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Cantor, Georg (1845–1918)


Georg Cantor, a mathematician who created set theory and a corresponding theory of transfinite numbers, revolutionized mathematics at the end of the nineteenth century with his ideas about the infinite, which were to be of profound significance not only for mathematics but for philosophy and many allied disciplines as well.

He was born on March 3, 1845, in St. Petersburg, Russia, to Georg Woldemar Cantor, a successful merchant and the son of a Jewish businessman from Copenhagen, and Maria Anna Böhm, who came from a family of notable musicians and was a Roman Catholic. But Cantor's father, raised in a Lutheran mission, was a deeply religious man and passed his own strong convictions on to his son. Later in life, Cantor's religious beliefs would play a significant role in his steadfast faith in the correctness of his controversial transfinite set theory, just as his mother's Catholicism may have made him particularly amenable to the substantial correspondence he undertook with Catholic theologians over the nature of the infinite from a theological perspective.

Early Mathematical Studies

Cantor received his doctorate in 1868 from the University of Berlin, where he had studied with Leopold Kronecker, Ernst Eduard Kummer, and Karl Weierstrass. His dissertation was devoted to number theory, as was his Habilitationsschrift. When Cantor began teaching as an instructor at the University of Halle, among his colleagues there was Eduard Heinrich Heine. Heine had been working on problems related to trigonometric series, and he urged Cantor to take up the challenging problem of whether or not, given an arbitrary function represented by a trigonometric series, the representation was unique. In 1870 Heine had established the uniqueness of such representations for almost-everywhere continuous functions, assuming the uniform convergence of the trigonometric series in question. Cantor succeeded in establishing increasingly general versions of the uniqueness theorem in a series of papers he published between 1870 and 1872, the most remarkable of which showed that even if an infinite number of exceptional points for the representation were allowed, the uniqueness could still be shown if such infinite sets of "exceptional" points were distributed in a particular way. Such sets of exceptional points constituted what Cantor called sets 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 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 ν the νth derived set Pν contains only a finite number of points, then its derived set will be empty, that is, Pν +1 = . It was for such first-species sets that he was able to establish the uniqueness of trigonometric series representations, even though there were an infinite number of exceptional points. Transfinite set theory would arise from Cantor's later consideration of point sets of the second species, all of whose derived sets were infinite. From these Cantor would eventually generate an endless hierarchy of what he came to call transfinite ordinal, and later their corresponding cardinal, numbers.

The Real Numbers

Cantor realized that to define the structure of point sets of the first species unambiguously required a rigorous definition of the real numbers, which he approached in terms of fundamental, convergent sequences of rational numbers in his last paper on trigonometric series of 1872. In the same year Richard Dedekind introduced his own rigorous definition of the real numbers in terms of "Dedekind cuts." Both approaches are concerned with the continuity of the real numbers in general, a subject that was to haunt Cantor for the rest of his life. In particular, he succeeded in proving just a few years later, in 1874, that the set of all real numbers was in fact nondenumerably infinite, that is, of a distinctly higher order of infinity than denumerably infinite sets like the whole, rational, or algebraic numbers. This fact soon led to the articulation of one of Cantor's most famous problems: his continuum hypothesis, that the infinite set of real numbers R is the next higher order of infinite sets following denumerably infinite sets like the set of all natural numbers N. Cantor became especially interested in the question of whether or not point sets of two and higher dimensions might furnish examples of increasingly infinite orders of infinity, something he answered negatively in 1877. This was another of Cantor's important early results, his proof (though faulty) of the invariance of dimension; the first correct proof was published by L. E. J. Brouwer in 1911.

Between 1879 and 1883 Cantor wrote a series of articles that culminated in an independently published monograph devoted to the study of linear point sets, Grundlagen einer allgemeinen Mannigfaltigkeitslehre: Ein mathematisch-philosophischer Versuch in der Lehre des Unendlichen (Foundations of a general theory of sets: A mathematico-philosophical investigation into the theory of the infinite). In addition to introducing such concepts as everywhere-dense sets, he showed that whereas everywhere-dense sets were necessarily of the second species, first-species sets could never be everywhere-dense.

