I 
 
i 
 
CONTRIBUTIONS TO 
 
 THE FOUNDING OF THE THEORY OF 
 
 TRANSFINITE NUMBERS 
 
/ Court Series of Classics of Science and 
 Thilosophy, ,?(o. i 
 
 TRIBUTIONS TO 
 
 INDING OF THE THEORY OF 
 
 Ik SFINITE NUMBERS 
 
 BY 
 
 GEORG CANTOR 
 
 TRANSLATED, AND PROVIDED WITH AN INTRODUCTION 
 AND NOTES, BY 
 
 PHILIP E. B. JOURDAIN 
 
 M. A. (Cantab.) 
 
 CHICAGO AND LONDON 
 PEN COURT PUBLISHING COMPANY 
 1915 
 
G2 
 
 Copyright in Great Britain under the Act of igii 
 
PREFACE 
 
 This volume contains a translation of the two very 
 iniportant memoirs of Georg Cantor on transfinite 
 numbers which appeared in the Mathematische 
 Annalen for 1895 ^^'^^ 1897* under the title: 
 ''Beitrage zur Begriindung der transfmiten 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^ theorie des ensembles), 
 — the elements of the sets being real or complex 
 numbers which are imaged as geometrical " points " 
 in space of one or more dimensions, — the title given 
 to them 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 1870. 
 It is, I think, necessary, if we are to appreciate the 
 full import of Cantor's work on transfinite numbers, 
 ■' -> ^ave thought through and to bear in mind Cantor's 
 
 ' researches on the theory of point-aggregates. 
 
 s in these researches that the need for the 
 
 ^ol, xlvi, 1895, pp. 481-512 ; vol. xlix, 1897, pp. 207-246. 
 
 334640 
 
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. I^ 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 at out 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 u idying 
 theory; but very many philosophers combaied 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 philosophcs and 
 mathematicians. 
 
 The three men whose influence on moder;i pure 
 mathematics — and indirectly modern logic and the 
 
PREFACE vii 
 
 philosophy which abuts on it — is most marked are 
 Karl Weierstrass, Richard Dedekind, 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 unci irrationale Zahlen 
 ar^d Was sind und was sollen die Zahlen P, 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, 1 
 am indebted to Messrs B. G. Teubner of Leipzig 
 and Berlin, the publishers of the Matheinatische 
 Annalen. 
 
 PHILIP E. B. JOURDALN. 
 
 * Essays on the Theory of Numbers (I, Continuity and Irrational 
 Numbers; II, The Nature arid Meaning- of Numbers), translated by 
 W. W. Reman, Chicago, 190T. I shall refer to this as Essays on 
 Number. 
 
 t By F. Marotte, Sur les fondements de la theorie des ensembles 
 trans/mis, Paris, 1899. 
 
TABLE OF CONTENTS 
 
 I'AGE 
 
 Preface v 
 
 Table of Contents ix 
 
 Introduction i 
 
 Contributions to the Founding of the Theory 
 OF Transftnite 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 
 
1 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 P^ourier 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 th^ 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 conaplex 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 me.moir 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 BernouUi 
 produced a solution in the form of an infnite 
 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 ^(;ir) were develops 
 able in a series of the form 
 
 * Cf. the references given in my papers in the Archiv der Matkematfk 
 liud Physik, 3rd series, vol. x, 1906, pp. 255-256, and his, vol. i, 
 1914, pp. 670-677. Much of this Introduction is taken frorr. my 
 account of "The Development of the Theory of Transfinite Number-i" 
 in the above-mentioned Archiv, 3rd series, vol, x, pp. 254-2S1 ; 
 vol. xiv, 1909, pp. 289-311; vol. xvi, 1910, pp. 21-43; vol. j tii, 
 1913, pp. 1-21. 
 
 t The arbitrary functions chiefly considered in this connexio i by 
 Euler were what he called "discontinuous" functions. This ^/ord 
 does not mean what we now mean (after Cauchy) by it. Cj\ my paper 
 in his, vol. i, 191 4, pp. 661-703. 
 
INTRODUCTION 
 
 vkx 
 
 
 That this was, indeed, the case, even when 0(;f) 
 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 b}' 
 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, 
 
 0(;ir) = J/^o + ^i cos^'H-<^2 ^os 2;f+ . . . 
 -\-a^?A\-\ x-\-a^^^\\\2x-\- . . ., 
 in the form 
 
 d^=- I (l>{a) cos vada, «,. = — I 0(a) sin vada. 
 it] ttJ 
 
 -n -7T 
 
 This determination was probably independent of 
 Euler's prior determination and Lagrange's analog- 
 ous determination of the coefficients of a Jinzte 
 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 des series trigonometriques qui servent a 
 representer une fonction arbitraire entre des liniites donnees,"y<?«;-«. 
 fur Math., vol. iv, 1829, pp. 157-169; Ges. Werke^ vol. i, 
 pp. 1 17-132, 
 
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 contem.plation 
 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- 1 857).* 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 18 14 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 
 prohfic 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.'Maih. (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 (185 i) 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- 1 873) 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 Hankel's 
 research was provided by reflexion on the founda- 
 tions of Riemann's theory of functions of a complex 
 variable. It was Hankel's 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 Riemann 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 
 Feststelkmg des Begriffes der Function liberhaupt " 
 of 1870 by a thorough examination of the various 
 possibihties 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 x'<^ 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 ;r 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 ^=;^ + i, where n is an integer (positive, 
 negative, or zero), and with \ and — \ for upper and 
 lower limits respectively. Further, (la-), where v is 
 an integer, is discontinuous at the points vx=n-\-\ 
 
 ox x—-{7i-\-\). Consequentl}-, the series 
 
 l'=l ^ 
 
 where the factor \\v^ is added to ensure convergence 
 for all values of ;r, may be supposed to be discon- 
 tinuous for all values of x of the form x=p\2n^ 
 where/ is an odd integer, relatively prime to ;^. 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, 
 
10 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 (i8i 5-1897) 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 18 39- 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 las yet no example, I regarded as one 
 
INTRODUCTION ii 
 
 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. " 
 
 VV^ith 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 — o^^ the 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 more 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 les fondements arithmetiques cle la theorie des fonctions 
 d'apres Weierstrass," Congees des Alath^ma/iques ct Shnk/iohn, 
 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 Riemann ; 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, qud 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 an 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 
 lriHhil;ely 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 Bolzano 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 defining 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 modern 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 us. 
 
INTRODUCTION 15 
 
 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 incomrtiensurables. 
 
 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 'Miumber" 
 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 " Hmit " 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 Hmit 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 lattei* 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 u^, u^, 
 u^y . . , to be convergent if the sum J„ = 2^0 + ^1 + ^2 
 + . . . +2^«-i, for values of n always increasing, 
 approaches indefinitely a certain limit s, that, "by 
 the above principles, in order that the series 
 ^oj ^^1' ^'i^ ' ' ' ^^y ^^ convergent, it is necessary 
 and sufficient that increasing values of n make the 
 sum s,, 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^, j^^+i, 
 s,,^2y ' • ' differ from the limit s, and consequently 
 from one another, by infinitely small quantities." 
 Hence it is necessary and sufficient that the different 
 sums u,, + u,,^i-{- . . . +2/,,+;;,, for different m'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,, 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 6, 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 Jias a Hmit — 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 j^, j,,+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 infijiite 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 " {b) 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 Ostwald^s A'iassiker, 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. 
 
 2 
 
V 
 
 > 
 
 ig INTRODUCTION 
 
 rational numbers can only be defined when we have 
 already defined the real numbers, of which b is one. 
 " I believe," said Cantor,* a propos 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, 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 aHquot parts, 
 Weierstrass called any aggregate whose elements 
 and the number (finite) of times each element 
 occurs in it f 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 <:^, 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 
 ^, 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 " b. 
 
 VVeierstrass called the numerical quantity c de- 
 fined by {i.e. identical with) the aggregate whose 
 elements are those which appear in a or ^, 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 '' stmi'' 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, ^, . . . was then defined to be the 
 aggregate {s) whose elements occur in one (at least) 
 of ^, ^, . . ., each of these elements e being taken 
 
20 INTRODUCTION 
 
 a number of times (;?) equal to the number of times 
 that it occurs in a, increased by the number of times 
 that it occurs in b^ 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^ . . . \s less than N. 
 , Such is the principal point of Weierstrass's theory 
 y/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 ' ' 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 
 Hne, 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- 
 
 * C/; Juurdain, Math. Gazette, ]iin. 1908, vol. 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 " {Hdufungss telle) in question. An 
 analogfous theorem holds for the two-dimensional 
 region of complex numbers. 
 
 Of real numerical magnitudes x^ 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 — (5), the end G being excluded. Ana- 
 logously for the " lower limit "^. 
 
 It must be noticed that, if we have ^ finite 
 aggregate of x's, one of these is the upper limit, 
 and, if the aggregate is infinite, one of them may 
 be the upper hmit. 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 ,r's " to "being 
 excluded" by " certain ;r's lie in the arbitrary small 
 interval (G, . . ., G — S), the end G being i^icluded.'' 
 
 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 o{ y=f{x)'^ which 
 belong to the values of x lying inside the interval 
 from a to h, there is, in this interval, at least one 
 point ;i = X such that the upper hmit of the j's 
 which belong to the ;r's in an arbitrarily small 
 neighbourhood of X is G ; and analogously for the 
 lower limit. 
 
 If the j^-value corresponding to ;r=X is G, the 
 upper limit is called the "maximum" of the j^'s 
 and, if j?^=/"(-^) is a continuous function of x^ the 
 upper limit is a maximum ; in other words, a con- 
 tinuo2is 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 \i y is finite for every single x of the interval a^x^b, all 
 these jj/'s need not be, in absolute amount, less than some finite number 
 (for example, f{x)=\\x for jf>0, /(o)=o, in the interval o^ ^^i), 
 but if they are (as in the case of the sum of a uniformly convergent 
 series), these j/'s have a finite upper and lower limit in the sense defined. 
 
 t There is another conception (due to Cauchy and P. du Bois- 
 Reymond), 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 " Limites" 
 
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 ^y 
 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. y 
 
 V 
 
 Georg Ferdinand Ludwig Philipp Cantor was 
 born at St Petersburg on 3rd March 1845, ^"d 
 lived there until 1856; from 1856 to 1863 he lived 
 in South Germany (Wiesbaden, Frankfurt a. M., 
 and Darmstadt); and, from autumn 1863 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 : Journ. fiir Alath.^ vols. Ixxvii and Ixxxiv, 1874 and 
 1878; Math. Ann., vol. iv, 1871, vol. v, 1872, vol. xv, 1879, vol. xvii, 
 j88o, 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 
 I 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^, a^, . . ., a^, . . . and b^, b., . . ., b^, . . . 
 
 are two infinite series such that the limit of 
 
 a^ sin vx-{-b^, 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 v, both a^ and b^ converge, with 
 increasing i/, to zero. This theorem leads to a 
 criterion for the convergence of a trigonometrical 
 series 
 
 \b^^a^^\Xix^b^QO'6x-\- . . . -\- a,,'=^\\\vx -\- b ^QO^vx -\- . . ., 
 
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 
 oi f{x) in the form of a trigonometrical series con- 
 vergent for every value of x, except, possibly, a 
 finite number of x's ; if the sums of two trigono- 
 metrical series differ for a finite number of ;tr's, the 
 forms of the series coincide. 
 
 In 1 87 1, Cantor gave a simpler proof of the 
 uniqueness of the representation, and extended this 
 theorem to : If we have, for every value of x, 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^ sin vx-\-b^ cos vx) = o for a<x<b, then both 
 lim a^, and lim b^ 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's in the / 
 interval o . . . 27r 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 
 
 (i) «i, a^, . . ., a,, . . , 
 
 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 a,,+,„-a,, I <e (^>^i), 
 
 whatever the positive integer m is.* This property 
 Cantor expressed by the words, ''the series (i) 
 has a determined limit ^," 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 d, we 
 must also attach different signs l?\ 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) d such that \b — a,\ 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 (a^) is a 
 '' number-series," or, as Cantor afterwards expressed 
 it, (a^) 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 (l) 
 should be a " finite numerical mai^nitude " in Weierstrass's sense ; and 
 consequently Cantor's theory of irrational numbers has been described 
 as a happy modification of Weierstrass's, 
 
INTRODUCTION 27 
 
 Let a second series 
 
 (i') a\, a\, . . ., a\,, . . . 
 
 have a determined limit d\ we find that (i) and 
 (i') have always one of the three relations, which 
 exclude one another: (a) a„ — a'„ becomes infinitely 
 small as 7i increases ; (d) from a certain n on, it 
 remains always greater than e, where e is positive 
 and rational ; (c) from a certain n on, it remains 
 always less than —e. In these cases we say, 
 respectively, 
 
 /; = ^', /;>//, or /?<d'. 
 
 Similarly, we find that (i) has only one of the 
 three relations with a rational number a : (a) a,^ — a 
 becomes infinitely small as n increases ; (d) from 
 a certain n on, it remains always greater than e ; 
 (c) from a certain n on, it remains less than — e. 
 We express this by 
 
 l? = a, d>a, or l?<a, 
 
 respectively. Then we can prove that /; — a„ becomes 
 infinitely small as n increases, which, consequently, 
 justifies the name given to I? of ''limit of the 
 series (i). " 
 
 Denoting the totality of the numerical magnitudes 
 d by B, we can extend the elementary operations 
 with the rational numbers to the systems A and B 
 united. Thus the formulae 
 
 l?±d' = y\ bb' = b'\ b\b' = b" 
 
28 INTRODUCTION 
 
 express that the relations 
 
 lim {a„^a ,, - a" ,^ = o, \\m (a,,a „ - 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) K ^2. • • •' ^v, . . . 
 
 be composed of numbers from A and B (not all 
 from A), and such that | b,,^„, — bn \ becomes in- 
 finitely small as n increases, whatever in is (this 
 condition is determined by the preceding definitions), 
 then (3) is said to have '*a determined limit ^." 
 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 <^ to a <^, but not inversely, 
 we can equate each ^ to a <;, 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^^ 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 b^ where b is the numerical 
 magnitude corresponding to the series (i). VVe 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 
 
30 INTRODUCTION 
 
 of the straight line, an axiom must still be added, 
 which runs : To each numerical magnitude belongs 
 
 falso, 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, and understand by the word " limit-point " 
 {Grenzpunkt) 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 VVeierstrass'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) V . 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 3I 
 
 process, a second derived system P'' from F ; and, 
 by V analogous operations, we arrive at the notion 
 of a i/th system P^"^ 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 . . . i), 
 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, . , ., \\v . . ., 
 
 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, P^") 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 ^^ vX\\ species 
 {Art);' and thus Y\ P", ... are of the (t/-i)th, 
 (i/— 2)th, . . . species respectively. 
 
 The extended trigonometrical theorem is now : 
 If the equation 
 
 = J^Q + <a:j sin. r -1-^1 cos ^'+ . . . -\-a^^\\\vX 
 
 + ^^cos vx-\- . . , 
 
 is satisfied for 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 . . . 27r), then 
 
 b^ = o, c\ = b, = o. 
 
32 INTRODUCTION 
 
 Further information as to the conthiuation of 
 these researches into derivatives of point-aggregates 
 was given in the series of papers which Cantor 
 began in 1879 under the title " Ueber unendHche, 
 lineare Punktmannichfaltigkeiten." Although these 
 papers were written subsequently to Cantor's dis- 
 covery (1873) of the conceptions of " enumerability " 
 {Abzdhlbarkeit) and "power" {Miichtigkeit), 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 
 1 87 1, we shall now extract from these papers -the 
 parts concerning derivatives. 
 
 A point-aggregate P is said to be of the ''first 
 kind" (Gattung) and i^th ''species" if P^''^ 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", Y" , ... 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, 
 necessary and logical," of the transfinite 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 31st August 1905, Professor 
 Cantor wrote: "Was die transfiniten Ordnungszahlen betriftt, ist es 
 mir wahrscheinlich, dass ich schon 1871 eine Voistellung vcn ihnen 
 gehal:)t habe. Den Begrifif der Abzahlbarkeit bildete ich mir erst 
 1873." 
 
INTRODUCTION 33 
 
 Some or all of the points of a continuous * interval 
 (a . . . ^), the extreme points being considered as 
 belonging to the interval, may be points of P ; if 
 none are, ? is said to be quite outside (a . . . /3). If 
 P is (wholly or in part) contained in (a . . . 0), a 
 remarkable case may present itself : every interval 
 (y . . . ^) in it, however small, may contain points 
 of P. Then P is said to be ' * everywhere dense " 
 in the interval (a . . . fi). For example, (i) the 
 point-aggregate whose elements are all the points 
 of (a . . . /3), (2) that of all the points whose 
 abscissa? are rational, and (3) that of all the 
 points whose abscissae are rational numbers of the 
 form zL(2n-\- i)/2"', where m and n are integers, are 
 everywhere dense in (a . . . ^)- It results from this 
 that, if a point-aggregate is not everywhere dense 
 in (a . . . ^), 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 . . . /3), not only is the same true for P', 
 but P' consists of all the points of (« . . . jS). We 
 might take this property of P' as the definition 
 of the expression : * ' P is everywhere dense in 
 {a...,8y 
 
 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 completest explanation respecting the determination of a continuum 
 rests " (see below). 
 
 3 
 
34 INTRODUCTION 
 
 second kind is everywhere dense in some intervals, 
 Cantor postponed it. y 
 
 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^ condense on nearing a finite 
 number of points k^ ; (2) the points k^ condense at 
 a finite number of points k^, . . . Thus, the 
 roots of o = sin i/jc condense near x=o, those of 
 o = sin I /sin i/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 {pantacJiiscJi) ; that 
 is to say, we can so distribute intervals D on the 
 interval ( — tt . . . + tt) that in any connected 
 portion, however small, of ( — tt . . . +7r) connected 
 intervals D occur. Let, now, ^(,r) be o in these 
 D's and i in the points of ( — tt . . . +7r) not covered 
 by D's ; then (^{x) is not integrable, although any 
 interval inside ( — tt . . . + 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 i 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/r, . . . ; in the interval (1/1/ . . . i/(i/+ i)) 
 take a point-aggregate of the first kind and j/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 
 
 pC'^-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'', . . ., P^"), . . ., 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(°°\ and called it 
 "the derived aggregate of P of order oo (infinity)." 
 The first derivative of P^°°) was denoted by p(~+i), 
 and so on for 
 
 p(cxD+2) p(co+3) ^ ^ ^ p(o 
 
 +v) 
 
 Again, P(~) may have a derivative of infinite order 
 which Cantor denoted by P^^°°^ ; and, continuing 
 these conceptual constructions, he arrived at de- 
 rivatives which are quite logically denoted by 
 p(woo+«)^ where in 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(°°'^), and so on without 
 end. Thus he got derivatives of indices 
 
 S3 
 
 1^0^ +1^100 i- . . . -tv^^y ... 00 , ... CO J . . . 
 
