 

l-.*332-37.*301-21.D 

I- : Hp . D . rep/(Z''+«^ | M'+''^) = rep/lrep/iL' \ M") \ rep/X | rep/i/} (2) 

I- . (2) . D J- : Hp . rep/CZ' | Jf") = rep/(i | ilf)" . D . 

rep/(L'+«i I iy'+'i) = rep/{rep/(Z | M)' \ rep/Z | rep/if} (3) 

»- . (3) . *333-32 . *332-37 . D 

h : Hp (3) . D . rep/(Z''+'=' | M''+'') = rep,'{(Z \My\L\M} 

[*301-21] = rep«'(Z \ My+-^ (4) 

h . (1) . (4) . Induct . D h . Prop 

*333-34. H : Hp *333-33 . D . rep/(Z' | M") = rep/{rep/(Z| Jlf )}''=rep/(Z| Jlf )' 
Dem. 

F . *330-626-6 . *332-31 . D 

I- : Hp . D . A ~ 6 Pot'(Z \M).±\ rep/(Z | il/) (1) 

h . (1) . *333-24 . D h : Hp . D . rep/{rep,'(Z j M)]' = rep/(Z | M)" (2) 
h . (2) . *333-33 . D h . Prop 

*333-41. h : . K e J?'Jf ap conx . Z, J/ e /t. . o- e NC ind - I'O . D : 

rep^'Z" = rep^'JIf" . = .L = M 
Dem. 

V . *333-34-22-2 . D I- : Hp . Z + il/ . D . rep/(Z°- 1 M") e K.g . 

[*333-21-32.*332-33] D . rep/{rep/Z°- 1 rep/i^"} e K.g . 

[*332-44.Transp] D . ~ {rep/Z" | rep/^" G / [^ s'a"«} . 

[*332-15-46.Transp] D . rep/Z' + rep/ilf' (1) 

h . (1) . Transp . D h . Prop 

*333-42. h :.Hp*333-41.D:Z'' = if". = .Z = ilf [*333-41] 



SECTION B] open FAMILIES 381 

*333-43. I- :. Hp *333-41 . D : g ! i' n ilf" . = . i = JIf 
Dem. • 

1- .*333-21 .*332-26 . D h : Hp . g ! i' n M' . D . rep/i' = rep/itf" . 
[*333-41] D.L = M (1) 

h.(l).*330-624.Dt-.Prop 

«333-44. \-:.Ke FMa^p conx .L,MeK,.p,a;Te NO ind - I'O . D : 

re^p^'L"^"^ = reTp/M"^"^ . = . rep^'i> = rep^'JIf"' 
Dem. 

l-.*301-5-.*333-24.D 

I- :. Hp . D : rep«'i>>^«'' = rep^'J/""*"'' . = . rep^'C^ep^'i^'')'' = rep,'(i'ep„'ilf'')'' - 

[*333-41-21] = . rep/Zp = rep/Jf" :. D I- . Prop 

*333-45. l-:.Hp*333-44.D:i>X'^^ = ilf""<-='-.D.rep/i> = rep.'Jlf<' [*333-44] 

*333-46. H :. Hp *333-44 .Dig! I/"""^ n if'X'^ . D . rep/i" = rep/il/' 
J5em. 

H . *332-26 . *333-21 . D 

h : Hp . a ! Li'^''' n Jj/'X"' . D . rep«'i>xcT = rep/Jif'^^'^ (1) 

h . (1) . *333-44 . D h . Prop 

*333-47. h :. K e ZMap conx .L,MeK,.p,a-e NO ind - t'O . D : 

rep«'J> = rep^'if" . = . g ! 2/" n J/' 
Dem. 

h . *333-46 . D h : Hp . a ! Z^ o Jlf" . D . rep/Z" = rep/JM' (1) 

h . *332-53 . *72-92 . D 

\-:'ELp.P,Q,R,Seic.L = P\Q.M=R\S.D.L' = (I'i'\Q'')\-a'D'. 

M' = iB' i S") r a'J/' . rep/Zp = P" | Q" . rep/Jf" = E' | /Sf" (2) 

h . (2) . *35-14 . D 

I- : Hp (2) . rep/Z" = rep/ilf-' . D . Z" r» ilf" = (P" | Qp) [- (Q'Z" n Q'M") . 

[*330-72] D . a ! Zp n M" (3) 

h . (1) . (3) . D I- . Prop 

^333*48. \-:.Ke FM ap conx .L,MeK,.p,a;T e NO ind - t'O . D : 

a ! Zp (S if' . = . a ! Zpxc^ n Jf'XcT 
i)em. 
h.*333-46. 3l-:Hp.a!-^''"J^"'-3-rep/ZP = rep/Jlf' (1) 

h . *330-624 . *332-61 . D h : Hp . D . A ~ e Pot'Zp . a ! Tep/L> . 
[*333-24] 3 . rep/Zpxo- = rep/(rep/Zp)' (2) 

Similarly h : Hp . D . rep/il/'Xcr =, rep/(rep/Jf' )» (3) 

h . (1) . (2) . (3) . D 1- : Hp . a ! Zp n ilf" . D . rep/Zpxo^ = rep/ilf'Xcr . 

[*333-47] 3 . a ! -^''°' ^ ■^'''°' (4) 

h . *333-46-47 . D h : Hp . a ! i>''°" " -3^'"'°' ■ 3 . 3 ! -^'' « ^'' (o) 

1- . (4) . (5) . D I- . Prop 



382 QUANTITY [PART VI 

*333-49. h : « e FM ap conx .L,MeK,.p,a-e NC ind - t'O . rep^'Z^ = rep^'M' . 
D . /.<■ I' a'M' = ¥'[ a'Z" . CD'M") 1 L" = (D'i>) 1 M' 
Bern. 
I-.*333-21 .*330-6 . D I- : Hp . D . g ! rep/X" . 
[*332-ll] D.i/Grep/X'. 

[*72-92] D . Z" = (rep/i>) p a'i> (1) 

Similarly h : Hp . D . Jlf" = (rep/J/") |^ aW' . 

[Hp] D.Jlf'' = (rep/i>)|^a'il/<' (2) 

1- . (1) . (2) . D h : Hp . D . Z" I' Q'il/' = (rep/Z") p (Q'Z' n a'i/") = M''\ <l'Ij> (3) 
Similarly I- : Hp . D . (D'J/") 1 Z^ = (D'Zo) 1 ilf" (4) 

1- . (3) . (4) . D 1- . Prop 

*333-5. V:.KeFM&p conx . P, Q e « . <t e NC ind - t'O . D : 

P' = Q" . = . a ! P' n Q' . = . P = Q [*333-42-43 . *331-24] 

*333-51. V :. K€ FM ap conx . M e ic^^ . p, a e NC ind . D : 

rep^'if'' = rep^'i/"' . = .p = a- 
Bern, 

h . *333-47 . D h :. Hp . rep/Jf" = rep/il/'' . D : g ! ilf " r> ilf -^ : 

[*301-23.*120-412-416] D : p ^ o- . 3 . g ! Jfc-"" A 7 . 

[*333-101] D . p = o- (1) 

Similarly h :. Hp(l) . D : o-^/j . D.p = o- (2) 

I- . (1) . (2) . D F . Prop 

*333-52. I- :. Hp *333-51 .D : M" = M' . = . p = a- [*333-51] 

*333-53. h : «: e Pil/ap conx . Z, ilf e /c,g . g ! Z"' n il/p . g ! Z" n Jf'' . D . 

liXga- = v Xap 
Dem. 

h . *333-48 . *301-16 . D H : Hp . D . g ! Z^^"' n Jf^X"" . g ! Z'"<'='> n ilf'^x^" . 

[*333-47] D . rep^'Z''^^"' = rep«'ilf''X'='' = rep/Z"'*'"' . 

[*333-.51] .D.yu,Xocr = vXo/3:DI-. Prop 



*334. SERIAL FAMILIES. 

Summary of *334. 

The purpose of the present number is to consider what properties of 
a family k will insure that s'/cg is serial, or has one or more of the properties 
characteristic of serial relations. Suppose, for example, that « consists of dis- 
tances on a line. Then Kg consists of those distances which are members of k 
and are not zero. Any selection of distances on the line may constitute k; thus 
e.g. K may consist of all distances which are integral multiples of a given distance, 
or of all which are rational multiples of a given distance,. or of all distances 
from left to right, or of all distances on the line in either direction. It is 
plain to begin with that if s'k^ is to be serial, k must not contain equal 
distances in opposite directions, since if it does, (s'k^Y will not be contained 
in diversity, i.e. s'icg will not be asymmetrical. We call a family « asym- 
metrical when no member of kq has a converse which is also a member of 
Kg. The definition is 

*334-05. FM asym = FMnK(Kn Cnv"« C Rl'/) Df 

It will be observed that s'k^ G J" in any connected family, by *331"23. If 
K e FM asym, we have also (s'/eg)" G J. 

In order to secure that i'/cg shall be transitive, we require that the field of 

K should contain at least one " transitive point," where a " transitive point " 

means a point a such that any point which can be reached from a* by two 

successive non-zero steps can also be reached by one non-zero step, i.e. such 

that 

— » — > 

(s'Ks)"s'Kg'a C s'Kg'a. 

The definition of transitive points is 

*334-01. tia'K = s'a"Kna{(s'Ks)"s'Ks'aCs'Ks'a} Df 

Thus if a is a transitive point, and R,8eKg, there is always a member of 
Kg, say T, such that R'S'a = T'a. It will be seen that if « is a connected 
family, the existence of a transitive point implies that the family is asym- 
metrical. Again, if there is a transitive point in a connected family, then 
E,/Sf6Kg. D.iJ|/SieKg, by *331"32; hence Kg is a group. The converse also 



38-1 QUANTITY [part VI 

holds, i.e. if Kg is a group, any member s'G."k is a transitive point (*334"11). 
Hence if there is any transitive point, every point of s'(1"k is a transitive 
point. 

The definition of a transitive family is 

*334-02. FM trs = iW n « (g ! trs'/c) Df 

By what has just been said, a connected transitive family is one in which Kg 
is a group, i.e. 

*33413. 1- : . K 6 FM conx . D : « e FM trs . s . s'/tg ["/cg C Kg 

A connected family is transitive when, and only when, s'/eg is a transitive 
relation, i.e. 

*33414. 1- : . K 6 FM conx . D : « e FM trs . = . s'Kg e trans 

In order to secure that s'k^ shall be a connected relation, it is not enough 
that K should be an FM conx, i.e. that s'Q."k should have at least one con- 
nected point. We require that every point of s'(I"k should be a connected 
point. This will be secured if there is a connected point which belongs to 
the field of every member of «,, i.e. if 

g ! coux'k np'C'K,. 

For suppose a e coux'k n p'C'Kt. Then if X e Ki, either L'a or L'a exists, and 

is of the form B'a or B'a, where jR e k. Hence, by *331'32, L is identical 

with R or with R; hence Ki = A;uCnv"K. Hence by *331"4, s'Kge connex. 
Conversely, if Kei'W conx and s'Kge connex, it follows from *331'32 that 
Ki=«uOnv"K; hencep'0"Ke=s'C["K, and therefore we have glconx'Knp'C'K,. 
Hence putting 

*334-03. FM connex = FMn^ ('3^1 conx' Knp'C'ic,) Df 

where ".FJf connex'' means "families having connexity," we have 

«334'26. h : . K e FM conx . D : k e FM connex . = . s'k^ e connex . 

= .K, = lc^J Cnv"K . = . 0"k. = a"K 
and 

*334-27. h . FM connex = FM n k (s'a"K = coux'k . k 4= I'A) 

I.e. a family having connexity is one whose field consists wholly of connected 
points and is not null. 

"We thus secure (1) s'k^ G J" by the hypothesis k e FM conx, (2) s'Kg e trans 
by the hypothesis k e.Fif conx n jPif trs, (3) s'Kg e connex by the hypothesis 
K e FM connex (which implies k e FM conx). Hence we secure s'Kg e Ser by 
the hypothesis Kei^'ilf trs n^ilf connex. When this hypothesis is fulfilled, 
we call K a " serial " family ; thus we put 



SECTION B] serial FAMILIES 385 

*334-04. FM sr = FM trs n FM connex Df 

and we have 

*334-3. h : « e J'ilf sr . D . s'Kg 6 Ser 

*334-31. \-:.K6FM.I[s'a"KeK.:i:iceFMsr. = .s'KseSeT-i'A 

An important special case, which is briefly considered in this number, is 
the case when the domains of members of k are the same as their converse 
domains, i.e. when 

This case is illustrated, e.g. by the family whose members are all relations of 
the form {+gX)^C'Hg, where XeC'H'. It is also illustrated by cyclic 
families, which are considered in the next Section but one. When D"k=Q."k, 
if « is a family, so is k u Cnv"«:(*334"4), and if k is a connected family, so is 
K u Cnv"« (*334'41). In the case of the above family, whose members are 
(+gX)lG'Hg where XeC'H', k^Cuv^k will consist of all relations 
i+gX) I. G'Hg where X e G'Hg, i.e. it will consist of all additions of positive 
or negative ratios to positive or negative ratios. 

A connected family in which D"/«: = Q."k is a family having con- 
nexity, i.e. 
*334-42. V-.Ke FM conx . D"/c = (I"/e .D.ke FM connex 

The definitions and propositions of this number are much used through- 
out the remainder of Part VI. ' 



*334-01. tTs'K = s'a"Kna{{s'Ks)"s'Kd'aCs'Ks'a] Df 

*33402. FMtrs = FMnlt{'3^1tvs'K) Df 

*33403. FM connex = FMn^{'g^l coux'k n p' G"k,) Df 

*33404. FM sr = FM trs n FM connex Df 

*33405. FM asym = FMn1i(Kn Cnv"/<: C Rl'7) Df 

*33409. h : « e FM conx . D . s'kq G J [*331 "23] 

*3341. \-::KeFM.D:.aetrs'K. = : 

a e s'a"K : E, fif 6 «3 . Djj,s . {'^T) .TeK^. R'S'a = Pa [(*334-01)] 

3|«334°11. H :.KeFM conx. D : a etrs'K . = . a e s'(I"k . s'K^\"KgC kq 

)} 

Bern. 

h . *331 •33-24 . Dl-:Hp.i2,/Sf6A:g.D.i?|^eK. (1) 

1- . (1) . *331-32 . D h : Hp . T 6 /eg . R'S'a = T'a.D.R\8=T (2) 

I- . (2) . *334-l . Dl-::Hp.D:. 

a€trs'K.= :a6s'a"K:R,SeKs.:>ji,s-{'aT).TeK^.R\S==T: 
[*13-195] = :aes'a."KiR,SeK^ . Ds.s ■ -B i/Se/eg :: D h . Prop 

H. & w. III. 25 



386 QUANTITY [part VI 

*33412. h :. «; 6 FM coDx .a,xe s'Q."k . D : 

a 6 trs*« .= .xe tis'/c . = . s'/cg |"/cg C /cg [*334"11] 

)) 

*33413. h :. « e J^JIf conx . D : k e FM trs . = . s'k^ \"Kg C Kg 

[*334-12 . *33112 . (*334-02)] 

*334-131. \-:KeFM conx n J'ilf trs . E e /eg . D . Pot'E C /eg [*334-13 . Induct] 

*334132. \-:k6FM conx n ^ilf trs . 3 . s'Pot"/c C k [*334131] 

*334-14. h : . K e i^if conx . D : /c e Fil/ trs . = . s'/cg e trans 

Dem. 
y . *41-51 . *334-13 . D h :. Hp . D : /c e^Jftrs . D . (s'/«:g)= G s'/cg (1) 

1- . *330-52 . D 1- :: Hp . D :. s'/tg e trans . D : 

iJ,6'6/eg .x€s'a"K . D^,s,^. (aT) . Te/eg . R'S'x= T'x. 
[*331-31-33-24] '^r.s.x- i'ST) .TeK^.R\S=T. 

[*13-195] D^,s,,.i?|fif6/.g (2) 

I- . (2) . *331'12 . D I- :: Hp . D :. s'/tg e trans . D : ii, /S e /eg . D2S,S' -^ | /S e /cg : 
[*33413] DzAceJfWtrs (3) 

1- . (1) . (3) . D h . Prop 

*33415. I- : /e e ^Jf conx n FM trs. . s'k\"k = k 

Dem. 
f- . *331-321-22 . D h :. Hp . i? 6/e - «g . D : R = I\s'a"K : 

[*50-62-63] D:SeK.:i.R\S,S\ReK (1) 

h . (1) . *33413 . DI-:Hp.D.s'A:|"/eC/e (2) 

h . *331-22 . *50-62-63. D h : Hp . D . /e C s'/e |"/e (3) 

h . (2) . (3) . D h . Prop 

*334-16. h : K 6 FM conx nFM trs. ReKf,.':i.R^„Q. J [*334-13109] 

— » 
*334161. V-.Ke FM conx nFMtis.ReK^.ae s'a"K . 3 . R^'a e «„ 

[*334-16.*123-191] 

*334162. h : a ! FM conx nFMtvs-l.:^. Infin ax [*334-161] 

*33417. h-.KeFM conx /^ 1 . 3 . /eg = A [*331-22] 

*33418. h : K 6 ^ilf conx - 1 . D . C's'kq = s'a"/e=s'a"«:g . a ! s'k^ . g ! /eg 

-Dem. 
h . *331-22-321 . D I- :. Hp . 3 : a ! /eg : 

[*330-52] D : a 6 s'a"/e . D . (a^?) . i? e /eg . a e Q'i? . 

[*40-4] D.aes'a"/tg. (1) 

[*41-45] D.aeC's'/eg (2) 

h. (1). (2). *331 12.31-. Prop 



SECTION B] serial FAMILIES 387 

*334-19. \-:K€FM.:>.C's'ic-^Cs'a"K [*41-45 . *330-52] 

*334-2. \- ::. KeFM .:>:: aep'G"K,. = :. L e K,. Ol-.EI L'a .V .'E.l L'a 
[*330-52] 

*334-21. I- : « e FM connex . D . «i = « u On v"« 

Dem. 
\- . *334-2 . *331-11 . D h :. Hp . a 6 codx'« np'C'K, .Lbk^.D: 

(giJ) :ReKU Cnv"« : L'a=R'a . v . L'a = R'a : 

[*331-42-24] D:(ai2):i2e«oCnv"«:Z = E.v.i = i2 (1) 

h . (1) . *331-24 . D I- . Prop 

*334-22. h-.KeFM connex. 0.p'C"K, = s'a"K [*334-21 . *330-52] 

*334-23. l-:/e6^Jlf connex. D.conx'/e = 5'a"« [*334-21 . *331-4] 

*334'24. h : « e ^Jlf connex . D . s'k^ e connex 
Dem. 

h . *334-21 . *331-4 . D 
t- :. Hp .x,ye s'(J"k .x^y."^: (a-R) : iJ e Kg : xRy . v . yRx :. D h . Prop 

*334-25. VzKeFM connex . D . (7"k. = Q"* [*334-21 . *330-52] 

*334-251. h : « e iW . «. = /e w Cnv"* . D .fG"K, = s'Q"* 
i)em. 

f- . *40-18 . *33-22 . D h : Hp . 3 .p'G"K, = p'<7"« (1) 

F . (1) . *330-52 . D I- . Prop 

^334*252. V -.Ke FM conx . s'wg e connex . D . Kj = k u Cnv"* 

Dem. 
V . *41-11 . D h : Hp . i e K. . a; = L'y . D . (gE) .ReK\j Cnv"« . ajJSy . 
[*331-42-24] D.ieKwCnv"* (1) 

h . (1) . *330-6 . *331-12 . D I- . Prop 

*334-253. h : « 6 i^W conx . C"k, = Q"* . D . /c e ^ilf connex 
Dem. 

I- . *330-52 . D I- : Hp . 3 .p'G'^K = s'C["ic . 

[*331-1] D . a ! p'G"k, n conx'« : D h . Prop 

*334'26. h : . K 6 ^Jf conx . D : « e ^if connex . = . sVg e connex . 

= .«. = «« Cnv"K . = . G"k, = a"K [*334-21-24-25-251-252-253] 

*334-27. h . FM connex = i^Jf n k {s'(1"k = conx'/e . k 4= t'A) 

Dem. 
h . *33ri . D I- : « 6 i^ilf . k =|= t'A . s'Q"* = conx'w . D . s'k^ e connex . 
[*334-26.(*331-02)] D.iceFM connex (1) 

h .*334-23 .(*334-03) . D h : « e 2?'if connex . D . s'a"« = conx'* . « + I'A (2) 
h . (1) . (2) . D I- . Prop 

25—2 



388 QUANTITY [part VI 

*334-3. h-.iceFMsr.::). s'k^ e Ser 

jDem. 

h.*334-09.Dh:Hp.D.s'/cgej' (1) 

h . *334-14 . D I- : Hp . D . s'Kg e trans (2) 

h . *334-24 . D h : Hp . 3 . s'kq e connex (3) 

l-.(l).(2).(3).DI-.Prop 

*334-31. l-:.«eZM.7|^s'a"«e«.D:«6^ilfsr. = .s'«9 6Ser-t'A 
Dem. 

i- . *41-11 . D I- :. Hp . s'kq e Ser - I'A . D : 

x,ye s'Q."k . D-t, j, . (gi?) .BeK.!Jc{RK/R)y: 
[*33iai] D:s'a."K = conx'K (1) 

I- .(l).*33414-26 . D h : Hp(l) . D . « e^if trs .KsFMconnex (2) 

I- . (2) . *334-3 . *331-12 . D 1- . Prop 

*334-32. b.FM srC iWap [*334-16-21 . *333-101] 

*334-4. \-:ic6FM. D"« = Q"* . D . « u Cnv"K e ^ilf 
Dem. 

I- . *33-2-21 . D h : Hp . D . D"(« « Cnv"/c) = a"(« u Cnv"«) = Q"* (1) 
I- . *330-561 . D h :. Hp . D : i?, 5f 6 « . 3 . E I <Sf = /Sf I ^ (2) 

I- . (1) . (2) . *330-52 . D h . Prop 

*334-41. \-:KeFMconx.I>"K = a"K.:>.KyjCnv"KeFMconx 
[*334-4.*33111] 

*334-42. h : K e ^Jlf conx . D"« = Q"* . D . « e ^Jf connex 
Dem. 

h . *37-323 . D I- :. Hp . D : iJ, ^f e « . D . a'(E | S) = a'S : 
[*330-4] D:a"«. = a"«: (1) 

I- . (1) . *334-26 . D h . Prop 

*334-43. V-.KeFM conx n ZM trs . D"k = Q"* . D . « e i^if sr 
[*334-42 . (*334-04)] 

*334-44. V-.Ke FM conx . D"« = a"« . i e «, . D . D'Z = Q'Z = (7'Z = s'Q"* 
Dem. 

I- . *37-323 .Dh:Rp.R,SeK.D.a'{R\S} = (I'S:D\-. Prop 



SECTION B] serial FAMILIES 389 

*334-45. y-.KeFM conx . T>"k = Q"*: .L.MeK^.D. a'(L \M) = s'a"K 
[*334-44] • 

*334-451. h : Hp *334-44 . /S e Pot'Z . D . D'>S = a'S = G'S = s'Q"* [*334-44] 

*334-46. h : . Hp *334-44 .M,NeK,.0:'g^lL\M nN . = .L\M=N 
[*334-45 . *331-45] 

*334-5. h : « 6 FM conx n FM asym . D . (i'«g)^ G / 
Bern. 

h.*332-46. D\-:B.p.R,S€K.R\S(lI.O.R = S. 
[(*33405)i D . iJ = 7 p s'a"« (1 ) 

I- . (1) . Transp . 3 h :. Hp .3 : iJ.^fe «g . D .~(i2|/SG7) . 
[*331-33-23] D.J2|<SG/:.Dh.Prop 



*335. INITIAL FAMILIES. 

Summary of *335. 

A family of vectors may or may not have a point in its field which is a 
starting-point but not an end-point of non-zero vectors. For example, the 
family of which a member is (+« X) l G'E', where X e G'H', has such a point 
in its field, namely 0, ; but the family of which a member is (-!-« X) ^ G'H, 
where X e G'H', has no such point in its field, and no more has the family of 
which a member is (+gX) I G'Hg, where X e G'H'. If such a point exists, it 
is a member of s'Q."k but not of s'D"*;g. Such a point, if it is also a con- 
nected point, must be unique, i.e. we have 

*33512. h-.iceFM.'^. conx'« - s'T>"ks e w 1 

When conx' K — s'D" K^ exists, we call its only member "the initial point 
of K," putting 

*33501. init'K = 7'(conx'K - s'D"A;g) Df 

If the initial point of k exists, we call k an " initial " family ; thus we put 

*33502. FM im.t=FMna'im.t Df 

An initial family is asymmetrical (*335"16) and transitive (*335"18), and 
forms a group (*335'17); and if its initial point is a member of p'C'iei, it is a 
serial family (*335-3). 



°'"'a 



*33501. init'« = i'(conx'K-s'D"«g) Df 
*33502. FM init = FMn a'init Df 

*33511. h-.KeFM.ae conx'/e - s'D"«g . D . s'a"K = s'ic'a . I'a = ¥k' 
Dem. 

h . *41-43 . *33-4 . D h : Hp . D . Picg'a = A (1) 

I- . *331 •23-22 . D h : Hp . D . s'k'u = s'xg'a w I'a (2) 

h . *331-1 •23-22 . D h : Hp . D . s'a"« = s Va u |^9'a (3) 

h . (1) . (2) . (3) . D h . Prop • 



SECTION B] initial FAMILIES 391 

*33512. hzKcFM.O. conx'« - s'D"«g e w 1 

Dem. « 

— > 
I- . *335*11 . D I- : Hp . a, 6 6 conx'/e - s'D"«g . D . 6 e sVa . 

[*32-182] D . a e s'«'6 . 

[*335-ll] D . a = & : D h . Prop 

*33513. h :. « e jPif . D : E ! init'«: . = . g ! conx'/e - s'D"«g 
[*33512 . (*335-01)] 

*33514. h-.ice FMinit . = . keFM.'s^ ! conx'«-s'D"«g [*335-13.(*33502)] 

*33515. l-:«6^il/init.D.s'a"/«: = sVinit'«r [*335-ll.(*335-01)] 

*33516. h . FMinit C J?'Jf asym 

Dem. 
h . *335-14 .:>[-:.KeFM init . D : 

(ga) : a e s'a"* :Reic.ae D'i? . Djj . ii e Rl'J (1) 
h . *330-52 , D h : /e e ZAf . a e s'a"K .Rexn Cnv"K . D . aeD'R (2) 

I- . (1) . (2) . D h :. « 6 JW init . D : i? e « n Onv"« ."Us. Re Rl'/ : 
[(*334-05)] D : K e ^Jl/asym :. D h . Prop 

*335-17. h : K e ^ilf init . D . s'« !"« = « 

Dewi. 
I- . *335-15 . 3 h :. Hp . D : iJ, ,Se K . D . (gT) . Te k . i2'>S'init'«; = T'init'« . 
[*331-24-33-32] D . (gr) . Te/c . i? | <S= T. 

[*13-195] D-JSI^Se/B (1) 

h.*331-22.DI-:Hp.D./«;Cs'«;|"« (2) 

h . (1) . (2) . 3 h . Prop 

*335-18. h . FM^ init C FM trs 

Dem. 
l-.*335-l7. Dh:.K6^ilfinit.D:i2,/Sf6/cg.3.J?|)SeK (1) 

1- . *334-5 . *335-16 . D h :. «6i?'ilf init . D :i2,/Se«g. D .E | -Sf G J" (2, 

I- . (1) .(2).*330-551 . D I- :. « e ^i/init . D : 22, )S e Kg . D . i? | -S e «g (3) 

h . (3) . *334-13 . D I- . Prop 

*33519. h :. « 6 i^lf init . D : « e FM connex . = . init'w ep'C'K, 
[*334-23 . (*334-03 . *335-02-01)] 

*335-21. h-.Ke FMinit . D . s'«g e trans . (s'k^Y G J [*335-1816 . *334-14-5] 

*335-22. h :. xeFM init . D : s'wg e connex . = . C"k, = (I"« . =.init'«; ep'C'K, 
[*334-26 . *335-19] 



392 QUANTITY [part VI 

*335-23. h : . K 6 FM init n FM connex . £ e /ic,g . D : 

init'/e e D'i . = . init'/c ~ e d'L 
Dem. 
V . *335-19 . D h :. Hp . D : init'« e D'Z . v . init'« e Q'i (1) 

I- . *334-21 . D f- : Hp . D . i e Kg u Ciiv"«g (2) 

I- . *335-ll . D h :. Hp . D : i e /(g . D . init'« ~ e D'X : 

ieCnv"A;g.D.init'«~ea'X (3) 

h . (2) . (3) . D h :. Hp . D : init'* ~ e D'i . v . initV ~ e Q'Zi (4) 

h . (1) . (4) . *5-l7 . D 1- . Prop 

*335-24 h :. K 6 J^Winit n ^if connex .B,SeK .R^8 ."2: 

E'init'/c e D'/S . = . /Sf'init'/c ~ e D'i? 
Z)em. 

1- . *71-162 . D h :. Hp . D : i?'imt'« e D'/S . = . init'« e a'(^ | i?) . 

[*333-l.*335-23] = . init'/e ~ e D'(S | i?) . 

[*71-162] = . S'miVK ~ e B'R :. D I- . Prop 

*335"25. h : : . « 6 ^1/ init . D : : s'/eg e connex . = : . 

E, /S e « . D^,g J D'i? C D'/S . V . D'/Sf C D'i? :. 
= :.a,;86D"K.D.,p:aC|8.v./3Ca 
Z)em. 
h . *202'135 . D h :: Hp . s'«g e connex . D :. s'/e e connex :. 
[*211-6.*330-542] D :.E,/Sf€«. D :D'iiCD'/S. v . D'/SCD'i? (1) 

h . *71-162 . D h : Hp . ii'init'« e D',S . D . init'* e a'(^ | S) (2) 

h . *71162 . D h : Hp . /S'init'« e D'i? . D . init'/e e D'(^ 1 8) (3) 

l-.(2).(3). DI-:.Hp.J?,S6/<;:D'i?CD'^.v.D'^CD'i?:D. 

init'«eC"(E|.Sf) (4) 
h . (4) . *330-4 . D h : : Hp :. ^, 5 e « . 3^,s : D'iZ C D'/S . v . D'/Sf C J)'R : . 3 . 

init'/e e p'C'K^ . 
[*335-22] D . s'/eg 6 connex (5) 

|-.(l).(5).*37-63.DI-.Prop 
*335-26. h-.Ke JW init n FM connex .D.D['/cel-*l 

Dem. 
h.*33-43. Dh:Hp.i?,/Se/«:.J?'init'K~eD'^.D.D'i?=f=D'/S (1) 

h .*335-24 . D h : Hp . J?, ,Sf e « . i? + /S . iJ'init'« e D'/Sf . D . fif'init'« ~ e D'i? . 
[*33-43] D.D'iJ + D'S (2) 

I- . (1) . (2) . D I- : Hp . i?,,Sf6« .i« + ;Sf . D . D'iJ + D'^f : D h . Prop 
*335-3. h:«6J?'if.init'«6^'C"«..D.s'«g6Ser [*335-21-22] 



*336. THE SERIES OF VEOTOES. 

Summary of *336. 

• In this number we consider a relation between members of k or of k^ 
which, with suitable limitations as to the nature of the family, may be 
identified with the relation of greater and less. If there is a member of k 
which takes us from a point ^^ to a point y, i.e. if y (s'kq) z, we say that z is an 
earlier point than y ; thus we regard s'/cg as the relation of later to earlier. 
If now M and N are two members of k„ and if, for some x, M'x is later than 
N'x, we shall say that M is "greater" than N with respect to k. This 
relation we denote by V^, where " V" is intended to suggest that the relation 
holds between vectors. The definition is : 

*336-01. V, = 'MN[M,N6Kr. (g^) . {M'x) (s'«g) {N'x)] Df 

For the same relation when confined to members of k, we use the notation 
U^ ; thus we put 

*336011. U^ = V^Ik Df 

In dealing with F^ and U^ it is desirable to be able to express M'x as a 
function of M. We wish to consider (say) a fixed origin a, and the various 
points R'a, 8'a, T'a, ... to which the various vectors which are members of k 
carry us from a. For this purpose we put 

B'a = Aa'B, 
where " A '' stands for " argument," and " A^'R " may be read " the value, 
for the argument a, of R." The definition is 

Aa = ^R(xRa) Df, 
whence we obtain 

*336101. \-:ElR'a.D.R'a = Aa,'R 

Then the points R'a, S'a, T'a, . . ., where R,S,T,... are the various members 

of K, form the class Aa"ic, which is thus the same class as s'x'a. The relation 
Aa\^ K correlates the point R'a with the vector R. The vector R is analogous 
to the coordinate of R'a when a is the origin ; thus .4„ f « is analogous to 
the relation of a point to its coordinate. A relation which is more exactly 
that of a point to its coordinate will be explained in Section 0, where, in 



394 QUANTITY [part VI 

addition to the above correlator A^ \ k, we shall also correlate a vector with 
its numerical measure in terms of an assigned unit. 

If « is a connected family, and a is any point of its field, Aa f «i is a one- 
one relation (*336'2). If k is an initial family, and a is its initial point, 
Aa\ K is a correlator of s'(1"k and k (*336"21), so that in an initial family 
the class of vectors is similar to the field (*336'22). If k is a connected, 
family, and a is any point of the field, and \ is those members L of /c, for 
which L'a exists, then ila f" X. correlates the field with \, so that X, is similar 
to the field (*336-24). 

By the definition of Aa, it Mek, and M'a exists, we have 
M'a = Aa'M = Aa\->cJM. 
Hence by the definition of F^, 

h -.'MV^N . = . (aa) . (Aa r >c,'M) (s'k^) (A^ [ kJN) . 

= . (aa).ilf(«:.1 Aa>s'Ks)N, by *150-41. 

Similarly \-:PU,Q. = . (ga) . P (« 1 AJs'ks) Q- 

Now in a connected family, if a and b are any two members of the field, and 
P,QeK, 

(P'a) (s'ks) (Q'a) . = . (P'b) (s'«9) (Q'b) (*336-38) ; 

hence fc '] AJs^k^ ^k"] A}}s^k^, 

and hence U^ = k'\ AJs'k^ (*336"43). 

Since /c"] -4a is one-one (by *336'2), the above gives an ordinal correlation of 
Ux with (s'«g) t Aa'K (*336'461), i.e. JJ^ is ordinally similar to s'«g with its 
field confined to those points which can be reached from a by vectors which 
are members of k. If « is an initial family, it follows that U^ is similar to 
s'/cg (*336"44) ; if not, U^ is in general only similar to a segment of s'/cg (in 
the sense of *213). 

