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Cantor's Discovery of Transfinite Numbers HISTORIA MATHEMATICA 22 (1995), 33-42 "What Fermented in Me for Years": Cantor's Discovery of Transfinite Numbers Jos~ FERREIROS History of Science and Technology, 543 Stephens Hall, Unioersity of California at Berkeley, Berkeley, California 94720 Transfinite (ordinal) numbers were a crucial step in the development of Cantor's set theory. The new concept depended in various ways on previous problems and results, and especially on two questions that were at the center of Cantor's attention in September 1882, when he was about to make his discovery. First, the proof of the Cantor-Bendixson theorem motivated the introduction of transfinite numbers, and at the same time suggested the "principle of limitation," which is the key to the connection between transfinite numbers and infinite powers. Second, Dedekind's ideas, which Cantor discussed in September 1882, seem to have played an important heuristic role in his decision to consider the "symbols of infinity" that he had been using as true numbers, i.e., as autonomous objects; to this end Cantor introduced in his work, for the first time, ideas on (well) ordered sets. © 1995 Academic Press, Inc. Los ntimeros (ordinales) transfinitos constituyeron un paso clave en el desarrollo de la teorfa de conjuntos de Cantor. La nueva idea dependi6 en varias formas de resultados y problemas previos, pero especialmente de dos cuestiones que ocuparon a Cantor en septiem- bre de 1882, estando a punto de realizar su descubrimiento. En primer lugar, el teorema de Cantor-Bendixson, cuya demostraci6n motiv6 la introducci6n de los ntimeros transfinitos, y a la vez sugiri6 el "principio de limitaci6n" que constituye la clave de la conexi6n entre ntimeros transfinitos y potencias infinitas. En segundo lugar, las ideas de Dedekind, que Cantor discuti6 en septiembre de 1882, parecen haber desempefiado un importante papel heurfstico en la decisi6n de considerar los "simbolos de infinitud" que Cantor venfa em- pleando como verdaderos mlmeros, esto es, como objetos aut6nomos; para ello, Cantor introdujo en su obra por vez primera consideraciones sobre conjuntos (bien) ordenados. © 1995 Academic Press, Inc. Die transfiniten (Ordnungs)Zahlen stellten einen wesentlichen Schritt in der Entwicklung der Cantorschen Mengenlehre dar. Der neue Begriff war yon verschiedenen frtiheren Prob- lemstellungen und Resultaten abhiingig, darunter besonders zwei Themen, die im Oktober 1882 im Mittelpunkt von Cantors Interesse lagen, als er auf die neue ldee kam. Erstens motivierte der Beweis des Cantor-Bendixsonschen Satzes die Einftihrung der transfiniten Zahlen und legte gleichzeitig das sogenannte Hemmungsprinzip nahe, welches die Verbin- dung zwischen transfiniten Zahlen und unendliche Miichtigkeiten herstellte. Zweitens scheinen Dedekinds Ideen, welche Cantor im September 1882 kennenlernte, eine wichtige heuristische Rolle bei dem Schritt gespielt zu haben, Cantors "Unendlichkeitssymbole" als wirkliche Zahlen, d.h. als selbst~ndige Objekte, zu behandeln. Cantor sah sich dadurch veranlasst, zum ersten Mal Ideen tiber (wohl)geordneten Mengen in seine Arbeiten einzuftihren. © 1995 Academic Press, Inc. MSC 1991 subject classification: 01A55, 01A70, 04-03. KEY WORDS: set theory, Cantor-Bendixson theorem, Richard Dedekind, chain theory, well-ordered sets. 33 0315-0860/95 $6.00 Copyright © 1995 by Academic Press, Inc. All rights of reproduction in any form reserved. 34 JOSI~ FERREIROS HM 22 Although Cantor's transfinite numbers have been discussed by several historians (e.g., [10, I 1, 18, 19, 25]), two aspects of their discovery have not received the attention they deserve. First, it can be shown that the context of Cantor's work at the time had an intimate connection with his discovery. By late 1882, Cantor was trying to prove a part of the Cantor-Bendixson theorem, a part that I will call Cantor's theorem. The proof of this theorem motivated the introduction of transfinite numbers, and, most importantly, it led directly to a restricted version of the principle of limitation, which was crucial to the new theory. (The principle of limitation defines "natural sections" of the unlimited series of transfinite numbers, the number-classes; these classes became central to Cantor's theory of infinite cardinalities [5, 166-167, 196-197].) Second, the introduction of transfinite numbers depended on a reconceptualiza- tion of Cantor's previous ideas in ordinal terms. Here, Cantor's interaction with Dedekind during September 1882 seems to have played an important heuristic role. 1. CANTOR'S THEOREM The 1883 Grundlagen, where Cantor presented his new numbers, was part of a series of papers that he published in Mathematische Annalen between 1879 and 1884, under the title "Ober unendliche, lineare Punktmannichfaltigkeiten" [On infinite, linear point-sets]. Most