CANTOR, GEORG FERDINAND LUDWIG PHILIP

(b. St. Petersburg, Russia, 3 March 1845;

d. Halle, Germany, 6 January 1918), mathematics, set theory, philosophy. For the original article on Cantor see DSB, vol. 3.

Cantor is best known as the creator of transfinite set theory, a theory of the mathematical infinite that revolutionized mathematics at the end of the nineteenth century. But the corresponding paradoxes of set theory discovered at the end of the century, including Russell’s and Burali-Forti’s paradoxes, proved to be inherent in the logic and substance of Cantor’s work. These raised serious questions about the consistency of set theory and prompted various approaches to secure rigorous foundations for mathematics that continued in the early 2000s to occupy mathematicians and philosophers of mathematics alike. So controversial were Cantor’s ideas—both mathematically and philosophically—that Leopold Kronecker once called him a scientific charlatan, a renegade, a “corrupter of youth.” Henri Poincaré considered set theory and Cantor’s transfinite numbers to be “a grave mathematical malady, a perverse pathological illness that would one day be cured” (Dauben, 1979, p. 1). Taking the opposite position, Bertrand Russell regarded Cantor as one of the greatest intellects of the nineteenth century, and David Hilbert believed Cantor had created a new paradise from which mathematicians would never be driven, despite the paradoxes of set theory. Meanwhile, Cantor was plagued by recurring nervous breakdowns and ongoing academic rivalries, and his religious convictions played a significant role in his steadfast faith in the correctness of his controversial transfinite set theory; he was convinced that, no matter what the opposition might say, transfinite set theory would eventually be vindicated and accepted by mathematicians as essential to their discipline.

Family History . When Herbert Meschkowski published the first book-length study of Cantor’s life and work in 1967, he included photographs of Cantor’s parents, noting that his father, Georg Woldemar Cantor, was born in Denmark in 1813 or 1814, the son of a successful businessman. He was raised an Evangelical Lutheran, and he conveyed his deeply held religious views to his son. Cantor’s mother, Marie Böhm, was from a family of virtuoso violinists, and she was a Roman Catholic. Meschkowski briefly addresses the significance of religion in Cantor’s life in a short section of his biography (Meschkowski, 1967, “Die Religion Cantors,” pp. 122–129).

One of the most contentious questions about Cantor’s own religious heritage and beliefs turns on the question of whether he was Jewish. In his widely read Men of Mathematics, Eric Temple Bell declared that Cantor was “of pure Jewish descent on both sides” (Bell, 1937, p. 558), and went on to make some of the most unfounded and scurrilous remarks about Cantor that have ever been published. In describing the bad blood between Cantor and Kronecker, whose incompatible views on the foundations of mathematics (see below) are legendary, Bell wrote: “there is no more vicious academic hatred than that of one Jew for another when they disagree on purely scientific matters” (1937, p. 562). Cantor had been included as Jewish in a number of earlier reference works, including The Jewish Encyclopedia (1901); a volume edited by Siegmund Kaznelson, Juden im Deutschen Kulturbereich (pp. 389–390); and in the Universal Jewish Encyclopedia, 3 (1969, pp. 18-19).

Ivor Grattan-Guinness, after consulting many archival and hitherto unpublished manuscript documents, concluded that “Georg Cantor was not Jewish, contrary to the view which has prevailed in print and in general opinion for many years” (Grattan-Guinness, 1971, p. 351). Grattan-Guinness based his conclusion on the fact that Cantor was given Christian names, which he took to imply “that the Cantors were not Jewish” (p. 351), and the results of a Danish scholar, Theodor Hauch-Fausbøll, whose research at the Danish Genealogical Institute in Copenhagen had come to the same conclusion (in a document of 1937).

Walter Purkert and Hans Joachim Ilgauds were even more adamant in their biography, in which Cantor’s Christian and non-Jewish racial profile is stressed. They cite the same certification reported by Grattan-Guinness from the Danish Genealogical Institute of 1937 saying that there is no record of Cantor’s father in any of the records of the Jewish community there (Purkert and Ilgauds 1987, p. 15). The fact that this document was produced for the Cantor family at the height of the German persecution of Jews prior to World War II, however, casts considerable doubt on the legitimacy of this testimony. Purkert and Ilgauds leave open the question of whether ancestors of Cantor’s may have converted to Christianity and say that the question is irrelevant to Cantor’s mathematics, except for the fact that transfinite set theory was condemned during the Nazi period as “Jewish” mathematics.