It should be observed that k, '\ A^'x is the member of /e. which takes us 

from a to X, and « 'j Aa'x (if it exists) is the member of k which takes us from 

a to X. Thus k'\ AJs'k^ is the series of vectors which take us from a to all 
the various points which can be reached from a by members of k, the order 
of the series being that of the points to which the various vectors take us 
from a. 

If « is a connected family, U^ is the relation which holds between two 
members of « when one of them is the relative product of the other and a 
third (other than the zero vector), i.e. 

*336-41. V : >c e FM couK .:> . U,^ PQ {P,Q e >c : (•^T) . T e Ks . P ='T\Q] 



SECTION B] THK series OF VECTORS 395 

This is for many purposes the most convenient formula for TJ^- If, in 
addition, we have D"/c^Q"/c, a similar formula holds for V^, i.e. 

*336-54. V-.KeFM conx . D"«; = <1"k . D . 

V, = MN{M,NeKr.{'sT)-TeK^.M=T\N] 

If «6^Jf conx, F, is contained in diversity (*3366); if « is also transitive, 
Fk is transitive (336"61) ; and if « has connexity, so has F, (*336"62). Hence 
if /c is a serial family, F^ and U„ are serial (*336'63'64). 

In addition to the above-mentioned propositions, the following propo- 
sitions in this number are important : 

*336'411. h -..KeFMcou^ . s'« !"«: C k . D : PU,Q .Rsk.D .(P\R)U,{Q\R) 
*336-511. \-:.KeFMsr.ve'!^Gmd-i'O.D:RU,S. = .R-'U.S-' 

*336-53. \-:.K€FM conx .M,NeK,.0'. MV^ N.= . NVJl 

The present number is important, since F, and TJ^ are the general 
relations from which greater and less are derived, and the subject of magni- 
tude is therefore intimately dependent upon them. 



*336-01. V,^MN[M,NeKr.{'K'«).{,M'x){s'K^){N'x)] Df 
*336011. Cr«=F,p« Df 

*33602. Aa = ^R{ooRa) Df 

*336-l. V : xAaR . = .xRa [(*336-02)] 

*336101. h:E!iJ'a.D.i2'a = ^„'i2 [*3361] 

*336-ll. Vix{Aa\ic)R. = .ReK. xRa [*336-l] 

*336-12. V.I^K'a = Aa"K = 'D%Aa\ic) 
Dem. 

V . *41-11 .31-. s'K'a = a {(gi?) .ReK. xRa] 

[*336-l] = ^ {(a-B) .ReK. xAaR] ^ 3 H . Pr'op 

*33613. V.J)'Aa\KCs'Ji"K 

Dem. 

V . *33612 . *33-15 . D H . D'J.« \kC D's'k .31-. Prop 

*33614. f-:A:Cl-»Cls.3.^„rKel->Cls 

Dem. 

V .^^^%-ll .':i\- : x{Aa[ k) R .y {Aa\ k) R ."^ . Re K . xRa .yRa (1) 
I- . (1) . *71-17 . 3 h : Hp . Hp (1) . 3 . « = 2/ (2) 

H . (2) . *71-17 . 3 h . Prop 



396 QUANTITY [part VI 

*33615. \- : kCct'u . aea .:^ .a'(Aa[ k) = K 
Dem. 

I- . *336-ll .Oh: Re a'(Aa fx) . = . (a*) .ReK.xRa (1) 

F . (1) . (*330'01) . D I- . Prop 

*33616. \-:a€Conx'K. = .aes'a"K.Aa"{K\jCny"K) = s'a"K 
Dem, 

[-.*331-1.*336-12.D 

h:ae conx'/e . = . a e s'Q"* . il„"« w 4„"Cnv"« = s'a"/c (1) 

h . (1) . *37-22 . D h . Prop 

*33617. h : K 6 ^ilf conx nFMtrs-l.P = s'k^ . 3 . Aa"ic = P^'a 
Dem. 

h .. *334-14-18 . D h : Hp . D . P^'a = P'a^I^ s'a^x'a 

[*331-22-23] =s'K'a 

[*336-12] = Aa"K : 3 I- . Prop 

*336-2. I- -.KsFM conx .aes'(I"K .D .Aa[ K,el-*1 

Dem. 
I- . *336-14 . D f- : Hp . D . ^„ I' «. e 1 -> Cls (1) 

h . *336-l 1 .Ol-:B.p.x(AaiK,)L.oi;{Aa[ic,)M.D.L,MeK,. xLa . xMa . 
[*33r42] D.£ = ilf (2) 

t- . (1) . (2) . D 1- . Prop 

*336-21. V-.Ke FM . a = init'« . 3 . ^„ [^ /« e (s'a."je) sm k 

Dem. 

l-.*336-2. DhiHp.O.^al'/cel-*! (1) 

b . *335-15 . *336-12 . D h : Hp . D . D'^« [k = s'a"K (2) 

[-.*336-15. Dh:Hp,D.a'^a|^/«; = /«; (3) 

I- . (1) . (2) . (3) . 3 h . Prop 

*336-22. h-.xeFM init . D . (s'a"«) sm « [*336-21] 

*336-23. h : kbFM conx . a es'Q"* . \= k.a i(aeC['Z) . D . 

Dem,. 

l-.*336-2. DI-iHp.D.^aPXel-^l (1) 

I- . *336-ll . 3 h : Hp . D . D'(Aa T \) = ^ {(gZ) .Le\. xLa] 
[Hp] = o5 {(gi) .LsK,. xLa] 

[*331-4] =s'a"« (2) 

h . *336-ll . D F : Hp . "0 .a\Aa \'^) = L[{'^x) .LeX. xLa] 

[Hp] =\ (3) 

I- . (1) . (2) . (3) . D h . Prop 



SECTION B] the series OF VECTORS 397 

*336-24. I- : Hp *336-23 . D . (s'a"«:) sm \ [*336-23] 

*336-25. h:KeFMcoux.a,be s'a"« .X = K,nL(ae O'i) . 

fi = K,nM{bea'M).:^.Xsmfi [*336-24] 

*336-26. h : «e^ilf . aeconx'« . \ = k w Cnv"E (ReK .aeD'R) . D . 

^„ 1^ \ 6 (s'a"«) sin \ [*336-23 . *331-48] 

*336-3. h :.kC1-*CIs. :^:R(K'\Aa'P)S.= .R,SeK.{R'a)P(S'a) 

Bern. 
V . *150-11 . D I- : i2 (k ^ A^'P) S.= . (ga;, y).R,8eK. xA^R . y^a-S . xPy . 
[*336"1] = . (a*, y).R,SeK . ocRa . ySa . xPy (1 ) 

h . (1) . *7l-36 . D h . Prop 

*336-31. h : « e FJlf conx . a e s'(1"k . D . Kg C D'(« 1 ^a's'/eg) 

Dem. 
F- . *336-3 . D 

I- :. Hp . D : i? 6 D'(/«: 1 J^^Js'^g) . = . (g/S, T) . JB,/Se« . Te/tg . R'a=T'8'a (1) 
I- . *331-22 . D h : Hp . i2 6 Kg . D . -R 6 Kg . /|^s'a"K6 k . ii'a = ii'(/ 1' 5'a"K)'a . 

[(1)] O.Re D'(k 1 Jf^Js'Kg) : D H . Prop 

*336-311. h : K 6 FJ/conx - 1 . a e s'a"ic . 3 . 1 [~ s'a^K e a'(K 1 laJs'Kg) 

Dem. 
h.*336-3.D 
h :. Hp . D : ^!ea'(Kl^<.'s'«3) ■ = ■ (H-K, T).R,8eic. Teic^ . R'a=T'8'a : 

[*331-22] D : / 1^ s'a"K e a'(K 1 AJs'xg) . = . (giZ, T).ReK.T eK^.R'a=^T'a. 
[*330-52] s . a ! Kg (1) 

h . (1) . *334-18 . D h . Prop 

*336-312. ViiceFM conx - 1 . D . C"(k 1 i^Js'Kg) = k [*336-31-311] 
«336-313. h : K e jfil/ conx n ^if asym . a e s'Q."k . D . D'(k ^ ^a^s'Kg) = Kg 



h . *336-3 . D 

h :. Hp . D : I[ s'a"Ke'D'{K^Aa's'K^) . = . (a^f,^) ./Sfe K.Te Kg. a = T'/S'a (1) 

h . (1) . *334-5 . Dh:Hp.D./|'s'a"K~6D'(Kl-4„;s'Kg) (2) 

I- . (2) . *336-31 . D h . Prop 



398 



QUANTITY 



[part VI 



*336-32. h : « e FM . a e conx'/c .\ = Kr\R{ae B'R) . D . 

C'[(k w Cnv"«) 1 AJs'k^} ^ksj Ciiv"\ 
De?w. 

h . *33616 . *334-18 . D h : Hp . D . O's'/^g = a'(« u Cnv"*) 1 ^„ . 

[*150-23] D . G'{(k w. Cnv"*;) 1 AJs'^g} = D'(« w Cnv"*) 1 la 

[*336-15-ll] = K w ^ {(ga;) . E e Cnv"« . xEa} 

[Hp] = « u Cnv"\ : D h . Prop 

*336-34. h : « 6 JW . a = init'* . D . (« 1 Aa's'K^) smor (s'/cg) 
i)em. 
I- .*336-21 .Dh:Hp.D.«;liL„6l->l. Q'* ^ 2„ = O's'/eg : D h . Prop 

j|(336°35. h : K 6 Flf . a e conx'« . 3 . {(« w Cnv"*) 1 AJs'k^} smor (s'Kg) 
[*336-2-16] 

*336-351. h : K 6 i^iiif conx . a e s'Q"* . 3 . (k ^ ^^Js'/cg) smor (s'/cg) ^ 4a"« 
Dem. 

h.*336-2. DhiHp.D.Kllael-*! (1) 

h . *150-37 . D 1- : Hp . D . K 1 i^a^s'/eg = « ^ i^aKs'^e) t ^a"« (2) 

I- . (1) . (2) . D h . Prop 

*336-36. hz.Ke FM conx .L,MeK,.a,be G'L n a'M. TeK.li: 

L'a = T'M'a . = .L'b = T'M'h : L'a = T'M'a . = .L'b = T'M'b 
Bern. 
I- .*13-12. D h :. Hp . iVe/e. .a = N'b.:i: L'a= T'M'a. = .L'N'b = T'M'N'b. 
[*330-63] = . N'L'b = N'T'M'b . 

[*71-56] =.L'b = T'M'b (1) 

I- . (1) . *331-4 . D I- :. Hp . D : Z'a = T'M'a . = .L'b = T'M'h (2) 

h . *71-362 . D h :. Hp . 3 : i'a = T'M'a . = .M'a= T'L'a . 
M,L~ 



(2)- 



V'L,M\ 

[*7l-362] 

h . (2) . (3) . D I- . Prop 



= .M'b = T'L'b. 
= .L'b= T'M'b 



(3) 



*336-37. h :. « e ^Jlf conx .L,M€K,.a,be a'L n G'M . D : 

(L'a) is' Kg) (M'a) . = . (L'b) (s'ks) (M'b) 
Bern. 
\- . *336-36 . D 
\- -..Rp . D :('g^T).T e Kg. L'a = T'M'a. = .('^T).T e Kg. L'b=T'M'b:.:i\- .Frop 



SECTION B] the series OF VECTORS 399 

*336-371. V'..iceFM conx . Z, il/ e ^. . a e d'L n a'M . D : 

tLV,M. = . (L'a) (s'k^) (M'a) [*336-37 . (*33601)] 

*336-38. h : . « 6 J?'Jlf conx . P, Q e « . a, 6 e s'a"K . D : 

(P'a) (s'«g) (Q'a) . = . (P'6) (s'«g) (Q'b) [*336-37 . *331 -24] 

*336-4. bzKeFM conx . a e s'a"K .D . U, = PQ{P,QeK . (P'a) (sVg) (Q'a)} 

Dem. 
h . *336-38 . D 

h :. Hp . D : 6 e s'a"* . (P'6) (s'xg) (Q'b) . = .be s'a"« . (P'a) (s'k^) (Q'a) : 
[*10-ll-281.Hp] D : (36) . & e s'a"K . (P'b)(s'Ks) (Q'b) . = . (P'a)(s'«g)(Q'a) (1) 
h . (1) . (*336-011) . D h . Prop 

*336-41. h-.KeFM conx . D . [;■, = PQ {P, Q e « : (gT) . T e Kg . P = T | Q} 

Dem. 
h.*41-ll.DI-:Hp.aes'a"«:.P,Q6«.re«:9.P=r|Q.D.(P'a)(s'Kg)(Q'a) (1) 
h .*41-11 . D h : Hp . a 6 s'Q"* . (P'a) (s'/cg) (Q'a) . D . (gT) . Te «g . P'a= T'Q'a . 
[*331-32-33-24] D.('3^T).TeKs.P = T\Q (2) 

h . (1) . (2) . *336-4 . D h . Prop 

*336-411. \-:.KeFM conx .s'k\"kCk.O: PU^Q.ReK .0 .(P\R)U^(Q\R) 

[*336-41] 

*336-412. h : Hp *336-411 . P.Q,R€h:.(P\R)U,(Q\R).D . PU^Q 
Dem. 

h . *336-41 . D h : Hp . D . (gT) .2'e«g.P|J? = T|Q|i2. 

[*330-5] :>.('^T).Te>cs.R\R\P = R\R\T\Q. 

[*330-31] 3.(a2').2'6Kg.P = y|Q. 

[*336-41] D . P f7,Q : D h . Prop 

*336-413. h :. Hp *336-411 .P,Q,Re k .0: PU^Q. = .(P\R)U,(Q\R) 
[*336-411-412] 

*336-42. h : « 6 Pilf conx . a ep'D"* . D . F^ = M {i, iW e «, . (i'a)(s'«9)(ilf' a)} 

Dem. 
h . *330-54 . D t- :. Hp . i,lf 6 K. . D : a 6 a'L n Q'ilf : 

[*336-37] D : (L'b) (s'k^) (M'b) . D . (L'a) (s'«g) (Jf 'a) : 

[(*336-01)] D : ZF^if . D . (Z'a) (s'/cg) (M'a) (1) 

h . (1) . (*336-01) . 3 h . Prop 

«336-43. h : « 6 Pilf conx . a e s'(1"k . D . U^= k^ Aa's'K^ 
Dem. 

y . *336-4101 . D H : Hp . D . CT, = PQ {P, Q e K . (2I/P) (s'^g) (A^'Q)} 

[*35-7] = pO {(^„ r «'-p) (s'«9) (4« r «'<?)} 

[*150-41.*336-2] = « 1 ii'aJs'/eg : D h . Prop 



P{F,p(«uCiiv"\)}Q.= 
[*14-21.Hp] = 

[*4<1-11] = 

[*336-3] = 



400 QUANTITY [part VI 

*336-44. h : « e FM init .O.U, sraor (s'/tg) 

Bern. 
h . *336-41 . D h : Hp . a = init'/e . D . f7« = k 1 Aa's'K^ (1) 

h . *336-21 . D 1- : Hp . a = init'« . D . /c ^ 1„ e 1 -> 1 . a'(K 1 Aa) = s'Q"* (2) 
1- . (1) . (2) . *334-19 . D h . Prop 

*336-45. h ZKeFM. a e coux'k .\ = k r, R{a eB'B) . D. 

F. C (« u Cnv"\) = (« w Cnv"«;) 1 AJs'ks 
Dem. 
1-.*4111.(*33601).D 

l-:.P{F,t(«uCnv"\)}Q. = :P,Qe««Ciiv"\:(a«,r).re«g.P'a;=r'Q'« (1) 
l-.(l).*336-36.DI-::Hp.D 

: P, Q 6 « u Cnv"X, : (gT) . Te «g . P'a = T'Q'a : 
.P,QeKyj Cnv"/c : (aT) . Te /«:g . P'a = T'Q'a : 
\P,Q6Kyj Cnv"« . (P'a) (s'/cg) {Q'a) : 

: P {(« u Cnv"A;) 1 i«5s'«g} Q :: D h . Prop 

*336-46. h:Hp*336-45.D.F4(«uCnv"\)smor(s'K9) [*336-45-2-16] 

*336-461. I- : K 6 Pilf conx . a e s'Q."k . D . fT'^ smor (s'«g) p (-4a"«:) 
[*336-351-43] 

*336-462. f- : « 6 Pilf conx n PM trs . a e s'a"/e . P = s'«g . D . !/;= smor (P ^ P^j^'a) 
[*33G-461-17.*334-17] 

*336-47. h : « e PM conx . D . Kg C D' D^ [*336-31-43] 

*336-471. \-:KeFM conx - 1 . D . « = C f/^ [*336-312-43] 

*336-472. \- ZKeFM conx n Pif asym . D . /cg = D' tT^ [*336-313-43] 

*336-51. h :. « ePilf sr . P, fi:e« . i; eNC ind - I'O . D : 

(P'a) (s'Kg) (5f'a) . = . (R'"a) (s'«g) (/S-'a) 

I- . *333-42 . *334-32 . *330-57 . *331-42 . D 

h :. Hp . D zTeic^.B'a = T'8'a . D . R'"a=T-"S-"a . 

[*334-131] D . (If'a) (s'/cg) (/S-"a) 

f- . (1) . *41-11 . D h :. Hp . D : (B'a) (s'/cg) (^'a) . D . (P'"a) (s'«g) (&'""») 



I- . (2) ^ . D I- : . Hp . D : (S"a) (s'/tg) (B'a) . D . (-S'-'a) (s'«g) (P-"a) 

I- . *331-42 . D h :. Hp . D : P'a = /Sf'a . D . P'-'a = S'-'a 
1- . (3) . (4) . *334-3 . D 

h :. Hp . D : ~ {(B'a) (s'«g) (fif'a)} . D . ~ {(P-'a) (s'«g) (,S'"a)} 
h . (2) . (5) . D h . Prop 



(1) 
(2) 

(3) 
(4) 



(5) 



SECTION B] the series OF VECTORS 401 

*336-511. \- :. K eFM sr . V el^C'md- I'O .■^ : RU,S . = . R-'U^S'' [*336-51-4] 

*336-52. !-:.«€ FM conx .Q,B,S,TeK .xe a'(Q | R) n a'(^ | T) . D : 

($ I i?) F, (S I r) . = . {S'R'x) (s'ks) (Q'T'x) 
Dem. 
h . *336-37l . D 

I- :. Hp . D : (Q I JS) F, (^| T). = . (gP) . P e /cg . Q'i2'a! = P'ST'a; (1) 

h . *330-56 . D I- :. Hp . P e «g . D : Q'iJ'a; = P'S'T'x . = . Q'R'co = S'P'T'x . 
[*7l-362] s . E'/B = Q'S'P'T'a; . 

[*330-54-56] =.R'x = S'Q'P'T'x . 

[*71-362.*330-5] = .S'R'a; = P'Q'T'a; (2) 

I- . (1).(2) . D h:.Hp.D:(Q|iJ)F,(S| r) . = . (^P).PeK^.S'R'x = P'Q'T'a;. 
[*41-11] = . {8'R'iJo)(s'KgXQ'r'a!):.0 \- . Prop 

*336-53. \-:.>ceFM conx . if, JV e /c. . D : MV,N . = . i^F«i^ 

Bern. 
h . *330-5-54 . D 
l-:Hp.Q.iJ,<S,r6K.M=Q|i2.JV = ^|r.ae s'Q"*: . « = Q'R'S'T'a . D . 

E ! J/'* . E ! iV'a; . E ! i^'a; . E ! ^'a; (1) 
h . (1) .*336-52 . D h :. Hp (1) . D : MV,N . = .(S'R'x)(s'ks)(Q'T'x) . 
[*330-5] s . (R'S'x) (s'Kg) (2"Q'a;) . 

[*336-52] =.(T\S)V,(R\Q). 

[Hp] = . NV^M (2) 

I- . (2) . *331-12 . D h . Prop 
*336-54. \-:KeFM conx . D"« = a"« . D . 

F. = M{J/,iV 6 «. : (gT) . Te «9 . Jl/ = r|i\r} 

Dem. 
h . *334-46 . D h :. Hp . ilf.iVe «. . D : 

{^T,x).TeKs.M'x = T'N'x. = .{'s^T).T6Ks.M=^T\N (1) 
h . (1) . (*336-01) . D h . Prop 

«336-6. \- iKeFM conx. O.V^QJ 
Dem. 
H . *331-23 . D h :. Hp . D : MV^N . D . (ga;) . il/'a; + iV'a; :. D h . Prop 

Observe that, by the conventions explained in *14, " M'x^N'x" implies 
E ! M'x . E ! N'x. From " (ga;) . ~ (ilf'a; = N'x) " we cannot infer ilf =j= N. 

R. & w. III. 26 



402 QUANTITY [PABT VI 

*336-61. h : « € FM conx trs . D . F, e trs 

Dem. 
\- . *330-612 . D 1- : Hp . i, ilf, iV"e«. . D . a ! a'Lna'M^a'N (1) 

I- .*336-37l . D I- : Hp .LV,M .MV.N' . aea'L n a'Mn O'iV. D . 

(L'a) (s'«9) (ilf' a) . (if 'a) (sVg) (iV'a) . 
[*334-14] D . (L'a) {s'icg) (N'a) . 

[(*336 01)] D.LV^N (2) 

F- . (1) . (2) . D h . Prop 

*336-62. h : « 6 ^il/ connex . D . F, e connex 

Dem. 
h . *330-61 . D h : Hp . i, il/6«, . D . a ! a'i n a'il/ (1) 

h . *334-24 . D h :. Hp . i, ilf e K, . a 6 Q'Z n Q'ilf . D : 

Z'a = JIf 'a . V . (i'a) (s'/cg) (if' a) . v . (M'a) (s'/cg) (Z'a) : 
[*331-42.(*33601)] ^:L=M.v. LV.M.v . MV,L (2) 

f-.(l).(2).DI-.Prop 

*336-63. hzKeFMsr.D.V^eSer [*3366-61-62] 

*336-64. y-:iceFMsi.D.U^eSeT [*336-63] 



*337. MULTIPLES AND SUB-MULTIPLES OF VECTORS. 

Summary of *337. 

In this number, we are concerned with the axiom of Archimedes and the 
axiom of divisibility. If k is a family of vectors, k obeys the axiom of 
Archimedes if, given any two points sc, a in the field of «, and any vector 
jB which, is a member of «, there is some power R" of R such that R'''a is 
later than x. That is, k obeys the axiom of Archimedes if, starting from 
any given point in the field, a sufficient finite number of repetitions of any 
given vector will take us beyond any other assigned point. A sufficient 
hj^othesis for this is that k should be serial and Cnv's'/cg should be semi- 
Dedekindian (cf. *214), i.e. we have 

*33713. h :. « e FM sr . P = s'k^. P e semi Ded .ReKg.ae C'F . D : 

xeG'P.D. (gi;) . i- e NC ind - I'O . xP (R'^a) 

' The hypothesis P = s'k^, which appears in the above proposition, is often 
notationally convenient. It will be observed that s'/cg gives us the series 
in the Opposite order to that in which it is usually wanted ; hence the intro- 
duction of the above relation P tends to avoid confusions. 

A family k is said to obey the axiom of divisibility when, given any 
member R of k, and any inductive cardinal v other than 0, there is a 
member L oi k such that L' = R. When this axiom holds, every vector 
can be divided into any assigned finite number of equal parts. We shall in 
the next Section (*351) define a family for which this holds as a " sub-multi- 
pliable family," denoted by " FM subm.'' For the present we are concerned 
to find a hypothesis as to s'/cg from which this property can be deduced. 
The hypothesis in question is that Cnv*s'«g is serial, compact, and semi- 
Dedekindian'; i.e. we have 

*337'27. V i.Ke FM sr . Cnv's'/eg e comp r> semi Ded . D : 

<S6«.i/eNCind-i'0.D.(ai).i6/«;.(S = i-' 

The proof proceeds by taking two points a, x in the field of k, of which a is 
earlier than x, and considering the class 

■K = K^r\R{{R'"a)Px], 

26—2 



404 QUANTITY [PAET VI 

i.e. the class of vectors such that v repetitions of them, starting from a, do 
not take us as far as x. It is easy to show that, when P is compact, this 
class has no maximum (*337"23), and therefore, when P is also semi-Dede- 
kindian, has a limit, whose vth. power is the vector which takes us from a to 
X (*337'26). Hence our result follows. 



*3371. h ■.KeFM.P = s'Kg.Reicg .aeC'P.O. R^'aCP"R^'a 

Bern. 
h . *9016 . *41-141 . D h : Hp . xR^a . y = R'x .1^ .ye R^'a . xPy . 
[*37i] D . a; 6 P"R^'a Oh. Prop 

^ -* — » 

*33711. h : « 6 FM connex asym . P = s'k^ .ReK^.ae C'P . D . seqp'R^'a = A 

Dem. 
h . *206-15 . D h : Hp , D . s'^p'R^'a=p'%'R^'a - P"p'P"R^'a (1) 

f- . *330-542 . *40-61 . D h : Hp . » ep^"R^'a .D.xe T>'R . 
[Hp] D . (go) .x = R'c. cPx (2) 

l-.*90-l72. DI-:cei25,t'a.D.E'cei2*'a (3) 

h . (3) . Transp . *200-5 . *334-5 . D I- : Hp (2) .« = iJ'c . D . c ~ e R^'a (4) 

1- . *37-l . D h : c 6 P"R^'a . D . (36) . b e E*'a . cPb (5) 

h . (5) . *208-2 . 3 h : Hp . ceP"R^'a .x^R'c.O. (36) . 6 eiiji^'a . xP (R'b) . 
[*90-l72] :i.x€P"R^'a (6) 

h . (6) . Transp . *200 53 . D I- : Hp (2). a; = iJ'c . D . c~eP"%'a (7) 

h . (4) . (7) . *202-502 . *334-24 . D h : Hp (2) .« = i?'c . D . c ep'P"R^'a (8) 

h.(2).(8). DhzHp (2). 0.x 6 P'Y'P"R^'a (9) 

h . (1) . (9) . D h . Prop 

*33712. b-.KeFMsr.P = i'«g .Pesemi Ded.i? e «g .aeG'P.'O.P"R^'a=C'P 

Bern. 

-> — » 

h . *337-l . D H : Hp . D . ~ a ! maxp'R^'a . 

[*205-7] 3 . ~ a ! iaa,Xp'P"R^'a (1) 

h . (1) . *206-33 . *33711 . D h : Hp . D . ~ g ! aeqp'P"R^'a (2) 

h . (1) . (2) . *214'7 . D h . Prop 

*33713. h :. « 6 Pil/ sr . P = s'«3 . P e semi Ded . i2 e «g . a e G^P . D : 

«; e C'P . D . (gi/) . 1/ e NC ind - t'O . a;P (R-'a) [*337-12 . *301-26] 



SECTION B] multiples AND SUB-MULTIPLES OF VECTORS 405 

*33714. t- : K e FM sr . P = s'«g . P e semi Ded .0 .U^e semi Ded 
[*336-462 . *ll4-r4-75] 

*337-2. I- : « 6i?'il/conx .LU.R . R=^I[s'a"K .O.LU,(R\ L) 
Bern. 

I- . *336-41 . D h : Hp . D . (gT) . Z, jB c « . r 6 «g . Z = T I i? . 

[*330-31] D . (gT) . Te/cg . B | Zi = T. i= T | fi . 

[*13-195] D.E|ie«:g.i = (^|Z)|i2. 

[*330-5.*336-41] D . Z CT^ (^ | i) : D I- . Prop 

*337-21. h : k eFM conx n ^ilf trs . E e«g . i/eNC ind - I'O-t'l . D.R-'U^R 

Dem. 

V . *334-162 . *301-23 . D I- : Hp . D . iJ" = R"-'^ \R (1) 

h.*334-131. Dh:Hp.D.iJ,iJ-',i?''-«i6«9 (2) 

h . (1) . (2) . *336-41 . D f- . Prop 

*337-22. h : K 6 JW sr . P = s'yeg . P e comp . aPx . i/ e NO ind - t'O . D . 

{'SR)'ReK .{R"a)Px 
Dem. 

h . *27011 . D h : Hp . D . (gy) . aPy . yPx . 

[*41-11] 0.(<^R,y).R€Ks.y = R'a.{R'a)Px (1) 

I- . (1)^ . D h : Hp . E e Kg . (i?-"a) Pa; . D . (gS) . S e «g . (,S"ii!-"a) Pa; (2) 

F . *336-64 . D h : . Hp (2) . /S e Kg . {S'R''a) Pa;.D:R = S .v .RU^S.w .S U^R : 
, [*336-511-4] D : B = >Sf . V . (E'+'i'tt) P ((Sf'iJ-"a) . v . (/S- +«i'a) P (S'R-'a) (3) 
f- . (2) . (3) . *334-3 . D h : Hp (2) . D . (a>S) . 5f e Kg . (/S''+«i'a) Px (4) 

h . (1) . (4) . Induct . D h . Prop 

*337-23. t- : Hp *337-22 . X = Kg n E {(iZ'-'a) Px}.D.X= U/'\ 

Bean. 
h .*336-511 . D h : Hp . JB e \ . SU^R . D . (;S''"a) P (i2-"a) . (iZ'-'a) P« . 
[*334-3.Hp] D.SeX (1) 

h . *337-22 .DI-:Hp.i2e\.3. (g^) . S e Kg . {S-'R'"a) Px . 
[*330-57-5.*334-13] D . (gfi') . P | /Sf e Kg . {(fl | ^f^'a} Pa; . 

[*336-41] D . (a;S) . P t 'S' 6 Kg . {(P I /S)'"a} Pa; . P [/■^(P | fif) . 

[*3r-l] ^.ReU,"\ (2) 

l-.(l).(2).Dh.Prop 



406 QUANTITY [PA«T VI 

*337-24. h:Hp*337-2&'.Z = tl(fr^y\.D.~{(i'"a)Pa;} 

Dem. 

V . *206-2 .Dh:Hp.D.Z~e\. 

[Hp] D . ~ {(L-'a) Pw} Oh. Prop 

*337-241. 1- : Hp *837-24 . D . ~ {a;P (Z-'a)} 
-Dem. 

I- . *337-2-23 .Df-:Hp.i26\.D.E|i6X. 

[*332-53-241 .*334-131] . R\ L eX .(R\ L)" = R" \ L" . 

[Hp] D.(P'"X'"a)P«. 

[*71-362.*41-11] D . (L-'a) P{R'"x) (1) 

I- . *337-23 . D I- : Hp . -B 6 «9 - X . D . ~ {Z f/,i2} . 

[*336-.511] D . ~ {{R-'a) P (Z-'a)} . 

[*330-5.Hp.*334-14] D . ~ {(J?"'*) P (I/'a)} (2) 

h . (1) . (2) . D h : Hp . D . ~(ai2) . P e «g . (R"a;) P {L"a) . 

[*337-22.Transp] D . ~ [a;P (/."'a)} Oh. Prop 

*337-25. h : Hp *337-24 .D.L- = k^ A^'x 

Dem. 

h . *337-24-241 . D h : Hp . D . L'''a = a; : D h . Prop 

*337-26. h : Hp *337-23 . Pe semi Ded . D . {tl (f7,)'\|- = k^IA^'x 

Bern. 

h . *337-21 . D h : . Hp . D : E 6 \ . Db ■ (M'a) Px : 

[*336-4] D : « 1 Aa'x€p''u,"X (1) 

h . (1) . *337-23-14 . D h : Hp . D . E ! tl ( U,y\ (2) 

h . (2) . *337-25 . D h . Prop 

*337-27. h :. K e PM sr . Cnv's'/cg e corap n semi Ded . D : 

/S 6 /t . z; 6 NC ind - t'O . D . (gX) .LeK.8 = L-' [*337-26] 



SECTION C. 

MEASUREMENT. 

SumTnary of Section C. 

In this Section, the " pure " theory of ratios and real numbers developed 
in Section A is applied to vector-families. A vector-family, if it has suitable 
properties, may be regarded as a kind of magnitude. In order to derive from 
the "pure" theory of ratio a theory of measurement having the properties 
which we should expect, it is necessary to confine ourselves to some one 
vector-family; that is, instead of considering the general relation X, where 
X is a ratio, we consider the relation X^ k, where k is the vector- family in 
question; or sometimes we consider X^k„ or sometimes X ^ (/t w Onv"*). 

Concerning ratios with their fields thus limited, which are what we may 
call " applied " ratios, we have to prove various propositions. 

(1) No two members of a family must have two different ratios. This 
is proved, for an open and connected family, in *350'44!. 

(2) All ratios except Og and oo g must be one-one relations when limited 
to a single family. This is proved, for an open and connected family, in 
*350'5; with the same hypothesis, Og is one-many (*35051). 

(3) The relative product of two applied ratios ought to be equal to the 
arithmetical product of the corresponding pure ratios with its field limited, 
i.e. if X, Y are ratios, we ought to have 

Z^« |7^«=(Zx,F)C« 
or XlK,\7lK, = (Xy,Y)lK,. 

That is to say, two-thirds of half a pound of cheese ought to be (2/3 Xg 1/2) 
of a pound of cheese; and similarly in any other ease. For any open connected 
family, we have (*350"6) 

XtK,\7tK,(l(Xx,Y)lK,, 
but in order to obtain an equation instead of an inclusion, it is necessary 
(*351"31) that K should be " submultipliable," i.e. that if R is any member 
of K, and V any inductive cardinal other than zero, there should be a member 
of K whose vth power is M. The class of such families is denoted by 
" FM suhm," and considered in *351. 



408 QUANTITY [part VI 

(4) If X, Y are ratios,, and T is a member of the family k, we ought 
to have 

{X i k'T) I (Ft: k'T) = (X +, Y) t k'T, 

that is, two-thirds of a pound of cheese together with half a pound of cheese 
ought to be (2/3 +« 1/2) of a pound of cheese, and similarly in any other 
instance. This property is shown, in *351"43, to hold for any open connected 
submultipliable family in which all powers of members are members. In any 
open connected family, if R, 8, Te k, we have 

RXT . SYT . D . (i2 1 *Sf) (Z +, F) T (*350-62). 

The remainder of the hypothesis of *351'43 is required in order to prove 
(a) that Xp/cT, F^AT'Tand (X+.F) ^kT exist,(6) that (Zp«'2')|(Fp«'r), 
which is the i2 1 /S of *350'62, is a member of k. As applied to «„ we have 
to take the representative (cf. *332) of the relative product; if i e /c., we have 
(*351-42) 

rep/{(Z I K,'L) i (Yt «.'£)} = (X +, F) t k,'L, 

provided k is open and connected and submultipliable. 

The fact that the above propositions can be proved for suitable vector-- 
families constitutes the reason for studying such families, as we did in 
Section B. The proof of the above propositions, together with other 
elementary properties of applied ratios, occupies the first two numbers of 
this Section. 