Transfinite Numbers

In his series of papers on linear point sets, and in the Grundlagen, Cantor introduced his new concept of transfinite numbers. At first, these were limited to the transfinite ordinal numbers that he generated from the point sets of the second species that he had introduced in 1872. Considering the entire sequence of derived sets Pν, none of which was empty (i.e., every derived set Pν contained an infinite number of limit points): P, P, , Pν, , Cantor defined the intersection of all these sets as P. This was an infinite set that in turn led to the next derived set P+1. If this set were infinite, and in fact every derived set thereafter, this led to an endless hierarchy of further infinite derived sets: P, P, , Pν, , P, P+1, , P+ ν, , P2,

At first, Cantor only regarded the superscripts as "infinite symbols," but early in the 1880s he began to distinguish these indexes as numbers independently of point sets of the second species. By the time he wrote the Grundlagen in 1883, these infinite symbols had become transfinite ordinal numbers.

Controversy and Criticism

Cantor understood that his new ideas would be controversial, and his work had already met with criticism, especially from Kronecker, his former teacher at the University of Berlin. Cantor was so concerned about the possible objections to his new ideas that he undertook a detailed analysis of the subject historically, which served his strategy in the Grundlagen to present a detailed analysis of the foundations of transfinite set theory from both a philosophical and theological perspective. It was in the Grundlagen that he made one of his most famous statements, that "the essence of mathematics lies precisely in its freedom" (1996, p. 182). As Cantor later confided to the mathematician David Hilbert, this statement was inspired by the negative criticism Kronecker had made of set theory and was a call for open-mindedness among mathematicians, especially in dealing with new and novel ideas proposed by younger mathematicians. But the opposition mounted by Kronecker served a useful purpose in stimulating Cantor's own philosophical reaction and his determination to provide the soundest possible foundations, both mathematically and philosophically, for transfinite set theory.

What Cantor did in the Grundlagen was to present the transfinite ordinal numbers as a direct extension of the real numbers. But because he generated these infinite real numbers as abstractions from sets of points, he rejected the possibility of there being actually infinitesimal numbers. He also knew that an important property of the transfinite ordinal numbers was their noncommutativity, that is:
2+ω = (1, 2, a1, a2, , an, an+1, )
(a1, a2, , an, an+1, , 1, 2) = ω+2,
2ω = (a1, a2, a3, ; b1, b2, b3, )
(a1, b1, a2, b2, a3, b3, ) = ω2.

Such distinctions brought new insights to the differences between finite and infinite sets. For finite sets and their corresponding ordinal numbers, addition and multiplication were commutative; infinite sets were more interesting because their corresponding ordinal numbers and transfinite arithmetic were not commutative. Cantor expected that understanding such differences would not only explain the seemingly paradoxical nature of the infinite but would also answer some of the long-standing objections to the infinite that historically had been so persuasive to mathematicians and philosophers alike.

Transfinite Cardinals and Cantor's Alephs

Although the Grundlagen offered a systematic presentation of Cantor's transfinite ordinal numbers, there was no mention of his best-known innovation: the transfinite cardinal numbers, or alephs. Indeed, nowhere in the Grundlagen was there any indication that the power of an infinite set was to be equated with the concept of a transfinite cardinal number, a step he first took in a lecture he delivered at Freiburg in September 1883. Over the next decade he used a number of different notations for transfinite cardinal numbers, but did not decide on a definite symbol until Giulio Vivanti, an Italian mathematician who was writing an introductory monograph on set theory, asked Cantor about notation. Only then did he finally choose the Hebrew aleph for the transfinite cardinal numbers. In "Beiträge zur Begründung der transfiniten Mengenlehre" (Contributions to the founding of the theory of transfinite numbers) he designated the least transfinite cardinal number as 0.

It was also in "Beiträge" that Cantor offered an algebraic interpretation of his continuum hypothesis, based on his proof of 1891 that given any infinite set P, the set of all its subsets was of a higher power than P. Since the cardinality of the set of all real numbers could be written as 20, and if 1 was the next largest cardinal following 0, then the continuum hypothesis could now be expressed as 20 = 1. Cantor hoped that with this new algebraic formulation of the hypothesis, he would soon manage to produce a proof that the power of the real numbers was indeed equal to 1. He never succeeded in doing so, for reasons that only became apparent in the twentieth century, thanks to the results of Kurt Gödel (who established that the continuum hypothesis was consistent with the basic axioms of Zermelo-Fraenkel set theory) and Paul Cohen (who showed, on the contrary, that the continuum hypothesis was independent of the same axioms), which meant that it was possible to conceive of consistent set theories in which Cantor's continuum hypothesis did not hold.