The truth of the matter of Cantor’s Jewish heritage, however, was acknowledged by Cantor himself in a letter to the French philosopher and historian of mathematics, Paul Tannery, in which he referred to his “israelitische” grandparents. And in a letter to the Jesuit priest Alex Baumgartner only recently published, Cantor wrote at even greater length about various pseudonyms he had adopted hinting at his “Portuguese Jewish origins.” He noted that one in particular, Vincent Regnäs, spelled backwards, was Sänger=Cantor, which was a reference to the fact that his father was born in Copenhagen and a member of the orthodox Portuguese Jewish community there (Tapp 2005, p. 129). Although clearly of Jewish ancestry, Cantor himself was baptized and confirmed as an Evangelical Lutheran, although in later life he was not an observant follower of any particular confession.

Early Works . Cantor wrote his dissertation at the University of Berlin on number theory (De aequationibus secudi gradus indeterminatis, 1867), but his early research after accepting a position at the University of Halle was devoted to the theory of trigonometric series. In 1872 he published a paper establishing the uniqueness of representations of arbitrary functions by trigonometric series in cases where even an infinite number of points might be excepted from the function’s domain of definition, so long as these happened to constitute what Cantor called a set of points of the first species. (An infinite set of points P was said to be of the “first species” if its set of limit points P' was finite; if not, then P', the first derived set of P, must contain an infinite number of points and also have a derived set, the second derived set of P, P”. If for some finite number υ the υ^th derived set P^ν contains only a finite number of points, then its derived set will be empty, i.e., P^{n +1}= Ø, and such infinite point sets were said to be of the first species. Infinite points sets for which none of its derived sets was finite were said to be of the second species).

Cantor’s early work on trigonometric series not only launched his early interest in point sets, which led to his later abstract development of set theory, but it also required him to introduce a rigorous theory of real numbers. This too proved to be a central element of Cantor’s transfinite set theory, for one of Cantor’s most famous conjectures that has yet to be solved is his Continuum Hypothesis, which in one form says that the set of all real numbers (which comprise the continuum) is the next largest infinite set after the set of all integers (which comprise a denumerably infinite set; denumerably infinite sets are the least in power or cardinality of all infinite sets, like the set of all integers).

Meschkowski covered the major details of Cantor’s early work in his DSB article, noting that the revolution in mathematics that Cantor launched can be dated to 7 December 1873, when he wrote to the mathematician Richard Dedekind to say he had found a way to prove that the set of all real numbers was non-denumerably infinite. Cantor had already proven (in a seminar with his teacher Karl Weierstrass at the University of Berlin) that the set of all algebraic numbers was denumerably infinite; if the set of all real numbers was non-denumerably infinite, this meant that there must be real numbers that were non-algebraic or transcendental. Joseph Liouville had proved the existence of such numbers in 1844; Cantor’s proof was an independent verification of this discovery, without identifying any transcendental numbers in particular (the two best-known transcendental numbers are φ, established by Charles Hermite in 1873, and e, proven transcendental by Ferdinand von Lindemann in 1882).

Cantor published his truly revolutionary discovery that the real numbers are non-denumerably infinite, establishing for the first time that “the infinite” was not some vast concept that simply included everything that was not finite, but that there were definite distinctions to be drawn between the relative sizes of infinite totalities, or sets. Sets such as the natural numbers, fractions, and algebraic numbers were denumerably infinite; the real numbers were non-denumerably infinite and, as Cantor conjectured, constituted a set of the next highest level of infinity after denumerably infinite sets, a conjecture he spent the rest of his life trying to prove without success (later, in the 1930s, Kurt Gödel would establish two results that explained why—although Cantor’s Continuum Hypothesis was consistent with the axioms of basic set theory, it was also independent of those axioms and could not be proven, or disproven, in the context of Zermelo-Fraenkel set theory).

Meschkowski noted that Cantor’s paper proving the non-denumerability of the real numbers was published in Crelle’s Journal in 1874: “Über eine Eigenschaf des Inbegriffes aller reellen algebraischen Zahlen,” a paper, Meschkowski explained, that “contained more than the title indicated.” But why should Cantor have titled his paper “On a Property of the Collection of All Real Algebraic Numbers,” when the clearly important, even revolutionary discovery was his proof that the real numbers were non-denumerably infinite? The “property” of the algebraic numbers that Cantor established in this paper was that they are only “countably infinite,” but this is a minor result compared to what he had discovered about the set of all real numbers.

Why Cantor gave this paper such a consciously deceptive title was no doubt due to his mathematical rival and former teacher at the University of Berlin, Leopold Kronecker. Kronecker was a well-known opponent of the school of analysis associated with Karl Weierstass, and he believed that the proper foundation for all of mathematics should rest on the integers alone. Kronecker rejected, for example, appeals to the Bolzano-Weierstrass theorem, upper and lower limits, and to irrational numbers in general. When Lindemann proved that e was transcendental, Kronecker asked what difference that made, because transcendental numbers did not exist (Weber, 1893, p. 15; Kneser, 1925, p. 221; Pierpont, 1928, p. 39; Dauben, 2005, p. 69). Worse for Cantor, Kronecker was a member of the editorial board of the journal to which he submitted his proof of the non-denumerabilty of the real numbers, and to disguise the true import of the paper was doubtless a strategic choice.