We proceed next (*352) to consider all the rational multiples of a given 
vector in a given family, i.e. all the members of a given family « which have, 
to a given vector T, a ratio which is a member of C'H', or, alternatively, all 
the members of k, which have to T a ratio which is a member of G'Hg. It 
will be observed that, in virtue of *307, if R and 8 have a ratio X which is 

a member of C'H', R and 8 have the corresponding negative ratio X \ Cnv. 
The members of k which have to T a, ratio which is a member of C'H' are 
those vectors R for which we have 

(gZ) . Z 6 C'H' . RXT, 
i.e. using the notation of *336, those for which we have 

(,^X).XeC'H'.RATX. 

Thus they constitute the class 

K n Aj."C'H'. 

Assuming that TeK, the vector which has the ratio Z to T is «'] ^.y'Z. 
This is the vector whose measure is Z when yis the unit. Thus k^ Af^G'H' 
is the correlator of a vector with its measure. It is easy to prove (*352"12) 
that k'\ Aj,\C'H' is one-one. 



SECTION C] MEASUREMENT 409 

We can arrange the vectors which are rational multiples of T in a series 
by correlation with thfir measures, putting vectors with smaller measures 
before those with larger measures. The ordering relation is T^, where 

T.^k^^At'^H' Df 

Similarly the members of k^ which are positive or negative rational multiples 
of T may be ordered by the relation T„, where 

T,, = K.'\A^iHg Df. 

We prove that change of units makes no difference to T,, i.e. if S is any 
member of k which is a rational multiple of T, then 8^ = T^ (*352-45). The 
corresponding proposition holds for T^, if S has a positive ratio to T, but if 8 

has a negative ratio, *S„ = T,, (*352-56-57). 

If K is a serial family, T^ is the converse of U^ (cf. *336) with its field 
limited to rational multiples of T (*352-72). This proposition connects the 
generalized form of greater and less represented by {/„ with the form of 
greater and less derived from greater and less among the measures of vectors, 
since it shows that, in a serial family, the vectors which have greater measures 

come later in the series C7'„, and those with smaller measures come earlier. 

We next proceed (*353) to consider "rational" families. These are 
families in which every member is a rational multiple of some one unit T, 
i.e. in which 

('S^T).T6Ks.kCAj,"G'H'. 

It is obvious that, given any family, the rational multiples of one of its 
members constitute a rational sub-family. In a rational family, rationals 
are sufficient for measurement, and irrationals are not required. If the 
family has connexity, it will be serial; in fact, if T is one of its vectors and 
a is a member of its field, we have (cf. *353"32'33) 

Uk = k'] Aj:' H' . s'Kg = AJ K'] Aji'H'. 

Thus both Ux and s'k^ are ordinally similar to H' ^ Ax"k. If k is sub- 

multipliable, U^ is ordinally similar to H' (*353'44). 

We proceed next (*354) to consider " rational nets," which are important 
in connection with the introduction of coordinates in geometry. A rational 
net is obtained from a given family, roughly speaking, by selecting those 
vectors which are rational multiples of a given vector, and then limiting their 
fields to the points which can be reached by means of them from a given 
point. In order to make this more precise, we proceed as follows: Let us 
define as the "connection" of a with respect to k the class Aa"ic„ i.e. all the 
points which can be reached from a by a member of k^. We will now define 
as the " a-connected derivative of k " the class of relations obtained by limiting 



410 QUANTITY [part VI 

the field of every member of k to the connection of a with respect to k. This 
class of relations we denote by cxa'/e, putting 

CX„'K=^(^a"«0"/C Df. 
Instead of k, we take, in order to obtain a rational net, all the rational 
multiples (in k) of a given member T of k, i.e. G'T^. Then cXa'G'T^ is a 
rational net, namely the rational net associated with the origin a and the 
unit vector T. 

In proving propositions concerning the rational net cXa'C'T^, we often 
require the hypothesis that « is a group. In order to avoid having to make 
this hypothesis concerning our original family, we construct a closely allied 
family, which is always a group when k is connected. This family, which we 
call Kg, is obtained from k by including the converses of those members of k, 
if any, whose domains are equal to their converse domains, i.e. we put 

«3 = « w Cnv"(« " D Va"K) Df. 
Then if « is a connected family, Kg is a connected family wliich is a group 
(*354!-14-16), and (Kg\ = «. (*354-15). Then putting X = Kg, we take cxa'G'Ti, 
rather than cXa'C'T^ as the rational net to be considered. If k is an open 
and connected family, this rational net is a family which is open, connected, 
rational, transitive and asymmetrical (*354'41). 

We proceed next (*356) to the application of real numbers to vector- 
families. For the application of real numbers, it is essential that our family 
should be serial. Given a serial family in which a given vector 8 is the limit 
(in the series U,,) of a set of vectors which are rational multiples of another 
vector R, it is natural to take as the measure of 8, with the unit R, the limit 
of the measures of the vectors whose limit is 8. It is convenient to take our 
real numbers in the relational form given in *314, i.e. if ^ is a segment of H, 
we take s'^ as the corresponding real number. Thus positive real numbers 
are the class s"G'@,. while positive and negative real numbers together with 
zero are the class s'^G'®g. If f eC©, a vector which has to i? a ratio which 
is a member of ^ has a measure which is less than s'^. The class of all such 

vectors is s'^'R, i.e. if X = s'f , it is X'R. The limit of such vectors in the 

series U,^, if it exists, will naturally be taken as the vector whose measure is 

X. Remembering that U,, proceeds from greater to smaller vectors, we see 

— > 
that the first vector which is greater than every member of X'R will be the 

— » 
lower limit of X'R with respect to TJ^- Hence, if we write X«'i? for the 

vector whose measure with the unit R is X, we have 

X/i2 = prec(C/^,)'Z'i2. 
Hence we may take as our definition of X, 

Z, = prec(f/'«)|X|'« Df. 
Then X« is an " applied " real number. 



SECTION C] MEASUREMENT 411 

The properties to be proved concerning applied real numbers almost all 
require that the familJ>to which they are applied should be serial and sub- 
multipliable, and most of them also require that Cnv's'/cg should be semi- 
Dedekindian. Assuming this, we can prove that, if X, Yes"G'@, X^^k is 
one-one, and, with various hypotheses, 

iXlK)\{YlK) = (XxrY)lK (*356-31), 

X,\7, = {XxrY), (*356-33), 

(X/R) I ( r^'R) = (Z +r YyR (*356-54). 

These are the essential properties required of measurement, as in the 
analogous case of ratios. 

We might proceed to consider "real" multiples of a given vector, and 
" real " nets. But these subjects have less importance than in the analogous 
case of rationals, and are therefore not discussed. 

The Section ends (*359) with a number on existence- theorems for vector- 
families. The most important of these are derived from rationals and real 
numbers. The family whose members are of the form {+gX)^G'H', where 
XeG'H', is initial, serial, and submultipliable (*359'21). The family whose 
members are of the form (-1-^ /i) ^ G'&, where ^ e G'@', is initial, serial, and 
submultipliable, and has Cnv's'«;g = @', so that Cnv's'«g e semi Ded (*359'31). 
Finally we prove that the properties of families are unaffected by the 
application of correlators, whence it follows that, given any series P whose 
relation-number is l+rj, or is 6' where 6' + 1 = 6, there is an initial serial 
submultipliable family k such that Cnv's'Kg = P. Such a family may be 
used for the measurement of distances in P. 

It is of some interest to observe that, given a suitable family k, ratios 
with their field limited to Kg form a family whose field is /eg. In this family, 
the zero vector is (1/1) ^«g, and the family is connected if k is a rational 
family. If we wish to obtain a serial family, we must limit ourselves to 
ratios not less than 1/1, i.e. to 

t:«g"^*'(l/l). 

This family is serial, and if we call it \, we have (with a suitable hypothesis) 

It is necessary, however, if we are to obtain a family, that our original family 
should be submultipliable, since otherwise we do not necessarily have 
CI'X^«:g = Kg. For this reason, we cannot use the family of ratios without 
a frequent loss of generality in the resulting theorems. 

The theory of measurement developed in this Section is only applicable 
to open families. The application of ratio to cyclic families is more complicated, 
and is considered separately in Section D. 



*350. RATIOS OF MEMBERS OP A FAMILY. 

Summary of *350. 

In this Dumber we introduce no new definitions, but merely bring together 
the propositions of *303 on the pure theory of ratio, and the propositions of 
*333 on powers of vectors in open connected families, especially *333'47'48. 
We thus find that, if k is an open connected family, and fi, v are inductive 
cardinals which are not both zero, 

M [iiilv) lK:\N. = .M,NeK,.^\M''hNi'. (*350-4) 

= .M,NeK,. rep/if " = rep/iV'' (*350-41), 
while if R, T are members of k, 

R (fi/v) T. = .R' = Ti' (*3.50-43). 

We prove also, by means of *333"53, that if L and M are members of w, other 
than I f" s'(l"ic, they cannot have more than one ratio, i.e. 

*350-44. h : « 6 -FM ap conx .X,Ye G'H' . g ! Z ^ K^g n F ^ /e.g . D . Z = F 

We next prove that any ratio other than 0, and oo g becomes one-one when 
its field is limited to /c, (*350"5), while 0, becomes one-many (*350'51) and 
X q becomes many-one (*350'511), Og being in fact the ratio of the zero vector 
r[s'(I"K to any member of /£„ and oo q being the converse of Og. 

We consider next the multiplication and addition of ratios, but in this 
subject we cannot obtain some of the main theorems without the hypothesis 
that our family is submultipliable (introduced in *351). In the present 
number, we prove that, if k is an open connected family, and /j,, v are inductive 
cardinals other than 0, 

(/i/l)t«.|(l/«.)p«.C(/./^)p«. (*350-53), 

{II v) I K, I (fjL/1) t «. = (fi/u) t /.. (*350-54), 

(ji/1) I K, I ivjl) I «. = {{ti x„ v)l\] I «. (*350-5o), 

and (l/yii) I K, I (1/z.) t «c = {l/(/i Xe v)] I «. (*350-56). 

Hence we find that, if Z, F are ratios other than Og and oo q, 

Z ^ K. I F t «. G (Z X, F) t /cc (*350-6), 

while if R, 8, T are members of k, 

RXT.SYT.-2.(R\S)(X+s7)T (*350-62), 
and if L, M, N are members of k^, 

LXN . MYN . D . {rep/(i; | M)} (Z +s F) N (*350-63). 



SECTION C] RATIOS OF MEMBERS OF A FAMILY 413 

We then prove similar results for subtraction, and thus arrive at the following 
proposition concerning ^neralized addition of positive or negative ratios : 

*350-66. hiKeFMa.^ conx . L, M, JTe «. . X, Ye G'Hg . LXN .MYN.D. 

iep.%L\M) = (X+gY)tKjN 



*3501. I- : /e e FM ap . D . /e^ C Rel num id . /c,g C Rel num 
Bern. 

H.*333-101. DI-:Hp.Z6«:.3.3.L6l-»l.ipoeJ (1) 

h . (1) . *300-3 . D h : Hp . D . /e^g C Relnum (2) 

h.*3331-101. Dh:Hp.2;e«.-/«:.g.D.ie/. 
[*300-325] D.ieRelnumid (3) 

1- . (2) . (3) . D I- . Prop 

ii^50'2. I- : K e FM ap conx . g ! K^g . D . Infin ax 

Bern. 
h . *330-624 . *333-15 . D h :. Hp . Z e /c^g . D : A~e finid'Z : 
[*121-1112] D : 1/ e NC induct . D, . (ga;, y) .L{xh-iy)ev+al: 

[*120-3] D : Infin ax :. D h . Prop 

*350-21. hialJWapconx-l.D.Infinax [*334a8 . *350-2] 

*350-31. h :. « e ^Jlf ap conx .fi,ve NC ind - 1<0 . ilf, iVe «:,g . D : 

■ M(/i/v)N. = .'3_lM''nNi' 
Bern. 
h . *3031 . (*3020203) . *1 13-602 . D 
h :: Hp . D :. M(jj,/v)N.= : (gp, a;r).p Prin v. re NC ind - I'O . 

[*333-48] = : (a/3, <r, t) . p Prm <r . T e NC ind - t'O . p + . o- + . 

/i = /3 Xe T . v = o- Xg T : a ! Jf" «S iV'' : 
[*I 13-602.(*302-0203)] = : (gp, a) . (p, o-) Prm (/i, v) : g ! Jlf " n iV'' : - 
[*302-36] = : a ! JIf " n iV'' : : 3 I- . Prop 

*350'32. I- : . Hp *350-31 .0:M (fi/v) N. = . rep.'ilf " = rep/i\/''* 
[*350-31 . *333-47] 

*350-33. I- :. « 6 FM ap conx ./M.ve'NG ind - t'O . M=I[s'<I"k . JVe «. . D : 

M(/ji/v)N. = .M = N. = .'3^lM-'nN'' 
Bern. 
h . *301-3 . *333-2 . D I- :. Hp. D : o-eNOind- I'O . D . M » = ilf (1) 

F- . (1) . *303-l . D 

h :. Hp . D : Jf (fi/v) N. = . (gp, ff) . (p, ff) Prm (/*, v) . g ! if A JV" . 
[*333-101] = . (ap, ff) . {p, ff) Prm (/i, v).M=N. 

[*302-36] =.M=N. (2) 

[(l).*331-42] =.a!il^''«-ZV'' (3) 

h . (2) . (3) . D h . Prop 



414 QUANTITY [part VI 

*350-331. h :. AC 6 Fif ap conx . /t, v e NO ind - I'O . ilf e k. . JV = / f' s'a"K . D : 
M (/t/i/) iV". = .Jf=iV. = .a!ilf-niV'' [*350-33 . *303-13] 

*350-34. \-:.Ke FMap conx . i/ e NC ind - I'O .M,NeK,.D: 

Bern. 
f- .*303-151 . :>h :. Hp . D : M{0/v)N. = . il/G/. g ! G'MnC'N. 
[*330-43-61] = . M = I[s'a"K :. D I- . Prop 

*350 35. h :. xeFMap conx . i/ e NC ind - I'O . M,NeK,. D : 

Bern. 

h . *301-2 . D h :. Hp . D : a ! Jtf - n iV» . = . a ! il/-' n / [^ s'(1"k . 
[*333-101.*331-12] =.M=Ils'a"K (1) 

h . (1) . *350-34 . D h . Prop 

*350-351. h :. /c 6 FMwp conx . fi e NC ind - t'O . D : 

i/ (/i/0) iV . = . iV= / I' s'a"/e [*350-35 . *303-13] 

*350-4. h:.«ei'il/apconx,/i, i/eNCind.~(/i = i/ = 0). D: 

M {(fi/v) lK,]N. = .M,N6K,.'s_\M^hNi' [*350-31-33-331-35-351] 

*350-41. h : . Hp *350-4 . D : il/ {{fijv) ^ «.) iV . = . if, iV e «■. . rep/if ^ = rep/if -^ 
Z)em. 
h . *332-243 . *301-3 . D h : Hp . ilf = / I' s'a"« . D . rep/ilf - = il/ (1) 
1- . (1) . *350-33-331-32 . D I- .Prop 

*350-42. h : . Hp *350-4 .Q,R,S,Teic.D: 

{Q \ R) i^ilv) {S\T). = .Q-\R- = h\Ti' [*350-41 . *332-53] 

*350-43. I- :. Hp*350-4 .R.TeK.'^iR (fi/v)T. = .R^=Ti' 

*350-44. I- : xeJ^ifapconx . X, YeC'E' . g ! Z p K^g n F^ «.g . D . Z = F 

Dem. 
h . *350-4 . D h : Hp . D . (gii, M, /j,, v, p,<T).L,Me K.g . 

a ! Z" n il/p . g ! Z- n if (^ . Z = ^/«; . F= p/<7 , 
[*333-53] D.fix^a- = vx^p.X = fi/v.Y=p/a. 

[*303-39] D.Z=F:DF. Prop 



SECTION C] RATIOS OF MEMBERS OF A FAMILY 415 

«350'5. h :«6^il/apconx./i, veNCind-t'O. D . (/it/i;) ^ «. e 1 -» 1 

Dem. • 

h.*350-41.Dl-:.Hp.D: 

L,M,NeK,.L {(i/v) N . M (fi/v) N.D. rep/i" = rep/iV" = rep/if " . 
[*333-41] ^.L = M (1) 

h.(l). Df-:Hp.D.(/i/j/)t«.6l->Cls (2) 

Similarly I- : Hp . D . (;n/j/) ^ k. e Cls -> 1 (3) 

F- . (2) . (3) . D h . Prop 

*350-51. h : K e i^lf ap conx . i; e NC ind - t'O . D . 

(O/i/) D /<. e 1 -> Cls . a'(0/j;) D «. = «. . D'(0/i') i «. = i'/ 1^ s'G"k [*350-34] 

*350-511. h:Hp*350-51.D. 

(v/0) C «. 6 01s -» 1 . D'(i;/0) D «. = «. . a'Cv/O) t «. = I'/ r «'a"ye 
[*350-51.*303-13] 

*350-52. 1- : « e ^M ap conx . X e G'H .D.X^K.el-*! 
[*350-5 . *304-34 . *383-2] 

*350-521. I- : « 6 JW ap conx . Z e G'H' . D . Z C «. e 1 -» Cls 
[*350-52-51 . *303-l] 

*350-53. h : Hp *350-5 . D . {(/t/l) t «4 I {(!/") D «•} ^ (W") D «' 

Dem. 

h .*350-4 . D h : Hp.i {(;tt/l)D «4 -^- -^RlMD ««} -^- ^ • 
L,M,NeK,.'3_lLnMi--.'^lN^M-'. 

[*333-48] D . i, ilf, iV 6 «, . a ! i- A Jf'^'^"'' . a ! i^" A M'^>^-' - 

[*333-47] D.L,M,NeK,. rep/Z" = rep/Jf^*^-^" = rep/iV^^ . 

[«350-41] D . i {(/*/") D «.} -^ : 3 •■ • ^'I'^P 

*350-54. h : Hp *350-5 . D . {(l/v) t «.} | {(/i/l) D «'! = (/*/") t «• 

Dem. 
h . *350-41 . *332-241 . D 

I- :. Hp . D : X [{(!/«;) D «.} | {(m/1) D «')] -^ ■ = ■ 

(gJlf ) . Z, if , JV e «. . rep/Z- = iW = rep/J\^'' - 

[*332-22] =.L,NeK,. rep/X' = rep/i\r'' - 

[*360-41] = ■ L (fijv) N :.:>}■ . Prop 



416 



QUANTITY 



[PAKT VI 



*350-55. h : Hp*3o0-5 . D . {(/t/1) t k.} \ {(v/1) f «.} = {(^ x„ v)/l} t «. 

= K''/i)D«^}|{Wi)D«.} 

h . *350-4 . D I- :. Hp . D : X [{(li/l) ^ «.} | {(k/I) p «.}] iV . = . 



[*333-47] 

[*333-21] 

[*333-47] 

[*333-24] 

[*350-41.*3015] 

f-. (1).*1 13-27. Dh. Prop 



. Z, iV e /e. . a ! i « (rep/i\r'')'' . 

.L,NeK,.L = rep<'{(rep^'i\7'''/} . 

.L,NeK,.L = vb^^^N^Y . 

.L[{(vx,^)/l]tK.]N (1) 



*350-56. h : Hp *350-5 . 3 . {(1//.) ^ «.] | {{1/v) t «.} = {l/(/. x^ i;)} t: «. 

= {(1/v) I «4 I {(l//i) p K.} [*350-55 . *303-13] 

*350-6. I- ZKeFMapconx.X, YeG'H.:) . (Z^ «OI(I^P«OG(X x, F) t «, 

Dem. 
h . *304-34 . D 

1- : Hp. D. (a/i,v,p,(7)./i,K, (0,0- 6 NC induct -i'O.Z = /t/i/. Y=p/<T (1) 
h . *350o4 . D f- : K 6 FM ap conx .fj,,v,p,cre NC induct - I'O . D . 

{(W*-) D «.} I {(pH t «4 = {(1/^) D «.} I (Wi) t «.} I {(i/<r) D «. }i {(/j/i) D «4 

[*350-53-54] G {(1/^) I «.} | {(l/<r) ^ «.} | {(/./l) D «.} | {(/,/!) ^ «.} 

[*3o0-56-55] G {l/(i/ x. a)} t «, | {(/. x„ p)/l} ^ «. 

[*350-54] G {(/.x„ /,)/(«; x„ ,7)} C«. 

[*305-14] Gl/./z;x,p/<r}D«. (2) 

I- . (1) . (2) . D I- . Prop 

*350-61. h -..KeFMapconx . Xe G'H.D :M=(XIk:)'N . = . iV = (ZC «,)'if 
[*350-52] 



Dem. 



*350-62. H : « ei^if ap conx . X, Ye G'H' .R,S,TeK. RXT.SYT. D . 

(R\S)(X+sY)T 
[-.*350-43.DI-:Hp.X = /i/i/. F=p/(r.D. 

[*301-5] D . JS"^"' = Ti^^"" . ,S'"<'='' = T^xcp . 

[*330-57] 3 ■ (-R I /S)''><°°- = ytf^x^-fJ+cC-xcp) . 

[*350-41.*306-14] 3 . (i? | fif) (Z +, F) T : D h . Prop 



SECTION C] RATIOS OF MEMBERS OF A FAMILY 417 

*350-63. hiiceFMa,^ conx . X, Ye G'H.L,M,Ne>c, . LXN.MYN. D . 

{rep/(X|Jl/)}(Z+,F)iyr 
Dem. 
V . *360-41 . D 

h : Hp . X = /i/v . F= p/o- . D . rep/i" = rep.'iV^^ . re^^'M' = rep/Nf . 
[*332-81] D . rep/i-X""' = rep^'i\f "><'='' . rep/i/''x«"-.= rep/iV-X"" . 
[*332-33] D . rep/(i/'"<"'' | M-""'") = rep.'iV<'^X'""+«<'"<"''i . 
[*332-8] D.rep/(X| Jf)''X'='' = rep,'i\r'^X'='^+'=('"<'=p) . 
[*332-82] D . rep/{rep/(i | ¥)}•"'•'' = rep/iV^xcoi+c (-xop) . 
[*350-41] D . {rep/(i | M)} [{(jjl x, <r) +„ (i- x„ p)}/{v x„ ,r)] J\r . 
[*306-14] D . {rep/(i | Jlf )} (Z +, F) iV : D h . Prop 

*350-64. h : Hp *350-63 . XHY . D . {rep/(Z | M)} {Y-,X)N 
Dem. 

h . *33215-81 . D f- : Hp . D , rep/L"'Xc<r = Cnv'(rep/i)''X"'' (1) 
Thence the proof proceeds as in *350"63. 

*350-65. f- : Hp *350-62 .0 .{R\8){Y -,X)T [*350-64 . *308-21] 

*350'66. V-.KeFMs.-^ conx .L,M,N eK,.X,Y e G'Hg . LXN . MYN . D . 

rep/(L|ilf) = (X+gF)^«/iV 
Dem. 
V . *350-63 . D 

h :. Hp . ir = rep/(X I M).:>:X, YeC'H . D . 1F= (Z +, F) ^ ic.'N (1) 

y.*S50-6i.O\-:Rf(l).XeG'H„.YeG'E.-^.W =(Z+jF)t«/iV (2) 
h . *350-63 . *3071 . D h : Hp (.1) • X, Ye G'H^ .:> .W={X +gY)l k,'N (3) 
h . *350-34 . D h : Hp . Z= Oj . D . rep/(Z | il/) = il/ 

[*308-51] = (X +g F) C «/iV (4) 

Similarly F : Hp . F= 0^ . D . rep,'(i \M) = {X +g F) I kJN (5) 

h . (1) . (2) . (3) . (4>. (5) . D h . Prop 



R. & W. IIL 27 



*351. SUBMULTIPLIABLE FAMILIES. 

Summary of *351. 

A " submultipliable " family is one in which any vector can be divided 
into V equal parts (where v is any inductive cardinal other than 0), i.e. in 
which, it Re K, there is a vector S which is a member of k and is such that 
S' = R. The definition is 

*351-01. FM suhm = 

FM n ii{Re K .V e'NCind- I'Q .Ds,..('a.S) . SeK . R = S''] Df 

In open families, such as we are considering in this Section, S will be unique 
when R and v are given. But in cyclic families, as we shall show in 
Section D, there will be v values of S. For example, let « be a family of 
angles. Then the vector-angle 2fnrjv has its vth power equal to 27r for any 
integral value of fi, since 2/i7r is the same vector as 27r ; and 'ifnrjv has v 
different values, since, considered as a vector, any angle 6 is identical with 
6 + 2ir. In the present Section, however, these complications are excluded, 
owing to the fact that we confine our attention to open families. 

In virtue of *337'27, a family is submultipliable if it is serial and 
Cnv*i'«g is compact and semi-Dedekindian (*35111). 

When « is a family which is open, connected, and submultipliable, if 
LeKi and fi e NC ind — t'O, we have 

(aJ/).ilfe«..rep/ilf'' = Z (*351-2). 
Hence if X is any ratio (excluding x g, now and always henceforth), we 
have 

E!Zt«;/X (*351-21). 

In order to obtain the same result for k, we have to assume that all powers 
of members of k are members of k (*351'22), but we can obtain the same 
result for k w» Cnv"* without this assumption (*351*221), because of *331'54, 
which shows that in any connected family all powers of members of «w»Cnv"« 
are members of k w Cnv"*. 

In virtue of the above propositions, the propositions on products and 
sums of ratios, which in *350 only stated inclusions, now state identities. 
Thus if X, YeC'H', we have 

(XlKdliYl K,) = (Z X, Y) t K, (*351-31), 
rep/{(Z I K,'L) \(Yl «.'i)} = (Z +, F) I k,'L (*35r42), 



SECTION C] SUBMULTIPLIABLE FAMILIES 419 

where Ze/e^; also 

rep,' {{X I «/4) \{Yl «/i)} = (Z -, 7) t «.'-£ (*351-45). 

The corresponding propositions for ratios confined to k instead of to k^ 
require the additional hypothesis s'Pot"« C k, because this hypothesis is 
required in *351-22; on the other hand, in the analogue of «351'42 "rep," 
does not appear, and we have (with the above hypothesis) 

(X I k'R) \(Tl k'R) = {X+,Y)l, k'B (*351-43), 

where Ren. For ratios confined to k w Cnv"« instead of to k, the corre- 
sponding result can be proved without the hypothesis s'Pot"* C k (*351"431). 
It will be observed that the hypothesis s'Pot"« C k is satisfied if k is a group, 
though it may also be satisfied when k is not a group. Since a transitive 
connected family is a group, a transitive connected family always satisfies 
s'Pot"* C K, as has been proved already (*334*132). 



*35101. FMs\xhm = 

FMn lc{ReK.vel>iCmd-i'0. Djj,..(a/S) .SeK.R = S'} Df 

*3511. h :. « 6 FM suhm . = : « e FM iReK.ve NC ind - t'O . Djb,, . 

(•S^S).8e>c.R = S' [(*35101)] 

*351101. h : a ! i?'if subm . D . lufin ax [*3511 . *30116 . *300-14] 

*351'11. \- :k€ FM sr . Cnv's'/cg e comp n semi Ded . D . « e FM subm 
[*337-27] 

«351'2. h :. « e FMap subm conx . D : /i e NO ind - I'O . Z e k. . D . 

(gilf ) .Meic,. rep^'Jlf" = L 
Dem. 

h . *3511 . 3 h : Hp . /iie NC ind - t'O . Q,-B6«: . i = Ql i? . D . 

[*332-53] D . (a<S, r ) . S, r e « . i = rep/(S | 7)^ : D h . Prop 

*351-21. h : Hp *351-2 . X e G'H' .Lbk,. O .ElX^ k,'L 

Dem. 
|-.*351-2.*332-61.D 

h : Hp . /x,i; 6 NO ind -t'O . Z = /i/i' . D . (gi/) . ilf e «. . rep^^'if" = rep/X" . 
[*350-41-5] D.ElXtK^'L ' (1) 

l-.*350-34.D 
|-:Hp./i = 0.i/eNOind-t'0.Z = /*/j;.D.XC«.'i/ = /ts'a"/e (2) 

I- . (1) . (2) . D I- . Prop 

27—2 



4,20 QUANTITY [part VI 

*351-22. \-:B.p*351-2. s'Pot">cCK. XeC'H' .BeK.D.ElXlK'R 

Dem. 
V . *301-22 . D h : Hp . /i,, i; € NO ind . V + . D . i2^ 6 « . 

[*351-1] D.(aS)-'Se«-^ = 'S''- . 

[*350-4.*331-12] 0.('3iS).SeK.8i/j,/v)B (1) 

I- . (1) . *350-521 . D f- . Prop 

*351-221. h : Hp *351-2 . X e G'H' . \ = « u Cnv"« . Re\.D .ElX^X'R 
[Proof as in *351-22, using *331-54,] 

*351-3. h : Hp*351-2 . /t, j; e NC ind . v + O . D . 

{(/./i)t«.}|{(i/'')D«4 = (W'')D«-' 

Dem. 
\- . *350-41 .DI-r.Hp./i + O.D: 

i {(fi/v) t K.} iV . = . X, iV e «. . rep/Z" = rep/iV" . 
[*351-2] = . (gi/) .L,M,NeK,.L = rep/il/" . rep/i- = rep/iV"" . 
[*333-24] = . (gJl/) .L,M,NeK,.L = rep/ J/" . rep/ilf ''^"- = rep/iV'^ - 
[*333-44] = . (ail/) .L,M,NeK,.L = rep/ J/" . rep/i/" = rep/iV . 
[*3.50-41] = . (^M) . L {(/./I) tK.}M.M {(1/v) [. «.} N (1) 

1- . *350-34 . D f- :. Hp . /* = . D : 

Z {(^/i;) C «.} iV^ . s . L = / 1^ s'a"«: . iV 6 «. (2) 

I- . *350-34 . *351-21 .DI-:.Hp./t = O.D: 

z KWi) t «4 1 {(!/") D /^a -^ • = ■ -^ = -^ r s'a"« ■ -zv" e «. (s) 

I- . (1) . (2) . (3) . D 1- . Prop 

*351-31. h : Hp *351-2 .X,Y€ G'H' .D .(XI k,)\{YI k,) = (X x,Y)l «, 
[Proof as in *350'6, using *351-3 instead of *350-53] 

*351"4. \- : Ke FM ap subm conx . fj.,v,p,ae NO ind .i'4=0-<''=fO.Ze«:. .3. 

rep/[{(^/i;) t k.'L} \ {(pja) ^ «/Z}] = {^^Iv +. pja) I k.'L 
Dem. 

h .*350-41 . D H : Hp . //,+ . p^^O . M = {n/v) [, k,'L . D . rep/lf " = rep/Z" - 

[*333-44] D . rep/ilf •"<-=■' = rep/Z^^-'' (1) 

Similarly 

|-:Hp./i + 0./3=|=0.iy^ = (|o/o-) t Kc'Z . D . rep/iV'X'^-' = rep/Z-x-^o (2) 

h . (1) . (2) . *333-34 . *332-33 . 3 

I- : Hp(l) . Hp(2) . D *rep/(ilf I iV)''X'='' = rep/{Z<^>'<"'' | Z""'"")} . 

[*301-23.*333-24] 3 . {rep/(il/ 1 N)]'"''"' = rep/Z»'><«"i +cCXcp) . 

[*306-14.*3o0-41] D.reTp,'(M\N) = (fi/v+,p/a)lic,'L (3) 

f- . (3) . *351-21 . *350-34 . D h . Prop 



SECTION C] SCBMULTIPLIABLE FAMILIES 421 

*351-41. h : « e FM ap subm conx . s'Pot"« C « . 

* jj,, V, p„ a e NC ind .v^O.a^O.ReK.D. 

[(^fv) t k'R} I {{p/a) t >c'R} = (/*/«/ +, p/a) t k'R 
Bern. 

h.*351-21-22.D 

I- : Hp . D . (fi/pyi k'R = (/./«/) I K,'R . {pi a) I k'R = (p/a) t kJR ' (1) 

I- . (1) . *332-241 . *331-24-33 . D 

h : Hp . D . {(j,/v) I k'R} \ {{p/ak'c'R} = rep,'[{(/./^) t k,'R\ \ {(p/a) t k^'R}] 

[*351-4.(1)] = (fj./v +, p/a) p K'i2 : D h . Prop 

*351-411. h : Hp *351-4 . \ = « w Cnv"« . /Sf 6 X . 3 . 

{(f./v) t X'S} I {(p/a) I \'S} = (W«' +s Pl<y) D ^'S 
[Proof as in *351-41, using *331'54] 

*351-42. h : K 6 jPilf ap subm conx .X,Ye &H' . i e «. . 3 . 

rep«'((X t ic,'L) \{Yl k.'L)] = {X +, Y) I k,'L [*351-4] 

*351-43. \-:KeFM&^ subm conx . s'Pot"/e C /c . Z, Fe 0'^' . i? e « . D . 

(Z t «'i?) I (Ft «'i?) = (X +, F) C «'E [*351-41] 

*351-431. V : Hp*351-42 . \= « w Cnv"* . SeX . D . 

(Z p V-S) I (Ft X'S) = (Z+, F) t X'/S [*351-4ll] 

*351'44. h : « 6 ^ilf ap subm conx . 

/i, V, p, (T e NC ind . v 4= . 0- + . {pja) H' {njv) . i e «. . D . 

rep/CKW") t «.'^} I {(p/<^) D «/^}] = (W" -^P/-^) D >c.'L 
Bern. 

As in *351-4, 

h : Hp . M = (,i/v) t icJL .N={pIit) I k,'L . D . 

{rep/(J!f I N)]'"''"' = rep/{i>>^'='' | L""'i'] (1) 

V . *301-23 . *308-13 . 3 h : Hp . t = (/* x^ o-) -„ (i; Xe p) . D . 

[*72-59.*332-25] = rep/X' (2) 

H. (1).(2).*350-41.D. 

I- : Hp (1) . Hp (2) . 3 . rep,'(i¥ | N) = [rjiv x„ a)} ^ k,'L (3) 

h . (3) . *308-24 .31-. Prop 



422 QUANTITY [PART VI 

*351-441. h : K e FM ap subm conx . 

fi, v,p,a 6 NC ind . v 4= ■ o- =t= . {fijv) H' (p/a) . X e /ci . D . 

i)em. 

|-.*33215.*303-19.D 

h : Hp . 3 . rep/[{(/./i.) t k,'L} \ {{pja) I «/£}] = ^ 

Cnv'rep.'[l(p/cr) ^ «/i} | {(W") D «.'^J] 

[*351-44] = Cnv'(p/o- -.p-lv) I k,'L 

[*303-19] = (p/o- -s fjLJv) I k/l 

[*308-21] = {fijv ->pI<t) t K.'i : D h . Prop 

*351-45. h : K 6 J^Jf ap subm conx . X, Ye G'H' . Z e /c. . D . 

rep/{(X t «/i) I (F p «/i)} = (Z -. F) D «/-t 
Dem. 