 "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 i88o], but when 
 exposing my theory of the number-conception I did not refer to it." 
 
. INTRODUCTION 37 
 
 We see that point-aggregates of the first kind 
 are characterized by the property that P^~^ has no 
 elements, or, in symbols, 
 
 p(-) = o^ 
 
 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 "derivative" and "everywhere 
 dense " to aggregates situated in continua of 71 
 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^°°^, 
 P^^~\ . . ., 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 
 possible 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 onl}' 
 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 of whole 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 w which is a root of a non-identical 
 equation of the form 
 
 (4) ^0^^^ + ^!^""^+ • • • +^« = 0' 
 
 where n, a^, a^, . . ., a„ are integers, is called a real 
 algebraic number ; we may suppose n and a^ positive, 
 ^0, a^, . . ., a,, to have no common divisor, and (4) 
 to be irreducible. The positive whole number 
 
 N = ?2 - I -F I ^0 I + I ^1 I + • • • + I ^« I 
 may be called the 'Mieight" of w ; 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 
 
 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 totahty of the real 
 numbers in the interval (a . . . ,8), where a</3, 
 could be arranged in the simply infinite series 
 
 (5) ^1, u.^, . . .^u,, . . . 
 
 Let a', /3' be the two first numbers of (5), different 
 from one another and from a, /3, and such that 
 a <^' \ similarly, let a\ j^'\ 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 p\ ^'\ ... of decreasing magnitude. 
 Each of the intervals (a . . . /3), (a . . . /3'), 
 {a" . . . j3"), . . . 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^"^ . . . /S^"^) ; then, since there is in this interval at 
 most one number of (5), we can take in it a number 
 rj which does not belong to (5) ; — or (d) there are 
 infinitely many intervals. Then, since a, a, a\ . . . 
 increase constantly without increasing ad infinituvi^ 
 they have a certain limit a^°°\ and similarly /3, /3', 
 /3'', . . . decrease constantly towards a certain 
 limit Z?^"'). If a^°°) = /3(°°) (which always happens 
 when applying this method to the system (o))), we 
 easily see that the number ri = a^"'^ cannot be in (5).* 
 If, on the contrary, a^°'^<^''~\ every number tj in 
 
 •■ For if it were, we would have y} = up, p being a determined index ; 
 but that is not possible, for up is not in (a^^^ . . . /S^-^^), whilst tj, by 
 definition, is. 
 
v-^ 
 
 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 
 (i/), 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, 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 " {Mdchtigkeit *) or to be 
 ' ' equivalent " {aequivalent). When an aggregate 
 is finite, the notion of power corresponds to that of 
 number {Anzahl), 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 projective 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 j/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 ;2-ply 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^^ x^, 
 . . ., ;ir„, and reciprocally to each admissible system 
 x^^ x^, . . ., 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,t we can prove 
 that there is a (i, i)-correspondence between the 
 elements of the linear continuum and those of a 
 n-p\y extended continuum. 
 
 This evidently follows from the proof of the 
 theorem: Let x^^, x^, . . ., x,, be real, independent 
 variables, each of which can take any value o<x< 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^^, x,^, . . ., x,„ and via versa. 
 To prove this, we set out from the known theorem 
 that every irrational number e between o and i can 
 be represented in one manner by an infinite con- 
 tinued fraction which may be written : 
 
 («!, otg, . . ., a^, . . .)» 
 
 * That is to say, an infinitely small variation in position of the element 
 imphes an infinitely 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 a's are positive integers. There is thus 
 a (i, i)-correspondence between the ^'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, 1, ai, 2j ' • •, cti, ^, . . .), 
 ^a~(^2, 1) «2, 2> •••) a2, I'j •••))•• •) 
 ^/; = (a„, 1, a„, 2> . . •> a«, V, . . •) 5 
 these n irrational numbers uniquely determine a 
 (/^+ i)th irrational number in (o . . . i), 
 
 ^=(A, ft, . . . ft, . . .), 
 
 if the relation between a and /3 : 
 
 (6) ft.-i)«+A^ = a^,v* (m= I, 2, . . ., n\ 1^=1, 2, . . .CO) 
 
 is established. Inversely, such a ^t' determines 
 uniquely the series of fts and, by (6), the series of 
 the a's, and hence, again of the ^'s. We have only 
 to show, now, that there can exist a (i, i)-corre- 
 spondence between the irrational numbers o<e<\ 
 and the real (irrational and rational) numbers 
 o<^'<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 
 
 ^1, 02J • • -J V-V) • • -t 
 
 * If we arrange the n series of o's in a double series with n rows, 
 this nieans that we are to enumerate the a's in the order aj ^ ^ a., , ^ 
 . . . a^, J, Oj o, a., o, . . . , and that the ^th term of this series 
 is /3r. ' 
 
 t This is done most simply as follows : Let plq be a rational number 
 of this interval in its lowest terms, and put / + ^ = N. To each plq 
 
44 INTRODUCTION 
 
 Then in (o . . .i) we take any infinite series of 
 irrational numbers tji, t]2y - - -y ^vy • - • (^o^ example, 
 tj^= J 212"), and let /i take any of the values of 
 O . . . i) except the 0's and >/s, so that 
 
 and we can also write the last formula : 
 
 Now, if we write a Oo d for ' ' the aggregate of the a's 
 is equivalent to that of the ^'s," and notice that aooa, 
 a c\j 3 and d cx) c imply a c\) 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 
 
 and 
 
 X cso e. 
 
 A generalization of the above theorem to the case 
 of x-^, x.^, . . ., x^, . . . 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^^ „}, where 
 
 ^;a = («^t, 1 , tt/x, 2, • . . , a/x, «" • • • ) ^^^' M = I J 2, ... CO 
 
 belongs a determined positive integral value of N, and to each such N 
 belong a finite number of fractions />/(/. Imagine now the numbers //(^ 
 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 a's is the union of those 
 of the ^'s, 7;»''s, and (pyS ; and analogously for that of the ^'s. 
 
INTRODUCTION 45 
 
 and the simple series {/3;y^}, a (i, i)-correspondence 
 can be established * by putting 
 
 \=fj.-^{fj,^V- l)(/x + j/-2)/2, 
 
 and the function on the right has the remarkable 
 property of representing all the positive integers, 
 and each of them once only, when /x 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 u can 
 receive all positive integral values ; and the other 
 containing all manifolds reducible to the ioxxn functio 
 ipsius ,r, where x can take all the real values in the 
 interval (o . . . i). " 
 
 In the paper of 1879 already referred to, Cantor 
 
 * Enumerate the double series |a^^ ^,\ diagonally, that is to say, 
 in the order 
 
 The term of this series whose index is (/i, v) is the Ath, where 
 A=I+2 + 3+ . . .+()U + l/-2) + /i = (/A + l'-2)(;i + |/-l)/2 + /x. 
 
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 them. 
 
 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 tt 
 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, 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 11 
 dimensions f {a) separated from one another and 
 most meeting at their boundaries ; the aggregate of 
 the rt:'s is enumerable. 
 
 For refer A by means of reciprocal radii vectores 
 to an ;^-ply extended figure B within a («+ i)- 
 dimensional infinite space /\', and let the points of 
 B have the constant distance i from a fixed point 
 of A'. To every a corresponds a /^-dimensional 
 part <^ of B with a definite content, and the ^'s are 
 enumerable, for the number of b's greater in con- 
 tent than an arbitrarily small number y is finite, for 
 their sum is less than 2'V + (the content of B). § 
 
 * Lindemann afterwards proved that it 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. 
 
 :;: In the French translation (1883) of Cantor's memoir, this num])er 
 was corrected to Q w (m I ly g /F ^fyr-MH/g). 
 
 § When n=i, the theorem is that every aggregate of intervals on a 
 
 2^ -i. cf,M,.tt..A..n t^Zl ^.fT'^P, 
 
 r(^-r) 
 
48 INTRODUCTION 
 
 Finally, Cantor made the interesting remark that, 
 if we remove from an ;^-dimensional continuum any 
 enumerable and everywhere-dense aggregate, the 
 remainder (21), if ^>2, does not cease to be con- 
 tinuously connected, in the sense that any two 
 points N, N' of 51 can be connected by a continuous 
 line composed of circular arcs all of whose points 
 belong to 51. 
 
 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 ^p{x) be a given function with the 
 single singularity x—O, and (w^,) any enumerable 
 
 aggregate ; put 
 
 00 
 
 v = l 
 
 where the ^^'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 their ends is 
 enumerable. The end-points are consequently enumerable, but not 
 always the derivative of this aggregate of end-points. 
 
 * Inaletter tome of 29th March 1905, Professor Cantor said : "Atthe 
 conception of enumerability, of which he [Weierstrass] heard from me at 
 Berlin in the Christmas holidays of 1873, l^e 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 A' = a)^ the same kind of singu- 
 larity as (^{x) at;ir=o, and at other points behaves, 
 in general, regularly. The singularity at x = {^^ is 
 due exclusively to the one term of the series in 
 which j/ = /u ; the aggregate (co^) 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 " (Jeber 
 unendliche, lineare Punktmannichfaltigkeiten " con- 
 tained six theorems on enumerable point-aggregatfes. 
 H^an aggregate O (in a continuum of n dimensions) 
 is such tha't none of its points is a limit-point,* it is 
 said to be " isolated." Then, round every point of 
 O 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. P'or let 
 
 ^(P, P')^R, P-R^O;t 
 
 then O 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 X'(Q> Q') — O- Q- Dedekind's Essays on 
 Nuf/iber, p. 48. 
 
 t If an aggregate B is contained in A, and E is the aggregate left 
 when B is taken from A, we write 
 
 E = A-B. 
 
50 INTRODUCTION 
 
 P^«>, where a is any one of the ' ' definitely defined 
 symbols of infinity {bestimmt definirte Unendlich- 
 keitssymbole),'' is enumerable, then P is. 
 
 If the aggregates 1\, V^, . . . have, two by two, 
 no common point, for the aggregate P formed by 
 the union of these (the '' Vereinigungsmenge'') Cantor 
 now used the notation 
 
 P^P, + P2+. .. 
 Now, we have the following identity 
 P'^(P'-P") + (P"-P"0+ . • • +(P(''-i)-P(''))+P^^^ 
 and thus, since 
 
 p/_p// Y" —V" . . p(''-i) — p('') 
 
 are all isolated and therefore enumerable, if P^'^^ is 
 enumerable, then P' is also. 
 
 Now, suppose that P(~> exists ; then, if any par- 
 ticular point of P' does not belong to P^°°), there is a 
 first one among the derivatives of finite order, P^"^), 
 to which it does not belong, and consequently P^^-^) 
 contains it as an isolated point. Thus we can write 
 P'^(P^_p-) + (p-_p-)+ . . . +(p(-i)-PM) 
 
 + . . . +P(~); 
 
 and consequently, since an enumerable aggregate of 
 enumerable aggregates is an enumerable aggregate 
 of the elements of the latter, and P^"^^ is enumerable, 
 then P' is also. This can evidently be extended to 
 P^*^), 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 P' 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 P, 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 as infinite 
 (or transhnite) tnanbers. 
 
52 INTRODUCTION 
 
 VII 
 
 VVe have thus seen the importance of Cantor's 
 "definitely defined symbols of infinity" in the 
 theorem that if P^"^ 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^"^ 
 vanishes. By defining these indices in an inde- 
 pendent manner as real, and in general transfinite, 
 integers. Cantor was enabled to form a conception 
 of the enumeral * {Anza/il) 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 woid to translate " Anzahl," to avoid confusion 
 with the word " number" [Zahl). 
 
INTRODUCTION 53 
 
 The significance of this distinction for the theor}' 
 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 tra7isfiuite 
 
 numbers of which the first had been denoted b}- the 
 symbol "00." Thus, although there is no greatest 
 
 * C/:, for example, pp. 1 1 3, 1 58-159 of the translations of Cantor's 
 memoirs of 1895 and 1897 given below. 
 
U' 
 
 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 '■^ MannichfaltigkeitsleJire^'" ^ which is 
 usually translated as " theory of aggregates." In a 
 note to the Gyundlagen^ 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, welches sick als Eines 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., vol. xxi, 1883, pp. 545-591 ; reprinted, 
 with an added preface, with the title : Griindlagen einer allgevieinen 
 Mamiichfaltigkeitslehre. Ein uiathematisch-philosophischer Versiich in 
 der Lehre des Unendiichen,'Lit\\:>z\g, 1883 (page n of the Grnndlagenh 
 page w + 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 
 Acta Math., ii, pp. 381-408, part of the Grundlagen was translated 
 into French. • 
 
 t Or •'' Manntgfaitigkeitslehre," or, more usually, " Mengenlehre " ; in 
 French, ^^ th^orte 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 '* od ," which was and is used also 
 for the improper infinite, was here replaced by ''w." 
 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 th^fiist principle of gene^'ation {Erzeugungs- 
 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, w, 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 iii finito, 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 o), we obtain with the help of the first 
 principle of generation the further numbers : 
 w+ I, (0+2, . . ., a) + j^, ... 
 
 Since again here we come to no greatest number, we 
 imagine a new one, which we may call 2a), and which 
 is to be the first which follows all the numbers v and 
 o) + 1^ hitherto formed. Applying the first principle re- 
 peatedly to the number 2co, we come to the numbers : 
 
 g^^ 2a) + I, 2a) +2, . . ., 2a) +i', ... 
 
 The logical function which has given us the 
 numbers a) and 2a) 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 
 number is created, which is defined as the next greater 
 number to them all. 
 
 By the combined application of both principles 
 we get, successively, the numbers : 
 30), 3a) + I , . . . , 3a) + 1/, . . . , . . . , /xa), . . . , //a) + j/, . . . 
 
 w as the li7nit to which the numbers v strive, if by that nothing else is 
 understood than that « is to be the first integer which follows all the 
 numbers v, that is to say, is to be called greater than every »/." Cf. 
 the next section. 
 
 If we do not know the reasons in the theory of derivatives which 
 prompted the introduction of a>, but only the grounds stated in the text 
 for this introduction, it naturally seems rather arbitrary (not apparently^ 
 useful) to create &j 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 fxw-\-v is greatest, we create 
 a new next number to all these, which may be 
 denoted by oo^. To this follow, in succession, 
 numbers : 
 
 Xo)'^ + fX(Jd + V, 
 
 and further, we come to numbers of the form 
 
 and the second principle then requires a new number, 
 which may conveniently be denoted by 
 
 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 /x, i/q, i/^, . . . , i/^ have to take all finite, 
 positive, integral values including zero and exclud- 
 ing the combination v^ = v^= . . . =v^=zO. 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 
 
 (II) o), CD+I, . . ., j.Qfo'^H-i/iw'^-H . . . +j/^_r+>, 
 
 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 (Q) 
 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 fl). 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. 
 
6o INTRODUCTION 
 
 of numbers preceding each one should be of a certain 
 power, was called by Cantor the third or limitation- 
 principle {Hemmungs- 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 
 {Absclmitte), 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,! Cantor under- 
 
 * "This principle (or requirement, or condition) circuaiscribes 
 {limits) each number-class." 
 
 t 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 transtinite numbers. In 
 fact, the above definition of a wejl-ordered aggregate simply indicates 
 
INTRODUCTION 61 
 
 stood any well-defined aggregate whose elements 
 have a given definite succession such that there is 
 2. 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 lo generate elements, not numbers. 
 
 An important property of a well-ordered aggregate,— indeed, a 
 characteristic property, — is that any series of terms in it, ^j , ao , . . ., 
 «^ , . . ., where «^+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 enumerals 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 
 {Erfasseti) of the Absolute,— I cannot doubt. The Absolute can 
 only be recognized {anerkannt), but never apprehended {erkannt), 
 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 Haller : ' ich zieh' 
 sie ab [die ungeheure ZahlJ und I^u [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 1 hold it to be an idea (not presentation) that can be appre- 
 hended as a vanishing nothing in comparison with the former. It also 
 seems 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^ be well-ordered aggregates 
 of enumerals a and ^, the aggregate which arises 
 when first M is posited and then M^^, following it, 
 and the two are united is denoted M -f M^ and its 
 enumeral is defined to be a-\- ^. Evidently, if a 
 and /3 are not both finite, a + /3 is, in general, 
 different from /^ + a. It is easy to extend the con- 
 cept of sum to a finite or transfinite aggregate of 
 summands in a definite order, and the associative 
 law remains valid. Thus, in particular, 
 
 a + (/3 + y)-(a+iS) + y. 
 
 If we take a succession (of enumeral /3) of equal 
 and similarly ordered aggregates, of which each is 
 of enumeral «, we get a new well-ordered aggregate, 
 whose enumeral is defined to be the product y8a, 
 
 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 has 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 c)f 
 the continuum is that of (11), as stated, of course in other words, in 1878. 
 See the Notes at the end oi^ this book. 
 
^4 INTRODUCTION 
 
 where ^ is the multiplier and a the multiplicand. 
 Here also /3a is, in general, different from a^ ; but 
 we have, in general, 
 
 a(,%) = (a/3)y. 
 
 Cantor also promised an investigation of the 
 ''prime number-property " of some of the transfinite 
 numbers * a proof of the non-existence of infinitely 
 small numbers,! and a proof that his previous 
 theorem on a point-aggregate P in an ^2-dimensional 
 domain that, if the derivate P^'^^ where « is any 
 integer of (I) or (11), vanishes, P', and hence P, 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('^> = 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) ai, 02, . . ., 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 a. = ^y is only possible when /8= i or j8 = a. 
 t See the next section. 
 