Kronecker had already tried to discourage Cantor’s colleague at Halle, Eduard Heine, from publishing a paper in Crelle’s Journal to which he objected, and Cantor could well have expected a very negative reaction from Kronecker had his paper carried a title like “Proof that the Collection of All Real Numbers is Non-Denumerably Infinite.” In fact, a year later Cantor discovered something he regarded as possibly even more remarkable, that the set of points in the two-dimensional plane could be corresponded in a one-to-one fashion with those on the one-dimensional line. So counterintuitive was this result that Cantor exclaimed in a note to his colleague Richard Dedekind, “I see it, but I don't believe it!” (Dauben, 1979, p. 55). Cantor must have hoped that the infinities of points in the plane and in three-dimensional space might prove to be distinctly higher levels of infinity than the one-dimensional continuum of real numbers, but his proof of the invariance of dimension showed that the number of points in spaces of any dimension was no greater than the points on the one-dimensional line.

Kronecker objected to Cantor’s proof, and for a time managed to delay its publication, something that so infuriated Cantor that he refused ever to publish in Crelle’s Journal again. Although Meschkowski does not mention any of this in his DSB article, he does characterize the remarkable nature of Cantor’s result: “It looked as if his mapping had rendered the concept of dimension meaningless” (p. 54). But as Dedekind soon pointed out to Cantor, although his correspondence between the points of the line and plane was one-to-one, it was not continuous. Cantor and others offered proofs that, indeed, a continuous mapping of points between dimensions was impossible, but a fully satisfactory proof establishing the invariance of dimension was not provided until the topologist L. E. J. Brouwer did so in 1910 (Brouwer, 1911). There was a positive side, however, to Kronecker’s early opposition to Cantor’s work, for it forced Cantor to evaluate the foundations of set theory as he was in the process of creating it. Such concerns prompted long historical and

philosophical passages in Cantor’s major publication of the 1880s on set theory, his Grundlagen einer allgemeinen Mannigfaltigkeitslehre of 1883.

Cantor’s Grundlagen . At the very beginning of this revolutionary monograph, Georg Cantor admitted how difficult it had been at first for him to accept the concept of actually infinite numbers, but he found they were absolutely necessary for the further development of mathematics:

As risky as this might seem, I can voice not only the hope, but my strong conviction, that in time this will have to be regarded as the simplest, most appropriate and natural extension [of the concept of number]. But I realize that in this undertaking I place myself in a certain opposition to views widely held concerning the mathematical infinite and to opinions frequently defended on the nature of numbers. (Cantor, 1883, p. 165; quoted from Dauben, 1979, p. 96)

The Grundlagen itself provided a systematic defense of Cantor’s new theory on mathematical, historical, and philosophical grounds, and made clear Cantor’s metaphysical justification for the new theory, which he knew would be controversial. Although the Grundlagen advanced Cantor’s thinking about the infinite from point sets to transfinite ordinal numbers, it did not include his later theory of transfinite cardinal numbers and the well-known alephs. Nevertheless, the Grundlagen was the earliest systematic treatise devoted to transfinite set theory and arithmetic. Along with Cantor’s later “Beiträge zur Begrundung der transfiniten Mengenlehre” (published in two parts, in 1895 and 1897), it had a profound effect on the further development of analysis and topology, and created a virtually new discipline, set theory.

Principles of a Theory of Order Types . Early in 1885 Cantor drafted an article for Acta Mathemaica, where a number of his early and seminal papers introducing the theory of point sets and transfinite arithmetic had just appeared in French translation, thanks to the efforts of Gösta Mittag-Leffler, the journal’s editor and an early champion of Cantor’s set theory. The “Principles of a Theory of Order Types” was a new effort to advance beyond well-ordered sets and their order types (the transfinite ordinal numbers) to a general theory of order types, but Mittag-Leffler declined to publish it. In explaining his grounds for rejecting the “Principles,” Mittag-Leffler said he thought Cantor was at least one hundred years ahead of his time, and added: “I am convinced that the publication of your new work, before you have been able to explain new positive results, will greatly damage your reputation among mathematicians” (Mittag-Leffler in a letter to Cantor, 9 March 1885; quoted from Cantor ed. Grattan-Guinness, 1970, p. 102).

In fact, the “Principles” was filled with new terminology and philosophical reflections that were not pleasing to Mittag-Leffler. Earlier, when he had prepared the French translation of Cantor’s Grundlagen for publication in Acta Mathematica, he deleted all of the historical and philosophical sections, leaving only those that dealt specifically with the theory of point sets and transfinite ordinal numbers. Readers of the French translations of Cantor’s Grundlagen thus learned nothing about the historical and philosophical arguments that Cantor regarded as important support for his treatment of the actual infinite mathematically. Similarly, his new general theory of order types did not appear in Acta Mathematica, and was unknown to Meschkowski when he wrote his DSB article on Cantor. The rejected “Principien” was published by Ivor Grattan-Guinness in 1970.