I- . *351-21 . *350-34 . *30812 .Dh:Hp.X=r.D. 

rep/{(Z C «/X) I ( F t «.'Z)} = / [ s'Q"* = (Z -, F) t «.'X (1) 

h . (1) . *351-44-441 . D h . Prop 

*351-46. I- : /c 6 FM ap subm conx . s'Pot"« C k . Z, Fe C'H' .ReK.O. 

(Cnv'Yl K'R)\{Xt K'R)e K, 
Dem. 
h . *351-22 . 3 h : Hp . D . Z p «'i? 6 « . F t; «'i? e K . 
[*37-62] D . Z t «'i2 6 « . Cnv'Ft ic'R e Cnv"*: Oh. Prop 

*351-47. h : Hp *351-46 . D . (Cnv'F ^ /e'E) | (Z p «'«) = (Z .-« F) p «/i2 
[*351-45-46] 



*352. EATIONAL MULTIPLES OP A GIVEN VECTOR. 

Summary of *352. 

By a " rational multiple " of a given vector in a family k we mean, if we 
are dealing with k, any vector in the family which has to the given vector 
a relation which is a member of G'H', and if we are dealing with «„ we mean 
any member of /c, which has to the given member of «, a relation which is 
a member of G'Hg. We will call the former "rational /c-multiples" and the 
latter " generalized rational multiples." It will be observed that if k contains 
pairs of members which are each other's converses, only one member of such 
a pair can be contained among the rational «-multiples of a given member 
of K, provided k is an open family. Hence the rational ^-multiples of a given 
vector all have one " sense," even if this was not the case with the original 
family. 

Rational multiples of a given vector T can be arranged in a series by 
correlation with their measures with T as unit. These measures are ordered, 
in the case of rational /e-multiples, by the relation H', and in the case of 
generalized rational multiples, by the relation Hg. Moreover if X is the 
measure of a gi,ven member of k with T as unit, the given member of k is 
k'\ Afp'X ; while if X is the measure of a given member of «., the given 
member of /c, is «, "1 AjfX. Hence the rational K-multiples of T are ordered 
by the relation k'\ Aj^'H', and the generalized rational multiples are ordered 
by the relation ic,'] Aj,>Hg. These two relations, therefore, are the relations 
we shall consider in this number. We put 

*35201. T,=k^At'H' Df 

*35202. T,, = K,^AT'Hg Df 

We assume throughout this number that k is open and connected. In 
dealing with T^, we assume TeKg, and in dealing with r„, we assume 
TeKg. We then prove the following propositions among others: 

k1Aj,[ G'H' el-*l (*35212), 

K. 1 J.r r G'Hg e 1 ^ 1 (*352-15), 
i.e. the relation of a rational multiple of T to its measure is one-one. 



424 QUANTITY [part VI 

T„ r„ e Ser (*352-16-17). 

Observe that this requires only that k should be open and connected. The 
serial property results from the correlation with H' or Hg. 

C'T, = /«: n Aj."G'H' . C'T,, = ic,n Ajf'G'H^ (*352-3-31). 

If (Sis any non-zero member of G'T^, C'S^ = C'T^ (*352'41), i.e. the rational 
K-multiples of T are the same as those of any rational «-multiple of T ; with 
a similar proposition for C'T^, (*352*42). 

RT^S. = : E./Se K n Ajf'G'H' : (a/i,i/)./i,i' e NO ind ./* <v.R' = -S" (*352-43). 

This is a convenient formula for T^, and leads immediately to 

T, = {s'H"{ll\)\ t: (« n Aj,"G'H') (*352-44). 

Observe that H''(l/1) is the class of rational proper fractions, including Oj. 
By *352-44 and *3.52-41-3, we see that, ii8^I\s'(l"K, 

SeG'T,.D.S. = T, (*352-45), 

i.e. the order of magnitude of a set of vectors which are rational /e-multiples 
of a given unit is independent of the choice of the unit. 

In order to establish the analogous property for T^, we first prove a 
formula analogous to *352'44, namely 

T„ = Cnv;{s'^'(l/1)} t («. n Aj."G'H) ^ 

{s'H"(l/l)} t («. A Ajf'G'H') (*352-54). 

Here the first term gives the series of negative multiples of T, while the 
second gives the series of positive multiples of T (including I \ s'Ql"k). 

From the above formula it follows, as in the case of T^, that if <Si is a 
positive multiple of T (not including I\s'Q."k), /S„= f^,. while if /S is a 

negative multiple of T, /S„=I'„ (*352-56-57). 

Finally we deal with the relation of U^ to T^. Here we have to assume 
that K is a serial family. We then find that U^ with its field confined to 
rational K-multiples of T is the converse of T^ , i.e. we have 

*352-72. ViKeFMsT.TeK^.-^.U, ^G'T, = « 1 A^'H' = T, 



*352-01. T, = K^Aj.'H' Df 

*35202. T,, = K.^A-p'>Hg Df 

*3521. \-:.RT,S. = :R,SeK:{'^X,Y).XH'Y.RXT.8YT [(*352-01)] 

*35211. \-:.RT,,8. = :R.S6>cr.{lX,r).XHgY.RXT.8YT [(*352-02)] 



SECTION C] RATIONAL MULTIPLES OF A GIVEN VECTOB 425 

*35212. h : K 6 FM&^ conx . Te /eg . D . « 1 ilr| C'H' e 1 ^ 1 

Bern. • 

l-.*336-l. Oh:R(K'\Ar[C'H')X. = .ReK.XeG'H' .RXT (1) 

l-.*350-521. D\-:Rf.R,SeK.X6C'H'.RXT.SXT.D.R = S (2) 
l-.*350-44. lihiHip. R6Ks.X,Y6G'H'.RXT.RYT.D.X=Y (3) 
h . *350-34-4 . D 

\-:'iiT>.R=:I[s'a">c.X.YeC'H'.RXT.SYT.-:i.X = Og.Y=0, (4) , 
\-.(S).{4>).0\-:HY>.Reic.X,YeC'H'.RXT.8YT.:3.X=Y (5) ' 

h . (1) , (2) . (5) . 3 h . Prop 

*35213. h : «; e ^ilf ap conx . T e K.g . D . «. n Aj,"G'H C /e.g 
Dem. 

h . *350-4 . D I- : Hp . ii e K. o Ajf'G'H . D . 

(a/x, v) . /[*, 1/ e NO ind - I'O . a ! i?" n J^' - 
[*333-101] D.i26«;,g:DI-.Prop 

*352 131. h : Hp *35213 . D . «. n At"G'H„ = Cnv"(K. n A/'G'H) [*3071] 
*352132. I- : Hp *352-13 .D.K.n At"G'H^ C /c^g [*352-13131] 

*35214. h : « e JW ap conx . T e /<:,g . D . k, n At"C'H' n Ajf'G'H^ = A 

De?w. 
|-.*307-l.*350-4.*352132.DH:Hp.ii,S6«..jBe^j."O'ir„.fif6^r"C"ir'.D. 
(H/*j ". p. o") • H;V,p,a-e NC ind .i'=^O.jO=j=0.o-4=0.i?6/<:jg. 

rep/J2- = rep/f" . rep/^S" = rep/rf . 
[*333-44] D .(a/t,z/,/),<7) . /t, i/,p, ffeNCind . 1/4=0 ./34=0 . O- + . i? 6«,9 . 

rep»'i?""««'' = rep/i^^"" = rep^'>S''^"'' . 
[*333-47] D.(a^,77).f,^6NCind.f+0.a!^fn;Sf''.iJe«.g. 
[*71-192] D.(af,7;).f,,76NCind.f + 0.a!7ni2«|,S''.i26«,g. 
[*333101.Transp] D . i2 =1= ,S : D h . Prop 

*352-15. l-:«eJWapconx. Te/c^g . D , k,1 ^rCCjET^el -♦I 

\-.*Sm-l.D\-:lIi>.R(K.^Aj,[G'Hg)X.R{H:,^Aj.\-G'H^)Y.':>. 

ReK,.X,YeG'Hg.RXT.RYT (1) 

h . (1) . *352-14 . D h :. Hp (1) . D : 

ReK,.X,YeG'H' .RXT.RYT.y.ReK,.X,YeG'Hn-RXT.RYT: 
[*3071.*350-44.*35213i32] ':>iX=Y, (2) 

h . *336-l . D I- : Hp . i? (/<:.1 4r r (?'^s)X . S'(«.1 ^r f G'Hg)X. D . 

R,S6K,.XeG'Hg.RXT.SXT. 
[*350-521.*307-l] D.i? = »S ' (3) 

I- . (2) . (3) . D h . Prop 



426 QUANTITY [part VI 

*35216. h : a: 6 ZAf ap conx . T e /eg . D . T. e Ser [*352-12 . *304-48] 

*352-17. h-.KeFMap conx . Te «,g . D . ?„ e Ser [*352-15 .*307-45 .*304'23] 

*35218. h-.Ke FMat^ conx . s'Pot"Kg CK^.Kgn Cnv"A;g = A . Te /cg . D . 

Z)cm. 
h . *350-43 . D 

h : . Hp . /i, j; 6 NO ind - t'O . Z = (fi/u) \Cnv.8eK.D: SXT . = .S' = Ti'. 
[Hp] D.>S-e/«:gnCnv"«g (1) 

h . (1) . Transp . D h : Hp . D . ~ (gZ, /S) . X e C"^„ .SeK.SXT: D h . Prop 

*352181. h : « 6 i^Minit . fe Kg . D . « n A/'C'H„ = A [*35218 . *335-21] 

*352-2. \-:KeFMsL-p conx . T e Kg . D . (/ [^ s'Q"*:) T^ T 
Bern. 

h . *350-34 . *331-22 . D h : Hp . D . (/ |^s'a"K) 0, T (1) 

h.*350-31. DI-:Hp.D.r(l/l)T (2) 

h . *304-45-48 . D h : Hp. D . 0,^^(1/1) (3) 

h . (1) . (2) . (3) . *352-l . D h . Prop 

*352-21. h : K e I'Jf ap conx . T e K,g . D . (/ 1^ s'a"«:) T,, T [Proof as in *352-2] 

*352-22. I- : K 6 iW ap conx . T e Kg . D . g ! T^ [*352-2] 

*352-23. l-:K6l'Mapconx .2'6K,g.D.a!7'„ [*352-21] 

*352-3. h : K 6 iW ap conx . Te Kg .D .C'T, = k r^ Aj,"G'H' 
Dem. 
h . *350-31 . *304-48 . D 

h : Hp . Z 6 G'H' . Z 4= 1/1 . D . Z (Zf ' c< H') (1/1) . Tiljl) T . 

[*3061] ^2.Xe{H'^:lH')"AJf'K (1) 

h . *350-34 . *331-22 . *304-45-48 . D 

h : Hp . Z = 1/1 . D . XH'Oq . {I\ s'(l"K)QqT .1 \s'(I"k e k . 

[*3061] D . Z e ^'"i'j,"« (2) 

h.(l).(2). Dh:Hp.D.a'fi:C(5''aS')"Ir"K (3) 

h . *150-201 . D h : Hp . D . C'T, = « 1 ^/'(fl"' a HJ'A/'k . 

[(3)] D . K 1 A/'C'H' C OT, (4) 

h. (4). *1 50-202. Dh. Prop 



SECTION C] RATIONAL MULTIPLES OF A GIVEN VECTOR 427 

*352-31. F- : K 6 JfM ap conx . T e «:,9 . 3 . O'T,, = «. n A/'C'Hg 
Dem. • 

Asin*352-3, h iKp.:) .C'H' C{HgK> Hg)"Aj."ic (1) 

H . *350-31 . (*307-05) . D h : Hp .XeC'Hn . D . XHg(l/l) . T (1/1)1. 
[*336-l] :i.X6Hg"Aj,"K (2) 

h.(l).(2). Dh:Hp.D.a'^gC(^gaff^)"^2,"« (3) 

I- . (3) . *150201-202 . D I- . Prop 

*352-32. h :. Hp*352-3 . Z, YeC'H' .R = Xl k'T.S= 7^ k'T.D : 

RT,S . = . XH'Y [*352-l . *350-521] 

*352-33. h :. Hp*35231 . X, Fe C'^^ . i? =Z ^ ic,'T.S= Y^ kJT . D : 

ijr„,S . = . XHg Y [*352-ll-15] 

*352-34. h :. Hp *352-3 .0:RT,T. = . (gZ) . XH' (1/1) .R = XIk'T 
[*352-l . *350-521-31] 

*352-341. I- :. Hp*352-3 . D : TT,R . = . (gZ) . (1/1) H'X . i? = Z f «'r 

*352-35. h :. Hp *352-31 . D : RT,, T. = . (gZ) . XHg (1 /I) . E = Z p k.'T 
[*35211-15] 

*352-351. I- :. Hp *352-31 . D : TT,,R . = . (gZ) . (1/1) HgX .R = X ^k,'T 

*352-36. h : Hp *352-3 . s'Pot"* C k . D . Pot'T - iT C K'^ 
Dew. 
I- . *350-43 . 3 I- : Hp . i; e NO ind - I'O - I'l . D . r- (iz/l) T . 
[*304-4.*352-341] D . IT, T" : D h . Prop 

*352-37. h : Hp*352-31 . Te* u Cnv"* . D . PotT- t'TC T^.'T 

Dem. 

V . *331-24-54 . D h : Hp . D . PotT C «. 

Hence as in *352"36. 

*352-38. I- : Hp *352-31 . 3 . rep«"(Pot'T -i"r)cV,,'T 

Dem. 

h . *332-61 . D h : Hp . D . rep/'(Pot'^- i-'^) C «. 
Hence as in *352'36. 

*352-41. \- -.KeFM&^conx.S.TeK^.SeG'T^.D. 

G'8^ = C'T^ = K n At"C'H' = Kf^ As"C'H 
Dem. 
h . *352-3 . *350-43 . D f- : Hp . D . (g/i, v).fi,veNC ind - I'O . (S" = T" . (1) 
[*352-3] -i.TeC'S, (2) 



428 QUANTITY ^ [PART VI 

h . (1) . *352-3 . *350-43 . I- : Hp . i? e 0'^« . D . 

(a/i, I/, p, 0-) . /x, I/, o- e NC ind - t'O . p e NC ind . (S" = y . -R' = /S" . 
[*301-504] D . (a^, v,p, a). fjL,v,ae NO ind -I'O.pe NO ind . R''"'"' = r-x-o . 
[*352-3.*350-43] D.EeCT, (3) 

h.(2).(3)|^,DI-:Hp.i?6C'r,.D.iieC"<S. (4) 

h . (3) . (4) . *352-3 . D h . Prop 

*352-42. \- : K eFM a.^coux . S,T 6 K^S ■ ^ ^ C"T„ . D . C'S^, = C"r„ 

Z)em. 
h . *352-3 . *350-4 . *307-l . D 

h :. Hp . D : (a/x, i.) : ^, i^ e NC ind - t'O : g ! /S" n r** . v . g ! /S- «S T" : (1) 

[*352-31]D:2'6C";Sf„ (2) 

h . (1) . *3o2-3 . *350-4 . *307-l . D 
h :. Hp . i? 6 0'/S„ . D : (g/i, v, />, a) : fi.v.ae NO ind - i'O . p e NO ind : 

a ! /Sf" n T^.v-a ! /S' n T" : a '. -R" A -S" ■ V . a ! -R" " -S" : 
[*333-48] D : (a/t, i-, p, a-) : /m, v, <7 e NO ind - I'O . p e NC ind : 

a ! R"^''' r, f^'f . V . a ! R"'"'!' n T-x"" : 
[*352-31] O-.ReC'T,, (3) 

l-.(2).(3)^.DI-:Hp.i?eO'2'...D.i260'/S., (4) 

h . (3) . (4) . D h . Prop 

*352-43. \-::KeFMa,pconx.TeKg.D:. 

RT,S. = :R,SeKnAj."C'H':{'^fj,,v)./i,vel!(Cind.fi<v.R' = S'' 
Bern. 
b . *3317 . D h : RT,S . = .R,SeC'T,. RT,S (1) 

h . (1) . *352-31 . *350-43 . D h :: Hp . D :. 

RT,S.= -.R.Sexn Aj."C'H' : (ap,o-,^,i?).cj-,^,7j eNC ind- I'O.p eNC ind . 

px„7;<<7X„f.i?- = rp.<S'' = Tf: 
[*333-5] = : ii, <Sf 6 « n .4 /'C'ff ' : (ap, 0-, ^ I?)- 0-, ^,i; 6 NO ind - I'O . p eNC ind . 

p x„ t; < o- Xe ^ . iJ-'X'f = Tf-X'^ = <Sf x«i : 
[*12614] ■DiR.SeKn Ajf'C'H' : (a//., i/) . /*, i- e NC ind . /i < i- . E" = /S" (2) 
h . *350-43 . *304-4 . D 
h :. iJ, /S e « n ^ /'C'ZT' : (a/ii, J') . /i, 1/ e NC ind . /4 < 1/ . jB- = /Sf" : D : 

R,SeKn Ajf'C'H' : (a^) . Zif ' (1/1) . -BZ/S 
[*336-l] :i'.R,SeK: (aZ; F, ^) . XH' (1/1) . F, Ze C'H' . iJZ<S . 22Fr . SZT 
[*350-6.*305-71-51] OiR,SeK: ('3_X,Z) . (Z x,Z)H'Z.R{X x,Z)T. SZT 
[*352-l] 0:RT,8 (3) 

h . (2) . (3) . D h . Prop 



SECTION' C] * RATIONAL MULTIPLES OF A GIVEN VECTOR 429 

*352-44. \-:keFMa.p conx . 2' e «g . D . T^ = {s'H"(lll)} ^ (« n Ar"C'H') 

Dem. • 

h . *352-43 . *304-4 . 3 1- :: Hp . D :. 

RT,S. = :B,8eKn A/'C'H' : (gX) . XH'il/l) . RXS ::0h . Prop 

*352-45. \-:KeFMa.pconx.S,TeKs.SeC'T^.O.S^ = T^ [*352-44-41] 
*352-5. h : « e i'Jlf ap conx . Te /f.g . D . G'k, 1 A^'H' =K,n A/'G'H' 
[Proof as in *352-3] 

*352-51. h : K e FM ap conx . T e /e^g . D . G'k, ^AjMI^^K^n A ^"G'Hn 
Dem. 

V . *150-202 . D h : Hp . D . a'Af, 1 At'E^, C «. n AT"G'Hn (1) 

V . *352-131 . D h : Hp . iJ 6 «. n Aj."G'Hn . D . (gX) . X e G'H. Rbk,. RXT (2) 
I- . *304-23 . D h : Hp .X eG'H-i\\ll) .ReK,. RXT ."^ . 

Z(irw ^)(1/1) . Ee/c. . EZr. ^(1/1)7. 
[*3071 .*3361] D . jB 6 O'/c, 1 J r'iT™ (3) 

I- . *352-38 . D h : Hp . Z = 1/1 . E e K. . EZf . D . ^ {k,'\ A/^H) (rep/T^ . 
[*307-l] D . i? e C"«. 1 ^r'i^n (4) 

h.(3).(4). Dh-.Hp.XeC'H.ReK^.RXT.Ii.ReG'K.'lAT'Hn (5) 

I- . (2) . (5) . D h : Hp . 3 . «;e n At"G'H„ C C'«, 1 ^riZT^ (6) 

h . (1) . (6) . D h . Prop 

*352-52. h : K e FM &]) conx . Te/c^g. 3 . T^, = K,'\Aj.'>Hn^K,']Ai.''H' 

Dem. 
h . *160-43 . (*307-05) . D 

I- . r.. = «. 1 4i.;^„ c; «, 1 ^ rJfi-' o («. 1 A/'G'Hn) t («t 1 Ajf'G'H') (1) 
h . (1) . *352-5-51 . *307-l . D I- . Prop 
*352-53. h : K e iW ap conx . T e «,g . D . 

«. 1 ^tSZT' = {s'fl^''(l/l)} D («t n Aj,"G'H') [Proof as in *352-44] 

*352-531. h : Hp*352-53 . D . k,^Aj.'H= {s'^'(1/1)} I (k, n At"G'H) 
[Proof as in *352-44] 

*352-54. h : Hp *352-53 . D . T„ = Cnv ;{s'^'(l/l)} ^ («, n Ajf'G'H) ^ 

{s'^"(l/l)} I («. ft Aj,"G'H') [*352-52-53-531] 
*352-55. h : K 6 i^Jf ap conx . /Sf, T e /e,g . (S e k. n Ajf'G'H . D . 

«. ft As"G'H' = K. ft Aj,"G'H' . K, ft As"G'H =K,nA j."G'H 
[Proof as in *352-41] 



430 QUANTITY [part VI 

*352-56. h : « e i^Tlf ap conx . ^, T e «.g . /S e «, n At"G'H . D . S„ = T,, 
[*352-54-55] 

*352-57. h : « e if'if ap conx . <Sf, T 6 «,9 . )Sf 6 «. A J. r"C/f„ . D . (Sf„ = T,. 
[*352o4-55 . *307-l] 

*352-7. V:.KeFMst.X, Ye G'H' . Te kq. P,QeK. PXT . QYT . 3 : 

PU,Q. = .XH'Y 
Bern. 

y . *:J5218 . DI-:Hp.Q!Pe«5.D.Q|P~e A/'G'Hn . 

[*350-65] D . X -, Fe G'H' (1) 

l-.*350-52. DI-:Hp(l).D.X+F (2) 

F- . (1) . (2) . *336-41 . D h : Hp . PU,Q . D . Z - Fe 0'^ . 

[*30812-19.Transp] D . Z^'F (3) 

h . *336-64 . D H :. Hp . ~ (PU^Q) .^•.P = Q.v .QU^P : 

[*350-44.(3)] D : Z = F . V - Fff 'Z : 

[*304-48] D:^(XH'Y) (4) 

h . (3) . (4) . 3 h . Prop 

*352-71. h -..iceFMsr .TeK'^.P,Qe G'T, . D : PU,Q . = .P(At'H')Q 
[*352-7-3] 

*352-72. l-i/ceJWsr.re^g.D. f7^tC''rK = «1^r5fi^' = 2^« [*352-71] 

*352-73. hz.KeFMsr subm . Z, Fe G'H' .TeK^.O: 

(Z t «'T) U, (Ft kT) . = . XH'Y [*352-7 . *351-22] 



*353. RATIONAL FAMILIES. 

Summary of *S53. 

A "rational family" is one which consists entirely of positive rational 
multiples of one of its members. We denote rational families by " FM rt " ; 
the definition is 

*353-01. FM rt = FMnii[(r^T).T€icg.icC A j."G'H'} Df 

It is obvious that, if k is any family, k r\ Ajf'G'H', which we considered 
in the last number, is a rational family. If k is a connected family, it does 
not follow that k n Ajf'G'H' is a connected family, but the proofs of its 
properties, as we saw in *352, make use of the fact that it is contained in 
a connected family. Many of the most important properties of connected 
families hold equally of sub-classes of connected families, notably the property 
that two members of k or k^ whose logical product exists are identical 
(*331'4224). In dealing with rational families, a good many propositions 
can be proved by merely assuming that they are contained in connected 
families. We put 

*35302. FM ex = FM n A, {(gw) . « e FM conx . \ C «} Df 

*35303. FM TtCK = FMTtnFMcx Df 

We will call a family " sub-connected " when it is contained in a connected 
family. When a family k is open, rational, and sub-connected, any member 
of Kg may be taken as the T of the definition *353'01 (this is proved in 
*353'13) ; and if 8, T are any two members of Kg, some power of 8 will be 
identical with some power of T (*353"12). An open rational sub-connected 
family is asymmetrical (*353"2) ; no power of a member, and no product of 
two members, is the converse of a non-zero member (*353"22'23). Hence by 
*331'54-33, if the family is connected", and not merely sub-connected, it is 
a group and transitive (*353'25"27). 

If X is a family which, besides being open and rational, has connexity, 
then if a is a member of the field and 7 e Kg we shall have 

s'\g = i4„;\ 1 At'H' . CT, = \ 1 At'H' (*353-32-33). 
That is, the series of points in the field and the series of vectors are both 



432 QUANTITY [part VI 

ordinally similar to part or the whole of the series of ratios ; they will be 
similar to the whole if X, is submultipliable (*353'44). But when \ is 
submultipliable, a smaller hypothesis suffices, for in that case we can prove 
that if \ is connected, then \ = X\j Cnv"\ (*353'41),so that \ has connexity, 
and is serial (*35342). Thus we have 

*353'44. f- : X e FM ap conx rt subm . D . s'Xg smor H' 

*353-45. f- . FM ap conx rt subm C FM sr 



*35301. FMri = FMn^ {('^T) .Tsk^./cC At"G'H'} Df 

*35302. FMcx=FMn\ {(g/e) . k e FM conx . \ C «) Df 

*35303. FMitcx = FMitnFMcK Df 

*3531. \-:.KeFMit. = :KeFM: (gT) . T e «rg . « C Aj."G'H' [(*353-01)] 

*35312. h : X 6 ^if ap rt ex . (Sf, r 6 \g . \ C Ajf'G'H' . D . 

(a/4, 1/) . /^, v 6 NC ind . i; 4= . /S" = T" [*350-43] 

*35313. 1- : \ 6 ZM ap rt ex . r e Xg . D . \ C Ajf'G'H' 

Bern. 
I-.*35312. Dh:B.p.SeXs .\CAs"G'H' .ReX.li. 

(a/i, v, /3, <r) . /t, I/, p, o- 6 NC ind . p + . 1/ + . o- 4= . ii" = /S" . T' = /S" . 
[*333-5] 

D . (a;i4, V, p, a-) .n,v,p,<Te NO ind.p=t=0.v=t=0-<^ + .-B"'^'"' = Si^^"" = T^^xo-r _ 
[*3o0-43] D . R eAj."G'H' : D h . Prop 

*35314. h : Hp *35313 . D . \. C Aj,"G'Hg 
Bern. 
I- . *353-13 . D I- : Hp . E, ;S 6 \ . D . (gZ, Y).X,Ye G'H' . RXT .SYT. 

[*3.50-65] D.(R\S){Y-,X)T. 

[*308-2] :>.R\8eA ^''G'Hg : 3 h . Prop 

*353-15. h : « e i^'if conx . T e /eg . D . « r. Aj."G'H' e FM it ox 
[*353-l . (*35302)] 

*353-2. h : \ 6 i?W ap rt ex . D . Xg rt Onv"\9 = A . \ e i^if asym 

Dem. 
h . *353-12-13 . D 

h : Hp . E, iJ 6 \g . D . (a^, z/) . /t, i; 6 NO ind - I'O . iZ^ = ij" (l) 

l-.(l).*301-23.DI-:Hp(l).D.(a/i,,v).^,i/eNCind-t'0.i2^+'=''G/ (2) 
h . *333-101 . D I- : Hp (1) . D . Pot'E C Rl'J" (3) 

h . (3) . (2) . Transp . (*334-05) . D h . Prop 



SECTION C] RATIONAL FAMILIES 433 

*353-22. h : Hp *353-2 . D . s'Pot"\g n Cnv"A.9 = A 
Dem, • 

h . *353-1213 . *301-5 . 3 h : Hp . ff e NO ind- t'O . E, E'eXg . D . 

(a/t, v) . /u,, v 6 NO ind - t'O . E'^'" = iZ" . 
[*301-23] D . (a/t, J/) . ju, i; 6 NO ind - I'O . iZ^'+cC'XcO q / (i) 

h . *333101 . *330-23 . D I- : Hp . D . Pot'B C J . A ~ e Fot'R (2) 

h . (2) . (1) . Transp . 3 h : Hp . i? e \g . D . ~ g ! Pot'ii r> Cnv"\ : D I- . Prop 

*353-23. h : Hp *353-2 . D . (s'X \ "\) n Cnv"X9 = A [Proof as in *353-22] 

*353-24. y : Hp *353-2 .XeFM conx . D . s'Pot"\ C X [*353-22 . *331-54] 

*353-25. l-:Hp*353-24.D.s'\i"\CX [*353-23 . *331-33] 

>j 

*353-26. h : Hp *353-24 . D . s'Xg | "\g C \g 

Dem. 
I- . *353-12-13\ D h : Hp . iJ, ,Sf e \g . D . (a/i, v) . /t, i; 6 NO ind - I'O . i?" = /S" . 
[*336-57] D . (a/i, I/) . /t, i; e NO ind - I'O . {R \ S)" = 8'-+'" . 

[*333101] D . a ! Pot'(i2 1 8)' n Rl'J". 

[*301-3.Transp.*331-23] D . JJ | S e Rl'J (1) 

h . (1) . *353-25 . D h . Prop 

*353-27. h : Hp *353-24 .D.XeFMtrs asym [*353-26-2 . *334-13] 

*353-3. h : . Hp *353-2 . v e NO ind - t'O . s'Pot"X, CX.:i:RU^S .0 .R' U^.S" 

Dem. 

h . *336-41 . D h : Hp . D . (a^) . Te \g . iJ = Tj ,S. 

[*330-57] D . (a?) . r e \g . i?" = T' 1 6^- . 

[*336-41.Hp] D . R' U^S' : D I- . Prop 

«353'31. I- :. XeJWaprt connex.iJ, »Si e\ . i/eNO ind - I'O . D : 

RU^S.= .R'Uk8'' 
Dem. 

h . *336-62 . D h : Hp . R^ 8 .'>-{RU^8) . D . 8U^R . 

[*353-3-24] D.8''V'kR''. 

[*336-6-61.*353-27] .'>^(B''UkS'') (1) 

|-.*336-6. D\-:'Ep.R = 8.'D.'^{R'U^8'') (2) 

I- . (1) . (2) . D h : Hp . ~ (RU,.8) . D . ~ (i2-fr;,fi(') (3) 
I- . (3) . *353-3 . D h . Prop 

R. & w. III. 28 



434 QUANTITY [part VI 

*353'32. t- : A, 6 FM a.Tp rt connex .TeX^.O . U^ = X'\ A^'H' 

Bern. 
h . *35312-13 . *350-5 . D h : Hp . E, -S e \ . -R + /S . 3 . ' 

('3^IM,v,p,a).fj,,v,p,a-e'NCmd.vJp0.a-^0.R'' = Ti'.S'=T''.fJL/v^plcr (1) 
h . (1) . *350-43 . D h :. Hp (1) . D : ii! {X^Aj^'H') S.v.S{X^ A^'H') R (2): 
h . *301-5 . 3 
h : Hp(l) ./i, i;, p,<7 eNC ind . 1-4=0 . O- + . iJ''= r''..;S''=r''.;u,XeO-<i'X„|0.D.; 

^■-XciT-l j§[»Xc<r= 2'("'Xop) -cdiiXco-) ^3^ 

V . *334-21 . D h : Hp(3) . D . E I ,Sf e\ w Cnv"\ . 

[*331-54.*332-241] 3 . (^ | <Sf)''x«-'= rep/(E | .Sf)'"<='' 

[*332-.53.(3)] =y(.xcp)-c(iixcT) (4) 

I- . (4) . *353-24-2 . D I- : Hp (3) . D . E | ;Sf e X . 

[*336-41] D . ,St^,J? (5) 

1- . (1) . (5) . *304-4 . D [- : Hp . ii(A, 1 Aj.->H')S.'D . 8U.R (6) 

h . (2) . *304-4 . D h : Hp(l) . ~ {R{\1Aj,'H')S} . D . S{X^At''H')R; 
[(6)] :>.RU,S. 

[*336-6-61.*353-27] :>.~(SU,R) (7) 

h.*336-6.Dh:Hp.i?=/S.D.~(;S'f/,ii) (8) 

h . (6) . (7) . (8) . D h . Prop 

*353-33. h : Rf *35S-32 .aes'a"X.D .s'Xs = Aa''X ] A j.iH' 
Bern. 

I- . *336-43 . D h : Hp . D .?/;, = \ 1 l^Js'Xg (1) 

I- . (1) . *336-2 . Dl-:Hp.D.s'Xg = AJ[7;, (2) 

I- . (2) . *353-32 . D h . Prop 

*353-34. h . ^itf ap rt connex C FM sr [*353-27] 

*353-4. \-:Xe FMa.p rt ex . s'Pot"\ C \ . Z e \,g . 3 . 

(go-) . o- 6 NC ind - t'O . repA'i" eX\j Cav"\ 
Dem. 

h . *35312'].3 .' D 

|-:Hp.D.(a/i,v,i?,/Sf).yi4,z^eNCind.E,/Sfe\.i.= E!,Sf./i + i-.ie'' = ^f'' (1) 
h.*301-23.D 

\- -..Rl) . ijL.v e'NCind. R,S eX . R" = Si' .1^ : fi <v .D .R'lS' ^S'-"!- . 
[*332-53] D.rep,'(R\SyeX (2) 

Similarly h :. Hp (2) . D : /i > z; . D . rep/(^ | /S)^ e Cnv"/«: (3) 

h . (1) . (2) . (3) . D I- . Prop 



SECTION C] RATIONAL FAMILIES 435 

*353-41. h : X, e FM ap conx rt subm . D . X^ = \ w Onv"\ 

Dem. • 

I- . *353-4 . D 

1- : Hp . i 6 X,,9 . D . (gii, a).Re\yj Cnv"\ . o- e NC ind - I'O . rep/Z" = R' . 
[*333-41] D . Z- 6 \ u Ciiv"\ : D h . Prop 

*353-42. h : Hp *353-41 . 3 . \ e i^Jf sr [*353-41 . *334-26 . *353-27] 

*353-43. f- : \ e FMa.^ ex rt subm . TeX^ . Potid'TC \ . D . G'H' C Aj."X 

Dem. 

I-.*351-1 . 3 I- : Hp./i,j;eNCind .i;4=0 . D.(a<S).,S6X,./Sf'' = r''. 

[*350-43] D.('^S).SeX.S{^/v)T (1) 

f- . (1) . *336-l . D h : Hp . Z e G'H' . D . (gS) . >SfeX . S4yZ : D h . Prop 

\j 
^353'44. I- : X, e FM ap conx rt subtn . D . s'Xg smor H' 

Dem. 

l-.*3o3-42-33. DI-:Hp.a6 5'a"X.D.s'Xg = ^„5Xl4r;5^' (1) 

h.*353-43. :>\-■.Rp{l).D.G'H'Ca'iAa\\^AJ,) (2) 

h . *336-2 . *352-15 . D h : Hp (1) . D . AJX^A^^G'S' e 1 -^ 1 (3) 

h . (1) . (2) . (3) . 3 h . Prop 

*353-45. h . I'il/ ap conx rt subm C ^if sr [*353-42] 



28—2 



*354. RATIONAL NETS. 

Summary of *354. 

