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Full text of "Contributions to the Founding of the Theory of Transfinite Numbers"

CONTRIBUTIONS TO 

THE FOUNDING OF THE THEORY OF 

TRANSFINITE NUMBERS 




GBOBG CANTOR (1846-1918) 



u nns-u '" ' "- '"- jf <t 

.n.i.l nr ~.^.^->...^^AW^.-*^>'"-**~ 1 **"" 1 ' - P W 



CONTRIBUTIONS TO 

THE FOUNDING OF THE THEORY OF 

TRANSFINITE NUMBERS 



HY 

GEORG CANTOR 



TRANSLATED, AND PROVIDED WITH AN INTRODUCTION 
AND NOTES, UY 

PHILIP E. B. JOURDAIN 

M. A. (CANTAB.) 



'-I 1-OCT1954 




NEW YOKK 

DOVER PUBLICATIONS, INC. 



/ 

IV) 



Unabridged and unaltered reprint 
of the English translation first published 
in 1915 < 



DOVER PUBLICATIONS, INC. 
Printed and bound in the United States of America 



PREFACE 

THIS volume contains a translation of the two very 
important memoirs , of Georg Cantor on transfinite 
numbers which appeared in the Mathematische 
Annalen for 1895 an d 1897* under the title: 
"Beitra^e zur Begrimdurig der transfiniten Mengen- 
lehre." It seems to me that, since these memoirs 
are chiefly occupied with the investigation of the 
various transfinite cardinal and ordinal numbers and 
not with investigations belonging to. what is usually 
described as ".the theory of aggregates" or "the 
theory of sets " (Mengenlehre , thtorie des ensembles), 
the elements, of 1 the sets being real or complex 
numbers wliich are imaged as 'geometrical ' f points " 
in space of one or more dimensions, the title given 
to tnem in this translation is more suitable. 

These memoirs are-the final and logically purified 
statement of many of the most, important results of 
the long series of memoirs begun by Cantor in 1 870. 
It is, I think, necessary, if we are to appreciate the 
full import of Cantor's work on transfinite numbers, 
to have thought through and to bear in mind Cantoris 
earlier researches on the theory of point-aggregates. 
It was in these researches that the need for the 
* Vol. xlvi, 1895, PP- 481-512 ; vol. xlix, 1897, pp. 207-246. 

V 



vi PREFACE 

transfinite numbers first showed itself, and it is only 
by the study of these researches that the majority 
of us can annihilate the feeling of arbitrariness and 
even insecurity about the introduction of these 
numbers. Furthermore, it is also necessary to trace 
backwards, especially through Weierstrass, the 
course of those researches which led to Cantor's 
work. I have, then, prefixed an Introduction tracing 
the growth of parts of the theory of functions during 
the nineteenth century, and dealing, in some detail, 
with the fundamental work of Weierstrass and others, 
and with the work of Cantor from 1870 to 1895. 
Some notes at the end contain a short account of the 
developments of the theory of transfinite numbers 
since 1897. ^ n these notes and in the Introduction 
I have been greatly helped by the information that 
Professor Cantor gave me in the course of a long 
correspondence on the theory of aggregates which 
we carried on many years ago. 

The philosophical revolution brought about by 
Cantor's work was even greater, perhaps, than the 
mathematical one. With few exceptions, mathe- 
maticians joyfully accepted, built upon, scrutinized, 
and perfected the foundations of Cantor's undying 
theory j but very many philosophers combated it. 
This seems to have been because very few under- 
stood it. I hope that this book may help to make 
the subject better known to both philosophers and 
mathematicians. 

The three men whose influence on modern pure 
mathematics and indirectly modern logic and the 



PREFACE vii 

philosophy which abuts on it is most marked are 
Karl Weierstrass, Richard Dedekincl, and Georg 
Cantor. A great part of Dedekind's work has de- 
veloped along a direction parallel to the work of 
Cantor, and it is instructive to compare with Cantor's 
work Dedekind's Stetigkeit und irrational?. Zahlen 
and Was sind und was sollsn die Zahlen ?> of which 
excellent English translations have been issued by 
the publishers of the present book. * 

There is a French translation f of these memoirs of 
Cantor's, but there is no English translation of them. 
For kind permission to make the translation, I 
am indebted to Messrs B, G. Teubner of Leipzig 
and Berlin, the publishers of the Mathematische 
Annalcn. 

PHILIP E. B. JOURDA1N. 

* Essays on the Theory of Numbers (I, Continuity and Irrational 
Numbers ', II, The Naiure and Meaning 1 of Nuwberx\ translated by 
W. W. Benian, Chicago, 1901. I shall refer to this as JSssaps on 
Number, 

t By F. Marotte, Sur les fonttemettts de la throne ties ensembles 
trans/inis, Paris, 1899, 



TABLE OF CONTENTS 

PAGE 

PREFACE v 

TABLE OF CONTENTS ix 

INTRODUCTION i 

CONTRIBUTIONS TO THE FOUNDING OF THE THEORY 
OF TRANSFINITE NUMBERS 

ARTICLE I. (1895) ...... 85 

ARTICLE II. (1897) 137 

NOTES 202 

INDEX 209 



CONTRIBUTIONS TO THE 

FOUNDING OF THE THEORY 

OF TRANSFINITE NUMBERS 

INTRODUCTION 

I 

IF it is safe to trace back to any single man the 
origin of those conceptions with which pure mathe- 
matical analysis has been chiefly occupied during 
the nineteenth century and up to the present time, 
we must, I think, trace it back to Jean Baptiste 
Joseph Fourier (1768-1830). Fourier was first and 
foremost a physicist, and he expressed very defin- 
itely his view that mathematics only justifies itself 
by the help it gives towards the solution of physical 
problems, and yet the light that was thrown on the 
general conception of a function and its ' ' con- 
tinuity," of the "convergence" of infinite series, 
and of an integral, first began to shine as a result 
of Fourier's original and bold treatment of the 
problems of the conduction of heat. This it was 
that gave the impetus to the formation and develop- 
ment of the theories of functions. The broad- 
minded physicist will approve of this refining 



2 INTRODUCTION 

development of the mathematical methods which 
arise from physical conceptions when he reflects 
that mathematics is a wonderfully powerful and 
economically contrived means of dealing logically 
and conveniently with an immense complex of data, 
and that we cannot be sure of the logical soundness 
of our methods and results until we make every- 
thing about them quite , definite. The pure mathe- 
matician knows that pure mathematics has an end 
in itself which is more allied with philosophy. But 
we have not to justify pure mathematics here : we 
have only to point out its origin in physical con- 
ceptions. But . we have also pointed out that 
physics can justify even the most modern develop- 
ments of pure mathematics, 

II 

During the nineteenth century, the two great 
branches of the theory of functions developed and 
gradually separated. The rigorous foundation of 
the results of Fourier on trigonometrical series, 
which was given by Dirichlet, brought forward as 
subjects of investigation the general conception of a 
(one-valued) function of a real variable and the (in 
particular, trigonometrical) development of functions. 
On the other hand, Cauchy was gradually led to 
recognize the importance of what was subsequently 
seen to be the more special conception of function of 
a complex variable ; and, to a great extent independ- 
ently of Cauchy, Weierstrass built up his theory of 
analytic functions of complex variables. 



INTRODUCTION 3 

These tendencies of both Cauchy and Dirichlet 
combined to influence Riemann ; his work on the 
theory of functions of a complex variable carried on 
and greatly developed the work of Cauchy, while 
the intention of his " Habilitationsschrift " of 1854 
was to generalize as far as possible Dirichlet's partial 
solution of the problem of the development of a 
function of a real variable in a trigonometrical 
series. 

Both these sides of Riemann's activity left a deep, 
impression on Hankel. In a memoir of 1870, 
Hankel attempted to exhibit the theory of functions 
of a real variable as leading, of necessity, to the 
restrictions and extensions from which we start in 
Riemann's theory of functions of a complex variable ; 
and yet Hankel's researches entitle him to be called 
the founder of the independent theory of functions 
of a real variable. At about the same time, Heine 
initiated, under the direct influence of Riemann's 
" Habilitationsschrift," a new series of investigations 
on trigonometrical series, 

Finally, soon after this, we find Georg Cantor 
both studying Hankel's memoir and applying to 
theorems on the uniqueness of trigonometrical de- 
velopments those conceptions of his on irrational 
numbers and the ''derivatives" of point- aggregates 
or number-aggregates which developed from the 
rigorous treatment of such fundamental questions 
given by Weierstrass at Berlin in the introduction to 
his lectures on analytic functions. The theory of 
point-aggregates soon became an independent theory 



4 INTRODUCTION 

of great importance, and finally, in 1882, Cantor's 
' ' transfinite numbers " were defined independently 
of the aggregates in connexion with which they first 
appeared in mathematics. 

Ill 

The investigations * of the eighteenth century on 
the problem of vibrating cords led to a controversy 
for the following- reasons. D'Alembert maintained 
that the arbitrary functions in his general integral 
of the partial differential equation to which this 
problem led were restricted to have certain pro- 
perties which assimilate them to the analytically 
representable functions then known, and which would 
prevent their course being completely arbitrary at 
every point. Euler, on the. other hand, argued for 
the admission of certain of these "arbitrary" 
functions into analysis. Then Daniel Bernoulli 
produced a solution in the form of an infinite 
trigonometrical series, and claimed, on certain 
physical grounds, that this solution was as general 
as d'Alembert's. As Euler pointed out, this was so 
only if any arbitrary f function <fr(x) were develop- 
able in a series of the form 

* Cf. the references given in my papers in the Archiv der Mathematik 
und Physik, 3rd series, vol. x, 1906, pp. 255-256, and fst's, vol. i, 
1914, pp. 670-677. Much of this Introduction is taken from ray 
account of " The Development of the Theory of Transfinite Numbers " 
in the above-mentioned Archiv ', 3rd series, vol. x, pp. 254-281 ; 
vol. xiv, 1909, pp. 289-311; vol. xvi, 1910, pp. 21-43; vol. xxii, 
1913, pp. l-2i. 

f The arbitrary functions chiefly considered in this connexion by 
Euler were what he called "discontinuous" functions. This word 
does not mean what we now mean (after Cauchy) by it. Cf. my paper 
in 7j, vol. i, 1914, pp. 661-703. 



INTRODUCTION 



That this was, indeed, the case, even when <^(x) 
is not necessarily developable in a power-series, was 
first shown by Fourier, who was led to study the 
same mathematical problem as the above one by 
his researches, the first of which were communicated 
to the French Academy in 1807, on the conduction 
of heat. To Fourier is due also the determination 
of the coefficients in trigonometric series, 



sin -tf-Mjs sin 2^+ , . ., 
in the form 

f-ir -l-ir 

b v =.~\ 0() cos va.da t a v = l <h(a) sin vada. 

"KJ 1TJ 

-v -w 

This determination was probably independent of 
Euler's prior determination and Lagrange's analog- 
ous determination of the coefficients of a finite 
trigonometrical series. Fourier also gave a geo- 
metrical proof of the convergence of his series, 
which, though not formally exact, contained the 
germ of Dirichlet's proof. 

To Peter Gustav Lejeune-Dirichlet (1805-1859) 
is due the first exact treatment of Fourier's series, * 
He expressed the sum of the first n terms of the 
series by a definite integral, and proved that the 

* "Sur la. convergence ties series trigonomitriquea qui servent A 
reprtfsenter unc fonction arbltrnlr* entre des limites donnien,"/""''^ 1 
ftir Math. t vol. iv, 18291, pp. 157-169; Gtt, Werk9 t voL i, 
pp. ny-rja. 



6 INTRODUCTION 

limit, when n increases indefinitely, of this integral 
is the function which is to be represented by the 
trigonometrical series, provided that the function 
satisfies certain conditions. These conditions were 
somewhat lightened by Lipschitz in 1864. 

Thus, Fourier's work led to the contemplation 
and exact treatment of certain functions which 
were totally different in behaviour from algebraic 
functions. These last functions were, before him, 
tacitly considered to be the type of all functions that 
can occur in analysis. Henceforth it was part of 
the business of analysis to investigate such non- 
algebraoid functions. 

In the first few decades of the nineteenth century 
there grew up a theory of more special functions of 
an imaginary or complex variable. This theory was 
known, in part at least, to Carl Friedrich Gauss 
(1777-1855), but he did not publish his results, and 
so the theory is due to Augustin Louis Cauchy 
(1789-1857).* Cauchy was less far-sighted and 
penetrating than Gauss, the theory developed 
slowly, and only gradually were Cauchy's prejudices 
against * ' imaginaries " overcome. Through the 
years from 1814 to 1846 we can trace, first, the 
strong influence on Cauchy's conceptions of Fourier's 
ideas, then the quickly increasing unsusceptibility to 
the ideas of others, coupled with the extraordinarily 
prolific nature of this narrow-minded genius. Cauchy 
appeared to take pride in the production of memoirs 

* Cf. Jourdain, "The Theory of Functions with Cauchy and Gauss," 
Bibl. Math. (3), vol. vi, 1905, pp. 190-207. 



INTRODUCTION 7 

at each weekly meeting of the French Academy,- and 
it was partly, perhaps, due to this circumstance that 
his works are of very unequal importance. Besides 
that, he did not seem to perceive even approximately 
the immense importance ,of the theory of functions 
of a complex variable which he did so much to 
create. This task remained for Puiseux, Briot and 
Bouquet, and others, and was advanced in the 
most striking manner by Georg Friedrich Bernhard 
Riemann (1826-1866). 

Riemann may have owed to his teacher Dirichlet 
his bent both towards the theory of potential 
which was the chief instrument in his classical 
development (1851) of the theory of functions of a 
complex variable and that of trigonometrical series. 
By a memoir on the representability of a function 
by a trigonometrical series, which was read in 1854 
but only published after his death, he not only laid 
the foundations for all modern investigations into the 
theory of these series, but inspired Hermann Hankel 
(1839-1873) to the method of researches from which 
we can date the theory of functions of a real variable 
as an independent science. The motive of HankePs 
research was provided by reflexion on the founda- 
tions of Riemann's theory of functions of a. complex 
variable. It was HankePs object to show how the 
needs of mathematics compel us to go beyond the 
most general conception of a function, which was 
implicitly formulated by Dirichlet, to introduce the 
complex variable, and finally to reach that con- 
ception from which Rietriann started in his inaugural 



8 INTRODUCTION 

dissertation. For this purpose Hankel began his 
' ' Untersuchungen liber die unendlich oft oscilli- 
renden und unstetigen Functionen ; ein Beitrag zur 
Feststellung des Begriffes der Function Uberhaupt " 
of 1870 by a thorough examination of the various 
possibilities contained in Dirichlet's conception. 

Riemann, in his memoir of 1854, started from 
the general problem of which Dirichlet had only 
solved a particular case : If a function is developable 
in a trigonometrical series, what results about the 
variation of the value of the function (that is to say, 
what is the most general way in which it can become 
discontinuous and have maxima and minima) when 
the argument varies continuously ? The argument 
is a real variable, for Fourier's series, as Fourier -had 
already noticed, may converge for real ;r's alone. 
This question was not completely answered, and, 
perhaps in consequence of this, the work was not 
published in Riemann's lifetime ; but fortunately 
that part of it which concerns us more particularly, 
and which seems to fill, and more than fill, the place 
of Dirichlet's contemplated revision of the principles 
of the infinitesimal calculus, has the finality obtained 
by the giving of the necessary and sufficient condi- 
tions for the integrability of a function f(x), which 
was a necessary preliminary to Riemann's investiga- 
tion. Thus, Riemann was led to give the process 
of integration a far wider meaning than that 
contemplated by Cauchy or even Dirichlet, and 
Riemann constructed an integrable function which 
becomes discontinuous an infinity of times between 



INTRODUCTION 9 

any two limits, as close together as wished, of the 
independent variable, in the following manner : If, 
where x is a real variable, (x) denotes the (positive or 
negative) excess of x over the nearest integer, or 
zero if x is midway between two integers, (x) is 
a one-valued function of x with discontinuities at 
the points x=n-\-$, where n is an integer (positive, 
negative, or zero), and with \ and \ for upper and 
lower limits respectively. Further, (vx), where v is 
an integer, is discontinuous at the points vx=n-\-$ 

or x= -(# + ). Consequently, the series 






where the factor i/j/ 2 is added to ensure convergence 
for all values of x t may be supposed to be discon- 
tinuous for all values of x of the form x=fl/2n, 
where/ is an odd integer, relatively prime to n. It 
was this method that was, in a certain respect, 
generalized by Hankel in 1870. In Riemann's 
example appeared an analytical expression and 
therefore a " function " in Euler's sense which, on 
account of its manifold singularities, allowed of no 
such general properties as Riemann's ' ' functions of 
a complex variable," and Hankel gave a method, 
whose principles were suggested by this example, of 
forming analytical expressions with singularities at 
every rational point. He was thus led to state, with 
some reserve, that every "function" in DIrichlet's 
sense is also a ' ' function " in Euler's sense, 
The greatest influence on Georg Cantor seems, 



io INTRODUCTION 

however, not to have been that exercised by 
Riemann, Hankel, and their successors though 
the work of these men is closely connected with 
some parts of Cantor's work, but by Weierstrass, 
a contemporary of Riemann's, who attacked many 
of the same problems in the theory of analytic 
functions of complex variables by very different and 
more rigorous methods. 

IV 

Karl Weierstrass (1815-1897) has explained, in 
his address delivered on the occasion of his entry 
into the Berlin Academy in 1857, that, from the 
time (the winter of 1839-1840) when, under his 
teacher Gudermann, he made his first acquaintance 
with the theory of elliptic functions, he was power- 
fully attracted by this branch of analysis. " Now, 
Abel, who was accustomed to take the highest 
standpoint in any part of mathematics, established 
a theorem which comprises all those transcendents 
which arise from the integration of algebraic differ- 
entials, and has the same signification for these as 
Euler's integral has for elliptic functions . . . ; and 
Jacobi succeeded in demonstrating the existence of 
periodic functions of many arguments , whose funda- 
mental properties are established in Abel's theorem, 
and by means of which the true meaning and real 
essence of this theorem could be judged. Actually 
to represent, and to investigate the properties of 
these magnitudes of a totally new kind, of which 
analysis has as yet no example, I regarded as one 



INTRODUCTION H 

of the principal problems of mathematics, and, as 
soon as I clearly recognized the meaning and sig- 
nificance of this problem, resolved to devote myself 
to it. Of course it would have been foolish even 
to think of the solution of such a problem without 
having prepared myself by a thorough study of the 
means and by busying myself with less difficult 
problems. " 

With the ends stated here of Weierstrass's work 
we are now concerned only incidentally : it is the 
means the ' ' thorough study " of which he spoke 
which has had a decisive influence on our subject in 
common with the theory of functions. We will, 
then, pass over his early work which was only 
published in 1894 on tne theory of analytic 
functions, his later work on the same subject, and 
his theory of the Abelian functions, and examine 
his immensely important work on the foundations 
of arithmetic, to which he was led by the needs of 
a rigorous theory of analytic functions. 

We have spoken as if the ultimate aim of Weier- 
strass's work was the investigation of Abelian 
functions. But another and ,m,ore philosophical 
view was expressed in his introduction to a course 
of lectures delivered in the summer of 1886 and 
preserved by Gosta.Mittag-Leffler * : "In order to 
penetrate into mathematical science it is indispens- 
able that we should occupy ourselves with individual 



"Sur lea fondements arithm&iques de la thorie des. /enactions 
Weierstrass," Congrts ties MeUhintaliqws h Stockholm^ 
1909, p. 10, 



12 INTRODUCTION 

problems which show us its extent and constitution. 
But the final object which we must always keep in 
sight is the attainment of a sound judgment on the 
foundations of science. " 

In 1859, Weierstrass began his lectures on the 
theory of analytic functions at the University of 
Berlin. The importance of this, from our present 
point of view, lies in the fact that he was naturally 
obliged to pay special attention to the systematic 
treatment of the theory, and consequently, to 
scrutinize its foundations. 

'In the first place, one of the characteristics of 
Weierstrass's theory of functions is the abolition of 
the method of complex integration of Cauchy and 
Gauss which was used by Rieimann ; and, in a 
letter to H. A. Schwarz of October 3, 1875, 
Weierstrass stated his belief that, in a systematic 
foundation, it is better to dispense with integration, 
as follows : 

"... The more I meditate upon the principles 
of the theory of functions, and I do this incessantly, 
the firmer becomes my conviction that this theory 
must be built up on the foundation of algebraic 
truths, and therefore that it is not the right way to 
proceed conversely and make use of the trans- 
cendental (to express myself briefly) for the establish- 
ment of simple and fundamental algebraic theorems ; 
however attractive may be, for example, the con- 
siderations by which Riemann discovered so many 
of the- most important properties of algebraic 
functions, That to the discoverer, quA discoverer, 



INTRODUCTION 13 

every route is permissible, is, of course, self-evident ; 
I am only thinking of the systematic establishment 
of the theory. " 

In the second place, and what is far more im- 
portant than the question of integration, the 
systematic treatment, ab initio, of the theory of 
analytic functions led Weierstrass to profound in- 
vestigations in the principles of arithmetic, and the 
great result of these investigations his theory of 
irrational numbers has a significance for all mathe- 
matics which can hardly be overrated, and our 
present subject may truly be said to be almost 
wholly due to this theory and its development by 
Cantor. 

In the theory of analytic functions we often have 
to use the theorem that, if we are given an infinity 
of points of the complex plane in any bounded 
region of this plane, there is at least one point of 
the domain such that there is a.n infinity of the 
given points in each and every neighbourhood round 
it and including it. Mathematicians used to express 
this by some such rather obscure phrase as : " There 
is a point near which some of the given points are 
infinitely near to one another." If we apply, for the 
proof of this, the method which seems naturally to 
suggest itself, and which consists in successively 
halving the region or one part of the region which 
contains an infinity of points,* we arrive at what is 
required, namely, the conclusion that there is a 
point such that there is another point in any neigh- 
* This method 'was first used by Bernard Bolzaao in 1817. 



14 INTRODUCTION 

bourhood of it, that is to say, that there is a so- 
called "point of condensation," when, and only 
when, we have proved that every infinite ' ' sum " 
such that the sum of any finite number of its terms 
does not exceed some given finite number defines a 
(rational or irrational) number. The geometrical 
analogue of this proposition may possibly be claimed 
to be evident ; but if our ideal in the theory of 
functions which had, even in Weierstrass's time, 
been regarded for long as a justified, and even as a 
partly attained, ideal is to found this theory on the 
conception of number alone,* this proposition leads to 
the considerations out of which a theory of irrational 
numbers such as Weierstrass's is built. The theorem 
on the existence of at least one point of condensa- 
tion was proved by Weierstsass by the method of 
successive subdivisions, and was specially emphasized 
by him. 

Weierstrass, in the introduction to his lectures on 
analytic functions, emphasized that, when we have 
admitted the notion of whole number, arithmetic 
needs no further postulate, but can be built up in a 
purely logical fashion, and also that the notion of a 

* The separation of analysis from geometry, which appeared in the 
work of Lagrange, Gauss, Cauchy, and Bolzano, was a consequence of 
the increasing tendency of mathematicians towards logical exactitude 
in denning their conceptions and in making their deductions, and, con- 
sequently, in discovering the limits of validity of their conceptions and 
methods. However, the true connexion between the founding of 
analysis on a purely arithmetical basis " arithmetization," as it has been 
called and logical rigour, can only be definitely and convincingly 
shown after the comparatively modem thesis is proved that all the con- 
cepts (including that of number) of pure mathematics are wholly logical. 
And this thesis is one of the most important consequences to which the 
theory whose growth we are describing has forced \ts. 



INTRODUCTION iS 

One-to-one correspondence is fundamental in count- 
ing. But it is in his purely arithmetical introduction 
of irrational numbers that his great divergence from 
precedent comes. This appears from a consideration 
of the history of incommensurables. . 

The ancient Greeks discovered the existence of in- 
commensurable geometrical magnitudes, and there- 
fore grew to regard arithmetic and geometry as 
sciences of which the analogy had not a logical 
basis. This view was also probably due, in part at 
least, to an attentive consideration of the famous 
arguments of Zeno. Analytical geometry practi- 
cally identified geometry with arithmetic (or rather 
with arithmetica universalis\ and, before Weier- 
strass, the introduction of irrational "number" 
was, explicitly or implicitly, geometrical. The 
view that number has a geometrical basis was taken 
by Newton and most of his successors. To come 
to the nineteenth century, Cauchy explicitly 
adopted the same view. At the beginning of his 
Cours d 'analyse of 1821, he defined a "limit" as 
follows : ' ' When the successive values attributed 
to a variable approach a fixed value indefinitely 
so as to end by differing from it as little as is 
wished, this fixed value is called the ' limit ' of all 
the others " ; and remarked that ' ' thus an irrational 
number is the limit of the various fractions which 
furnish more and more approximate values of it." 
If we consider as, however, Cauchy does not 
appear to have done, although many others have- 
the latter statement as a definition, so that an 



1 6 INTRODUCTION 

"irrational" number is defined to be the limit of 
certain sums of rational numbers, we presuppose 
that these sums have a limit. In another place 
Cauchy remarked, after defining a series , 1> 
u z) . . . to be convergent it the sum s n =u Q + u 1 -{-u z 
+ . . +*-i, for values of n always increasing, 
approaches indefinitely a certain limit .vthat, "by 
the above principles, in order that the series 
w o u v u t> ' mav k convergent, it is necessary 
and sufficient that increasing values of n make the 
sum s n converge indefinitely towards a fixed limit 
s ; in other words, it is necessary and sufficient that, 
for infinitely great values of n, the sums $, s n +i, 
S M +Z, . . . differ from the limit s, and consequently, 
from one another, by infinitely small quantities." 
Hence it is necessary and sufficient that the different 
sums # + #+! 4- . . . +#+ for different ;#'s, end, 
when n increases, by obtaining numerical values con- 
stantly differing from one another by less than any 
assigned number, 

If we know that the sums s n have a limit s, we 
can at once prove the necessity of this condition ; 
but its sufficiency (that is to say, if, for any assigned 
.positive rational e, an integer n can always be found 
such that 



where r is any integer, then a limit s exists) re- 
quires a previous definition of the system of real 
numbers, of which the supposed limit is to be one. 
For it is evidently a vicious circle to define a real 



INTRODUCTION 17 

number as the limit of a "convergent" series, as 
the above definition of what we mean by a ' ' con- 
vergent" series a series which lias a limit in- 
volves (unless we limit ourselves to rational limits) 
a previous definition of what we mean by a " real 
number." * 

It seems, perhaps, evident to "intuition", that, 
if we lay off lengths S M , j w+ i, . . ., for which the 
above, condition is fulfilled, on a straight line, that 
a (commensurable or incommensurable) "limiting" 
length s exists ; and, on these grounds, we seem to 
be justified in designating Cauchy's theory of real 
number as geometrical. But such a geometrical 
theory is not logically convincing, and Weierstrass 
showed that it is unnecessary, by defining real 
numbers in a manner which did not depend on a 
process of "going to the limit." 

To repeat the point briefly, we have the following 
logical error in- all would-be arithmetical f pre- 
Weierstrassian introductions of irrational numbers : 
we start with the conception of the system of 
rational numbers, we define the ' ' sum " (a limit of 
a sequence of rational numbers) of an infinite series 
of rational numbers, and then raise ourselves to the 
conception of the system of real numbers which are 
got by such means. The error lies in overlooking 
the fact that the ' ' sum " () of the infinite series of 

* On the attempts of Bolzano, Hankel, and Stolz to prove arithmetic- 
ally, without an arithmetical theory of real numbers, the sufficiency of 
the above criterion, see stwa.lt? s Klassiker, No. 153, pp. 42, 95, 107. 

t It must be remembered that Cauchy's theory was not one of these. 
Cauchy did not attempt to define real numbers arithmetically, but 
simply presupposed their existence on geometrical grounds. 



18 - INTRODUCTION 

rational numbers can only be defined when we have 
already defined the real numbers, of which b is one. 
" I believe," said Cantor,* Apropos of Weierstrass's 
theory, ' ' that this logical error, which was first 
avoided by Weierstrass, escaped notice almost 
universally in earlier times, and was not noticed on 
the ground that it is one of the rare cases in which 
actual errors can lead to none of the more important 
mistakes in calculation." 

Thus, we must bear in mind that an arithmetical 
theory of irrationals has to define irrational numbers 
not as "limits" (whose existence is not always 
beyond question) of certain infinite processes, but 
in a manner prior to any possible discussion of the 
question in what cases these processes define limits 
at all. . - 

With Weierstrass, N a number was said to be ' 
"determined" if we know of what elements it is 
composed and how many times each element 
occurs in it. Considering numbers formed with 
the principal unit and an infinity of its aliquot parts, 
Weierstrass called any aggregate whose elements 
and the number (finite) of times each element 
occurs in it j- are known a (determined) ' ' numerical 
quantity " (Zahlengrosse). An aggregate consisting 
of a finite number of elements was regarded as equal 
to the sum of its elements, and two aggregates of a 
finite number of elements were regarded as equal 
when the respective sums of their elements are equal. 

* Math. Ann., vol. xxi, 1883, p. 566. 

t It is not implied that the given elements are finite in number. 



INTRODUCTION 19 

A rational number r was said to be contained in 
a numerical quantity a when we can separate from 
a a partial aggregate equal to r. A numerical 
quantity a was said to be "finite" if we could 
assign a rational number R such that every rational 
number contained in a is smaller than R. Two 
numerical quantities a, b were said to be "equal," 
when every rational number contained in a is con- 
tained in b, and vice versa. When a and b are not 
equal, there is at least one rational number which 
is either contained in a without being contained in 
3, or vice versa : in the first case, a was said to be 
' ' greater than " b \ in the second, a was said to be 
"less than " . 

Weierstrass called the numerical quantity c de- 
fined by (i.e. identical with) the aggregate whose 
elements are those which appear in a or b, each of 
these' elements being taken a number of times equal 
to the number of times in which it occurs in a 
increased by the number of times in which it occurs 
in b, the "sum" of a and b. The "product" of 
a and b was defined to be the numerical quantity 
defined by the aggregate whose elements are ob- 
tained by forming in all possible manners the product 
of each element of a and each element of b. In the 
same way was defined the product of any finite 
number of numerical quantities. 

The ' ' sum " of an infinite number of numerical 
quantities a, b, . , . was then defined to be the 
aggregate (s) whose elements occur in one (at least) 
of a, 6) . . ,., each of these elements e being taken 



20 INTRODUCTION 

a number of times (n) equal to the number of times 
that it occurs in a, increased by the number of times 
that it occurs in 6, and so on. In order that s be 
finite and determined, it is necessary that each of the 
elements which occurs in it occurs a finite number of 
times, and it is necessary and sufficient that we can 
assign a number N such that the sum of any finite 
number of the quantities a, b, . . . is less than N. 

Such is the principal point of Weierstrass's theory 
of real numbers. It should be noticed that, with 
Weierstrass, the new numbers were aggregates of 
the numbers previously defined ; and that this view, 
which appears from time to time in the better text- 
books, has the important advantage which was first 
sufficiently emphasized by Russell. This advantage 
is that the existence of limits can be proved in 
such a theory. That is to say, it can be proved by 
actual construction that there is a number which is 
< the limit of a certain series fulfilling the condition 
of f ' finiteness "or " convergency. " When real 
numbers are introduced either without proper defini- 
tions, or as "creations of our minds," or, what is 
far worse, as "signs,"* this existence cannot be 
proved. 

If we consider an infinite aggregate of real 
numbers, or comparing these numbers for the sake 
of picturesqueness with the points of a straight 
line, an infinite "point-aggregate," we have the 
theorem : There is, in this domain, at least one point 
such that there is an infinity of points of the aggre- 

* Cf. Juurdain, Math, Gazette^ Jan. 1908, vpl. iv, pp. 201-209, 



INTRODUCTION 21 

gate in any, arbitrarily small, neighbourhood of it. 
Weierstrass's proof was, as we have mentioned, 
by the process, named after Bolzano and him, 
of successively halving any one of the intervals 
which contains an infinity of points. This process 
defines a certain numerical magnitude, the "point 
of condensation " (Htiufungsstelle) in question. An 
analogous theorem holds for the two-dimensional 
region of complex numbers. 