The Grundlagen had only used well-ordered sets to define the transfinite ordinal numbers, but in the “Principles” Cantor presented a new and independent theory of ordered sets in general (see Cantor, 1970). While the sequence of natural numbers 1, 2, 3, … in their natural order represented a well-ordered set, Cantor had begun to consider the properties of “simply ordered” sets, like the rational numbers in their natural order, which he designated by the order-type η (between any two numbers of type η there was always another number, i.e., they were said to be “everywhere dense”), or the natural order of the real numbers, which he designated by the order-type θ (in addition to being everywhere dense, simply ordered sets of type θ were also continuous). The properties of simply ordered sets were later published by Cantor in his “Beiträge” of 1895 and 1897.

Transfinite Cardinal Numbers: The Alephs . Although Meschkowski in his DSB entry for Cantor goes into considerable detail about the mathematics of transfinite set theory, he has little to say about their most famous element, the transfinite cardinal numbers, or alephs (these are only mentioned once, and as Meschkowski explains, “in all of Cantor’s works we find no usable definition of the concept of the cardinal number,” Meschkowski, 1971, p. 56). Indeed, transfinite cardinal numbers were not presented in the Grundlagen, and the evolution of Cantor’s thinking about them is curious. Although the alephs are probably the best-known legacy of Cantor’s creation, they were the last part of his theory to be given either rigorous definition or a special symbol. Cantor first introduced notation for sequences of derived sets P of the second species in 1879. (A set of points P was said to be of the second species if there was no finite index υ such that P^υ was empty; this meant that the intersection of all derived sets P^υ of P would be an infinite set of points, which Cantor designated P^∞, and this in turn would have a derived set P^∞+1; this, in fact, let to an entire sequence of transfinite sets of the second species.) These point sets of the second species served to extend Cantor’s idea well beyond the limitation he had earlier set himself to sets of the first species in his study of trigonometric series. However, in the early 1880s he only referred to the indexes ∞, ∞+1, … as “infinite symbols,” with no hint that they might be regarded as numbers.

By 1883, when he wrote the Grundlagen, the transfinite ordinal numbers had finally achieved independent status as numbers, ω being the first transfinite ordinal number following the entire sequence of finite ordinal numbers, that is, 1,2,3, …,ω. Although no explicit mention was made in the Grundlagen of transfinite cardinal numbers, Cantor clearly understood that it is the power of a set that establishes its equivalence (or lack thereof) with any other set, and upon which he would base his concept of transfinite cardinal number.

In September 1883, in a lecture to mathematicians at a meeting in Freiburg, Cantor defined the concept of transfinite cardinal number, but as yet without any particular symbol. Because he had already adopted the symbol ω to designate the least transfinite ordinal number, when Cantor finally introduced a symbol for the first transfinite cardinal number (in correspondence, as early as 1886), he represented the first transfinite cardinal as and the next as This notation was not very flexible, and within months he began to use fraktur o', derivatives from his omegas, to represent the sequence of cardinal numbers o₁, o₂, o₃, …. For a time, he used an assortment of notations, including superscripted stars, bars, and his fraktur o’s interchangeably for transfinite cardinal numbers. (For a detailed discussion of the evolution of Cantor’s notation for the transfinite cardinal numbers, see Dauben, 1979, pp. 179–183.)

However, when the Italian mathematician Giulio Vivanti was preparing a general introduction to set theory in 1893, Cantor realized it would be timely to decide on a standard notation. He chose the Hebrew alephs (N) for transfinite cardinal numbers because the Greek and Roman alphabets were already widely used in mathematics. Cantor believed his new numbers deserved something distinctive, and the Hebrew alphabet had the advantage that it was readily available among the type fonts of German printers. Moreover, this choice was particularly clever because the Hebrew aleph was also a symbol for the number one. Since the transfinite cardinal numbers were themselves infinite unities, the alephs represented a new beginning for mathematics. When Cantor introduced his transfinite cardinal numbers for the first time in the “Beiträge” in 1895, he used N₀ to represent the first and least transfinite cardinal number, after which there followed an unending, well-defined sequence of transfinite cardinal numbers (for details, see Cantor, 1895, pp. 292–296; 1915, pp. 103–109; and Dauben, 1979, pp. 179–183, 194–218).