INTRODUCTION 65 
 
 in the form (7), and therefore has not the power of 
 (I). Let a^ be the first number of (i) which is 
 greater than ai, a^ the first greater than a^ , and 
 
 i 2 
 
 so on ; so that we have 
 
 and 
 
 I < /Cg < /C3 < . . 
 
 
 ai<a^^<aK^< . 
 
 and 
 
 
 
 au<a^^ if v<kk' 
 
 Now it may happen that, from a certain number 
 a^ on, all following it in the series (7) are smaller 
 
 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 ^a^ and 
 > a„ , then the series of integers > a« and < a^ , 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 jS of (II) which is 
 greater than all of these numbers. Therefore /3> a^ 
 
 and thus also ^ > a^, and also every number /3' <fi 
 is surpassed in magnitude by certain numbers a^ . 
 
 If there is a greatest Uk =y, then the number y+ i 
 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 7iext greater to 
 that of (I), so that no other powers lie between 
 
 5 
 
66 INTRODUCTION 
 
 them, for any aggregate of numbers of (I) and (II) 
 is of the power of (I) or (II). In fact, this aggregate 
 Zj, when arranged in order of magnitude, is well- 
 ordered, and may be represented by 
 
 (a/s), (/5 = w, o) + I , . . . a, . . . ) 
 
 where we always have P<Q, where Q is the first 
 number of (III); and consequently (a^) is either 
 finite or of the power of (I) or of that of (II), 
 quartuin 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 : 
 
 (a + /3)y = ay + ^y, 
 
 where a + ^, 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 o) of the transfinite numbers were then dealt with 
 
 * From the occurrence of this theorem on p. 484 of the Math. Ann., 
 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|itreated 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 *' 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 problem ; 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 pu7'e 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. In 
 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 Kummer'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, Kirchhoft" pro- 
 posed ( Vorlesungen iiber mathemaiische Physik, 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, 191 5), and Popular Scientijic Lecitires, 3rd ed., Chicago 
 and London, 1898, pp. 236-258. 
 
70 INTR OD UCTION 
 
 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 — notions 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 Xh^ 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 aritJinietical 
 space G,„ that is to say, the totality of systems of 
 values 
 
 in which every x can receive any real value from 
 — 00 to -f 00 independently of the others. Every 
 such system is called an ''arithmetical point" of 
 G„, the "distance" of two such points is defined 
 by the expression 
 
 + x/{(-^'i-n)'+(-^-'2-^.)'+ • • • +(;^„-^,y), 
 
 and by an ''arithmetical point-aggregate" P con- 
 tained in G„ 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), 
 V 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) = o,— 
 
72 INTRODUCTION 
 
 and S is such that derivation does not alter it. Then 
 
 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 (zusaminenhdngend), that is to say, if t 
 and f are any two of its points and e a given arbi- 
 trarily small positive number, a finite number of 
 points /j, ^2? • • •) ^^ of P exist such that the dis- 
 tances /^i, t^t^^ . . ., K^' 3.re 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 (185 1) definition of a continuum is 
 certainly not correct, for it expresses only 07ie 
 property of a continuum, which is also possessed by 
 
INTRODUCTION 73 
 
 aggregates which arise from G„ 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 
 rt;;/^///^;- 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, Broden, Osgood, Baire, 
 Arzela, Schoenflies, and many others, — and will now 
 trace the development, in Cantor's hands, of the 
 theory of the transfinite cardinal and ordinal numbers 
 from 1883 to 1895. 
 
 VIII 
 
 An account of the development that the theory 
 of transfinite numbers underwent in Cantor's mind 
 from 1883 to 1890 is described in his articles 
 published in the ZeitscJirift fiir PJiilosophie tmd 
 pJiilosophiscJie Kritik for 1887 and 1888, and 
 collected and published in 1890 under the title Zur 
 Lehre vojh 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. U. 
 
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 {unuvi 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 ' 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 
 * Cy. § 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 1 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 
 Grundlage7i 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 
 
^6 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 
 *' i?imilarity," and in the laws of multiplication of 
 
INTRODUCTION 77 
 
 two ordinal numbers he departed from the custom 
 followed in the Gnindlagen 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^ .a^ = a^^'^ ; whereas we would 
 have to write, in the notation of the Grundlagen : 
 
 At the end of this letter, Cantor remarked that 
 a) may, in a sense, be regarded as the limit to which 
 the variable finite whole number v tends. Here " w is 
 the least transfinite ordinal number which is greater 
 than all finite numbers ; exactly in the same way 
 that J 2 \s the limit of certain variable, increasing, 
 rational numbers, with this difference : the difference 
 between J 2 and these approximating fractions be- 
 comes as small as we wish, whereas w — v is always 
 equal to w. But this difference in no way alters the 
 fact that 0) is to be regarded as as definite and com- 
 pleted as J 2, and in no way alters the fact that 00 
 has no more trace of the numbers v which tend to it 
 than J 2 has of the approximating fractions. The 
 transfinite numbers are in a sense new h^rationalities, 
 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^ where v is 
 a variable, finite, whole number, which increases 
 above all finite limits. In the first place the 
 difference \—z^ 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^. 
 Each of these two properties characterizes the finite 
 number i as limit of the variable magnitude z^ . 
 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 w the limit of the 
 increasing, finite, whole numbers v , because to is the 
 least of all numbers which are greater than all the 
 finite numbers. But co — 1/ is always equal to co, and 
 therefore we cannot say that the increasing numbers 
 
 V come as near as we wish to w ; indeed any number 
 
 V however great is quite as far off from co 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 w 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 Gvundlagen. 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 
 n different directions, an intellectual picture, a general 
 concept, is generated in us, and I call this the /^-ple 
 ordinal type." The definition of the " similarity of 
 ordered aggregates " is : 
 
 "Two ;/-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 8i 
 
 E to E' in the jyth direction inside the aggregate M 
 is exactly the same as the relation of rank of F to F' 
 in the v\\\ 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 Ipublished in 1892. In this paper 
 we can see the origins of the conception of ' * cover- 
 ing" {Belegmig) 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 transfinite 
 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 transfinite 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 ^"^ 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 
 
[48i] CONTRIBUTIONS TO THE 
 FOUNDING OF THE THEORY OF 
 TRANSFINITE NUMBERS 
 
 (First Article) 
 
 " Hypotheses non fingo." 
 
 " Neque enim leges intellectui aut rebus damus 
 ad arl)itrium nostrum, sed tanquam scribal 
 fideles ab ipsius naturae voce latas et prolatas 
 excipimus et describimus." 
 
 "Veniet tempus, quo ista quae nunc latent, in 
 lucem dies extrahat et longioris xvi diligentia." 
 
 The Conception of Power or Cardinal Number 
 
 By an ^ ' aggregate " {Menge) we are to understand 
 any collection into a whole {Zusaimnenfassung zu 
 einem Ganzen) M of definite and separate objects ;;/ 
 of our intuition or our thought. These objects are 
 called the ' ' elements " of M. 
 In signs we express this thus : 
 
 (i) 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 
 
m 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 
 3-ggi'egate " of an aggregate M any other aggregate 
 Mj whose elements are also elements of M. 
 
 If M2 is a part of M^ and M^ is a part of M, then 
 M2 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 in 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) M. 
 
 Since every single element vi, 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 00 N or N fV) M, 
 
 if it is possible to put them, by some law, in such a 
 relation to one another that to every element of each 
 onelof them corresponds one and only one element 
 
OF TRANSFINITE NUMBERS 87 
 
 of the other. To every part Mj 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 
 in^ of M a special element n^ of N corresponds. For 
 if, according to the original law, the elements ni^ 
 and Uq do not correspond to one another, but to the 
 element m^ of M the element n^^ of N corresponds, 
 and to the element n^ of N the element in^ of M 
 corresponds, we take the modified law according to 
 which m^ corresponds to n^ and m^ to n^ 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 00 M. 
 
 If two aggregates are equivalent to a third, they are 
 equivalent to one another ; that is to say : 
 
 (6) from M 00 P and N 00 P follows M 00 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 M 00 N we get M = N, 
 and 
 
 (8) from M = N we get M 00 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 in of M other things are substituted. If, 
 now, M 00 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 in of M corresponds the element n of N. 
 Then we can imagine, in the place of every element 
 m of M, the corresponding element /^ of N substi- 
 tuted, and, in this way, M transforms into N without 
 alteration of cardinal number. Consequently 
 
 M = N. 
 
 The converse of the theorem results from the re- 
 mark that between the elements of J^I 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 ;;/ of M a special unit of M arises. Thus 
 we can say that 
 
 (9) M 00 M. 
 
 In the same way N 00 N. If then M = N, we have, 
 by (6), M 00 N. 
 
 We will mention the following theorem, which 
 results immediately from the conception of equival- 
 
OF TRANSFINITE NUMBERS 89 
 
 ence. If M, N, P, . . . are aggregates which have 
 no common elements, M', N", P', . . . are also aggre- 
 gates with the same property, and if 
 
 M 00 M', N 00 N; P cv) P^ . . . , 
 
 then we always have 
 
 (M, N, P, . . .) fNJ (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^ of N, suCh that N^ 00 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] P'urther, the equivalence of M and N, and 
 thus the equality of a and b, is excluded ; for if we 
 had M 00 N, we would have, because N^ f\j M, the 
 equivalence N^ po N, and then, because M 00 N, 
 there would exist a part Mj of M such that M^ oo M, 
 and therefore wg should have Mj 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 vi ; for if 
 
90 THE FOUNDING OF THE THEORY 
 
 in ia) 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 ip) by saying: a is "less" than b or b is 
 "greater " than a ; in signs 
 
 (i) a<b or \i>(x. 
 
 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 ?i is part of an aggregate P, from a<Pi 
 
 follows a < P and from P < b follows P^ < b. 
 We have seen that, of the three relations 
 
 a = b, (x<^, \i<(x, 
 
 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. 
 
 P>om 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^ of N and N to a part Mj of 
 M, then M and N are equivalent ; 
 
 C. If Mj is a part of an aggregate M, Mg is a 
 part of the aggregate Mj, and if the aggregates 
 M and Mg are equivalent, then Mj is equivalent to 
 both M and Mg ; 
 
 D. If, with two aggregates M and N, N is 
 equivalent neither to M nor to a part of M, there is 
 a part Nj of N that is equivalent to M ; 
 
 E. If two aggregates M and N are not equivalent, 
 and there is a part N^ 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 elemeijts was denoted in § i, (2), 
 by (M, N). We call it the ''union-aggregate 
 ( Vereinigungsinenge) of M and N. " 
 
 If M' and N' are two other aggregates without 
 common elements, and if M 00 M' and N 00 N', we 
 saw that we have 
 
 (M, N) 00 (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 vi and b. 
 We put ' 
 
 (I) a + b = (M7>^). 
 
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 vi 
 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 (w, n)^ and call 
 it the ''aggregate of bindings {Verbindungsniejige) 
 ofMandN." Thus 
 
 (4) (M.N) = {K;.)}. 
 
 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 in, in' 
 and ft, n' as corresponding elements, then the 
 aggregate 
 
 (M'.N') = {(^^/, n)) 
 
 is brought into a reciprocal and univocal corre- 
 spondence with (M . N) by regarding {in, n) and 
 {in' , n') as corresponding elements. Thus 
 
 (5) (m;no<^^(m.n). 
 
 We now define the product vi . b by the equation 
 
 (6) a.b = (M.N). 
 
OF TRANSFINITE NUMBERS 93 
 
 [486] An aggregate with the cardinal number 
 a . 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„ fNJ M ; if, then, we collect the elements of all 
 these aggregates M„ to a whole S, we see that 
 
 (7) S ro (M . N), 
 and consequently 
 
 S = a.b. 
 
 For, if, with any given law of correspondence of the 
 two equivalent aggregates M and M,,, we denote 
 by in the element of M which corresponds to the 
 element m^ of M„, we have 
 
 (8) S^ {;;/.}; 
 
 and thus the aggregates S and (M . N) can be re- 
 ferred reciprocally and univocally to one another by 
 regarding ni^ and {in, n) as corresponding elements. 
 From our definitions result readily the theorems : 
 
 (9) a.b = b.a, 
 
 (10) a.(b . c) = (a. b). c, 
 
 (11) a(b + c) = ab + ac; 
 because : 
 
 (M.N)r>o(N.M), 
 
 (M.(N.P)) 00 ((M.N). P), 
 (M . (N, P)) fNj ((M . N), (M . P)). 
 Addition and multiplication of powers arc 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{}i) ; it is 
 called a '' covering function of n.'' The correspond- 
 ing covering of N will be called /(N). 
 
 [487] Two coverings /i(N) and/2(N) are said to 
 be equal if, and only if, for all elements ;^ of N the 
 equation 
 
 (I) /lW=/2(«) 
 
 is fulfilled, so that if this equation does not subsist 
 for even a single element n^n^, f^{^) and/2(N) are 
 characterized as different coverings of N. For ex- 
 ample, if vi^ is a particular element of M, we may 
 fix that, for all n'?> 
 
 f{n) = m^; 
 
 this law constitutes a particular covering of N with 
 M. Another kind of covering results if ui^ and m^ 
 are two different particular elements of M and n^ a 
 particular element of N, from fixing that 
 
OF TRANSFINITE NUMBERS 95 
 
 /(«o) = ^'^o 
 
 for all ;2's which are different from n^. 
 
 The totality of different coverings of N with M 
 forms a definite aggregate with the elements /(N) ; 
 we call it the ** covering-aggregate (yBelegungsvienge) 
 of N with M " and denote it by (N | M). Thus : 
 
 (2) (N |M)={/(N)}. 
 
 If M 00 M' and N Oo N', we easily find that 
 
 (3) (N I M) 00 (N' I 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* : 
 
 (4) a» = (N^r). 
 
 For any three aggregates, M, N, P, we easily prove 
 the theorems: 
 
 . (5) ((N |M).(P|M))f\j((N, P)|M), 
 
 (6) ((P| M).(P|N))<N;(P|(M.N)), 
 
 (7) (P|(N| M))'\^((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, b, and c : 
 
 (8) 
 
 at.rt' = a''+^ 
 
 (9) 
 
 a'.b^ = (a.b)f, 
 
 (10) 
 
 (a*/ = '!'•'• 
 
96 THE POUNDING 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 0, we easily 
 see that it may be represented by, amongst others, 
 the formula : 
 
 (II) r o = 2^\ 
 
 where § 6 gives the meaning of Nq. In fact, by (4), 
 2^0 is the power of all representations 
 
 (-) .=^+f>+...+f>+... 
 
 (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= <i, which are represented twice over, we 
 
 have, if we denote the "enumerable" totality of 
 the latter by {s^], 
 
 2«o = ({j4, X). 
 
 If we take away from X any " enumerable " aggre- 
 gate {t^} and denote the remainder by X^, we have : 
 
 ({sA, X)=({^4, {4}, X,), 
 
 so 
 
 Xco(K}, X), 
 
OF TRANSFINlTli NUMBERS 97 
 
 and thus (§ i) 
 
 2^" = X = 0. 
 
 From (11) follows by squaring (by § 6, (6)) 
 
 and hence, by continued multiplication by 0, 
 
 (13) o''=o, 
 
 where v is any finite cardinal number. 
 
 If we raise both sides of (11) to the power* ^j^ 
 we get ^ 
 
 But since, by § 6, (8), No.^^o = ^^o, we have 
 
 (14) o^''*° = o. 
 
 The formulae (13) and (14) mean that both the 
 jy-dimensional and the {s?Q-dimensional continuum have 
 the power of the one-dimensional continuum. Thus 
 the whole contents of my paper in Crelle's Journal^ 
 vol. Ixxxiv, i878,t 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 ambiguity.] 
 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 e^, if we subsume it under the 
 concept of an aggregate Eo = (^o)' corresponds as 
 cardinal number what we call ''one" and denote by 
 I ; we have 
 
 (1) i = Eo. 
 
 Let us now unite with E^ another thing e^, and 
 call the union-aggregate E^, so that 
 
 (2) t:i = (Ko, e^ = {e^, e^). 
 
 The cardinal number of E^ is called "two" and is 
 denoted by 2 : 
 
 (3) 2=1,. 
 
 By addition of new elements we get the series of 
 aggregates 
 
 E2 = (Ei, ^2)' E3 = (E2, ^3), . . ., 
 
 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 b"een defined as a cardinal number. 
 We have, if by i/— i is understood the number im- 
 mediately preceding v in the above series, 
 
 (4) j/ = E,_i, 
 
 (5) E,-(E,_i, O = (^o' ^D • • • ^^)' 
 
OF TRANSFINITE NUMBERS 99 
 
 From the definition of a sum in § 3 follows : 
 
 (6) E,-E,_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 
 
 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 rest 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, e), 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 ^z, and N^ is any part of N, the cardinal 
 
lOO THE FOUNDING OF THE THEORY 
 
 number of Nj, is equal to one of the preceding 
 numbers i, 2, 3, . . . , ly— i. 
 
 Pr^^/^/D.— Suppose that the aggregate (M, e) 
 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 : 
 
 {(i) The aggregate N contains e as element ; let 
 N = (Ml, e) ; then M^ 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^, 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 M^ 
 are referred reciprocally and univocally to one 
 another. But this contradicts the supposition that 
 M is not equivalent to its part M^. 
 
 {U) The part N of (M, e) 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 / of the 
 latter correspond. Let N = (Mi, /) ; then the 
 aggregate M is put in a reciprocally univocal relation 
 with Mj. But Ml 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 v\- i which immediately 
 follows, in the following manner : — We start from 
 the aggregate E„ = (^o. ^1, • • -, ^.) as an aggregate 
 
OF TRANSFINITE Ntl'MBERTy (ot 
 
 with the cardinal number v-\-\. If the theorem is 
 true for this aggregate, its truth for any other 
 aggregate with the same cardinal number v-\-\ 
 follows at once by § i. Let E' be any part of E^ ; 
 we distinguish the following cases : 
 
 {a) E' does not contain e^ as element, then E is 
 either E^_i [491] or a part of E^,_i, and so has as 
 cardinal number either v or one of the numbers 
 I, 2, 3, . . ., I/— I, because we supposed our theorem 
 true for the aggregate E^_i, with the cardinal 
 number v. 
 