Of real numerical magnitudes #, all of which are 
less than some finite number, there is an "upper 
limit," which is defined as : A numerical magnitude 
G which is not surpassed in magnitude by any x and 
is such that either certain x's are equal to G or 
certain x's lie within the arbitrarily small interval 
(G, . . . , G - <S), the end G being excluded. Ana- 
logously for the " lower limit "g. 

It must be noticed that, if we have a finite 
aggregate of ^s, one of these is the upper limit, 
and, if the aggregate is infinite, one of them may 
be the upper limit. In this case it need not also, 
but of course may, be a point of condensation. If 
none of them is the upper limit, this limit (whose 
existence is proved similarly to the existence of a 
point of condensation, but is, in addition, unique^ 
is a point of condensation. Thus, in the above 
explanation of the term "upper limit," we can 
replace the words ' ' either certain x*s "to " being 
excluded "by " certain x*s lie in the arbitrary small 
interval (G, . . ., G <$), the end G being included* 

The theory of the upper and lower limit of a 



22 INTRODUCTION 

(general or " Dirichlet's ") real one-valued function 
of a real variable was also developed and emphasized 
by Weierstrass, and especially the theorem : If G is 
the upper limit of those values of y=f(x)* which 
belong to the values of x lying inside the interval 
from a to b, there is, in this interval, at least one 
point ^=X such that the upper limit of the j/'s 
which belong to the x's in an arbitrarily small 
neighbourhood of X is G ; and analogously for the 
lower limit. 

If the y- value corresponding to x="K is G, the 
upper limit is called the "maximum" of the jp's 
and, if yf(^f) is a continuous function of #, the 
upper limit is a maximum ; in other words, a con- 
tinuous function attains its upper and lower limits. 
That a continuous function also takes at least once 
every value between these limits was proved by 
Bolzano (1817) and Cauchy (1821), but the Weier- 
strassian theory of real numbers first made these 
proofs rigorous, f 

It is of the utmost importance to realize that, 
whereas until Weierstrass's time such subjects as 
the theory of points of condensation of an infinite 
aggregate and the theory of irrational numbers, 
on which the founding of the theory of functions 



* Even if y is finite for every single x of the interval a^x^b, all 
these ys need not be, in absolute amount, less than some finite number 
(for example, f(x) =lfx for je>o, /(o)=o, in the interval o^ *^i), 
but if they are (as in the cose of the sum of a uniformly convergent 
series), these ys have a finite upper and lower limit in the sense defined. 

t- There is another conception (due to Cauchy and P. du Bois- 
Reymnnd) allied to that of upper and lower limit, With every infinite 
aggregate, there are (attained; upper and lower points of condensation, 
which we may call by the Latin name " Z.imiffs," 



INTRODUCTION 23 

depends, were hardly ever investigated, and never 
with such important results, Weierstrass carried 
research into the principles of arithmetic farther 
than it had been carried before. But we must also 
realize that there were questions, such as the nature 
of whole number itself, to which he made no valuable 
contributions. These questions, though logically 
the first in arithmetic, were, of course, historically 
the last to be dealt with. Before this could happen, 
arithmetic had to receive a development, by means 
of Cantor's discovery of transfinite numbers, into a 
theory of cardinal and ordinal numbers, both finite 
and transfinite, and logic had to be sharpened, as 
it was by Dedekind, Frege, Peano and Russell to 
a great extent owing to the needs which this theory 
made evident. 

V 

Georg Ferdinand Ludwig Philipp Cantor was 
born at St Petersburg on 3rd March 1845, and 
lived there until 1856; from 1856 to 1863 he lived 
in South Germany (Wiesbaden, Frankfurt a. M., 
and Darmstadt) ; and, from autumn 1 863 to Easter 
1869, in Berlin. He became Privatdocent at Halle 
a.S. in 1869, extraordinary Professor in 1872, and 
ordinary Professor in 1879.* When a student at 
Berlin, Cantor came under the influence of Weier- 
strass's teaching, and one of his first papers on 

* Those memoirs of Cantor's that will be considered here more 
particularly, and which constitute by far the greater part of his writings, 
are contained in : Joum.fiir Math,, vols. Ixxvii and Ixxxiv, 1874 and 
1878; Math. Ann t > vol, iv, 1871, vol. v, 1873, vol. xv, 1879, vol, xvii, 
1880, vol. xx, 1882, vol, xxi, 1883. 



24 INTRODUCTION 

mathematics was partly occupied with a theory of 
irrational numbers, in which a sequence of numbers 
satisfying Cauchy's condition of convergence was 
used instead of Weierstrass's complex of an infinity 
of elements satisfying a condition which, though 
equivalent to the above condition, is less convenient 
for purposes of calculation. 

This theory was exposed in the course of Cantor's 
researches on trigonometrical series. One of the 
problems of the modern theory of trigonometrical 
series was to establish the uniqueness of a trigono- 
metrical development. Cantor's investigations re- 
lated to the proof of this uniqueness for the most 
general trigonometrical series, that is to say, those 
trigonometrical series whose coefficients are not 
necessarily supposed to have the (Fourier's) integral 
form. 

In a paper of 1870, Cantor proved the theorem 
that, if 

a lt <z a , . . ., , . . . and x , & , ...,,... 

are two infinite series such that the limit of 



a v sn vx+ v cos vx, 

for every value of x which lies in a given interval 
(a<x<b} of the domain of real magnitudes, is zero 
with increasing j/, both a v and b v converge, with 
increasing v t to zero. This theorem leads to a 
criterion for the convergence of a trigonometrical 
series 

J +#! sin x -\-d l cosx-{- . . . +a v sin vx +b v cos wr-f ... 



INTRODUCTION 25 

that Riemann proved under the supposition of the 
integral form for the coefficients. In a paper im- 
mediately following this one, Cantor used this 
theorem to prove that there is only one representation 
viflx} in the form of a trigonometrical series con- 
vergent for every value of x t except, possibly, a 
finite number of x's ; if the sums of two trigono- 
metrical series differ for a finite number of ^s, the 
forms of the series coincide. 

In 1871, Cantor gave a simpler proof of the 
uniqueness of the representation, and extended this 
theorem to : If we have, for every value of #, a 
convergent representation of the value o by a 
trigonometrical series, the coefficients of this re- 
presentation are zero. In the same year, he also 
gave a simpler proof of his first theorem that, if 
lim (a v sin vx-^-b v cos n#) = o for a<x<b> then both 
lim a v and lim b v are zero. 

In November 1871, Cantor further extended his 
theorem by proving that the convergence or equality 
of the sums of trigonometrical series may be re- 
nounced for certain infinite aggregates of X'B in the 
interval . . . 2ir without the theorem ceasing to 
hold. To describe the structure that such an 
aggregate may have in this case, Cantor began 
with ' ' some explanations, or rather some simple 
indications, intended to put in a full light the 
different manners in which numerical magnitudes, 
in number finite or infinite, can behave," in order 
to make the exposition of the theorem in question 
as short as possible. 



26 INTRODUCTION - 

The system A of rational numbers (including o) 
serves as basis for arriving at a more extended 
notion of numerical magnitude. The first general- 
ization with which we meet is when we have an 
infinite sequence 

(1) a ly a z , ...,#,... 

of rational numbers, given by some law, and such 
that, if we take the positive rational number e as 
small as we wish, there is an integer n such that 

(2) I *+*- | < (>i), 

whatever the positive integer m is.* This property 
Cantor expressed by the words, "the series (i) 
has a determined limit b" and remarked particularly 
that these words, at that point, only served to 
enunciate the above property of the series, and, 
just as we connect (i) with a special sign b, we 
must also attach different signs b', b", . . ., to 
different series of the same species. However, 
because of the fact that the "limit" may be 
supposed to be previously defined as : the number 
(if such there be) b such that \b a v \ becomes in- 
finitely small as v increases, it appears better to 
avoid the word and say, with Heine, in his ex- 
position of Cantor's theory, the series (#) is a 
"number-series," or, as Cantor afterwards expressed 
it, (aj) is a " fundamental series. " 

* It may be proved that this condition (2) is necessary and sufficient 
that the sum to infinity of the series corresponding to the sequence (i) 
should be a "finite numerical magnitude" in Weierstrass's sense ; and 
consequently Cantor's theory of irrational numbers has been described 
as a happy modification of Weierstrass'a, 



INTRODUCTION 27 

Let a second series 
(l') a' 1} a' z , . . ., a' v) . . . 

have a determined limit b\ we find that (i) and 
(i') have always one of the three relations, which 
exclude one another: (a) a n a' n becomes infinitely 
small as increases ; (b) from a certain n on, it 
remains always greater than e, where e is positive 
and rational ; (c) from a certain on, it remains 
always less than e. In these cases we say, 
respectively, 

b = b'> b>b'> or b<b'. 

Similarly, we find that (i) has only one of the 
three relations with a rational number a : (a) a n a 
becomes infinitely small as n increases ; (b) from 
a certain n on, it remains always greater than e ; 
(c) from a certain on, it remains less than e. 
We express this by 

6 = a, b>a, or b<a f 

respectively. Then we can prove that b a n becomes 
infinitely small as n increases, which, consequently, 
justifies the name given to b of "limit of the 
series (i)." 

Denoting the totality of the numerical magnitudes 
b by B, we can extend the elementary operations 
with the rational numbers to the systems A and B 
united. Thus the formulae 



28 INTRODUCTION 

express that the relations 

lim (a n a'-a"J = o, lim (a n a' n -a" n ) = o, 
lim (*/'- a",) = o 

hold respectively. We have similar definitions 
when one or two of the numbers belong to A. 

The system A has given rise to B ; by the same 
process B and A united give rise to a third system 
C. Let the series 

(3) #1. # a , ...,,- 

be composed of numbers from A and B (not all 
from A), and such that | b n + m b n \ becomes in- 
finitely small as n increases, whatever m is (this 
condition is determined by the preceding definitions), 
then (3) is said to have "a determined limit c." 
The definitions of equality, inequality, and the 
elementary operations with the members of C, or 
with them and those of B and A, are analogous to 
the above definitions. Now, whilst B and A are such 
that we can equate each a to a b t but not inversely, 
we can equate each b to a c, and inversely. ' ' Although 
thus B and C can, in a certain measure, be regarded 
as identical, it is essential in the theory here ex- 
posed, according to which the numerical magnitude, 
not having in general any objectivity at first,* only 
appears as element of theorems which have a certain 
objectivity (for example, of the theory that the 
numerical magnitude serves as limit for the corre- 
sponding series), to maintain the abstract distinction 

* This is connected with Cantor's formalistic view of real numbers 
(see below). 



INTRODUCTION 29 

between B and C, and also that the equivalence of 
b and b' does not mean their identity, but only 
expresses a determined relation between the series 
to which they refer." 

After considering further systems C, D, . . . , L 
of numerical magnitudes which arise successively, 
as B did from A and C from A and B, Cantor dealt 
with the relations of the numerical magnitudes with 
the metrical geometry of the straight line. If the 
distance from a fixed point O on a straight line has 
a rational ratio with the unit of measure, it is 
expressed by a numerical magnitude of the system 
A ; otherwise, if the point is known by a con- 
struction, we can always imagine a series such as 
(i) and having with the distance in question a 
relation such that the points of the straight line to 
which the distances a lt a 2 , . . ., a v , ... refer 
approach, ad infinitum, as v increases, the point to 
be determined. We express this by saying : The 
distance from the point to be determined to the 
point O is equal to 6, where b is the numerical 
magnitude corresponding to the series (i). We can 
then prove that the conditions of equivalence, 
majority, and minority of known distances agree 
with those of the numerical magnitudes which 
represent these distances. 

It now follows without difficulty that the numerical 
magnitudes of the systems C, D, . . ., are also 
capable of determining the known distances. But, 
to complete the connexion we observe between the 
systems of numerical magnitudes and the geometry 



5/4*7 



30 INTRODUCTION 

of the straight line, an axiom must still be added, 
which runs : To each numerical magnitude belongs 
also, reciprocally, a determined point of the straight 
line whose co-ordinate is equal to this numerical 
magnitude.* This theorem is called an axiom, for 
in its nature it cannot be demonstrated generally. 
It also serves to give to the numerical magnitudes 
a certain objectivity, of which, however, they are 
completely independent. 

We consider, now, the relations which present them- 
selves when we are given a finite or infinite system of 
numerical magnitudes, or "points," as we may call 
them by what precedes, with greater convenience. 

If we are given a system (P) of points in a finite 
interval, arid understand by the word ' ' limit-point " 
(Grenzpunkf) a point of the straight line (not 
necessarily of P) such that in any interval within 
which this point is contained there is an infinity of 
points of P, we can prove Weierstrass's theorem 
that, if P is infinite, it has at least one limit-point. 
Every point of P which is not a limit-point of P 
was called by Cantor an ' ' isolated " point. 

Every point, then, of the straight line either is or 
is not a limit-point of P ; and we have thus defined, 
at the same time as P, the system of its limit-points, 
which may be called the ' ' first derived system " 
(erste Ableitung) P'. If P' is not composed of a 
finite number of points, we can deduce, by the same 



* To each numerical magnitude belongs a determined point, but to 
each point are related as co-ordinates numberless equal numerical 
magnitudes. 



INTRODUCTION 31 

process, a second derived system P" from P ; and, 
by v analogous operations, we arrive at the notion 
of a j/th system PW derived from P. If, for 
example, P is composed of all the points of a line 
whose abscissae are rational and comprised between 
o and i (including these limits or not), P' is com- 
posed of all the points of the interval (o . . . l), 
including these limits ; and P', P", ... do not 
differ from P. If P is composed of the points whose 
abscissae are respectively 

I, 1/2, 1/3, ...,!/.. ., 

P' is composed of the single point o, and derivation 
does not give rise to any other point. It may 
happen and this case alone interests us here 
that, after v operations, PM is composed of a finite 
number of points, and consequently 'derivation does 
not give rise to any other system. In this case 
the primitive P is said to be of the " vth species 
(Art)," and thus P', P", ... are of the (i/-i)th,' 
(v 2)th, . . . species respectively. 

The extended trigonometrical theorem is now : 
If the equation 

cos #4- .- . . 



is satisfied for 1 all values of x except those which 
correspond to the points of a system P of the i/th 
species, where v is an integer as great as is pleased, 
in the interval (o . . . 271-), then 



32 INTRODUCTION 

Further information as to the continuation of 
these researches into derivatives of point-aggregates 
was given in the series of papers which Cantor 
began in 1879 under the title " Ueber unendliche, 
lineare Punktmannichfaltigkeiten. " Although these 
papers were written subsequently to Cantor's dis- 
covery (1873) f tne conceptions of " enumerability " 
(Abzdhlbarkeit) and "power" (Machtigkeit), and 
these conceptions formed the basis of a classification 
of aggregates which, together with the classification 
by properties of the derivatives to be described 
directly, was dealt with in these papers, yet, since, 
by Cantor's own indications,* the -discovery even of 
derivatives of definitely infinite order was made in 
1871, we shall now extract from these papers the 
parts concerning derivatives. 

A point-aggregate ? is said to be of -the "first 
kind" (Gattung) and pth "species" if PC") consists 
of merely a finite aggregate of points ; it is said to 
be of the ' : second kind " if the series 

P', P'", . . . P, . . . 

is infinite. All the points of P", P'", . . . are always 
points of P', while a point of P' is not necessarily a 
point of P. 

* In 1880, Cantor wrote of the " dialectic generation of conceptions, 
which always leads farther and yet remains free from all arbitrariness 1 , 
necessary and logical," of the transfinile series of indices of derivatives. 
"I arrived at this ten years ago [this was written in May 1880]; on 
the occasion of my exposition of the number-conception, I did not 
refer to it." And in a letter to me of 3lst August 1905, Professor 
Cantor wrote: "Was die transfiniten Ordnungszahlen betrifft, 1st es 
mir wahrscheinlidi, dass ich schon 1871 eine vorstellung von ihnen 
gehabt habe. Den Begriff der AbzShlbarkeit bildete ich mir erst 
1873." 



INTRODUCTION 33 

Some or all of the points of a continuous * interval 
(a ... j8), the extreme points being considered as 
belonging to the interval, may be points of P ; if 
none are, P is said to be quite outside (a . . . @)> If 
P is (wholly or in part) contained in (a . . . #), a 
remarkable case may present itself : every interval 
(y ... 8) in it, however small, may contain points 
of P. Then P is said to be " everywhere dense " 
in the interval (a ... /3). For example, (i) the 
point-aggregate whose elements are all the points 
of (a ... /3), (2) that of all the points whose 
abscissae are rational, and (3) that of all the 
points whose abscissae are rational numbers of the 
form (2# + i)/2 w , where m and n are integers, are 
everywhere dense in (a ... )8). It results from this 
that, if a point-aggregate is not everywhere dense 
in (a . . . /3), there must exist an interval (y . . . ) 
comprised in (a . . . /3) and in which there is no 
point of P. Further, if P is everywhere dense 
in (a ... /S), not only is the same true for P', 
but P' consists of all the points of (a ... j8). We 
might take this property of P' as the definition 
of the expression : c ' P is everywhere dense in 
(a ... /3). 

Such a P is necessarily of the second kind, and 
hence a point- aggregate of the first kind is every- 
where dense in no interval. As to the question 
whether inversely every point-aggregate of the 

* At the beginning of the first paper, Cantor stated : " As we shall 
show later, it is on this notion [of derived aggregate] that the simplest 
and completes! explanation respecting the determination of a continuum 
rests " (see below). 



34 INTRODUCTION 

second kind is everywhere dense in, some intervals, 
Cantor postponed it. 

Point-aggregates of the first kind can, as we have 
seen, be completely characterized by the notion of 
derived aggregate, but for those of the second kind 
this notion does not suffice, and it is necessary to 
give it an extension which presents itself as it 
were of its own accord when we go deeper into the 
question. It may here be remarked that Paul du 
Bois-Reymond was led by the study of the general 
theory of functions to a partly similar development 
of the theory of aggregates, and an appreciation of 
its importance in the theory of functions. In 1874, 
he classified functions into divisions, according to, 
the variations of the functions required in the theory 
of series and integrals which serve for the repre- 
sentation of "arbitrary" functions. He then 
considered certain distributions of singularities. 
An infinite aggregate of points which .does not form 
a continuous line may be either such that in any 
line, however small, such points occur (like the points 
corresponding to the rational numbers), or in any 
part, a finite line in which are none of those points 
exists. In the latter case, the points are infinitely 
dense on nearing certain points ; "for if they are 
infinite in number, all their distances cannot 'be 
finite. But also not all their distances in an 
arbitrarily small line can vanish ; for, if so, the 
first case would occur. So their distances can be 
zero only in points, or, speaking more correctly, 
in infinitely small lines." Here we distinguish: 



INTRODUCTION 35 

(i) The points k l condense on nearing a finite 
number of points k z ; (2) the points a condense at 
a finite number of points 8 , . . . Thus, the 
roots of o = sin \\x condense near #=0, those of 
o = sin i /sin \\x near the preceding roots, . . . 
The functions with such singularities fill the space 
between the " common " functions and the functions 
with singularities from point to point. Finally, 
du Bois-Reymond discussed integration over such 
a line. In a note of 1879, he remarked that 
Dirichlet's criterion for the integrability of a function 
is not sufficient, for we can also distribute intervals 
in an everywhere dense fashion (pantachisch) ; that 
is to say, we can so distribute intervals D on the 
interval ( TT . . . + TT) that in any connected 
portion, however small, of ( TT . . . + TT) connected 
intervals D occur. Let, now, 0(#) be o in these 
D's and i in the points of ( TT . . . + IT) not covered 
by D's ; then <j>(x) is not integrable, although any 
interval inside ( w. . . +TT) contains lines in 
which it is continuous (namely, zero). ' ' To this 
distribution of intervals we are led when we seek 
the points of condensation of infinite order whose ex- 
istence 1 announced to Professor Cantor years ago." 
Consider a series of successive intervals on the 
line like those bounded ,by the points i, 1/2, 1/3, 
. . ., i/v, . , . ; in- the interval (i/v . . . i/(i/+i)) 
take a point-aggregate of the first kind and i/th 
species. Now, since each term of the series of 
derivatives of P is contained in the preceding ones, 
and consequently each P<"> arises from the preceding 



36 INTRODUCTION 

p(v-i) by the falling away (at most) of points, that 
is to say, no new points arise, then, if P is of the 
second kind, P' will be composed of two point- 
aggregates, Q and R ; Q consisting of those points 
of P which disappear by sufficient progression in 
the sequence P', P", . . ., PM, . . ., and R of the 
points kept in all the terms of this sequence. In 
the above example, R consists of the single point 
zero. Cantor denoted R by P^ 00 *, and called it 
' ' the derived aggregate of P of order oo (infinity). " 
The first derivative of P (oo) was denoted by pf 08 * 1 ), 
and so on for 

p(oo+2) p(+8) f p(+i>) 

Again, P (ec > may have a derivative of infinite order 
which Cantor denoted by P (2oo) ; and, continuing 
these conceptual constructions, he arrived at de- 
rivatives which a 1 re quite logically denoted by 
p(>+*) j w here m and n are positive integers. But 
he went still farther, formed the aggregate of 
common points of all these derivatives, and got a 
derivative which he denoted p(" a ), and so on without 
end. Thus he got derivatives of indices 

^ , ^-1 , , . co 00 

v <x> +VIOQ + . . . +Vn, . . . oo , . . . oo , . , . 

" Here we see a dialectic generation of concep- 
tions,* which always leads yet farther, and remains 
both free from every arbitrariness and necessary 
and logical in itself." 

* To this passage Cantor added the note : " I was led to this genera- 
tion ten years ago [the note was written in May 1880], but when 
exposing my theory of the number-conception I did not refer to it." 



INTRODUCTION 37 

t 

We see that point-aggregates of the first kind 
are characterized by the property that P (00) has no 
elements, or, in symbols, 



and also the above example shows that a point- 
aggregate of the second kind need not be every- 
where dense in any part of an interval. 

In the first of his papers of 1882, Cantor extended 
the conceptions c c derivative " and ( ' everywhere 
dense" to aggregates 'situated in continua of n 
dimensions, and also gave some reflexions on the 
question as to under what circumstances an (infinite) 
aggregate is well defined. These reflexions, though 
important for the purpose of emphasizing the 
legitimacy of the process used for defining P (oo) , 
P( 2oo >, . . ., are more immediately connected with 
the conception of "power," and will thus be dealt 
with later. The same applies to the proof that it is 
passible to remove an everywhere dense aggregate 
from a continuum of two or more dimensions in 
such a way that any two points can be connected 
by continuous circular arcs consisting of the re- 
maining points, so that a continuous motion may 
be possible in a discontinuous space. To this 
Cantor added a note stating that a purely arith- 
metical theory of magnitudes was now not only 
known to be possible, but also already sketched out 
in its leading features. 

We must now turn our attention to the develop- 
ment of the conceptions of " enumerability " and 



38 INTRODUCTION 

"power," which were gradually seen to have a 
very close connexion with the theory of derivatives 
and the theory, arising from this theory, of the 
transfinite numbers. 

In 1873, Cantor set out from the question 
whether the linear continuum (of real numbers) 
could be put in a one-one correspondence with the 
aggregate o f w hple numbers, and found the rigorous 
proof that this is not the case. This proof, together 
with a proof that the totality of real algebraic 
numbers can be put in such a correspondence, and 
hence that there exist transcendental numbers in 
every interval of the number -continuum, was 
published in 1874. 

A real number (a which is a root of a non-identical 
equation of the form 

(4) a Q < a + a 1 (a-' l + . . . +a n = o,' 

where , a Q) a v . . ., a n are integers, is called a real 
algebraic number ; we may suppose n and a positive, 
#0. a i> , a n to have no common divisor, and (4) 
to be irreducible. The positive whole number 

N = -i+ | * I + 1 i I + + I I 
may be called the ' ' height " of a? ; and to each 
positive integer correspond a finite number of real 
algebraic numbers whose height is that integer. 
Thus we can arrange the totality of real algebraic 
numbers in a simply infinite series 

MI> w a , . . , )., . . . , 
by arranging the numbers corresponding to the 



INTRODUCTION 39 

height N in order of magnitude, and then the 
various heights in their order of magnitude. 

Suppose, now, that the totality of the real 
numbers in the interval (a ... /8), where a < /#, 
could be arranged in the simply infinite series 

(5) #1, *, ...,... 

Let a, ($' be the two first numbers of (5), different 
from one another and from a, j3, and such that 
a'</3'; similarly, let ", /3", where a"</3", be the 
first different numbers in (a' ... /3')> and so on. 
The numbers a', a", . . . are members of (5) whose 
indices increase constantly ; and similarly for the 
numbers /3', j3", ... of decreasing magnitude. 
Each of the intervals (a ... ), (a 7 ... '), 
(a" . . . ft"), . ' . includes all those which follow. 
We can then only conceive two cases : either (a) 
the number of intervals is finite ; let the last be 
(a M . . . jSPy) ; then, since there is in this interval at 
most one number of (5), we can take in it a number 
tj which does not belong to (5) ; or (b) there are 
infinitely many intervals. Then, since o, a, a", , . 
increase constantly without increasing ad infinitum, 
they have a certain limit a (oo >, and similarly ft, /3', 
$", . . . decrease constantly towards a certain 
limit /3< B> >. If a (co) =/5 (oo) (which always happens 
when applying this method to the system ()), we 
easily see that the number q a (oo) cannot be in (5),* 
If, on the contrary, a**^^"^, every number 17 in 



* For if it were, we would have ??-/, p being a determined index ; 
but that is pot possible, for ity is not in (a^ . . . !&)), whilst ij, by 
definition, is. 




40 INTRODUCTION 

the interval (a<> .../*">) or equal to one of its 
ends fulfils the condition of not belonging to (5). 

The property of the totality of real algebraic 
numbers is that the system (co) can be put in a one- 
to-one or (i, i ^correspondence with the system 
(), and hence results a new proof of Liouville's 
theorem that, in every interval of the real numbers, 
there is an infinity of transcendental (non-algebraic) 
numbers. 

This conception of (i, ^-correspondence between 
aggregates was the fundamental idea in a memoir 
of 1877, published in 1878, in which some import- 
ant theorems of this kind of relation between various 
aggregates were given and suggestions made of a 
classification of aggregates on this basis. 

If two well-defined aggregates can be put into 
such a (i, i ^correspondence (that is to say, if, 
element to element, they can be made to correspond 
completely and uniquely), they are said to be 
of the same ' ' power " (Mdcktigkeit *) or to be 
"equivalent" (aequivalenf). When an aggregate 
is finite, the' notion of power corresponds to that of 
number (Angaht)> for two such aggregates have the 
same power when, and only when, the number of 
their elements is the same. 

A part (Bestandteil ; any other aggregate whose 
elements are also elements of the original one) of a 
finite aggregate has always a power less than that 

* The word "power" was borrowed from Steiner, who used it in a 
quite special, but allied, sense, to express that two figures can be put, 
element for element, in protective correspondence, 



INTRODUCTION 41 

of the aggregate itself, but this is not always the 
case with infinite aggregates,* for example, the 
series of positive integers is easily seen to have the 
same power as that part of it consisting of the even 
integers, and hence, from the circumstance that 
an infinite aggregate M is part of N (or is equiva- 
lent to a part of N), we can only conclude that the 
power of M is less than that of N if we know that 
these powers are unequal. 

The series of positive integers has, as is easy to 
show, the smallest infinite power, but the class of 
aggregates with this power is extraordinarily rich 
and extensive, comprising, for example, Dedekind's 
"finite corpora," Cantor's "systems of points of 
the i/th species," all -ple series, and the totality of 
real (and also complex) algebraic numbers. Further, 
we can easily prove that, if M is an aggregate of 
this first infinite power, each infinite part of M has 
the same power as M, and if M', M", ... is a finite 
or simply infinite series of aggregates of the first 
power, the aggregate resulting from the union of 
these aggregates has also the first power. 

By the preceding memoir, continuous aggregates 
have not the first power, but a greater one ; and 
Cantor proceeded to prove that the analogue, with 
continua, of a multiple series a continuum of many 
dimensions has the same power as a continuum of 

* This curious property of infinite aggregates was first noticed by 
Bernard Bolzano, obscurely stated (" , . . two unequal lengths [may 
be said to] contain the same number of points") in a paper of 1864 in 
which Augustus De Morgan argued for a proper infinite, and was used 
as a definition of "infinite" by Dedekind (independently of Bolzano 
and Cantor) in 1887. 



42 INTRODUCTION 

one dimension. Thus it appeared that the assump- 
tion of Riemann, Helmholtz, and others that the 
essential characteristic of an n-p\y extended con- 
tinuous manifold is that its elements depend on n 
real, continuous, independent variables (co-ordin- 
ates), in such a way that to each element of the 
manifold belongs a definite system of values x lt x^ 
. . . , x n , and reciprocally to each admissible system 
*i> x z, - ) x * belongs a certain element of the 
manifold, tacitly supposes that the correspondence 
of the elements and systems of values is a continuous 
one. * If we let this supposition drop, f we can prove 
that there is a (i, ^-correspondence between the 
elements of the linear continuum and those of a 
-ply extended continuum. 

This evidently follows from the proof of the 
theorem : Let x^ x^ . , . , x n be real, independent 
variables, each of which can take any value o^tr^ I ; 
then to this system of n variables can be made to 
correspond a variable *(o<.*^i) so that to each 
determined value of t corresponds one system of 
determined values of x lt x v . . ., x, n and vice versa. 
To prove this, we set out from the known theorem 
that every irrational number e between and I can 
be represented in one manner by an infinite con- 
tinued fraction which may be written : 



* That is to say, an infinitely small variation in position of the element 
implies an infioitely small variation of the variables, and reciprocally. 

T In the French translation only of this memoir of Cantor s is added, 
here : "and this happens very often in the works of these authors 
(Riemann and Helmholtz)." Cantor had revised this translation. 



INTRODUCTION 43 

where the <i's are positive integers. There is thus 
a (i, i ^correspondence between the l s and the 
various series of a's. Consider, now, n variables, 
each of which can take independently all the ir- 
rational values (and each only once) in the interval 
(o. . . i): 

*i = (ai,i, ai, 2, . . ., ai, , ) 

e i (<*2, 1 OB, 2i O2, w )' 



these n irrational numbers uniquely determine a 
(n+ i)th irrational number in (o ... i), 

<*=(&, A, . . . &,, . . .), 

if the relation between a and : 

(6) j8 fr -i>,+ M =a M|F * OA=I, 2, . . ., n; y=l, 2, . . .00) 

is established. Inversely, such a d determines 
uniquely the series of fi's and, by (6), the series of 
the a's, and hence, again of the <r's. We have only 
to show, now, that there can exist a (i, i ^corre- 
spondence between the irrational numbers o < e < i 
and the real (irrational and rational) numbers 
o<.x<.i. For this purpose, we remark that all 
the rational numbers of this interval can be written 
in the form of a simply infinite series 



* If we arrange ihe n aeries of o'a iu a double series with n rowi, 
this meauB ihat we are to enumerate the a's in the order a^ , ( u. Jt 1 1 
. . . M( t , 1( 2 , a.^ 3, , . . , and that the yth term of ['hi* series 
is ftv, 

f This ia done most simply as follows : Let ply be a rational number 
of this interval in its lowest terms, and put/+4'=JN', To each f\q 



44 INTRODUCTION 

Then in (o . . . i) we take any infinite series of 
irrational numbers i^, j/ a> . . ., >?,... (for example, 
ij v s= /s/2/2"), and let h take any of the values of 
o . . . i) except the 0's and jj's, so that 



and we can also write the last formula : 



Now, if we write a c\j b for c ' the aggregate of the a's 
is equivalent to that of the 's," and notice that a Oo, 
a .r\j b and b c\> c imply <z oo c, and that two aggre- 
gates of equivalent aggregates of elements, where the 
elements of each latter aggregate have, two by two, 
no common element, are equivalent, we remark that 

h 00 A, 

and 



A. generalization of the above theorem to the case 
of #!, Xfr . . ., x v , . , . being a simply infinite series 
(and thus that the continuum may be of an infinity 
of dimensions while remaining of the same power 
as the linear continuum) results from the observa- 
tion that, between the double series {a M , }, where 
^ = (a^,i, 0,1,1, . . ., ap |V , . . .) for /u-i, 2, . . .00 

belongs a determined positive integral value of N, and to each such N 
belong a finite number of fractions p(q. Imagine now the numbers plq 
arranged so that those which belong to smaller values of N precede 
those which belong to larger ones, and those for which N has the same 
value are arranged the greater after the smaller. 