The subject of " rational nets," which is to be considered in this number, 
is of importance for the introduction of coordinates in geometry. We have 
three stages in the construction of a rational net. First, taking any vector 
r in a family k, we construct C'T^, i.e. the positive rational multiples of T, 
as in ^352. The result is, as a rule, a family which is not connected, even 
when the family « is connected. For if there are in k any vectors other 
than G'Tic, any point of the field which is reached from a given point a by 
one of these " irrational " vectors cannot be reached from a by a member of 
C'T^, though it will be in the field of G'T,^. Thus in order to obtain from 
CT^ a connected family, we shall have to limit the fields of its members to 
the points which can be reached from a given point a by one or more 
rational steps backwards or forwards, i.e. to the points Aa"(C'T^\. It will 
be observed that whereas, in the construction of C'T^, only positive vectors 
are used, negative vectors, i.e. the converses of positive vectors, are also 
admitted in constructing what we may call the "rational points" with 
respect to a and T. Having constructed these points, i.e. the class 
■Aa'(G'T^\, we then proceed to the third and last stage in constructing a 
rational net, by limiting the field of every member of C'Ti^ to Aa"(G'T^\. 

Many of the propositions concerning rational nets require the hypothesis 
that the family concerned is a group. If this is not the case with the 
family k from which we start, we replace k by k^, where Kg is formed by 
adding to k the converses of those members of k (if any) whose domains 
are identical with the common converse domain of members of k. The 
definition is 

*354-01. Kg = lc^J Cnv"(/t n D's'a"«) Df 

We put also 
*354-03. FM grp = FMn/c (s'k \"kCk) Df 

We then easily prove that if k is connected. Kg is a group (*35414), and 
if K is open and connected, Kg is open and connected and a group (*354'17). 
If K is connected, (Kg\ = Kt (*354'15), so that properties only dependent on 
K„ like that of openness, always hold for Kg when they hold for k. 



SECTION C] RATIONAL NETS 437 

Next, we prove that if k is open, connected, and a group, G'T^ is open, 
rational, sub-connectec^and a group (*354'22). Hence if k is open and 
connected, and \ = Kg, G'T^ is open, rational, sub-connected and a group 
(*354-24). 

The " rational points " with respect to a and T are Aa"{G'T^\. In order 
to study them, we consider A^'X^, where X, is a family concerning which we 
make hypotheses which will be fulfilled in the case of G'T^. We prove that 
if \ is a family which is a group, and SeX.ae s'G."\ then 

A^"X, C ^8"Aa"\ (*354-31), 
whence 8l{Aa"\.) = {Aa"\)^8^S\(A„."\,) (*354-312). 

Next we prove that, with the same hypothesis, if b is any other member of 
Aa"X, then 

^„"\. = ^6"\. (*354-33). 

Thus the rational points with respect to a and T are the same as the 
rational points with respect to 6 and T, if 6 is one of these rational points. 

The "rational net" is the family l{Aa"{C'T,)]"G'T^. Writing \ for 
G'T^, this becomes ^{Aa"Xi)"\. In order to obtain the properties of the 
rational net, we therefore continue to consider a family \, concerning which 
we make hypotheses which are verified in the case of OT,, and we put 

*35402. oxa'\ = t{Aa"X)"'K Df 

Thus cXa'G'Ti, is the rational net defined by «, T, and a. We prove 
(*3o4-4) that if \ is a group, cx^'X is a family whose field is -4^'%. We 
prove that if \ is a family, and a a member of its field such that any 
member L of \, for which L'a exists is a member of \ o Cnv"X, then a is a 
connected point of cx^'X, i.e. 

*354-32. h : \ e FM . a e s'Q."\ . \ n aM„ C \ w Gnv"X . D . a e conx'cx„'X, 

The hypothesis X,. n C[M„C\ wCnv"\ would be verified if \ were a 
connected family and a were a connected point of \. But we want to be 
able to replace X, by G'T^, which is in general not connected. The above 
hypothesis, unlike \ e^fWconx, is satisfied by C'T^, provided k is open and 
a group and a is a connected point of k (*354"34). Hence it follows that if k 
is a family which is open, connected, and a group, and a is a connected point 
of K, csiaG'T^ is open and connected, and a is a connected point of cHa'G'T^ 
(*354"401). Again, in virtue of *354312, if \ is a family which is a group, 
and a is any member of its field, cx„'X is a group (*354"313) ; hence when 
/t is a family which is open, connected, and a group, cXa'G'T^ is a group. 
(*354"402); and it is easy to prove that it is also a rational family 
(*354-403). Hence, by *353-27, cXa'G'T^ is a family which is open, 
connected, rational, a group, transitive, and asymmetrical (*354"404). If our 
original family is open and connected but not a group, we only have to 



438 QUANTITY [PABT VI 

substitute Kg for «, i.e. putting \ = Kg, we only have to take cXa'G'Tx, in 
order to obtain a rational net with all the above properties. This is stated 
in the proposition 

*354'41. \- : K6 FMa.p conx . T e k^ . a e conx'/c .'\.= Kg.D . 

cx/G'T), e FM ap conx rt trs asym 



*35401. Kj = « u Cnv"(«: n DVa"/c) Df 
*35402. cx„'X = I {A „"\.)"\ Df 

*35403. FM grp = FMnlc (s'k \ "k C k) Df 

*3541. \-:.ReKg.= :ReK.v.IteK.a'R = s'a"K [(*354-01)] 
*35411. \-:KeFMcoQK.R,Seic.:i.R\S6Kg [*331-33 .*3541] 

*35412. V : Hp *35411 . D'R = s'a"K .'D .R\S = S\R .R\S€Kg 
Dem. 
I- . *330-52 . D h : Hp . a 6 conx'* . D . E ! R'S'a . a'(R \ S) = s'O"* . 
[*33111-42] D.R\S€K\J Cnv"« . a'{R | S) = s'a"« . 

[*354-l.*330-561] O.R\SeKg.S\R=:R\S::^\-. Prop 

*35413. t- : Hp *354-l 1 . D'i? = D'/S = s'a"K . D . ^ | ^ e «^ 
Dem. 

\- . *33r33 . D h : Hp . D .^1 6'6/«: u Cnv"*: (1) 

I- . *37-323 . D I- : Hp . D . a'(R \ S) = s'Q"*; (2) 

h . (1) . (2) . *3541 . D h . Prop 

*35414. \-:>ceFM conx. D.s'Kg\"KgCKg [*35411-12-13-1] 

*35415. h : « 6 FM conx . D . {Kg\ = k, 

Dem. 
I- . *3541 . D i- :. Hp . E, S 6 /cj . D : 

ii,,sf6«.v.^.(S6K.v.i?,SeK.v.^,^6«;.a'ij=a'/S=s'a:"/«; (i) 

F.*330-4. Dh:Hp.ii,S6/«:.D.E|,S6A;. (2) 

h . *33r33-24 .D\-:.'ilY>:R,SeK.v.R,SeK:D.R\SeK, (3) 

l-.*354-12. ■D\-:B.^.R,Seic.a'R = a'S = s'a"K.'D.R\8€K, (4) 
h . (1) . (2) . (3) . (4) . D h . Prop 

*35416. hiKeJ^il/conx.D./tje^il/conx [*354-l-12] 

*35417. I- : K 6 FM ap conx .D .Kge FM ap conx grp 
[*354-16-15-14 . *333-101] 



SECTION C] RATIONAL NETS 439 

*35418. h:.KeFMgrp. = :K6FM:R,8eK.-2s^s-R\SeH: [(*354-03)] 
*354-19. h-.KeFMgrp^O.s'Fof'KCK [*354-l 8 . Induct] 

f 

*354-2. I- : K e ^Jl/ ap conx .Te k^.O . G'T^eFMa.p rt ex ' 

[*353-15 . *352-3] 

*354-22. h : K 6 FM ap conx grp . y e Kg . D . C"T« e FM ap rt ex grp 

Dem. 
h . *350-62 . *354-18 . D h : Hp . iJ, .S, Te /<: . Z, Fe C'H' . RXT.SYT. D . 

(iJ|;S)(Z+,F)r.i2|S6«. 
[*306-67.*352-3] D . ii | /Sf e C'T. (1) 

1- . (1) . *352-3 . D I- : Hp . iJ, ,S 6 CT, . D . E | S e OT, (2) 

h.(2).*354-2.DI-.Prop 

*354-23. l-:«:eJPifrtconx.7'e«:9.D.GX = «: [*353-13 . *352-3] 

*354-24. h : K 6 ^If ap conx . Te /tg . A. = /c, . D . C'Ta e FM ap rt ex grp 
[*354-2217] 

*354-31. ViXeFMgt^.ae s'(l"\ . /S e \ . D . ^„'% C S"4„'% 
Dew. 

h . *336-l . D h :. Hp . D : « e 4,,'% . D . (gP, Q) . P, Q e k . « = P'Q'a . 
[*330-56] D . (gP, Q).P,QeK .S'x = P'S'Q'a . 

[*354-18] ~ D . (gP, R).P,ReK. S'x = P'R'a . 

[*336-l] D./Sf'a;e4<,'%. 

[*37106] D.xe S"Aa"X, :. D I- . Prop 

*354-311. F:Hp*354-31.D.S"4/%C4/% [*3o4-31] 

*354-312. h : Hp *354-31 .D.Sl iAa"X) = (Aa"\) 1 /Sf = >Sf p (4„'%) 
[*354-31-311] 

*354-313. h : \ e Pilf grp . a e s'a."\ . fi = cx^'X .D .s'fj,\"fiCfj. 

Dem. 
I- . *354-312 . D 

\-:R-p.R,S€\.0.{RtiA/'\)}\{St(A,"\)]={R\8)t{A^"X,) (1) 
I- . (1) . *354-18 . D 
. 1- : Hp . E, Se\ . D . {i2 p (A/'K)} \ [8 C (Aa"X.)} ecx^'X : D h . Prop 

*354-32. hzXeFM.ae s'G'^X . \. n aM„ C \ u Cnv"\ . D . a e conx'cx„'\ 

Dem. 
J- .*336-l . D h :. Hp . D : a;64„"\ . D . (gZ) . ZeX . a; = Z'a. Xea'4„ . 
[Hp] D.(aX).Le\wCnv"\.a; = 2;'a. 

[*330-43] D.(aM).ilf6Cx„'\uCnv"cx/\.a;=if'a: 

[*331-llO - ■■ ~ D:a6Conx'cx«'\:.DI-.Prop '..•-• 



440 QUANTITY [part VI 

*354-33. I- : \ 6 ^il/ grp . a 6 s'a"X . b e Aa"X, ■ 3 . Aa"\, = Ai"\ 
Dem. 
V . *336-l . D 

h : Hp . c 6 Ab"\, . D . (gP, Q,R, S) .P,Q,R,Se k .c = R'S'P'Q'a . 
[*330-56] D . (gP, Q,R,S).P,Q,R,S6h:.c = R'P'S'Q'a . 

[*354-18] D . (a J/, N).M,NeK.c = M'N'a . 

[*3361] 0.ceAa"X (1) 

Similarly t- : Hp . c e Aa"X, .D.ceA b"\, (2) 

l-.(l).(2).DI-.Prop 

*354'34. 1- : K e ^il/ ap conx grp . Te Kg . \ = C'T^ . a e conx'/c . D . 

\.naM„C\wCnv"\ 

|-.*354-22. DI-:Hp.D.\e^ilfaprtcx. 

[*35314] D.\.n(«:uCnv"/«;)C\wCnv"X, (1) 

h . *33111-32 . D h : Hp . i e\. A QM,, . D . Z e « u Cnv"*: . 

[(1)] D . Z 6 \ u Cnv"\ : D h . Prop 

^354-35. h : « 6 FMsip conx .TeK^. fi = Kg.X = C'T^ . a e coux'/e . D . 

\, rt a'^a C X. v^ Cnv"X, [*354-34l7] 

*354-4. h : X 6 Pi/grp . a e s'a"\ . D . cx<,'X, e FM . s'a"cx„'X = Aa"K 

Bern. 
h.*330o2. DI-:Hp.D.cx/\Cl->l (1) 

h . *354-311 . D h :. Hp . D : E 6 \ . 3 . a'R = ^«'% . B'R C Q'iJ (2) 

l-.*354-312.Dl-:Hp.i?,/S6\.D.{i?p(^/%)}|lSt(^„"X.)}=('B|'S)D(^»'%) 
[*330-5-52] = (/S I -R) D (^»"X0 

[*354-312] = {St (Aa"\)} I {i? D {^a"\)} (3) 

h . (3) . *330-5 . D h : Hp . D . cx„'\ e Abel (4) 

I- . (1) . (2) . (4) . *330-52 . D 1- . Prop 

3K354°401. 1- : «6i^^apconxgrp.aeconx'«. Te/Kg. D . 

cxa'C'T^ 6 FM ap conx . a e conx'cXa'CT, 
Dem. 

I- . *354-4-22 . D h : Hp . D . cXa'G'T, e FM (1) 

h . *354-34-32-2 . D h : Hp . D . a e conx'cXa'C'T^ (2) 

h . (1) . (2) . *333101 . D h . Prop 

*3.')4-402. h : Hp *354-401 . D . cx<.'C"T, e PJlf grp [*354-313-22-401] 



SECTION C] RATIONAL NETS 441 

*354-403. h : Hp *354-401 . D . cx^'OT, e FM rt 

Dem. • 

V . *353-12 . *354-2 . D 

h : Hp . S e OT, . \ = 0'r« . D . (g/*, k) . /*, i^e NCind . i; + . «"= Z^* ■ 
[*354-312.Induct] 3 . (g^, ,-)./*, v e NC iad . j/ + . 

[*350-43.*354-401] 

D . (a/., i;) . ,., i; e NC ind . ,; + . {SD (^e'%)} (/^/i') {^ D (^a"5^0} (1) 
h.(l).*353-l.Dh.Prop 

«354'404. h : « e ^Jlf ap conx grp . a e conx*« . Te Kg . 3 . 

CKa'C'T^ e FM ap conx rt grp trs asym [*354-401-402-403 . *353-27] 

*354'41. h : K e FM ap conx .TeK^.ae conx'« .X = Kg.O . 

cXa'C'Tx 6 i^Jf ap conx rt trs asym [*354-l7-404] 



*356. MEASUREMENT BY REAL NUMBERS. 

Swmmary of *356. 

In this number we consider the application of real numbers to the 
measurement of vectors in a family. The principle of this application is 
as follows: If a given set of vectors, all of which are rational multiples of a 
given vector R, have a limit with respect to U^, and if their measures 
determine a segment of H, then we take the real number represented by 
this segment as the measure of the limit of the given set of vectors. 
For the sake of homogeneity with rational measures, it is well to take our 
real numbers in the relational form given in *314 ; i.e. if feC©, we take 
s'f as the corresponding real number. With a suitable hypothesis, the 
result of the above principle for applying real numbers is, where rational 
multiples of the unit -B are concerned, to replace the ratio X by the 

rational real number s'H'X, as the measure of the vector X ^ k'R 
(cf. *356'63). Then the measure of the limit of a set of rational vectors 
will be, by our principle, the limit of their measures. Thus our principle is 
conformable to what is required for an application of real numbers. 

It should be observed that, if any application of irrationals is to be 
possible, it is necessary that the vectors of the family concerned should 
have a serial or quasi-serial order, independently of the order generated by 
their measures. The order generated, among rational multiples of T, by the 
ratios which are measures of these multiples, is Tic (cf. *352). A vector 
which is not a member of O'T,, cannot be the limit of any set of vectors 
with respect to Tn. But we saw (*352'72) that if k is a serial family, 

Hence when k is a serial family, a vector which is not a member of O'Tn 
may be the limit of a set of members of OT^ with respect to U^. It is the 
existence of an independent series [/„, not generated by measurement, which 
makes the application of irrationals as measures possible. 

The following phraseology may be found convenient. Taking a unit T 
in a family k, and an origin a in its field, if X e C'H' and S = Xl k'T and 
a! = 8'a = {X[,K'Tya, we call X the "rational measure" of S and the 
' rational coordinate " of so. We have, in the same circumstances, 
S= K 1 ^y'Z . X = Aa'8= Aa'ic'lAr'X. 



SECTION C] MEASUREMENT BY REAL NUMBERS 443 

We will call S the vector of X, and a; the point of X ; and the same 
phraseology will be eigployed for the vectors and points obtained by 
measures which are real numbers. We may now state the principle 
according to which we apply real numbers as measures as follows. Given 
a segment f of H, take all the vectors of f s : these form the class k n Ajf'^. 
Then the real number s'f is to be the measure of the limit (with respect 
to U^) of the class « n Aj,"^. Since CT, has the opposite sense to that of T^, 
I.e. U^ pl"oceeds from the vectors with bigger measures to those with smaller 
ones, the limit we shall have to take will be the lower limit with respect to 
U„. Thus the vector whose measure is s'^ will be 

prec (?/,)'(«: n^j,"^). 

Now if we put X = s% At"^ = X'T, and Z is a relational real number. 

Hence using *206-131, the vector whose measure is X is ^rec{U^yX'T. 
Hence if " X^'T" represents the vector whose measure is X (unit T), 
we put 

*35601. X, = i>Tec(U,)\X[K Df 

Assiiming now that « is a serial submultipliable family, in which we take 
R as the unit and 'a as the origin, and putting, for notational convenience, 

we have first a set of preliminary propositions (*356'1 — ■191), of which the 
most important are 

//' = {C'H') 1 As'P=(0'H') 1 As'Aa'Q (*356-13), 

P I G'R, = K 1 As'H' (*356-14), 

giving the relations between the series of ratios, the series of their vectors, 
and the series of their points. 

We proceed next (*356-2— -26) to the proof that X^keI-*!. . This 
requires, in addition to our previous hypothesis, that Q should be semi- 
Dedekindian. With this hypothesis, we first prove that if X, Y are 
relational real numbers, 

a'X, = a' F. = «9 : Z« = F« . H . Z = F (*3.56-21). 
We then prove, by the help of some arithmetical lemmas, that the lower 
limit of the submultiples of a given vector is the zero vector, i.e. 

tip's {Seic: (gi;) .R = S'}=I\- C'Q (*356-22). 

Hence we easily prove that, if R is any non-zero vector, and \ is a class 
of vectors having a lower limit L, the lower limit of the relative products of 
R and members of X is the relative product df R and L, i.e. 

\Ck.L = t\p'X .ReKs.D.R\L= tlp'R \ "\ (*356-221). 



444 QUANTITY [part VI 

Remembering that the relative product is represented arithmetically by 
the sum, we may express the above proposition by saying that the limit 
of the sums of a given vector and a set of vectors is the sum of the given 
vector and the limit of the set. From this proposition we easily deduce that 
i{RPS, Z/iJ + X/S, whence it follows that 

X^[-K6l-*1 (*356-26). 

Our next set of propositions (*356'3 — 'SS) is concerned in connecting the 
relative product of X^ and Y^ with the arithmetical product X Xr Y, where 
" Xr" has the meaning defined in *314. Here we only require that k should 
be serial and submultipliable, and we obtain 

Z, I F, = (X Xr Y), (*356-33). 

This proposition is the analogue of *351'31 (except that k. is replaced by k) ; 
it has a similar importance, and calls for similar remarks. 

Our next set of propositions (*356'4 — '43) is concerned in proving that 
the limit of the points of a segment of ratios is the point of their limit, in 
other words, that the limit of a set of points whose coordinates are a segment 
of rationals is the point whose coordinate is the limit of the segment. Here 
we again require that our family should be semi-Dedekindian ; then if ^ is 
a segment of ratios, and X = s'^, the above proposition is 

(X/Rya = se(iQ'A„,"Ai,"^ = seqe'^„"Z'i2 (*356-43). 
Here X/R is the vector of X, {X^'Rya is the point of X ; A^"^ = X'R, 

and each is the class of vectors of members of ^; and Aa"A^'^ or Aa"X'R 
is the class of points of members of ^. Moreover X is a relational real 
number. Thus the above proposition states that the point of X is the 
segment (i.e. the limit) of the points of the ratios contained in X ; i.e. of the 
ratios which may be considered less than X. 

We next proceed (*356*o — '54) to connect the relative multiplication of 
vectors with the addition of their measures. Here we require that k should 
be semi-Dedekindian as well as serial and submultipliable. We then find 
that if X, Y are relational real numbers, and J? is a non-zero vector, 

(Z/i?) I {YJR) = (Z +, Y\'R (*356'54). 

This proposition is the analogue of *351*43, and calls for similar remarks. 
The proof proceeds without much difficulty by means of *356*43. 

Finally we have a set of propositions (*356'6 — •63) to prove that the real 
number which measures a rational vector is the real number corresponding 
to the ratio which is its measure ; i.e. if Z is a ratio, the vector which has 

the ratio Z to the unit has the real number s'H'X for its measure. It is to 
be remembered that rational real numbers must not be identified with ratios, 



SECTION C] MEASUKEMENT BY REAL NUMBERS 445 

any more than integral ratios {i.e. ratios of the form v/l) must be identified 

with cardinals. The reaftiumber corresponding to a ratio X is s'H'X ; this 
is what we call a "rational real number." In measurement, when we are 
measuring by ratios, if R is our unit, X will be the measure of X ^ k'R ; but 
when we are measuring by real numbers, the measure of Z ^ k'R must be a 
real number. The real number which is the measure of X ^ k'R will, by our 
definition, be a real number Z such that 

XI K'R = ^veo{U;)'Z'R. 

Thus we have to prove that, if X is a ratio, the above equation is satisfied if 

we put Z= s'H'X, This requires that k should be serial, submultipliable 
and semi-Dedekindian ; we then have 

X e C'H . D . {s'H'X ). = Z ^ « (*356-63). 

Thus although the "pure" real number s'H'X is not identical with the 

— > 
" pure " ratio X, yet the " applied " real number {s'H'X\ is identical with 

the " applied " ratio X^ k. This fact explains why the results of the habitual 

confusion between a ratio and a rational real number have not been even 

more disastrous. 



*35601. Z« = prec(i7,)|Xp«i Df 

*3561. V:.ReK.'^:S = X^'R. = .8=-pvec{U:)'X'R [(*356-01)] 

*356H. h :. i? e « . D : /Sf = {s'^\'R . = .8 = prec ( U,)'Ak"^ 
[*356-l.*33612] 

*35612. 1- : . K e FM sr subm . 

X,¥eG'H'.ReH:s.aes'a"K.Q = s'Ks.P=U,.D: 

XH'Y. = .{Xl k'R) P{Yt k'R) . = .{{Xt K'R)'a] Q {( F t k'R)' a] 

[*352-73 . *336-4] 

*35613. h : vei^'ilf srsubm . Ji!e«g .aes'<l"K .Q = s'kq . P= CTk . D . 

H' = {C'H') 1 Aj,'P={C'H') 1 Aj^'AjQ [*356-12] 

*356-14. \-:Rp*356lS. O.PtO'R. = K^AR'H' [*352-72] 

-> -» -> 

*35615. h : Hp *356-13 . X C C'H .X = s'\.0. ma.xp'X'R = « 1 As"ma,XB'X 

Dem. 
h.*352-41. :)\-■.Rp.D.Kf^X'RCC'R..X'R = AJ,"X (1) 

I- . (1) . *356-14 . D h : Hp . D . r^xp'X'R = max (P^ C'R,)'X'R 
[*356-14] = « 1 -A/'max/X : D h . Prop 



446 QUANTITY [PAKT VI 

*35616. l-:Hp*356-13.XeC"@.X = s'\.D.maxp'X'jB = A [*356-15] 
*356-17. h : Hp *356-16 . D . X« =.ltp | X[ G'P [*356-16] 

*356-18. h : K e.FM conaex . D . Z» e 1 ^ Cls 
[*20616l . *336-62 . (*353-01)]. 

*35619. V:.iceFMsv.P= U,.D : ZeG'H .D . Z^ h:'>P Q P 

Dem. 
K. *336-511 . D h :. Hp . ii, >S6«. /i, I'eNC ind-i'O ..Z=,jl/v . D : 

RPS . = . iJ^PiSf" . 
[*356-43] D : RPS . M = (fi/v) ^ k'R . W = (/i/i;) Ik'S.D. M-PN" . 
[*336-511] 0.ilfPiV:.Dt-.Prop 

*356191. 1- : Hp *35619 . X e s"C'® .:i . XIk\P Q.P\Xl k 
Dem. 

\: . *35.6-19 . D 
h :.,Hp . D : X e a'® . Z = sa . ZeX . D . ^^ « I P GP I ^ t « :■ 3 I- • Prop 

*356-2. h : YL^mbQ-U .fieC® .LeX- (i. D . «;1^/i;ep'P"^B'V 
1- . *310-11 . b h : Hp . D . Lep^"iM . 



[*206'6.*352-12] D . k 1 Aj^'L ep'n: 1 As'H"Ar"/j, . 

[*356'14] D . «; 1 ^jj'Z ep'P"^jj"/i : D I- . Prop 

*356-21. 1- : . «: 6 Pi/ sr subm . Cnv's'«g e semi Ded .X,Ye s"G'@ . D : 

Dem. 

1- . *356-16 . *214-7 . D 
h : Hp . X,/it6 C© . X = s'\ . F=sV ■ -B e«g . 3 . E ! Z/E . E ! Y,'R (1) 

|-.(l).*356-2. DI-:Hp(l).P=t7-,.a!\-/..D.(7/ii)P(X/i2) (2) 

Similarly h : Hp(l).P= P, . g !^-X. 3 .(X/P)P(7/ii) (3) 

1- . (1) . (2) . (3) . D h : Hp (1).X,'R= Y.'R . D . X = /t. 
[Hp] D.X=Y (4) 

h . (1) . (4) . D I- . Prop 

*356-211. h : 0-, T e NC ind - t'O . i- e NO ind - I'O - t'l . D . 

(o- +e t)- > a" +0 {v X„ o-"--^! X„ t) 
Bern. 
h .*113-43-66.*116-34.D I- .(<7 x„ t)^ = ff" +„ (2 x„o- XeT)+oT^ (1) 

I- . *126-5 . D h :. Hp . D : (ff +„ t)" > o-" +o (v x„ o-"-"' Xo t) . D . 

(o- +, t)"+«' > a- '+'' +e (i/ x„ a- x„ t) +« (a" x^ r) (2) 
1- . (1) . (2) . Induct . D f- . Prop 



SECTION C] MEASUREMENT BY REAL NUMBERS 447 

*356-212. I- : p > <7 . p, ff, ?6 NC ind . D . (gi;) . i; € NC ind . p" > ff" x^ ^ 

Dem. '"' • 

l-.*356-211,D 

h : Hp . V 6 NC ind . p = o- +„ r . D . p" > o-"""! x„ {o- +„ (7; x„ t)} (1) 

h . (1) . *126-51 . D h : Hp (1) . ff +„ (,; x„ t) > <r x„ r . D . p' > t7-' x„ ? (2) 
h . (2) . *113-4:3 . *120-416 . *126-5 . D 
I- : Hp (1) . ,; Xe T > o- X,, (5--„ 1) . D . p' > ff" Xo f : D 1- . Prop 

*356-213. h:p>o-.p,o-,f,9?eNCind.7;=t=0.D. 

(gv) . i^ 6 NC iud . p' Xo 1? > ff" x„ ^ 
i)em. 

h . *356-212 . D I- : Hp . D . (gi;) . v e NC ind . p- > a" x^ ^ : D h . Prop 

*356-214. l-:p,o-6N0ind-t'0.p>ff.ZeO'jH'.D. 

(av).i/6NCind.(p/ff)-^Z [*356-213] 

*356-215. h : \ e C"0 . p, o- 6 NC ind - I'O . p > ff . D . 

(gX) . X e X, . X Xj p/o- <%> e X 
Dem. 

h . *805-142 . Induct . D h :. \ C O'/f. g ! \ . v e NC ind - t'O : 

X e \ . D^ . X Xj pja eXiDiXeX.D^.XXg p'/ff" e X : 

[*356-214] :^:H"X = &H . (1) 

1- . (1) . Tran'sp . D h . Prop 

*356-22. t- : Hp *356-13 . Q e semi Ded . D . 

tip's [SeKZ i'^v) . R = 3"} =I[ C'Q 
Dem-. 

h . *336-.511 . D h :. Hp . L = tlp'S {(gv) . i? = /Sf"} . p,, j; e NC ind -I'O.D: 

SeK.S''''''' = B.y.L''P8'': 

[*301-5] 0:TeK.Ti^ = R.D.L''FT: 

[Hp] Dri-P^i ' (1) 

f- . *337-21 . D h : Hp . i; 6 NC ind - I'O - I'l . Z e «g . D . LPL" (2) 

h . (1) . (2) . D h : Hp . 3 . Z~e Kg : D I- . Prop 

*356-221. h : Hp *356-19 . Q = s'k^ .XC.k.L = tip'X . E e «g . D . 

R\L = tlp'R I "X 
Dem. 
\- . *334-16 . *336-411 . D h : . Hp . 3 : LPM .0 .{R\L)P(R\M): 
[Hp] D:ilfe\.D.(iJ|Z)P(i2|4f.): 

[*37-61] D:i2]"XCP'(P|Z) (1) 



448 QUANTITY [part VI 

h . *336-41 . D I- : Hp . (ii I i) Pilf . D . (giV) . N eKs.M=R\L\N . 

[*330-31] D.(aiV).iV6«9.E|J/ = Z|iV. 

[*336-41.*334l:3] :i .LP{R\M).R\MeK^. (2) 

[Hp] :i.{'^N).Ne\.NP{R\M). 

[*336-411.(2)] D . (aiV) .Ne\.{R\N)PM. 

[*37-l] ■^.MeP"R\"X (3) 

l-.(l).(3).*207-21.Dh.Prop 

*356-23. V : Hp*356-22.EP,S.D.(ai;).i/eNCmd-i'0.[Ki/+el)/i'} D«'i?]P/S 

Dem. 
h . *356 22-221 . 3 I- : Hp .X = T{TeK: (gi/) . iJ = 7"} . D . tlp'Ej "\ = iJ. 
[Hp] D.(ar).Te\.(ii|T)P/Sf. 

[Hp] D . (gi;) . ,; 6 NC ind - I'O . {R \ (l/v) ^ k'R} PS . 

[*350-62.*334-32] D . (gi;) . i; e NO ind - t'O . [{(v +„ l)lv} t k'R] P8:0\-. Prop 

*356-231. I- : Hp *356-23 . D . (gv) . j; e NC ind - t'O . SP [{(v -„ l)/v} t ic'R] 
[Proof as in *356-23] 

*356-24. h : Hp *3o6-23 . X e s"G'& . D . X/i? + XJS 

Dem. 
I- . *356-23 . D h : Hp . \ 6 Ce . Z = s'\ . D . 

(ap, 0-) . ,0, <T € NC ind - t'O . p > o- . {(p/tr) ^ k'R} PS . 
[*356-215] D .(ap, 0-, F) . /3, o- e NC induct - I'O . p > cr . FeX. Fx,jo/ff~6\. 

KpMD«'i?|P^. ^ 

[*336-511] D.(a/3,o-,I').p,o-eNCind-i'0.p>o-.F6\.Fx,/j/o-6^'ff"\. 

{Ft«'(p/<7)t«'i?}P{Ft«'5f}. 
[*3ol-31.*356-13] D . (ap. ff, F) . F t ^'(/a/ff) I k'R 6p'*P"X'R rx P"X'S . 
[*3561] D . X/ii + Y,'R .Oh. Prop 

*356-25. I- : Hp *356-22 . X e s"G'@ . D . X,'R G Q 

I- . *356-l-21 . D h : Hp . D . Z/E e Kg (1) 

h . (1) . *41-13 . D h . Prop 

*356-26. h:Hp*356-25.D.Z«|^«:el->l 

h . *356-24 . Transp . D 1- : Hp . i2, >§ e /cg . Z/i? = Z//Sf . D . ^ = 5* (1) 
h . (1) . *35618-21 . D h . Prop 



SECTION C] MEASUREMENT BY REAL NUMBERS 449 

*356-3. l-:.«rei^Jlfapconxsubm . s'Pot"«: C « . /i, i/ e C"@ . ii, 5f e « . D : 

• R (s'fjL Xr s'v) S. = .B {{s'/i) [■ K I (s'v)] S 

Dern. 

V . *314-14 . *313-21 . D f- : Hp . D . sV Xrs'y = s's'fj, x, "v (1) 

f- . (1) . D h :. Hp . D :B(s'iJ. XrS'v)S. = . {'^M,N) . Me fi.N ev.R{Mx, N)S. 
[*351-31-22] = .{'^M,N) . M e ii.N ev . R{M^ k\N) S i.:iV . Prop 

*356-31. ViKeFMa,^ conx subm . s'Pot"« C « . Z, F e s"G'® . D . 

(Xx,y)C« = (Zt«)|(7^«) [*356-3] 

*356-32. h-.KeFMsr subm . X, Yes"G'® . i? e/cg . D . X,'Y/R={X \ Y\'R 
Dem. 

h .*356191 . D h :. Hp • D : (Se/i; A Y'R . D . « n 'x'SCP"X'Y,'R : 
[*37-63] D:Z"(«n F'i?)CP"Z'F/E (1) 

h . *305-6 . D h : Hp . \ 6 C'@ . Z = s'\ . ^,^'6\ . ^^Z' . D . 

^ t «'F/E = Z'l k'(Z I i') t ic'Y.'R . 
[*356-12] D.Zl «'F/J? 6 ^' t «"P'F.'i? . 

[*35617] D . ^t «'I^«'^ e ^' I k'T'^Y'R . 

[*35619] D . ^ t «' ^''^ e P"^' ^ «"PP . 

[Hp] :^.ZlK'Y,'Re P"X"'Y'R (2) 

h . (1) . (2) . D h : Hp . D . P"X"'Y'R = P"X'Y,'R . 
[*3561] D . (X I F)/E = X/F/P : D h . Prop 

*356-33. h : Hp *356-32 . D . Z, | F^ = (X x^ FX [*356-31-32] 

*356-4. h : K e Pif conx . Q = Criv's'«g ./Se/e.aCO'Q.aia.E! seq^'a . D . 

(S'seqg'a = seqQ'S"a 
Bern. 

h . *330-o63 . D I- : Hp . D . S'seq^'a ep'Q"S"a (1) 

h . *37-l . 0\-::Rp.D:.S'ze Q"p^"S"a . = : 

(ay) : « e a . Da, . /Sf'a;Qy : yQ;S'^: : 
[*330-542} = : (aw) : « e a . D,; ■ -S'* Q -Sf' w : S'w Q S'z : 

[*208-2] = : (aw) : « 6 a . Da . ajQw : wQz : 

[*37-l] =:^eQ'yV"a ^ (2) 

h . (2) . Transp . D h :. Hp . D : ^ ~ e Q"p'Q"<x . = .S'zr^e Q"p^"S"a (3) 
h . (1) . (3) . *330-542 . D h . Prop 

E. & w. III. 29 



450 QUANTITY [part VI 

*356-41. I- : . « 6 FM conx trs. P=U,.Q = s'kq . a e G'Q . X C k . g ! \ . D : 

N = seqp'X .= .NeK. seqg'Aa''^ = N'a 
Bern. 
I- . *336-43-2 . *206-61 . D 

I- :. Hp . D : iV= seqp'\ . = . iVe « . Aa'N= seq (Q I A/'k) 'Aa"\ (1) 

h . *206-21 1 . D I- : Hp . 6 = seqe'^/'X . D . (gE) . E e \ . B'aQb . 
[Hp] D.(a/S).>Sfe«.6;Sra. 