 {b) E' consists of the single element ^„, then 
 
 E'=i. 
 
 ic) E' consists of e^ and an aggregate E'', so that 
 E' = (E'', e^, E'' is a part of E^_i and has there- 
 fore by supposition as cardinal number one of the 
 numbers i, 2, 3, . . ., 1/— i. But now E' = E''4-i, 
 and thus the cardinal number of E' is one of the 
 numbers 2, 3, . . . , i/. 
 
 Proof of K. — Every one of the aggregates which 
 we have denoted by E^ has the property of not 
 being equivalent to any of its parts. Eor if we 
 suppose that this is so as far as a certain i/, it follows 
 from the theorem D that it is so for the immediately 
 following number v-\-\. For 1/= i, we recognize at 
 once that the aggregate Ei = (^o, e^ is not equivalent 
 to any of its parts, which are here {e^ and {e^. 
 Consider, now, any two numbers fx and v of the 
 series 1,2, 3, . . . ; then, if jx is the earlier and v 
 the later, E^_i is a part of E^_i. Thus E^_i and 
 
i02TliE FOUNDING OF THE THEORY 
 
 E^_i are not equivalent, and accordingly their 
 
 cardinal numbers ^=:E^_i and i/=E^_i are not 
 equal. 
 
 Proof of ^. — If of the two finite cardinal numbers 
 fj. and V the first is the earlier and the second the 
 later, then fj. < v. For consider the two aggregates 
 M = E^_i and N = E^,_i; 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^_i can only have one of the cardinal 
 numbers i, 2, 3, . . ., /x— i, and therefore, by 
 theorem A, cannot be equivalent to the aggregate 
 N=E^_i. The condition (b) 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 E^, can only have one 
 of the cardinal numbers i, 2, 3, . . . , i/. Thus a is 
 equal to one of the cardinal numbers I, 2, 3, . . ., i/. 
 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, /c^, amongst them 
 which is smaller than the rest, and therefore the 
 smallest of all. 
 
 [492] Proof — The aggregate K either contains 
 
OF TRANSFINITE NUMBERS 103 
 
 the number i, in which case it is the least, «:i=i, 
 or it does not. In the latter case, let J be the 
 aggregate of all those cardinal numbers of our series, 
 I, 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 
 i/j such that J'l+i, 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 {AbscJinitt) (1,2, 3, . . ., j/j). 
 The number v^-\-\=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 
 
 such that 
 
 K = (/Cp /C2, /C3, . . .) 
 
 /Ci *^ /C9 "^ '^Sj • •' • 
 
 §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 ; 
 
I04 THE FOUNDING OF THE THEORY 
 
 we call its cardinal number (§ i) " Aleph-zero " and 
 denote it by j^q ; thus we define 
 
 (i) 5^0= {i^}- 
 
 That j^o is a transfinite number, that is to say, is 
 not equal to any finite number /x, follows from the 
 simple fact that, if to the aggregate {v} is added a 
 new element e^, the union-aggregate ({i/}, e^ is 
 equivalent to the original aggregate {j^}. 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-\-\ oi 
 the other. By § 3 we thus have 
 
 (2) ^?o+I=^^o• 
 
 But we showed in § 5 that // -f- i is always different 
 from /x, and therefore ^^ is not equal to any finite 
 number fx. 
 
 The number j^^ is greater than any finite number ix : 
 
 (3) So>M- 
 
 [493] This follows, if we pay attention to § 3, 
 from the three facts that />t = (i, 2, 3, . . ., /x), that' 
 no part of the aggregate (i, 2, 3, . . ., ^) is equiva- 
 lent to the aggregate {v}, and that (i, 2, 3, . . ., yu) 
 is itself a part of {v}. 
 
 On the other hand, h^^ is the least transfinite 
 cardinal number. If a is any transfinite cardinal 
 number different from i^^, then 
 
 (4) ^0<<'^' 
 
OF TRANSFINITE NUMBERS 105 
 
 This rests on the following theorems : 
 
 A. Every transfmite aggregate T has parts with 
 the cardinal number ^^Q. 
 
 Proof. — If, by any rule, we have taken away a 
 finite number of elements t^, t^^ . . .,/,,_i, there 
 always remains the possibility of taking away a 
 further element t^. The aggregate {/,,}, where v 
 denotes any finite cardinal number, is a part of T 
 with the cardinal number t^^, because {/^}f\j{i'} (§ i). 
 
 B. If S is a transfinite aggregate with the cardinal 
 number j^^, and S^ is any transfinite part of S, then 
 
 Proof. — We have supposed that S 00 {v\. Choose 
 a definite law of correspondence between these two 
 aggregates, and, with this law, denote by s^ that 
 element of S which corresponds to the element v of 
 {i/}, so that 
 
 The part S^ of S consists of certain elements s^ 
 of S, and the totality of numbers k forms a trans- 
 finite part K of the aggregate [v). By theorem G 
 of § 5 the aggregate K can be brought into the 
 form of a series 
 
 where 
 
 consequently we have 
 
io6 THE FOUNDING OF THE THEORY 
 
 Hence follows that Sj oo S, and therefore Si = j^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) ^*o + ^ = «o- 
 We have also 
 
 (6) «o + «o = ^*o- 
 
 [494] For, by (i) of § 3, No + «o is the cardinal number 
 ({«J, {by)) because 
 
 Now, obviously 
 
 {.}=({2.-l}, {2.}), 
 ({2.- I}, {2.})00(K}, {b^\ 
 
 and therefore 
 
 The equation (6) can also be written 
 
 t^o- 2 = ^^0 5 . 
 and, by adding j^q repeatedly to both sides, we 
 find that 
 
 (7) ^^o•^ = I^• «o = ^^o- 
 We also have 
 
 (8) ^^o•No = ^^o• 
 
OF TRANSFIX/TIL NUAIBFKS 107 
 
 Proof. — By (6) of § 3, Nq • t^o ^-^ ^^^^ cardinal 
 number of the aggregate of bindings 
 
 {{p., y)], 
 
 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 {\}, (fj.), 
 and {v) are only different notations for the same 
 aggregate o( all finite numbers, we have to show 
 that 
 
 {(m, ^)}fX^{X}. 
 
 Let us denote ^ + 1/ by p\ then ^ takes all the 
 numerical values 2, 3, 4, . . ., and there are in all 
 p— I elements (/x, v) for which ^-|-j/ = p, namely : 
 
 (1,^-1), (2,p-2),.. , (p-I, I). 
 
 In this sequence imagine first the element (i, 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 
 (juL, p) in a simple series : 
 
 (I, i);(i, 2),(2, i);(i,3),(2, 2),(3, i);(i,4),(2, 3) 
 
 and here, as we easily see, the element {ju, v) comes 
 at the Xth place, where 
 
 (9) X = /x+ 2 
 
 The variable X takes every numerical value i, 2, 3, 
 . . ., once. Consequently, by means of (9), a 
 
To8 THE FOUNDING OF THE THEORY 
 
 reciprocally univocal relation subsists between the 
 aggregates {v} and {(/x, v)}. 
 
 [495] If both sides of the equation (8) are multi- 
 plied by j^o> we get ^^^ = 'i:^^^ = ^^^ and, by repeated 
 multiplications by ^^^ we get the equation, valid 
 for every finite cardinal number v : 
 
 (lo) ^^o" = «o• 
 
 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 transfinite aggregates : 
 
 D. Every transfinite aggregate T is such that it 
 has parts T^ which are equivalent to it. 
 
 Pi'oof. — By theorem A of this paragraph there is 
 a part S={^4 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 
 C Let us put Si = {/,+i}, Ti = (Si, U) ; then T^ is 
 a part of T, and, in fact, that part which arises out 
 of T if we leave out the single element t^. Since 
 S 00 Si, by theorem B of this paragraph, and 
 UooU, we have, by § i, T rx; T^. 
 
 In these theorems C and D the essential differ- 
 ence between finite and transfinite aggregates, to 
 which I referred in the year 1877, in volume Ixxxiv 
 [1878] of Crelle's Journal, p. 242, appears in the 
 clearest way. 
 
 After we have introduced the least transfinite 
 
OF TR A XS FINITE NUMJUIRS loo 
 
 cardinal number Nq and derived its properties tliat 
 lie the most readily to hand, the question arises 
 as to the higher cardinal numbers and how they 
 proceed from h^^. 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 ^^q pro- 
 ceeds, by a definite law, the next greater cardinal 
 number j^^, out of this by the same law the next 
 greater n^j ^'^'^^ so o^"*- ^^t even the unlimited 
 sequence of cardinal numbers 
 
 No' «i' «2' • • •) «.', ... 
 
 does not exhaust the conception of transfinite 
 cardinal number. We will prove the existence of 
 a cardinal number which we denote by ^^^ and 
 which shows itself to be the next greater to all 
 the numbers ^^ ; out of it proceeds in the same 
 way as i^^ out of ^? a next greater ^,^+1^ ^^^^ ^^ o^''» 
 without end. 
 
 [496] To every transhnite 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 allgemeitjen MannicJifaltigkeits- 
 lehre (Leipzig, 1883) and in volume xxi of the 
 
110 THE FOUNDING OF THE THEORY 
 
 MatJiematische AnnaUn, 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" {Rmigordnung) rules 
 over its elements m, so that, of every two elements 
 m^ and ?;^2> ^^^ takes the " lower " and the other the 
 ' ' higher " rank, and so that, if of three elements ni^, 
 jn^, and Wg, nt-^, say, is of lower rank than z//^, and 
 7n.2^ is of lower rank than m^y then m^ is of lower 
 rank than m^. 
 
 The relation of two elements ///j and ;;/^, in which 
 m^ has the lower rank in the given order of pre- 
 cedence and m^ the higher, is expressed by the 
 formulae : 
 
 (i) m^ -< m^, in^ >- m^ 
 
 Thus, for example, every aggregate P of points 
 defined on a straight line is a simply ordered 
 aggregate if, of every two points /^ 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, witl^i the aggregate R of 
 
OF TRANSFINITE NUMBERS \ 1 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 Rq) so that, of two numbers /^/^^ and 
 A/^- for which the sums /i + ^i and p^ + g^ have 
 different values, that number for which the corre- 
 sponding sum is less takes the lower rank, and, if 
 A + ^i=/2 + ^2' 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 oi p-\-q only a finite number of rational 
 numbers//^ belongs, evidently has the form 
 
 -•^^0 \'l> '2' • • ■' «" • • '/ V2' 3' 4' ."' 5 J «» 5' 4' • • VJ 
 
 where 
 
 r, < /-,+!. 
 
 Always, then, when we speak of a "simply 
 ordered " aggregate M, we imagine laid down a 
 definite order or precedence of its elements, in the 
 sense explained above. 
 
 There are doubly, triply, j/-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 
 "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. 
 
112 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 in^ 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" {dhnlich) if they can be put into a bi- 
 univocal correspondence with one another in such 
 a manner that, if ?n-^ and ni^ are any two elements 
 of M and n^ and n^ the corresponding elements of N, 
 then the relation of rank of in^ to m^ in M is the 
 same as that of n-^ to n^ in N. Such a correspond- 
 ence of similar aggregates we call an '* imaging" 
 {Abbildimg) of thes'e aggregates on one another. In 
 such an imaging, to every part — which obviously 
 also appears as an ordered aggregate — M^ of M 
 corresponds a similar part N^ of N. 
 
 We express the similarity of two ordered aggre- 
 gates M and N by the formula : 
 
 (3) MooN. 
 
 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 
 formula: 
 
OF TRANSFINITE NUMBERS 113 
 
 (4) M = N, MOON, 
 
 one is always a consequence of the other. 
 
 If, with an ordinal t}'pe 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) « 
 
 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 i/, 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, . . ., Vy . . . for them, although they are 
 conceptually different from the cardinal numbers. 
 The case is quite different with the transfinite 
 ordiiial types ; for to one and the same cardinal 
 
 8 
 
114 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 [^^o] which embraces 
 all the types with the least transfinite cardinal 
 number ^^q. 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] ^^ determmed 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 ;/. 
 
 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 "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 ri is 
 similar to itself in an infinity of ways. 
 
 We will make this difference clear by two simple 
 examples. By o) we understand the type of a well- 
 ordered aggregate 
 
 in which 
 
 and where p represents all finite cardinal numbers in 
 turn. Another well-ordered aggregate 
 
 with the condition 
 
 of the same type od can obviously only be imaged 
 
Ti6 THE FOUNDING OF THE THEORY 
 
 on the former in such a way that e^, and /, are 
 correspondhig elements. For e^, 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 {e^) to/g, the next after/^, 
 and so on. [500] Every other bi-univocal corre- 
 spondence of the two equivalent aggregates {e^} and 
 {/,} 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 
 
 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 vq a 
 definite one of the numbers /, to the element e,' of 
 
OF TRANSFINITE NUMBERS ti; 
 
 the first the element /j,^,'_^_,/ of the second corresponds. 
 Since j/q' is arbitrary, we liave 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 {Anzahlmdssige) 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 Gnuidziige der 
 Geoinetrie (German by A. Schepp, Leipzig, 1894) 
 is the non-recognition of this point. 
 
 On page 30 the ' ' number {Anrjahl oder ZaJd) 
 of an ordered group " is defined in exactly the same 
 way as what we have called the "ordinal type of 
 a simply ordered aggregate " {Ztir Lehre vorn 
 Transfiniten, Halle, 1890, pp. 68-75 i reprinted 
 from the 'ZeitscJir. fiir Pliilos. 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 
 
ii8 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 Fontenelle's Geo7netrie de IFnfifii (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 lectionuni of the Miinster Academy for 1895- 
 i896.t 
 
 * In the original Italian edition (p. 27) this passage runs : " Numeri 
 le unita dei quali si corrispondono univocamente e nel medesimo ordine, 
 e di cui 1' uno non e parte o uguale ad una parte dell' altro, sono uguali." 
 
 t [Veronese replied to this in Math. Ann., vol. xlvii, 1897, pp. 423- 
 432. Cf. Killing, ibid., vol. xlviii, 1897, pp. 425-432.] 
 
OF TRANSFINITE NUMBERS 119 
 
 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 
 N fV) N', [502] then (M, N) cx> (M^ N') ; so the 
 ordinal type of (M, N) depends only on the ordinal 
 types M = a and N = /3. Thus, we define: 
 
 (1) a + /3 = (M, N). 
 
 In the sum ct + /3 we call a the ''augend " and /3 the 
 "addend." 
 
 For any three types we easily prove the associa- 
 tive law : 
 
 (2) a + (/3 + y)=:(a4./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 o) is the type, already mentioned in § 7, of 
 the well-ordered aggregate 
 
I20 THE FOUNDING OF THE THEORY 
 
 then I +w is not equal to (o+ i. For, if/ is a new 
 element, we have by (i) : 
 
 l+a) = (/E), 
 a,+ l=(E;7)- 
 But the aggregate 
 
 (/E)=(y;^i, ^2. • •-^.'•- •) 
 
 is similar to the aggregate E, and consequently 
 
 I +co = a). 
 
 On the contrary, the aggregates E and (E, f) are 
 not similar, because the first has no term which is 
 highest in rank, but the second has the highest 
 term /! Thus coH- i is different from a)= i +a). 
 
 Out of two ordered aggregates M and N with 
 the types a and /3 we can set up an ordered 
 aggregate S by substituting for every' element n of 
 N an ordered aggregate M,, which has the same 
 type a as M, so that 
 
 (3) M, = a; 
 and, for the order of precedence in 
 
 (4) S = {M„} 
 
 we make the two rules : 
 
 (i) Every two elements of S which belong to 
 one and the same aggregate M^, are to retain in 
 S the same order of precedence as in M,, ; 
 
 (2) Every two elements of S which belong to two 
 different aggregates M^^ and M^,.^ have the same 
 relation of precedence as n^ and 71^ have in N. 
 
OF TRANSFfNITE NUMBERS 121 
 
 The ordinal type of S depends, as we easily see, 
 only on the types a and /5 ; 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„ let in^ be the 
 element of M„ that corresponds to the element m 
 of M; we can then also write 
 
 (6) S = {/«„}. 
 
 Consider a third ordered aggregate V = [p] with 
 the ordinal type P = y, then, by (5), 
 
 a.(/5-y) = {^^')}. 
 But the two ordered aggregates {(m,,) } and {^^^(n.)} 
 are similar, and are imaged on one another if we 
 regard the elements (^z^^) ^^^^ ?''^/«.\ as correspond- 
 ing. Consequently, for three types a, ^8, and y 
 the associative law 
 
 (7) (a.^).y = a.(/5.y) 
 
 subsists. From (i) and (5) follows easily the dis- 
 tributive law 
 
 (8) a.(^-hy) = 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, 2.0) and w .2 are 
 different types ; for, by (5), 
 
 while 
 
 W. 2 = {e^, ^2' • • •> ^.', ...; /l,/2, ••.,/',-. •) 
 
 is obviously different from o). 
 
 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 r] 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 q being relatively 
 prime) which >o and < i, in their natural order 
 of precedence, where the magnitude of a number 
 
 } 
 
 / 
 
OF TRANSFINITR NUMBERS 123 
 
 determines its rank. We denote the ordinal t)'pe 
 of R by >y : 
 
 (1) >/ = R. 
 