* This notation means : the aggregate of the x'a is the union of those 
of the A's, ?;p's, and ^ F 's ; and analogously for thai of the e'a. 



INTRODUCTION 45 

and the simple series {/3J, a (i, ^-correspondence 
can be established * by putting 



and the function on the right has the remarkable 
property of representing all the positive integers, 
and each of them once only, when p. and v inde- 
pendently take all positive integer values. 

"And now that we have proved," concluded 
Cantor, "for a very rich and extensive field of 
manifolds, the property of being capable of corre- 
spondence with the points of a continuous straight 
line or with a part of it (a manifold of points con- 
tained in it), the question arises . . .: Into how 
many and what classes (if we say that manifolds of 
the same or different power are grouped in the same 
or different classes respectively) do linear manifolds 
fall ? By a process of induction, into the further 
description of which we will not enter here, we are 
led to the theorem that the number of classes is two : 
the one containing all manifolds susceptible of being 
brought to the form : functio ipsius v, where v- can 
receive all positive integral values ; and the other 
containing all manifolds reducible to the form functio 
ipsius x, where x can take all the real values in the 
interval (o . . . i).' 1 

In the paper of 1879 already referred to, Cantor 

* Enumerate the double series (a^ } diagonally, that is to say, 
in the order 

"i, n *i, > a, i "i, 8> "a, n "a, n 
The term of this series whose index is (/u. v) is the \th, where 



46 INTRODUCTION 

considered the classification of aggregates* both 
according to the properties of their derivatives and 
according to their powers. After some repetitions, 
a rather simpler proof of the theorem that the con- 
tinuum is not of the first power was given. But, 
though no essentially new results on power were 
published until late in 1882, we must refer to the 
discussion (1882) of what is meant by a "well- 
defined" aggregate. 

The conception of power f which contains, as a 
particular case, the notion of whole number may, 
said Cantor, be considered as an attribute of every 
"well-defined" aggregate, whatever conceivable 
nature its elements may. have. "An aggregate of 
elements belonging to any sphere of thought is said 
to be ' well defined ' when, in consequence of its 
definition and of the logical principle of the excluded 
middle, it must be considered as intrinsically deter- 
mined whether any object belonging to this sphere 
belongs to the aggregate or not, and, secondly, 
whether two objects belonging to the aggregate 
are equal or not, in spite of formal differences 
in the manner in which they are given. In fact, 
we cannot, in general, effect in a sure and precise 
manner these determinations with the means at our 
disposal ; but here it is only a question of intrinsic 
determination, from which an actual or extrinsic 

* Linear aggregates alone were Considered, since all the powers of 
the continua of various dimensions are to be found in then). 

t "That foundation of the theory of magnitudes which we may 
consider to be the most general genuine moment in the case of 
manifolds." 



INTRODUCTION 47 

determination is to be developed by perfecting the 
auxiliary means." Thus, we can, without any 
doubt, conceive it to be intrinsically determined 
whether a number chosen at will is algebraic or 
not ; and yet it was only proved in 1874 that e is 
transcendental, and the problem with regard to v 
was unsolved when Cantor wrote in 1882.* 

In this paper was first used the word "enumer- 
able " to describe an aggregate which could be put 
in a (i, ^-correspondence with the aggregate of 
the positive integers and is consequently of the first 
(infinite) power ; and here also was the important 
theorem : In a ^-dimensional space (A) are defined 
an infinity of (arbitrarily small) continua of n 
dimensions t (a) separated from one another and 
most meeting at their boundaries ; the aggregate of 
the a's is enumerable. 

For refer A by means of reciprocal radii vectores 
to an -ply extended figure B within a (+i)- 
dimensional infinite space A', and let the points of 
B have the constant distance i from a fixed point 
of A'. To every a corresponds a -dimensional 
part b of B with a definite content, and the 3's are 
enumerable, for the number of d's greater in con- 
tent than an arbitrarily small number y is finite, for 
their sum is less than 2V J (the content of B). 

* Lindemann afterwards proved that -K is transcendental. In this 
passage, Cantor seemed to agree. with Dedekind. 

t With every a the points of its boundary are considered as belong- 
ing to it. 

t In the French .translation (1883) of Cantor's memoir, this number 
was corrected to an^n + l^tlT^n + i)^. 

When =i, the theorem is that every aggregate of intervals on a 



48 INTRODUCTION 

Finally, Cantor made the interesting remark that, 
if we remove from an ^-dimensional continuum any 
enumerable and everywhere*dense aggregate, the 
remainder (91), if ^2, does not cease to be con- 
tinuously connected, in the sense that any two 
points N, N' of 91 can be connected by a continuous 
line composed of circular arcs all of whose points 
belong to 1. 

VI 

An application of Cantor's conception of enumera- 
bility was given by a simpler method of condensation 
of singularities, the construction of functions having 
a given singularity, such as a discontinuity, at an 
enumerable and everywhere-dense aggregate in a 
given real interval. This was suggested by Weier- 
strass, and published by Cantor, with Weierstrass's 
examples, in 1882.* The method may be thus 
indicated : Let $(x) be a given function with the 
single singularity x=o t and (CD,,) any enumerable 
aggregate ; put 



where tne c v 's are so chosen that the series and 
those derived from it in the particular cases treated 
converge unconditionally and uniformly, Then 

(finite or infinite) straight line which at most meet at theft ends is 
enumerable. The end-points are consequently enumerable, but not 
always the derivative of this aggregate of end-points. 

* In a letter to me of 29th March 1905, Professor Cantor said : " At the 
conception of enuraerabilily, of which he [ Weierstrass] heard from me at 
Berlin in the Christmas holidays of 1873, he was at first quite amazed, 
but one or two days passed over, [and] it became his own and helped him 
to an unexpected development of his wonderful theory of functions." 



INTRODUCTION 49 

f(x) has at all points x w^ the same kind of singu- 
larity as 0(ar) at jr=o, and at other points behaves, 
in general, regularly. The singularity at x=& tL is 
due exclusively to the one term of the series in 
which v fj. ; the aggregate (&>) may be any enumer- 
able aggregate and not only, as in Hankel's method, 
the aggregate of the rational numbers, and the 
superfluous and complicating oscillations produced 
by the occurrence of the sine in Hankel's functions 
is avoided. 

The fourth (1882) of Cantor's papers " CJeber 
unendliche, lineare Punktmannichfaltigkeiten " con- 
tained six theorems on enumerable point-aggregates. 
If an aggregate Q (in a continuum of n dimensions) 
is such that none of its points is a limit-point,* it is 
said to be " isolated. " Then, round every point of 
Q a sphere can be drawn which contains no other 
point of Q, and hence, by. the above theorem on 
the enumerability of the aggregate of these spheres, 
is enumerable. 

Secondly, if P' is enumerable, P is. For let 

3>(P, P>R, P-RsQjt 

then Q is isolated and therefore enumerable, and R 
is also enumerable, since R is contained in P'; so 
P is enumerable. 
The next three theorems state that, if P ( ">, or 

* Cantor expressed this )(Q, Q')so. Cf, Dedekind's Essays on 

Nuwber> p. 48. 

f If an aggregate B is contained in A, and E is the aggregate left 
when B is taken from A, we write 

EsA-B. 



50 INTRODUCTION 

P (tt >, where a is any one of the ' ' definitely defined 
symbols of infinity (bestimmt definirte Unendlich- 
keitssymbole)" is enumerable, then P is. 

If the aggregates P I} P 2J . . . have, two by two, 
no common point, for the aggregate P formed by 
the union of these (the " Vereinigungsmenge") Cantor 
now used the notation 



Now, we have the following identity 



and thus, since 

P' P" p"_p'" p^-D pM 

are all isolated and therefore enumerable, if P (l/) is 
enumerable, then P' is also. 

Now, suppose that P (eo) exists ; then, if any par- 
ticular point of P' does not belong to P^ 00 ), there is a 
first one among the derivatives of finite order, P ( ">, 
to which it does not belong, and consequently P*"- 1 ) 
contains it as an isolated point. Thus we can write 



and consequently, since an enumerable aggregate of 
enumerable aggregates is an enumerable aggregate 
of the elements of the latter, and P (a ' ) is enumerable, 
then P' is also. This can evidently be extended to 
P (a) , if it exists, provided that the aggregate of all 
the derivatives from P' to P^ is enumerable. 

The considerations which arise from the last 



INTRODUCTION 51 

observation appear to me to have constituted the 
final reason for considering these definitely infinite 
indices independently* on account of their con" 
nexion with the conception of "power," which 
Cantor had always regarded as the most funda- 
mental one in the whole theory of aggregates. 
The series of the indices found, namely, is such 
that, up to any point (infinity or beyond), the 
aggregate of them is always enumerable, and yet 
a process exactly analogous to that used in the 
proof that the continuum is not enumerable leads 
to the result that the aggregate of all the indices 
such that, if a is any index, the aggregate of.all the 
indices preceding a is enumerable, is not enumer- 
able, but is, just as the power of the series of 
positive integers is the next higher one to all finite 
ones, the next greater infinite power to the first. 
And we can again imagine a new index which is the 
first after all those defined, just as after all the finite 
ones. We shall see these thoughts published by 
Cantor at the end of 1882. 

It remains to mention the sixth theorem, in 
which Cantor proved that, if F is enumerable, P 
has the property, which is essential in the theory 
of integration, of being "discrete," as Harnack 
called it, " integrable, " as P. du Bois-Reymond did, 
"unextended," or, as it is. now generally called, 
"content-less." 

* When considered independently of F, these indices form a series 
beginning with the finite numbers, but extending beyond them ; so 
that it suggests itself that those other indices be considered a infinite 
(or transfinitej numbers. 



52 INTRODUCTION 

VII 

We have thus seen the importance of Cantor's 
"definitely defined symbols of infinity" in the 
theorem that if P (a > vanishes, P', and therefore P, is 
enumerable. This theorem may, as we can easily 
see by what precedes, be inverted as follows : If 
P' is enumerable, there is an index a such that P (a) 
vanishes. By defining these indices in an inde- 
pendent manner as real, arid in general transfinite, 
integers, Cantor was enabled to form a conception 
of the enumeral * (Anzahl) of certain infinite series, 
and such series gave a means of defining a series of 
ascending infinite "powers." The conceptions of 
"enumeral" and "power" coincided in the case of 
finite aggregates, but diverged in the case of infinite 
aggregates ; but this extension of the conception of 
enumeral served, in the way just mentioned, to 
develop and make precise the conception of power 
used often already. 

Thus, from the new point of view gained, we get 
new insight into the theory of finite number ; as 
Cantor put it : "The conception of number which, 
in finito, has only the background of enumeral, 
splits, in a manner of speaking, when we raise our- 
selves to the infinite, into the two conceptions of 
power . . . and enumeral . . . ; and, when I again 
descend to the finite, I see just as clearly and 
beautifully how these two conceptions again unite 
to form that of the finite integer." 

* I have invented this word to translate " Anaahl" to avoid confusion 
with the word " number " (Zakl). 



INTRODUCTION 53 

The significance of this distinction for the theory 
of all (finite and infinite) arithmetic appears in 
Cantor's own work* and, above all, in the later, 
work of Russell. 

Without this extension of the conception of 
number to the definitely infinite numbers, said 
Cantor, "it would hardly be possible for me to 
make without constraint the least step forwards in, 
the theory of aggregates," and, although "I was 
led to them [these numbers] many years ago, 
without arriving at a clear consciousness that 1 
possessed in them concrete numbers of real signi- 
ficance," yet " I was logically forced, almost against 
my will, because in opposition to traditions which 
had become valued by me in the course of scientific 
researches extending over many years, to the 
thought of considering the infinitely great, not 
merely in the form of the unlimitedly increasing, 
and in the form, closely connected with this, of 
'convergent infinite series, but also to fix it mathe- 
matically by numbers in the definite form of a 

completed infinite.' I do not believe, then, that 
any reasons can be urged against it which I am 
unable to combat." 

The indices of the series of the derivatives can 
be conceived as the series of finite numbers 

i^ 2 , , followed by a series of transfinite 

numbers of which the first had been denoted by the 
symbol "oo." Thus, although there is no greatest 

* Cf.^ for example, pp. 113, 158-159 of the translations of Cantor's 
memoirs of 1895 and W given below. 



54 INTRODUCTION 

finite number, or, in other words, the supposition 
that there is a greatest finite number leads to con- 
tradiction, there is no contradiction involved in 
postulating a new, non-finite, number which is to be 
the first after all the finite numbers. This is the 
method adopted by Cantor * to define his numbers 
independently of the theory of derivatives ; we shall 
see how Cantor met any possible objections to this 
system of postulation. 

Let us now briefly consider again the meaning of 
the word " Mannichfaltigkeitslehre," f which is 
usually translated as " theory of aggregates." In a 
note to the Gtundlagen, Cantor remarked that he 
meant by this word ' ' a doctrine embracing' very 
much, which hitherto 1 have attempted to develop 
only in the special form of an arithmetical or 
geometrical theory of aggregates (Mengenlehre). 
By a manifold or aggregate I understand generally 
any multiplicity which can be thought of as one 
(jedes Viele, welcJus sich als Bines denken lasst), that 
is to say, any totality of definite elements which 
can be bound up into a whole by means of a law. " 

* " Ueber unendliche, lineare Punktmannichfaltigkeiten. V." 
[December 1882], Math. Ann.,vo\. xxi, 1883; pp. 545-591 ; reprinted, 
with an added preface, with the title : Grwidlagen einer allgtmrinen 
Mannichfaltigkeitslekre. Sin mctfhematisch-philosopkischtr Versuch in 
derLehre des Unendtich.cn, Leipzig, 1883 (page n of the Grwidlagen is. 
page + 544 of the article in the Math. Ann. ). This separate publica- 
tion, with a title corresponding more nearly to its 'contents, was made 
" since it carries the subject in many respects much farther and thus is, 
for the most part, independent of the earlier essays" (Preface). In 
Ada Math., ri, pp. 381-408, part of the Grundlagen was translated 
into French. 

t Or " Mtinnigfaltigkeitslehre, " or, more usually, ' ' Mengenlehre " ; in 
French, "thiorit des ensembles" The English " theory of manifolds " 
has not come into general usage. 



INTRODUCTION 55 

This character of unity was repeatedly emphasized 
by Cantor, as we shall see later. 

The above quotations about the slow and sure 
way in which the transfinite numbers forced them- 
selves on the mind of Cantor and about Cantor's 
philosophical and mathematical traditions are taken 
from the Grundlagen. Both here and in Cantor's 
later works we constantly come across discussions 
of opinions on infinity held by mathematicians and 
philosophers of all times, and besides such names as 
Aristotle, Descartes, Spinoza, Hobbes, Berkeley, 
Locke, Leibniz, Bolzano, and many others, we find 
evidence of deep erudition and painstaking search 
after new views on infinity to analyze. Cantor has 
devoted many pages to the Schoolmen and the 
Fathers of the Church. 

The Grundlagen begins by drawing a distinction 
between two meanings which the word "infinity" 
may have in mathematics. The mathematical 
infinite, says Cantor, appears in two forms ! Firstly, 
as an improper infinite (Uneigentlich-Unendliches), 
a magnitude which either increases above , all limits 
or decreases to an arbitrary smallness, but always 
remains finite ; so that it may be called a variable 
finite. Secondly, as a definite, a proper infinite 
(Eigentlich~Unendliches\ represented by certain 
conceptions in geometry, and, in the theory of 
functions, by the point infinity of the complex plane. 
In the last case we have a single, definite point, 
and the behaviour of (analytic) functions about this 
.point is examined in exactly the same way as it is 



56 INTRODUCTION 

about any other point.* Cantor's infinite real 
integers are also properly infinite, and, to emphasize 
this, the old symbol " oo ," which was and is used also 
for the improper infinite, was here replaced by "o>." 
To define his new numbers, Cantor employed the 
following considerations. The series of the real 
positive integers, 

(I) i, 2, 3, . . ., V) . . ., 

arises from the repeated positing and uniting of 
units which are presupposed and regarded as equal ; 
the number v is the expression both for a definite 
finite enumeral of such successive positings and for 
the uniting of the posited units into a whole. Thus 
the formation of the finite real integers rests on the 
principle of the addition of a unit to a number 
which has already been formed ; Cantor called this 
moment the first principle of generation (Ereeugungs- 
princip). The enumeral of the number of the class 
(I) so formed is infinite, and there is no greatest 
among them. Thus, although it would be contra- 
dictory to speak of a greatest number of the class (I), 
there is, on the other hand, nothing objectionable 
in imagining a new number, , which is to express 
that the whole collection (I) is given by its law in 
its natural order of succession (in the same way as 
v is the expression that a certain finite enumeral of 
units is united to a whole), f By allowing further 

* "The behaviour of the function in the neighbourhood of the 
infinitely distant point shows exactly the same occurrences as in that 
of any other point lying infinite, so that hence it is completely justified 
to think of the infinite, in this case, as situated in a point" 

t " It is even permissible to think of the newly and created number 



INTRODUCTION 57 

positings of unity to follow the positing of the 
number , we obtain with the help of the first 
principle of generation the further numbers : 



Sfrnce again here we come to no greatest number, we 
imagine a new one, which we may call 2to, and which 
is to be the first which follows all the numbers v and 
w + v hitherto formed. Applying the first principle re- 
peatedly to the number 20, we come to the numbers : 

2co+I, 2ftj+2, . . ., 2ft> + y, . . . 

The logical function which has given us the 
numbers to and 2o> is obviously different from the 
first principle ; Cantor called it the second principle 
of generation of real integers, and defined it more 
closely as follows : If there is defined any definite 
succession of real integers, of which there is no 
greatest, on the basis of this second principle a new 
numberis created, which is defined as the next greater 
number to them all. 

By the combined application of both principles 
we get, successively, the numbers : 

3o>, 30+ I, . . . 



w as the limit to which the numbers v strive, if by that nothing else is 
understood than that u is to be the first integer which follows all the 
numbers y, that is to say, is to be called greater than every p." Cf. 
the next section. 

If we do not know the reasons in the theory of derivatives which 
prompted the introduction of u, but only the grounds stated in the text 
for this introduction, it naturally seems rather arbitrary (not apparently, 
useful) to create u because of the mere fact that it can apparently be 
defined in a manner free from contradiction. Thus, Cantor discussed 
(see below) such introductions or creations, found in them the dis- 
tinguishing mark of pure mathematics, and justified them on historical 
grounds (on logical grounds they perhaps seem to need no justification). 



58 INTRODUCTION 

and, since no number fjna+v is greatest, we create 
a new next number to all these, which may be 
denoted by cu a . To this follow, in succession, 
numbers : 



and further, we come to numbers of the form 



and the second principle then requires a new number, 
which may conveniently be denoted by 

0)". 

And so on indefinitely. 

Now, it is seen without difficulty that the 
aggregate of all the numbers preceding any of the 
infinite numbers and hitherto defined is of the 
power of the first number-class (I). Thus, all the 
numbers preceding w" are contained in the formula : 



where /*, v 0) v lt . . ., ^ have to take all finite, 
positive, integral values including zero and exclud- 
ing the combination v^v^^-... =^ = 0. As is 
well known, this aggregate can be brought into the 
form of a simply infinite series, and has, therefore, 
the power of (I). Since, further, every sequence 
(itself of the first power) of aggregates, each of 
which has the first power, gives an aggregate of the 
first power, it is clear that we obtain, by the con- 
tinuation of our sequence in the above way, only 
such numbers with which this condition is fulfilled. 



INTRODUCTION 59 

Cantor defined the totality of all the numbers a 
formed by the help of the two principles 

. . ., w", . . ., a, . . ., 

such that all the numbers, from I on, preceding a 
form an aggregate of the power of the first number- 
class (I), as the "second number-class (II)." The 
power of (II) is different from that of (I), and is, 
indeed, the next higher power, so that no other 
power lies between them. Accordingly, the second 
principle demands the creation of a new number (fl) 
which follows all the numbers of (II) and is the 
first of the third number-class (III), and so on.* 

Thus, in spite of first appearances, a certain 
completion can be given to the successive formation 
of the numbers of (II) which is similar to that 
limitation present with (I). There we only used 
the first principle, and so it was impossible to 
emerge from the series (I) ; but the second principle 
must lead not only over (II), but show itself indeed 
as a means, which, in combination with the first 
principle, gives the capacity to break through every 
limit in the formation of real integers. The above- 
mentioned requirement, 'that all the numbers to 
be next formed should be such that the aggregate 

* It is particularly to be noticed that the second principle will lake 
us beyond any class, and is not merely adequate to form numbers which 
are the limit-numbers of some enumerable series (so that a "third 
principle" is required to form 1). The 'first and second principles 
together form all the numbers considered, while the "principle of 
limitation" enables us to define the various number-classes, of un- 
brokenly ascending powers in the series of these numbers. 



60 INTRODUCTION 

of numbers preceding each one should be of a certain 
power, was called by Cantor the third or limitation*- 
principle (Hemwungs- oder Beschrdnkungsprincip\* 
and which acts in such a manner that the class (II) 
defined with its aid can be shown to have a higher 
power than (I) and indeed the next higher power to 
it. In fact, the two first principles together define 
an absolutely infinite sequence of integers, while the 
third principle lays successively certain limits on 
this process, so that we obtain natural segments 
(Akschnitte), called number-classes, in this sequence. 

Cantor's older (1873, 1878) conception of the 
"power" of an aggregate was, by this, developed 
and given precision. With finite aggregates the 
power coincides with the enumeral of the elements, 
for such aggregates have the same enumeral of 
elements in every order. With infinite aggregates, 
on the other hand, the transfinite numbers afford a 
means of defining the enumeral of an aggregate, if 
it be "well ordered," and the enumeral of such an 
aggregate of given power varies, in general, with 
the order given to the elements. The smallest 
infinite power is evidently that of (I), and, now for 
the first time, the successive higher powers also 
receive natural and simple definitions ; in fact, the 
power of the yth number class is the yth. 

By a "well-ordered" aggregate, f Cantor under- 

* "This principle (or requirement, or condition) circumscribes 
(limits} each number-class.*' 

f The origin of this conception can easily be seen to be the defining 
of such aggregates as can be "enumerated" (using the word in the 
wider sense of Cantor, given below) by the transfinite numbers. In 
fact, the above definition of a well-ordered aggregate simply indicates 



INTRODUCTION 61 

stood any well-defined aggregate whose elements 
have a given definite succession such that there is 
a. first element, a definite element follows every one 
(if it is not the last), and to any finite or infinite 
aggregate a definite element belongs which is the 
next following element in the succession to them 
all (unless there are no following elements in the 
succession). Two well-ordered aggregates are, now, 
of the same enumeral (with reference to the orders 
of succession of their elements previously given for 
them) if a one-to-one correspondence is possible 
between them such that, if E and F are any two 
different elements of the one, and E' and F' the 
corresponding elements (consequently different) of 
the other, if E precedes or follows F, then E' 
respectively precedes or follows F'. This ordinal 
correspondence is evidently quite determinate, if it 
is possible at all, and since there is, in the extended 
number-series, one and only one number a such that 
its preceding numbers (from i on) in the natural 
succession have the same enumeral, we must put a 
for the enumeral of both well-ordered aggregates, if 
a is infinite, or a I if a is finite. 

The essential difference between finite and infinite 
aggregates is, now, seen to be that a finite aggregate 
has the same enumeral whatever the succession of 

the construction of any aggregate of the class required when the first 
two principles are used, but to generate elements, not numbers. 

An important property of a well-ordered aggregate, indeed, a 
characteristic property, is that any series of terms in it, a^. eh, . . ., 
tf^, . . ., where a v +i precedes av , must be finite. Even if the well- 
ordered aggregate in question is infinite, such a series as that described 
can never be infinite. 



62 INTRODUCTION 

the elements may be, but an infinite aggregate has, 
in general, different enumerate under these circum- 
stances. However, there is a certain connexion 
between enumeral and power an attribute of the 
aggregate which is independent of the order of the 
elements. Thus, the enumeral of any well-ordered 
aggregate of the first power is a definite number of 
the second class, and every aggregate of the first 
power can always be put in such an order that its 
enumeral is any prescribed number of the second 
class. Cantor expressed this by extending the 
meaning of the word ( ' enumerable " and saying : 
Every aggregate of the power of the first class is 
enumerable by numbers of the second class and only 
by these, and the aggregate can always be so 
ordered that it is enumerated by any prescribed 
number of the second class ; and analogously for 
the higher classes. 

From his above remarks on the "absolute"* 

* Cantor said "that, in the successive formation of number-classes, 
we can always go farther, and never reach a limit that cannot be sur- 
passed, so that we never reach an even approximate comprehension 
(Erfassen) of the Absolute, I cannot doubt. The Absolute can 
only be recognized (arurkannt} r but never apprehended (crkannt), 
even approximately. For just as inside the first number-class, at any 
finite number, however great, we always have the same 'power' of 
greater finite numbers before us, there follows any transfinite number 
of any one of the higher number-classes an aggregate of numbers and 
classes which has not in the least lost in ' power ' in comparison with the 
whole absolutely infinite aggregate of numbers, from i on. The state 
of things is like that described by Albrecht von Holler : ' igh zieh' 
sie ab [die ungeheure Zahl] und Du [die Ewigkeit] liegst ganz vor mir.' 
The absolutely infinite sequence of numbers thus seems to me to be, in 
a certain sense, a suitable symbol of the Absolute ; whereas the infinity 
of (I), which has hitherto served for that purpose, appears to me, just 
because I hold it to be an idea (not presentation) that can be appre- 
hended as a vanishing nothing in comparison with the former. It also 
seemi to me remarkable that every number-class and therefore every 



INTRODUCTION 63 

infinity of the series of ordinal numbers and that of 
powers, it was to be expected that Cantor would 
derive the idea that any aggregate could be arranged 
in a well-ordered series, and this he stated with a 
promise to return to the subject later.* 

The addition and multiplication of the transfinite 
(including the finite) numbers was thus defined by 
Cantor. Let M and M t be well-ordered aggregates 
of enumerals a and ft, the aggregate which arises 
when first M is posited and then M^ following it, 
and the two are united is denoted M + M x and its 
enurneral is defined to be a + /S. Evidently, if a 
and ft are not both finite, a-\-($ is, in general, 
different from {3-\-a. it is easy to extend the con- 
cept of sum to a finite or transfinite aggregate of 
summands in a definite order, an4 the associative 
law remains valid. Thus, in particular, 



If we take a succession (of enurneral ft) of equal 
and similarly ordered aggregates, of which each is 
of enurneral a, we get a new well-ordered aggregate, 
whose enurneral is defined to be the product ^a, 

power corresponds to a definite number of the absolutely infinite 
totality of numbers, and indeed reciprocally, so that corresponding to 
any transfinite number 7 there is a (7th) power ; so that the various _ 
powers also form an absolutely infinite sequence. This is so much the 
more remarkable as the number 7 which gives the rank of a power 
(provided that 7 ha& an immediate predecessor) stands, to the numbers 
of that number-class which has this power, in a magnitude-relation 
whose smallness mocks all description, and this the more 7 is taken to 
be greater." 

*^ With this is connected the promise to prove later that the power of 
the continuum is that of (II), as stated, of course in other words, in 1878. 
See the Notes at the end of this book. 



64 INTRODUCTION 

where ft is the multiplier and a the multiplicand. 
Here also fta is, in general, different from aft ; but 
we have, in general, 



Cantor also promised an investigation of the 
' c prime number-property " of some of the transfinite 
numbers * a proof of the non-existence of infinitely 
small numbers, f and a proof that his previous 
theorem on a point-aggregate P in an ^-dimensional 
domain that, if the derivate P (a) , where a i$ any 
integer of (I) or (II), vanishes, P', and hence P s is 
of the first power, can be thus inverted : If P is 
such a point-aggregate that P' is of the first power, 
there is an integer a of (1) or (II) such that P (tt) = o, 
and there is a smallest of such a's. This last 
theorem shows the importance of the transfinite 
numbers in the theory of point-aggregates. 

Cantor's proof that the power of (II) is different 
from that of (I) is analogous to his proof of the 
non-enumerability of the continuum. Suppose that 
we could pUt (II) in the form of a simple series : 

(7) <*!> a 2 > a " 

we shall define a number which has the properties 
both of belonging to (II) and of not being a member 
of the series (7) ; and, since these properties are 
contradictory of one another if the hypothesis be 
granted, we must conclude that (II) cannot be put 

* The property in question is: A "prime-number" a is such that 
the resolution *=Py is only possible when jS = I or 0=a. 
t See the next section. 



INTRODUCTION 65 

in the form (7), and therefore has not the power of 
(I). Let a K be the first number of (i) which is 
greater than a 1} a Ka the first greater than a K)| , and 
so on ; so that we have 

i< K 2 < K 3 < . . . 

and 



and 

a v <a K ^ if 

Now it may happen that, from a certain number 

a K on, all following it in the series (7) are smaller 
p 

than it ; then it is evidently the greatest. If, on 
the other hand, there is no such greatest number, 
imagine the series of integers from i on and smaller 
than ai , add to it the series of integers ^ fll and 
> a,. , then the series of integers .> a^ and < a v and 
so on ; we thus get a definite part of successive 
numbers of (I) and (II) which is evidently of the 
first power, and consequently, by the definition of 
(II), there is a least number ft of (II) which is 
greater than all of these numbers. Therefore /3 > a A 

and thus also /3>a v , and also every number /3'</3 
is surpassed in magnitude by certain numbers a x . 

If there is a greatest a* p = y, then the number y+ l 
is a member of (II) and not of (7) ; and if there is 
not a greatest, the number /3 is a member of (II) 
and not of (7), 

Further, the power of (II) is the next greater to 
that of (I), so that no other powers lie between 



66 INTRODUCTION 

them, for any aggregate of numbers of (I) and (II) 
is of the power of (I) or (II). In fact, this aggregate 
Z 1} when arranged in order of magnitude, is well- 
ordered, and may be represented by 



where we always have /8<fl, where fi is the first 
number of (III) ; and consequently (op) is either 
finite or of the power of (I) or of that of (II), 
quartum non datur. From this results the theorem : 
If N is any well-defined aggregate of the second 
power, M' is a part of M and M" is a part of M', 
and we know that M'' is of the same .power as M, 
then M' is of the same power as M, and therefore 
as M" ; and Cantor remarked that this theorem is 
generally valid, and promised to return to it. * 

Though the commutative law does not, in general, 
hold with the transfinite numbers, the associative 
law does, but the distributive law is only generally 
valid in the form : 



where a+/3, a> and /3 are multipliers, "as we im- 
mediately recognize by inner intuition. " 

The subtraction, division, prime numbers, and 
addition and multiplication of numbers which can 
be put in the form of a rational and integral function 
of a) of the transfinite numbers were then dealt with 

* From the occurrence of this theorem on p. 484 of the Afath. Attn. t 
xlvi, 1895, which we now know (see the note on p, 204 below) to have 
been a forestalling of the theorem that any aggregate can be well-ordered, 
we may conclude that this latter theorem was used in this instance. 



INTRODUCTION 67 

much in the same way as in the memoir of 1897 
translated below. In the later memoir the subject 
is treated far more completely, and was drawn up 
with far more attention to logical form than was the 
Grundlagen. 

An interesting part of the Grundlagen is the 
discussion of the conditions under which we are to 
regard the introduction into mathematics of a new 
conception, such as w, as justified. The result of 
this discussion was already indicated by the way in 
which Cantor defined his new numbers : * ' We may 
regard the whole numbers as ' actual ' in so far as 
they, on the ground of definitions, take a perfectly 
determined place in our understanding, are clearly 
distinguished from all other constituents of our 
thought, stand in definite relations to them, and 
thus modify, in a definite way, the substance of 
our mind." We may ascribe f ' actuality " to them 
"in so far as they must be held to be an expression 
or an image (Abbild) of processes and relations in 
the outer world, as distinguished from the intellect. " 
Cantor's position was, now, that while there is no 
doubt that the first kind of reality always implies 
the second,* the proof of this is often a most 
difficult metaphysical proulem ; but, in pure mathe- 
matics, we need only consider the first kind of 
reality, and consequently "mathematics is, in its 
development, quite free, and only subject to the 

* This, according to Cantor, is a consequence of "the unity of the 
All, to which we ourselves belong," and so, in pure mathematics, we 
need only pay attention to the reality of our conceptions in the first 
sense, as stated in the text. 