[*336-ll] 0.beAa"K (2) 

h . (1) . (2) . D h :. Hp . D : i\r= seqp'X, . = . iVe « . A^'N = seq<2'J.»"X. (3) 
h . (3) . *336-ll . D h . Prop 

*356-42. l-:Hp*356-41.E!seqp'\.D.(seqp'\)'a = seqQ'^„"\ [*356-41] 

*356-43. h : Hp *356-22 .^eG'® .X = s'^ .aeC'Q.D . 

(X.'Rya = seqQ' A/'Ai,"^ = seqQ'Aa"X'R 
[*356-4211-21.*336-12] 

*356-5. l-:Hp*356-22. 

Z, F 6 s"C"0 . a e C'Q . jR 6 « . X = « r. X'iJ . /i = « n Y'R . D . 

(X J By {Y^' By a = seqg's'X'seqe'sV'a 
Dem. 

I- . *356-43 . *33612 . D h : Hp . D . (X,'By(Y,'Bya = seqg's'\'(F/i?)'a 

[*356-43.*336-12] = seqg's'X'seqg's'/i'a Oh. Prop 

' *356-51. f- : Hp *356-5 . D . (Z +^ Y)^'B = seqp's'X | 'V 

i)em. 
I-.*35611 .*31413.DI-:Hp.f,7;ea'©.Z=s'^. F=s'77.D. 

(X +r Y\'B = seqp'^^"(f +» '?) 
[*312-32.*31111.*308-32] = seqp'4B"s'?+/''7 

[*336-ll] = seqp'i^ {(gi, if ) . Z e ? . if e ^y . iV = (Z +, ilf ) f k'R] 
[*351-43] = seqp'^ {(gi, M) .Le^.Mer, .N={Ll k'B) \ (M t ic'B)} 
[Hp] =seqp'#l(af7, F) . fTeX. Wefi.N= U\ TF} : D h . Prop ' 

*356-52. h : Hp *356-5 . D . {(X +^ Y^'Bya = seqQ'(s'^')"sV'a 

h . *356-51 . D h : Hp . D . {(X +^ F)«'i?}'a = (seqp's'X, | "/*)'« 

[*356-42] = seqQ'4a"s'\| 'V 

[*336-ll] = seqg'S {(gZ, F) . Z e \ . Fe /i . a; = (Z | Y)'a} 

[*41-11] = seqe'^ {(gZ) .XeX.xe X"s'ii'a} 

[*41-11] = seqe'(s'X,)"s'/i'a : D f- . Prop 



SECTION C] MEASUREMENT BY REAL NUMBERS 451 

— > — > — > 

*356-53. t- : Hp *356-.5 . D . seqg's'X'seqg'sV'" = seqQ'(s'\y's'fi'a 

Bern. 

-> -> -» -» 

V . *356-16 . D 1- : Hp . D . seqg's'X'seqe's'/i'a = Itg's'X'seqg's'/a''* 

— > 
[*4111] = Hq'B {(gi) .LeX.x = L'seqq's'fi'a} 

[*356-4] = Itg'^ {(gi) .Le\.w = seqQ'i"sV'«} 

[*356-16.Hp] = Itg'^ {(gi) .LeX.x = ltg'Z"sVa} 

[*207-55] = kg va Kai) . i e X . « = X"j5'a} 

[*41-11] =ltg'(s'X)"s>'a 

[*356161; = seqQ'(s'X)"sVa, : D h , Prop 

*356-54. h : « e J^W sr subm . Cnv's'/Kg e semi Ded . Z, Yes"G'& .ReK^.D. 

(X/R) I (F/E) = {X+r Y\'R [*356-5-53-52] 

*356-6. \-:KeFMsT.ReKs.P^U,.Q = s'Ks.X6G'H.O. 

icnAs"H'XC^'Xlic'R 
Dem. 

h . *37-6 . D h :. Hp . D : ilf e Aj,"H'X . = . (aF) . FiTZ . MYR . 

[*352-7] D . ifP (Z t «'iJ) : . 3 I- . Prop 

*356-61. h : Hp *356-6 .iceFM subm . Q e semi Ded . /SP (Z ^ k'R) . D . 

(aF).FffZ.>SP(FC«'i?) 
Dem. 

h . *356-231 . D h : Hp . D . (gf ) . i/ e NC ind - I'O . SP [{(i^ -„ iVi'} i k'X I «'i2] 
[*351-31] 3 . (ai') . z/ 6 NO ind - t'O . >SP [{(i; -„ l)/i; x,Z} t «'iJ] 

[*305-7l-51] D . (gF) . Ffi^Z . SP{Yl k'R) : D h . Prop 

*356-62. I- : Hp *356-6 . k e FM siibm . Q e semi Ded . D . 

^'X l k'R C P"A^"H'X [*356-61] 

*356-63. h : Hp *356-62 . D . (s'H'X ). = Z ^ « 
Z)em. 

I- . *356-6-62 . D h : Hp . 3 . Z ^ /^'i?; = ltp'u4B"-H''Z . 

[*356-ll] :>.XlK'R = {s'H'XyR (1) 

|-.(l).*356-21.DI-.Prop 



29—2 



*359. EXISTENCE-THEOEEMS FOR VECTOR-FAMILIES. 

Summary of *359. 

In this number we prove that, assuming the axiom of infinity, there are 
vector-families of the various kinds considered in previous numbers. 

If P is any well-ordered series having no last term, the converses of the 

interval-relations, I.e. the class finid'P, form an open family of C'P(*359"11). 
If P is a progression, this family is serial and initial (*35912). 

The family consisting of additions of positive ratios to positive ratios 
(including Oj), i.e. consisting of all terms of the form {-\-gX)^G'H' , where 
X e G'H', is initial, serial, open, and submultipliable (*359"21), assuming the 
axiom of infinity. The family consisting of generalized additions of positive 
ratios to generalized ratios is serial, open, and submultipliable, but not initial 
(*359-25). 

The family consisting of multiplications of positive ratios not 0, by positive 
ratios not Og is open and connected, but not serial or submultipliable (*359"22); 
if we confine the multipliers to ratios not less than 1/1, the family becomes 
serial (*359-25). 

The family consisting of additions of positive real numbers to positive 
real numbers (including I'Og) is serial, initial, and submultipliable (*359"31); 
the family consisting of generalized additions of positive real numbers (including 
I'Oq) to generalized real numbers is serial and submultipliable, but not initial 
(*359'32). Similar propositions hold for multiplication, provided I'Og is 
omitted; but the resulting families will not be serial. In the case where 
the field is confined to positive real numbers, however, the family becomes 

serial if the multipliers are confined to such as are not less than H'iljV), 
which is the real number 1. 

The last set of propositions in this number (*359"4— "44) are concerned 
in proving that, given a family k whose field is yS, if (S is a correlator of 
a and jS, ;St"« is a family whose field is a, and which has the same properties 
of being connected, open, etc. as the original family k. Hence if « is a family 
whose field is the real numbers, and we are given any class « similar to the real 
numbers (in other words the field of any continuous series), if 8 is the correlator 



SECTION C] EXISTENCE-THEOREMS FOR VECTOR-FAMILIES 453 

of this class with the real numbers, Sf'ie gives a family whose field is a. Hence 
from our previous exis|gnce-theorems we derive the existence, for a, of an 
initial serial family, giving us a system of measurement for a. Similarly 
if a is similar to the rationals. 



*3591. I- : P e fl . ~ E ! 5'P . D . finid'P e CI ex'cr'C'P 
Dem. 

h . *260-23-28 . D h : Hp . D . finid'P C 1 -♦ 1 (1) 

h.*121-302. DI-:Hp.D.D'P„ = C'P (2) 
H . (2) . *121-302-35 . *260-28 . D 

h : Hp . 1, 6 NC ind . D'P„ ==G'P.:i. D'P„+a = O'P (3) 

^ . (2) . (3) . Induct . D h : Hp . i? 6 finid'P . D . D'iJ = O'P (4) 

f-.*121-322. Dh:i?e finid'P. D.a'^C(7'P (5) 

I- . (1) . (4) . (6) . *330-l . D h . Prop 

*35911. h: Pen. ~E!£'P.D. finid'P efmap'C'P 
Dem. 

V . *260-28 . *121-352 . D h : Hp . D . finid'P e Abel (1) 

l-.*7l-19. Dh:Hp./^,i;6NCind.a!P;.|P./S/.D.At + j' (2) 

t-.*121-35. DI-:Hp(2).^>i;.D.P^|P,GP^_,„. 

[*91-6.*121-36] D.(P^|P,)p„G/ (3) 

Similarly h :Hp(2) . i/ >/i. D .(P^|P„)poG/ (4) 

h . (2) . (3) . (4) . D h : Hp . i 6 (finid'PXg . D . ip,, G J (5) 

l-.(l).(5).*359-l.DI-.Prop 

*35912. h : P 6 ft) . « = finid'P . D . k e fm sr init'C'P . s'k^ = P 
Dem. 

b . *263-14141 . *1221 . D h : Hp . D . sV£'P = G'P (1) 

h . *26314-141 . Dt-:Hp.D.s'«9 = P. (2) 

[*334-31.*3.59-ll] D.KeFMsr (3) 

h . (1) . (2) . (3) . *335-14 . D h . Prop 

*359-2. h : Infia ax . « = ^ {(gZ) . X e G'H' . iJ = (+« Z) ^ G'B'} . D . 

Dem. 

I- . *306'54-26 . *304-49 . D I- : Hp . D . « C 1 -> 1 (1) 

I- . *306-25 . *304-49 . DI-:Hp.i2e«.D. Q'P =G'H'.'D'RC G'H' (2) 

l-.*30611-31. D\-:Rp.R,S6K.:).R\S = S\R (3) 

h.*306-52. DI-:Hp.D.s'«g = J^' (4) 

I- . (1) . (2) . (3) . (4) . D h . Prop 



454 QUANTITY [part VI 

*359-21. 1- : Hp *359-2 . D . k e FM iuit sr subm . s'/cg = R' 
Dem. 

l-.*306-24.DI-:Hp.D.i^'05 = O'Zr' (1) 

h.*306-41.D 

I- : . Hp . X 6 G'H' . /i. 1/ 6 NC ind - I'O . .S = {+, (X X, l/i/)j I C'H' . D : 

S" = {+, (X Xs fj./v)\ t C'H' . D . >Sf''+«i = {+, (X X, ]:^1 /v)} t C'H' : 
[Induct] D:Si'=^{+,(Xx,t^/i,)}tC'H': 

[*m5-51]D:S'' = (+,X)lC'H' (2) 

H . (2) . *351-1 . *359-2 . D h : Hp . 3 . /c e JW subm (3) 

h . (1) , (3) . *8.59-2 . *334-31 . D h . Prop 

*359-22. f- : Infin ax . « = E {(gX) . X e C'H' .R = {+gX)t CH,] . D . 

« 6 FM sr subm . s'«g = if^ 

The proof proceeds as in *359'21, but in this case there is no origin. 

Every member of k is a connected point, i.e. a member of conx'/c. This 

results from *308'64. If, in *359'21, we substitute H for H', the proposition 

holds except that k has no origin. 

*359-23. h : Infin ax . « = E {(•gX) . X e C'H .R = (x,X)l C'H} . D . 

KeJWapconx 

The proof proceeds as in *359'21. We have to take H instead of H', 
because {xsOg)lC'H' is not 1 — > 1. We do not get KeFMsahm, because 
not every rational has a rational vth root. 

*359-24. h : Infin ax . 

K = R {(aX) . X 6 C'Hg - t'O, .R = (XgX)\: {CH, - 1'%)\ . D . 
K 6 FM ap conx 
The proof proceeds as in *359'23. 

*359-25. h : Infin ax . « = ^ {(.gX) . (1/1) H^X . ii = (x, X) ^ C'H} . 3 . 

KeFMsT.s'Ks = H 
The proof proceeds as in *359"21. 

*359-31. h : Infiu ax . «: = ^ {(g/i,) . yx e C'@' .R^(+pfi)l G'&} . D . 

v./ 

K e FM sr init subm . s'k^ = @' 
Dem. 

t-.*311-74. Dh:Hp.D./cCl->l (1) 

f-.*311-27. D\-:Rp.ReK.D.a'R = C'& .D'RCC'®' (2) 

l-.*311-43. Dh:Hp.D.t'0jt(7'@' = init'« (3) 

F.*311-12-121.DI-:Hp.D.«6Abel (4) 

l-.*311-65. DI-:Hp.D.s'A:g = ^' (5) 



SECTION C] EXISTENCE-THEOREMS FOR VECTOR-FAMILIES 455 

f-.(l).(2).(3).(4).(5). 0\-:R^.D.K€FMsrimt.s'Ks = & (6) 
H . (6) . *310-151 .*t51-ll . D h : Hp . D . « 6 FM snhm (7) 

f- . (6) . (7) . 3 h . Prop 

*359-32. I- : Infin ax.K = R {(g^) . ^ e C'& .R = (+a^)l G'@g] . D . 

K e FM sr subm . s'«g = ©^ 

The proof proceeds as in *359-22. Similarly the analogues of *359-23-24-25 
can be proved for real numbers ; the resulting families, in these cases, will be 
submultipliable, but it will be necessary to omit 1% from their fields. 

*359-4. h : K 6 CI ex'cr'/3 . <S' e a sm /3 . D . /Sff"* e CI ex'cr'a 

Bern. 
l-.*330-l.*7l-252. Dh:Hp.D.St"«Cl^l (1) 

I- . *150-21-211 . *3301 . D h : Hp . i? e 8f"ic . D . a'R = S"^ . B'R C a'R . 
[*^303] D.a'i? = a.D'i?Ca (2) 

I- . (1) . (2) . *330-l . D t- . Prop 

*359-401. I- : « 6 Abel . S e Cls -> 1 . s'a"« Ca'S.D. Sf'K e Abel 
i)em. 

\-.*72-G01.D\-:.Kp.D:F,Q6K.D.P\S\S = P.Q\S\S=Q. (1) 

[*150-1] D.(StP)|(-Ste) = 'S|P|Q|S 

[*330-5] =S\Q\P\S 

[(1).*150-1] =(8fQ)\{SfB) (2) 

h . (2) . *330o . D I- . Prop 

*359-41. h:«6fm'y8.*Sea&m/3.D./S't"«6fm'« [*359-4-401 . *330-51] 

*359-411. \-:KeFM.ae coux'k .Sel^l. s'a"K = a'S.D. S'a e conx'S't"* 
Dem. 
h . *151-11 . D h : Hp . P = /S; s'/c . D . iS e P slEof (s'«) . 

[*151-33] D . "P'S'a u Ip'S'a = ,Sf"s'«'a u iS'^i^'a 

[*3311] =,Sf"s'a"« 

[*330-13.*160-211] =a'S'>s'K 

[Hp] =a'P (1) 

t- . *150-16 . 3 f- : Hp (1) . D . P = i'Sf"* (2) 

l-.(l).(2).*331-l.DI-.Prop 

*359-412. I- : « 6 fm conx'^ . iSf e a sm yS . D . Sfx e fm conx'a [*359-41-411] 



456 QUANTITY [part VI 

*359-413. 1- : « 6 JW ap . /S 6 1 -> 1 . s'a"K = a'S.D. Sf'ie e FMap 

Dem. 
l-.*72-601. DI-:Hp.P,Qe«.D.(fif;P)|(*Sf;Q) = /S;(P|Q) (1) 

t- . (1) . *i50-4 . D I- : Hp (1) . a ! (s; P) I (/S; Q) n J . D . a ! P I Q n J . 

[*333101] D.(P|Q)p„GJ-. 

[*200-21] D.^;(P|Q)p„GJ-. 

[*150-83] D.fS-JCPIQMpoGJ (2) 

f-.(l).(2). DI-:.Hp.D:Z, F6St"«.a!X|FnJ.D.(X|7)poGJ"(3) 
l-.*359-4. DI-:Hp.D.>Sft"«6PJf (4) 

F . (3) . (4) . *333-101 . D F . Prop 

*359-414. l-iKeFM.Sel^l. s'a"« = a'S.a = init'« . D . /Sf'a = lait'S f'lc 
[Proof as in *359-411] 

*359-415. h : « e Pilf subm . >Sf e 1 -> 1 . a'>Sf = s'a"K . D . <Sft"« e FM subm 

Dem. 
f-.*30r21. DI-:Hp.F6«.veNCind.D.F-'+'i=F-'|F (1) 

I- . (1) . *72-601 . D h : Hp . S5 F" = (S; F)" . D . ^Sf? F'+'i = (S^J F^+^i (2) 

h . (2) . Induct . Dh:Hp(l).D.S;F'' = (S5F)'' (3) 

h.*351-l. Di-:Hp.i'eNCind-i'0.Z6«.D.(aF).X=F''.Fe«. 

[(3)] D.(^Y).YeK.8>X = (S>Yy (4) 

I- . (4) . *351-1 . *359-41 . D h . Prop 

*359'42. h : a ! fm conx ap subm'/3 . a sm /3 . D . g ! fm conx ap subm'a 
[*3S9-41-412-413-415] 

*359-43. hiPei+v'^-'SL- Pit/ init sr subm r. ^(s'«g = P) 

[*359-42-21-414 . *274-44 . *123-18 . *304-47 . *273-4] 

*359-44. l-:Nr'P+l = ^.3.a! FM init sr subm n ^ (s'«g = P) 
[*359-42-31-414 . *275-3 . *310-16 . *204-47] 



SECTION D. 

CYCLIC FAMILIES. 

Summary of Section D. 

The theory of measurement hitherto developed has been only applicable 
to open families. But in order to be able to deal with such cases as the angles 
at a point, or the elliptic straight Hue, we require a theory of measurement 
applicable to families which are not open. This theory is given briefly in the 
present Section. 

When a family is not open, two vectors which have one ratio will usually 
also have many others, i.e. we shall not have ^\X^Kf\Y\,K.'^.X=Y, 
where X, Y are ratios. Also a ratio confined to the family will not usually 
be one-one. Under these circumstances, it is necessary, if measurement is to 
be possible, that there should be some way of distinguishing one among the 
ratios of two vectors as their " principal " ratio, and of then showing that, by 
confining ourselves to principal ratios, the requisite properties of ratios re- 
appear. 

The case of angles will serve to illustrate our procedure. Considered 
geometrically, not kinematically, a vector which is a multiple of 27r is identical 
with the null-vector, and if 6 is any angle, 6 = 2vit + 6, where v is any integer 
positive or negative. We are here considering an angle as a vector whose field 
is all the rays in a given plane through a given point. Thus there will be two 
angles which are half of the null-vector, namely ir and 27r, and four angles 
Which are a quarter of the null-vector, namely 7r/2, tt, 37r/2 and 27r; and 
so on. The ratio of 7r/2 to ir is any number of the form {2)1 + l)/(4j/ + 2) ; 
thus two terms may have many difierent ratios. 

In order to evade this difficulty, we first arrange angles in a series ending 
with 27r, and having no first term, but proceeding from smaller to greater 
angles. Then the angles which have a given ratio fijv to a given angle will 
be finite in number, and therefore one of them will be the smallest. We take 
this as the "principal" angle having the ratio iijv to the given angle, and 
define "(jj,/v)k" to mean the relation between two angles consisting in the 
fact that the first is the "principal" angle having the ratio (j,/v to the second. 
Then of all the ratios between the two angles, the ratio fi/v may be regarded 



458 QUANTITY [PAET VI 

as the "principal" ratio. It will be found that, with suitable hypotheses, 
(/a/i^), has the properties required in order to make measurement possible. 

In order to make the above method feasible, certain properties must be 
assumed to hold concerning the family k. (These properties are all verified 
in the cases that arise in practice.) We shall therefore only speak of a family 
as cyclic when it fulfils the following conditions : 

(1) It must be connected. 

(2) It must contain a non-zero member which is identical with its 
converse. This is the property which makes the family cyclic. In the 
case of angles, the member in question is tt. 

(3) It must be such that Kg 1 U„ is transitive. This is the property 
which enables us to arrange the field in a series. It will be observed that 
U^ cannot be transitive, since, if K^ is the member which is its own converse, 
we have 

(/ \s'a"K) U, K, . K, U, (I [ s'a^K), 

but we do not have (/ [^ s'Q"k) U^ {I f s'(I"k), because U^ is contained in 
diversity (by *336'6). It is, however, possible that U^ should be transitive 
so long as we do not start from / f s'Q"/c, and this we assume as part of the 
definition of cyclic families. 

(4) In order to avoid trivial exceptions, we assume that k does not have 
only two members, since otherwise it might consist only of/ [a'd'^/c and K^. 

We are thus led to the following definition : 
FM eycl = (FM conx - 2) n ^ {/cg ^ fT, e trans : (gjf ) .KeKs.K = K} Df. 
We prove that there is only one such relation as K, and therefore put 

K, = (iK){K6Ks.K = K) Df. 
Also for the sake of brevity we put 

I^ = I\-s'a"K Df. 

We then prove that k is a family having connexity, and satisfying the 
condition 

D"K = a"/e, 

i.e. having the domain of a member always identical with the common 
converse domain. Thus by *334'21, /Cj = « u Cnv"«;. 

In a cyclic family, k u Cnv"« consists of two mutually exclusive parts, 
namely vg and Kic\"k^. (In the case of angles, Kk\R would be tt + R. 
Thus Kg would be the angles from (exclusive) to tt (inclusive), and Kg | "Kg 
would be the angles from tt (exclusive) to 27r (inclusive).) Also Kic \ "Kg 
consists of the converses of k — i'K^. 

We take up next (*371) the question of arranging k u Cnv"K in a series. 
For this purpose, in order to avoid circularity, we have to erect a barrier at 
some point ; we choose I^ as this point. By the definition of cyclic families. 




SECTION D] CYCLIC FAMILIES 459 

/cg1 CT, is transitive; hence, since the family has connexity, P^t/eg is serial. 

This relation therefore s^anges all the members of «g in a series, beginning 

with K^ and proceeding towards /„. In order to extend our series to 

Kk I "k^, we only have to make K^ j R precede K^\8 ii R precedes 8, where 

R and S are members of Kg. That is, we arrange K^ \ "/eg in the order 

^k\'U^I,kq. This gives a series which begins 

with I^ and proceeds towards K^ without 

reaching it. Thus taking the sum of the 

above two series (in the' sense of *160), we 

get a series whose field is « u Cnv"/i:, which 

begins with /«, travels through K,, \ 'Vg to K^, 

and on through Kg towards /,, without quite 

reaching I^ again. This relation we call W^; 

the definition is 

F, = ^,|;Cr,^Kg4L?7.pKg Df. 

Taking an arbitrary origin, a vector may be indicated by the point to which 
it carries the origin. Thus in the figure, J^ is at the origin, K^ is opposite 
the origin ; the upper semi-circle, including both ends, is k ; not including 
the right-hand end, it is k^; the lower semi-circle, including both ends, is 
Cnv"*; including Ki^ but not /„, it is Cnv"Kg; including /« but not K^, 
it is K^\"kq. Then W^ starts from I^, and proceeds through the lower 
semi-circle first, and afterwards through the upper semi-circle, stopping just 
short of 7k- 

If K is cyclic, W^ is a series. Under most circumstances, if Rex, we 
shall have 

PW.Q.O.(P\R)W.{Q\R). 
The investigation of the various cases in which this holds occupies a large 
part of *371. 

In the remainder of this Section, our work becomes more full of ordinary 
arithmetic than it has been hitherto. We shall therefore, where cardinals 
are concerned, abandon the explicit notatipn we have hitherto employed, and 
substitute the ordinary notation. Thus we shall write fi+v in place of fi +„ v, 
and ij,v in place of fi x^ v. We shall, however, retain fi—gV for subtraction, 
in order to avoid confusion with the sign of negation of a class. 

We proceed next (*372) to consider what is in effect the class of vectors 
not greater than the vth part of a complete revolution {e.g: in the case of 
angles, not greater than 27r/v). We define this by means of the relation W^. 
It will be seen from the figure that if iJ is a non-zero vector, we shall have 
iJo'+i WkR", unless R" belongs to the lower semi-circle and R'^^ to the upper, 
in which case R" W^R'^'^^- The first time this happens is the first time that 
R'^^ becomes greater than one complete revolution. Hence if, for every 
number o" less than i^ and not zero, R'^^ W^R', it follows that R" is not greater 



460 QUANTITY [part VI 

than one complete revolution, and therefore R is not greater than the uth 
part of a complete revolution. The class of such relations we call v^ ; thus 
we put 

v. = {ku Cnv"«) n^(o-<i-.o- + O.D,. iJ'+i W^R') Df. 

The main propositions to be proved in this subject are 

Pev..PW,Q.:>.P'W.Q' 

and (what is an immediate consequence) . 

P,Qev..:>:P'=Q-'. = .P = Q. 

This latter proposition is the foundation of the theory of principal ratios. 

Another important property of v^ is 

so that z/< is an upper section of W^. 

We proceed next (*373) to consider submultiples of identity, i.e. vectors 
R such that R'' = Ig, where v is a cardinal. We assume here, and almost 
always henceforth, that « is a submultipliable family. ,We first consider 
vectors which can be reached from /^ by successive bisections. We know 
that K^^l^; if R^ = K^, then R^K^, because K^^^K^. Hence by con- 
tinuing the same process we arrive at the existence of a vector Q such that 

Q^'' = /,:/3<2-.p=t=0.Dp.Q''=t=/,. 
Hence we easily arrive at the result that, if v is any inductive cardinal, 
there is a non-zero vector whose vth power is /,. (This does not follow 
from KeFMsnhm alone, because /»"=/,, so that from the definition of 
.Fif subm we cannot know that there is any vector except /, whose vth power 
is lie.) Thence we prove that there are non-zero vectors whose vth power is 
Ik, and which are such that no earlier power is I^, i.e. we prove 

(gii) : i? 6 «g . i?" = /« : o- < 1/ . (7 4= . D, . J?-' =)= /,. 

The class of such vectors we call {I,,, v). If R is such a vector, the number 
of different vectors which are powers of R is v. Hence the powers of R have 
a maximum in the order Wk ', since W^ proceeds from greater to smaller 
vectors, this will be the smallest vector, other than /„, which is a power oi R. 
Concerning this vector, we show that it is a member of i',, i.e. it is such that, 
if o- < 1/ . 0-4= 0, R"^^ Wi^R". Finally we prove that there is only one member 
of Vk whose vth power is /,. This will be what we may call the " principal " 
j/th submultiple of /,< ; in the case of angles, it will be the angle 27r/i'. It 
will be observed that 27r/i/i' always has identity for its vth power, and has no 
lower power equal to identity if /x is prime to v. Thus the uniqueness of the 
" principal " vth submultiple depends upon the fact that it is a member of i/, , 
so that, by what has been proved in the previous number, no other member 
of z/j has the same vth power. 



SECTION d] cyclic FAMILIES 461 

We next, in a short number (*374), extend the last of the above results 
to any vector, proving that, if R is any member of « w Cnv"«r, there is a 
unique member of v^ whose vth power is B. We may call this the "principal" 
j/th submultiple of R. We prove also in this number that, if S is the principal 
vth submultiple of /», v^ consists of all vectors not earlier than S in the order 
Wk, i.e. of all vectors not greater than S. 

Finally (*375) we define " principal ratios " and show that they are one- 
one and mutually exclusive. We denote the '' principal ratio " corresponding 
to fi/v by "{iilv\." This is defined as the relation holding between R and 8 
when the principal /ith submultiple of R is identical with the principal vth 
submultiple of S; that is, we put 

{,ilv\ = RS{(^T).Te^L.r^v,.R = T'^.S=T'} Df. 

It is obviou>s that (fi/v% G (fi/v) ^ «. ; and there is no difficulty in showing 
that principal ratios are one-one and mutually exclusive. 

We have not thought it necessary to carry the development of this subject 
any farther, since, from this point onwards, everything proceeds as in the case 
of open families. We have given proofs rather shortly in this Section, 
particularly in the case of purely arithmetical lemmas, of which the proofs 
are perfectly straightforward, but tedious if written out at length. 



*370. ELEMENTARY PROPERTIES OP CYCLIC FAMILIES. 

Summary of *370. 

In this number, after the definition of cyclic families already cited, we 
proceed first to prove that only one non-zero vector is equal to its converse 
(*370'23). This one we define as K^. Next we prove that, if i2 is a non- 
zero vector other than K^, R\Kk is the converse of a non-zero vector, and 

R\Kx is a non-zero vector (*3'70'31"311), whence it follows that 

'D'R = a'R = s'a"K (*370-32), 

whence further we obtain 

D"« = a"«; . K e FMcomi&x (*370-33). 

Hence further, since hy definition /eg 1 f/^ is transitive, it follows that Kg 1 U^ 
is a series (*370"37). The remaining propositions (*370'4 — •44) are concerned 
with the relations of the two semi-circles «g and K^ \ "«g (cf. figure, p. 459). 

We have 

Gnv"K = K^\"K (*370-4), 

K n Cnv"/c = i'/, u I'K, (*370-42), 

K^ I "«g = Cnv"/e - I'K^ (*370-43), 

and ic-^rxK^\ "Kg = A (*370-44). 



*370-01. .fWcycl = 

(^i/conx-2)n^{«:g1f/'«6trans:(aZ).ifeKg .K = K] Df 

*37002. K, = {iK){KeK^.K = K) Df 

*37003. /, = Jrs'a"K Df 

*3701. h : . K e FM cycl . = : 

KeFM conx - 2 . Kg 1 fT, e trans : (^Z) . KeK^.K = K [(*370-01)] 

*37011. h : K 6 FM conx . D . Kg 1 P. G / [*336-6 . (*336-011)] 

*37012. I- : KeZM conx . k^ 1 Z7,, etrans . R,SeK^ . RU^S . SU^T . D.R=\=T 
[*370-ll] 

*37013. V:KeFM.KeK.K = K.D.K^ = I^ [*330-31] 



SECTION D] elementary PROPERTIES OF CYCLIC FAMILIES 463 

*370-2. \-:.KeFMconx.Ks'\ U^etr&nB.K 6Kq. K=K .0 : 

* Reicg.n\Keic.D.RU,{R\K).{R\K)U^B 
Bern. 

I-.*37013. Oh:B.^.-D.R\K' = R (1) 

I- . *336-41 . (1) . D 1- : Hp . D . E [T, (i? I Z) . (E I Z) IT'.E : D h . Prop 

*370-21. l-:Hp*370-2.i2 6/«;g.jB|^6K.D.i?|ir = 7, 

Bern. 
h . *37012 . Transp . D 1- : Hp . iJfZ, (ii | Z) . (i? | ^) CT.i? . D . i? |Z ~ e «g (1) 
h . (1) . *370-2 . D h . Prop 

*370-22. l-:Hp*370-2.E6A;9-i'^.D.i?|ir~e« 
Bern. 

h . *370-21 . *330-32-5 . D h : Hp *370-21 .':>.R = K (1) 

F . (1) . Transp . D h . Prop 

*370-23. V : Hp*370-2 .i?6«;g. i2 = E. 3 .5 = ^ 

Bern. 

i-.*331-33. DI-:Hp.D.i?|Ze«uCnv"K (1) 

h . *330-5-52 . *34-2 . D h : Hp . D . jB | ^= Cnv'(i? | K) (2) 

l-.(l).(2). DV:Hp.D.R\KeK. 

[*370-22.Transp] D.R = K:D\-. Prop 

*37024. hzKeFMcycl.D.ElK, [*370-l-23.(*370-02)] 

*370-25. \-:.KeFMcycl.D:ReKs.R = R. = .R=K^ [*370-24 . (*370-02)] 

*37026. h-.KeFM cycl .D .K.e Kg. K, = K,. K,^^ 7. [*370-24-2513] 

*370-3. h-./ceFM cycl . RU^K^ .D.R = L 

Bern. 

b .*S36-4>1 .D\- i.B.p .D : Re K :('3^S) .8 e Kg. R = K^\8 (1) 

I- . (1) . *370-21-24 . D h . Prop 

*370-31. \-:KeFMcycl.ReKs-i'K^.'2.R\K^6 Cnv"«:g 

[*331-33 . *370-22] 
*370-311. H : Hp *370S1 . D . E j i<r« e «g 
Bern. 

\- . *370-31 . D h : Hp . D . ^4Ee«g . 

[*330-5.*370-26] O . ^ | Z. e «g : D h . Prop 



464 QUANTITY [part VI 

*370-32. \- : K 6 FMcycl . Re K .D .D'R = a'R = s'a^K 
Dem. 

V . *50-5-52 . 3 h . D'/^ = d'l^ = s'Q'V (1) 

I- . *370-26 . *:330-52 . D f- : Hp . D . D'^« = d'K, = s'a"* (2) 

V . *370-31 . *330o2 . D h : Hp . iJ e «g - i'K^ . D . D'(i? j ^,) = s'a"« . 
[*330-52.*34-36] D . D'J? = s'a"« (3) 

I- . (1) . (2) . (3) . D h . Prop 

*370-33. h : K e i^if cycl . D . D"« = Q"* .ksFM connex 
[*370-32 . *334-42] 

*370-34. h : « e iWcycl . D . Jf^ e connex [*370-33 . *336-62 . (*336-011)] 

*370-35. h : Hp *370-31 . D . ^, UJi .'^{RJJ^ K^) 

[*370-3 . Transp . *370-34] 
*370-36. V-.Ks FM cycl . D . Kg ^ /7^ e connex . (7'«g '\U^ = k 

Dem. 
l-.*336-41. DI-:Hp.D.C"K3lf/^C«r (1) 

h.*370-34. DI-:.Hp.i2,/SeKa.ii4='Sf.D: 

i?(«g1f7,)S.V.;S(«g1Z7.)i? (2) 

f-.*336-41. DI-:Hp.iJe«;g.,Sf = /,.D.E(/i;g1C7,),Sf (3) 

l-.*336-41. DI-:Hp.S6A;9.ii; = /«.D.S'(«g1fr,)i2 (4) 

h . (2) . (3) . (4) . D H : . Hp . E, »S e K . R 4= <S . D : 

iJ(«g1i7,)>S.V.S(«g1i7,)iJ (6) 

I- . (1) . (5) . D I- . Prop 

*370-37. h : K 6 J^if cycl . D . «9 1 f/, e Ser [*37011-l-36] 

*370-38. h:«;6JWcycl.ii,/Se«.D.^|/S = 5|^ [*330-561 . *370-32] 
*370-4. V-.KeFM cycl . D . Cnv "« = iT, | "« 

h . *370-31 . *330-5 . D h : Hp . D . ^, | "(xg - I'K,) C Cnv"« (1) 

I- . (1) . *370-26 . D h : Hp . D . ^. I "« C Cnv"« (2) 

h . *370-311-26 . Dh:Hp.ii6/c.D.^|^«6/<;. 
[*370-26] D.(a5f).-S'eK.^ = >Sliir,. 