 But we have put the same aggregate in another 
 order of precedence in which we call it Rq. This 
 order is determined, in the first place, by the 
 magnitude of p-\-q, and in the second place — for 
 rational numbers for which p -\- q has the same value 
 — by the magnitude of pjq itself. The aggregate 
 Rq is a well-ordered aggregate of type w : 
 
 (2) Ro = (>'i, ^'2, . . ., i^.M . • •), where r,<r,+i, 
 
 (3) Ro = ^- 
 
 Both R and Rq have the same cardinal number 
 since they only differ in the order of precedence 
 of their elements, and, since we obviously have 
 
 Ro = j^o' ^^'^ ^^^° \\di\^ 
 
 (4) R = 7/ = No- 
 
 Thus the type ri belongs to the class of types [^^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" {uberalldicht), 
 
 VVe will now show that these three properties 
 characterize the type >/ 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 = «o; 
 
 {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 ^ : 
 
 Pjvof. — Because of the condition {a), M can be 
 brought into the form [505] ^^ ^ well-ordered 
 aggregate of type w ; having fixed upon such a 
 form, we denote it by Mq and put 
 
 (5) Mo = (?//i, Wg, . . ., ;//„, . . .). 
 We have now to show that 
 
 (6) MooR; 
 
 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 7\ in R be correlated to the 
 element m^ in M. The element rg h^s a definite 
 relation of precedence to 7\ in R. Because of the 
 condition {b), there are infinitely many elements 
 vi^, of M which have the same relation of precedence 
 in M to m-^ as r^^ to 7\ in R ; of them we choose 
 that one which has the smallest index in M^, let it 
 be ifii and correlate it to r^. The element r, has 
 
 in R definite relations of precedence to i\ and r^ ; 
 because of the conditions (b) and (c) there is an 
 
OF TRANSFINITIi NUMBERS 125 
 
 infinity of elements in^, of M which have the same 
 
 relation of precedence to m^ and nii in M as rg to }\ 
 
 and r.y to R ; of them we choose that — let it be ;;/, 
 z 5 
 
 — which has the smallest index in Mq, and correlate 
 it to Tg. According to this law we imagine the 
 process of correlation continued. If to the v 
 elements 
 
 ^1' '''2' ^3) • • • ) ^i' 
 
 of R are correlated, as images, definite elements 
 
 m^, nu, in^, . . ., in, 
 
 which have the same relations of precedence amongst 
 one another in M as the corresponding elements in 
 R, then to the element ;v+i of R is to be correlated 
 that element ni,^ of M which has the smallest 
 index in Mq of those which have the same relations 
 of precedence to 
 
 ;;/i, m,^, in,^, . . . , m,^ 
 
 in M as r^+i to r^, r^, . . ., r^ in R. 
 
 In this manner we have correlated definite 
 elements m. of M to all the elements }\ of R, and 
 the elements m,^ have in M the same order of pre- 
 cedence as the corresponding elements i\ in R. But 
 we have still to show that the elements m,^ include 
 all the elements in^ of M, or, what is the same 
 thing, that the series 
 
 I ) '2' '3' • • • ' ^f ' • • • 
 
 [506] is only a permutation of the series 
 I, 2, 3, ....',.. . 
 
126 THE FOUNDING OF THE THEORY 
 
 We prove this by a complete induction : we will 
 show that, if the elements m^, m^, . . ., m^ appear 
 in the imaging, that is also the case with the 
 following element m^+i. 
 
 Let X be so great that, among the elements 
 
 w,, nil, nil , . . ., lUu, 
 
 ■■■as '^ 
 
 the elements 
 
 m^, in,, . . ., ;;/,, 
 
 which, by supposition, appear in the imaging, are 
 contained. It may be that also ?;/,,+i is found 
 among them ; then /;2^+i appears m the imaging. 
 But if /;^^+i is not among the elements 
 
 m^, in,, m,, . . ., vi,^, 
 
 then ///^+i 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 i\, rg, . . . , r^, amongst which let 
 /'x+a be that with the least index in Rq. Then ni^^i 
 has, as we can easily make sure, the same ordinal 
 position with respect to 
 
 m^, m,, m,, . . ., nh^^^_^ 
 in M as r^j^„ has with respect to 
 
 ^'d ''2) • • •) ^^A + a--l 
 
 in R. Since in^, in.^^, • • • , ^f^u have already appeared 
 in the imaging, ni^^i is that element with the smallest 
 index in M which has this ordinal position with 
 respect to 
 
OF TRANSFJNITE NUMBERS 127 
 Consequently, according to our law of correlation, 
 
 ''^^,+^ = '^'^+1- 
 
 Thus, in this case too, the element ;;/^.|.i appears in 
 the imaging, and r^+^ 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 r^. 
 
 The ordinal type of the aggregate of all rational 
 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 <b, is rj. 
 
 The ordinal type of the aggregate of all real alge- 
 braic numbers in their natural order of precedence is tj. 
 
 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<b, is ;/. 
 
 P^or all these ordered aggregates satisfy the three 
 conditions required in our theorem for M (see 
 CygWq' s /ou7^nal, vol. Ixxvii, p. 258).* 
 
 If we consider, further, aggregates with the types 
 — according to the definitions given in § 8 — written 
 
 [* C/. Section V of the Introduction.] 
 
128 THE FOUNDING OF THE THEORY 
 
 '/ + ';, nn^ (i+>7)'7, ('?+i)'7. (!+>?+ i>/, we find that 
 those three conditions are also fulfilled with them. 
 Thus we have the theorems : 
 
 (7) 
 
 >; + ') = '?, 
 
 (8) 
 
 m = i> 
 
 (9) 
 
 (I +,,),, = ,,, 
 
 (10) 
 
 (>;+ !)>; = >;, 
 
 (11) 
 
 (l+,,+ l)ri = >i. 
 
 The repeated application of (7) and (8) gives for 
 every finite number u : 
 
 (12) t1'V = t1, 
 
 (13) n" = ^- 
 
 On the other hand we easily see that, for j/> I, the 
 types 1+^, i]-^i, v.r], i+>?+i are different both 
 from one another and from t]. We have 
 
 (14) ;^+I+,y = ^, 
 
 but r] + i/ + t], for ^/> I, is different from rj. 
 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 *a) appear to 
 be especially valuable ; we call them " fundamental 
 series of the first order contained in M," and the 
 former — of type w — we call an ''ascending" series, 
 the latter — of type *o) — a * ' 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 
 
 (i) {a,}, where ^,<^,+i; 
 
 a " descending fundamental series " is of the form 
 
 (2) {b^}, where b^)^ b^+i. 
 
 The letter v, as well as /c, X, and /x, 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^} and 
 {a\) in M "coherent" {zusammengehbrig), in signs 
 
 (3) {^.} 11 {<}) 
 
 if, for every element a^ there are elements a\ such 
 that 
 
 and also for every element a\ there are elements a^ 
 such that 
 
 9 
 
I30 THE FOUNDING OF THE THEORY 
 
 Two descending fundamental series {by} and {b' ^ 
 in M are said to be "coherent," in signs 
 
 (4) {K) II {b'.\. 
 
 if for every element b^ there are elements b\ such 
 that 
 
 b. > b\, 
 
 and for every element b\ there are elements b^j, such 
 that 
 
 K > b,. 
 
 An ascending fundamental series {a^} and a 
 descending one {b^} are said to be "coherent," in 
 signs 
 
 [509] (5) WlllW, 
 
 if (a) for all values of v and /n 
 
 and (<^) in M exists at most one (thus either only 
 one or none at all) element m^^ such that, for all i/'s, 
 
 ^. <^ f^h < ^>" 
 
 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 [a^] that : 
 {a) for every v 
 
 {b) for every element m of M that precedes m^ 
 there exists a certain number vq such that 
 
 a„ >^ m, for v'^Vft, 
 
 then we will call iUq a "limiting element (Grenz- 
 element) of {^J in M " and also a " principal element 
 {Haupt element) of M." In the same way we call 
 niQ a "principal element of M " and also " Hmiting 
 element of [b^ in M" if these conditions are 
 satisfied : 
 
 {a) for every v 
 
 ip) for every element m of M that follows m^ 
 exists a certain number v^ such that 
 
 b^ >^ m, for v^Vf^, 
 
 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 THE 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 Mengey If to every funda- 
 mental series in M there is a limiting element in M, 
 we call M a "closed {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," 
 "perfect types," and also "everywhere-dense 
 types " (§ 9). 
 
 For example, >; 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 
 ft) and *ft) have no principal elements, but w-\-v and 
 i/ + *ft) each have a principal element, and are 
 "closed" types. The type 0^.3 has two principal 
 elements, but is not "closed"; the type w.3-f^ 
 has three principal elements, and is "closed." 
 
 §11 
 
 The Ordinal Type Q 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>^o and < i, in their natural order of pre- 
 cedence, so that, with any two of its elements x 
 
 and x\ 
 
 x^x\ if x<x'. 
 
 Let the notation for this type be 
 
 (I) x = e. 
 
 [511] From the elements of the theory of rational 
 and irrational numbers we know that every funda- 
 mental series {x^} in X has a limiting element Xq in 
 X, and that also, inversely, every element x of X 
 
134 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 Q 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 >] investi- 
 gated in § 9, and in such a way that, between any 
 two elements x^ and x^ of X, elements of R lie. 
 
 We will now show that these properties, taken 
 together, characterize the ordinal type Q 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 = h?o ^^^ which bears 
 such a relation to M that, between any two elements 
 Wq and m^ of M elements of S lie, then M = 0. 
 
 Proof. — If S had a lowest or a highest element, 
 these elements, by {b), 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 t]. ¥or since S is a part of M, 
 between any two elements Sq and s^ of S other 
 elements of S must, by {b), lie. Besides, by {b) we 
 have S = «o' Thus the aggregates S and R are 
 "similar" to one another. 
 
 (2) S ro R. 
 
OF TRANSFINITE 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 li Xq is an element of X which does not belong 
 to R, Xq may be regarded as a limiting element of 
 a fundamental series {x^] contained in X, and this 
 series can be replaced by a coherent fundamental 
 series {r^J contained in R. To this [512] corre- 
 sponds as image a fundamental series [s-^^] in S and 
 M, which, because of {a), is limited by an element 
 niQ 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^} and [r^^] are replaced by others limited by the 
 same element x^ in X) be the image of x^^ in X. 
 Inversely, to every element m^ 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 x^ of X which does not belong 
 to R ; let the corresponding elements of M be j 
 and niQ. If r<XQ, there is an ascending funda- 
 mental series {r^,,}, which is limited by x^ and, from 
 
 a certain v^ on, 
 
 r<r^^ for v^v^. 
 
 The image of {r^J in M is an ascending funda- 
 mental series {^a^}j which will be limited by an uZq 
 of M, and we have (§ lo) s^^ ^ in^ for every j/, and 
 •$■ < ^K f°^ '^^ ^0- Thus (§ 7) J < m^. 
 
 U r>XQ, we conclude similarly that s >- 7/1^. 
 
 Let us consider, finally, two elements Xq and x'q 
 not belonging to R and the elements m^ and m'^ 
 corresponding to them in M ; then we show, by 
 an analogous consideration, that, if Xq <x'q, then 
 
 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) 
 
 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 i/ (§ 6). 
 
 We call a simply ordered aggregate F (§ 7) 
 "well-ordered" if its elements /ascend in a definite 
 succession from a lowest f^ in such a way that : 
 
 I. There is in F an element /^ 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 
 
 137 
 
138 THE FOU AWING OF THE THEORY 
 
 that no elements in rank between f and F' occur 
 in F.* 
 
 In particular, to every single element / 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 / 
 Further, if, for example, an infinite series of con- 
 secutive elements 
 
 e' < e" < e'" < . . . < e^^^ -< ^(^+i) . . . 
 
 is contained in F in such a way, however, that there 
 are also in F elements of [2o8] higher rank than all 
 elements e^"^, then, by the second condition, putting 
 for F' the totality {^^"^}, there must exist an element 
 f sugh that not only 
 
 f > e^^^ 
 
 for all values of v, but that also there is no element 
 g in F which satisfies the two conditions 
 
 g > e"^ 
 
 for all values of v. 
 
 Thus, for example, the three aggregates 
 (^1, a^, . . ., a^, . . .), 
 («i, a^, . . ., a^, . . ., dj^, i?2- • -y ^fj^y • ' -^ 
 
 where 
 
 * This definition of "well-ordered aggregates," apart from the 
 wording, is identical with that which was introduced in vol. xxi of the 
 A/at/i. Attn., p. 548 {Grundlas:en einer allgcfneinen Maimichfaltig- 
 keitslehre, p. 4). [See Section VII of the Introduction.] 
 
OF TRANSFINITE NUMBERS 139 
 
 are well-ordered. The two first have no highest 
 element, the third has the highest element ^3; in 
 the second and third b^ immediately follows all 
 the elements a^, in the third i\ immediately follows 
 all the elements a^ 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^ of a well-ordered aggregate F 
 has a lowest element. 
 
 Proof. — If the lowest element /^ of F belongs to 
 Fj, then it is also the lowest element of F^. In 
 the other case, let F' be the totality of all elements 
 of F"" which have a lower rank than all elements F^, 
 then, for this reason, no element of F lies between 
 F' and F^. Thus, if/' follows (II) next after F, 
 then it belongs necessarily to I^ 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. 
 
 \20()\ 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^ be the totality of all these 
 elements and /' the lowest element of F^, 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 rsj 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' 00 F'. 
 
 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 141 
 
 its elements g well-ordered aggregates are sub- 
 stituted in such a way that, if F^^ and F"^' are the 
 well-ordered aggregates which occupy the places 
 of the elements g and g' and g -<^ g\ then also 
 F^^ -< F^^', then the aggregate H, arising by com- 
 bination in this manner of the elements of all the 
 aggregates F^, is well-ordered. 
 
 Proof. — Both H and every part H^ 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 Hj of H, its elements 
 belong to definite aggregates F^ which form, when 
 taken together, a part of the well-ordered aggre- 
 gate {F^}, which consists of the elements F^ and 
 is similar to the aggregate G. If, say, F^ is the 
 lowest element of this part, then the lowest element 
 of the part of H^ contained in F^ is at the same 
 time the lowest element of H. . 
 
 [210] § 13 
 
 The Segments of Well-Ordered Aggregates 
 
 If / is any element of the well-ordered aggre- 
 gate F which is different from the initial element y^^, 
 then we will call the aggregate A of all elements 
 of F which precede /"a " segment {AbscJinitt) 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 /, 
 is a ''remainder of F," and, more fully, ''the 
 remainder which is determined by the element /!" 
 The aggregates A and R are, by theorem C of 
 § 12, well-ordered, and we may, by § 8 and § I2, 
 write : 
 
 (1) F = (A, R), 
 
 (2) R = (/, R'), 
 
 (3) A < R. 
 
 R' is the part of R which follows the initial element 
 / and reduces to o if R has, besides /, no other 
 element. 
 
 For example, in the well-ordered aggregate 
 
 the segment 
 
 and the corresponding remainder 
 
 (<^3, ^4, . . . «^ + 2, . . . ^1, <^2' • • • ^H^ ' ' ' ^V ^2' ^'3) 
 
 are determined by the element a^ ; the segment 
 
 (^1, a^, . . ., a,, . . .) 
 and the corresponding remainder 
 
 {b^, b^, . . ., b^, . . . q, ^2, ^3) 
 
 are determined by the element b-^ ; and the segment 
 
 (^1, «2) • • • . ^^^» . . . b^, b^, , . ., b^, . . . q) 
 
 
OF TRANSFINITE NUMBERS 143 
 
 "remainder 
 
 and the corresponding gogmcnt 
 
 by the element c.^. 
 
 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. 
 
 [211] 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 ^ 
 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 in must correspond, by § 7, an element 
 
144 THE FOUNDING OF THE THEORY 
 
 n' of N that precedes n^ 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 oo 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 F, so that A' oo A. Thus also we would have 
 A' rsj F and A'< A. From A' would result, in the 
 same manner, a smaller segment A'' of F, such that 
 A'' oo F and A" < A' ; and so on. Thus we would 
 obtain an infinite series 
 
 A>A'>A". . . A(^)>A('^+i). . . 
 
 of segments of F, which continually become smaller 
 and all similar to the aggregate F. We will 
 denote by /, /', f'\ . . . , /^"^ . . . the elements of 
 F which determine these segments ; then we would 
 have 
 
 />/' >/" >■■■ >/<-* > A+i) . . . 
 
 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 
 
 F = (^i, a^, . . ., «,,, . . .) 
 
 is similar to every one of its remainders 
 
 Consequently, it is important that we can put by the 
 side of theorem B the following : 
 
 C. 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' 00 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 Y" of F' ; let this 
 segment be determined by the element f of F'. 
 The element /' 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' 00 F leads us consequently to a part F" 
 of a segment A' of A such that Y" 00 A. The same 
 manner of conclusion gives us a part Y'" of a 
 segment A" of A' such that F"' 00 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". . . A(''>>A<''+i>. . ., 
 
 10 
 
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' 00 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 
 in G correspond. Let A be the segment of F that 
 is determined by/, and B and B' the segments of G 
 that are determined by g and g . By theorem A, 
 both A 00 B [213] and A 00 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- 
 
OP TRANSFJNITR NUMBERS 147 
 
 ordered aggregates F and G, for every smaller 
 segment A'<A of F there is a similar segment 
 B' < ]^ of G and for every smaller segment W < 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 
 aerereeate F there is a similar see^ment B of another 
 well-ordered aggregate G, and also inversely, for 
 every segment B of G a similar segment A of F, 
 then F 00 G. 
 
 Proof. — We can image F and G on one another 
 according to the following law : Let the lowest 
 element f^ of F correspond to the lowest element g^ 
 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 ^ 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 
 
148 THE FOUNDING OF THE THEORY 
 
 segment A of F belongs. Let the element /"which 
 determines this segment A be the image of ^. 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' [2 1 4] the corresponding elements of 
 G, A and A' the segments determined by/ and /', 
 B and B' those determined by g and g\ 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^ of G such that B^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 B^ of G determined by that element is the 
 least of that totality. By theorem I, every segment 
 
OF JRANSFINITE NUMBERS 149 
 
 of G which is greater than B^ 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 Bj, and to every segment 
 B < B^ belongs a similar segment A of F, because 
 Bj 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^, and for every segment B of B^ there is a similar 
 segment A of F. By theorem K, we thus have 
 
 FooB^. 
 
 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^ 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^, would be the least. For 
 every segment of Aj^ would then exist a similar 
 segment of B^^, and also for every segment of B^^ a 
 similar segment of A^. Thus, by theorem K, we 
 would have 
 
 B^ no A^. 
 
 [215] But this contradicts the datum that for B^ 
 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. 
 
ISO 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 
 
 {b) there is a definite segment Bj of G to which 
 F is similar, or 
 
 (<:) there is a definite segment A^ 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 cNj G. 
 
 In the second case there is, by theorem L, a definite 
 segment V>^ of B such that 
 
 BiOoF; 
 
OF TRANSFINITE NUMBERS 151 
 
 and in the third case there is a definite segment A^ 
 of F such that 
 
 Ai fxj G. 
 