68 INTRODUCTION 

self-evident condition that its conceptions are both 
free from contradiction in themselves and stand 
in fixed relations, arranged by definitions, to 
previously formed and tested conceptions. In 
particular, in the introduction of new numbers, it 
is only obligatory to give such definitions of them 
as will afford them such a definiteness, and, under 
certain circumstances, such a relation to the older 
numbers, as permits them to .be distinguished from 
one another in given cases. As soon as a number 
satisfies all these conditions, it can and must be 
considered as existent and real in mathematics. 1 n 

V 

this I see the grounds on which we must regard the 
rational, irrational, and complex numbers as just as 
existent as the positive integers." 

There is no danger to be feared for science from 
this freedom in the formation of numbers, for, on 
the one hand, the conditions referred to under which 
this freedom can alone be exercised are such that 
they leave only a very small opportunity for arbi- 
trariness ; and, on the other hand, every mathe- 
matical conception has in itself the necessary 
corrective, if it is unfruitful or inconvenient, it 
shows this very soon by its unusability, and is 
then abandoned. 

To support the idea that conceptions in pure 
mathematics are free, and not subject to any 
metaphysical control, Cantor quoted the names 
of, and the branches of mathematics founded by, 
some of the greatest mathematicians of the nineteenth 
century, among which an especially instructive 



INTRODUCTION 69 

example in Rummer's introduction of his ' ' ideal " 
numbers into the theory of numbers. But " applied " 
mathematics, such as analytical mechanics and 
physics, is metaphysical both in its foundations 
and in its ends. ' ' If it seeks to free itself of this, 
as was proposed lately by a celebrated physicist,* 
it degenerates into a ( describing of nature,' which 
must lack both the fresh breeze of free mathematical 
thought and the power of explanation and grounding 
of natural appearances. " 

The note of Cantor's on the process followed in 
the correct formation of conceptions is interesting. 
In his judgment, this process is everywhere the same ; 
we posit a thing without properties, which is at first 
nothing else than a name or a sign A, and give it 
in order different, even infinitely many, predicates, 
whose meaning for ideas already present is known, 
and which may not contradict one another. By 
this the relations of A to the conception already 
present, and in particular to the allied ones, are 
determined ; when we have completed this, all the 
conditions for the awakening of the conception A, 
which slumbers in us, are present, and it enters 
completed into "existence" in the first sense; to 
prove its < ( existence " in the second sense is then 
a matter of metaphysics. 

This seems to support the process by which Heine, 

* This is evidently Kirchhoff. As is well known, Kirchhoff pro- 
posed (Vorlesungen iiber mathcmalische Physift, vol. i { Mechanik, 
Leipzig, 1874) this. Cf. E. Mach in his prefaces to his Mechanics 
(3rd ed., Chicago and London, 1907 ; Supplementary Volume, Chicago 
and London, 1915), and Popular Scientific Lectures, 3rd ed., Chicago 
aqd London, 1898, pp. 236-258. 



70 INTRODUCTION 

in a paper partly inspired by his discussions with 
Cantor, defined the real numbers as signs, to which 
subsequently various properties were given. But 
Cantor himself, as we shall see later, afterwards 
pointed out emphatically the mistake into which 
Kronecker and von Helmholtz fell when they started 
in their expositions of the number-concept with the 
last and most unessential thing' the ordinal words 
or signs in the scientific theory of number ; so. that 
we must, I think, regard this note of Cantor's as 
an indication that, at this time (1882), he was a 
supporter of the formalist theory of number, or at 
least of rational and real non-integral numbers. 

In fact, Cantor's notions as to what is meant 
by ' ' existence " in mathematics nptions which 
are intimately connected with his introduction of 
irrational and transfinite numbers were in substance 
identical with those of Hankel (1867) on "possible 
or impossible numbers." Hankel was a formalist, 
though not a consistent one, and his theory was 
criticized with great acuteness by Frege in 1884. 
But these criticisms mark the beginning of the 
logical theory of mathematics, Cantor's earlier work 
belonging to the formal stage, and his later work to 
what may be called the psychological stage. 

Finally, Cantor gave a discussion and exact de- 
termination of the meaning of the conception of 
"continuum." After briefly referring to the dis- 
cussions of this concept due to Leucippus, Demo- 
critus, Aristotle, Epicurus, Lucretius, and Thomas 
Aquinas, and emphasizing that we cannot begin, in 



INTRODUCTION 71 

this determination, with the conception of time or 
that of space, for these conceptions can only be 
clearly explained by means of a continuity-concep- 
tion which must, of course, be independent of them, 
he started from the -dimensional plane arithmetical 
space G w , that is to say, the totality of systems of 
values 

(*!> ^jii > *)> 

in which every x can receive any real value from 
oo to +00 independently of the others. Every 
such system is called an ' ' arithmetical point ;J of 
G H , the "distance" of two such points is defined 
by the expression 



and by an ' ' arithmetical point-aggregate " P con- 
tained in G n is meant any aggregate of points 
G selected Out of it by a law. Thus the investi- 
gation comes to the establishment of a sharp and 
as general as possible a definition which should 
allow us to decide when P is to be called a "con- 
tinuum. " 

If the first derivative P' is of the power of (I), 
there is a first number a of (I) or (II)* for which 
p() vanishes ; but if P' is not of the power of (I), 
P' can be always, and in only one way, divided into 
two aggregates R and S, where R is " reducible, " 
that is to say, such that there is a first number y 
of (I) or (II) such that 

R(v)= 0j 



72 INTRODUCTION 

and S is such that derivation does not alter it. Then 

SsS' 
and consequently also 



and S is said to be " perfect. " No aggregate can 
be both reducible and perfect, "but, on the other 
hand, irreducible, is not so much as perfect, nor 
imperfect exactly the same as reducible, as we 
easily see with some attention. " 

Perfect aggregates are by no means always every- 
where dense ; an example of such an aggregate 
which is everywhere dense in no interval was given 
by Cantor. Thus such aggregates are not fitted 
for the complete definition of a continuum, although 
we must grant that the continuum must be perfect. 
The other predicate is that the aggregate must be 
connected (zusammenhangend), that is to say, if t 
and t are any two of its points and e a given arbi- 
trarily small positive number, a finite number of 
points t lt t t) . . ., t v of P exist such that the dis- 
tances tt lf f]/ a , . . ., tjf are all less than e. 

"All the geometric point-continua known to us 
are, as is easy to see, connected ; and I believe, 
now, that I recognize in these two predicates 
'perfect' and 'connected* the necessary and sufficient 
characteristics of a point-continuum. " 

Bolzano's (1851) definition of a continuum is 
certainly not correct, for it expresses only one 
property of a continuum, which is also possessed by 



INTRODUCTION 73 

aggregates which arise from G H when any isolated 
aggregate is removed from it, and also in those 
consisting of many separated continua. Also 
Dedekind * appeared to Cantor only to emphasize 
artother property of a continuum, namely, that which 
it has in common with all other perfect aggregates. 
We will pass over the development of the theory 
of point-aggregates subsequently to 1882 Ben- 
dixson's and Cantor's researches on the power of 
perfect aggregates, Cantor's theory of ' ' adherences " 
and "coherences," the investigations , of Cantor, 
Stolz, Harnack, Jordan, Borel, and others on the 
' ' content " of aggregates, and the applications of 
the theory of point-aggregates to the theory of 
functions made by Jordan, Brod6n, Osgood, Baire, 
Arzela, Schoenflies, and many others, and will now 
trace the development, in Cantor's hands, of the 
theory of the transfmite cardinal and ordinal numbers 
from 1883 to 1895. 

VIII 

An account of the development that the theory 
of transfmite numbers underwent in Cantor's mind 
from 1883 to 1890 is described in his articles 
published in the Zeitschrift filr Philosophie und 
philosophised Kritik for 1887 and 1888, and 
collected and published in 1890 under the title Zur 
Lehre vom Transfiniten. A great part of this little 
book is taken up with detailed discussions about 
philosophers' denials of the possibility of infinite 

* Essays on Number, p. IT. 



74 INTRODUCTION 

numbers, extracts from letters to and from philo- 
sophers and theologians, and so on.* "All so- 
called proofs of the impossibility of actually infinite 
numbers," said Cantor, "are, as may be shown in 
every particular case and also on general grounds, 
false in that they begin by attributing to the 
numbers in question all the properties of finite 
numbers, whereas the infinite numbers, if they are 
to be thinkable in any form, must constitute quite 
a new kind of number as opposed to the finite 
numbers, and the nature of this new kind of number 
is dependent on the nature of things and is an object 
of investigation, but not of our arbitrariness or our 
prejudice." 

In 1883 Cantor had begun to lecture on his view 
of whole numbers and types of order as general 
concepts, or universals (unum versus alia) which 
relate to aggregates and arise from these aggregates 
when we abstract from the nature of the elements. 
' ' Every aggregate of distinct things can be regarded 
as a unitary thing in which the things first mentioned 
are constitutive elements. If we abstract both from 
the nature of the elements and from the order in 
which they are given, we get the c cardinal number ' 
or 'power' of the aggregate, a general concept in 
which the elements, as so-called units, have so 
grown organically into one another to make a 
unitary whole that no one of them ranks above the 
others. Hence results that two different aggregates 
have the same cardinal number when and only when 
* C/. VII, near the beginning. 



INTRODUCTION 75 

they are what I call ' equivalent ' to one another, 
and there is no contradiction when, as often happens 
with infinite aggregates, two aggregates of which 
one is a part of the other have the same cardinal 
number. I regard the non-recognition of this fact 
as the principal obstacle to the introduction of 
infinite numbers. If the act of abstraction referred 
to, when we have to do with an aggregate ordered 
according to one or many relations (dimensions), is 
only performed with respect to the nature of the 
elements, so that the ordinal rank in which these 
elements stand to one another is kept in the general 
concept, the organic whole arising is what I call 
' ordinal type, ' or in the special case of well-ordered 
aggregates an ' ordinal number. ' This ordinal 
number is the same thing that I called, in my 
Grundlagen of 1883, the ' enumeral (Anzahl) of a 
well-ordered aggregate.' Two ordered aggregates 
have one and the same ordinal type if they stand 
to one another in the relation of 'similarity,' 
which relation will be exactly defined. These are 
the roots from which develops with logical necessity 
the organism of transfinite theory of types and in 
particular of the transfinite ordinal numbers, and 
which I hope soon to publish in a systematic form. " 

The contents of a lecture given in 1883 were also 
given in a letter of 1884. In it was pointed out 
that the cardinal number of an aggregate M is the 
general concept under which fall all aggregates 
equivalent to M, and that : 

"One of the most important problems of the 



76 INTRODUCTION 

theory of aggregates, which I believe I have solved 
as to its principal part in my Grundlagen> consists 
in the question of determining the various powers 
of the aggregates in the whole of nature, in so far 
as we can know it. This end I have reached by 
the development of the general concept of enumeral 
of well-ordered aggregates, or, what is the same 
thing, of the concept of ordinal number." The 
concept of ordinal number is a special case of the 
concept of ordinal type, which relates to any simply 
or multiply ordered aggregate in the same way as. 
'the ordinal number to a well-ordered aggregate. 
The problem here arises of determining the various 
ordinal numbers in nature. 

When Cantor said that he had solved the chief 
part of the problem of determining the various 
powers in nature, he meant that he had almost 
proved that the power of the arithmetical continuum 
is the same as the power of the ordinal numbers of 
the second class. In spite of the fact that Cantor 
firmly believed this, possibly on account of the fact 
that all known aggregates in the continuum had 
been found to be either of the first power or of the 
power of the continuum, the proof or disproof of 
this theorem has not even now been carried out, 
and there is some ground for believing that it 
cannot be carried out. 

What Cantor, in his Grundlagen, had noted as the 
relation of two well-ordered aggregates which have 
the same enumeral was here called the relation of 
"similarity," and in the laws of multiplication of 



INTRODUCTION 77 

two ordinal numbers he departed from the custom 
followed in the Grundlagen and wrote the multiplier 
on the right and the multiplicand on the left. The 
importance of this alteration is seen by the fact 
that we can write : a /J .o? = a' J+ T ; whereas we would 
have to write, in the notation of the Grundlagen : 
a".ay = a? + 0. 

At the end of this letter, Cantor remarked that 
W may, in a sense, be regarded as the limit to which 
the variable finite whole number v tends. Here " is 
the least transfinite ordinal number which is greater 
than all finite numbers ; exactly in the same way 
that ,J2 is the limit of certain variable, increasing, 
rational numbers, with this difference : the difference 
between J2 and these approximating fractions be- 
comes as small as we wish, whereas co v is always 
, equal to &>. But this difference in no way alters the 
fact that <o is to be regarded as as definite and com- 
pleted as J2, and in no way alters the fact that o> 
has no more trace of the numbers v which tend to it 
than ^2 has of the approximating fractions. The 
transfinite numbers are in a sense new irrationalities, 
and indeed in my eyes the best method of defining 
finite irrational numbers is the same in principle as 
my method of introducing transfinite numbers. We 
can say that the transfinite numbers stand or fall 
with finite irrational numbers, in their inmost being 
they are alike, for both are definitely marked off 
modifications of the actually infinite." 

With this is connected in principle an extract from 



78 INTRODUCTION 

a letter written in 1886: "Finally I have still to 
explain to you in what sense I conceive the minimum 
of the transfinite as limit of the increasing finite. 
For this purpose we must consider that the concept 
of ' limit ' in the domain of finite numbers has two 
essential characteristics. For example, the number 
i is the limit of the numbers z v =i ij Vt where v is 
a variable, finite, whole number, which increases 
above all finite limits. In the first place the 
difference i ss v is a magnitude which becomes in- 
finitely small ; in the second place i is the least of 
all numbers which are greater than all magnitudes z v . 
Each of these two properties characterizes the finite 
number i as limit of the variable magnitude s v . 
Now if we wish to extend the concept of limit to 
transfinite limits as well, the second of the above 
characteristics is used ; the first must here be 
allowed to drop because it has a meaning only for 
finite limits. Accordingly 1 call co the limit of the 
increasing, finite, whole numbers v , because o> is the 
least of all numbers which are greater than all the 
finite numbers. But o v is always equal to o>, and 
therefore we cannot say that the increasing numbers 
v come as near as we wish to CD ; indeed any number 
v however great is quite as far off from a> as the least 
finite number. Here we see especially clearly the 
very important fact that my least transfinite ordinal 
number <o, and consequently all greater ordinal 
numbers, lie quite outside the endless series i, 2, 3, 
and so on. Thus o> is not a maximum of the finite 
numbers, for there is no such thing, " 



INTRODUCTION 79 

In another letter written in 1886, Cantor empha- 
sized another aspect of irrational numbers. In all 
of the definitions of these numbers there is used, 
as is indeed essential, a special actually infinite 
aggregate of rational numbers. In both this and 
another letter of 1886, Cantor returned in great 
detail to the distinction between the ' ' potential " 
and ' ' actual " infinite of which he had made a great 
point under other names in his Gj-undlagen. The 
potential infinite is a variable finite, and in order 
that such a variable may be completely known, we 
must be able to determine the domain of variability, 
and this domain can only be, in general, an actually 
infinite aggregate of values. Thus every potential 
infinite presupposes an actually infinite, and these 
' ' domains of variability " which are studied in the 
theory of aggregates are the foundations of arith- 
metic and analysis. Further, besides actually infinite 
aggregates, we have to consider in mathematics 
natural abstractions from these aggregates, which 
form the material of the theory of transfinite 
numbers. 

In 1885, Cantor had developed to a large extent 
his theory of cardinal numbers and ordinal types. 

In the fairly long paper which he wrote out, he 
laid particular stress on the theory of ordinal types 
and entered into details which he had not published 
before as to the definition of ordinal type in general, 
of which ordinal number is a particular case. In 
this paper also he denoted the cardinal number of 
an aggregate M by M, and the ordinal type of 



8o INTRODUCTION 

M by M ; thus indicating by lines over the letter 
that a double or single act of abstraction is to 
be performed. 

In the theory of cardinal numbers, he defined the 
addition and multiplication of two cardinal numbers 
and proved the fundamental laws about them in 
much the same way as he did in the memoir of 
1895 which is translated below. It is characteristic 
of Cantor's views that he distinguished very sharply 
between an aggregate and a cardinal number that 
belongs to it : "Is not an aggregate an object out- 
side us, whereas its cardinal number is an abstract 
picture of it in our mind ? " 

In an ordered aggregate of any number of 
dimensions, such as the totality of points in space, 
as determined by three rectangular co-ordinates, or 
a piece of music whose dimensions are the sequence 
of the tones in time, the duration of each tone in 
time, the pitch of the tones, and the intensity of the 
tones, then "if we make abstraction of the nature 
of the elements, while we retain their rank in all the 
different directions, an intellectual picture, a general 
concept, is generated in us, and 1 call this the -ple 
ordinal type." The definition of the " similarity of 
ordered aggregates " is : 

"Two tf-ply ordered aggregates M and N are 
called similar if it is possible so to make their 
elements correspond to another uniquely and com- 
pletely that, if E and E' are any two elements of 
M and F and F' the two corresponding elements of 
N, then for i/= i, 2, . , , n the relation of rank of 



INTRODUCTION 81 

E to E' in the v th direction inside the aggregate M 
is exactly the same as the relation of rank of F to F' 
in the vth direction inside the aggregate N. We 
will call such a correspondence of two aggregates 
which are similar to one another an imaging ,of the 
one on the other." 

The addition and multiplication of ordinal types, 
and the fundamental laws about them, were then 
dealt with much as in the memoir of 1895 which is 
translated below. The rest of the paper was devoted 
to a consideration of problems about #-ple finite 
types. 

In 1888, Cantor, who had arrived at a very clear 
notion that the essential part of the concept of number 
lay in the unitary concept that we form, gave some 
interesting criticisms on the essays of Helmholtz and 
Kronecker, which appeared in 1887, on the concept 
of number. Both the authors referred to started 
with the last and most unessential feature in our 
treatment of ordinal numbers : the words or other 
signs that we use to represent these numbers. 

In 1887, Cantor gave a more detailed proof of the 
non-existence of actually infinitely small magnitudes. 
This proof was referred to in advance in the Grund- 
lagen, and was later put into a more rigorous form 
by Peano. 

We have already referred to the researches of 
Cantor on point-aggregates published in 1883 and 
later; the only other paper besides those already 
dealt with that was published by Cantor on an 
important question in the theory of transfinite 



82 INTRODUCTION 

numbers was one published in 1892. In this paper 
we can see the origins of the conception of ' ' cover- 
ing" (Belegung) defined in the memoir of 1895 trans- 
lated below. In the terminology introduced in this 
memoir, we can say that the paper of 1892 contains 
a proof that 2, when exponentiated by a transfmite 
cardinal number, gives rise to a cardinal number 
which is greater than the cardinal number first 
mentioned. 

The introduction of the concept of "covering" is 
the most striking advance in the principles of the 
theory of transfmite numbers from 1885 to 1895, 
and we can now study the final and considered form 
which Cantor gave to the theory in two important 
memoirs of 1895 and 1897. The principal advances 
in the theory since 1897 will be referred to in the 
notes at the end of this book. 



CONTRIBUTIONS TO THE 

FOUNDING OF THE THEORY OF 

TRANSFINITE NUMBERS 



[481] CONTRIBUTIONS TO THE 
FOUNDING OF THE THEORY OF 
TRANSFINITE NUMBERS 

(FIRST ARTICLE) 

" Hypotheses non fingo." 

"Neque enim leges intellectui aut rebus damus 
ad arbitrium nostrum, sed tanquam scribse 
fideles ab ipsius naturae voce latas et prolatas 
excipimus et describimus." 

"Veniet tempus, quo ista quse nunc latent, in 
lucem dies extrafiat et longioris sevi diligentia." 

1 

The Conception of Power or Cardinal Number 

BY an ' ' aggregate " (Menge) we are to understand 
any collection into a whole (Zusammenfassung su 
einem Ganeen) M of definite and separate objects m 
of our intuition or our thought. These objects are 
called the ' ' elements " of M. 
In signs we express this thus : 

(1) M = {**}. 

We denote the uniting of many aggregates M, N, 
P, . . ., which have no common elements, into a 
single aggregate by 

(2) (M, N, P, . . .) 

85 



86 THE FOUNDING OF THE THEORY 

The elements of this aggregate are, therefore, the 
elements of M , of N, of P, . . . , taken together. 

We will call by the name "part" or "partial 
aggregate " of an aggregate M any* other aggregate 
A^ whose elements are also elements of M. 

If M a is a part of M x and Mj is a part of M, then 
M a is a part of M. 

Every aggregate M has a definite "power," which 
we will also call its ' ' cardinal number. " 

We will call by the name " power" or "cardinal 
number " of M the general concept which, by means 
of our active faculty of thought, arises from the 
aggregate M when we make abstraction of the 
nature of its various elements m and of the order 
in which they are given. 

[482] We denote the result of this double act of 
abstraction, the cardinal -number or power of M, by 

(3) 5. 

Since every single element m, if we abstract from 
its nature, becomes a "unit," the cardinal number 

M is a definite aggregate composed of units, and 
this number has existence in our mind as an intel- 
lectual image or projection of the given aggregate M. 
We say that two aggregates M and N are ' ' equi- 
valent," in signs 

(4) M c\> N or N r\J M, 

if it is possible to put them, by some law, in such a 
relation to one another that to every element of each 
one of them corresponds one and only one element 



OF TRANSFINITE NUMBERS 87 

of the other. To every part M x of M there corre- 
sponds, then, a definite equivalent part N of N, and 
inversely. 

If we have such a law of co-ordination of two 
equivalent aggregates, then, apart from the case 
when each of them consists only of one element, we 
can modify this law in many ways. We can, for 
instance, always take care that to a special element 
m Q of M a special element of N corresponds. For 
if, according to the original law, the elements m 9 
and do not correspond to one another, but to the 
element m Q of M the element n of N corresponds, 
and to the element of N the element m of M 
corresponds, we take the modified law according to 
which '/# corresponds to and m^ to n t and for the 
other elements the original law remains unaltered. 
By this means the end is attained. 

Every aggregate is equivalent to itself : 

(5) M oo M. 

If two aggregates are equivalent to a third, they are 
equivalent to one another ; that is to say : 

(6) from M oo P and N oo P follows M oo N. 

Of fundamental importance is the theorem that 
two aggregates M and N have the same cardinal 
number if, and only if, they are equivalent : thus, 

(7) from MooN we get M fT, 
and 

(8) from M = N we get M oo N, 

Thus the equivalence of aggregates forms the neces- 



88 THE FOUNDING OF THE THEORY 

sary and sufficient condition for the equality of their 
cardinal numbers. 

483] In fact, according to the above- definition of 

power, the cardinal number M remains unaltered if 
in the place of each of one or many or even all 
elements m of M other things are substituted. If, 
now, M oo N, there is a law of co-ordination by 
means of which M and N are uniquely and recipro- 
cally referred to one another ; and by it to the 
element m of M corresponds the element n of N. 
Then we can imagine, in the place of every element 
m of M, the corresponding element n of N substi- 
tuted, and, in this way, M transforms into N without 
alteration of cardinal number. Consequently 

S=N. 

The converse of the theorem results from the re- 
mark that between the elements of M and the 
different units of its cardinal number M a recipro- 
cally univocal (or bi-univocal) relation of correspond- 
ence subsists. For, as we saw, M grows, so to 
speak, out of M in such a way that from every 
element m of N. a. special unit of M arises. Thus 
we can say that 

(9) M no M. 

In the same way N oo N. If then M N, we have, 
by (6), M oo N. 

We will mention the following theorem, which 
results immediately from the conception of equival- 



OF TRANSFINITE NUMBERS 89 

euce. If M, N, P, . . . are aggregates which have 
no common elements, M', N', P', . . . are also aggre- 
gates with the same property, and if 

MooM', NooN', P oo P' 

then we always have 

(M, N, P, . . .) ro (M', N', P', . . .) 

2 
"Greater" and "Less" with Powers 

If for two aggregates M and N with the cardinal 
numbers a = M and b = N, both the conditions : 

(a) There is no part of M which is equivalent to N, 

(b) There is a part N x of N, such that N x oo M, 

are fulfilled, it is- obvious that these conditions still 
hold if in them M and N are replaced by two 
equivalent aggregates M' and N'. Thus they ex- 
press a definite relation of the cardinal numbers 
a and b to one another. 

[484] Further, the equivalence of M and N, and 
thus the equalky of rt and b, is excluded ; for if we 
had M oo N, we would have, because Nj. oo M, the 
equivalence N x oo N, and then, because M oo N, 
there would exist a part M x of M such that M x oo M, 
and therefore we should have M x oo N ; and this 
contradicts the condition (a). 

Thirdly, the relation of a to b is such that it 
makes impossible the same relation of b to a ; for if 



90 THE FOUNDING OF THE THEORY 

in (a) and (b) the parts played by M and N are 
interchanged, two conditions arise which are con- 
tradictory to the former ones. 

We express the relation of a to 'b characterized by 
(a) and () by saying : a is ' ' less " than b or b is 
' ' greater " than a ; in signs 

(1) a<b or b>a. 
We can easily prove that, 

(2) if a < b and b < c, then we always have a < c. 
Similarly, from the definition, it follows at once 
that, if P x is part of an aggregate P, from a<Pj 
follows a < P and from P < b follows i\ < b. 

We have seen that, of the three relations 

a=b, a<b, b<a, 

each one excludes the two others. On the other 
hand, the theorem that, with any two cardinal 
numbers a and b, one of those three relations must 
necessarily be realized, is by no means self-evident 
and can hardly be proved at this stage. 

Not until later, when we shall have gained a 
survey over the ascending sequence of the transfinite 
cardinal numbers and an insight into their connexion, 
will result the truth of the theorem : 

A, If a and b are any two cardinal numbers, then 
either a = b or a < b or a > b. 

From this theorem the following theorems, of 
which, however, we will here make no use, can be 
very simply derived ; 



OF TRANSFINITE NUMBERS 91 

B. If two aggregates M and N are such that M is 
equivalent to a part N x of N and N to a part M x of 
M, then M and N are equivalent ; 

C. If M x is a part of an aggregate M, M a is a 
part of the aggregate M lf and if the aggregates 
M and M a are equivalent, then M is equivalent to 
both M and M a ; 

D. If, with two aggregates M and N, N is 
equivalent neither to M nor to a part of M, there is 
a part N x of N that is equivalent to M ; 

E. If two aggregates M and N are not equivalent, 
and there is a part N 1 of N that is equivalent to M, 
then no part of M is equivalent to N. 

[485] 3 

The Addition and Multiplication of Powers 

The union of two aggregates M and N which 
have no common elements was denoted in I, (2), 
by (M, N). We call it the "union -aggregate 
(Vereinigungsmenge) of M and N." 

If M' and N' are two other aggregates without 
common elements, and if M t\> M' and N ro N', we 
saw that we have 

(M, N) oo (M', N'). 

Hence the cardinal number of (M, N) only depends 
upon the cardinal numbers M = a and N = b. 

This leads to the definition of the sum of a and b. 
We put 

(i) 



92 THE FOUNDING OF THE THEORY 

Since in the conception of power, we abstract from 
the order of the elements, we conclude at once that 

(2) a+b = b+a; 

and, for any three cardinal numbers a, b, c, we have 

(3) a+(b+c) = (a+b)+c. 

We now come to multiplication. Any element m 
of an aggregate M can be thought to be bound up 
with any element n of another aggregate N so as 
to form a new element (m, n) ; we denote by (M . N) 
the aggregate of all these bindings (m, n\ and call 
it the "aggregate of bindings (Verbindungsmenge) 
ofMandN." Thus 

(4) (M.N) = {(*,)}. 

We see that the power of (M . N) only depends on 

the powers M == a and N = b ; for, if we replace the 
aggregates M and N by the aggregates 

W = {m'} and N' = {'} 

respectively equivalent to them, and consider m, m' 
and , n' as corresponding elements, then the 
aggregate 

(M'.N') = {(**', ')} 

is brought into a reciprocal and univocal corre- 
spondence with (M.N) by regarding (m t n) and 
(m' t if) as corresponding elements. Thus 

(5) (M'. N') ou (M.N). 

We now define the product a . b by the equation 



(6) a.b = (M.N). 



OF TRANSFINITE NUMBERS 93 

[486] An aggregate with the cardinal number 
n . b may also be made up out of two aggregates M 
and N with the cardinal numbers a and b according 
to the following rule : We start from the aggregate 
N and replace in it every element n by an aggregate 
M <>J M ; if, then, we collect the elements of all 
these aggregates M w to a whole S, we see that 

(7) Soo(M.N), 
and consequently 

=a.b. 

For, if, with any given law of correspondence of the 
two equivalent aggregates M and M H , we denote 
by m the element of M which corresponds to the 
element m n of M tt , we have 

(8) SHM; 

and thus the aggregates S and (M.N) can be re- 
ferred reciprocally and univocally to one another by 
regarding m H and (m, ) as corresponding elements. 
From our definitions result readily the theorems : 

(9) a.b=b.a, 

(10) .a.(b,c)=(a.b).c, 
(n) a(b+c)-ab+ac; 

because : 

(M . N) no (N . M), 

(M . (N . P)) oo ((M . N) . P), 
(M . (N, P)) CNJ ((M . N), (M . P)). 
Addition and multiplication of powers are subject, 



94 THE FOUNDING OF THE THEORY 

therefore, to the commutative, associative, and dis- 
tributive laws. 

4 
The Exponentiation of Powers 

By a "covering of the aggregate N with elements 
of the aggregate M," or, more simply, by a "cover- 
ing of N with M," we understand a law by which 
with every element n of N a definite element of M 
is bound up, where one and the same element of M 
can come repeatedly into application. The element 
of M bound up with n is, in a way, a one-valued 
function of n, and may be denoted by f(n) ; it is 
called a ' ' covering function of n." The correspond- 
ing covering of N will be called /(N). 

[487] Two coverings /^(N) and./j,(N) are said to 
be equal if, and only if, for all elements of N the 
equation 

(i) fM=fM 

is fulfilled, so that if this equation does not subsist 
for even a single element , ./i(N) and^(N) are 
characterized as different coverings of N. For ex- 
ample, if #z is a particular element of M, we may 
fix that, for all n's 

f(n) = m Q \ 

this law constitutes a particular covering of N with 
M. Another kind of covering results if m and m t 
are two different particular elements of M and a 
particular element of N, from fixing that 



OF TRANSFINITE NUMBERS 95 

/() = z 
f(n) = m v 

for all 's which are different from . 

The totality of different coverings of N with M 
forms a definite aggregate with the elements /"(N) ; 
we call it the "covering-aggregate (Belegungsmenge) 
of N with M " and denote it by (N | M). Thus : 

(2) (N|M) = {/(N)}. 

If M fX> M' and N ru N', we easily find that 

(3) (N|M)oo(N'|MO. 

Thus the cardinal number of (N | M) depends only 
on the cardinal numbers M = a and N = b ; it serves 
us for the definition of a 6 : 



(4) a=(N|M). 

For any three aggregates, M, N, P, we easily prove 
the theorems : 

(5) ((N lM).(P|M))oo((N, P)|M), 

(6) ((P|M),(P|N))cs J (P|(M.N)), 

(7) (P|(N|M))r\>((P.N)|M), 

from which, if we put P = c, we have, by (4) and by 
paying attention to 3, the theorems for any three 
cardinal numbers, a, 6, and c : 

(8) c^.a' 

(9) a'.b 

(10) (a*) e = 



96 THE FOUNDING OF THE THEORY 

[488] We see how pregnant and far-reaching 
these simple formulae extended to powers are by the 
following example. If we denote the power of the 
linear continuum X (that is, the totality X of real 
numbers x such that x>. and :<i) by o, we easily 
see that it may be represented by, amongst others, 
the formula : 

(II) = 2 No , 

where 6 gives the meaning of N O . In fact, by (4), 
2 N is the power of all representations 



.. 

(where f(v)=o or i) 

of the numbers x in the binary system. If we pay 
attention to the fact that every number x is only.? 
represented once, with the exception of the^numbers 

x= v ^ <i, which are represented twice over, we 

have, if we denote the "enumerable" totality of 
the latter by {s v }, 



3 *-(W, X). 

If we take away from X any " enumerable " aggre- 
gate {t v } and denote the remainder by X x , we have : 

X-((M. X 1 ) = ({^_ 1 }, {*,}, 
({*}, X) = (M, {*}, X,), 

{^-i}^{jv}, {ffcjoofc}. 
K }, X), 



OF TRANSFINITE NUMBERS 97 
and thus ( i) 

2*-X-0. 

