CONTRIBUTIONS TO
THE FOUNDING OF THE THEORY OF
TRANSFINITE NUMBERS
GBOBG CANTOR (1846-1918)
u nns-u '" ' "- '"- jf <t
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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.