 We cannot have F c\j G and F 00 B^ simultaneously, 
 for then we would have G 00 Bj, contrary to theorem 
 B ; and, for the same reason, we cannot have both 
 F rv) G and G 00 A^. Also it is impossible that 
 both F 00 Bj and G 00 A^, for, by theorem A, 
 from F 00 B^ would follow the existence of a 
 segment B'^ of B^^ such that A^ 00 B\. Thus we 
 would have G 00 B'j, 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'^ of F' which is similar to F. But F'^ 
 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. 
 
 § 14 
 
 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 
 {Einsefi) 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 /3 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 = a, G = /3, 
 
 then, by theorem N of § 13, three mutually ex- 
 clusive cases are possible : 
 
 {a) F 00 G ; 
 
 {b) There is a definite segment Bj of G such that 
 
 FooB^; 
 (c) There is a definite segment Aj of F such that 
 
 Goo A^. 
 
 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 )8 to one another. In the first case a = /3; in 
 the second we say that a</5; in the third we say 
 that a>/3. Thus we have the theorem : 
 
OF TRANSFINITE NUMBERS 153 
 
 A. If a and ^ are any two ordinal numbers, we 
 have either a = l3 or a<^ or a> /3. 
 
 From the definition of minority and majority 
 follows easily : 
 
 B. If we have three ordinal numbers a, /5, y, and 
 if a < 18 and ^ <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 + /3 = (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 a + l3, a is called the "augend" and 
 ^ 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) ^<«+/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 + ^ = a + y, we always 
 have ,8 = y. 
 
 In general a-\-^ and /3 + a are not equal. On 
 the other hand, we have, if y is a third ordinal 
 number, 
 
 (4) (a + ^) + y = a + (/3 + 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 /3 an aggregate 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 /3 and will be called the product 
 
 (5) " i^^^=«^ 
 
 (6) a.^ = H. 
 
 F. The product of two ordinal numbers is also 
 an ordinal number. 
 
 In the product a .^8, a is called the '' multiplicand " 
 and /3 the "multiplier." 
 
 In general a./^and /S.aare not equal. But we 
 have (§ 8) 
 
OF TRANS FINITE NUMBERS 155 
 
 (7) (a.^).y = a.(/3.y). 
 
 That is to say : 
 
 [218] G. In the multipHcation 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.^ + a.y. 
 
 With reference to the magnitude of the product, 
 the following theorem, as we easily see, holds : 
 
 H. If the multipHer 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 a.^ = a.y, then it always 
 follows that /8 = y. 
 
 On the other hand, we evidently have 
 
 (9) a . I = I . a = a. 
 
 We have now to consider the operation of sub- 
 traction. If a and /3 are two ordinal numbers, and 
 a is less than /5, there always exists a definite 
 ordinal number which we will call /3 — a, which 
 satisfies the equation 
 
 (10) a + (/3-a) = ^. 
 
 For if G = ^, G has a segment B such that B = a ; 
 we call the corresponding remainder S, and have 
 
 G = (B, S), 
 
 /3 = a + S; 
 
156 THE FOUNDING OF THE THEORY 
 
 and therefore 
 
 (ii) /5-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 (lo) : 
 
 (12) (7 + /3)-(y + a) = ^-a, 
 
 (13) y(/3-a) = y^-ya. 
 
 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) ^v=^v^ 
 
 [219] then, by theorem E of § 12, 
 
 (15) .G = (Gi, G2, . . ., G^, . . .) 
 
 is also a well-ordered aggregate whose ordinal 
 number represents the sum of the numbers /3,,. 
 We have, then, 
 
 (16) A + ^2+ • • • +A.+ . . . =G = /3, 
 
 and, as we easily see from the definition of a 
 product, we always have 
 
OF TRANSFINITE NUMBERS 157 
 
 (17) y.(/5i + ^,+ ... +/'i+...) 
 
 If we put 
 
 (18) «^^^^ + /3^ + . . . +/^^, 
 then 
 
 (19.) a, = (Gi, G2, . . . G,). 
 
 We have 
 
 (20) a.+i>a., 
 
 and, by (10), we can express the numbers P^, by 
 the numbers a^ as follows : 
 
 (21) i^i=ai; /5^+i = a^+i — a^. 
 
 The series 
 
 «!, aa, . . ., 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). 
 Betw^een it and ^ subsists a relation which can be 
 expressed m the following manner : 
 
 {a) The number /5 is greater than a, for every 
 J/, because the aggregate (G^, Gg, . . ., G„), whose 
 ordinal number is a^, is a segment of the aggregate 
 G which has the ordinal number ^ ; 
 
 {b) If /3' is any ordinal number less than ^, then, 
 from a certain v onwards, we always have 
 
 For, since S' < jS, there is a segment B' of the 
 
158 THE FOVNDING OF THE THEORY 
 
 aggregate G which is of type /3'. The element of 
 G which determines this segment must belong to 
 one of the parts G^ ; we will call this part G^,^. But 
 then B' is also a segment of (G^, Gg, . . ., G^ ), and 
 consequently /3' < a^, . Thus 
 
 for v^v^. 
 
 Thus /3 is the ordinal number which follows next 
 in order of magnitude after all the numbers a^ ; 
 accordingly we will call it the "limit" {Grenze) of 
 the numbers a^ for increasing v and denote it by 
 Lim a^, so that, by (i6) and (21) : 
 
 (22) Lim a^ = ai + («2 - «!> + •" • • + (a^+i -«.) + • • . 
 
 [220] We may express what precedes in the 
 following theorem : 
 
 1. To every fundamental series {a,J of ordinal 
 numbers belongs an ordinal number Lim a,, which 
 
 V 
 
 follows next, in order of magnitude, after all the 
 numbers a„ ; it is represented by the formula (22). 
 
 If by y we understand any constant ordinal 
 number, we easily prove, by the aid of the formulae 
 (12), (13), and (17), the theorems contained in the 
 formulae : 
 
 (23) Lim (y + a„) = y + Lim a, ; 
 
 V V 
 
 (24) Lim y . ay = y . Lhn a^. 
 
 V V 
 
 We have already mentioned in § 7 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 other than finite ordinal numbers. But 
 two different ordinal numbers a and ^ cannot belong 
 to the same finite cardinal number v. For if, say, 
 a</3 and G = /3, 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(h?Q), 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. 
 
i6o THE FOUNDING OF THE THEORY 
 [221] % 15 
 
 The Numbers of the Second Number- Class Z({^o) 
 
 The second number-class Z(}^q) is the totality {a} 
 of ordinal types a of well-ordered aggregates of 
 the cardinal number «o (§ 6). 
 
 A. The second number-class has a least number 
 0) = Lim V. 
 
 V 
 
 Pi'oof. — By o) we understand the type of the 
 well-ordered aggregate 
 
 (1) Fo = (/i, /„ . . ., /,, . . .), 
 
 where v runs through all finite ordinal numbers and 
 
 (2) /.-</.+i^ 
 Therefore (§ 7) 
 
 (3) w=Fo> 
 
 and (§ 6) 
 
 (4) ^ = f^o- 
 
 Thus 0) is a number of the second number-class, 
 and indeed the least. For if y is any ordinal 
 number less than o), it must (§ 14) be the type of 
 a segment of Fq. But F^ has only segments 
 
 A = (/i,/2, . . .,/.), 
 with finite ordinal number v. 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^ given in § 14, we 
 
 V 
 
 obviously have a)=Lim v. 
 
OF TRANSFINITE NUMBERS i6i 
 
 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 Nq : 
 
 (5) F = a, 
 
 (6) a=No- 
 
 We have, where by g is understood a new element, 
 
 (7) a+i=(F, ^). 
 Since F is a segment of (F, g), we have 
 
 (8) a+i>a. 
 We also have 
 
 a+i=a+i=No+i=«o (§^)- 
 
 Therefore the number a+i belongs to the second 
 number-class. Between a 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 ai, aa, . . ., a„ . . . is any fundamental series 
 of numbers of the first or second number-class, then 
 the number Lim a, (§ H) following them next in 
 
 V 
 
 order of magnitude belongs to the second number- 
 class. 
 
 Proof. — By § 14 there results from the funda- 
 
 II 
 
1 62 THE FOUNDING OF THE THEORY 
 mental series {a,,} the number Lim a^ if we set up 
 
 V 
 
 another series ^j, /^g, • . ., /8^, . . ., where 
 
 If, then, Gj, Gg, . . ., G^, . . . are well-ordered aggre- 
 gates such that 
 
 then also 
 
 G = (^i) ^2, . . ., G^, . . .) 
 
 is a well-ordered aggregate and 
 Lim a^ = G. 
 
 V 
 
 It only remains to prove that 
 
 Since the numbers /^j, ^82, . . ., ft, • • . belong to 
 the first or second number-class, we have 
 
 and thus 
 
 G<h?o • No = ^^o- 
 But, in any case, G is a transfinite aggregate, and 
 so the case G<^?q is excluded. 
 
 We will call two fundamental series {a^} and {a^\ 
 of numbers of the first or second number-class (§ lO) 
 "coherent," in signs : 
 
 (9) {«.} 11 {«'.}. 
 
 if for every v there are finite numbers Xq and ^^ 
 such that 
 
 (10) aK>OLy, X>Xo> 
 
OF rRANSFINTTE NUMBERS 163 
 
 and 
 
 (11) a^>a\, At^Mo- 
 
 [223] D. The limiting numbers Lim a^ and Lim a\ 
 
 V V 
 
 belonging respectively to two fundamental series 
 {ay} and [a\} are equal when, and only when, 
 {a.} II {«;}. 
 
 Proof. — For the sake of shortness we put 
 Lim a^, = /3, Lim a'^ = y. We will first suppose 
 
 that {a^ II {a'4 ; then we assert that /3 = y. For 
 if /3 were not equal to y, one of these two numbers 
 would have to be the smaller. Suppose that /8<y. 
 From a certain v onwards we would have a'v>/3 
 (§ 14), and consequently, by (11), from a certain 
 /x onwards we would have a^>/5. But this is 
 impossible because /3=Lim a^. Thus for all ^'s 
 
 V 
 
 we have a^</3. 
 
 If, inversely, we suppose that /3 = y, then, because 
 oiv<y^ we must conclude that, from a certain X 
 onwards, a\>a^, and, because a\<l3, we must 
 conclude that, from a certain ^ onwards, afj^> a\. 
 That is to say, {aj || {a\]. 
 
 E. If a is any number of the second number- 
 class and vq any finite ordinal number, we have 
 j/^-|-a = a, and consequently also a — VQ = a. 
 
 Proof. — We will first of all convince ourselves of 
 the correctness of the theorem when a = no. We 
 have 
 
 ^0 = Uv <^2' • • • <r J' 
 
1 64 THE FOUNDING OF THE THEORY 
 and consequently 
 
 But if a>a), we have 
 
 a = CO + (a — w), 
 
 i/Q + a = (I'o + w) + (« "" w) = ^ + (« ~ ^^ = «• 
 F. If j/Q is any finite ordinal number, we have 
 
 I/q . ft) = ftj. 
 
 Ptoof, — In order to obtain an aggregate of the 
 type i/Q . o) we have to substitute for the single 
 elements /, of the aggregate (/i, Z^, ...,/„...) 
 aggregates {g,^ i, ^v, 2> • • • > gv, v) of the type v^. We 
 thus obtain the aggregate 
 
 (^1. 1' ^1, 2) • • -5 S\, V ^2. i» • • • ' ^2, "o' • • • ' «^*'. 1' 
 
 which is obviously similar to the aggregate {/,}. 
 Consequently 
 
 I/q . 0) = ft). 
 
 The same result is obtained more shortly as follows. 
 By (24) of § 14 we have, since ft)=Lim j/, 
 
 1/q 0)= Lim j^Q I/. 
 On the other hand, 
 
 and consequently 
 
 Lim j/Qy=Lim t/ = ft) ; 
 1/ I' 
 
 so that 
 
OF TRANSFINITE NUMBERS 165 
 
 [224] G. We have always 
 
 (a + ^0)0) = aw, 
 
 where a is a number of the second number-class 
 d i/q a number of 1 
 Proof. — We have 
 
 and Vq a number of the first number-class. 
 
 Lim i/ = ft). 
 
 V 
 
 By (24) of § 14 we have, consequently, 
 
 (a + i/o)a) = Lim {a-\-v^v. 
 
 But 
 
 I 2 V 
 
 (a4-i/o>= (a + i/o) + (« + »'o)+ • • • +(« + t^o) 
 I 2 v—l 
 
 = a + (^^o + «) + K + «) • • • (t^o + «) + ^'o 
 I 2 1/ 
 
 = a + a+...+a + i/o 
 
 = aj/ + i/q. 
 
 Now we have, as is easy to see, 
 
 {av\-VQ} II {av}, 
 and consequently 
 
 Lim (a + i^o)i/= Lim (ai^ + i/o) = Lim a»/ = aaj. 
 
 V V V 
 
 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 
 
 Proof. — Let F be a well-ordered aggregate such 
 that F = a, and let/^ 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 _ 
 
 A' = a, 
 
 and inversely every segment A' determines by its 
 type h! = 0! a number a <a of the first or second 
 
 number-class. For, since F = ^?Q, A' can only be 
 either a finite cardinal number or ^q. The segment 
 A' is determined by an element/' >^/i of F, and 
 inversely every element/' >-/i of F determines a 
 segment A' of F. If/' and/" are two elements of 
 F which follow / 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 = (/, F'), we obtain, when we make the 
 element/' of F' correspond to the element a of {a'}, 
 an imaging of these two aggregates. Thus we have 
 
 R}^F'. 
 
 ButF' = a— I, and, by theorem E, a— i=a. Con- 
 sequently 
 
 {a)=a. 
 
 Since a = «o, we also have {a'} = t^o; thus we have 
 the theorems : 
 
 I. The aggregate {a'} of numbers a of the 
 first and second number-classes which are smaller 
 
OF TRANSFINITE NUMBERS 167 
 
 than a number a of the second number-class has 
 the cardinal number j^^. 
 
 K. Every number a of the second number-class 
 is either such that {a) it arises out of the next 
 smaller number a.^ by the addition of i : 
 
 or {ii) there is a fundamental series {a^ of numbers 
 of the first or second number-class such that 
 
 a — Lim a^. 
 
 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, 
 
 a = A-f i=a_iH- I. 
 
 There exists, therefore, a next smaller number 
 which is that called %. 
 
 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 d, consequently, none is greatest. By 
 theorem I, the aggregate {a\ can be brought into 
 the form {ay\ of a simply infinite series. If we set 
 out from a'l, the next following elements ai^ a'3, . . . 
 in this order, which is different from the order of 
 magnitude, will, in general, be smaller than a^ ; 
 but in every case, in the further course of the 
 
1 68 THE FOUNDING OF THE THEORY 
 
 process, terms will occur which are greater than a'^ ; 
 
 for a'l cannot be greater than all other terms, 
 
 because among the numbers {a'^} there is no 
 
 greatest. Let that number a^ which has the least 
 
 index of those greater than a^ be a'p. Similarly, 
 
 let a p be that number of the series {a'^} which has the 
 
 least index of those which are greater than a ^ . By 
 
 proceeding in this way, we get ani infinite series of 
 
 increasing numbers, a fundamental series in fact, 
 
 /ft I 
 
 a 1, Op^, Qp^, . . ., Op^, . . . 
 
 [226] We have 
 
 I < /)2 < Pa < • • ' <pv< pv+i . . . , 
 a\ < a'p^ < ap^ < . . . < a^^ < Up^^^ . . . , 
 
 a\<ap^ always if /ui<p,; 
 
 and since obviously v ^ p^,, we always have 
 
 a, < a'p^. 
 
 Hence we see that every number a'^, 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 ap . If, 
 therefore, we put a\ = a-^, Op^^^ = «(,+!, we have 
 
 a= Lim a^. 
 
 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 1 
 according to the formula 
 
 a = a-i+ I ; 
 
 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 a_i, but they 
 arise from fundamental series [a^] as limiting 
 numbers according to the formula 
 
 a= Lim a^. 
 
 V 
 
 Here a is the number which follows next in order 
 of magnitude to all the numbers a^. 
 
 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 Transfinite 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 answ^er the question as to the 
 
 * [Cf. Section VII of the Introduction,] 
 
I/O THE FOUNDING OF THE THEORY 
 
 cardinal number which is possessed by the aggregate 
 Z(h?o)={a} 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„ the totality of 
 numbers of the second number-class which are 
 smaller than a given number a, arranged in order 
 of magnitude, then A„ is a well-ordered aggregate 
 of type a — o). This results from theorem H of § 14. 
 The aggregate of numbers d of the first and second 
 number-class which was there denoted by {a'}, is 
 compounded out of {v] and A<^, so that 
 
 Thus 
 and since 
 we have 
 
 {a1 = ({.}, A J. 
 {7} = M + A,; 
 
 A„ = a — w. 
 
 Let J be any part of {«} 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 of A^^+i is greater 
 than all the numbers of J. Since A^^+i is a well- 
 ordered aggregate, by § 12 a number d of A^^+i, 
 and therefore also of {a}, must follow next to all 
 the numbers of J. Thus the condition II of § 12 is 
 
OF TRANSFINITE NUMBERS 171 
 
 fulfilled in the case of {a} ; the condition 1 of § 12 
 is also fulfilled because {a} has the least number co. 
 
 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 of the first, which 
 
 is No- 
 
 D. The power of the totality {a} of all numbers 
 a of the second number-class is not equal to ^^q. 
 
 Proof. — If {a} were equal to j^q, we could bring 
 the totality {a} into the form of a simply infinite 
 series 
 
 Vp y2> • • •> 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^ be the term of the series 
 which has the least index of those greater than y^, 
 yp the term which has the least index of those 
 greater than y^, and so on. We get an infinite 
 series of increasing numbers, 
 
 yi> yp> • • •' ypv' • • •> 
 
1/2 THE FOUNDING OF THE THEORY 
 such that 
 
 yi<yp.<yp.r • • <rp„<rp,+i< • • •> 
 r^ ^ yp,- 
 
 By theorem C of § 15, there would be a definite 
 number S of the second number-class, namely, 
 
 ^ = Limy,^, 
 
 V 
 
 which is greater than all numbers y^ . Consequently 
 
 we would have 
 
 S>y. 
 
 for every v. But {y^} contains all numbers of the 
 second number-class, and consequently also the 
 number S ; thus we would have, for a definite v^, 
 
 ^=yv 
 
 which equation is inconsistent with the relation 
 ^ > y^ . The supposition {a} = t^o consequently leads 
 to a contradiction. 
 