From (n) follows by squaring (by 6, (6)) 

. = 2 So . 2 No = 2 No+Wo = 2 No =sO, 

and hence, by continued multiplication by o, 
(13) o" = o, 

where v is any finite cardinal number. 

If we raise both sides of (n) to the power* N O 
we get 



But since, by 6, (8), O .N O = N O > we have 
(14) 0*0 = 0. 

The formulae (13) and (14) mean that both the 
i/-dirnensional and the N -dimensional continuum have 
the power of the one-dimensional continuum. Thus 
the whole contents of my paper in Crelle's Journal^ 
vol. Ixxxiv, i878,f are derived purely algebraically 
with these few strokes of the pen from the fundamental 
formulae of the calculation with cardinal numbers. 

[489] 5 

The Finite Cardinal Numbers 

We will next show how the principles which we 
have laid down, and on which later on the theory 
of the actually infinite or transfinite cardinal numbers 

* [In English there Is an nmbiguity,] 
t [See Section V of the Introduction.] 



98 THE FOUNDING OF THE THEORY 

will be built, afford also the most natural, shortest, 
and most rigorous foundation for the theory of 
finite numbers. 

To a single thing , if we subsume it under the 
concept of an aggregate E = (), corresponds as 
cardinal number what we call "one" and denote by 
i ; we have 

(1) i = E . 

Let us now unite with E another thing e lt and 
call the union-aggregate E I} so that , 

(2) E I = ( E O> *i) = (*o *i> 

The cardinal number of Ej is called "two" and is 
denoted by 2 : 

(3) 2 = E r 

By addition of new elements we get the series of 
aggregates 

E 2 = ( E 1> e s)> E 3 = ( E 2> *s)> ' ' 

which give us successively, in unlimited sequence, 
the other so-called "finite cardinal numbers" de- 
noted by 3, 4, 5, ... The use which we here 
make of these numbers as suffixes is justified by 
the fact that a number is only used as a suffix 
when it has been defined as a cardinal number. 
We have, if by v i is understood the number im- 
mediately preceding v in the above series, 

(4) v=iUi, 

(5) E, =(&-!, * v ) = (* , *!,... *). 



OF TRANSFINITE NUMBERS 99 
From the definition of a sum in 3 follows : 

(6) R-lL-i+i; 

that is to say, every cardinal number, except I, is 
the sum of the immediately preceding one and i. 

Now, the following three theorems come into the 
foreground : 

A. The terms of the unlimited series of finite 
cardinal numbers 

i, 2, 3, . . ., v, . . . 

are all different from one another (that is to say, 
.the condition of equivalence established in i is 
not fulfilled for the corresponding aggregates). 

[490] B. Every one of these numbers v is greater 
than the preceding ones and less than the following 
ones ( 2). 

C. There are no cardinal numbers which, in 
magnitude, lie between two consecutive numbers 
v and j/+i ( 2). 

We make the proofs of these theorems r.est on 
the two following ones, D and E. We .shall, then, 
in the next place, give the latter theorems rigid 
proofs, 

D. If M is an aggregate such that it is of equal 
power with none of its parts, then the aggregate 
(M, *), which arises from M by the addition of a 
single new element e, has the same property of 
being of equal power with none of its parts. 

E. If N is an aggregate with the finite cardinal 
number v, and N x is any part of N, the cardinal 




TOO THE FOUNDING OF THE THEOR Y 

number of N a is equal to one of the preceding 
numbers I, 2, 3, . . ., v I. 

Proof of D. Suppose that the aggregate (M, <?) 
is equivalent to one of its parts which we will call 
N. Then two cases, both of which lead to a con- 
tradiction, are to be distinguished : 

(a) The aggregate N contains e as element ; let 
N = (M ls e) ; then M l is a part of M because N is 
a part of (M, e). As we saw in I, the law of 
correspondence of the two equivalent aggregates 
(M, e) and (M 1} e) can be so modified that the 
element e of the one corresponds to the same 
element e of the other ; by that, then, M and Mj 
are referred reciprocally and univocally to one 
another. But this contradicts the supposition that 
M is not equivalent to its part Mj. 

(b) The part N of (M, <?) does not contain e as 
element, so that N is either M or a part of M. In 
the law of correspondence between (M, e) and N, 
which lies at the , basis of our supposition, to the 
element e of the former let the element f of the 
latter correspond. Let N = (M x , /) ; then the 
aggregate M is put in a reciprocally univocal relation 
with M r But MJ is a part of N and hence of M. 
So here too M would be equivalent to one of its 
parts, and this is contrary to the supposition. 

Proof of E. We will suppose the correctness 
of the theorem up to a certain v and then conclude 
its validity for the number j/+ I which immediately 
follows, in the following manner : We start from 
the aggregate E p = (* , * lf . . ., <?) as an aggregate 



OF TRANSFINITE NUMBERS 101 

with the cardinal number v+i. If the theorem is 
true for this aggregate, its truth for any other 
aggregate with the same cardinal , number v + 1 
follows at once by I. Let E' be any part of E F ; 
we distinguish the following cases : 

(a) E' does not contain e v as element, then E is 
either _]. [491] or a part of _!, and so has as 
cardinal number either v or one of the numbers 
J > 2 i 3> "I, because we supposed our theorem 
true for the aggregate _!, with the cardinal 
number v. 

(b) E' consists of the single element e n then 

E'=i. 

(c) E' consists of e v and an aggregate E", so that 
E' = (E", *). E" is a part of _! and has there- 
fore by supposition as cardinal number one of the 

numbers I, 2, 3, . . ., >/ i.. But now E > s=E"+i, 
and thus the cardinal number of E' is one of the 
numbers 2, 3, . . ., v. 

Proof of A. Every one of the aggregates which 
we have denoted by E v has the property of not 
being equivalent to any of its parts. For if we 
suppose that this is so as far as a certain v, it follows 
from the theorem D that it is so for the immediately 
following number v+i. For v I, we recognize at 
once that the aggregate E t = (* , e^) is not equivalent 
to any of its parts, which are here <V ) and (ej. 
Consider, now, any two numbers /* and v of the 
series i, 2, 3, . . . ; then, if p is the earlier and v 
the later, E^j is a part of _!. Thus E^-i and 



102 THE FOUNDING OF THE THEORY 

E v _! are not equivalent, and accordingly their 

cardinal numbers / u = E fl _i and !/=_! are not 
equal. 

Proof 'of ft. If of the two finite cardinal numbers 
p and v the first is the earlier and the second the 
later, then /* < v. For consider the two aggregates 
M = E ft _i and N = _!; for them each of the two 

conditions in 2 for M < N is fulfilled. The con- 
dition (a) is fulfilled because, by theorem E, a part 
of M = E M _i can only have one of the cardinal 
numbers i, 2, 3, . . ., yu i, and therefore, by 
theorem A, cannot be equivalent to the aggregate 
N = _!. The condition () is fulfilled because M 
itself is a part of N. 

Proof of C. Let a be a cardinal number which 
is less than v+i. Because of the condition (b) of 
2, there is a part of E,, with the cardinal number 
a, By theorem E, a part of can only have one 
of the cardinal numbers I, 2, 3, . . ., j/. Thus a is 
equal to one of the cardinal numbers I, 2, 3, . . ., v. 
By theorem B, none of these is greater than v. 
Consequently there is no cardinal number a which 
is less than v + i and greater than v. 

Of importance for what follows is the following 
theorem : 

F. If K is any aggregate of different finite 
cardinal numbers, there is one, K V amongst them 
which is smaller than the rest, and therefore the 
smallest of all. 

[492] Proof. The aggregate K either contains 



OF TRANSF1NITE NUMBERS 103 

the number I, in which case it is the least, <c l = I, 
or it does not. In the latter case, let J be the 
aggregate of all those cardinal numbers of our series, 
!} 2 > 3> which are smaller than those occurring 
in K. If a number v belongs to J, all numbers less 
than v belong to J. But J must have one element 
v 1 such that v + 1 , and consequently all greater 
numbers, do not belong to J, because otherwise 
J would contain all finite numbers, whereas the 
numbers belonging to K are not contained in J. 
Thus J is the segment (Abscknitt) (i, 2, 3, . . ., j/j). 
The number ^ + I = K^ is necessarily an element of 
K and smaller than the rest. 

From F we conclude : 

G. Every aggregate K = {/c} of different finite 
cardinal numbers can be brought into the form of 
a series 

IV. = (jfj, /C 2 > K$> ) 

such that 



6 

The Smallest Transfinite Cardinal Number 
Aleph-Zero 

Aggregates with finite cardinal numbers are called 
"finite aggregates," all others we will call "trans- 
finite aggregates" and their cardinal numbers 
"transfinite cardinal numbers." 

The first example of a transfinite aggregate is 
given by the totality of finite cardinal numbers v ; 



104 THE FOUNDING OF THE THEORY 

we call its cardinal number ( i) " Aleph-zero" and 
denote it by M O ; thus we define 

(I) NO=R. 

That N O is a transfinite number, that is to say, is 
not equal to any finite number /m, follows from the 
simple fact that, if to the aggregate {v} is added a 
new element e Q) the union-aggregate ({j/}, e ) is 
equivalent to the .original aggregate {v}. For we 
can think of this reciprocally univocal correspond- 
ence between them : to the element e of the first 
corresponds the element I of the second, and to the 
element v of the first corresponds the element v+ i of 
the other. By 3 we thus have 

(2) N O +I=NO- 

But we showed in 5 that ^ + 1 is always different 
from fi, and therefore N O is not equal to any finite 
number /*. 

The number N O is greater than any finite number /* : 

(3) . Mo>M- 

[493] This follows, if we pay attention to 3, 

from the three facts that yu = (i, 2, 3, , . ., ^,), that 
no part of the aggregate (i, 2, 3, . . ., ^) is equiva- 
lent to the aggregate {j/}, and that (i, 2, 3, . . ., /tt ) 
is itself a part of {v}. 

On the other hand, N O is the least transfinite 
cardinal number. If a is any transfinite cardinal 
number different from N O , then 

(4) 



OF TRANSFINITE NUMBERS 105 

This rests on the following theorems : 

A. Every transfinite aggregate T has parts with 
the cardinal number N O . 

Proof. If, by any rule, we have taken away a 
finite number of elements t lt t z ,...,t v . lt there 
always remains the possibility of taking away a 
further element t v . The aggregate {}, where v 
denotes any finite cardinal number, is a part of T 
with the cardinal number N O , because {t v }c\j{v} ( i). 

B. If S is a transfinite aggregate with the cardinal 
number M O , and Sj is any transfinite part of S, then 

S! = N O . 

Proof. We have supposed that S oo {v}. Choose 
a definite law of correspondence between these two 
aggregates, and, with this law, denote by s v that 
element of S which corresponds to the element v of 
{j/ so that 



The part S x of S consists of certain elements j* 
of S, and the totality of numbers K forms a trans- 
finite part K of the aggregate {/}. By theorem G 
of 5 the aggregate K can be brought into the 
form of a series 



where 

consequently we have 



106 THE FOUNDING OF THE THEORY 

Hence follows that S a ro S, and therefore S x = M O . 

From A and B the formula (4) results, if we have 
regard to 2. 

From (2) we conclude, by adding I to both sides, 



and, by repeating this 

(5) N + " = <) 
We have also 

(6) MO + NO = NO- 

[494] For, by (i) of 3, N O + N O is tlie cardinal number 



Now, obviously 



and therefore 



The equation (6) can also be written 

MO . 2 = NO J 

and, by adding M O repeatedly to both sides, we 
find that 

(7} -v=v, NO- NO- 

We also have 

(3) No MO = NO- 



OF TRANSFINITE NUMBERS 107 

Proof. By (6) of 3, N O . N O is the cardinal 
number of the aggregate of bindings . 



where /& and v are any finite cardinal numbers which 
are independent of one another. If also X repre- 
sents any finite cardinal number, so that {X}, {p}, 
and {v} are only different notations for the same 
aggregate o'f all finite numbers, we have to show 
that 

(Ox, )} no {X|. 

Let us denote ^ + v by p ; then p takes all the 
numerical values 2, 3, 4, . . ., and there are in all 
p i elements (/A, v) for which /j. + v = p, namely : 



(I.p-l), (2,p-2),..., (p-I, I). 

In this sequence imagine first the elemental, i), 
for which p = 2, put, then the two elements for 
which p 3, then the three elements for which 
p = 4, and so on. Thus we get all the elements 
(fji, v) in a simple series : 



and here, as we easily see, the element (/*, v) comes 
at the Xth place, where 



(9) X 

The variable X takes every numerical value i, 2, 3i 
, . ., once, Consequently, by means of (9), a 



io8 THE FOUNDING OF THE THEORY 

reciprocally univocal relation subsists between the 
aggregates {j/} and {(/*, v)}. 

[495] If both sides of the equation (8) are multi- 
plied by N , we get No 8 = a = Noj and, by repeated 
multiplications by N O , we get the equation, valid 
for every finite cardinal number v : 

(10) N " = No- 

The theorems E and A of 5 lead to this theorem 
on finite aggregates : 

C; Every finite aggregate E is such that it is 
equivalent to none of its parts. 

This theorem stands sharply opposed to the 
following one for transfmite aggregates : 

D. Every transfinite aggregate T is such that it 
has parts T 1 which are equivalent to it. 

Proof. By theorem A of this paragraph there is 
a part S = {t v } of T with the cardinal number . 
Let T = (S, U), so that U is composed of those 
elements of T which are different from the elements 
/,. Let us put S 1 = {^ +1 } ) ^-(Si, U) ; then T x is 
a part of T, and, in fact, that part which arises out 
of T ff we leave out the single element t^. Since 
S PVJ S lf by theorem B of this paragraph, and 
UroU, we have, by i, T no T x . 

In these theorems C and D the essential differ- 
ence between finite and transfinite aggregates, to 
which I referred in the year 1 877, in volume Ixxxiv 
[1878] of Crelle's Journal, p. 242, appears in the 
clearest way. 

After we have introduced the least transfinite 



OF TRANSFINITE NUMBERS 109 

cardinal number NO and derived its properties that 
lie the most readily to hand, the question arises 
as to the higher cardinal numbers and how they 
proceed from . We shall show that the trans- 
finite cardinal numbers can be arranged according 
to their magnitude, and, in this order, form, like 
the finite numbers, a "well-ordered aggregate" in 
an extended sense of the words. Out of pro- 
ceeds, by a definite law, the next greater cardinal 
number & lt out of this by the same law the next 
greater N 2 , and so on. But even the unlimited 
sequence of cardinal numbers 



does not exhaust the conception of transfinite 
cardinal number. We will prove the existence of 
a cardinal number which we denote by M and 
which shows itself to be the next greater to all 
the numbers N V ; out of it proceeds in the same 
way as M X out of M a next greater M W+ I, and so on, 
without end. 

[496] To every transfinite cardinal number a 
there is a next greater proceeding out of it accord- 
ing to a unitary law, and also to every unlimitedly 
ascending well-ordered aggregate of transfinite 
cardinal numbers, {a}, there is a next greater pro- 
ceeding out of that aggregate in a unitary way. 

For the rigorous foundation of this matter, dis- 
covered in 1882 and exposed in the pamphlet 
Grundlagen einer allgemeinen Mannichfaltigkeits- 
lehre (Leipzig, 1883) and in volume xxi of the 



no THE FOUNDING OF THE THEORY 

Mathematiscke Annalen, we make use of the so- 
called ' ' ordinal types " whose theory we have to 
set forth in the following paragraphs. 

7 

The Ordinal Types of Simply Ordered 
Aggregates 

We call an aggregate M ' ' simply ordered " if a 
definite ' ' order of precedence " (Rangordnung) rules 
over its elements m t so that, of every two elements 
m t and m^ one takes the "lower" and the other the 
' ' higher " rank, and so that, if of three elements m 1} 
7# 2 , and m& m v say, is of lower rank than m t , and 
m% is of lower rank than m t , then m is of lower 
rank than m s . . 

The relation of two elements m^ and m v in which 
m i has the lower rank in the given order of pre- 
cedence and *# a the higher, is expressed by the 
formulae : 

(i) nti<m z , m^m v 

Thus, for example, every aggregate P of points 
defined on a straight line is a simply ordered 
aggregate if, of every two points / x and p % belong- 
ing to it, that one whose co-ordinate (an origin and 
a positive direction having been fixed upon) is the 
lesser is given the lower rank. 

It is evident that one and the same aggregate can 
be "simply ordered" according to the most different 
laws. Thus, for example, with the aggregate R of 



OP TRANSPINITE NUMBERS 1 I 1 

all positive rational numbers // (where/ and q are 
relatively prime integers) which are greater than o 
and less than I, there is, firstly, their "natural" 
order according to magnitude ; then they can be 
arranged (and in this order we will denote the 
aggregate by R ) so that, of two numbers pj^ and 
A/?J f r which the sums # 1 + g 1 and / a + a have 
different values, that number for which the corre- 
sponding sum is less takes the lower rank, and, if 
A + Q\ ~Pi + a> then the smaller of the two rational 
numbers is the lower. [497] In this order of 
precedence, our aggregate, since to one and the 
same value of p + q only a finite number of rational 
numbers p\q belongs, evidently has the form 

RO = (^I. *2 ..... r m .. .) = (! fc, -h fr, *, *, T. *. ) 
where 



Always, then, when we speak of a "simply 
ordered' 1 aggregate M, we imagine laid down a 
definite order or precedence of its elements, in the 
sense explained above. 

There are doubly, triply, i/-ply and a-ply ordered 
aggregates, but for the present we will not consider 
them. So in what follows we will use the shorter 
expression "ordered aggregate" when we mean 
1 ' simply ordered aggregate. " 

Every ordered aggregate M has a definite "ordinal 
type," or more shortly a definite "type," which we 
will denote by 

(2) M. 



ii2 THE FOUNDING OF THE THEORY 

By this we understand the general concept which 
results from M if we only abstract from the nature 
of the elements m, and retain the order of precedence 
among them. Thus the ordinal type M is itself an 
ordered aggregate whose elements are units which 
have the same order of precedence amongst one 
another as the corresponding elements of M, from 
which they are derived by abstraction. 

We call two ordered aggregates M and N 
"similar" (ahnlich) if they can be put into a bi- 
univocal correspondence with one another in such 
a manner that, if m^ and /# a are any two elements 
of M and x and a the corresponding elements of N, 
then the relation of rank of ; x to m z in M is the 
same as that of x to a in N. Such a correspond- 
ence' of similar aggregates we call an "imaging" 
(Abbildung) of these aggregates on one another. In 
such an imaging, to every part which obviously 
also appears as an ordered aggregate Mj of M 
corresponds a similar part N x of N. 

We express the similarity of two ordered aggre- 
gates M and N by the formula : 

(3) M rvj N. 

Every ordered aggregate is similar to itself. 

If two ordered aggregates are similar to a third, 
they are similar to one another. 

[498] A simple consideration shows that two 
ordered aggregates have the same ordinal type if, 
and only if, they are similar, so that, of the two 
formulae 



OF TRANSFINITE NUMBERS 113 

(4) M = N, MooN, 

one is always a consequence of the other. 

If, with an ordinal type M we also abstract from 
the order of precedence of the elements, we get ( i) 

the cardinal number M of the ordered aggregate M, 
which is, at the same time, the cardinal number of 
the ordinal type M. From M = N always follows 
M = N, that is to say, ordered aggregates of equal 
types always have the same power or cardinal 
number ; from the similarity of ordered aggregates 
follows their equivalence. On the other hand, two 
aggregates may be equivalent without being similar. 
We will use the small letters of the Greek alphabet 
to denote ordinal types. If a is an ordinal type, 
we understand by 

(5) 5 

its corresponding cardinal number. 

The ordinal types of finite ordered aggregates 
offer no special interest. For we easily convince 
ourselves that, for one and the same finite cardinal 
number v t all simply ordered aggregates are similar 
to one another, and thus have one and the same 
type. Thus the finite simple ordinal types are 
subject to the same laws as the finite cardinal 
numbers, and it is allowable to use the same signs 
I, 2, 3, . . ., j/,... for them, although they are 
conceptually different from the cardinal numbers. 
The case is quite different with the transfinite 
ordinal types ; for to one and the same cardinal 



H4 THE FOUNDING OF THE THEORY 

number belong innumerably many different types of 
simply ordered aggregates, which, in their totality, 
constitute a particular ' ' class of types " (Typenclasse). 
Every one of these classes of types is, therefore, 
determined by the transfinite cardinal number a 
which is common to all the types belonging to the 
class. Thus we call it for short the class of types [a]. 
.That class which naturally presents itself first to us, 
and whose complete investigation must, accordingly, 
be the next special aim of the theory of transfinite 
aggregates, is the class of types [N O ] which embraces 
all the types with the least transfinite cardinal 
number M O . From the cardinal number which 
determines the class of types [a] we "have to dis- 
tinguish that cardinal number a' which for its part 
[499] & determined by the class of types [a]. The 
latter is the cardinal number which ( i) the class 
[a] has, in so far as it represents a well-defined 
aggregate whose elements are all the types a with 
the cardinal number a. We will see that a' is 
different from a, and indeed always greater than a. 

If in an ordered aggregate M all the relations of 
precedence of its elements are inverted, so that 
' ' lower " becomes ' ' higher " and ' ' higher " becomes 
"lower" everywhere, we again get an ordered 
aggregate, which we will denote by 

(6) *M 

and call the "inverse" of M. We denote the 
ordinal type of *M, if a= M, by 

(7) 



OF TRANSFINITE NUMBERS 115 

It may happen that *a = a, as, for example, in the 
case of finite types or in that of the type of the 
aggregate of all rational numbers which are greater 
than o and less than I in their natural order of 
precedence. This type we will investigate under 
the notation rj. 

We remark further that two similarly ordered 
aggregates can be imaged on one another either in 
one manner or in many manners ; in the first case 
the type in question is similar to itself in only one 
way, in the second case in many ways. Not only 
all finite types, but the types of transfinite c ' well- 
ordered aggregates," which will occupy us later 
and which we call transfinite "ordinal numbers," 
are .such that they allow only a single imaging on 
themselves. On the other hand, the type r\ is 
similar to itself hi an infinity of ways. 

We will make this difference clear by two simple 
examples. By & we understand the type of a well- 
ordered aggregate 

\ e \t e z> ' ' ^vt )> 

in which 

e v -^ e v+i> 

and where v represents all finite cardinal numbers in 
turn. Another well-ordered aggregate 



with the condition 

of the. same type to can obviously only be imaged 



n6 THE FOUNDING OF THE THEORY 

on the former in such a way that e v and f v are 
corresponding elements. For e lt the lowest element 
in rank of the first, must, in the process of imaging, 
be correlated to the lowest element /[ of the second, 
the next after e^ in rank (<? 2 ) to/ 2 , the next afterj^, 
and so on. [5] Every other bi-univocal corre- 
spondence of the two equivalent aggregates {e v } and 
{f v } is not an "imaging" in the sense which we 
have fixed above for the theory of types. 

On the other hand, let us take an ordered 
aggregate of the form 

K), 

where v represents all positive and negative finite 
integers, including o, and where likewise 



This aggregate has no lowest and no highest 
element in rank. Its type is, by the definition of 
a sum given in 8, 



It is similar to itself in an infinity of ways. For 
let us consider an aggregate of the same type 

{/*}, 
where 



Then the two ordered aggregates can be so imaged 
on one another that, if we understand by v ' a 
definite one of the numbers i/, to the element e^ of 



OF TRANSFINITE NUMBERS 117 

the first the element f^'^ of the second corresponds. 
Since j/ ' is arbitrary, we have here an infinity of 
imagings. 

The concept of ' ' ordinal type " developed here, 
when it is transferred in like manner to "multiply 
ordered aggregates," embraces, in conjunction with 
the concept of "cardinal number" or "power" 
introduced in I, everything capable of being 
numbered (Anzahlmassige) that is thinkable, and 
in this sense cannot be further generalized. It 
contains nothing arbitrary, but is the natural ex- 
tension of the concept of number. It deserves to 
be especially emphasized that the criterion of 
equality (4) follows with absolute necessity from 
the concept of ordinal type and consequently 
permits of no alteration. The chief cause of the 
grave errors in G. Veronese's Grundziige der 
Geometrie (German by A. Schepp, Leipzig, 1894) 
is the non-recognition of this point. 

On page 30 the "number (Ansahl oder Zahl) 
of an ordered group " is defined in exactly the same 
way as what we have called the "ordinal type of 
a simply ordered aggregate" (Zur Lehre vom 
Transfiniten, Halle, 1890, pp. 68-75 ; reprinted 
from the Zeitschr. fiir Philos. und philos. Kritik 
for 1887). [501] But Veronese thinks that he 
must make an addition to the criterion of equality. 
He says on page 31: "Numbers whose units 
correspond to one another uniquely and in the 
same order and of which the one is neither a part 
of the other nor equal to a part of the other are 



n8 THE FOUNDING OF THE THEORY 

equal."* This definition of equality contains a 
circle and thus is meaningless. For what is 
the meaning of ' ' not equal to a part of the 
other " in this addition ? To answer this question, 
we must first know when two numbers are equal 
or unequal. Thus, apart from the arbitrariness 
of his definition of equality, it presupposes a 
definition of equality, and this again presupposes 
a definition of equality, in which we must know 
again what equal and unequal are, and so on ad 
infinitum. After Veronese has, so to speak, given 
up of his own free will the indispensable foundation 
for the comparison of numbers, we ought not to 
be surprised at the lawlessness with which, later 
on, he operates with his pseudo-transfinite numbers, 
and ascribes properties to them which they cannot 
possess simply because they themselves, in the 
form imagined by him, have no existence except 
on paper. Thus, too, the striking similarity of his 
"numbers " to the very absurd "infinite numbers " 
in Foritenelle's Gtomttrie de Plnfini (Paris, 1727) 
becomes comprehensible. Recently, W. Killing 
has given welcome expression to his doubts con- 
cerning the foundation of Veronese's book in the 
Index lectionum of the Munster Academy for 1895- 



* In the original Italian edition (p. 27) this passage runs : " Numeri 
le unitfc, deiquali si corrispondono univocamente e nel medesimo ordine, 
e di cui 1' uno non e parte o uguale ad una parte deli' altro, sono uguali." 

t [Veronese replied to this in Math. Arm., vol. xlvii, 1897, pp. 423- 
432. Cf. Killing, ibid., vol. xlviii, 1897, pp. 425-432.] 



OF TRANSFINITE NUMBERS 119 

8 
Addition and Multiplication of Ordinal Types 

The union-aggregate (M, N) of two aggregates 
M and N can, if M and N are ordered, be conceived 
as an ordered aggregate in which the relations of 
precedence of the elements of M among themselves 
as well as the relations of precedence of the elements 
of N among themselves remain the same as in M 
or N respectively, and all elements of M have a 
lower rank than all the elements of N. If M' and 
N' are two other ordered aggregates, M oo M' and 
NooN', [502] then (M, N) c\> (M', N')T'so the 
ordinal type of (M, N) depends only on the ordinal 
types M = a and N /3. Thus, we define: 



(1) a+/8-(M, N). 

In the sum a + /3 we call a the "augend" and $ the 
"addend." 

For any three types we easily prove the associa- 
tive law : 

(2) a + (/9 + y) = (a + /3) + y. 

On the other hand, the commutative law is not 
valid, in general, for the addition of types. We 
see this by the following simple example. 

If a) is the type, already mentioned in 7, of 
the well-ordered aggregate 



120 THE FOUNDING Of THE THEORY 



then i + eo is not equal to + I. For, if /is a new 
element, we have by (i) : 

i+a^/E), 
+i=(E77). 
But the aggregate 

(/,E) = (/*i,' 2 ,. ,*, ...) 
is similar to the aggregate E, and consequently 



On the contrary, the aggregates E and (E, /) are 
not similar, because the first has no term which is 
highest in rank, but the second has the highest 
term/! Thus w+ I is different from o>= I +. 

Out of two ordered aggregates M and N with 
the types a and we can set up an ordered 
aggregate S by substituting for every element n of 
N an ordered aggregate M w which has the same 
type a as M, so that 

(3) M* = a; 
and, for the order of precedence in 

(4) S = {MJ 

we make the two rules : 

(1) Every two elements of S which belong to 
one and the same aggregate M M are to retain in 
S the same prder of precedence as in M w ; 

(2) Every two elements of S which belong to two 
different aggregates M W1 and M^ have the same 
relation of precedence as x and a have in N. 



OF TRANSFtNITE NUMBERS 121 

The ordinal type of S depends, as we easily see, 
only on the types a and f$ \ we define 

(5) -/3 = S. 

[503] In this product a is called the " multiplicand " 
and /3 the "multiplier." 

In any definite imaging of M on M w let m n be the 
element of M w that corresponds to the element m 
of M; we can then also write 

(6) S -{}. 

Consider a third ordered aggregate P = {p} with 
the ordinal type P = y, then, by (5), 

a . 18= {*,}, /3 . y = {^}, (a . /3) . y = {K", 



But the two ordered aggregates {(#z rt ) } and (fi^)} 
are similar, and are imaged on one another if we 
regard the elements (?), and MfnA as correspond- 

ing, Consequently, for three types o, /8, and y 
the associative law 



subsists. From (i) and (5) follows easily the dis- 
tributive law 

(8) a.(/3+y) = a./3+a.y; 

but only in this form, where the factor with two 
terms is the multiplier. 

On the contrary, in the multiplication of types 
as in their addition, the commutative law is not 



122 THE FOUNDING OF THE THEORY 

generally valid. For example, 2 . o> and w . 2 are 
different types ; for, by (5), 



2.u = (e lt f 1 ] e z ,f z ; ...; e vi f v \ . . .) = ; 
while 

a) . 2 = (*!, * a , ...,,...;/!., / z , ...,/...) 

is obviously different from <. 

If we compare the definitions of the elementary 
operations for cardinal numbers, given in 3, with 
those established here for ordinal types, we easily 
see that the cardinal number of the sum of two 
types is equal to the sum of the cardinal numbers 
of the single types, and that the cardinal number 
of the product of two types is equal to the pro- 
duct of the cardinal numbers of the single types. 
Every equation between ordinal types which pro- 
ceeds from the two elementary operations remains 
correct, therefore, if we replace in it all the types 
by their cardinal numbers. 

[504] 9 

The Ordinal Type ^ of the Aggregate R of all 
Rational Numbers which are Greater than 
o and Smaller than i, in their Natural 
Order of Precedence 

By R we understand, as in 7, the system of 
all rational numbers p\q (p and g being relatively 
prime) which >o and <i, in their natural order 
of precedence, where the magnitude of a number 



OF TRANSFIN1TR NUMBERS 123 

determines its rank. We denote the ordinal type 
of R by 17 : 

(1) fl=R. 

But we have put the same aggregate in another 
order of precedence in which we call it R . This 
order is determined, in the first place, by the 
magnitude of p + g, and in the second place for 
rational numbers for which f + g has the same value 
-r-by the magnitude of p\q itself. The aggregate 
R is a well-ordered aggregate of type : 

(2) Ra-Oi, r a> . . ., r f , . . .), where r,<y v +i, 

(3) RO = *>- 

Both R and R have the same cardinal number 
since they only differ in the order of precedence 
of their elements, and r since we obviously have 

R^N,,, we also have 

(4) R = ij = Mo . 

Thus the type n belongs to the class of types [M O ]. 

Secondly, we remark that in R there is neither 
an element which is lowest in rank nor one which 
is highest in rank. Thirdly, R has the property 
that between every two of its elements others lie. 
This property we express by the words : R is 
' ' everywhere dense " (tiberalldichf). 

We will now 'show that these three properties 
characterize the type 17 of R, so that we have the 
following theorem : 



124 THE FOUNDING OF THE THEORY 

If we have a simply ordered aggregate M such 
that _ 

(a) M - N ; 

(b) M has no element which is lowest in rank, 

and no highest ; 

(c) M is everywhere dense ; 
then the ordinal type of M is ^ : 



Proof. Because of the condition (a), M can be 
brought into the form [505] of a well-ordered 
aggregate of type w ; having fixed upon such a 
form, we denote it by M and put 

(5) M = (/!, m^ . . ., m v , . . .). 
We have now to show that 

(6) MroR; 

that is to say, we must prove that M can be imaged 
on R in such a way that the relation of precedence 
of any and every two elements in M is the same 
as that of the two corresponding elements in R. 