Cantor’s Nervous Breakdowns . In his DSB article of 1971, Meschkowski had little to say about Cantor’s famous nervous breakdowns but their role in Cantor’s defense of his mathematics may have been crucial, as was his deeply held religious faith, which was also connected, at least in his mind, with his nervous breakdowns. It was in May 1884 that Cantor suffered the first of a recurring series of episodes that were to plague him for the rest of his life. The mathematician Arthur Schoenflies, when he chronicled Cantor’s “mathematical crisis” over failure to resolve the Continuum Hypothesis in the 1880s, suggested that this no doubt triggered Cantor’s first major breakdown (Schoenflies, 1927). Cantor’s lack of progress resolving the Continuum Hypothesis or stress from Kronecker’s ongoing attacks may have contributed to the breakdown, but as Ivor Grattan-Guinness concluded, based on evidence from Cantor’s records at the Nervenklinik in Halle where he was treated, mathematics probably had little to do with his mental illness. Cantor suffered from acute manic depression, which was only remotely— if at all—connected to his career.

The manic phase took over with no warning and lasted somewhat more than a month (for details, see Grattan-Guinness, 1971, and Charraud, 1994). When Cantor “recovered” at the end of June 1884 and entered the depressive phase of his illness, he complained that he lacked energy and had no interest in returning to rigorous mathematical thinking. Instead, he took up the study of English history and literature, seriously advocating a popular theory of his day that Francis Bacon was the true author of Shakespeare’s plays. Cantor also tried his hand without success at teaching philosophy, and about this time began to correspond with Roman Catholic theologians who had taken an interest in the philosophical implications of transfinite set theory. This correspondence was of special significance to Cantor because he was convinced that he was the messenger of the divinely inspired transfinite numbers.

Cantor and Catholic Theologians . Although Meschkowski later published a collection of Cantor’s letters, a number of which reflect exchanges between Cantor and various theologians, including Cardinal Johannes Franzelin, he made only passing reference to their correspondence in his DSB article on Cantor. The significance of this correspondence was the subject of Christian Tapp’s doctoral thesis at the Ludwig Maximilians Universität (Munich) published in 2005, which explores what Tapp calls Cantor’s “dialogue” with Catholic theologians of his time.

What emerges from Cantor’s letters to theologians is a much clearer picture of his understanding of the prehistory of his theory and the difficulties he knew the reception of set theory would face. In his correspondence, the philosophical foundations of set theory are discussed candidly, including the concept of infinity, the problem of the potential infinite, and Cantor’s criticism of so-called proofs of the impossibility of actually infinite numbers. Cantor was especially concerned with combating objections that theologians raised in opposition to any “actual” concept of infinity apart from God’s absolute infinite nature, which Cantor’s transfinite numbers seemed to challenge directly. Cantor approached these matters by affirming the existence of sets as abstractions, and through a systematic critique of philosophical works, especially with respect to scholasticism and, much later, Naturphilosophie.

Tapp evaluates the rather eccentric interest Cantor had in Baconian studies, various claims that Bacon was a crypto-Catholic, and the relevance of the Bacon-Shakespeare question, all of which he uses to better understand Cantor’s personality, if not his mathematics. He also considers a rather odd pamphlet that Cantor published privately at his own expense, Ex Oriente Lux (1905), in which Cantor argued that Christ was the natural son of Joseph of Arimathea (see Dauben, 1979, p. 289; Tapp, 2005, pp. 157–159). Tapp makes good use as well of information concerning often obscure individuals, some of whom no one has written about previously in relation to Georg Cantor. The new information Tapp provides leads to a very rich analysis of the “Catholic” connection in Cantor’s attempts to promote and defend his transfinite set theory, especially from attacks by philosophers and theologians. In turn, Cantor’s interest in “saving the Church” from mistakenly opposing transfinite mathematics for somehow being in conflict with the absolute infinite nature of God also plays a role in Cantor’s thought. In addition to the various pseudonyms Cantor adopted, Tapp also considers other very original and interesting information from the correspondence, including an analysis of differences in Cantor’s handwriting, to shed new light on aspects of his character and personality (for details, see Tapp, 2005).

Evaluating Cantor’s Manic Depression . Much has been written about Cantor’s unfortunate history of mental illness, which some such as Schoenflies have linked to his distress at not being able to prove his Continuum Hypothesis and the relentless criticism of transfinite set theory by Kronecker (Schoenfliess, 1927). The mathematician E. T. Bell explained the root of Cantor’s many tribulations in completely Freudian terms, as stemming from what Bell characterized as a disastrous relationship with his father. According to Bell, it was his father’s initial opposition to Cantor’s wish to become a mathematician that was the source of Cantor’s later mental problems (Bell, 1937, chap. 29). In 1994, Nathalie Charraud, a Lacanian psychoanalyst, after examining the records of Cantor’s treatment at the neurological clinic in Halle, offered a very different interpretation of the very positive role that Cantor’s father played in his son’s life. She suggested that his father was a constructive force, and that the deeply religious sensibility Cantor inherited from his father prompted a connection that Cantor felt to his transfinite numbers, which he took to have been communicated to him from God directly. This, in fact, was crucial to the unwavering support Cantor always gave transfinite set theory, no matter what criticisms might be directed against it. (For details of how his religious convictions and periods of manic depression may actually have played constructive, supportive roles in the battle to establish transfinite set theory as a fundamental part of modern mathematics, see Dauben, 2005.)