 E. Any totality {/3} of different numbers ^ of 
 the second number-class has, if it is infinite, either 
 the cardinal number «(, or the cardinal number {a} 
 of the second number-class. 
 
 Proof. — The aggregate {^}, 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 Aa , which is the totality 
 
OF TRANSFINITE NUMBERS 173 
 
 of all numbers of the same number-class which are 
 less than a©, 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 {^)—a^ — w or {/3} = {a}, and 
 
 consequently {^} is either equal to Qq — w or {a). 
 But oq — ft) is either a finite cardinal number or is 
 equal to «o (theorem I of § 15). The first case is 
 here excluded because {/3} is supposed to be an 
 infinite aggregate. Thus the cardinal number {^} 
 is either equal to j^q 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^ and less than {a}. For if 
 not, there would have to be, by § 2, an infinite part 
 
 {/3} of {a} such that {/3}=a. But by the theorem 
 E just proved, the part {/3} has either the cardinal 
 
 number js^q or the cardinal number {a}. Thus the 
 cardinal number {a\ is necessarily the cardinal 
 number which immediately follows «o in magnitude ; 
 we call this new cardinal number «j. 
 
 In the second number-class Z(«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 a)% + a)'""\+ . . . +j/^. 
 
 It is convenient to make ourselves familiar, in the 
 first place, with those numbers of Z({^o) which are 
 whole algebraic functions of finite degree of w. 
 Every such number can be brought — and brought 
 in only one way — into the form 
 
 (I) = a,% + a)'^-V+ . . . +.^, 
 
 where ;x, v^ are finite and different from zero, but 
 j^i, 1^2' • • •> '^i'* "^^y ^^ zero. This rests on the fact 
 
 4, we 
 
 and, by theorem E of § 15, 
 
 Thus, in an aggregate of the form 
 
 . . . +w'^V + w^i/+ . . ., 
 
 all those terms which are followed towards the right 
 by terms of higher degree in « may be omitted. 
 This method may be continued until the form given 
 in (i) is reached. We will also emphasize that 
 
 that 
 
 
 
 
 
 
 
 (2) 
 
 
 
 of'v -\r 
 
 Iffv = 61)' 
 
 V, 
 
 
 if m'</x 
 
 and 
 
 i/>0, />o. 
 
 For, 
 
 by (8) 
 
 of 
 
 have 
 
 
 
 
 
 
 
 
 
 iJd^ V 
 
 + ^^^1/ = 
 
 i/{v^. 
 
 ^^-^v\ 
 
 
 (3) w'^J/ + ft)V = a)'^(t/ + 0- 
 
OF TRANSFINITE NUMBERS 175 
 
 Compare, now, the number with a number \p- of 
 the same kind: 
 
 (4) V^ = coVo + ^')^"Vi+ • • • +P^' 
 
 If yu and X are different and, say, iu<Xy we have by 
 (2) + \^ = -v/r, and therefore (p<\fr. 
 
 [230] If jUL = \, vq, and Pq are different, and, say, 
 i'o<Poy ^^^ h^^'^ t)y (2) 
 
 + (^Xpo-^o) + ^^"Vi+ • • • +M = \^i 
 and therefore 
 
 If, finally, 
 
 /x = X, V(i = Po, I'l — Pv • ' • ^<T-l = PcT-l^ (rKfXy 
 
 but i;^ is different from p^ and, say, v^<p„, we 
 have by (2) 
 
 and therefore again 
 
 Thus, we see that only in the case of complete 
 identity of the expressions ^ and yp- can the numbers 
 represented by them be equal. 
 
 The addition of and \/r leads to the following 
 result : 
 
 {a) If ^<A, then, as we have remarked above, 
 
 {b) If ^ = X, then we have 
 
176 THE FOUNDING OF THE THEORY 
 {c) \i fx>\ we have 
 
 In order to carry out the multiplication of and i/r, 
 we remark that, if p is a finite number which is 
 different from zero, we have the formula : 
 
 (5) ^yo = a)%/D + w'^-Vi+ ... +1/^. 
 
 It easily results from the carrying out of the sum 
 consisting of p terms + ^+ . . . +0. By means 
 of the repeated application of the theorem G of 
 § 15 we get, further, remembering the theorem F 
 of §15, 
 
 (6) 0CO = ft)'^+S 
 
 and consequently also . 
 
 (7) 9^0)^ = 60^ + ^. 
 
 By the distributive law, numbered (8) of § 14, 
 we have 
 
 0\/^ = 0a)^iOo + 0w^'Vi+ • • • +V^^Px-i + 0/3a. 
 
 Thus the formulae (4), (5), and (7) give the following 
 result : 
 
 {a) if p;^ = o, we have 
 
 {b) If Pa 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.) = a)% + a)'^'/ci+ . . . -^(*r--K,, 
 
 where 
 
 and /Cq, /ci, . . ., act 3-re finite numbers which are 
 different from zero. Then we have 
 
 ^ = (co'^'/ci + w^^/ca + . . . + ur-^K,){s^^ - '^•/f + I ). 
 
 By the repeated application of this formula we get 
 
 = fo'^r/c^(w'"T-l-MT/f^_^+ l)(ft)'^T-2-'^T_l^^_2+ l). . . 
 
 (co'^-'^'/Co+l). 
 But, now, 
 
 a)\+ I =(co^+ l)/c, 
 
 if /c is a finite number which is different from zero ; 
 so that : 
 
 . . . (ft)'^-'^'+I>o. 
 
 The factors 0)^+ i which occur here are all irre- 
 soluble, and a number can be represented in this 
 product-form in only one way. If fXj — o^ then 
 is of the first kind, in all other cases it is of the 
 second kind. 
 
 The apparent deviation of the formulae of this 
 paragraph from those which were given in Math. 
 Ann., vol. xxi, p. 585 (or Grundlagen, p. 41), is 
 merely a consequence of the different writing of the 
 product of two numbers : we now put the multi- 
 
 12 
 
178 THE FOUNDING OF THE THEORY 
 
 plicand on the left and the multipHcator on the 
 right, but then we put them in the contrary way. 
 
 §i8 
 
 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 (5 be two constants belong- 
 ing to the same domain, and let 
 
 ^>0, y>l. 
 We can then assert the following theorem : 
 
 A. There is one wholly determined one-valued 
 function /(^) of the variable ^ such that : 
 
 {a) /(o) = ^. 
 
 {b) \i ^' and ^' are any two values of ^, and if 
 
 then 
 
 /(f)</(f)- 
 
 [232] ic) For every value of ^ we have 
 
 /(#+i)=/(ar- 
 
 {d) If {^^} is any fundamental series, then {/(f^)} 
 is one also, and if we have 
 
 ^=Lim ^„ 
 
 V 
 
 then 
 
 /(f) = Lini/(,^„). 
 
 V 
 
 * [Here obviously it is Potenz and not MdchHgkeit.'\ 
 
OF TRANSFINITE NUMBERS 179 
 
 Proof. — By {a) and {c), we have 
 
 /(l) = <5y, f{2) = Syy, f{z) = Syyy, . . ., 
 and, because 5>0 and y> i, we have 
 
 /(l)</(2)</(3)< . . . </(„)</(.+ I)< . . . 
 
 Thus the function f{^) is wholly determined for the 
 domain $<w. Let us now suppose that the theorem 
 is valid for all values of f which are less than a, 
 where a is any number of the second number-class, 
 then it is also vaUd for f <a. For if a is of the 
 first kind, we have from {c) : 
 
 /(a)=/(a-Oy>/(a.i); 
 
 SO that the conditions {b), (c), and (d) are satisfied 
 for f^a. But if a is of the second kind and {a^} is 
 a fundamental series such that Lim a^ = a, then it 
 
 follows from (d) that also {/(ay)] is a fundamental 
 series, and from (d) that /(a) = Lim /(a^). If we 
 
 take another fundamental series {a^} such that 
 Lim a\, = a, then, because of (d), the two funda- 
 
 mental series {/(a,,)} and {/(a\)} are coherent, and 
 thus also /(a) = Lim /(a'„). The value of /(«) is, 
 
 consequently, uniquely determined in this case also. 
 If a' is any number less than a, we easily convince 
 ourselves that /(a) < /(a). The conditions (d), (c)y 
 and (d) are also satisfied for ^<'a. Hence follows 
 the validity of the theorem /<?r all values of ^. For 
 if there were exceptional values of f 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 f<a, 
 but not for ^^a, and this would be in contradiction 
 with what we have proved. Thus there is for the 
 whole domain of f one and only one function /(f) 
 which satisfies the conditions {a) to {d). 
 
 [233] If we attribute to the constant ^ the value i 
 and then denote the function f{£) 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 f such that : 
 
 {a) y«=i; 
 
 W If f <rthen y^'<y^"; 
 {c) For every value of f we have y^+i = y^y ; 
 id) If {f^} is a fundamental series, then {y^"} 
 is such a series, and we have, if f=Lim f^, the 
 
 V 
 
 equation 
 
 y^ = Lim y^". 
 
 V 
 
 We can also assert the theorem : 
 
 C. If /(f) is the function of f which is characterized 
 in theorem A, we have 
 
 /(f) =V- 
 
 Proof. — If we pay attention to (24) of § 14, 
 we easily convince ourselves that the function ^y^ 
 satisfies, not only the conditions {a), {b), and {c) 
 of theorem A, but also the condition {d) of this 
 
OF TRANSFINITE NUMBERS i8i 
 
 theorem. On account of the uniqueness of the 
 function /(f), it must therefore be identical with ^y^. 
 D. If a and ^ are any two numbers of the first 
 or second number-class, including zero, we have 
 
 y»+^ = y*y^. 
 
 Proof.— W& consider the function 0(f) = y«+^. 
 Paying attention to the fact that, by formula (23) 
 
 of § 14, 
 
 Lim (a + f,) = a + Lim f„ 
 
 V V 
 
 we recognize that ^(f) satisfies the following four 
 conditions : 
 
 {a) 0(0) = y*; 
 
 {b) Iff <r, then 0(f) <0(r); 
 
 {c) For every value of f we have V'(f + = <^(f )y 5 
 
 (d) If {f } is a fundamental series such that 
 Lim f = f , we have 
 
 0(f) = Lim 0(f). 
 
 V 
 
 By theorem C we have, when we put (5 = y^ 
 
 0(a = yy- 
 
 If we put f = ^ in this, we have 
 
 E. If a and /3 are any two numbers of the first or 
 second number-class, including zero, we have 
 
 [234] Proof. — Let us consider the function 
 ■^(f) = y«f and remark that, by (24) of § 14, we 
 
1 82 THE FOUNDING OF THE THEORY 
 always have Lim af^ = a Lim ^^, then we can, by 
 
 V V 
 
 theorem D, assert the following : 
 
 {a) iA(o)=i ; 
 
 ib) Iff<r, thenV^(0<V^(r); 
 
 ic) For every value of f we have ^^(f + i ) = V^(f)y* ; 
 
 {d) If {^^} is a fundamental series, then {V^(^v)} is 
 also such a series, and we have, if ^=Lim^j,, the 
 
 V 
 
 equation '\/r(^) = Lim ■^{^^. 
 
 V 
 
 Thus, by theorem C, if we substitute in it i for (5 
 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 f, 
 
 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, 
 
 a_i<y«-S 
 and consequently 
 
 «-iy < y''"'y = r''- 
 
 Hence 
 
 y'^>a_i + a_i(y-l). 
 
 Since both a_^ and y— i are at least equal to i, and 
 a_i+ I =a, we have 
 
 y" > a. 
 
OF TRANSFINITE NUMBERS 183 
 
 If, on the other hand, a is of the second kind and 
 a = Lim a„, 
 
 
 then, because a^,<a, we have by supposition 
 
 Consequently 
 
 Lim a„ S. 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] i^y'; 
 
 but 
 
 a>y% 
 
 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 second number-class. 
 The power 00^ will be, for sufficiently great values 
 
1 84 THE FOUNDING OF THE THEORY 
 
 of ^, greater than a. By theorem F of § i8, this is 
 always the case for f >a ; but in general it will also 
 happen for smaller values of f. 
 
 By theorem B of § i6, there must be, among the 
 values of ^ for which 
 
 d)^> a, 
 
 one which is the least. We will denote it by /3, and 
 we easily convince ourselves that it cannot be a 
 number of the second kind. If, indeed, we had 
 
 /5 = Lim /5j,, 
 
 V 
 
 we would have, since ^^ < /3, 
 
 CO " < a, 
 
 and consequently 
 
 Lim 0)^' ^ a. 
 
 V 
 
 Thus we would have 
 
 o)^ < a, 
 whereas we have 
 
 Therefore /3 is of the first kind. We denote /3_i 
 by Oq, so that ^ = ao+i, and consequently can 
 assert that there is a wholly determined number a^ 
 of the first or second number-class which satisfies 
 the two conditions : 
 
 (l) (jo^<^a, o)'^o)>a. 
 
 From the second condition we conclude that 
 
OF TRANSFINITE NUMBERS 185 
 
 does not hold for all finite values of v, for if it did 
 we would have 
 
 Lim a)% = co"«a) < a. 
 
 V 
 
 The least finite number v for which 
 {£) ^v> a 
 
 will be denoted hy kq-\-i. Because of (i), we have 
 /Co > o. 
 
 [236] There is, therefore, a wholly determined 
 number k^ of the first number-class such that 
 
 (2) a)''oA:o< a, a)%(/Co + l) > a. 
 If we put a — a)%/co = a', we have 
 
 (3) a = a)%/Co + a' 
 and 
 
 (4) 0<a'<a)%, 0</fo<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 ao=|8_i 
 satisfies the conditions (i), and by the conditions 
 (2) the finite number /Cq 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 = w^o/Cq + a , 
 where 
 
 O <.a' < co%, o < /Co < Wj 
 
 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' = coVi + a'', 
 
 where 
 
 0<^a <i>f-h 0<Ki<(j», 
 and 
 
 ai < Oq, a^ < a. 
 
 In general we get a further sequence of analogous 
 equations : 
 
 (6) a' = of-'^K2 + a^'j 
 
 (7) a''' = a)'^3/c3 + a^\ 
 
 But this sequence cannot be infinite, but must 
 necessarily break off. For the numbers a, a, a\ . . . 
 decrease in magnitude : 
 
 a> a> a'> a" . . . 
 
 If a series of decreasing transfinite numbers were 
 infinite, then no term would be the least ; and this 
 is impossible by theorem B of § i6. Consequently 
 we must have, for a certain finite numerical value r. 
 
 «^^+^;=o- 
 
OF TRANSFINITR 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 Oq, ai , . . . a^ are numbers of the first or 
 second number-class, such that : 
 
 a^ > Oj > ag > . . . > ctj ^ O, 
 
 while /Co, /ci, . . . /c^, r+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 " ; a^ is called the ' ' degree " and a^ the 
 ''exponent" of a. For t = 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 ^ in the normal 
 form : 
 
 (8) ^ = a)^«Xo + a)^'Xi+ . • . +co^'^Xa. 
 The formulae : 
 
 (9) w'^'/c' -I- w'^'/c = w'^X'c' + /c), 
 
 (10) ■ a)*"'/ + co*V' = w'^V, a <d\ 
 
 where /c, k\ k' here denote finite numbers, serve 
 both for the comparison of a with ^ and for the 
 
1 88 THE FOUNDING OF THE THEORY 
 
 carrying out of their sum and difference. Tiiese are 
 generalizations of the formulae (2) and (3) of § 17. 
 
 For the formation of the product a^, the following 
 formulae come into consideration : 
 
 (11) aX = (o'^^X + w-^'/ci + . . . +w*X, 0<X< 
 (12) 
 
 Od , 
 
 = /.>«o+l 
 
 aa)= o) 
 
 (13) aw^' = a)'^o+^; /3'>0. 
 
 The exponentiation a^ can be easily carried out 
 on the basis of the following formulae : 
 
 (14) a^ = a)*o%+ . . ., 0<X<a). 
 
 The terms not written on the right have a lower 
 degree than the first. Hence follows readily that 
 the fundamental series {a^} and {co'"^^} are coherent, 
 so that 
 
 (15) a" = 0)^0-, ao>0. 
 
 Thus, in consequence of theorem E of § 18, we 
 have : 
 
 (16) a"^' = w"''"^ ao>0, ^'>0. 
 
 By the help of these formulae we can prove the 
 following theorems : 
 
 [238] C. If the first terms w^o/cq, w^oXq of the 
 normal forms of the two numbers a and /3 are not 
 equal, then a is less or greater than /3 according as 
 a)«o/c-Q is less or greater than co^^Xq. But if we have 
 
 and if w'^'^+Vp+i is less or greater than w^p-^\+i, then 
 a is correspondingly less or greater than ^. 
 
OF TRANSFINITE NUMBERS 189 
 
 D. If the degree a^ of a is less than the degree 
 i8o of ^, we have 
 
 a + /3 = /3. 
 If ao = /5o, then 
 
 a + /3 = a)^o(;,^ + Xo) + a)^*Xi+ . . . +oAA,. 
 But if 
 
 then 
 
 a + ^ = w'^/CoH- . . . +a)"''/Cp + a)^«Xo + ^'^i+ • • • +^"X^- 
 
 E. If ^ is of the second kind (/3^>o), then 
 a/3 = (o'^o+^oXQ + a)«o+^^Xi+ . . . + w'^o+^-X^ = w*o/3 ; 
 
 But if /3 is of the first kind (/3, = o), then 
 
 a^ = (o«o+^oX(, + a)«^^+^^Xi+ . . .+co'^'>+^'--iX^_i + oj'^*'/CoX^ 
 
 4- w'^'/ci + . . . + o/^/c^. 
 