Let the element r 1 in R be correlated to the 
element m t in M. The element r z has a definite 
relation of precedence to r in R. Because of the 
condition (.), there are infinitely many elements 
* of M which have the same relation of precedence 
in M to m : as r z to r in R ; of them we choose 
that one which has the smallest index in M 0> let it 
be m t and correlate it to r z . The element r a has 
in R definite relations of precedence to r and r t ; 
because of the conditions () and (c) there is an 



OF TRANSFINITE NUMBERS 125 

infinity of elements m v of M which have the same 
relation of precedence to m and m t in M as r 8 to ^ 
and r z to R ; of them we choose that let it be m l 
which has the smallest index in M 0) and correlate 
it to r 3 . According to this law we imagine the 
process of correlation continued. If to the v 
elements 

r i> r %> r a> i r v 
of R are correlated, as images, definite elements 



which have the same relations of precedence amongst 
one another in M as the corresponding elements in 
R, then to the element r y+i of R is to be correlated 
that element m tv+l of M which has the smallest 
index in M of those which have the same relations 
of precedence to 



in M as r v+1 to r l} r t , . . , , r v in R. 

In this manner we have correlated definite 
elements m lv of M to all the elements r v of R, and 
the elements mi v have in M the same order of pre- 
cedence as the corresponding elements r v in R. But 
we have still to show that the elements m lv include 
all the elements m v of M, or, what is the same 
thing, that the series 

*> 'a> 's> > l v) 
[506] is only a permutation of the series 

i, 2, 3, ...',... 



126 THE FOUNDING OF THE THEOR\ 

We prove this by a complete induction : we will 
show that, if the elements m lt m z , . . . , m v appear 
in the imaging, that is also the case with the 
following element m v+1 . 

Let X be so great that, among the elements 



the elements 

; 1( 7. 2 , . . ., m v , 

which, by supposition, appear in the imaging, are 
contained. It may be that also ;/VH is found 
among them ; then m v+l appears in the imaging. 
But if m v+l is. not among the elements 



then ;+! has with respect to these elements a 
definite ordinal position in M ; infinitely many 
elements in R have the same ordinal position in R 
with respect to r lt r zt , , ., r^ amongst which let 
;- A+0 . be that with the least index in R . Then m v+l 
has, as we can easily make sure, the same ordinal 
position with respect to 



in M as r^ ff has with respect to 



in R. Since m v m 2 , . . . , m v have already appeared 
in the imaging, w,, +1 is that element with the smallest 
index in M which has this ordinal position with 
respect to 



OF TRANSFINITE NUMBERS 127 
Consequently, according to our law of correlation, 



Thus, in this case too, the element ;+! appears in 
the imaging, and r^ +a . is the element of R which is 
correlated to it. 

We see, then, that by our manner of correlation, 
the whole aggregate M is imaged on the -whole 
aggregate R ; M and R are similar aggregates, 
which was to be proved. 

From the theorem which we have just proved 
result, for example, the following theorems : 

[507] The ordinal type of the aggregate of all 
negative and positive rational numbers, including 
zero, in their natural order of precedence, is y. 

The ordinal type of the aggregate of all rational 
numbers which are greater than a and less than #, 
in their natural order of precedence, where a and b 
are any real numbers, and a < d, is 97. 

The ordinal type of the aggregate of all real alge- 
braic numbers in their natural order of precedence is q. 

The ordinal type of the aggregate of all real alge- 
braic numbers which are greater than a and less 
than b, in their natural order of precedence, where 
a and b are any real numbers and a < d, is y. 

For all these ordered aggregates satisfy the three 
conditions required in our theorem for M (see 
Crelle's Journal, vol. Ixxvii, p, 258).* 

If we consider, further, aggregates with the types 
according to the definitions given in 8 written 

[* Cf. Section V of the Introduction.] 



128 THE POUNDING OF THE THEORY 



(1+1)1, ( r +?+i>/> we find that 
those three conditions are also fulfilled with them. 
Thus we have the theorems : 

(?) 1 + 1 = 1, 

(8) 11 = 1, 

(9) (i+l)l = 1, 

(10) (1+1)1 = 1, 
(n) (1+11+1)*] = ,). 

The repeated application of (7) and (8) gives for 
every finite number v : 

(12) i-v = ri, 

(13) 1" = ')- 

On the other hand we easily see that, for v> i, the 
types 1+1, 1+1, v.i, i + >7+i are different both 
from one another and from ;. We have 

(14) . 17+1 + 17 = 91 

but >; + )/ + 17, for v> I, is different from y. 
Finally, it deserves to be emphasized that 

(15) **, = *,. 

[508] 10 

The Fundamental Series contained in a 
Transfinite Ordered Aggregate 

Let us consider any simply ordered' transfinite 
aggregate M. Every part of M is itself, an ordered 
aggregate. For the study of the type M, those 



OF TRANSFINITE NUMBERS 129 

parts of M which have the types w and *< appear to 
be especially valuable ; we call them " fundamental 
series of the first order contained in M," and the 
former of type o> we call an "ascending" series, 
the latter of type *o> a ' c descending " one. Since 
we limit ourselves to the consideration of funda- 
mental series of the first order (in later investiga- 
tions fundamental series of higher order will also 
occupy us), we will here simply call them "funda- 
mental series. " Thus an ' ' ascending fundamental 
series " is of the form 

(1) {a v }> where a, -<,+!; 

a ' ' descending fundamental series " is of the form 

(2) {}, where b v > b v+l . 

The letter >/, as well as K, A, and /*, has everywhere 
in our considerations the signification of an arbitrary 
finite cardinal number or of a finite type (a finite 
ordinal number). 

We call two ascending fundamental series {a v } and 
(af v \ in M "coherent" (ftusammengehorig), in signs 

(3) M II W, 

if, for every element a v there are elements a\ such 
that 



and also for every element a' v there are elements 
such that 



130 THE FOUNDING OF THE THEORY 

Two descending fundamental series {b v } and {'} 
in M are said to be "coherent," in signs 

(4) {*,}\\{V,}, 

if for every element b v there are elements b\ such 
that 



and for every element b' v there are elements b^ such 
that 

*',>V 

An ascending fundamental series {a v } and a 
descending one {d v } are said to be "coherent," in 
signs 

[509] (5) M\\M, 

if (a) for all values of v and /*, 



and (6) in M exists at most one (thus either only 
one or none at all) element m Q such that, for all v's, 



Then we have the theorems : 

A. If two fundamental series are coherent to a 
third, they are also coherent to one another. 

B. Two fundamental series proceeding in the 
same direction of which one is part of the other are 
coherent. 

If there exists in M an element m which has 



OF TRANSFINITE NUMBERS 131 

such a position with respect to the ascending funda- 
mental series {#} that : 
(a) for every v 

a* -< M O , 

() for every element m of M that precedes m 
there exists a certain number j/ such that 

a v >- w, for v> 



then we will call m a "limiting element (Grenz- 
elemenf) of {a v } in M " and also a ' ' principal element 
(Hauptelemenf) of M." In the same way we call 
m Q a " principal element of M " and also " limiting 
element of {} in M" if these conditions are 
satisfied : 

(a) for every v 

b* > W , 

(A) for every element m of M that follows ?;z 
exists a certain number j/ such that 

*# for i/ 



A fundamental series can never have more than 
one limiting element in M ; but M has, in general, 
many principal elements, 

We perceive the truth of the following theorems : 

C. If a fundamental series has a limiting element 
in M, all fundamental series coherent to it have the 
same limiting element in M. 

D. If two fundamental series (whether proceeding 
in the same or in opposite directions) have one and 
the same limiting element in M, they are coherent. 



132 1HE FOUNDING OF THE THEORY 

If M and M' are two similarly ordered aggregates, 
so that 

(6) M = M', 

and we fix upon any imaging of the two aggregates, 
then we easily see that the following theorems 
hold: 

[510] E. To every fundamental series in M 
corresponds as image a fundamental series in M', 
and inversely ; to every ascending series an ascending 
one, and to every descending series a descending 
one; to coherent fundamental series in M corre- 
spond as images coherent fundamental series in M', 
and inversely. 

F. If to a fundamental series in M belongs a 
limiting element in M, then to the corresponding 
fundamental series in M' belongs a limiting element 
in M', and inversely ; and these two limiting 
elements are images of one another in the imaging. 

G. To the principal elements of M correspond as 
images principal elements of M', and inversely. 

If an aggregate M consists of principal elements, 
so that every one of its elements is a principal 
element, we 'call it an " aggregate which is dense 
in itself (insichdichte Menge)* If to every funda- 
mental series In M there is a limiting element in M, 
we call M a "dosed (abgeschlossene) aggregate." 
An aggregate which is both "dense in itself" and 
"closed" is called a "perfect aggregate." . If an 
aggregate has one of these three predicates, every 
similar aggregate 'has the same predicate ; thus 



OF TRANSFINITE NUMBERS 133 

these predicates can also be ascribed to the corre- 
sponding ordinal types, and so there are ' ' types 
which are dense in themselves," "closed types," 
1 ' perfect types, " and also ' ' everywhere-dense 

types" (9). 

For example, rj is a type which is "dense in 
itself," and, as we showed in 9, it is also "every- 
where-dense," but it is not "closed." The types 
to and *a> have no principal elements, but <a + v and 
v-\-*(o each have a principal element, and are 
' ' closed " types. The type o> . 3 has two principal 
elements, but is not ' ' closed " ; the type o> . 3 4- v 
has three principal elements, and is "closed," 



The Ordinal Type of the Linear 
Continuum X 

We turn to the investigation of the ordinal type 
of the aggregate X = [x] of all real numbers x, such 
that X>.Q and <!, in their natural order of pre- 
cedence, so that, with any two of its elements x 



if x<x t . 
Let the notation for this type be 
(i) X-0. 

[511] From the elements of the theory of rational 
and irrational numbers we know that every funda- 
mental series {x v } in X has a limiting element x^ in 
X, and that also, inversely, every element x of X 



i 3 4 THE FOUNDING OF THE THEORY 

is a limiting element of coherent fundamental series 
in X. Consequently X is a "perfect aggregate" 
and is a "perfect type." 

But is not sufficiently characterized by that ; 
besides that we must fix our attention on the 
following property of X. The aggregate X contains 
as part the aggregate R of ordinal type rj investi- 
gated in 9, and in such a way that, between any 
two elements X Q and x v of X, elements of R lie. 

We will now show that these properties, taken 
together, characterize the ordinal type 6 of the linear 
continuum X in an exhaustive manner, so that we 
have the theorem : 

If an ordered aggregate M is such that (a) it is 
"perfect," and (b) in it is contained an aggregate S 

with the cardinal number S^NO and which bears 
such a relation to M that, between any two elements 
w and m t of M elements of S lie, then M = 0. 

Proof. If S had a lowest or a highest element, 
these elements, by (), would bear the same character 
as elements of M ; we could remove them from S 
without S losing thereby the relation to M ex- 
pressed in (b}. Thus, we suppose that S is without 
lowest or highest element, so that, by 9, it has 
the ordinal type rj. For since S is a part of M, 
between any two elements s and s 1 of S other 
elements of S must, by (b\ lie. Besides, by () we 
have S = N O . Thus the aggregates S and R are 
' ' similar " to one another. 

SroR. 



OF TRANSFJN1TE NUMBERS 135 

We fix on any ' ' imaging " of R on S, and assert 
that it gives a definite " imaging " of X on M in the 
following manner : 

Let all elements of X which, at the same time, 
belong to the aggregate R correspond as images to 
those elements of M which are, at the same time, 
elements of S and, in the supposed imaging of 
R on S, correspond to the said elements of R. 
But if X Q is an element of X which does not belong 
to R, x may be regarded as a limiting element of 
a fundamental series {x v } contained in X, and this 
series can be replaced by a coherent fundamental 
series {r Ki/ } contained in R. To this [512] corre- 
sponds as image a fundamental series {j Xv } in S and 
M, which, because of (a), is limited by an element 
m Q of M that does not belong to S (F, 10). Let 
this element m of M (which remains the same, by 
E, C, and D of 10, if the fundamental series 
{x v } and {r Kv } are replaced by others limited by the 
same element x in X) be the image of x ti in X. 
Inversely, to every element # of M which does not 
occur in S belongs a quite definite element x of X 
which does .not belong to R and of which m is the 
image. 

In this manner a bi-univocal correspondence 
between X and M is set up, and we have now 
to show that it gives an ' ' imaging " of these 
aggregates. 

This is, of course, the case for those elements of 
X which belong to R, and for those elements of M 



136 TRANSFINITE NUMBERS 

which belong to S. Let us compare an element r 
of R with an element .% of X which does not belong 
to R ; let the corresponding elements of M be s 
and T# O . If r<x 0i there is an ascending funda- 
mental series {r Kv } t which is limited by X Q and, from 
a certain V Q on, 

r<r Kv for V >v . 

The image of {?} in M is an ascending funda- 
mental series {s^}, which will be limited by an m 
of M, and we have ( 10) s^ -< m for every v, and 
s < J X( , for i/^/ . Thus ( 7) s -< m . 

If r>x Q , we conclude similarly that s ^>- m . 

Let us consider, finally, two elements x^ and x' Q 
not belonging to R and the elements m Q and m' Q 
corresponding to them in M ; then we show, by 
an analogous consideration, that, if x Q <x f Q> then 
w < w' . 

The proof of the similarity of X and M is now 
finished, and we thus have 



HALLE, March 1895, 



[207] CONTRIBUTIONS TO THE 
FOUNDING OF THE THEORY OF 
TRANSFINITE NUMBERS 

(SECOND ARTICLE) 

12 

Weil-Ordered Aggregates 

AMONG simply ordered aggregates "well-ordered 
aggregates " deserve a special place ; their ordinal 
types, which we call ' ' ordinal numbers, " form the 
natural material for an exact definition of the 
higher transfinite cardinal numbers or powers, a 
definition which is throughout conformable to that 
which was given us for the least transfinite cardinal 
number Aleph-zero by the system of all finite 
numbers v ( 6). 

We call a simply ordered aggregate F ( 7) 
' ' well-ordered " if its elements f ascend in a definite 
succession from a lowest f^ in such a way that : 

I. There is in F an element /i which is lowest in 
rank. 

II. If F' is any part of F and if F has one or 
many elements of higher rank than all elements 
of F', then there is an element /' of F which 
follows immediately after the totality F', so 



138 THE FOUNDING OF THE THEORY 

that no elements in rank between /' and F' occur 
in F.* 

In particular, to every single element f of F, if 
it is not the highest, follows in rank as next higher 
another definite element f ; this results from the 
condition II if for F' we put the single element f. 
Further, if, for example, an infinite series of con- 
secutive elements 



is contained in F in such a way, however, that there 
are also in F elements of [208] higher rank than all 
elements e^\ then, by the second condition, putting 
for F' the totality [e^} } there must exist an element 
f such that not only 

f > * (I 

for all values of v, but that also there is no element 
g in F which satisfies the two conditions 



for all values of v . 

Thus, for example, the three aggregates 



where 



* This definition of "well-ordered aggregates," apart from the 
wording, is identical with that- which was introduced in vol. xxi of the 
Math. Ann., p. 548 (Gnmdlagm einer allgemeinen Mannichfaltig- 
Juitslehre, p. 4). [See Section VII of the Introduction.] 



OF TRANS FINITE NUMBERS 139 

are well-ordered. The two first have no highest 
element, the third has the highest element c d ; in 
the second and third b immediately follows all 
the elements a v) in the third ^ immediately follows 
all the elements a v and b^. 

In the following we will extend the use of the 
signs -< and >- , explained in 7, and there used 
to express the ordinal relation of two elements, to 
groups of elements, so that the formulae 

M<;N, 
M>N 

are the expression for the fact that in a given order 
all the elements of the aggregate M have a lower, 
or higher, respectively, rank than all elements of 
the aggregate N. 

A. Every part F x of a well-ordered aggregate F 
has a lowest element. 

Proof. If the lowest element^ of F belongs to 
F t , then it is also the lowest element of F x . In 
the other case, let F' be the totality of all elements 
of F' which have a lower rank than all elements F 1? 
then, for this reason, no element of F lies between 
F' and F r Thus, if/' follows (II) next after F', 
then it belongs necessarily to F and here takes the 
lowest rank. 

B. If a simply ordered aggregate F is such that 
both F and every one of its parts have a lowest 
element, then F is a well-ordered aggregate. 

[209] Proof. Since F has a lowest element, 
the condition I is satisfied. Let F' be a part of F 



140 THE FOUNDING OF THE THEORY 

such that there are in F one or more elements 
which follow F' ; let F 1 be the totality of all these 
elements and f the lowest element of F lt then 
obviously/' is the element of F which follows next 
to F'. Consequently, the condition II is also satis- 
fied, and therefore F is a well-ordered aggregate. 

C. Every part F" of a well-ordered aggregate F 
is also a well-ordered aggregate. 

Proof. By theorem A, the aggregate F' as well 
as every part F" of F' (since it is also a part of F) 
has a lowest element ; thus by theorem B, the 
aggregate F' is well-ordered. 

D. Every aggregate G which is similar to a well- 
ordered aggregate F is also a well-ordered aggregate. 

Proof. If M is an aggregate which has a lowest 
element, then, as immediately follows from the 
concept of similarity ( 7), every aggregate N 
similar to it has a lowest element Since, now, 
we are to have G \j F, and F has, since it is a 
well-ordered aggregate, a lowest element, the same 
holds of G. Thus also every part G' of G has a 
lowest element ; for in an imaging of G on F, to 
the aggregate G' corresponds a part F' of F as 
image, so that 

G'ooF'. 

But, by theorem A, F' has a lowest element, and 
therefore also G' has. Thus, both G and every 
part of G have lowest elements. By theorem B, 
consequently, G is a well-ordered aggregate. 

E. If in a well-ordered aggregate G, in place of 



OF TRANSFINITE NUMBERS 14 1 

its elements g well-ordered aggregates are sub- 
stituted in such a way that, if F f and F^ are the 
well-ordered aggregates which occupy the places 
of the elements g and g' and ff-^g 1 ', then also 
Fg. -< F^, then the aggregate H, arising by com- 
bination in this mariner of the elements of all the 
aggregates F^, is well-ordered. 

Proof. Both H and every part H a of H have 
lowest elements, and by theorem B this characterizes 
H as a well-ordered aggregate. For, if g^ is the 
lowest element of G, the lowest element of F^. is 
at the same time the lowest element of H. If, 
further, we have a part H 1 of H, its elements 
belong to definite aggregates F^. which form, when 
taken together, a part of the well-ordered aggre- 
gate {Fj.}, which consists of the elements F^. and 
is similar to the aggregate G. If, say, F f is the 
lowest element of this part, then the lowest element 
of the part of H t contained in F^ is at the same 
time the lowest element of H, 



[210] 13 

The Segments of Well-Ordered Aggregates 

If f is any element of the well-ordered aggre- 
gate F which is different from the initial element f^ 
then we will call the aggregate A of all elements 
of F Which precede /a "segment (Abschnitf) of F," 
or, more fully, ' ' the segment of F which is defined 
by the element./" On the other hand, the aggre- 



142 THE FOUNDING OF THE THEORY 

gate R of all the other elements of F, including f, 
is a "remainder of F," and, more fully, "the 
remainder which is determined by the element f." 
The aggregates A and R are, by theorem C of 
12, well-ordered, and we may, by 8 and 12, 
write : 

(1) F = (A, R), 

(2) R = (/, R'), 

(3) A < R. 

R' is the part of R which follows the initial element 
f and reduces to o if R has, besides f, no other 
element. 

For example, in the well-ordered aggregate 



the segment 

(*i> *a) 

and the corresponding remainder 

(<z 3 , a 4) . . . a v+z , . . . b^ b 2 , ... b^ . . . lt c t , c 9 ) 
are determined by the element a s ; the segment 

(u a > -..,<*,. ) 
and the corresponding remainder 

(*D ^a> - > &p - - - c\> c > c a) 
are determined by the element b ; and the segment 



OF TRANSFINITE NUMBERS 143 
and the corresponding segment 

ta, c a) 

by the element c 2 . 

If A and A' are two segments of F,/and/' their 
determining elements, and 

(4) /' </, 

then A' is a segment of A. We call A' the "less," 
and A the ' ' greater " segment of F : 

(5) A'<A. 

Correspondingly we may say of every A of F that 
it is "less" than F itself : 

A<F. . 

[21 1] A. If two similar well-ordered aggregates 
F and G are imaged on one another, then to every 
segment A of F corresponds a similar segment B of 
G, and to every segment B of G corresponds a 
similar segment A of F, and the elements / and g 
of F and G by which the corresponding segments 
A and B are determined also correspond to one 
another in the imaging. 

Proof. If we have two similar simply ordered 
aggregates M and N .imaged on one another, m and 
n are two corresponding elements, and M' is the 
aggregate of all elements of M which precede m 
and N' is the aggregate of all elements of N which 
precede n, then in the imaging M' and N' correspond 
to one another. For, to every element m' of M 
that precedes m must correspond, by 7> an element 



144 THE FOUNDING OF THE THEORY 

n' of N that precedes #, and inversely. If we apply 
this general theorem to the well-ordered aggregates 
F and G we get what is to be proved. 

B. A well-ordered aggregate F is not similar to 
any of its segments A. 

Proof. Let us suppose that F r\j A, then we will 
imagine an imaging of F on A set up. By theorem 
A the segment A' of A corresponds to the segment 
A of Fj so that A' (\j A. Thus also we would have 
A' r\j F and A' < A. From A 7 would result, in the 
same manner, a smaller segment A" of F, such that 
A" X) F and A" < A' ; and so on.' Thus we would 
obtain an infinite series 



of segments of F, which continually become smaller 
and all similar to the aggregate F. We will 
denote by f t /', /", . . . , /<*>, ... the elements of 
F which determine these segments ; then we would 
have 

f>f >/"> >/ M >/ ( " +1) . 
We would therefore have an infinite part 



of F in which no element takes the lowest rank. 
But by theorem A of 12 such parts of F are not 
possible. Thus the supposition of an imaging F on 
one of its segments leads to a contradiction, and 
consequently the aggregate F is not similar to any 
of its segments. 



OF TRANSFINITE NUMBERS 145 

Though by theorem B a well-ordered aggregate 
F is not similar to any of its segments, yet, if F is 
infinite, there are always [212] other parts of F to 
which F is similar. Thus, for example, the aggregate 



is similar to every one of its remainders 



Consequently, it is important that we can put by the 
side of theorem B the following : 

Q. A well-ordered aggregate F is similar to no 
part of any one of its segments A. 

Proof. Let us suppose that F' is a part of a 
segment A of F and F' oo F. We imagine an 
imaging of F on F' ; then, by theorem A, to a 
segment A. of the well-ordered aggregate F corre- 
sponds as image the segment F" of F' ; let this 
segment be determined by the element f of F'. 
The element f is also an element of A, and de- 
termines a segment A' of A of which F" is a part, 
The supposition of a part F' of a segment A of F 
such that F 7 oo F leads us consequently to a part F" 
of a segment A' of A such that F" oo A. The same 
manner of conclusion gives us a part F'" of a 
segment A" of A' such that F'" oo A'. Proceeding 
thus, we get, as in the proof of theorem B, an 
infinite series of segments of F which continually 
become smaller : 

A>A'>A". . , ' 



146 THE FOUNDING OF THE THEORY 

and thus an infinite series of elements determining 
these segments : 



in which is no lowest element, and this is impossible 
by theorem A of 12. Thus there is no part F' 
of a segment A of F such that F' r\j F. 

D. Two different segments A and A' of a well- 
ordered aggregate F are not similar to one another. 

Proof, If A'<A, then A' is a segment of the 
well-ordered aggregate A, and thus, by theorem B, 
cannot be similar to A. 

E. Two similar well-ordered aggregates F and G 
can be imaged on one another only in a single 
manner. 

Proof. Let- us suppose that there are two different 
imagings of F on G, and let /"be an element of F to 
which in the two imagings different images g and g f 
in G correspond. Let A be the segment of F that 
is determined byf, and B and B' the segments of G 
that are determined by g and g f . By theorem A, 
both Aj^B [213] and A r\j B', and consequently 
BooB', contrary to theorem D. 

F. If F and G are two well-ordered aggregates, 
a segment A of F can have at most one segment 
B in G which is similar to it. 

Proof. If the segment A of F could have two 
segments B and B' in G which were similar to it, B 
and B' would be similar to one another, which is 
impossible by theorem D. 

G. If A and B are similar segments of two well- 



OF TRANSFIN1TR NUMBERS 147 

ordered aggregates F and G, for every smaller 
segment A'<A of F there is a similar segment 
B' < B of G and for every smaller segment B' < B of 
G a similar segment A' < A of F. 

The proof follows from theorem A applied to the 
similar aggregates A and B. 

H. If A and A' are two segments of a well- 
ordered aggregate F, B and B' are two segments 
similar to those of a well-ordered aggregate G, and 
A'<A, then B'<B. 

The proof follows from the theorems F and G. 

I. If a segment B of a well-ordered aggregate G 
is similar to no segment of a well-ordered aggregate 
F, then both every segment B' > B of D and G itself 
are similar neither to a segment of F nor F itself. 

The proof follows from theorem G. 

K. If for any segment A of a well-ordered 
aggregate F there is a similar segment B of another 
well-ordered aggregate G, and also inversely, for 
every segment B of G a similar segment A of F, 
then F c\j G. 

Proof. We can image F and G on one another 
according to the following law : Let the lowest 
element./^ of F correspond to the lowest element ^ 
of G. If f^-fi is any other element . of F, it 
determines a segment A of F. To this segment 
belongs by supposition a definite similar segment 
B of G, and let the element g of G which determines 
the segment B be the image of F. And if g is any 
element of G that follows g^ it determines a 
segment B of G, to which by supposition a similar 



H8 THE FOUNDING OF THE THEORY 

segment A of F belongs. Let the element / which 
determines this segment A be the image of g. It 
easily follows that the bi-univocal correspondence of 
F and G defined in this manner is an imaging in the 
sense of 7. For if/ and/' are any two elements 
of F, g and g f [214] the corresponding elements of 
G, A and A' the segments determined by / and /', 
B and B' those determined by g and g f , and if, say, 



then 

A'<A. 

By theorem H, then, we have 

B'<B, 
and consequently 



L. If for every segment A of a well-ordered 
aggregate F there is a similar segment B of another 
well-ordered aggregate G, but if, on the other hand, 
there is at least one segment of G for which there is 
no similar segment of F, then there exists a definite 
segment B x of G such that Bj^ooF. 

Proof. Consider the totality of segments of G for 
which there are no similar segments in F. Amongst 
them there must be a least segment which we will call 
B!. This follows from the fact that, by theorem A 
of 12, the aggregate of all the elements determin- 
ing these segments has a lowest element ; the 
segment E 1 of G determined by that element is the 
least of that totality. By theorem 1, every segment 



OF TRANSFINITE NUMBERS 149 

of G which is greater than B x is such that no segment 
similar to it is present in F. Thus the segments 
B of G which correspond to similar segments of F 
must all be less than B,, and to every segment 
B<B X belongs a similar segment A of F, because 
B is the least segment of G among those to which 
no similar segments in F correspond. Thus, for 
every segment A of F there is a similar segment B of 
B 1} and for every segment B of B x there is a similar 
segment A of F. By theorem K, we thus have 



M. If the well-ordered aggregate G has at least 
one segment for which there is no similar segment 
in the well-ordered aggregate F,, then every segment 
A of F must have a segment B similar to it in G.. 

Proof. Let B x be the least of all those segments 
of G for which there are no similar segments in F. * 
If there were segments in F for which there were no 
corresponding segments in G, amongst these, one, 
which we will call A x , would be the least. For 
every segment of A x would then exist a similar 
segment of Bj, and also for every segment of B x a 
similar segment of A r Thus, by theorem K, we 
would have 

B!OJA V 

[215] But this contradicts the datum that for B x 
there is no- similar segment of F. Consequently, 
there cannot be in F a segment to which a similar 
segment in G does not correspond. 

* See the above proof of L. 



150 THE FOUNDING OF THE THEORY 

N. If F and G are any two well-ordered aggre- 
gates, then either : 

(a) F and G are similar to one another, or 

(#) there is a definite segment E 1 of G to which 
F is similar, or 

(c) there is a definite segment A x of F to which 

G is similar ; 
and each of these three cases excludes the two others. 

Proof. The relation of F to G can be any one of 
the three : 

(a) To every segment A of F there belongs a 
similar segment B of G, and inversely, to every 
segment B of G belongs a similar one A of F ; 

(b) To every segment A of F belongs a similar 
segment B of G, but there is at least one segment 
of G to which no similar segment in F corresponds ; 

(c) To every segment B of G belongs a similar 
segment A of F, but there is at least one segment of 
F to which no similar segment in G corresponds. 

The case that there is both a segment of F to 
which no similar segment in G corresponds and a 
segment of G to which no similar segment in F 
corresponds is not possible ; it is excluded by 
theorem M. 

By theorem K, in the first case we have 

F f\jG. 

In the second case there is, by theorem L, a definite 
segment Bj of B such that 

B 1 ojF; 



OF TRANSFINITE NUMBERS 151 

and in the third case there is a definite segment A x 
of F such that 

A x oo G. 

We cannot have F c\j G and F oo B x simultaneously, 
for then we would have G 00 B.,, contrary to theorem 
B ; and, for the same reason, we cannot have both 
F r\j G and G c\> A x . Also it is impossible that 
both F oo B a and G oo Aj, for, by theorem A, 
from F oo B! would follow the existence of a 
segment B' x of Bj such that A oo B'j. Thus we 
would have G oo B'.,, contrary to theorem B. 

O. If a part F' of a well-ordered aggregate F is 
not similar to any segment of F, it is similar to F 
itself. 

Proof. By theorem C of 12, F' is a well-ordered 
aggregate. If F' were similar neither to a segment 
of F nor to F itself, there would be, by theorem N, 
a segment F' x of F' which is similar to F. But F f 1 
is a part of that segment A of F which [216] is 
determined by the same element as the segment F\ 
of F'. Thus the aggregate F would have to be 
similar to a part of one of its segments, and this 
contradicts .the theorem C. 



The Ordinal Numbers of Weil-Ordered 
Aggregates 

By 7, every simply ordered aggregate M has a 
definite ordinal type M ; this type is the general con- 



152 THE FOUNDING OF THE THEORY 

cept which results from M if we abstract from the 
nature of its elements while retaining their order of 
precedence, so that out of them proceed units 
(Einsen) which stand in a definite relation of pre- 
cedence to one another. All aggregates which are 
similar to one another, and only such, have one and 
the same ordinal type. We call the ordinal type of 
a well-ordered aggregate F its " ordinal number. " 

If a and ft are any two ordinal numbers, one can 
stand to the other in one of three possible relations. 
For if F and G are two well-ordered aggregates 
such that _ _ 

F-o, G=& 

then, by theorem N of I3> three mutually ex- 
clusive cases are possible : 

(a) FooG; 

(V) There is a definite segment Bj of G such that 



(c) There is a definite segment A of F such that 
Gr\j A r 

As we easily see, each of these cases still .subsists 
if F and G are replaced by aggregates respectively 
similar to them. Accordingly, we have to do with 
three mutually exclusive relations of the types a 
and $ to one another. In the first case a = /3; in 
the second we say that ct<j8; in the third we say- 
that a>j& Thus we have the theorem ; 



OF TRANSFINITE NUMBERS 153 

A. If a and ft are any two ordinal numbers, we 
have either a = ft or a < /8 or a > ft. 

From the definition of minority and majority 
follows easily : 

B. If we have three ordinal numbers a, /&, y, and 
if a < and ft < y, then a < y. 

Thus the ordinal numbers form, when arranged 
in order of magnitude, a simply ordered aggregate ; 
it will appear later that it is a well-ordered aggre- 
gate. 

[217] The operations of addition and multipli- 
cation of the ordinal types of any simply ordered 
aggregates, defined in 8, are, of course, applicable 
to the ordinal numbers. If a = F and /3 = G, where 
F and G are two well-ordered aggregates, then 



(1) a+/8-(F,G). 

The aggregate of union (F, G) is obviously a 
well-ordered aggregate too ; thus we have the 
theorem : 

C. The sum of two ordinal numbers is also an 
ordinal number. 

In the sum n+/9, a is called the "augend" and 
ft the "addend." 

Since F is a segment of (F, G), we have always 

(2) a<a + /3. 