of the results in them deal with the powers of point-sets (subsets of R or ~n) studied through an analysis of properties of their derived sets; quite clearly, Cantor's aim was to find a way to prove the continuum hypothesis (but, in spite of some progress, the series had to end with a new announcement of a proof never found [6, 244; cf. 24]). Among those results, one can be said to have structured Cantor's work from 1882 to 1884: the well-known Cantor-Bendixson theorem, first announced in a paper dated 1 September 1882 [4]. Derived sets had been introduced by Cantor in 1872, and they led him to "symbols of infinity" that he would later recognize as transfinite numbers. ~ The first derived set of a point-set P, denoted P', is the set of all limit points of P [2, 98]. Since P' will also, in general, have limit points, Cantor considered iterations of the derivation procedure: Pn is the set of all limit points of pn-1. Most impor- tantly, the first derived set of an infinite order was defined in 1880 as the intersection of all derived sets of finite order: Even granting this connection, and the fact that the "symbols of infinity" carried a germ of the arithmetic of transfinite numbers, it would be too optimistic to consider that early idea as representing a clear grasp of the ordinals. Cantor had seen the possibility of unlimited continuation of the process of set derivation, but all the crucial details that made the transfinite numbers a part of his set theory were lacking. Only in 1882 did Cantor come to regard the symbols as true numbers, making them correspond to well-ordered sets and defining rigorously the arithmetical operations. And only in 1882 did he see the possibility of establishing their crucial connection with infinite powers, through the "principle of limitation" which clustered them into number-classes. HM 22 CANTOR'S TRANSFINITE NUMBERS 35 po,= n(p,,p,,,. " .p~ .... )= n(e~,en+l .... ). Sets of higher orders, P~, can be defined similarly, either by reference to an immediately preceding set P~-~, or as the intersection of an infinite family. The Cantor-Bendixson theorem establishes a (unique) decomposition valid for all derived sets, P' = R U S, where S is a perfect set (S = S'), and R is a denumerable set such that, for some transfinite number of the second class a, R N R ~ = ~b (R is a separated set) [6, 221-225]. This result grew out of the generalization to nondenumerable sets P' of a previous result, which I will call Cantor's theorem. If P' is denumerable, S = ~b, and we obtain the part of the Cantor-Bendixson theorem that Cantor proved first: Cantor's Theorem. If a point-set P is such that P' is denumerable, then there exists some index a such that P~ = ~b; and conversely, if P~ = ~b for some index or, then P' and P are denumerable [4, 160; 5, 171]. In fact, the indices o~ turn out to be either natural numbers or transfinite numbers of the second number-class (the set of all denumerable ordinals). But when he first announced the theorem in the paper of 1 September 1882, Cantor still lacked basic results on the indices ~, so that he really could not prove it yet. Those results were provided in his Grundlagen [5], and the theorem itself was finally proven in [6, 218-221]. While working on this theorem in 1882, Cantor was well aware of its importance, particularly in relation to Mittag-Leffler's Theorem for the construction of analytic functions with an infinite set of singular points. This line of research generalized Weierstrass' Product Theorem, and Mittag-Leffler himself emphasized in his let- ters to Cantor that Cantor's theorem would give a natural conclusion to his own work. 2 Therefore, in this case Cantor was aware of a very important application of one of his results to central mathematical problems of the time. In his paper of September 1882, Cantor only proved a very restricted version of the above theorem. His strategy was based on the notion of an isolated set (a point set P is isolated iff P n P' = ~b), and exploited the fact that isolated sets are denumerable [4, 158-159]. This fact had far-reaching consequences. First, if a derived set P' is denumerable, then P itself is denumerable, because P is the union of the isolated set P\(P n P') and a subset of P', both denumerable. In this way, Cantor could reduce the proof of Cantor's theorem to the characteristics of P'. Second, since for derived sets it holds that P~ C_ P', Cantor was able to establish the following decomposition [4, 158]: P' -- (P'\P") U (P"\P') U . UP'. 2 Cf. their correspondence of 17 and 22 October 1882, in [20, 88-89]. After seeing the result of this "conclusion" [21], Weierstrass wrote to its author (Mittag-Leffler)that the main problem of the theory of the representation of analytic functions, which seemed a matter for the future, had found its most general solution [22, 25].
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