 F. If /3 is of the second kind (/3<^>o), then 
 
 But if (3 is of the first kind (^^ = 0), and indeed 
 8 = ^' + Xo-, where (3' 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 : 
 
 a = a)V(w^'+ l)Kr-lW'+ l)/Cr-2 • • • (o)^" + iK, 
 
 and we have 
 
 70 = «T) yi = ar-i — aT5 y2 = Wr-2 — «T-1, . . •>yT = «o'~«i' 
 
190 THE FOUNDING OF THE THEORY 
 
 whilst /cq, /cj, . . . K^ have the same denotation as 
 in the normal form. The factors (0"^+! 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 
 
 a — L'd'^a i 
 
 where yo>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 ^ of 
 the second number-class should satisfy the relation 
 
 it is necessary and sufficient that they should have 
 the form 
 
 a = yyw, /3 = yv^ 
 
 where /x and v are numbers of the first number-class. 
 K. In order that two numbers a and ^ of the 
 second number-class, which are both of the first 
 kind, should satisfy the relation 
 
 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 
 
 a + /3 = /3 + a 
 
 we first conclude that the degree oq of a must be 
 equal to the degree ^^ of ^. For if, say, Qq </3o, we 
 would have, by theorem D, 
 
 a + ^ = A 
 and consequently 
 
 which is not possible, since, by (2) of § 14, 
 
 Thus we may put 
 
 a = (jo^fA + a', |8 = M^'ov + 13', 
 
 where the degrees of the numbers a and /3' are less 
 than ao, and ju and v are infinite numbers which are 
 different from zero. Now, by theorem D we have 
 
 a + P = w''ifjL + v) + ^\ P + a-=co'^(^ + p) + a, 
 
 and consequently 
 
 (^''Klii + 1/) + /3' = co'^ifjL + 1/) + a. 
 
 By theorem D of § 14 we have consequently 
 
 13' = a. 
 Thus we have 
 
 a = (jo"-"/ul -\-a, 13 = iff--v + a\ 
 
192 THE FOUNDING OF THE THEORY 
 
 [240] and if we put 
 
 uf-'' + a' = 7 
 we have, by (11): 
 
 a = y/x, /5 = yv. 
 
 Let us suppose, on the other hand, that a and ^ are 
 two numbers which belong to the second number- 
 class, are of the first kind, and satisfy the condition 
 
 and we suppose that 
 
 We will imagine both numbers, by theorem G, in 
 their product-form, and let 
 
 where a and /3' are without a common factor (besides 
 i) at the left end. We have then 
 
 a>/3\ 
 and 
 
 All the numbers which occur here and farther on 
 are of the first kind, because this was supposed of 
 a and /3. 
 
 The last equation, when we refer to theorem G, 
 shows that a and /3' 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 S would be transfinite, and, if k is 
 the finite factor at the left end of S, we would have 
 
OP TRANSPINITE NUMBERS 193 
 and thus 
 
 Thus there remains only the possibility that 
 
 a >o), P' < CO. 
 But the finite number /3' must be t : 
 
 because otherwise it would be contained as part in 
 the finite factor at the left end of «'. 
 
 We arrive at the result that /3 = (5, consequently 
 
 a = (3a , 
 
 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 ^ the relation 
 
 afi = l3a 
 subsists. 
 
 [241] Consequently if also a>l3, we conclude in 
 the same way the existence of a transfinite number 
 of the first kind a' which is less than a and such that 
 
 a=Pa\ a '13 = /3a'. 
 
 If also a' is greater than (3, there is such a number 
 a"' less than a\ such that 
 
 a =pa , a p = /5a , 
 
 and so on. The series of decreasing numbers, a, a', 
 a\ a"\ . . ., must, by theorem B of § 16, break 
 
 \ 
 
 i3 
 
194 THE FOUNDING OF THE THEORY 
 
 off. Thus, for a definite finite index ^q, we must 
 have 
 
 (p. < 
 
 a^'^o 
 
 /3. 
 
 If 
 
 we have 
 
 the theorem K would then be proved, and we would 
 have 
 
 But if 
 
 then we put 
 
 and have 
 
 Thus there is also a finite number p^ such that 
 
 In general, we have analogously : 
 
 and so on. The series of decreasing numbers /3i, 
 /^a, I3q, . . . also must, by theorem B of § i6, break 
 off. Thus there exists a finite number k such that 
 
 If we put 
 then 
 
 
OF TRANSFTNITR NUMBERS 195 
 
 where /x and v are numerator and denominator of 
 the continued fraction: 
 
 
 +- 
 p. 
 
 [242] § 20 
 
 The e-Numbers of the Second Number- Class 
 
 The degree a© of a number a is, as is immediately 
 evident from the normal form : 
 
 (1) a = w'^o/Cq + (o*i/Ci + . . . , «(, > ai > . . . , 0<k„<w, 
 
 when we pay attention to theorem F of § 18, never 
 greater than a ; but it is a question whether there 
 are not numbers for which a^ = a. In such a case 
 the normal form of a would reduce to the first term, 
 and this term would be equal to w*, that is to say, 
 a would be a root of the equation 
 
 (2) a,^ = f. 
 
 On the other hand, every root a of this equation 
 would have the normal form w" ; 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 " f-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 
 
 71 = ^^ y2 = w^S • • 'J y^ = w^''"S . . • 
 The limit Lim y^ = E(y) of this fundamental series 
 
 V 
 
 is always an e-number. 
 
 Proof. — Since y is not an e-number, we have 
 a)>'>y, that is to say, 7i>y. Thus, by theorem B 
 of § 1 8, we have also w'^'i > w'>', that is to say, y2>yi ; 
 and in the same way follows that y3>y2, 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 : 
 
 ^E(v) _ Lin^ ^v^^ Lin^ y^^^ — E(y). 
 
 V V 
 
 Consequently E(y) is an e-number. 
 
 B. The number eo = E(i) = Lim w^,, where 
 
 V 
 
 is the least of all the e-numbers. 
 
 [243] Proof. — Let e be any e-number, so that 
 
 Since e'>w, we have w''>w'", that is to say, e > w^. 
 Similarly t/xo'*'', that is to say, e'xo^, and so on. 
 We have in general 
 
OF TRANSFINITE NUMBERS 197 
 
 and consequently 
 
 e'>.Lim a)„, 
 
 that is to say, 
 
 Thus eo = 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 
 
 e' < y <: e' 
 follows 
 
 that is to say, 
 
 e' < yi < e' . 
 
 Similarly we conclude 
 
 e' < ya < 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 
 
 E(y) = e". 
 
 Since e'+i is not an e-number, simpl}- because all 
 6-numbers, as follows from the equation of definition 
 
198 THE FOUNDING OF THE THEORY 
 
 i = w^, are of the second kind, e -{- i is certainly less 
 than e\ and thus we have the following theorem : 
 
 D. If e' is any e-number, then E(e'+ i) is the next 
 greater e-number. 
 
 To the least e-number, €q, follows, then, the next 
 greater one: 
 
 ei = E(eo+i), 
 
 [244] to this the next greater number : 
 
 ^2 = E(6i+l), 
 
 and so on. Quite generally, we have for the 
 (1/+ i)th e-number in order of magnitude the formula 
 of recursion 
 
 (3) e, = E(e,_i-|-l). 
 
 But that the infinite series 
 
 ^0' ^1) • • • 6,., . . . 
 
 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^^^ < e^-^+i^ < . . ., 
 
 then Lim e^''^ is an e-number, and, in fact, the 
 
 I' 
 e-number which follows next in order of magnitude 
 to all the numbers eH 
 Proof.— 
 
 Lim eW . ,) 
 
 (0 " = Lim w* = Lim e^''^. 
 
OF TRANSFINITE NUMBERS 199 
 That Lim e^''^ is the e-number which follows next 
 
 V 
 
 in order of magnitude to all the numbers e^"^ results 
 from the fact that Lim e^"^ is the number of the 
 
 second number-class which follows next in order of 
 magnitude to all the numbers e^^^ 
 
 F. The totality of e-numbers of the second 
 number-class forms, when arranged in order of 
 magnitude, a well-ordered aggregate of the type Vt 
 of the second number-class in its order of magnitude, 
 and has thus the power Aleph-one. 
 
 Proof. — The totaHty 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) €q, e^, . . ., e^,, . . . €0,+!, . . . ea' . . ., 
 
 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 Q, the type of (4) is 
 
 w + Q = w + o.)2 + (C2-w-^). 
 
 [245] But since a) + a)^ = w^, we have 
 
 w + il = il\ 
 
200 THE FOUNDING OF THE THEORY 
 
 and consequently 
 
 o) + n = Q = ^^^. 
 
 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 : 
 
 a + e = e, a^ — e^ a^ — e. 
 
 Proof. — If ao is the degree of a, we have ao^a, and 
 consequently, because of a<e, we also have ao<e. 
 But the degree of e = w^ is e; thus a has a less 
 degree than e. Consequently, by theorem D of 
 § 19, 
 
 and thus 
 
 On the other hand, we have, by formula (13) of 
 
 § 19, 
 
 ae = ao)" = isf-^^^ — co^ = e, 
 and thus 
 
 a^e = €. 
 
 Finally, paying attention to the formula (16) of 
 
 §19, 
 
 .H. If a is any number of the second number-class, 
 the equation 
 
 ''=i 
 
 has no other roots than the e-numbers which are 
 greater than a. 
 
OF TRANSFINITE NUMBERS 201 
 Proof. — Let /3 be a root of the equation 
 
 af = |. 
 SO that 
 
 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, 
 and consequently 
 
 [246] By theorem F of § 19, we have 
 and thus 
 
 But /3 cannot be greater than ao/3 ; consequently 
 
 ao/3 = ^, 
 and thus 
 
 Therefore ^ 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 
 Grundlageti der Arithmetik of 1884, defined the 
 " number {Anzahl) of a class u " as the class of all 
 those classes which are equivalent (in the sense of 
 PP- 75) S6 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 Grundgesetze der AritJunetik^ 
 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 u 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 ^n^y t)e gathered 
 from A. Schoenflies's reports : Die Entwickelung 
 der Lehre von den Pmiktmannigfaltigkeiten, Leipzig, 
 1900; part ii, Leipzig, 1908. A second edition 
 of the first part was published at Leipzig and Berlin 
 in 191 3, in collaboration with H. Hahn, under the 
 title : Entwickelung der Mengenlehre und iJirer 
 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, 111-116. Cf. Whitehead, Amer. Jotirn. of Mai h., vol. 
 xxiv, 1902, p. 378. For a more modern form of the doctrine, see 
 Whitehead and Russell, Frincipia Mathetnatica^ vol. ii, Cambridge, 
 1912, pp. 4, 13. 
 
 t Principles, pp. 262, 321 ; and Frtncipta, 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 : 
 
 (i) 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 
 A. N. Whitehead {Ainej\ Journ. of Math., vol. 
 xxiv, 1902, pp. 367-394). Cf. Russell, Principles, 
 pp. 1 1 7- 1 20. A more modern form is given in 
 Whitehead and Russell's Principia, 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; 191 3, p. 170; and p. 61 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, 19 10, 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. 1 6 1- 1 62 above,* unconsciously used this 
 axiom of infinite selection. Also G. H. Hardy 
 in 1903 (1908, pp. 22-23) used this axiom, un- 
 consciously at first, in a proof that it is possible to 
 have an aggregate of cardinal number j^^ 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 /3+ i is greater than f^. 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 (Phitosophical 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, and 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 Xo- Xo = Ko, 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. 
 
2o6 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 21/ 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 ze/ is a 
 member of w. The treatment and final solution of 
 
 * This argument was discovered in 1900 (see Monisf, 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, f and successfully by Russell {cf. Principles, 
 pp. 523-528; Principia, vol. i, pp. 26-31, 39-90)- 
 
 The theorem A on p. 105 is required (see theorem 
 D on p. 108) in the proof that the two definitions 
 of infinity coincide. On this point, cf. Principles, 
 pp. 1 21-123 ; Principia, 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 J (191 3» 
 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 
 exponentiation for cardinal numbers on p. 95 
 above. § Cf. 1913, pp. 75-80; 1908, pp. 4^-45- 
 
 (6) Theorems due to J. Konig (1904) on the 
 
 * Ephnenides 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." 
 
 + Just as in the proof that the multiplication of Ny by itself gives Kq, 
 the more general theorem here considered involves the multiplicative 
 axiom. 
 
 § Cf. Jourdain, Mess, of Math. (2), vol. xxxvi, May 1906, pp. 1 3- 1 6. 
 
2o8 NOTES 
 
 inequality of certain cardinal numbers ; and 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) Hausdorff's contributions from 1906 to 1908 
 to the theory of linear ordered aggregates (19 13, 
 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 191 3 (191 3, pp. 85-87). 
 
INDEX 
 
 Abel, Niels lienrike, lO, 
 Abelian functions, lO, ii. 
 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, 
 153 ff-> 175 ff-> 206. 
 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, 155. 
 
 Baire, Rene, ']t,. 
 Bendixson, Ivar, ']i. 
 Berkeley, George, 55. 
 Bernoulli, Daniel, 4. 
 Bernstein, Felix, 204. 
 Bois-Reymond, Pauldu, 22,34,51. 
 Bolzano, Bernard, 13, 14, 17, 21, 
 
 22,41, 55. 72. 
 Borel, Emile, 73. 
 Bouquet, 7. 
 Briot, 7. 
 Broden, 73. 
 Brouwer, 208. 
 Burali-Forti, Cesare, 20s, 206, 
 
 Cantor, Georg, v, vi, vii, 3, 9, 10, 
 13, 18, 22, 24, 25, 26, 28, 
 
 29, 30, 32, 11. 34, 35, 36, 
 . 37, 38, 41, 42, 45, 46, 47, 
 
 48, 49, 51, 52, 53, 54, 55, 
 56, 57, 59, 60, 62, 63, 64, 
 68, 69, 70, 72, 73, 74, 76, 
 77, 79, 80, 81, 82, 202, 
 204, 205, 206, 208. 
 Dedekind axiom, 30. 
 Cardinal number {see also Power), 
 74, 79 ff., 85 ff., 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, T32. 
 
 types, 133. 
 Coherences, 73. 
 Coherent series, 129, 130. 
 Commutative law with transfinite 
 numbers, 66, 92, 93, 
 119 ff., 190 ff. 
 Condensation of singularities, 3, 
 
 9, 48, 49. 
 
 Connected aggregates, 72. 
 
 Content of aggregates, ']1. 
 
 Content-less, 51. 
 
 Continuity of a function, i. 
 
 Continuous motion in discon- 
 tinuous space, 37. 
 
 Continuum, 33, yj , 41 ff., 47, 48, 
 
 64, 70 ff., 96, 205. 
 Contradiction, Russell's, 206, 207. 
 Convergence of series, i, 15, 16, 
 
 17, 20, 24. 
 Cords, vibrating, problem of, 4. 
 
 209 
 
 14 
 
2IO 
 
 INDEX 
 
 D'Alembert {see Alembert, 
 
 J. L. d'). 
 Dedekind, Richard, vii, 23, 41, 
 
 . 47, 49, IZ- 
 Definition of aggregate, 37. 
 Democritus, 70. 
 De Morgan, Augustus, 41. 
 Density in itself, 132. 
 Derivatives of point- aggregates, 
 
 3, 3off-.,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, 5ofif., 
 
 62. 
 Enumeral, 52, 62. 
 Epicurus, 70. 
 Epimenides, 207. 
 Equivalence of aggregates, 40, 
 
 75, 86 ff. 
 Euler, Leonhard, 4, 5, 9, 10. 
 Everywhere-dense aggregates, 33, 
 
 35. 37, 123. 
 types, 133. 
 Exponentiation of transfinite 
 
 numbers, 82, 94 ff,, 207. 
 
 Fontenelle, 118. 
 
 Formalism in mathematics, 70, 81. 
 
 Fourier, Jean Baptiste Joseph, I, 
 
 2, 5, 6, 8, 24. 
 Freedom in mathematics, d'] ff. 
 Frege, Gottlob, 23, 70, 202, 
 Function, conception of, i. 
 Functions, theory of analytic, 2, 
 
 6, 7, 10, II, 12, 13, 22, 
 
 73. 
 arbitrary, 4, 6, 34. 
 theory of real, 2, 8, 9, 73. 
 Fundamental series, 26, 128 ff. 
 
 Gauss, Carl Friedrich, 6, 12, 14. 
 Generation, principles of, 56, 57. 
 Gudermann, lO. 
 
 Hahn, H., 203. 
 
 Haller, Albrecht von, 62. 
 
 Hankel, Hermann, 3, 7, 8, 9, 17, 
 
 49, 70. 
 Hardy, G. H., 205, 206, 
 Harnack, Axel, 51, 73. 
 Hausdorff, Felix, 207 208. 
 Heine, H. E., 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 of, 8. 
 Integrable aggregates, 51. 
 Inverse types, 114. 
 Irrational numbers, 3, 14 ff. , 26 ff. 
 analogy of transfinite numbers 
 with, 77 ft". 
 Isolated aggregate, 49. 
 point, 30. 
 
 Jacobi, C. G. J., 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. 
 
 Konig, Julius, 207. 
 
 Kronecker, L., 70, 81. 
 
 Kummer, E. E. , 69. 
 
 Lagrange, J. L., 5, 14. 
 Leibniz, G. W. von, 55. 
 Leucippus, 70. 
 Limitation, principle of, 60. 
 Limiting element of an aggregate, 
 
 Limit-point, 30. 
 
 Limits with transfinite numbers, 
 
 77 ff., 131 ff., 58 ff 
 Liouville, J., 40. 
 Lipschitz, R., 6. 
 Locke, J., 55. 
 Lucretius, 70. 
 
INDEX 
 
 Mach, Ernst, 69. 
 Maximum of a function, 22. 
 Mitlag-Leifler, Gosta, ii. 
 Mutliplication of cardinal num- 
 bers, 80, 91 ff. 
 of ordinal types 81, 119 ff, 154. 
 of transfmite 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, 151 ff. 
 numbers, finite, 113, 158, 159. 
 type, 75, 79 ff., no ff. 
 type of aggregate of rational 
 
 numbers, 122 ff. , 202. 
 types of multiply ordered aggre- 
 gates, 81, 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 modern 
 
 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 ff., 60, 62. 
 Prime numbers, transfinite, 64, 66. 
 Principal element of an aggregate, 
 
 131- 
 Puiseux, v., 7. 
 
 Reducible aggregates, 71. 
 Relation numbers, 203. 
 Riemann, G. F. B., 3, 7, 8, 9, 10, 
 12, 25, 42. 
 
 Riess, 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 ff. 
 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, 112 ff. 
 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, 
 II, 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. 
 
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