On the other hand, G is not a segment but a re- 
mainder of (F, G), and may thus, as we saw in 
| 13, be similar to the aggregate (F, G). If this 



154 THE FOUNDING OF THE THEORY 

is not the case, G is, by theorem O of 13, similar 
to a segment of (F, G). Thus 

(3) <a+/3. 

Consequently we have : 

D. The sum of the two ordinal numbers is always 
greater than the augend, but greater than or equal 
to the addend. If we have a + /8 = a + y, we always 
have jQ = y. 

In general a + ft and fi+a are not equal. On 
the other -hand, we have, if y is a- third ordinal 
number, 

(4) (a+/3) + y = a + C6-r-y). 

That is to say : 

E. In the addition of ordinal numbers the associa- 
tive law always holds. 

If we substitute for every element g of the 
aggregate G of type ft an aggregate F f of type a, 
we get, by theorem E of 12, a well-ordered 
aggregate H whose type is completely determined 
by the types a and ^6 and will be called the product 
a. ft: 

(5) F,-* 

(6) a.j8 = H. 

F. The product of two ordinal numbers is also 
an ordinal number. 

In the product a .{3, a is called the " multiplicand " 
and ft the "multiplier." 

In general a . ft and ft . a are not equal. But we 
have ( 8) 



OF TRANSFINITE NUMBERS 155 

(7) (a./3).y = a.08.y). 

That is to say : 

[2l8] G. In the multiplication of ordinal numbers 
the associative law holds. 

The distributive' law is valid, in general ( 8), 
only in the following form : 

(8) a.(/3 + y) = a./3 + a.y. 

With reference to the magnitude of the product, 
the following theorem, as we easily see, holds : 

H. If the multiplier is greater than I, the product 
of two ordinal numbers is always greater than 
the multiplicand, but greater than or equal to the 
multiplier. If we have ct./3 = a.y, then it always 
follows that /3 = y. 

On the other hand, we evidently have 

(9) a. 1 = 1 .a = a. 

We have now to consider the operation of sub- 
traction. If a and ft are two ordinal numbers, and 
a is less than , there always exists a definite 
ordinal number which we will call $ a, which 
satisfies the equation 

(10) a+08-a)-/3. 

For if Gs=$, G has a segment B such that B = a ; 
we call the corresponding remainder S, and have 



156 THE FOUNDING OF THE THEORY 

and therefore 

(11) -a = S. 

The determinateness of a appears clearly from 
the fact that the segment B of G is a completely 
definite one (theorem D of 13), and consequently 
also S is uniquely given. 

We emphasize the following formulae, which 
follow from (4), (8), and (10) : 

(12) 



It is important to reflect that an infinity of 
ordinal numbers can be summed so that their sum 
is a definite ordinal number which depends on the 
sequence of the summands. If 



is any simply infinite series of ordinal numbers, and 
we have 

(14) A,= G,, 
[219] then, by theorem E of I2, t 

(15) G = (G lf G,, . . ., G,, . . .) 

is also a well-ordered aggregate whose ordinal 
number represents the sum of the numbers fi v . 
We have, then, 

(16) ft+&+ - . . + & + . . . =G = fr 

and, as we easily see from the definition of a 
product, we always have 



OF TRANS FINITE NUMBERS 157 
07) y- < 

If we put 

(18) 
then 



(19) a, = (G ls G |f . . . G,,). 

We have 
(20) 



and, by (10), we can express the numbers f$ v by 
the numbers a v as follows : 

(21) /3i = aj; /8+i a^+i a v . 

The series 

i> a a > ^> 

thus represents any infinite series of ordinal numbers 
which satisfy the condition (20) ; we will call it a 
"fundamental series" of ordinal numbers ( 10). 
Between it and ($ subsists a relation which can be 
expressed in the following manner : 

(a) The number /3 is greater than <* for every 
/, because the aggregate (G^ G 2 , . . ., G,,), whose 
ordinal number is a v) is a segment of the aggregate 
G which has the ordinal number /6 ; 

(b) If ft is any ordinal number less than ft, then, 
from a certain v onwards, we always have 



For, since &' < /3, there is a segment B' of the 



158 THE FOUNDING OF THE ' THEORV 

aggregate G which is of type (3 f . The element of 
G which determines this segment must belong to 
one of the parts G v ; we will call this part G VQ . But 
then B' is also a segment of (G lt G a , . . ., G^), and 
consequently /3' < a v . Thus 



for v^v . 

Thus fi is the ordinal number which follows next 
in order of magnitude after all the numbers a v ; 
accordingly we will call it the "limit" (Grense) of 
the numbers a v for increasing v and denote it by 
Lim a v , so that, by (16) a*nd (21) : 

V 

(22) Lim a ,= a 1 + ( 2 -a 1 ) + . . . + (a,,+i -a,) + . . . 

V 

[220] We may express what precedes in the 
following theorem : 

1. To every fundamental series {<*} of ordinal 
numbers belongs an ordinal number Lim a v which 

V 

follows next, in order of magnitude, after all the 
numbers a v ; it is represented by the formula (22). 

If by y we understand any constant ordinal 
number, we easily prove, by the aid of the formula: 
( I2 )> (13)1 and (i7), the theorems contained in the 
formulae : 

(23) Lim (y + a v ) = y + Lim a,, ; 

V V 

(24) Lim y . a v =y . Lim a v . 

V ' v 

We have already mentioned in ; that all simply 



OF TRANSFINITE NUMBERS 159 

ordered aggregates of given finite cardinal number 
v have one and the same ordinal type. This may 
be proved here as follows. Every simply ordered 
aggregate of finite cardinal number is a well-ordered 
aggregate ; for it, and every one of its parts, must 
have a lowest element, and this, by theorem B 
of 12, characterizes it as a well-ordered aggregate. 
The types of finite simply ordered aggregates are 
thus none otlier than finite ordinal numbers. But 
two different ordinal numbers a and /3 cannot belong 
to the same finite cardinal number j/. For if, say, 
a</3 and G = j3, then, as we know, there exists a 
segment B of G such that B = a. Thus the aggre- 
gate G and its part B would have the same finite 
cardinal number v. But this, by theorem C of 6, 
is impossible. Thus the finite ordinal numbers 
coincide in their properties with the finite cardinal 
numbers. 

The case is quite different with the transfinite 
ordinal numbers ; to one and the same transfinite 
cardinal number a belong an infinity of ordinal 
numbers which form a unitary and connected 
system. We will call this system the "number- 
class Z(a)," and it is a part of the class of types 
[a] of 7. The next object of our consideration is 
the number-class Z(N O ), which we will call ( ' the 
second number-class." For in this connexion we 
understand by ' ' the first number-class " the- totality 
{v} of finite ordinal numbers. 



160 THE FOUNDING OF THE THEORY 

[221] 15 

The Numbers of the Second Number-Class Z( MO ) 

The second number-class Z(M O ) is the totality {a} 
of ordinal types a of well-ordered aggregates of 
the cardinal number N O ( 6). 

A. The second number-class has a least number 



V 

Proof. By a> we understand the type of the 
well-ordered aggregate 

(1) FO = (/I. f^ ...,/ ). 

where v runs through all finite ordinal numbers and 

(2) /,-</,+! 
Therefore ( 7) 

(3) "> = F , 
and ( 6) 

(4) S^No- 

Thus o) is a number of the second number-class, 
and indeed the least. For if y is any ordinal 
number less than w, it must ( 14) be the type of 
a segment of F . But F has only segments 

A = (fit f& > fv)> 

with finite ordinal number j/. Thus y = v. There- 
fore there are no transfinite ordinal numbers which 
are less than w, and thus w is the least of them. 
By the definition of Lim a v given in 14, we 

V 

obviously have o)=Lim v. 



OF TRANSPINITE NUMBERS 161 

B. If a is any number of the second number-class, 
the number a+i follows it as the next greater 
number of the same number-class. 

Proof. Let F be a well-ordered aggregate of 
the type a and of the cardinal number M O : 

(5) F = , 

(6) a = Mo . 

We have, where by g is understood a new element, 

Since F is a segment of (F, g\ we have 

(8) o+i>a. 

We also have 



Therefore the number a+l belongs to the second 
number-class. Between o and a+ I there are no 
ordinal numbers ; for every number y [222] which 
is less than a+i corresponds, as type, to a segment 
of (F, g), and such a segment can only be either 
F or a segment of F. Therefore y is either equal 
to or less than a. 

C, If a X j a a , . . ., a v , . . . is any fundamental series 
of numbers of the first or second number-class, then 
the number Lim <* ( 14) following them next in 

V 

order of magnitude belongs to the second number- 
class. 

Proof. By 14 there results from the funda- 



1 62 THE FOUNDING OF THE THEORY 
mental series {<*} the number Lim <* if we set up 

V 

another series jS lt /?...,&,..., where 

$i = i5 Pa <*a ~ i> -i @v+i Gtp+i , . . . 

If, then, Gj, G a , . . ., G v , . . . are well-ordered aggre- 
gates such that ' 

G, = #,, 
then also 

G(G lf G a , ...,&,.. .) 

is a well-ordered aggregate and 
Lim a v = G. 

V 

It only remains to prove that 



Since the numbers {3^ &,...,$,,... belong to 
the first or second number-class, we have 

G,< i 

and thus 



But, in any case, G is a transfinite aggregate, and 
so the case G < is excluded. 

We will call two fundamental series {a v } and {a' v } 
of numbers of the first or second number-class ( 10) 
' ( coherent, " in signs : 

(9) {o,}ll{a',}, 

if for every v there are finite numbers X and ^ 
such that 

(10) a\>a v , X>X OJ 



OF TRANSFINITE NUMBERS 163 

and 

(n) <v>a',,, /A^LjUo- 

[223] D. The limiting numbers Lim a v and Lim d v 

V V 

belonging respectively to two fundamental series 
{a v } and {d v } are equal when, and only when, 
{a,} II {a',}. 

Proof, For the sake of shortness we put 
Lim a v = /3, Lim a' v = y. We will first suppose 

V\ V 

that {a,,} || {a' v } ; then, we assert that /3 = y. For 
if j8 were not equal to y, one of these two numbers 
would have to be the smaller. Suppose that /3 < y. 
From a certain v onwards we would have d v >{$ 
( 14), and consequently, by (n), from a certain 
IL onwards we would have a )X >/S. But this is 
impossible because jS = Lim a v . Thus for all /x's 

V 

we have a^fB. 

If, inversely, we suppose that /3 = y, then, because 
a v <y, we must conclude that, from a certain \ 
onwards, a\>a tf , and, because a' v <|8, we must 
conclude that, from a certain /* onwards, e^. ><*'. 
That is to say, {a,} || fa' v }. 

E. If a is any number of the second number- 
class and j/ any finite ordinal number, we have 
i/ + ct = ct, and consequently also a j/ = a. 

Proof. We will first of all convince ourselves of 
the correctness of the theorem when a = <a. We 
have 



164 THE FOUNDING OF THE THEORY 
and consequently 



But if a > w, we have 

a a> + (a CD), 

j/o + a = (j/ + u) + (a to) w + (a <) = a. 
F. If i/o is any finite ordinal number, we have 

|/ . 0) = ft>. 

Pi oof. In order to obtain an aggregate of the 
type i/o . (a we have to substitute for the single 
elements / of the aggregate (/ x , / a , ...,/....) 
aggregates (g v , v g v , 2 , . . . , g v> VQ ) of the type VQ , We 
thus obtain the aggregate 

CSi, u ^i, ai S\, v ^"a, u > a. v 0> . ^"v, i 

^"v, a> > ^V, > > /> 

which is obviously similar to the aggregate {f v }. 
Consequently 

>/ . o> = w. 

The same result is obtained more shortly as follows. 
By (24) of 14 we have, since o>=Lim y, 

V 

v ft) = Lim j/ >/. 

V 

On the other hand, 

{"<,*} IIM> 
and consequently 

Lim i/o v= Lim y = a> ; 

f ^ 

so that 



OF TRANSFfNITE NUMBERS 165 
[224] G. We have always 



where a is a number of the second number-class 
and i/ a number of the first number-class. 
Proof. We have 

Lim v = to. 

V 

By (24) of 14 we have, consequently, 

(a + >/) = Lim (a + V Q )V. 
But 

12 

( + )= (a + i/oHCa + VoH.. 
I 2 



12 

**-S \^f ' 

a-f-a+ . . 



Now we have, as is easy to see, 



and consequently 

Lim (a+v >=Lim (av+j/ )-Lim av- 



H. If a is any number of the second number- 
class, then the totality {a'} of numbers a' of the 
first and second number-classes which are less than 
a form, in their order of magnitude, a well-ordered 
aggregate of type a, 



1 66 THE FOUNDING OF THE THEORY 

^J' 'Let F be a well-ordered aggregate such 
that F = a , and let/j be the lowest element of F. If 
a is any ordinal number which is less than a, then, 
by 14, there is a definite segment A' of F such 
that 

AW, 

and inversely every segment A' determines by its 
type A' = ct' a number a'<a of the first or second 
number-class. For, since F = N O , A' can only be 
either a finite cardinal number or M O . The segment 
A' is determined by an element/' >/ x of F, and 
inversely every element/' >^/ x of F determines a 
segment A' of F. If/' and/" are two elements of 
F which follow /j in rank, A' and A" are the 
segments of F determined by them, a ' and a " are 
their ordinal types, and, say/' </", then, by 13, 
A'<A" and consequently a ' < a". [225] If, then, 
we put F = (j^, F'), we obtain, when we make the 
element f of F' correspond to the element a of {a'}, 
an imaging of these two aggregates. Thus we have 



a= 



But F' =s a I , and, by theorem E, a I = a. Con- 
sequently 



= a. 



Since a = M , we also have {a'} = ; thus we have 
the theorems : 

I. The aggregate {a'} of numbers a' of the 
first and second number-classes which are smaller 



OF TRANSFIN1TE NUMBERS 167 

than a number a of the -second number-class has 
the cardinal number N O . 

K. Every number a of the second number-class 
is either such that (a) it arises out of the next 
smaller number a_j by the addition of I : 



or (b) there is a fundamental series {<*} of numbers 
of the first or second number-class such that 

a = Lim a v , 

V 

Proof. Let a = F. If F has an element g which 
is highest in rank, we have F = (A, g), where A is 
the segment of F which is determined by g. We 
have then the first case, namely, 



There exists, therefore, a next smaller number 
which is that called 'a r 

But if F has no highest element, consider the 
totality {a'} of numbers of the first and second 
number-classes which are smaller than a. By 
theorem H, the aggregate {a'}, arranged in order of 
magnitude, is similar to the aggregate F ; among 
the numbers a', consequently, none is greatest. By 
theorem I, the aggregate {a'} can be brought into 
the form {<'} of a simply infinite series. If we set 
out from a'u the next following elements a' 2 , a' 8 , . 
in this order, which is different from the order of 
magnitude, will, in general, be smaller than a' x ; 
but in every case, in the further course of the 



168 THE FOUNDING OF THE THEORY 

process, terms will occur which are greater than a\ ; 
for a'x cannot be greater than all other terms, 
because among the numbers {a' v } there is no 
greatest. Let that number a' v which has the least 
index of those greater than a' x be a' Pa . Similarly, 
let a' p be that number of the series {<*'} which has the 
least index of those which are greater than a^. By 
proceeding in this way, we get an infinite series of 
increasing numbers, a fundamental series in fact, 



[226] We have 

K p 9 < p 8 < . . . < p 



V < a'p v always if p < p' v ; 
and since obviously v ^ p v , we always have 
OL'V ^i &p v - . 

Hence we see that every number <*', and conse- 
quently every number a'<a,'is exceeded by numbers 
a' p for sufficiently great values of v. But a is the 
number which, in respect of magnitude, immediately 
follows all the numbers a', and consequently is also 
the next greater number with respect to all a' Pv . If, 
therefore, we put a\ = a 1 , a/> I , +1 BB! <VH) we h ave 

a=Lim a v . 

V 

From the theorems B, C, . . ., K it is evident 
that the numbers of the second number-class result 



OF TRANSFINITE NUMBERS 169 

from smaller numbers in two ways. Some numbers, 
which we call ' ' numbers of the first kind (Art), " are 
got from a next smaller number a-i by addition of i 
according to the formula 



The other numbers, which we call ' ' numbers of the 
second kind," are such that for any one of them 
there is not a next smaller number o_ l5 but they 
arise from fundamental series {a v } as limiting 
numbers according to the formula 



Here a is the number which follows next in order 
of magnitude to all the numbers <*. 

We call these two ways in which greater numbers 
proceed out of smaller ones "the first and the 
second principle of generation of numbers of the 
second number-class."* 



16 

The Power of the Second Number- Class is equal 
to the Second Greatest Transfmite Cardinal 
Number Aleph-One 

Before we turn to the more detailed considera- 
tions in the following paragraphs of the numbers of 
the second number-class and of the laws which 
rule them, we will answer the question as to the 

* ICf. Section VII of the Introduction.] 



170 THE FOUNDING OF THE THEORY 

cardinal number which is possessed by the aggregate 
Z(MO)={} of all these numbers. 

[227] A. The totality {a} of all numbers a of 
the second number-class forms, when arranged in 
order of magnitude, a well-ordered aggregate. 

Proof. If we denote by A tt the totality of 
numbers of the second number-class which are 
smaller than a given number a, arranged in order 
of magnitude, then A a is a well-ordered aggregate 
of type a a>. This results from theorem H of 14. 
The aggregate of numbers a of the first and second 
number-class which was there denoted by {a'}, is 
compounded out of {v} and A a) so that 

{<*'} = ({}, A.). 
Thus 



and since 

{a}=a, { !>}=&>, 
we have 

A a =rt CD. 

Let J be any part of {a} such that there are 
numbers in {a} which are greater than all the 
numbers of J. Let, say, a be one of these numbers. 
Then J is also a part of A^+i, and indeed such a 
part that at least the number a o of A ao+1 is greater 
than all the numbers of J. Since A^+i is a well- 
ordered aggregate, by 12 a number a' of A afl+1 , 
and therefore also of {a}, must 'follow next to all 
the numbers of J. Thus the condition II of 12 is 



OF TRANSFIN1TE NUMBERS 171 

fulfilled in the case of {a} ; the condition I of 12 
is also fulfilled because {a} has the least number &>. 

Now, if we apply to the well-ordered aggregate 
{a} the theorems A and C of 12, we get the 
following theorems : 

B. Every totality of different numbers of the first 
and second number-classes has a least number. 

C. Every totality of different numbers of the first 
and second number-classes arranged in their order of 
magnitude forms a well-ordered aggregate. 

We will now show that the power of the second 
number-class is different from that o'f the first, which 
is N O . 

D. The power of the totality {a} of all numbers 
a of the second number-class is not equal to M O . 

Proof. If {c[y were equal to N O , we could bring 
the totality {a} into the form of a simply infinite 
series 

Vi. V* . . ., y,, . .. 

such that {y,,} would represent the totality of 
numbers of the second [228] number-class in an 
order which is different from the order of magni- 
tude, and {y,,} would contain, like {a}, no greatest 
number. 

Starting from y^ let y p be the term of the series 
which has the least index of those greater than y x , 
y p the term which has the least index of those 
greater than y p , and so on. We get an infinite 
series of increasing numbers, 



172 THE FOUNDING OF THE THEORY 

such that 

. . <p v <p v+ i< . . ., 



By theorem C of 15, there would be a definite 
number 8 .of the second number-class, namely, 



which is greater than all numbers y p . Consequently 
we would have 



for every v. But {y v } contains all numbers of the 
second number-class, and consequently also the 
number 8 ; thus we would have, for a definite v , 



which equation is inconsistent with the relation 
S > y Vo . The supposition {a} = M O consequently leads- 
to a contradiction. 

E. Any totality {($} of different- numbers {3 of 
the second number-class has, if it is infinite, either 
the cardinal number N O or the cardinal number {a.} 
of the second number-class. 

Proof. The aggregate {ft}, when arranged in its 
order of magnitude, is, since it is a part of the well- 
ordered aggregate {a}, by theorem O of 13, 
similar either to a segment A 0o , which is the totality 



OF TRANSFrNITE NUMBERS 173 

of all numbers of the same number-class which are 
less than oo, arranged in their order of magnitude, 
or to the totality {a} itself. As was shown in the 
proof of theorem A, we have 



Thus we have either {/8} = a w or {$} = {a}, and 
consequently {$} is either equal to c^ a> or {a}. 
But ft> is either a finite cardinal number or is 
equal to N O (theorem I of 15). The first case is 
here excluded because {fi} is supposed to be an 

infinite aggregate. Thus the cardinal number {ft} 

is either equal to N O or {a}. 

F. The power of the second number-class {a} is 
the second greatest transfinite cardinal number 
Aleph-one. 

[229] Proof. There is no cardinal number a 

which is greater than N O and less than {a}. For if 
not, there would have to be,, by 2, an infinite part 

{ft} of {a} such that{} = a. But by the theorem 
E just proved, the part \/3] has either the cardinal 

number N O or the cardinal number {a}. Thus the 
cardinal number {a} is necessarily the cardinal 
number .which immediately follows M D in magnitude ; 
we call this new cardinal number HI* 

In the second number-class Z(M O ) we possess, 
consequently, the natural representative for the 
second greatest transfinite cardinal number Aleph- 
one. 



174 THE FOUNDING OF THE THEORY 

17 
The Numbers of the Form 0^1/0+ a M "V+ + v 

It is convenient to make ourselves familiar, in the 
first place, with those numbers of Z(N O ) which are 
whole algebraic functions of finite degree of a. 
Every such number can be brought and brought 
in only one way into the form 

(i) f/ J =^"o + w^"\+ +V 

where /u., j/ are finite and different from zero, but 
i^ " a , . . ., v^ may be zero. This rests on the fact 
that 



(2) 
if P</JL and v>o ) i/>o. For, by. (8) of 14, we 



have 

af'v 

and, by theorem E of 15, 

V + W* 1 ~ ^ V = tiP ~ M V' 

Thus, in an aggregate of the form 



all those terms which are followed towards the right 
by terms of higher degree in <a may be omitted. 
This method may be continued until the form given 
in (i) is reached. We will also emphasize that 

(3) 



OF TRANSFINITE NUMBERS i;S 

Compare, now, the number with a number \ff of 
the same kind : 

(4) V r = a> % + 0)A ~ 1 / J i+ +/*- 

If /* and X are different and, say, /*<X, we have by 
(2) + \ff = i/r, and therefore <! ^. 

[230] If / a = X, v c , and /> are different, and, say, 
"o < PQ> we have by (2) 



and therefore 
If, finally, 

fJ. = \, VO=PQ> J/ 1 =/0 1 , 

but j/o. is different from />, and, say, v 9 <p a) we 
have by (2) 



and therefore again 



Thus, we see that only in the case of complete 
identity of the expressions and i/r can the numbers 
represented by them be equal. 

The addition of and \js leads to the following 
result : 

(a) If /x<X, then, as we have remarked above, 



If/zX, then we have 



176 THE POUNDING OF THE THEORV 
(c) If fji > X, we have 



In order to carry out the multiplication of and ^, 
we remark that, if p is a finite number which is 
different from zero, we have the formula : 



(5) <j>p = (* 

It easily results from the carrying out of the sum 
consisting of p terms + 0+ . . . + 0. By means 
of the repeated application of the theorem G of 
15 we get, further, remembering the theorem F 
of 15, 

(6) 
and consequently also 

(7) 



By the distributive law, numbered (8) of 14, 
we have 



Thus the formulae (4), (5), and (7) give the following 
result : 

(a) Ifp x =o, we have 



If p x is not equal to zero, we have 



OF TRANSFINITE NUMBERS 177 

[231] We arrive at a remarkable resolution of 
the numbers in the following manner. Let 

(8) ^ 

where 



and K Q) K V . . ., K r are finite numbers which are 
different from zero. Then we have 



By the repeated application of this formula we get 

= ftAK T (a/ lT - 1 ~'* T *r-l-f- l)(o)^-3-^-lK T _2+l). . . 

(*-'%+ 1). 

But, now, 



if * is a finite number which is different from zero ; 
so that : 



<9) ^ = ufTKfai+'i-*' + iX-iCw^-a-^-i + I K-s - - . 



The factors o^+i which occur here are all irre- 
soluble, and a number ^ can be represented in this 
product-form in only one way, If ^=0, then <J> 
is of the first kind, in all other cases it is of th 
second kind. 

The apparent deviation of the formulae of this 
paragraph from those which were given in Math. 
Ann., vol. xxi, p. 585 (or Gruwttage*, p, 41), is 
merely a consequence of the different writing of the 
product of two numbers; we now put the multi- 



178 THE FOUNDING OF THE THEORY 

plicand on the left and the multiplicator on the 
right, but then we put them in the contrary way. 



18 

The Power * y* in the Domain of the Second 
Number- Class 

Let be a variable whose domain consists of the 
numbers of the first and second number-classes in- 
cluding zero. Let y and S be two constants belong- 
ing to the same domain, and let 



We can then assert the following theorem : 

A. There is one wholly determined one-valued 
function f(g) of the variable such that : 

(a) /(o) = 6\ 

(b) If g and f" are any two values of , and if 

f<r, 

then 

/(f)</(f)- 
[232] (c) For every value of we have 



(d) If {,} is any fundamental series, then 
is one also, and if we have 



then 

* [Here obviously it is Potenz and not McUhtigkeit,] 



OF TRANSFINITE NUMBERS 179 
Proof. By (a) and (c\ we have 



and, because 8>o and y> I, we have 



Thus the function ./() is wholly determined for the 
domain < . Let us now suppose that the theorem 
is valid for all values of which are less than a, 
where a is any number of the second number-class, 
then it is also valid for <.a. For if a is of the 
first kind, we have from (c) : 

/(a)=/(a-!)y>/(a-i); 

so that the conditions (If), (c), and (d) are satisfied 
for ^a. But if a is of the second kind and {a,,} is 
a fundamental series such that Lim a v = a, then it 

V ' 

follows from (b) that also {/(a,,)} is a fundamental 
series, and from' (d) that /(a) = Lim /(a v ). If we 

V 

take another fundamental series {<*'} such that 
Lim a v = a, then, 'because of (b\ the two funda- 

V 

mental series (f(a v )} and {/(</)} are coherent, and 
thus also /( a ) = Lim f(d v ). The value of /(a) is, 

V 

consequently, uniquely determined in this case also. 
If a' is any number less than a, we easily convince 
ourselves that f(a)<f(a). The conditions (b\ (c), 
and (d} are also satisfied for <f a. Hence follows 
the validity of the theorem for all values of For 
if there were exceptional values of for which it 
did not hold, then, by theorem B of 16, one of 



i8o THE FOUNDING OF THE THEORY 

them, which we will call a, would have to be the 
least. Then the theorem would be valid for < a, 
but not for ^a, and this would be in contradiction 
with what we have proved. Thus there is for the 
whole domain of one and only one function f(g) 
which satisfies the conditions (a) to (*/). 

[233] If we attribute to the constant S the value i 
and then denote the function /() by 

**. 

we can formulate the following theorem : 

B. If y is any constant greater than I which 
belongs to the first or second number-class, there 
is a wholly definite function y* of such that : 

M/-I; 

(b) Iff <f then yf<yf; 

(c) For every value of we have y* + 1 = y*y ; 

(d) If {} is a fundamental series, then {y f *} 
is such a series, and we have, if =Lim , the 

V 

equation 



We can also assert the theorem i 
C. If /() is the function of which is characterized 
in theorem A, we have 



Proof. If we pay attention to (24) of 14, 
we easily convince ourselves that the function <Jy* 
satisfies, not only the conditions (a\ (), and (c) 
of theorem A, but also? the condition (</} of this 



OF TRANSFINITE NUMBERS 181 

theorem. On account of the uniqueness of the 
function /(), it must therefore be identical with <5yf 
D. If a and j3 are any two numbers of the first 
or second number-class, including zero, we have 



Proof, We consider the function #() y* + *. 
Paying attention to the fact that, by formula (23) 

of 14, 

Lim (a + & = a 



we recognize that 0() satisfies the following four 
conditions : 
(a) 0(0) = y; 



(c) For every value of we have </(+ 1) = 

(d) If {g v } is a fundamental series such that 
Lim = , we have 

= Lim 



By theorem C we have, when we put 5 = y, 

^(|) = y y. 
If we put =/3 in this, we have 



E. If a and /8 are any two numbers of the first or 
second number-class, including zero, we have 



[234] Proof. Let us consider the function 
s=:y tt * and remark that, by (24) of 14, we 



182 THE FOUNDING OF THE THEORY 
always have Lim a, = o Lim ,, then we can, by 

V V 

theorem D, assert the following : 
(a) V<o)=i; 



(c) For every value of we have ^(+ i) 
(J) If {,} is a fundamental series, then 
also such a series, and we have, if =Lim,, the 

V 

equation /() = Lim V r (^)- 
v 

Thus, by theorem C, if we substitute in it I for 8 
and y*.for y, 



On the magnitude of y* in comparison with ^ we 
can assert the following theorem : 

F. If y > i, we have, for every value of , 



Proof. In the cases =o and =i the theorem 
is immediately evident. We now show that, if it 
holds for all values of ^ which are smaller than a 
given number a> I, it also holds for =a. 

If a is of the first kind, we have, by supposition, 



and consequently 

a -i7 <y a -'y-y"- 
Hence 



Since both a_ x and y i are at least equal to i, and 
a.j+issa, we have 

y>a. 



OF TRANSFINITE NUMBERS 183 

If, on the other hand, a is of the second kind and 
a = Lim a v , 

V 

then, because <* < a, we have by supposition 

,</". 

Consequently 

Lim a v ^ Lim y"", 

V V 

that is to say, 

a<y. 

If, now, there were values of for which 



one of them, by theorem B of 16, would have to 
be the least. If this number is denoted by a, we 
would have, for <a, 

[235] ir^y f ; 

but 



which contradicts what we have proved above. 
Thus we have for all values of 



19 

The Normal Form of the Numbers of the 
Second Number-Class 

Let a be any number of the sec9nd number-class. 
The power o>* will be, for sufficiently great values 



1 84 THE FOUNDING OF THE THEORY 

of , greater than a. By theorem F of 1 8, this is 
always the case for > a ; but in general it will also 
happen for smaller values of 

By theorem B of 16, there must be, among the 
values of for which 



one which is the least. We will denote it by ft, and 
we easily convince ourselves that it cannot be a 
number of the second kind. If, indeed, we had 



we would have, since J3 V < (3, 

> <CGl| 

and consequently 

Lim to <f a. 

V 

Thus we would have 

o>^ a, 
whereas we have 



Therefore /3 is of the first kind. We denote /3_j 
by OQ, so that /3 = a +i, and consequently can 
assert that there is a wholly determined number oo 
of the first or second number-class which satisfies 
the two conditions : 



(l) 

From the second condition we conclude that 
a 



eoj' . a 



OF TRANS FINITE NUMBERS 185 

does not hold for all finite values of v, for if it did 
we would have 

Lim w a j/ = %> <. a. 

V 

The least finite number v for which 

a> a v>a 
will be denoted by + 1. Because of (i), we have 



[236] There is, therefore, a wholly determined 
number K Q of the first number-class such that 

(2) ft)"o/r ^a, flfftr +!)> 
If we put a w a o/f = a', we have 

(3) a*= a ox: + a' 
and 

(4) O <.a < <o a o, o < K O < (a . 

But a can be represented in the form (3) under the 
conditions (4) in only a single way. For from (3) 
and (4) follow inversely the conditions (2) and thence 
the. conditions (i). But only the number a = /8_i 
satisfies the conditions (i), and by the conditions 
(2) the finite number : is uniquely determined. 
From (i) and (4) follows, by paying attention to 
theorem F of 18, that 

a' < a, Oo^a. 

Thus we can assert the following theorem : 

A. Every number a of the second number-class 



1 86 THE FOUNDING OF THE THEORY 

can be brought, and brought in only one way, into 
the form 

a = tt a <Mc + a '> 
where 

O<.d' < ft)0, O < KQ < ft), 

and a is always smaller than a, but a is smaller 
than or equal to a. 