[*330-5.*37-6] :i.ReK^\"K (3) 

I- . (2) . (3) . D 1- . Prop 

*370-41. V:.KeFMcyA.R,SeK.-^:{K,\R)V,{K,\S). = .RU,S 
Dem. 
V . *336-54 . *370-33 . D 
|-:.Hp.D:(^.|ii)F,(Z.|S). = .(ar).Te«9.if,|ii = r|irjS. 

[*330o.*370-26] = . (gT) .TeK„.R=T\S. 

[*336-41] =.Et/',5(:.Dh.Prop 



SECTION D] elementary PROPERTIES OF CYCLIC FAMILIES 465 

*370-42. h-.KeFMcycl.D.Kn Cn\" ic = I'L " (-'K^ 
Dem. . ^ 

1- . *370-22 . D I- : Hp . ^ 6 «:g - I'K, . D . E | ^ ~ e « . 
[*.370-311.Transp] D . E ~ e /eg - I'K^ (1) 

l-.(l). DI-:.Hp.D:E,Ee«:.D.-Bei'/.ui'-ff. (2) 

l-.(2).*370-26.DI-.Prop 

*370-43. hzKeFM cycl . D . ^. | "k^ = Ca v"a: - I'K, [*370-4] 
*370-44. h-.KeFMcycl.D.Ksr^K^l "kq = A [*370-42-43] 



E. & w. III. 30 



*371. THE SEEIES OF VECTORS. 

Summary of *371. 

In this number, we begin by defining the relation W^, which takes the 
place, for cyclic families, of the relation F, defined in *336. The definition 
is 

*371-01. W^ = K,\'U,Iks^ U.Iks Df 

Then if k is a cyclic family, W. is a series (*37l"12), and its field is « u Cnv"/c 
(*37l"14), which = «. since k has connexity. It will be observed that F, is 
not a series if « is a cyclic family ; we have e.g. I^V^Kg . K.V.I. . The above 
relation W. is constructed so as to make a barrier at I., thereby preventing 
the relation W. from being cyclic. 

If P, Q are both members of /eg or both members of K. \ "«g, 
PW.Q. = .{'^T).TeKs.P = Q\T (*371-15-151). 
Most of the properties of W. depend upon the fact that «g "j U. is transitive, 
in virtue of the definition of cyclic families. If k is any connected family, we 
have 

«g1 C7,etrans.= :P,Q,Q|i2,P|e|i26/<:g.i?6«.Dp,Q,B.P|Qe«g (*37l-2). 
This proposition is required for most of the subsequent proofs in this number. 
It leads at once to 

*371-21. f-:«6Pilfcycl.P,Q,Q|P,P|Q|Ee«g.Pe«:.D.P|g6K9 

Most of the propositions of this number are concerned with the circum- 
stances under which we can infer (P | R)W. (Q | R) from PW.Q. We have 

*371'31. 1- :. « 6 PM cycl .i26«;g:Pe«g.v.P|i2~6«g:D: 

PW.q.:^.{P\R)W.{Q\R) 
Another useful proposition is 

*371-27. V z.KeFMayd .P.Qe/cg.'^: PW.Q . = . QW.P 



*37r01. W. = K.\''U.tKs^U.tKs Df 

*3711. h ::. KeFMcycl . :: PW.Q . = :.P,QeK.\ "k^ : 

{•^R,^.R,8eKs.RU.S.P = K.\R.Q = K.\S'.yi: 
P,QeKs.PU.Q:v:P€K.\"Ks.Q6Ks 
[*202-55 . *370-34 . (*37l-01)] 



SECTION D] the series OF VECTORS 467 

*37111. \-:KeFM.K€ic.-:i.(K\)[Kel-H 
Dem. ^ 

I- . *330-31 .Dh:Hp.i2,Se-«:.^|iJ = Z|/Sf.D.JS = S:Dh. Prop 
*37112. f-i/eeiWcycl.D.F^eSer [*3r0-37-44 .*371-11 .*204-21-5] 
*371-13. t- : /. 6 Jf'ilf cycl . D . F, = F. t (Cnv"« - I'K,) ^ U, ^k^ [*370-41-43] 
*37114. V-.kbFM cycl . D . C F« = « u Cnv"«: = k^^K^\ "wg 
Dem. 
V . *202-55 . *370-34 . *1 60-14 . D f- : Hp . D . C" TT, = ^, | "k^ yj «a 
[*370-43] = K u Cnv"* : D I- . Prop 

*37115. h :. K e J?'ilf cycl . P, Q e Kg . D : PTf,^ . = . (aT) . Te «g . P = Q | T 
[*370-44 . *336-41 . (*371-01)] 

*37ri51. h :. « 6 FM cycl .P,QeK.\ "wg . D : PW^Q . = . (^T).TeKg.P=Q\T 

Dem. 
h . *370-44 . *336-41 . D I- :. Hp . D : 

PW,Q. = .{-^R,S,T).R,S,TeKs.R = S\T.P = K,\R.Q = K,\S. 
[*370-26] = . (gT) . Te «g . P = Q | T :. D h . Prop 

*371152. h : « 6 l-Jlf cycl . P 6 if, I "«g . Q 6 «g . D . P r«Q [*37 1 -1] 

*37ri6. l-:K6J'ilfcycl.P6«g.PTr.Q.3.Qe«g [*370-44.*37ll] 

*371161. 1- : /tePilf cycl , QeiT, | "«g.PTf.Q . D .Pe^.] "«g 
[*370-44.*3711] 

*37117. I- : « 6 Pil/ cycl . Q, y 6 «g . D . (Q I T) W.Q .(Q\T) WJ! 
[*37115152] 

*37118. h : « e Pilf cycl . D . WJK, = Z, | "«g . W.'K, = «g - t'Z. 
[*37115-152 . *370-311-22] 

*37119. h :. « 6 Pilf cycl . P + /« . D : PPT^Z, . = . KJ17J' 
[*371-18.*370-43] 

*371'2. I- :: « e Pilf conx . D :. Kg ^ i7« e trans . = : 

P,Q,Qii?,P|Q|i2e«g.i?6K.Dp,c,ie.P|Q6Kg 

i)em. 
I- .*336-41 . D I- :. Hp . D : ^(Kgl Cr,)S. S(«g1 U,)R.~. 

(^P,Q).P,Q,S,TeKs.Re>c.T = P\S.8=Q\R (1) 
I- . (1) . *13-21 . D h :: Hp . D :. Kg1 C7« etrans . = : 

P,Q,Q|i«,P|Q|^eKg.i26K.Dp.Q,je.(P|Q|ii)f7'.iJ (2) 
f-.*330-31-5.D 
h:.lip.P,Q,Re>c.M6>cs.P\Q\R = M\R.:i.P\Q = M (3) 

h . (3) . *336-41 .Dh:.Ep.P,Q,R,P\Q\ReK.D: 

(P\Q\R)U,R, = .P\QeKs (4) 

l-,(2).(4).DI-.Prop 

30—2 



468 QUANTITY [PABT VI 

*371-21. [■■.KeFMcyc\.P,Q,Q\R,P\Q\R6Ks.ReK.D.P\QeKs 
[*37l-2 . *370-l] 

*371-22. h:KeFMcyc[.P,R,P\ReKs.PW.Q.0.Q\R6Ks 
Dem. 

h . *371-15-16 . D h : Hp . D . (gT) . Q, Te /cg . P = Q | T (1) 

l-.(l). 0\-:Rp.D.('^T).Q,R,T,Q\T,Q\T\R6Ks. 

[*37r21] D.QIfie/fgOI-.Prop 

*371-23. \-:k€FM cycl . TW.S . D . TW, (S \ T) 
Bern. 

h .*330-31.*370-38.D h : Hp . D . 2'=,Sf|(^| 2') (1) 

h . (1) . *37ri5-16 . ■^V:R^.T,S\TeK^.:i.TW,{S\T) (2) 

h . *371-15-16 . DhiHp.Te/tg.D.^ITe/tg (3) 

|-.(2).(3). DhiHp.re/.g.D.rif.C^ir) (4) 

h.*37l-152. Dh:Hp.r~6/«;g.S|T6«g.D.rTr,(,S|r) (5) 

l-.*37l-151-161. Dh:Hp.S~6/cg.D.T~e«g./Sj2'e«g (6) 

l-.(5).(6). DI-:Hp.<S~e«9.D.rF«(,S|7') (7) 

l-.(l).*37l-151. D(-:Hp.2',,S|r~6K9.5f6«g.D.rr,(iS|r) (8) 

h.(5).(8). Dh:Hp.y~e«g./SeA:g.D.rTr«(S|y) (9) 

h . (4) . (7) . (9) . D h . Prop 

*371-24. V:KeFMcyd.P,R,P\Reic^.PW,Q.-^.{P\R)W,{Q\R) 
Dem. 

h . *371-15-16 . D I- : Hp . D . (aT) .P,Q,R,P\R,T eK^.P=Q\T . 
[*37l-21.*330-5] D . (gT) .P\R,Q\R,TeK^.P\R = Q\R\T . 

[*371-15] D.(P|i?)F,(Q|E):DI-.Prop 

*371-241. l-:«6Pilfcycl.P,ii6«g.P|iJ~e«g.PF,Q.D.(P|iS:)F,(Q|JB) 
i)em. 

h . *371-152 . D h : Hp . Q I P e «g . D . (P I i?) F. (Q 1 E) (1) 

h . *37115 . D 

l-:Hp.Q|P~e«g.D.(ar).r€«g.P|P,QlP~6/«:g.P|P = Q|P|r. 
[*37ri51] D.(P|P)F,(Q|P) (2) 

I- . (1) . (2) . D h . Prop 

*371-25. h : « 6 Pif cycl .P,Reic^ . PF,Q . D . (P | P) F. (Q | P) 
[*37l-24-241] 



sECTibN d] the series of vectors 469 

*371-251. h : « 6 FMcyoX .R,R\QeK^..PW,Q.-^ .{R\P)W,(R\Q) 
Dem. • 

f- . *37l-25 . Transp . *371-12 . D 

h : /c e J'jl/ cycl . P, iJ 6 «g . ( Q I i?) F« (P I jR) . D . Q W,P ( 1 ) 

, . . R\q,R\P ^. p 

*371-26. h i.KeFMcyd '.P,QeK^.v .P,Q'^eic^'.:i: 

PW.Q.= .{K,\P)W.{K,\Q) 
Dem. 

l-.*37l-25.*370-26. D h : Hp.Pe^g.PPT^Q. D .(^,|P)Tf,(^,| Q) (1) 

h . *37l-251 . *370-26 . D I- : Hp . Q e «g . (iT, | P) PT, (Z^. | Q) . D . PTT.Q (2) 

l-.(l).(2). Df-:.Hp.P,Q6«9.D:PF.Q. = .(Z»|P)Tr,(Z.|(3) (3) 

l--(3)'^^^^^.*37l-14.D 

h:.Hp.P,Q~e«3.D:PTf,Q. = .(Z,|P)Tr,(Z,|Q) (4) 

h . (3) . (4) . 3 h . Prop 

*371-27. I- :. KeFMcyd .P,QeK^.:i: PW,Q . = . QWJP 
Dem. 

I- . *371-15 . D I- :. Hp . D : PWM ■ = ■ (a^) . Te /cg . P= Q | T . 

[*37o-33] =.(ar).ye«g.Q=Piy. 

[*37l-15119.*370-43] s . QTT^P :. D h . Prop 

*37r3. \-:K6FMcyc\.ReK^.P\R'^eKg.PW^Q.0.(P\R)WAQ\R) 
Dem. 

I- . *371-27 . D f- : Hp . D . QW^P . 

[*371-251] D.(R\Q)W,(R\P). 

[*37r27] D.{P\R)W, (Q \R) : D h . Prop 

*371-31. h :. « € FM cyc\ .i2e«g:i'e«g.v.P|E~e«g:D: 

P F«Q . D . (P I i?) F« (Q I R) [*371-25-3] 



*372. INTEGRAL SECTIONS OF THE SERIES OF VECTORS. 

Sicmmary of *372. 

The subject of this number is that section of TF» which consists of 
vectors not greater than the rth part of the whole circumference of the 
cycle. This is defined by means of W^, as consisting of those vectors which 
(taking W^ as " greater than ") are such that R''+^ is greater than R' so long 
as a <.v. It will be seen that so long as iJ" and all earlier powers of R 
do not exceed /,, R satisfies this condition; but if R" eKK\"K^, while 
ii°^' 6 Kg, we shall have R''WkR'''^\ Thus our definition selects those vectors 
which, starting from any origin, do not, by v repetitions, take us farther than 
once round the cycle. The definition is 

*372-01. K^ = (« w Cnv"«:) r. ^ (o- < 7/ . o- + . D, . E'+i W^R") Df 

We then have l» = «uCnv"K (*37211),2«=«:g(*372-13), /*<!'. D.v^C/t,, 
i.e. Vg diminishes as v increases (*372 15) ; i/ > 1 . D . i/^ C Kg (*372"16). 

An alternative formula for Vk, sometimes more convenient than the one 
given in the definition, is (assuming v > 1) 

i/« = «:gnP(Ai<i'./t=|=0.P''+'6«g.D^.P''6«;g) (*372-17); 

i.e. so long as /jl<.v, either P** comes in the upper semi-circle, or P''+i comes 
in the lower semi-circle ; that is to say, the step from P" to P''^^ does not 
cross /». For an even number (not zero), this leads to a simpler formula, 
namely 

(2j»)« = «g rx P (/i < i; . /t =)= . D^ . P" e Kg) (*372-18). 

We have next a set of propositions leading up to 

*37227. I- :. KePJf cycl . j; eNCind- I'O . Pe j;« . PW^Q . D : 

/t<i'./*=|=O.D.P''Tf,Q^ 
whence, since W^ is a series, we obtain 

*372-28. h : . K e FM cycl . i/ e NO ind - t'O . P, Q e i;« . D : P" = Q- . = . P = Q 

It is largely owing to this proposition that v^ is important. In virtue 
of this proposition, there is in v« at most one vector which is the vth sub- 
multiple of a given vector. We shall show later that, if « is a submultipliable 



SECTION D] integral SECTIONS OF THE SERIES OF VECTORS 471 

cyclic family, there is at least one such vector ; hence there is a unique vector 
in v« which is the i/th »ibmultiple of a given vector. This does not hold in 
general for larger classes than v^. 

A specially useful case of the above proposition is obtained by putting 
v = 2, which gives, in virtue of *372-13, 
*372-29. I- : . /c 6 iW cycl . P, Q e Kg . D : P" = Qi" . = . P = Q 

The remaining propositions of this number are concerned in proving that 
Vk is an upper section of Wk, i.e. 

*372-33. h : AC 6 Pif cycl . i; 6 NO ind . D . W^"v^ C v. 



*37201. i;« = («uCnv"«)nE(<7<i/.«r + 0.D,.-B''+'Tr«i?'') Df 

*3721. \-:.Rev^. = :ReKyj Cnv"* : o- < v . o- + . D„ . R'+^W^R' 
[(*872-01)] 

*37211. h.l. = /euCnv"« [*3721 . *ll7-53] 

*37212. h : K e PJlf cycl . iJ 6 /f I "Kg . D . jB W^R" 
Bern, 
h . *371152 . D f : Hp . iJ» 6 «g . D . P W,R' (1) 

l-.*370-44. DI-:Hp.P2~eKg.D.P,P^~e/c3.-B6/<:g.P = P|P=- 
[*371-151] :^.RW,R\ (2) 

I- . (1) . (2) . D h . Prop 

*372-121. > : « 6 PJM" cycl . P e Kg . D . P'' W^R [*371-l7] 

*372122. h:.K6PJfcycl.D:P6Kg. = .P»r«P [*372-12-121 . *37l-12] 

*37213. h : K e Pif cycl . D . 2, = Kg [*372-122] 

*37214. I- : K 6 PJ/ cycl . D . ^, ~ e 3, 

Bern. h . *371152 . D h : Hp . D . iT.^ F«Z/ Oh. Prop 

*372-15. \-:fi^v.0.v,Cti, [*372-l] 

*37216. h : K e Pilf cycl . K > 1 . D . v« C Kg [*372-l 5-13] 

*37217. f- : K e Pilf cycl . i/ > 1 . D . 

i;« = K9nP(/*<l'./*4=0.P^+l6Kg.D^.P''6Kg) 

Dem. 
1-.*3721-16.*371-16.3 

l-:Hp.D.i'«CK9ftP(/i<i;./i + 0.P''+>eKg.D^.P''eKg) (1) 

l-.*37115. ^^-:Hp.P,P^P'•+^eK^.D.P''«T^.P'' (2) 

t-.*37l-152. D 1- : Hp. P, P" e Kg. P''«~e Kg. D.P''+^ If > (3) 

h . *371-151 . D h : Hp . P e Kg . P", P''+' ~ 6 Kg . D . P''+^ W.P'^ (4) 
l-.(2).(3).(4).DI-:.Hp.PeKg:P''eK9.v.P''+"~eKg:D.P''+'F.P'' (5) 

t- . (6) . *3721 . 3 h : Hp . D . Kg A ^ (/* < I' . /t+0 . P^+'eKg . D^ . P''eKg)C i», (6) 
h . (1) . (6) . D h . Prop 



4)72 QUANTITY [part VI 

*37218. h-.KeFM cycl . «/ > . D . (2i/)« = /eg n P (/^ < j/ . /i + . D^ . P" e /cg) 

Dem. 
h . *372-l . *37112 . 3 h : Hp . P 6 (2i/)« . D . P^'WJ" . 
[*372122] D.P-e/tg (1) 

I- . (1) . *372-l7 . D h : Hp . D . (2v), C A;g n P (yit < i- ./*=!= . D^ . P^e «g) (2) 
l-.*37115152. Dh:Hp.P,P''6«g.D.P''+'F.P'' (3) 

t- . (3) . *371-25 . OI-:Hp.P,P''+SP''e«;g.D.P''+''+'F«P''+'' (4) 

H.(4). D \- :. P e Kg: fi^v . fi=^0 .D^. Pi" 6 KgzD : 

/i + 1 < 1/ . p < K . D^,p . P^+p+iTF^P^+p : 
[*117 561] D : o- < 2i; . D„ . P''+' PF«P' 

[*372-l] D:Pe(2i/), (5-) 

h . (2) . (5) . D f- . Prop 

*37219. h : « 6 PM cycl . /*, i/ e NC ind -I'O.Pe (/jlv\ . D . P" e i/« 
[*372-l . *371-12] 

*372-2. h : /cePikf cycl . i/eNOind . Pe i/« . /i< v . o- < /* . <7+0.D.P''F«P'' 
[*372-l . *371-12} 

*372-21. l-:«6PJ/cycl.i/6NOind.P6v,.2/i<i/./i4=0.D. 

P»'pr,P''.P''e«:g 
i)em. 

f-.*372-2.DI-:Hp.D.P^Tf,P''. (1) 

[*372-122] D.P^e/cg (2) 

h . (1) . (2) . D h . Prop 

*372-22. i- : « e Pilf cycl . P W^Q .P,P''eKg. P" W^Q' . D . P^+i TT^Q^+i 
i)em. 

I- . *371-25 . D h : Hp . D . Pi'+'W.P [ Q" (1) 

h . *371-16 . D h : Hp . 3 . 0^6 Kg . 

[*371-25] D.P|Q^FkQ^+i (2) 

h . (1) . (2) . *371-12 . D I- . Prop 

*372-23. \-:KeFMcyc\.ve'NCmd.P€v^.2fi^v.fi:^0.PW^Q.O. 

P''+' F«Q^+' [*372-21-22 . Induct] 

*372-24. f- :. « 6 PJlf cycl . o- e NC ind - t'O . P e (2o-). . P TF^Q . D : 

/i<2(r./i=t=0.D.P»'F«Q» 
Z>em. 

F . *372-21-23 . 3 h : Hp . f< o- . i; < o- . D . P', Q* e Kg . P^ F«Qf . P'TF^Q" . : 

[*37 1 -25] D . Pf +" TF^P^ | Q* . P" | Qf F.Qf +" . • 

[*37l-12] D.pf+'']f.Qf+'':Dh'.Prop 



SECTION D] integral SECTIONS OF THE SERIES OF VECTORS 473 

*372-25. V:.KeFMcya[..ae^C ind -t'O.Pe (2o- + 1)« . P W^Q . D : V- 

• /i<2<r./i + 0. D.P^Tf^Q" [*372-24-15] 

*372-26. h : « ePJlf cycl . o- e NO ind'. P e (2<t + 1). . PF«Q . D '. P^^'Tf.Q^^i 
Dem. 
h.*372-25. DI-:.Hp.D:P"'«F,Q''+': (1) 

[*37l-3] D : P*'+i ~ e /fg . D . P='+' TF.P' | Q^' (2) 

h . *371-31 . (1) . D>:.Hp:P''|Q'^'~6*:g.v.Q'+>6«g:D. 

h . *372-21 . *371-15151-152 . D h :. Hp . D : P'^ | Q'+^F «P' : 

[*371-1 6] D : P' I 0'+' 6 «;g . D . Q'+' e ^g (4) 

h . (3) . (4) . D h : Hp . D . P' I Q'+' W,Q"'+' (5) 

h . (2) . (5) . *371-12 . D h : Hp . P^-+i ~ e ^g . D . P*h-i if ^Q^^+i (6) 

h . *372-22 . D h : Hp . P"' 6 «g . D . P'*+' TT^Q^'+i (7) 

l-.*37l-16.*372-l. DhiHp.P^+^e/eg.D.P^e^g (8) 

h.(6).(7).(8).Dh.Prop 

*372-27. h :. « ePJl/cycl . i/ eNCind- t'O .Pev, .PW,Q . D : 

/i<v./*4=0.D.P'' TT.Q" [*372-24-25-26] 

*372-28. \-:.KeFMcycl.v€'i^G'md-i'0.P,Qev,.O:P''=Qr. = .P = Q 

Dem. 

l-.*371-12.Dh:.Hp.P + Q.D:PF,Q.v.QF.P: 

■ [*372-27] DiP'iT^Q-.v.Q-'F.P'': 

[*371-12] DiP' + Q' (1) 

I- . (1) . Transp . D h . Prop 
*372-29. h :.« 6 Pif cycl.P.Q 6 «g.D:P^=(2^ = .P = <2 [*372-28-13] 
*372-3. h : « 6 Pilf cycl . o- e NO ind - I'O . P e (2<r), . PF.Q . D . Q e (2<r), 

X)em. 

h .*372-18-27 . D I- :. Hp . D :/*< o- ./i + . 3;.. P'-e/cg . P'-Tf^Q" . 

[*37ri6] Dm -Q" 6*9: 

[*37218] D:Qei/«:.DK.Prop 

*372-31. h :. «ePMcycl . <7 e NC ind - t'O . Pe^g . D : PTT^P'*. D . P^'+>6«9 
Bern. 

I- .,*371-16 . D h : Hp . PWj'' . D . P^' e /cg (1) 

l-.*301-23. Dl-:Hp.D.P = P-|P-+^ (2) 

h . (1) , (2) . *371-15 . D I- : Hp . PF,P^ . D . P=^' 6 «9 : D h . Prop 



*74 QUANTITY- [part VI 

*372-32. hiKeFMcycl.ffe'NCind.Pe (2<r + 1)« . PF«Q . 3 . Q e (2«r + 1)« 
Z)em. 

l-.*372-315-17. 3l-:.Hp.D:^<2<r.(^e«g.D.(?'-ie«9 (1) 

H . *371-16 . *372-27-l . D h : Hp . Q="~ e ACg . D . i^+i ~ e Kg . 
[*372-31 .Transp] 3 . par W^P - 

[*37l-27] > D.Q^F^P. 

[*372-31 .Transp] 3 . Qsw+i ^ ^ ^^ (2) 

h . (1) . (2) . Transp . D 1" : . Hp . D : /* < 2o- + 1 . Q^ e Kg . D^ . Q^-' e Kg : 
[*372-17] D:Q6(2<r + l).:.DI-.Prop, 

*372-33. i-:KeFMcycl.v€'NCind.O.W,"v,Cv, [*372-3-32] j 



? 



*373. SUBMULTIPLES OF IDENTITY. 

Summary of *373. 

The purpose of this number is to prove that, in a cyclic submultipliable 
family, there exists a unique vector which is a member of v^ and satisfies 
E'' = Ik. This we call the "principal" i/th submultiple of /«. It is the 
smallest vector (other than /») which satisfies R' = I^. The proof of its 
existence proceeds by several stages; the problem is analogous to that of 
the construction of a regular polygon. Suppose the cycle divided into v 
equal parts. Then a vector which takes us from any one point of division 
to any other is a vth submultiple of identity. If v is prime, every such 
vector will have every power less than the vth different from I^; but if v 
has factors, say p and a, if R' = Iit, {BP)'' = I^ ; thus R", which is one of the 
vth submultiples of identity, has a power less than the vth which is equal 
to /,. We define (/«, v) as the class of those j;th submultiples of /» which 
have no power less than the I'th equal to /« ; more generally, we put 

*37303. (5,i;) = P(P-' = S:a-<i;.<r + 0.D,.P''=|=/S) Dft 

We then have first to prove the existence of «g n (/, , v) when k is cyclic 
and submultipliable. For this purpose, we put 

*373-01. il/„ = OP(Qe«g.Q'' = P) Dft 

I.e.M„^ is the relation of a vth submultiple of P to P, when the submultiple 
of P is a member of Kg. It is to be observed that although k is submultipliable, 
we do not know to begin with that /« has submultiples which are members 
of «g, except in the case of K^, which is half of /„. Owing to this, we proceed 
first by bisection, i.e. by means of the relation M^,,- We prove that the 
process of bisection can be applied endlessly to any member of Kg, and always 
gives new terms (*373'14-13), hence it gives a progression starting from any 
member of Kg (*378'141), and therefore the existence of a cyclic submultipliable 
family implies the axiom of infinity (*373"142) ; also we prove that v bisections 
starting from a member of Kg give a member of {2"+% (*373-15). Hence, 
taking K„ as the member of Kg to be bisected, we arrive at 

/* = 2"+^ 3 . a ! Kg ft (/„ /t) (*373-l7). 
In order to extend this result to numbers not of the form 2'"'"\ we have 



476 QUANTITY [part VI 

first to prove that there are ytith submultiples of identity. This we prove 
first for numbers of the form 2" + 1, then for .(2a- + 1) 2" + 1, and then for 
2<T (*373"21'22"23) ; hence it holds generally, i.e. we have 

*373-25. h -.KeFM cyclsuhm . /le'NCmd- L'O-l'l .D.('S^Q).Qe Kg. 0^=1^ 

Next, we prove that, if Re Kg and R'^ = R''= F^, then fi, v have some 
common factor p such that Re{I^,p), i.e. such that RP is the earliest power 
of R which is /, (*373'3). Hence if ft, is prime, and Rl^ = I^, it follows 
that no earlier power of R is /„, i.e. Re(I^,fi) (*373"32), and that, if 
Re(I^,p) and R/^=I^, then /i is a multiple of p (*373'33). 

We now make a fresh start with the general relation M^^. Owing to 
*373'25, we know that I^eQ-'M^^. Also since « is submultipliable, 
KgCd'My^. Hence if a is any inductive cardinal, I^ed'M^^'^ (*373"404). 
Also it is easy to show that if v is a prime, and Qif „«"/«, Q"" is the first power 
of Q which is /«. Hence when v is prime. Kg r\ (!„, v') exists (*373'43). In 
order to extend this result to numbers which are not powers of primes, we 
prove 

*373-45. I- : «:6^1f cycl . pFiraa- .Re(I^,p) .Se(I^, <r).D .R\Se{I^,pa) 

Hence by the help of a little elementary arithmetic we arrive at 

*373-46. \-:KeFM cycl subm . /> e NO ind - t'O - t'l . D . g ! Kg n (/^, p) 

Having now proved that there are j/th submultiples of I^ which have no 
power short of the vth equal to I„, we have still to show that there is one 
among them which is a member of v^. For this purpose, we take any one 
of them and consider its powers. It is obvious that it has only v different 
powers (*3735), since after reaching /^ the previous values repeat.themselves. 
It is this fact which makes it easier to deal with' submultiples of J^ than with 
submultiples of other vectors. 

Now let R be any vth submultiple of identity, and assume that S, T are 

powers of R, but T is not a power of S, and TW^S. Then /S | T is a power of 

R but not of 8, and TW^iS\T) (*373-53). Hence T is not the maximum, 
in the series W^, of the class Pot'i? — Pot'/S. Hence by transposition, if T is 
the maximum of Pot'i? — PofyS, we must have SWiJC. Now since Pot'iJ is 
a finite class, Pot'i2 — Pot'jS must have a maximum if it exists ; but since 8 
has the relation W^: to this maximum, 8 is not the maximum of Pot'iJ. 
Hence by transposition, if 8 is the maximum of Pot'i?, Pot'i? — Pot'^ is 
null, and therefore Pot'JS = Pot'/S (*373'54). Hence it follows easily that, 
if ReKgr\{I^,v), the maximum of the powers of iJ is a member of 
Kg n (/^, v) (*373'55), and further that it is a member of v^ (*373"56). Since 
we have already proved (*373"46) the existence of Kg n (7^, v), we thus have 

*373-6. f- : « 6 FM cycl subm . y e NO ind - I'O . D . g ! v« n ^ (zS- = I^) 



SECTION D] SUBMULTIPLES OF IDENTITY 477 

The uniqueness of v^nS (S' = /«) follows from *372-28, and tbus the 
principal vth submultijie of /« exists. Hence also it immediately follows 
that the other i/th submultiples of /« are powers of the principal vth sub- 
multiple, and that the total number of i/th submultiples is v (*373-63-64). 

*37301. M,, = QP{Q€Kg.Q'' = P) Dft [*373— 5] 

*37302. Prime = NCindr.J*0=o-x„T.D„,^:o- = l.v.o- = )Lt) Df 

*373-03. (S,v) = P(P' = S:ir<v.a-^0.:i,.P'^S) Dft [*373— 5] 

*373-l. \-:QM^P. = .Q6Ks.Q' = P > [(*373-01)] 

*37311. \- : K e FMcyd .D . M^el ^1 [*372-29] 

*37312. h-.Ke FM cycl . D . ilf^, G ^. [*372-121] 

*37313. V: K€FM cycl. D.(M^\„<1W,.(M^X„ Q J [*373-12 .*37l-12] 

*37314. \-:Ke.FM cycl subm . P e «g . v e NO ind - t'O . D . E ! M^"P 

Bern. 

h . *372-29 . *351-1 . D h :. Hp . D : Q e /cg . D . E ! M^'P (1) 

I- . (1) . Induct . D h . Prop 

*373141. \-:KeFM cycl subm .PeK^.O.M^i {M^\'P e Prog 

[*373-ll-13-14] 
*373142. hralMfcyclsubm.D.Infinax [*373-141] 

*37315. h : « 6 FM cycl subm . P e Kg . v e NO ind . D . M^*'P e {2'+% 
Dem. 
l-.*373-ri4. :)\-:B.^.Q = M,,''-^'P.R = M,,-"P.:i.Q- = R'' (1) 
I-. (1). *372-18. D h :. Hp (1). Qe (2'').. 3 :2<7< 2-. D.i?=- 6 Kg ■ (2) 
h . (2).*373-l . D h :. Hp(2). D : 2<r< 2". D . R^,R^+\R\ReK^. 
[*37l-2] D.E^'+'e.Kg (3) 

h.(2).(3). Dh:.Hp(2).D:/i<2''.D.iJ''6Kg: 

[*372-18] D:E6(2>'+iX W 

|-.*372-13. DI-:Hp.i' = 0.D.ilf,/'P62, (5) 

h . (4) . (5) . Induct .31-. Prop 

*37316. h :. K e i^'ilf cycl subm . v e NO ind . Q = M^''K^ . D : 

^^•'+' = /, : p < 2>'+> . p + . D^ . Qp + /, 

Bern. 
|-.*373-l. DI-:Hp.D.r = ir.. 

[*371-26] D.Q^''^' = /« (1) 

|-.*373-15.*372-2.(l).Dh:.Hp.D:p<2''+^p + 0.3.Q''Tr./. (2) 

h.(l).(2).D(-.Prop 



478 QUANTITY [PABT VI 

*37317. I- : K 6 FM cycl subm .ve'NCmd./i = 2'+"^ . D . g ! «g n (/«, /*) 
[*373-1614 . (*373-03)] 

*37318. \-:Q6Cnv"Kg.Q'' = I,.D.Q6Ks.Q" = L [*50-5-51] 
*37319. h : (aQ) . Q e Kg w Cnv"/Kg .Q" = !,. = . (gQ) .QeK^.Qr^I. 

[*373-18] 
*373-2. h : . « 6 ^if cycl subm . i/ e NC ind . P = M^^'K^ . 

;S e Kg . 5?^"+' = P . S^-^' = Q . D . Q=''+' = /, . Q + /« 
i)em. 

I- . *30r5 . D h : Hp . D . Qf+^ = P^'^' = /. (1) 

h.*373-l. DF:Hp.D.P'"'+' = Z,|P. 

[*370-22] D.P''''+'4=P. 

[Hp] D'.P2''+i + S^"'+>. 

[*30-37] D.P + S (2) 

h . *301-5-23 . D h : Hp . D . Q = (^''''+>)2 1 S'' 

[Hp] =I^\h 

[(2).*372-29] +/, (3) 

I- . (1) . (3) . D h . Prop 

*373-21. h : K e Plfcycl subm . v e NC ind . /n = 2" + 1 . 3 . 

(aQ)-Q6«a ■<?* = /« [*373-2-i9] 

*373-22. F : K € FM cycl subm . v, o- e NC ind . /i = (2o- + 1) 2" + 1 . D . 

(aQ).QeKg.(2'' = /. 

[The proof proceeds as in *373'2'21] 

«373-23. h:/eePJ>/cyclsubm.o-6NCind./i = 2<r.D.(aQ).Q6Kg.Q'' = J« 

Dem. 

h .*370-26 . D h : Hp . D . ^6Kg . ^" = 7, : D h . Prop 

*373-231. h :. T 6 NC ind . D : (go-) : o- e NC ind : t= 2o- . v . t= 2o- + 1 [Induct] 

*373-24. l-:/36NCind.p=)=0.D. 