Cantor and the Professionalization of Mathematics . In addition to stimulating the vigorous defense that Cantor mounted on behalf of his set theory from the outset, the opposition to Cantor’s work as a mathematician had another constructive result, namely the effort he made to establish the Deutsche Mathematiker-Vereinigung (German Mathematical Society). His motives are reflected in one of his most famous pronouncements about mathematics, that “the essence of mathematics lies precisely in its freedom” (Cantor, 1883, p. 182). This was largely motivated in response to Kronecker’s opposition to his work; Cantor had argued in the Grundlagen that if a theory could be shown to be not contradictory, mathematicians should be free to pursue it; posterity would show whether its results might be fruitful or not. It was in the same spirit of freedom, hoping to promote a forum where mathematics could be discussed openly, that Cantor put considerable effort into establishing the German Mathematical Society. He was elected its first president in 1891.

Cantor’s creation of transfinite set theory, despite opposition from some of the most prominent mathematicians of his day, eventually persisted, thanks in no small measure to the unwavering faith he had in the importance and correctness of the theory itself. His defense of set theory was as much historical and philosophical as it was technical, mathematically; on a very personal level, it was also religious. As Cantor himself once wrote about why he was so certain that his theory must be true:

My theory stands as firm as a rock; every arrow directed against it will return quickly to its archer. How do I know this? Because I have studied it from all sides for many years; because I have examined all objections which have ever been made against the infinite numbers; and above all, because I have followed its roots, so to speak, to the first infallible cause of all created things. (Cantor in a letter of 21 June 1888 to Carl Friedrich Heman, professor of theology at the University of Basel; quoted from Dauben, 1979, p. 298)

Cantor suffered the last of his nervous breakdowns in the spring of 1917. He was hospitalized against his wishes, and repeatedly asked for his family to take him home. As World War I raged on, food was scarce, and a surviving photograph of Cantor shortly before his death shows a face gaunt and tired (Dauben, 1979, p. 273). On 6 January 1918, he died, apparently of heart failure. But as Edmund Landau wrote when he heard the news, Cantor and all that he represented would never die. One had to be thankful for a Georg Cantor, from whom later generations of mathematicians would learn: “Never will anyone remain more alive” (Landau, in a letter of 8 January 1918; quoted from Meschkowski, 1967, p. 270). Indeed, Cantor’s creation of transfinite set theory has not only inspired mathematicians and philosophers, but the writers of poems, novels (Borges; see Hernández, 2001), and even an opera, Cantor: Die Vermessung des Unendlichen, by Ingomar Grünauer (Wilkening, 2006; Grattan-Guinness, 2007).

SUPPLEMENTARY BIBLIOGRAPHY

The major archival collections of Cantoriana are to be found in Germany in the archives of Halle University, and three surviving letter books now preserved in the Handschriftenabteilung of the Niedersächsische Staats- und Universitätsbibliothek, Göttingen. A substantial collection of correspondence between Cantor and the editor of Acta Mathematica, Gösta Mittag-Leffler, is preserved in the archives of the Institut Mittag-Leffler, Djursholm, Sweden, along with letters Cantor exchanged with the English mathematician and logician Philip Jourdain. For detailed discussion of other documents relevant to Cantor’s life and works, see the “List of manuscript sources” in Grattan-Guinness, 1971; and Tapp, 2005.

WORKS BY CANTOR

Gesammelte Abhandlungen mathematischen und philosophischen Inhalts, edited by Ernst Zermelo. Berlin: Springer, 1932. Reprint, Hildesheim: Olms, 1966; Berlin: Springer, 1980. The Springer reprint includes an appendix compiled by Joseph W. Dauben, “Weitere Arbeiten von Georg Cantor,” a list of works by Cantor that were not included in the Gesammelte Abhandlungen edited by Zermelo, as well as a list of book reviews Cantor had written, and works in which letters of Cantor have been published (pp. 487–489).

“Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen.” Journal für die reine und angewandte Mathematik 77 (1874): 258–262. Reprinted in Cantor, 1932, pp. 115–118; French translation, Acta mathematica 2 (1883): 205–310.

Grundlagen einer allgemeinen Mannigfaltigkeitslehre. Ein mathematisch-philosophischer Versuch in der Lehre des Unendlichen. Leipzig: Teubner, 1883. Also published (without the preface) as “Über unendliche, lineare Punktmannichfaltigkeiten” (Part 5). Mathematische Annalen23 (1884): 453–488. Reprinted in Cantor, 1932, pp. 165–208. English translation by W. B. Ewald.