Cantor's last major publication appeared in two parts in the journal Mathematische Annalen in 1895 and 1897. "Beiträge" not only offered a complete account of both his transfinite ordinal and cardinal numbers but also his theory of order types, which investigated in detail the different properties of the sets of natural, rational, and real numbers, respectively. The well-ordered set of integers, taken in their natural order, he designated (ω the set of rational numbers in their natural order, which were everywhere-dense but not continuous, he designated η; sets like the real numbers that were continuous he designated by the order-type θ. But the result he hoped to achieve in "Beiträge" but failed to produce, namely, proof of his continuum hypothesis, remained illusive.

Cantor's Manic Depression

Much has been written about Cantor's unfortunate history of mental illness, which some writers have linked with the heavy criticism of Cantor's transfinite set theory from Kronecker. But recent studies suggest that what Cantor suffered from was manic depression, which would have afflicted him regardless of the controversies surrounding his mathematical work (see Grattan-Guinness 1971, Dauben 1979, Charraud 1994). Whereas the earliest serious breakdown occurred in 1884, as Cantor was encountering his first disappointments in trying to prove the continuum hypothesis (for a detailed account of what happened, see Schoenflies 1927), the manic depression became more serious as he grew older, and after 1900 he spent increasingly long periods under professional care, often at the Nervenklinik in Halle. Also, following the first attack in 1884, Cantor began to take up interests other than mathematics, including the idea that Francis Bacon was the real author of writings attributed to William Shakespeare and that Joseph of Arimathea was the natural father of Jesus. Cantor also began an extensive correspondence with Catholic theologians, and even wrote to Pope Leo XIII directly, in hopes that a correct understanding of the infinite mathematically, in terms of his transfinite set theory, would help the church avoid making any incorrect pronouncements on the subject, especially where the absolutely infinite nature of God was concerned, which Cantor took to be consistent with but wholly different from the concepts of transfinite set theory.

The mathematician Eric Temple Bell (1986) offers a Freudian analysis of Cantor's relationship with his father, whose initial opposition to Cantor's wish to become a mathematician Bell takes to be the source of his son's later mental problems; more recently, Nathalie Charraud (1994), a French psychoanalyst, examined the records of Cantor's treatment at the neurological clinic in Halle and offers a different, Lacanian assessment of the role Cantor's father played in his son's life. Equally important in understanding Cantor's tenacious defense of his controversial set theory is the role that religion played with respect to the transfinite numbers, which he took to have been communicated to him from God directly. For details of how his religious convictions and periods of manic depression may actually have played important, supportive roles in the battle to establish transfinite set theory as a fundamental part of modern mathematics, see Joseph Warren Dauben (2005).

One final aspect of Cantor's career as a mathematician deserves brief mention, because he was primarily responsible for the creation of the Deutsche Mathematiker-Vereinigung (German Mathematical Society), of which Cantor was elected its first president in 1891. He was also instrumental in promoting the idea of the first International Congresses of Mathematicians, beginning with Zürich in 1897, and then Paris in 1900 (Dauben 1979, pp. 163165).

The Paradoxes of Set Theory

To conclude with an assessment of Cantor's significance for philosophy, he was above all responsible for making the infinite a central part of modern mathematics. From the time of the Greeks, Zeno's discovery of the paradoxes of motion and Aristotle's opposition to the concept of completed infinities (as opposed to the potential infinite) led most mathematicians to avoid using the infinite in their work. Cantor faced the subject head-on and argued that there was nothing inherently contradictory in considering actually infinite collections of point sets or the infinite sets of integers, rational, and real numbers as unified, completed objects of thought. His contemplation of these eventually led to his development of transfinite set theory, transfinite arithmetic, and his fundamental concepts of transfinite ordinal and cardinal numbers. His greatest contribution was understanding the roles these played in establishing a proper foundation for mathematics, which he approached essentially on formalist terms. Consistency, for Cantor, was the only test a new mathematical theory needed to pass before he considered it legitimate as a subject for study and application.