If a' is a number of the second number-class, we 
can apply theorem A to it, and we have 

(5) a'-dlKi + a", 

where 

o^a" < w a i, o < *! 
and 



In general we get a further sequence of analogous 
equations : 

(6) *"-**, + a'", 

(7) a"' 



But this sequence cannot be infinite, but must 
necessarily break off. For the numbers a, a', a", . . . 
decrease in magnitude : 



If a series of decreasing transfinite numbers were 
infinite, then no term would be the least ; and this 
is impossible by theorem B of 16. Consequently 
we must have, for a certain finite numerical value r, 



- 0, 



OF TRANSFINITE NUMBERS 187 

[237] If we now connect the equations (3), (5), 
(6), and (7) with one another, we get the theorem : 

B. Every number a of the second number-class 
can be represented, and represented in only one 
way, in the form 



where a,,, a x , . . . a r are numbers of the first or 
second number-class, such that: 

OQ > tt > Og > . . . > 0, >. O, 

while K O| K V . . . * TJ T+I are numbers of the first 
number-class which are different from zero. 

The form of numbers of the second number-class 
which is here shown will be called their ' ' normal 
form"; OQ is called the "degree" and a,, the 
"exponent" of a. For r=o, degree and exponent 
are equal to one another. 

According as the exponent a,, is equal to or greater 
than zero, a is a number of the first or second kind. 

Let us take another number /Q -in the normal 
form : 



(8) 0= 

The formulae : 

(9) 

(10) w-V + ^Y'^Y', a'<a", 



where K, K', K" here denote finite numbers, serve 
both for the comparison of a with ft and for the 



1 88 THE FOUNDING. OF THE THEORY 

carrying out of their sum and difference. These are 
generalizations of the formulae (2) and (3) of 17. 

For the formation of the product a/3, the following 
formulae come into consideration : 



(11) aX = to-OffoXH- aF/dH- . . . + ar jc T> O<X<a>; 

(12) ao>=ft) a +1 ; 

(13) aaP^a'o+P, /3'>0. 

The exponentiation c^ can be easily carried out 
on the basis of the following .formulae : 

(14) a A> = to a x /c + . . ., o<\<eo. 

The terms not written on the right have a lower 
degree than the first. Hence follows readily that 
the fundamental series {a x } and {w ^} are coherent, 
so that 

(15) a^eo""", ao>0. 

Thus, in consequence of theorem E of 18, we 
have : 

(16) 0*^ = 0)^ o >o, ft'>o. 

By the help of these formulae we can prove the 
following theorems : 

[238] C. If the first terms ta*K , w^Xo of the 
normal forms of the two numbers a and ft are not 
equal, then a is less or greater than ft according as 
is less or greater than oA\ . But if we have 



and if w^+^p+i is less or greater than o/P +l \ f+lt then 
a is correspondingly less or greater than ft. 



OF TRANS FINITE NUMBERS 189 

D. If the degree c^ of a is less than the degree 
A> of /3, we have 

If 00 = , then 
But if 



, . . . , a p > /8 , a p +i < /3 , 
then 



E. If /3 is of the second kind (jS ff >o), then 
a / S = w a o + ' J ox 4- c ,'o+^X 1 + . . . +w a o+^X (r 
But if is of the first kind (/3o. = o), then 



F. If is of the second kind (/^XD), then 



But if is of the first kind (^ = 0), and indeed 
where ]8' is of the second kind, we have: 



G. Every number a of the second number-class 
can be represented, in only one way, .in the product- 
form : 



and we have 



190 THE FOUNDING OF THE THEORY 

whilst K 0i K I} . . . K T have the same denotation as 
in the normal form. The factors aP+i are all 
irresoluble. 

H. Every number a of the second kind which 
belongs to the second number-class can be repre- 
sented, and represented in only one way, in the 
form 



where y >o and a is a number of the first kind 
which belongs to the first or second number-class. 

[239] I. In order that two numbers a and ft of 
the second number-class should satisfy the relation 



it is necessary and sufficient that they should have 
the form 

a yfi, ft = yv t 

where p and v are numbers of the first number-class. 
K. In order that two numbers a and ft of the 
second number-class, which are both of the first 
kind, should satisfy the relation 

aft=fta, 

it is necessary and sufficient that they should have 
the form 



where ^ and v are numbers of the first number-class, 

In order to exemplify the extent of the normal 

form dealt with and the product-form immediately 

connected with it, of the numbers of the second 



OF TRANSFINITE NUMBERS 191 

number-class, the proofs, which are founded on 
them, of the two last theorems, I and K, may here 
follow. 

From the supposition 



we first conclude that the degree a of a must be 
equal to the degree $ of $. For if, say, a < /# , we 
would have, by theorem D, 



and consequently 

which is not possible, since, by (2) of 14, 

Thus we may put 



where the degrees of the numbers a' and /3' are less 
than oo> and /x. and v are infinite numbers which are 
different from zero. Now, by theorem D we have 



and consequently 

o^Cu + v) + $' = O>^(M + v) + a'. 
By theorem D of 14 we have consequently 

/? = a'. 
Thus we have 



192 THE FOUNDING OF THE THEORY 

[240] and if we put 

a) a +a' = 7 
we have, by (n) : 

a = y/uL, ft = yv. 

Let us suppose, on the other hand, that a and (3 are 
two numbers which belong to the second number- 
class, are of the first kind, and satisfy the condition 

a/3 = (3a, 
and we suppose that 

a>p. 

We will imagine both numbers, by theorem G, in 
their product-form, and let 

JJ ' O 'SQ r 

a. = Co. } p op , 

where a and ft' are without a common factor (besides 
i) at the left end. We have then 



and 



All the numbers which occur here and farther on 
are of the first kind, because this was supposed of 
a and ft. 

The last equation, when we refer to theorem G, 
shows that a and ft' cannot be both transfinite, 
because, in this case, there would be a common 
factor at the left end. Neither can they be both 
finite ; for then 8 would be transfinite, and, if K is 
the finite factor at the left end of 8, we would have 



OF TRANSFINITE NUMBERS 193 

and thus 

a'-?. 

Thus there remains only the possibility that 

a >&) (y<ta. 
But the finite nutnber f# must be i : 

/3'=i, 

because otherwise it would be contained as part iti 
the finite factor at the left end of a '. 

We arrive at the result that /3 = <S, consequently 

a = pa', 

where a' is a number belonging to the second 
number-class, which is of the first kind, and must 
be less than a : 

a'<a. 

Between a and /3 the relation 

a'ft=pa 
subsists, 

[241] Consequently if also a>f$, we conclude in 
the same way the existence of a transfmite number 
of the first kind a" which is less than a and such that 



If also a" is greater than j6, there is such a number 
a" less than a", such that 



and so on, The series of decreasing numbers, a, a', 
a", a'", . . ., must, by theorem B of 16, break 



194 THE FOUNDING OF THE THEOR Y 

off. Thus, for a definite finite index p , we must 
have 

a<"o <j3. 
If 

a<*>-A 
we have 



the theorem K would then be proved, and we would 
have 

y=& M = A+i, v=i. 
But if 

then we put 
and have 



Thus there is also a finite number such that 



In general, we have analogously : 



and so on. The series of decreasing numbers $ 1} 
/Q> ^3, also must, by theorem B of 16, break 
off. Thus- there exists a finite number K such that 



If we put 
then 



OF TRANSFINITE NUMBERS 195 

where y. and v are numerator and denominator of 
the continued fraction. 



_ 

v 



P 



[242] 20 

The e-Numbers of the Second Number-Class 

The degree oo of a number a is, as is immediately 
evident from the normal form : 



(l) a = o> a /c + a> a i/c 1 -}-. . ., a >a 1 >..., O<K V <(O, 



when we pay attention to theorem F of 1 8, never 
greater than a ; but it is a question whether there 
are not numbers for which oo = a. In such a case 
the normal form of a would reduce to the first term, 
and this term would be equal to Q>*, that is to say, 
o would be a root of the equation 

(2) <o* = 

On the other hand, every root a of this equation 
would have the normal form eo a ; its degree would 
be equal to itself. 

The numbers of the second number-class which 
are- equal to their degree coincide, therefore, with 
the roots of the equation (2). It is our problem to 
determine these roots in their totality. To dis- 
tinguish them from all other numbers we will call 
them the "e-numbers of the second number-class." 



196 THE FOUNDING OF THE THEORY 

That there are such e-numbers results from the 
following theorem : 

A. If y is any number of the first or second 
number-class which does not satisfy the equation 
(2), it determines a fundamental series {y} by means 
of the equations 



The limit Lim y v = E(y) of this fundamental series 

V 

is always an e-number. 

Proof. Since y is not an e-number, we have 
o)>>y, that is to say, 7 r >y. Thus, by theorem B 
of 1 8, we have also a>n>u?, that is to say, y a >yj ; 
and in the same way follows that y s > y a , and so 
on. The series {y,,} is thus a fundamental series. 
We denote its limit, which is a function of y, by 
E(y) and have : 

w E<y) = Lim o>? = Lim y F+1 = E(y). 

V V 

Consequently E(y) is an e-number. 

B. The number e = E(i) = Lim <, where 

V 

! ** 0, ft)a = w<l "> ^3 == G> " a > * > &v = ft)""- 1 ) . j 

is the least of all the e-numbers. 

[243] Proof. Let e' be any e-number, so that 

of 1 = e'. 

Since e' > <a, we have of 1 > fo", that is to say, e' > o^. 
Similarly of > a)** 1 , that is to say, e" > w a , and so on. 
We have in general 



OF TRANS FINITE NUMBERS 197 
and consequently 

that is to say, 

e^e . 

Thus <r = E(i) is the least of all e-numbers. 

C. If e is any e-number, e" is the next greater 
e-number, and y is any number which lies between 

them : 

e'<y<e", 
then E(y) = e". 
Proof. From 

<='<y<e" 
follows 

uf f <uff<af / ' t 
that is to say, 



Similarly we conclude 
e' 
and so on. We have, in general, 

e'<y,,<e", 
and thus 

e '<E(y)<e". 

By theorem A, E(y) is an e-number. Since e" is 
the e-number which follows e next in order of mag- 
nitude, E(y) cannot be less than e", and thus we 
must have 



Since e'+ I is not an e-number, simply because ail 
e-numbers, as follows from the equation of definition 



198 THE FOUNDING OF THE THEORY 

=10*, are of the second kind, e'+ I is certainly less 
than e", and thus we have the following theorem : 

. D. If e r is any e-number, then E(e' + i) is the next 
greater e-number. 

To the least e-number, e , follows, then, the next 
greater one: 



[244] to this the next greater number : 



and so on. Quite generally, we have for the 
(i/+ i)th e-number in order of magnitude the formula 
of recursion 

(3) e^E^.x+i). 

But that the infinite series 

e , e t , . . . e v , . . . 

by no means embraces the totality of e-numbers 
results from the following theorem : 

E. If e, e, e", ... is any infinite series of 
e-numbers such that 



e<e'<e". . . e 
then Lim e*" J is an e-number, and, in fact, the 

V 

e-number which follows next in order of magnitude 
to all the numbers e (l<) . 
Proof. 

Lim<"> () 

w " = Lim W* = Lim e (l , 



OF TRANS FINITE NUMBERS 199 
That Lim <?<"> is the e-number which follows next 

V 

in order of magnitude to all the numbers ^ results 
from the fact that Lim e<"> is the number of the 

V 

second number-class which follows next in order of 
magnitude to all the numbers e (v \ 

F. The totality of e-numbers t>f the second 
number-class forms, when arranged in order of 
magnitude, a well-ordered aggregate of the type 
of the second number-class in its order of magnitude, 
and has thus the power Aleph-one. 

Proof. The totality of e-numbers of the second 
number-class, when arranged in their order of magni- 
tude, forms, by theorem C of 16, a well-ordered 
aggregate : 

(4) fy, e x , ...,,... e u +ij . . . Co? . . ., 

whose law of formation is expressed in the theorems 
D and E. Now, if the index a did not successively 
take all the numerical values of the second number- 
class, there would be a least number a which it did 
not reach. But this would contradict the theorem 
D, if a were of the first kind, and theorem E, if a 
were of the second kind. Thus a takes all numerical 
values of the second number-class. 

If we denote the type of the second number-class 
by ft, the type of (4) is 



[245] But since u> + o> a = o> 2 , we have 




200 THE FOUNDING OF THE THEORY 
and consequently _ _ 



G. If e is any e-number and a is any number of 
the first or second number-class which is less than e : 

a<e, 

then e satisfies the three equations : 



Proof. If ao is the degree of a, we have oo^la, and 
consequently, because of a < e, we also have a < e. 
But the degree of e = <o* is e ', thus a has a less 
degree than e. Consequently, by theorem D of 

19, 

a + e = e, 
and thus 

a +e = e. 

On the other hand, we have, by formula (13) of 
19, 



and thus 



ae = aw* = a> +e = of = e, 



Finally, paying attention to the formula (16) of 

19, 

a e = aw' = u> a u? ! = w" ' = a)' = e. 

H. If a is any number of the. second number-class, 
the equation 

< = f 

has no other roots than the e-numbers which are 
greater than a. 



OF TRANSFINITE NUMBERS 201 
Proof. Let ft be a root of the equation 

** = , 
so that 

o*-0. , 

Then, in the first place, from this formula follows 
that 



On the other, hand, /3 must be of the second kind, 
since, if not, we would have 



Thus we have, by theorem F of 19, 

a P = oP 

and consequently 

w^ = iQ. 

[246] By theorem F of 19, we have 
and thus 

But (3 cannot be greater than oo/3 ; consequently 

a oj# = fr 
and thus 

^ 
Therefore j3 is an e-number which is greater than a. 

HALLE, March 1897. 



NOTES 

IN a sense the most fundamental advance made in 
the theoretical arithmetic of finite and transfinite 
numbers is the purely logical definition of the 
number-concept. Whereas Cantor (see pp. 74, 
86, 112 above) defined "cardinal number" and 
' ' ordinal type " as general concepts which arise by 
means of our mental activity, that is to say, as 
psychological entities, Gottlob Frege had, in his 
Grttndlagen der Arithmetik of 1884, defined the 
" number (Ansahf) of a class u " as the class of all 
those classes which are equivalent (in the sense of 
PP- 75i 86 above) to u. Frege remarked that his 
"numbers" are the same as what Cantor (see pp. 
40, 74, 86 above) had called "powers," and that 
there was no reason for restricting ' ' numbers " to 
be finite. Although Frege worked out, in the first 
volume (1893) of his Grundgesetise der Arithmetik^ 
an important part of arithmetic, with a logical 
accuracy previously unknown and for years after- 
wards almost unknown, his ideas did not become at 
all widely known until Bertrand Russell, who had 
arrived independently at this logical definition of 
" cardinal number," gave prominence to them in his 



NOTES 203 

Principles of Mathematics of 1903.* The two chief 
reasons in favour of this definition are that it 
avoids, by a construction of "numbers" out of the 
fundamental entities of logic, the assumption that 
there are certain new and undefined entities called 
' ' numbers " ; and that it allows us to deduce at 
once that the class defined is not empty, so that 
the cardinal number of u "exists" in the sense 
defined in logic : in fact, since u is equivalent to 
itself, the cardinal number of has u at least as a 
member. Russell also gave an analogous definition 
for ordinal types or the more general "relation 
numbers. " f 

An account of much that has been done in the 
theory of aggregates since 1897 ma 7 De gathered 
from A. Schoenflies's reports : Die Entwickelung 
der Lehre von den Punktmannigfaltigkeiten, Leipzig, 
1900 ; 'part ii, Leipzig, 1908. A second edition 
of the first part was published at Leipzig and Berlin 
in 1913, in collaboration with H. Harm, under the 
title : Entwickelung der Mengenlehre und ihrer 
Anwendungen. These three books will be cited 
by their respective dates of publication, and, when 
references to relevant contributions not mentioned 
in these reports are made, full references to the 
original place of publication will be given. 



* Pp. 519, 1 1 1-116. Cf. Whitehead, Amer. foum. of Math. , vol. 
xxiv, 1902, p. 378. For a more modern form of the doctrine, see 
Whitehead and Russell, Principia McUhematica, vol. ii, Cambridge, 
1912, pp. 4, 13. 

t Principles, pp. 262, 321 ; and Principia^ vol. ii, pp. 330, 
473-510. 



204 NOTES 

Leaving aside the applications of the theory of 
transfinite numbers to geometry and the theory of 
functions, the most important advances since 1897 
are as follows : 

(1) The proof given independently by Ernst 
Schroder (1896) and Felix Bernstein (1898) of the 
theorem B on p. 91 above, without the supposition 
that one of the three relations of magnitude must 
hold between any two cardinal numbers (1900, pp. 
16-18; 1913, pp. 34-41 ; 1908, pp. 10-12). 

(2) The giving of exactly expressed definitions 
of arithmetical operations with cardinal numbers 
and of proofs of the laws of arithmetic for them by 
Ai N. Whitehead (Amer. Journ. of Math., vol. 
xxiv, 1902, pp. 367-394). Cf. Russell, Principles, 
pp. 117-120. A more modern form is given, in 
Whitehead . and Russell's Prindpia, vol. ii, pp. 
66-186. 

(3) Investigations on the question as to whether 
any aggregate can be brought into the form of 
a well-ordered aggregate. This question Cantor 
(cf. 1900, p. 49; 1913, p. 170; and p. 63 above) 
believed could be answered in the affirmative. 
The postulate lying at the bottom of this theorem 
was brought forward in the most definite manner 
by E. Zermelo and E. Schmidt in 1904, and 
Zermelo afterwards gave this postulate the form of 
an "axiom of selection" (1913* pp. 16, 170-184; 
1908, pp. 33-36). Whitehead and Russell have 
dealt with great precision with the subject in their 
Principia> vol. i, Cambridge, 1910, pp. 500-568. 



NOTES 205 

It may be remarked that Cantor, in his proof of 
theorem A on p. 105 above, and in that of theorem 
C on pp. 161-162 above,* unconsciously used this 
axiom of infinite selection. Also G. H. Hardy 
in 1903 (1908, pp. 22-23) use d this axiom, un- 
consciously at first, in a proof that it is possible to 
have an aggregate of cardinal number M X in the 
continuum of real numbers. 

But there is another and wholly different question 
which crops up in attempts at a proof that any 
aggregate can be well ordered. Cesare Burali-Forti 
had in 1897 pointed out that the series of all ordinal 
numbers, which is easily seen to be well ordered, 
must have the greatest of all ordinal numbers as its 
type. Yet the type of the above series of ordinal 
numbers followed by its type must be a greater 
ordinal number, for ft+ 1 is greater than /3. Burali- 
Forti concluded that we must deny Cantor's funda- 
mental theorem in his memoir of 1897. A different 
use of an argument analogous to Burali-Forti's was 
made by Philip E. B. Jourdain in a paper written in 
1903 and published in 1904 (Philosophical Magazine, 
6th series, vol. vii, pp. 61-75). The chief interest 
of this paper is that it contains a proof which is 
independent of, but practically identical with, that 
discovered by Cantor in 1895, an d of which some 



* Indeed, we have here to prove that any enumerable aggregate of 
any enumerable aggregates gives an enumerable aggregate of the 
elements last referred to. To prove that No Ko= No, it is not enough to 
prove the above theorem for particular aggregates. And in the general 
case we have to pick one element out of each of an infinity of classes, 
no element in each class being distinguished from the others. 



206 NOTES 

trace is preserved in the passage on p. 109 above 
and in the remark on the theorem A of p. 90. 
This proof of Cantor's and Jourdain's consists of 
two parts. In the first part 'it is established that 
every cardinal number is either an Aleph or is greater 
than all Alephs. This part requires the use of 
Zermelo's axiom; and Jourdain took the "proof" 
of this part of the theorem directly from Hardy's 
paper of 1903 referred to above. Cantor assumed 
the result required, and indeed the result seems very 
plausible. 

The second part of the theorem consists in the 
proof that the supposition that a cardinal number 
is greater than all Alephs is impossible. By a slight 
modification of Burali-Forti's argument, in which 
modification it is proved that there cannot be a 
greatest Aleph, the conclusion seems to follow that 
no -cardinal number can be other than an Aleph. 

The contradiction discovered by Burali-Forti is 
the best known to mathematicians ; but the simplest 
contradiction was discovered * by Russell (Principles,. 
pp. 364-368, 101-107) from an application to "the 
cardinal number of all things " of Cantor's argument 
of 1892 referred to on pp. 99-100 above. Russell's 
contradiction can be reduced to the following : If 
w is the class of all those terms x such that x is not 
a member of x > then, if w is a member of w, it is 
plain that w is not a member of w ; while if w is 
not a member of w, it is equally plain that w is a 
member of w. The treatment and final solution of 

* This argument was discovered in 1900 (see Mortis t t Jan. 1912). 



NOTES 207 

these paradoxes, which concern the foundations of 
logic and which are closely allied to the logical 
puzzle known as "the Epimenides, " * has been 
attempted unsuccessfully by very many mathe- 
maticians, -j- and successfully by Russell (cf. Principles, 
PP- 5 2 3-5 2 8 ; Principia, vol. i, pp. 26-31, 39-90). 

The theorem A on p. 105 is required (see theorem 
D on p. 1 08) in the proof that the two definitions 
of infinity coincide. On this point, cf. Principles^ 
pp. 121-123 ; Principle vol. i, pp. 569-666; vol. ii, 
pp. 187-298. 

(4) Investigation of number-classes in general, 
and the arithmetic of Alephs by Jourdain in 1904 
and 1908, and G. Hessenberg in 1906^ (1913, 
pp. 131-136; 1908, pp. 13-14)- 

(5) The definition, by Felix Hausdorff in 1904- 
1907, of the product .of an infinity of ordinal types 
and hence of exponentiation by a type. This 
definition is analogous to Cantor's definition of 
exponeritiation for cardinal numbers on p. 95 
above. Cf. 1913, pp. 75-80 ; 1908, pp. 42-45- 

(6) Theorems due to J. Konig (1904) on the 

* Epimenides was a Cretan who said that all Cretans were liars. 
Obviously if his statement were true he was a liar. The remark of a 
man who says, " I am lying," is even more analogous to Russell's w. 

t Thus Schoenflies, in his Reports of 1908 and 1913, devotes an 
undue amount of space to his "solution " of the paradoxes here referred 
to. This " solution" really consists in saying that these paradoxes do 
not belong to mathematics but to "philosophy." It may be remarked 
that Schoenflies seems never to have grasped the meaning and* extent of 
Zermelo's axiom, which Russell has called the " multiplicative axiom." 

t Just as in the proof that the multiplication of K by itself gives &*, 
the more general theorem here considered involves the multiplicative 
axiom. 

Cf, Jourdain, Mess, of Math. (2), vol. xxxvi, May 1906, pp, 13-16. 



208 . NOTES 

inequality of certain cardinal numbers ; aiid the 
independent generalization of these theorems, 
together with one of Cantor's (see pp. 81-82 
above), by Zermelo and Jourdain in 1908 (1908, 
pp. 16-17; 1913, PP. 65-67). 

(7) HausdorfPs contributions from 1906 to 1908 
to the theory of linear ordered aggregates (1913* 
pp. 185-205; 1908, pp. 40-71). 

(8) The investigation of the ordinal types of 
multiply ordered infinite aggregates by F. Riess 
in 1903, and Brouwer in 1913 (1913, pp. 85-87). 




INDEX 



Abel, Niels Henrike, 10. 
Abelian functions, 10, it. 
Absolute infinity, 62, 63. 
Actuality of numbers, 67. 
Addition of cardinal numbers, So, 
91 ff. 

of ordinal types, 81, 119 ff. 

of transfinite numbers, 63, 66, 



Adherences, 73. 

Aggregate, definition of, 46, 47, 

54, 74, 85- 
of bindings, 92. 
of union, 50, 91. 
Alembert, Jean Lerond d', 4. 
Algebraic numbers, 38 ff., 127. 
Aquinas, Thomas, 70. 
Aristotle, 55, 70. 
Arithmetic, foundations of, with 

Weierstrass, 12. 
with Frege and Russell, 202, 

203. 

Arzela, 73. 
Associative law with transfinite 

numbers, 92, 93, 119, 121, 

154, ISS- 

Baire, Rend, 73. 
Bendixson, Ivar, 73. 
Berkeley, George, 55. 
Bernoulli, Daniel, 4, 
Bernstein, Felix, 204. 
Bois-Reymond, Pauldu, 22,34, 51. 
Bolzano, Bernard, 13, 14, 17, 21, 

2>, 41. 55. 72- 
Borel, Emile, 73. 
Bouquet, 7. 
Briot, 7. 
Brodte, 73. 
Brouwer, 208. 
Burali-Forti, Cesare, 205, 206. 



Cantor, Georg, v, vi, vii, 3, 9, 10, 
13, 18, 22, 24, 25, 26, 28, 
29i 3, 3 2 . 33. 34, 3S, 3, 
37, 38, 41, 42, 45, 40, 47, 
48, 49, 5 1 , 52, 53, 54, 55, 
56, 57, 59, 60, 62 ' 6 3, 64. 
68, 69, 70; 72, 73, 74, 76, 
77, 79, 80, 8r, 82, 202, 
204, 205, 206, 208, 
Dedekind axiom, 30. 

Cardinal number ( see also Power), 

74, 79 ff., 85 ft, 202. 
finite, 97 ff. 
smallest transfinite, 103 ff. 

Cardinal numbers, operations 

with, 204. 
series of transfinite, 109. 

Cauchy, Augustin Louis, 2, 3, 4, 6, 
8,12, 14,15, 16,17,22,24. 

Class of types, 114. 

Closed aggregates, 132. 

types, 133- 

Coherences, 73. 

Coherent series, 129, 130. 

Commutative law with transflnite 
numbers, 66, 92, 93, 
119 ff., 190 ff. 

Condensation of singularities, 3, 
9, 48, 49- 

Connected aggregates, 72. 

Content of aggregates, 73. 

Content-less, 51. 

Continuity, of a function, i. 

Continuous motion in discon- 
tinuous space, 37. 

Continuum, 33, 37, 41 ff., 47, 48, 
64, 70 ff., 96,203. 

Contradiction, Russell's, 206, 207. 

Convergence of series, i, 15, 16, 
17, 20, 24. 

Cords, vibrating, problem of, 4. 
209 



2IO 



INDEX 



D'Alembert (see Alembert, 

J. L. d'). 
Dedekind, Richard, vii, 23, 41, 

,. 47, 49, 73- 
Definition of aggregate, 37. 
Democritus, 70. 
De Morgan, Augustus, 41. 
Density in itself, 132. 
Derivatives of point- aggregates, 

3. 30 ff., 37- 
Descartes, Rene, 55. 
Dirichlet, Peter Gustav "Lejeune, 

2, 3, 5, 7, 8, 9. 22, 35- 
Discrete aggregates, 51. 
Distributive law with transfinite 

numbers, 66, 93, 121, 155. 

Enumerability, 32, 38ff.,47, 50 ff., 

62. 

Enumeral, 52, 62. 
Epicurus, 70. 
Epirnenides, 207. 
Equivalence of aggregates, 40, 

75, 86 ff. 

Euler, Leonhard, 4, 5, 9, 10. 
Everywhere-dense aggregates, 33, 

35. 37, 3. 
types, 133. 

Exponentiation of transfinite 
numbers, 82, 94 ff., 207. 

Fontenelle, 118. 

Formalism in mathematics, 70, 81. 

Fourier, Jean Baptiste Joseph, i, 

2, 6, 8, 24. 

Freedom in mathematics, 67 ff. 
Frege, Gottlob, 23, 70, 202. 
Function, conception of, I. 
Functions, theory of analytic, 2, 

6, 7, 10, 11, 12, 13, 22, 

73- 

arbitrary, 4, 6, 34. 
theory of real, 2, 8, o, 73. 
Fundamental series, 26, 128 ff. 

Gauss, Carl Friedrich, 6, 12, 14. 
Generation, principles of, 56, 57. 
Gudermann, 10. 

Haim, H., 203. 

Haller, Albrecht von, 62. 



Hankel, Hermann, 3, 7, 8, 9, 17, 

49, 7- 

Hardy, G. H., 205, 206. 
Harnack, Axel, 51, 73. 
Hausdorff, Felix, 207, 208. 
Heine, H. E. f 3, 26, 69. 
Helmholtz,. H. von, 42, 70, 81. 
Hessenberg, Gerhard, 207. 
Hobbes, Thomas, 55. 

Imaginaries, 6. 

Induction, mathematical, 207. 

Infinite, definition of, 41, 61, 62. 

Infinitesimals, 64, 81. 

Infinity, proper and improper, 55, 

79- 

Integrability, Riemann's con- 
ditions or, 8. 

Integrable aggregates, 51. 
Inverse- types, 114. 
Irrational numbers, 3, 14 ff., 26 ff. 
analogy of transfinite numbers 

with, 77 ff. 

Isolated aggregate, 49. 
point, 30. 

Tacobi, C. G. T., 10. 
Jordan, Camille, 73. 
Jourdain, Philip E. B., 4, 6, 20, 
32, 52, 205, 206, 207, 208, 

Killing, W., 118. 

Kind of a point-aggregate, 32. 

Kirchhoff, G., 69. 

Ko'nig, Julius, 207. 

Kronecker, L., 70, 81. 

Kummer, E. E. , 69. 

Lagrange, J. L..S, 14- 
Leibniz, G. W. von, 55. 
Leucippus, 70. 
Limitation, principle of, 60. 
Limiting element of an aggregate, 

131- 

Limit-point, 30. 

Limits with transfinite numbers, 

77 ff., 131 ff., 58 ff. 
Liouville, L, 40. 
LipschitZj R., 6. 
Locke, J., 55. 
Lucretius, 70, 



INDEX 



211 



Mach, Ernst, 69. 
Maximum of a function, 22. 
Mittag-Leffler, Gosta, 11. 
Mutliplication of cardinal num- 
bers, 80, 91 fE 

of ordinal types 81, 119 ft, 154. 
of transfinite numbers, 63, 64, 
66, 176 ff. 

Newton, Sir Isaac, 15. 
Nominalism, Cantor's, 69, 70. 
Number-concept, logical definition 
of, 202, 203. 

Ordinal number (see also 

Enumeral), 75, ii ff. 
numbers, finite, 113, 158, 159. 
type, 75, 79 ff., noff. 
type of aggregate of rational 

numbers, I22ff., 202. 
types of multiply ordered aggre- 
gates, 8 1, 208. 
Osgood, W. F., 73. 

Peano, G., 23. 

Perfect aggregates, 72, 132. 

types, 133. 
Philosophical revolution brought 

about by Cantor's work, 

vi. 
Physical conceptions and modem 

mathematics, I. 
Point-aggregates, Cantor's early 

work on, v, vi. 
theory of, 3, 20 ff., 30 ff., 64, 

73- 

Potential, theory of, 7. 
Power, second, 64 ff. , 169 ff. 
of an aggregate, 32, 37, 40, 

52 ft, 60, 62. 

Prime numbers, transfinite, 64, 66.. 
Principal element of an aggregate, 

13*- 
Puiseux, V., 7. 

Reducible aggregates, 71. 
Relation numbers, 203. 
Riemann, G. F. B., 3, 7, 8, 9, 10, 
T2, 25, 42. 



Kiess, F., 208. 

Russell, Bertrand, 20, 23, 53, 202, 
203, 204, 206, 207. 

Schepp, A., 117. 
Schmidt, E., 204. 
Schoenflies, A., 73, 203, 207. 
Schwarz, 8, 12. 

Second number-class, cardinal 
number of, 169 ff. 

epsilon-numbers of the, 195 ft 

exponentiation in, 178 ff. 

normal form of numbers of, 
183 ff. 

numbers of, 160 ff. 
Segment of a series, 60, 103, 

141 ff. 

Selections, 204 ff. 
Similarity, 76, H2ff. 
Species of a point-aggregate, 31. 
Spinoza, B., 55. 
Steiner, J., 40. 
Stolz, O., 17, 73- 
Subtraction of transfinite numbers, 
66, 155, 156- 

Teubner, B. G., vii. 

Transfinite numbers, 4, 32, 36, 

50 ff., 52 ff. 
Trigonometrical developments, 2, 

3,4, 5,6, 7, 8, 24ff., 31. 

Unextended aggregates, 51. 
Upper limit, 21. 

Veronese, G., 117, 118. 

Weierstrass, Karl, vi, vii, 2, 3, 10, 
H, 12, 13, 14, 17, 18, 19, 
20, 21, 22, 23, 24, 26, 30, 
48. 

Well-ordered aggregates, 60, 61, 
75 ff., 137 ff. 

Well-ordering, 204 ff. 

Whitehead, A. N., 203, 204. 

Zeno, 15. 

Zermelo, E., 204, 206, 207, 208. 

Zermelo's axiom, 204 ff.