“Foundations of a General Theory of Manifolds: A Mathematico-philosophical Investigation into the Theory of the Infinite.” In From Kant to Hilbert: A Source Book in the Foundations of Mathematics, vol. 2, edited by William B. Ewald. New York: Oxford University Press, 1996. This work is also analyzed in detail in Joseph W. Dauben, “Georg Cantor, Essay on the ‘Foundations of General Set Theory,’ 1883.” In Landmark Writings in Western Mathematics, 1640–1940, edited by Ivor Grattan-Guinness. Amsterdam: Elsevier, 2005.

“Beiträge zur Begründung der transfiniten Mengenlehre.” Mathematische Annalen46 (1895): 481–512; 49 (1897): 207–246. Reprinted in Cantor, 1932, pp. 282–356. English translation by Philip E. B. Jourdain (Cantor 1915).

Contributions to the Founding of the Theory of Transfinite Numbers. Translated by Philip E. B. Jourdain. Chicago: Open Court, 1915.

“Principien einer Theorie der Ordnungstypen” (Erste Mittheilung). Edited by Ivor Grattan-Guinness: Acta Mathematica 124 (1970): 65–107. This paper was discovered by Ivor Grattan-Guinness among unpublished papers at the Insitut Mittag-Leffler; it was set in type but never printed, and was dated 6 November 1884.

With Richard Dedekind. Briefwechsel Cantor-Dedekind. Edited by Emmy Noether and Jean Cavaillès. Paris: Hermann, 1937.

OTHER SOURCES

Bandmann, Hans. Die Unendlichkeit des Seins. Cantors transfinite Mengenlehre und ihre metaphysischen Wurzeln. Frankfurt am Main: Lang, 1992.

Bell, Eric Temple. “Paradise Lost: Georg Cantor.” In Men of Mathematics. New York: Simon and Schuster, 1937. Reprint, 1986, chap. 29, pp. 555–579. To be used with extreme caution; although widely read and available in numerous reprintings, it has been described as “one of the worst” books on history of mathematics, and “can be said to have done considerable disservice to the profession” (Grattan-Guinness 1971, p. 350).

Brouwer, Luitzen E. J. “Beweis der Invarianz der Dimensionenzahl.” Mathematische Annalen 70 (1911): 161–165.

Charraud, Nathalie. Infini et inconscient: Essai sur Georg Cantor. Paris: Anthropos, 1994.

Dauben, Joseph Warren. Georg Cantor: His Mathematics and Philosophy of the Infinite. Cambridge, MA: Harvard University Press, 1979. Reprint, Princeton, NJ: Princeton University Press, 1990.

———. “Review of Walter Purkert and Hans Joachim Ilgauds: Georg Cantor, 1845–1918(Vita Mathematica 1).” Basel: Birkhäuser, 1987, in Isis 79, no. 4 (1988): 700–702.

———. “The Battle for Cantorian Set Theory.” In Mathematics and the Historian’s Craft: The Kenneth O. May Lectures, edited by Michael Kinyon and Glen van Brummelen. New York: Springer Verlag, Canadian Mathematical Society Books in Mathematics, 2005.

Epple, Moritz. “Georg Cantor.” In Modern Germany: An Encyclopedia of History, People, and Culture, 1871–1990, edited by D. K. Buse and J. C. Doerr. 2 vols. New York: Garland Publishing, 1998.

Fraenkel, Abraham A. “Georg Cantor.” Jahresbericht der Deutschen Mathematiker-Vereinigung 39 (1930): 189–266.

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———. Review of “Cantor: Die Vermessung des Unendlichen.” Annals of Science 64 (2007).

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———. Georg Cantor 1845–1918. Basel: Birkhäuser, 1987.

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Joseph W. Dauben

Georg Ferdinand Ludwig Philipp Cantor

1845-1918

German Mathematician

Georg Cantor is regarded as the founder of set theory. He also introduced the concept of transfinite numbers. This is the idea that there is not just one quantity of "infinity," but many quantities that are distinct from one another but all indefinitely large.

Cantor was born in St. Petersburg, Russia, on March 3, 1845. His father was a well-to-do merchant, and his mother was an artistic woman from a family of musicians. In 1856, the family moved to Germany. Cantor's mathematical talents soon became apparent at gymnasien, or secondary schools, in Darmstadt and Wiesbaden.

As an undergraduate student at the University of Berlin, Cantor specialized in mathematics, physics, and philosophy. There, he first met the mathematician Leopold Kronecker (1823-1891), who was later to become his archrival. In 1867, Cantor received his doctorate from the University of Göttingen, with a thesis entitled In re mathematica ars propendi pluris facienda est quam solvendi, or, "In mathematics the art of asking questions is more valuable than solving problems." The thesis addressed an unsettled question from Disquisitiones Arithmeticae, an 1801 work by Carl Friedrich Gauss (1777-1855).