When Cantor himself first realized the contradictions inherent in trying to decide the ordinal number of the set of all transfinite ordinal numbers, or the cardinality of the set of all transfinite cardinal numbers, his solution was to simply ban such "collections" from mathematics, saying they were too large to be considered legitimately as "sets." But as others like Cesare Burali-Forti and Jules Richard began to consider the antinomies of set theory, Bertrand Russell discovered a logical paradox at the heart of set theory involving the set of all sets that are not members of themselves. One solution to this dilemma was advanced by Ernst Zermelo, who sought to axiomatize set theory in such a way that the paradoxes would be excluded. Further developments along such lines were made by Russell and Alfred North Whitehead in their monumental Principia Mathematica ; alternative axiomatizations were also advanced by Abraham Fraenkel and John von Neumann, among others.

By the end of his life, Cantor was a mathematician honored by the Royal Society with its Copley Medal for his outstanding contributions to mathematics. He was also granted an honorary degree by the University of St. Andrews (Scotland). Today, the highest award conferred by the German Mathematical Society is a medal honoring its first president, Georg Cantor.

See also Infinity in Mathematics and Logic; Set Theory.


Cantor, Georg, and Richard Dedekind. Briefwechsel Cantor-Dedekind, edited by E. Noether and J. Cavaillès. Paris: Hermann, 1937.

works by cantor

Gesammelte Abhandlungen mathematischen und philosophischen Inhalts, edited by Ernst Zermelo. Berlin: Springer, 1932.

"Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen." In Gesammelte Abhandlungen mathematischen und philosophischen Inhalts, edited by Ernst Zermelo, 115118. Berlin: Springer, 1932. This was originally published in the Journal für die reine und angewandte Mathematik in 1874.

"Foundations of a General Theory of Manifolds: A Mathematico-Philosophical Investigation into the Theory of the Infinite." Translated by W. B. Ewald. In From Kant to Hilbert: A Source Book in the Foundations of Mathematics, edited by W. B. Ewald, 2:878920. New York: Oxford University Press, 1996. This was originally published under the title "Grundlagen einer allgemeinen Mannigfaltigkeitslehre: Ein mathematisch-philosophischer Versuch in der Lehre des Unendlichen" in 1883.

"Beiträge zur Begründung der transfiniten Mengenlehre." In Contributions to the Founding of the Theory of Transfinite Numbers. Translated by Philip E. B. Jourdain. Chicago: Open Court, 1915.

works about cantor

Bell, Eric Temple. "Paradise Lost: Georg Cantor." In Men of Mathematics, 555579. New York: Simon and Schuster, 1986.

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.

Dauben, Joseph Warren. "The Battle for Cantorian Set Theory." In Mathematics and the Historian's Craft. The Kenneth O. May Lectures, edited by Micahel Kinyon and Glen van Brummelen. New York: Springer Verlag, Canadian Mathematical Society Books in Mathematics, 2005.

Fraenkel, Abraham A. "Georg Cantor." Jahresbericht der Deutschen Mathematiker-Vereinigung 39 (1930): 189266.

Grattan-Guinness, Ivor. "Towards a Biography of Georg Cantor." Annals of Science 27 (1971): 345391.

Hallett, Michael. Cantorian Set Theory and Limitation of Size. Oxford, U.K.: Clarendon Press, 1984.

Lavine, Shaughan. Understanding the Infinite. Cambridge, MA: Harvard University Press, 1994.

Meschkowski, Herbert. Probleme des Unendlichen: Werk und Leben Georg Cantors. Braunschweig, Germany: Vieweg and Sohn, 1967.

Purkert, Walter, and Hans Joachim Ilgauds. Georg Cantor. Leipzig, Germany: Teubner, 1985.

Purkert, Walter, and Hans Joachim Ilgauds. Georg Cantor, 18451918. Basel, Switzerland: Birkhäuser, 1987.

Schoenflies, Arthur. "Die Krisis in Cantor's mathematischem Schaffen." Acta Mathematica 50 (1927): 123.

Joseph W. Dauben (2005)

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