(ai/, <r) . 1/, (7 e NC ind . 2p + 1 = (2o- + 1) 2- + 1 
Dem. 

f-.*l 17-661,3 

I- : . Hp . \ = i) {(gr) . t e NC ind - t'O . p = t2''} .D:z;e\.D./3>«/ (1) 

I- . *116-301 . 3 I- : Hp (1) . 3 . p = />2» - 

[*10-24] 3.G6\ (2) 

h . (1) . (2) . *261-26 . *263-47 . 3 h :. Hp (1) . 3 : 

(gi') : veX : /(i > v. 3^./i~e\, (3) 

l-.*116-52-321 .3 l-:/3 = T2-'.T = 2ff.3./3 = o-2''+' (4) 



SECTION D] SUBMULTIPLES OF IDENTITY 479 

t- . (3) . (4) . D h : . Hp . 3 : (a;v, t) : i;, T e NC ind . /o = t2"' :/[*>«/. 3^ . 

• ~(gT)./>=T2'':~(ao-).T=2<7: 

[*373-231] D : (gw, a) .v,ae NC ind . p = (2ff + 1) 2" : 
[*116-52-321] D : (gy, tr) . i/, o- « NC iud . 2/j + 1 = (2o- + 1) 2'+' + 1 :. D h . Prop 
*373-25. \-:KeFM cycl subm . /* e NC ind - t'O - t'l . D . 

(aQ)-Qe«S-Q^=^« [*373-22-24-23-14] 
*373-3. \-:k€FM cycl. ^=^0.v:^0. Re Kg. Ii^ = R'==I,.D. 

(a/>, a,/S)-/o + 0.(0=1=1./* = a/). i/ = /3p.^e(/„p) 

h.*300-23.DI-:.Hp.D:(ap)./, + 0.22p = 7,:«r</).«7=j=0.3,.iJ'=^/. (1) 
l-.*301-2. Dh:Hp. /&> = /,. D./,=|=l (2) 

l-.*302-25.DI-:Hp./)eNCind-i'0.D. 

|-.*301-23-504.D 'r /- f a' w 

h:Hp(3).i?^ = 7«./t = a/j + /3.i/ = 'yp + S.i&' = B-' = /,.D.JJs=Bs = /, (4) 
h . (4) . D I- :. Hp(4) : o-< p . o-=|=0 . D„ .i?''=t=/, : 

/t = a/) + /3.K = 7p + S:D.y3=i=0.S=0 (5) 
I- . (3) . (6) . D h :. Hp : p=|=0 .i&' = /, : cr< p . <r=t=0 . D,. iJ' + Z, : D. 

(a«. y)-ti = ap.v = yp (6) 
f- . (1) . (2) . (6) . (*373-03) . 3 h .Prop 

*373-31. l-:/<:e^il/cycl.J2e«g./*=|=0.i/=|=0.i^ = ^' = /,.D.~(/iPrmi/) 

[*373-3] 
*373-32. [■:KeFMcyc\.ReKs.fieTnme.Si^ = I^.'^.Re(I^,lj.) 

[*373-31 . Transp . (*373-03)] 

We assume here that a prime number is prime to all numbers less than 
itself except 1. This follows at once from the definition. 

*373-33. l-:«€^Jfcycl.Ee*:gn(/«, />).£"=/«. D.(aT)./t=/3T [*373-3] 

*373-4. h:QM,,P. = .QeKs.P = Q'' [(*373-01)] 

*373-401. h:/e€^ilfcyclsubm.i/eNCind-t'0.3./«ea'Jlf,, [«373-25] , 

*373'402. h-.KeFM subm . i; e NC ind - I'O . D . «g C a'Mi,^ [*373-4] 

*373-403. h:z/6NCind-t'0.D.D'if,,CKg [*373-4] 

*373-404. \-:iC€ FM cyc\ subm . i/, a e NO ind - 1'0 . 3 . /« e a'.¥„« 
[*373-401-402-403 . Induct] 

*373-405. \-:v,aeNCmd-i'0.QM,^'I^.O.Q'' = I^ [*373-4 . Induct] 



480 : QUANTITY [part VI 

*373-406. h : i;, « 6 N C ind - I'O . R e D'ilf,/ . D . M,,-^'R = i?'" 
[*373'4. Induct] 

*373-407. \-:v,a,ye NO ind - I'O . EM „»+»/. . D . E-'i/^T/, [*373-406] 

*373-41. I- : 7., «, ;S 6 NC ind -I'O . QM,,'I, . RMJI, . a < ;8 . D . Q + i? 
Bern. 

h . *373-405-4p7-403 . 3 h : Hp . D . Q"" = /, . i?"" e «g : D h . Prop 

*373-42. h-.Ke FM cycl . v e Prime - I'l . a e NC ind . 

Qilf„«/, . <r < i;« . <7 + . D . Q'=i=/. 

h.*373-405.*300-23.D 

l-:.Hp.D:(a/3):/> + 0.Q^ = /.:<r</3.<7 + 0.D„.Q' + /, (1) 

h.*373-33-405.D 

h :. Hp : p + . ^" = 7. : (7 < /3 . 0-4=0 . D, . Q' + /. : D . (gr) .V^ = pT. 

[Hp] D.(a^).(Q = i;3 (2) 

h . *373-407 . 3 h : Hp . ;8 < a . D . Q"^ =1= /« (3) 

l-.(2).(3). DI-:Hp(2).D./3 = z/' (4) 

l-,(l).(4).Dh.Prop 

In obtaining (2) of the above proof, we assume that if j/ is a prime, and 
pT is a power of v, then p is a, power of v. This is easily proved. 

*373-43. h : K e FM cycl subm . k e Prime — I'l.cte NC ind - t'l . D, . 

a ! «9 n (7, , j/») [*373-404-405-42] 

*373'44. I- : 7 Prm p . 7 Prm o- . D . 7 Prm /so- 

Dem. 
h . *302-l . D I- :. 7 Prm /) . ~ (7 Prm pa).(re NO ind . D . 

(a^. a, /8) . T e NC ind - t'O -I'l .y = ar . pa = ^t (1) 
l-.,*303-39. DI-:Hp(l).TeNCind-t'0-i'1.7 = aT.(0o- = j8T.D. 

7/p = ao/^ (2) 
I- . (2) . *308-341 . D h : Hp (2) . ao- Prm /3 . D . 7 = ao- (3) 

l-.(3).*302-l. DI-:Hp(3). 0-4=1. D.~(7Prmo-) (4) 

1-.*113-621. :>l-:/3 6NC.o- = l.~(7Prm/)o-).D.~(7Prm/>) (5) 

I- . (5) . Transp . DH : Hp(l). >.o-4=l : 

[(4)] Di-:Hp(3).D.~(7Prmff) (6) 

I- .*302-36 . D h : Hp(2) . ~(ao-Prmi8) , D . 

(a?,'?. • ?Prm ,, . 2:4= 1 . «a = ^r. ^ = ^f ^ (7) 
h .*303-39 . D h : Hp(7) . f PrmT? . ^4= 1 . aa- = ^?. ;8 = »;?. D . aa/^ = ^/i, . 
[(2).*303-341] O . .7 = f . p = ^ . 

[Hp] D . a/30- =0y=ttp^T. 

[*126-41] D . o- = S't (8) 



SECTION D] SUBMULTIPLES OF IDENTITY 481 

l-.(7).(8). DI-:Hp(7).D.(ar).7 = «T.<7=rT- 

[*302-l.Hp] • D.~(7PrmCT-) (9) 

l-.(6).(9). Dl-:Hp(2).D.~(7Prm<7) (10) 

l-.(l).(10).Dh:7Prm/3.~(7Prm/3<7-).o-6N0ind.D.~(7Prmo-) (11) 
h . (11) . Transp . D h . Prop 

*373-441. 1- :. jo Prm o- : (gS) . /3/3 = So- : D . (g^) .^=^a- 

Bern. 
h . *126-41 . D 

I- : Hp . /3^ = So- . /a = fw . S = 7/OT . f Prm i; . D . |^/3 = tjo- . ^ Prm rj . f Prm o- - 
[*373-44] D . ^^ = 170- . ^ Prm t/o- (1) 

l-.(l). Dl-:Hp(l).f + l.D.Hl-?=?Xcl-'7'^ = rXc/3- 

[*302-l] 3 ■ ~ (? Prm.i?ff) (2) 

F . (2) . Transp . (1) . D h : Hp(l) . D . ^= 1 (3) 

h . (1) . (3) . D h . Prop 

*373-45. h : K e FMcyd . pYrm (T . B e(T^, p) . S e{I^,a) .0 . R\S €(1^, pa-) 
Bern. 

h.*370-33. DI-:Hp.D.(^|/Sf)'>"- = /, (1) 

H . (1) .*373-31 . D h :. Hp . (^ j <Sf)i' = /, . 7=1=0 . D : ~ (YPrm/jo-) : 

[*373-44] D:~(7Prm;(»).v.~(7Prmo-) (2) 

h . *370-33 . *301-504 . D 

l-:Hp(2).p = aT.7 = /3T.D./, = (^|<Sf)"P^ = -S-3- = S'"'. 

[*373-33] D.(aS)./}/3 = So-. 

[*373-441] D.jaf).y8 = ?o- (3) 

f- .(3) . D I- : Hp(3) . D . (^| >Sf)^- = /. . S^^ = I, . 

[*370-33] •^.BP- = I,. 

[*3Jr3-33] 3 . (a/i) ■ ;St = IJMT . 

[Hp] 3-(a/<*)-7 = W-/* + (4) 

h . (3) . (4) . D h : Hp (3) . D . {'Sy) . 7 = 1//00- . 1/ 4= (5) 

Similarly h : Hp.~(7Prm o-) . D .(ai').7=i^po-. v=t=0 (6) 

I- . (2) .(6) . (6) , D h : Hp(2) . D . {'Sy) . «; =t= . 7 = i^po- (7) 

l-.(l).(7),*ll7-62.DI-.Prop 

*373-451. I- :. /) 6 NC ind - I'O : ~ (gv, a) . 1/ e Prime . /» = i;" : D . 

(SA*' ^) ■ /* ■P'"'^ v.fji<.p.v<C.p-p=iJi'V 
Bern. 
f-.*261-26.*263-47.D 

h : Hp . 3 . (37, a) . 7 « Prime . ,0 6 D'Xe 7" . p ~ 6 D'Xe 7"'+^ p =t= 7' . 
[*373-44.Induct] D . (37, «, ^8) . 7 e Prime . p = 7-/8 . /3 Prm 7" , /3 4= 1 : D h . Prop 
R. &w. III. 31 



482 QUANTITY [part VI 

*373'452. h :.v€ Prime . a e NC ind . D,_. . (v") : /j, Prm i; . <^/t . ^v . D^, „ . 

^ (/ii;) : D : /3 6 NO ind - I'O . D^ . ^ (jo) [*373-451] 

*373-46. h : « 6 ^Jlf cycl subm . |0 e NC ind - t'O - t'l . D . g ! Kg n (7^ , /a) 
[*373-43-4518-452] 

*373-5. \-:Ke FM cycl . y e NC ind . i? e Kg n (/, , i;) . D . Pot'i? e v 
Bern. 

h.*302-25.*301-504.D 

h : Hp . a 6 NC ind . D . (gf , i?) . a = ^z/ + 17 . ■»? < i/ . iJ" = iJi . 
[*120-57] D . Nc'Pot'^ < V (1) 

I- . *301-23 . D\-:RY>-P<v.(r<p.D.R'\B' = B/^'" . 
[Hp] D-E'lEo + Z.. 

[*330-32] D.iJo + jB' (2) 

h . (2) . Transp . "^V :Yi^ . p<v .a <v . BJ' = R'' ."^ . p = v (3) 

I- . (3) . *120-57 . D f- : Hp . D . Nc'Pot'i? > v (4) 

h . (1) . (4) . D 1- . Prop 

*373-51. h : K e^iW cycl . ii e Kg n (/«, /ii/) . D . i?^ e (7^, z^) . Pot'^ e v 
Bern. 

h.*301 -504.3 

I- :. Hp . D : (i^)-' = 7^ : o- < z/ . <r + . D, . (iJ")' + 7. :. D h . Prop 

*373-52. V-.KeFM cycl .Be K^ry{I^,v) . fi Prm i/ . D . 

i?" e (7«, J/) . Pot'i^ = Pot'J? 
Bern. 

l-.*373-33. DI-:Hp.i?''6(7,,jo).D.(aT)./i/9 = z/T. 

[*373-441] D.(a?)./p = i;^ (1) 

l-.*301-504. D\-:R-p(l). D.iRi^y^I^. 

[Hp] D.p^z. (2) 

1-.(1).(2). DI-:Hp.D.i^6(7<,i;) (3) 

I- . (3) . *373-51 . D h : Hp . D . Nc'Pot'Z^ = Nc'Pot'ii = v (4) 

l-.*91-6. D I- : Hp . D . Pot'iJy C Pot'E (5) 

h . (4) . (5) . *120-426 . Transp . D h : Hp . D . Pot'i?^ = Pot'JJ (6) 
h . (3) . (6) . D h . Prop 

*373-521. h : KeiW cycl . EeC^g u Cnv"Kg) . veNCind . B''=I,.D.Be'Pot'Ii 
Bern. 

l-.*301-2.*13-14.DI-:Hp.D.i/ + (1) 

I- . (1) .*301-21 . D h : Hp . D.B = B'-'' : D I- . Prop 



SECTION D] SUBMULTIPLES OF IDENTITY 483 

*373-522. I- : Hp *373-521 . S, Te Pot'iJ .D.S\ TeTot'R 
Bern. • 

h . *373-521 . D h : Hp . D .SePot'S. 

[*91-6] D.SeVot'R. 

[*91-343] D . S I TeFot'R Oh. Prop 

*373-53. h : Hp *373-521 . ,S, TeFot'R .T~e Fot'S . TW,8 . D . 

TW.(S\T).S\TeFot'R- Pot'S 
Bern. 

l-.*371-23. ■D\-'.H.Yi.:i.TW,(S\T) (1) 

l-.*373-522. D\-:Rp.O.S\TePot'R (2) 

I- . *91-36 . Transp .Df-:Hp.D.,S|r~e Pot',Sf (3) 

f- . (1) . (2) . (3) . D F . Prop 

*373-531. I- : Hp *373-53 . D . ~ {T = max ( Tf,)'(Pot'ii - Pot^^f)} [*373-53] 

*373-532. h : Hp *373-521 . ,Sf 6 Pot'i? . T = max ( F,)'(Pot'E - Pot'S) . D . 

SW^T [*373-531. Transp. *371-12] 

*373-533. h : Hp *373-52 1 . ^f e Pot'iJ . E ! max ( W,y{Pot'R - Pot'S) . D . 

~ {S = max ( W;)'Pot'R} [*373-532] 

*373-54. I- : Hp *373-521 . S = max ( F,)'Pot'i2 . D . Pot'i? = Pot'S 
Dem. 
f- . *373-533 . Transp . D I- : Hp . D . ~ E ! max ( F,)'(Pot'E - Pot'/S) (1) 
h . (1) . *373-3-5 . *261-26 . Transp . D h : Hp . D . Pot'E - Pot'*S= A (2) 
I- . (2) . *91-6 . 3 h . Prop 

*373-55. \-\K6 FMcycl . i; e NO ind - t'O . i2 e «g n (/«, v) . 

S' = max ( W.yPot'R .D.Se{I,,v) 
Dem. 

|-.*373-3-5. DI-:Hp.>.(a|o)./3 6NCind-i'0.<Sfe(7„,/3).Pot',Sfe/3 (1) 
h . *373-54-5 . D h :. Hp . D : Pot'»S e v : 

[*100-34] D:/36N0.Pot'^6/3.D.|0 = i' . (2) 

h . (1) . (2) . D h . Prop 

*373-56. f- : Hp *373-5o . D . >S e v, 

Dem. 
l-.*205-21. DI-:Hp.QePot'i?-i'S.D.QPr,fif (1) 

|-.(1).*301'21. DI-:.Hp.«6NCind.^''+'=|='S.3:^"+'W^«'S./S«+i = ,S-|,S: 
[*37l-15] 3 : ^"+' 6 «g . D . >Sf« 6 «9 (2) 

I- . (2) . *373-55 . DI-:.Hp.D:a + 0.a<i;.-Sf''+'6«:g.D.fif'>6«:g (3) 

|-.*371'16. Dl-:Hp.D./S6Kg (4) 

|-.*301-2.*13-14.DI-:Hp.D.i'>l (5) 

t- . (3) . (4) . (5) . *37217 . 3 h . Prop 



484 QUANTITY [part VI 

*373-6. b-.Ke FM cyd subm . v e NC ind - t'O . D . g ! v« n ^ ((S' = 7«) 
[*373-46-56-5 . *261-26 . *37211] 

*373-61. l-:Hp*373-6.D.i',n^(fif = /,)6l [*372-28 . *373-6] 

*373-62. h : Hp *373-6 .Sev^.S'' = I,.D . 

S e (/„ v) . Pot'-S = P(P' = I,) n (« u Cnv"/c) 
Dem. 

h . *373-55-56-61 . D h : Hp . D . /S e (7„ z^) (1) 

h .*373-56-54 . D h : Hp . i? eC/^.i/) n Kg . r=max(FO'Pot'i? . D . 

<S,Tey„.<Sf'' = r''.i?ePot'r. 
[*372-28] D.;S=y.i?ePot'r. 

[*13-12] D.i? 6 PofyS (2) 

l-.*373-33. Dh:Hp.iJ6(/.,/i)n«g.i?-' = /^.D.(aT).i' = /AT (3) 

h . *37219 . 3 t- :. Hp . D : I' = /iT . D . /Sf V /A, . 

[(2)] D.^ePot'/S- (4) 

h.(3).(4). Dh:Hp(3).D.EePot'<S (5) 

I- . (1) . (2) . (5) . D I- . Prop 

*373-63. biKeFM cycl subm . i/ e NO ind - I'O . D . 

P (P- = /,) n (« u Cnv"«) = Pot'(7/S) (Sev,.S' = /«) [*373-61-62] 

*373-64. I- : /e e iW cycl subm . v e NO ind - I'O . D . 

Nc'(P (P- = 7.) n (« w Cnv"*)} = j; [*373-63-5] 



*374. PRINCIPAL SUBMULTIPLBS. 

Summary of *374. 

In this number we prove for any vector what was proved for /„ in *373, 
namely that, if v is any inductive cardinal not zero, and R is any vector, 
there is just one member of v„ whose vth power is R. This one' we call the 
" principal " i/th submultiple of R. The proof of its existence is as follows. 

Assume i2 is a non-zero vector, and Q is a vih submultiple of R. (Q exists 
provided we assume that k is submultipliable.) Let T be the principal I'th 
submultiple of /k, whose existence has been proved at the end of *373. We 
wish to prove that there is a I'th submultiple of R which is a member of v^. 
By *372-33, Q is a member of v^ if TW^Q. But if QW^T, then T must have 
a last power T" such that QW^T", and for this value of a we shall therefore 
have T'+^TT^Q. (We cannot have T'^^=Q, because if Q were a power 
of T, we should have Q' = Ik, whereas by hypothesis Q" = R.) Now if 

T'+'^WkQ . QWkT", the vector T"] Q must be less than T, i.e. we shall have 

TWk (T' I Q), and therefore T" \ Q will be a member of v^, by *372-33. More- 

over since T'^I^, we have (T" \ Q)" = Q" = iJ by hypothesis. Hence 2" | Q is 
a vth submultiple of R and a member of v^. In virtue of *372-28, it is the 
only i;th submultiple of R which is a member of v^. Thus the existence of 
the principal vth submultiple of any vector is proved, assuming the family 
concerned to be cyclic and submultipliable. 

We prove also in this number that v^ consists of all non-zero vectors 
not greater than the principal I'th submultiple of /«, which is therefore the 
greatest member of Vg ; that is, we have 



*374-21. hiieeFM cycl subm . D . i/^ = ( W^h'iiR) (RevK.R' = L) 



*374-l. I- :.H:eFMcyc[ .R,QeKs.Q' = R-TevK.T-' = lK.D: 

TWkQ.D.Qsvk [*372-33] 

The above hypothesis is not all necessary for the conclusion, but is 
adopted because it gives the construction with which we shall be con- 
cerned. 

31—3 



486 QUANTITY [part VI 

*374 11. I- : Hp *374-l . QW.T.I) .{-^a-) .T'+'W^Q.QW, T' 
Bern. 

I- . *301-504-3 . D I- : Hp . o- 6 NC ind . D . Q =t= ?" (1) 

h.*373 62-5. D h : Hp . D . Pot'Tei/. 

[*261-26] D . E ! min ( W,y(Pot'T n W^'Q) (2) 

h.(l).(2).*372-l.Dh.Prop 

*37412. h : Hp*37411 . T'^+^W.Q . QW.T'^ .P=-T''\Q.D .P ev, 

Dem. 

I- . *371-2316 . D h :. Hp . D : P 6 /eg . T' 6 Kg : 

[#371-25] D:PW,T.::>.P\T'W,T'+^. 
[Hp] D.QW^T'^': 

[Tiansp.Hp] D : TW^P : 

[*372-33] D:P6i;«:.DI-.Prop 

*37413. h : KeFM eye] subm .Reic^.D. (gP) .Pev^.P' = R 
Dem. 

l-.*374-l. DI-:Hp*374-l.rF«Q.D.Q6i;«.Q'' = i? (1) 

l-.*374-12. DI-:Hp*374-12.D.Pei/«.P"' = E (2) 

I- . (1) . (2) . *37411 . D h : Hp*374-1 . D . (gP) .Pev..P' = R (3) 
l-.*373-6. Dh:Hp.D.(ar).T6v..T'' = /« (4) 

I- . (3) . (4) . D h . Prop 

*37414. V-.KeFM cycl subm . ii e « w Cnv"K . D . (gP) .Pev,.P'' = R 
Dem. 

\- . *374-13 . *373-6 .Dh :R^.8€k^.R = S .0 . 

i^T, Q) . T,Q6v..T' = I,.Q^ = S.R = S. 

[*372-27] D . (ar, Q) . r, Q 6 1;. . TTf , e . (Q \Ty =S=R. 

[*37ll6.*372-33] D . (aT, Q) . T, Qei;, . Q | Te ,;, . (Q| T)- = i? (1) 
I- . (1) . *374-13 . *373-6 . D I- . Prop 

*374-2. h : K 6 PJf cycl subm . i2 e « u Cnv"« . D . i/« o P (P" = i?) e 1 

[*374-14.*372-28] 

*374-21. h : « 6 .PW cycl subm . D . j;« = ( Tf ,)j,j'(?i?) (R6v^.R' = /«) 

Dem. 

l-.*374-2. DI-:Hp.D.E!(7i2)(JS;6i;^.i2'' = /^) (1) 

l-.*372-33. D\-:B.^.Rev^.R'' = I^.D.(Wj^'RCv^ (2) 

h . *372-152 . D I- : Hp . 2? 6 1/« . ii" = /« . P e v« . D . iJ- ( TF^)^^ P" . 
[*372-27] D.E(F«)*P (3) 

h . (1) . (2) . (3) . D I- . Prop 



*375. PRINCIPAL RATIOS 

Summary of *375. 

In this number we define a relation (li/v),,, which is contained in 
(fi/v)^ «.*) hut has the advantage of being one-one, and of excluding (p/f\ 
unless /jl/v = p/a. The relation (/m/v), is defined as holding between i2 and S 
when the principal /itth submultiple of R is identical with the principal vth 
submultiple of 8, i.e. we put 

*375-01. (fi/v\ = M {(gr) .Tefji,nv,.R=Ti^.S=T'} Df 
(Here /j,,r\Vic = fiKiifi'^ v, and = v^ if i; ^ /it, by *37215.) 
The properties of (fi/v\ result from *374"2. We find that, except when 

/j, = v = or ^=i; = 0, 

^l/v = ^Iv. = . (fi/v). = i^/v). (*375-27). 
If/i<i;, a'(fj./v)^=KyjCav"K (*37514.1), 

and I>'ip,/v), = (Wj^'(jjL/vyi, (*375-22). 

The principal vth submultiple of S is {1/v)k'S, and its /tth power is 
(fi/v)K'S. Also we have 

il/p\'(l/v).'S = (l/pv).'8 (*375-15), 
Nev^.D.(l/pyNe(pv)^ (*375-16), 
(/tH = (Wl).|(lR (*375-2). 

The propositions 

(j^/v)k I ip/<r)K = (/i/i' X, p/a% 

and {(W^X'-B} I {(PH'^'R} = (/*/" +» p/°-)«'-K 

do not hold without limitation. The former requires either 

/ji'^v .V . cr'^ p, 
or that the converse domain should be limited to 

m*'(a/p\'i., 

i.e. to D'(o-//j),. 

The latter requires either 

p-lv +s pl<T <r 1/1, 
or R€Q.'{/j,/v+gp/a-%. 



Except in the trivial case when /I =0. >' = 0. In this ease, {(tt/i>)f/fi = A but (/ii/i')« = r/c jl« 



488 QUANTITY [part VI 

*375-01. (^L/v), = BS{{-^T).Tefi,nv,.R=Ti-.S=T''} Df 

*375-l. \-:R(^/v\S. = .('s,T).T6fj,^nv,.B = T>-.S=T'' [(*375-01)] 

*37511. \-:KeFMcycl.fi,v6 NC ind - I'O . D . (fi/v\ e 1 ^ 1 
Dem. 

h . *372-28 . D 

\-:Rp.ReK^Cn\"K.T,We/jL,f^v,.R = Ti'= W.D.T^W (1) 
I- ..(1) . *375-l . D h : Hp . iJ (ij./v% S . R (,i,/v% >Sf' . D . /Sf = -S' (2) 

Similarly [■ :Rf .R(fx./v%S . R (fj,/v%S .D .R=R' (3) 

I- . (2) . (3) . D h . Prop 

*37512. \-:KeFMcyc\.'^(ij, = v = 0).D.(jj,/v%C(fi/v)lic, [*370-33] 

*37513. \-.(v/^i% = Gn\'(iJL/v% [*375-l] 

*37514. 1- : /i > V . « e FM cycl subrn . D . T>'{ix./v% = « u Ciiv"k 
[*374-2 . *372-15] 

*375141. h:ya<j;.«e^7lfcyclsubm.D.aV/i')-c = «uCnv"« [*375-13-14] 

*37515. h : K 6 FM cycl subm . /S e k u Cnv"K . p, i; e NC ind - t'O . D . 

(l/p)/(l/^)/^=(l/p^)/S 
Dem. 

h . *375-14 . D I- : Hp . D . E ! {l/p)/(l/v)/S . E ! il/pv),'S (1) 

I- . (1) . *375-l . D I- :. Hp . D : if = {l/py{l/vyS . = . 

{'SiN).Nev,.Mep,.N'' = S.Mi- = N (2) 

I- . (1) . *375-l . D h :. Hp . D : ilf = (l/pv),'S . = . if e (pv). . if p" = S . 

[*372-19] D.M€p,.M''ev,.{M'-y = 8. 

[(2)] D.if=(l//,)/(l/^)/^f (3) 

h . (1) . (3) . D h . Prop 

*375151. \-:KeFMcyc\.N€v,.D.N=(lJvyN' [*375-l] 

*37516. hiKe FM cycl subm . iV e v^ . /o e NC ind - I'O . D . (l//3)/iV e (;oj^X 
i)em. 

I- . *375-15-151 . D 1- : Hp . D . {\lp\'N={llpv\'N'' . 
[*375-l] D . {IjpyN 6 (/3i/)« : D h . Prop 

*375-2. f- :« 6 J^if cycl . /x, v e NC ind - t'O . D . (/t/i/X = (/*/!)« I (1/")-= 
Z)em. 

h . *375-l . D 1- :. Hp . D : J2 {(ju/l), | (l/i/).} »Sf . = . 

(gT) . y e/i. « V. . ii! = T** . >Sf = T" :. D I- . Prop 



SECTION D] principal RATIOS 489 

*375-21. \-:KeFM cycl subm . g ! (fi/v)^ n (p/a% .D.fi/v = pja 

Dem. « 

h . *3751 . D h : Hp .Pi^lv\ Q .P(p/a), Q . D / 

(:3.S,T).86,jL,nv,.Tep,na,.P = Si- = Ti'.Q = 8' = T' (1) 
h . (1) . *374-2 . *375-16 . D I- : Hp (1) . D . ('^R, S,T).Se^L,f^v,. 

[*301-504] D . (aiJ, S,T).Sefi,nv^. 

Tep^ n o-« . Re(jjLa\ a (va-)^ . P = 81"= T" = RT' . Q^S" = T" = R"' . 
[*372-28] 3 . (giJ, ,Sf, T) . S e /.. n ,., . 

Te/j, n a^.Re{iia\ n {va\ . P == 8>' = T' = R)-^ . T = R* '. 
[*301-504] D . (gii) . iJ e (/^o-), n (»//,), . iJ'c = i&" (2) 

h.*372-2.(2).DI-:Hp(l)./iff>i;,o.D./4o- = z;/3 (3) 

Similarly h : Hp (1) . v^ ^ /io- . D . /io- = i;/3 (4) 

h . (3) . (4) . D h : Hp . D . /io- = i;;o : D h . Prop 



*375-22. h : « 6 ^ilf cycl subm . /t < v . D . 'D'ifilv), = ( W,)^'(ji/v%'I^ 
Dem. 

I- . *375-l . D 

\-:.Rp.:):R6'D'(ji/v\.= .{'^8,T).T6fi,nv,.R = T'^.S=T'. 

= .(:^T).Tev,.R = T<^. 
= .{'^S,T).8ev,.S' = I,.8{W.)^T.R = Ti-. 
= .(^8,T).Sev..8' = I,.8'^(W.)^T'^.R=T>^ 
= .(^8).Sev..8'' = I,.8'^iW,)^R. 
= . {{f,/vymW,)^R:.Dh .-Prop 



[*37215.*21-2] 

[*374-21J 

[*372-27] 

[Hp] 

[*375-l-ll] 



*375-221. I- : « e FM cycl subm .fi^v.O. d'ifi/v^ = ( W,)^'(v/fj.yiK 

r*375-22 ?^ . *375-13l 
L /*'" J 

*375-23. h : « e FM cycl subm . /i, v e NC ind . ~ (/i = j; = 0) . D . g ! (/t/i;)« 
[*375-14-141] 

*375-24. h:K6FMcyclsuhm.{fi/v% = (p/(r)^.'D.fi/v = p/(r [*375'21-23] 

The cases when we do not have fi, v, p,eT€ NO ind — t'O require separate 
treatment in obtaining *375"24, but they offer no diflSculty. 



490 QUANTITY [part VI 

*375-25. \-:k6 FM cycl subm . p Prm a . njv = pja . D . (/i/z')^ = {pjo^K 

Dem. 
h . *303-39 . *302-35 . D h : Hp . 3 . (gr) .fi = pT.v = (TT (1) 

I- . *3 7 2- 1 9 . D h : Hp . /I = (OT . i; = ff T . r 6 /i, n i^. . iJ = I'" . <Sf = y - . P = T^ . D . 

P€p,na,.R = Pi'.S=P' (2) 
1- . (1) . (2) . *375-l . D I- : Hp . D . (i^/v\ G (,o/crX (3) 

l-.*375-15.D 
I- : Hp (1) . /i = ^T . j; = o-T . P 6 /)« ft 0-, . ii = P" . *Sf = P" . T= (1/tVP . D . 

TeiJ..nv,.R = T>-.S=T'' (4) 
h . (1) . (4) . *375-l . D h : Hp . D . (/)/<7), G (/./«.), (5) 

I- . (3) . (5) . 3 h . Prop 

*375-26. I- : K 6 FM cycl subm . ~ (/^ = v := 0) . ~ (^ = j; = 0) . /t/i' = ^/t; . D . 

Dem. 

h . *303-39 . *302-34 . D 

1- : Hp . /i, !», f , 7? 6 NO ind . D . (a,o, o-) . (/a, o") Prm (fi, v) . (p, a) Prm (?, i?) . 

[*375-25.*303-211] D . (a|0, o") . (/a/o-), = (ji/v), . (|o/o-)« = C^^). ■ 

[*13-171] :>.{fjL/v\ = (p/a), (1) 

h.*3751.*303-1114-182.D 

h : Hp . ~ (;n, V, I v) 6 NC ind . D . {,i/v\ = A . (p/a), = A (2) 

I- . (1) . (2) . D h . Prop 

*375-27. h :. « ePilf cycl subm . ~(^ = i; = 0) . ~(^ = i; = 0) . D : 

yct/i. = r/'7 ■ = • (H^\ = (?/'?)« [*375-24-26] 

*375-3. h : « e PM cycl subm .fjL,v,p,a-e NC ind - t'O . D . 

i)em. 
h .*375-l . D h : Hp .P(jilv\ Q .Q{p/a%R. D . 

('^S,T).S6fi,nv,.P = S'^.Q^8'.Tep,na,.Q = T''.B'=T' (1) 

h.*375-141-15.D 

h : Hp . >Se/i. ft J/. . P = 6> . Q = <S'' . Te/a. « <7« . Q= T" . P= T". D . 

(ailf) . iW ={\lpyS.P = Mi-^ .Q = M'i'=Ti' .R = T''.MeiiJ,p% . 
[*372-28] D . (gM) . Me (jip), .P^Mi^ .T=M^ .R = T' (2) 

t-.. (2) . *375-l . 3 I- : Hp (2) . /i|0 > i/ff . D . P {hpIv<t\ R (3) 

h.(l).(3). DI-:Hp(l).//,/3>i'o-.D.P(/i|o/j/<7).ii (4) 

Similarly b ■.B.^{1) .va^/jup.O .P(jjiplvcT\R (5) 

I- . (4) . (5) . D h . Prop 



SECTION D] principal RATIOS 491 

*375-31. h :. K 6 FM cycl subm . /i, v, ;o, o- e NC ind - t'O : /i^ ^ z/ . v . o- > /a : D . 

If P {fipjvcr)^ R, we have 

(gif ) . M e (up), n {v<r), .P = Mi-^.R = M"'. 
The result follows by putting Q = M"". 
Without the hypothesis /j.'^v .v .a-^ p, we have 
(/^p/va-V-R = (^/vy{p/<T\'R, 
if -R is sufficiently small to ensure (l/v<r)/Re{vp%, i.e. if 

(<r/pVI.iW,hR, 
^-e-if Rea'(p/a),. 

*37532. \-:k6FM cycl subm . fi/v +» p/o- <, 1/1 . i? e « w Cnv"* . D . 

Mv\'R} I {(/t)/<7)/i?} = {(/./„ +, p/^)/iJ} 

The proof follows immediately from the definitions. 

The same result follows without the hypothesis /i/v +, /j/o- <, 1/1 pro- 
vided R is sufficiently small to ensure 

il/va-yRe(fip + v<r),, 
i-e. Rea\p,jv+,pl<T\. 



CAMBRIDGE : PKINTEE BY JOHN CLAY, M.A. AT THE DNIVEBSITY PRESS.