Cantor spent a short time teaching at a girls' school in Berlin, and then joined the faculty at the University of Halle. He remained there for the rest of his life. Between 1869 and 1873, he published a series of papers on number theory and trigonometric series. It was in the 1870s that he began working on set theory, which led him to the concept of transfinite numbers.

A set is a group of objects or numbers that retain their individuality while having some property in common. The group can be either finite or infinite. For example, the set of students in a class is finite. The set of integers is infinite. Cantor concerned himself in particular with one-to-one correspondences. That is, the sets {a,b,c} and {1,2,3} are in a one-to-one correspondence because a can be paired with 1, b can be paired with 2, and c can be paired with 3. However, for infinite sets, one cannot simply compare the number of members, since the sets are indefinitely large.

Cantor developed a theory of countability based on one-to-one correspondences between infinite sets. Rational numbers, for example, are countable even though infinite. They can be placed in a one-to-one correspondence with integers. The real numbers, consisting of the set of rational and irrational numbers taken together, are uncountable. This led him to the realization that some infinite sets are larger than others, and thus to the idea of transfinite numbers.

Cantor's first paper on set theory was at first refused for publication by Crelle's Journal due to the vehement opposition of Leopold Kronecker. Most scholarly journals ask professionals in the field to referee papers before publishing, and Kronecker was serving in that capacity. The paper was published the next year, but the antagonism between the two mathematicians remained. Its basis was essentially philosophical. Cantor drew on his childhood religious training and Platonic metaphysics to come to the conclusion that infinite numbers had an actual existence. Kronecker was more limited in his view. "God made the integers," he asserted, "and all the rest is the work of man."

As for the work of Cantor, it became the basis for an entire field of study, the mathematics of the infinite. It was fundamental to the development of function theory, analysis, and topology. It also changed educators' ideas about the foundations of mathematical thought. Basic set theory and functions are part of today's elementary mathematics curriculum, first introduced in the 1960s as part of the "new math."

In 1874, Cantor married Vally Guttman. They had five children. The mathematician suffered from bouts of depression from about 1884, but continued to work as his health permitted. In 1897 he was involved in organizing the first international mathematical congress, which took place in Zurich. He died in Halle on January 6, 1918.

SHERRI CHASIN CALVO

Georg Ferdinand Ludwig Philipp Cantor

The German mathematician Georg Ferdinand Ludwig Philipp Cantor (1845-1918) was noted for his theory of sets and his bold analysis of the "actual" infinite, which provoked a critical examination of the foundations of mathematics and eventually transformed nearly every branch.

Georg Cantor was born in St. Petersburg, Russia, on March 3, 1845. He was German only by tenure. His father was a Danish Jew who converted to Protestantism and made his fortune in merchandising; his mother was a Roman Catholic whose family was richly endowed with musicians and artists. The Cantor family moved to Frankfurt, Germany, when Georg was 11. When he entered the Wiesbaden gymnasium at the age of 15, his mathematical talent was already evident.

In 1862 Cantor entered the University of Zurich. After a year he transferred to Berlin, where he studied mathematics and received his doctorate in 1867. Two years later he joined the faculty of the University of Halle as a privatdozent, or unpaid lecturer, and in 1872 he was appointed assistant professor. He married Vally Guttman in 1874, and they had six children.

Up to this time Cantor's mathematical work was uniformly excellent but with no harbinger of the explosive things to come. Then, while pursuing a problem on trigonometric series originally raised by Georg Riemann, he was led to investigate the nature of infinite sets of numbers. Fascinated by what he saw and driven almost against his will, he dug deeper and came up with some outrageously paradoxical results. Among other things, he proved that the number of points within a square is no more "numerous" than the number of points on one of its sides. "I see it but I don't believe it," he wrote to a friend in 1877. Neither did many others, and some were quick to say so. Leopold Kronecker attacked Cantor personally and used his prestige to prejudice others against Cantor's ideas. What made these ideas so pernicious in Kronecker's eyes was Cantor's insistence on treating infinite sets as completed entities.

Cantor developed his theory of sets and transfinite numbers in a long series of papers from 1874 to 1897. In 1879 he was promoted to full professor, and after a time his work came to be appreciated by some mathematicians. But the general hostility caused him to suffer frequent nervous breakdowns and long periods of depression. In 1908 even the great Henri Poincaré remarked that later generations would regard Cantor's set theory "as a disease from which one has recovered." Poincaré was wrong. Today the power of set theory to illuminate and unify vast areas of mathematics is generally recognized.

Cantor died on Jan. 6, 1918, in a mental hospital at Halle.

Further Reading

The best biography of Cantor is in E. T. Bell, Men of Mathematics (1937). See also Ganesh Prasad, Some Great Mathematicians of the Nineteenth Century: Their Lives and Their Works (2 vols., 1933-1934). For an introduction to set theory and transfinite numbers see Richard Courant and Herbert Robbins, What Is Mathematics (1941). □