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Georg Cantor

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Georg Cantor

Georg Ferdinand Ludwig Philipp Cantor (March 3, 1845, St. Petersburg, Russia – January 6, 1918, Halle, Germany) was a German mathematician. He is best known as the creator of set theory. Cantor established the importance of one-to-one correspondence between sets, defined infinite and well-ordered sets, and proved that the real numbers are "more numerous" than the natural numbers. In fact, Cantor's theorem implies the existence of an "infinity of infinities." He defined the cardinal and ordinal numbers, and their arithmetic. Cantor's work is of great philosophical interest, a fact of which he was well aware.

Cantor's work encountered resistance from mathematical contemporaries such as Leopold Kronecker and Henri Poincaré, and later from Hermann Weyl and L.E.J. Brouwer. Ludwig Wittgenstein raised philosophical objections. His recurring bouts of depression from 1884 to the end of his life were once blamed on the hostile attitude of many of his contemporaries, but these bouts can now be seen as probable manifestations of a bipolar disorder.

Today, the vast majority of mathematicians who are neither constructivists nor finitists accept Cantor's work on transfinite sets and arithmetic, recognizing it as a major paradigm shift. In the words of David Hilbert: "No one shall expel us from the Paradise that Cantor has created."

Life

Cantor was born in 1845 in St. Petersburg, Russia, and brought up in a Lutheran German mission in St. Petersburg. Cantor's father, Georg Woldemar Cantor, was a Dane of Lutheran religion born in either 1809 or 1814 and a broker on the St Petersburg Stock Exchange. His mother, Maria Anna Böhm, was born in St. Petersburg and came from an Austrian Roman Catholic family. She had converted to Protestantism upon marriage; his father was of Jewish and Danish descent.

Cantor was the eldest of six children. His father was very devout and instructed all his children thoroughly in religious affairs. Throughout the rest of his life, Cantor held to the Lutheran faith. He was also an outstanding violinist, having inherited his parents' considerable musical and artistic talents.

When Cantor's father became ill, the family moved to Germany in 1856, first to Wiesbaden then to Frankfurt, seeking winters milder than those of St. Petersburg. In 1860, Cantor graduated with distinction from the Realschule in Darmstadt; his exceptional skills in mathematics, trigonometry in particular, were noted. In 1862, following his father's wishes, Cantor entered the Federal Polytechnic Institute in Zurich, today the ETH Zurich and began studying mathematics.

After his father's death in 1863, Cantor shifted his studies to the University of Berlin, attending lectures by Weierstrass, Kummer, and Kronecker, and befriending his fellow student Hermann Schwarz. He spent a summer at the University of Göttingen, then and later a very important center for mathematical research. In 1867, Berlin granted him the Ph.D. for a thesis on number theory, De aequationibus secundi gradus indeterminatis. After teaching one year in a Berlin girls' school, Cantor took up a position at the University of Halle, where he spent his entire career. He was awarded the requisite habilitation for his thesis on number theory.

In 1874, Cantor married Vally Guttmann. They had six children, the last born in 1886. Cantor was able to support a family despite modest academic pay, thanks to an inheritance from his father. During his honeymoon in Switzerland, Cantor spent much time in mathematical discussions with Richard Dedekind, whom he befriended two years earlier while on another Swiss holiday.

Cantor was promoted to Extraordinary Professor in 1872, and made full Professor in 1879. To attain the latter rank at the age of 34 was a notable accomplishment, but Cantor very much desired a chair at a more prestigious university, in particular at Berlin, then the leading German university. However, Kronecker, who headed mathematics at Berlin until his death in 1891, and his colleague Hermann Schwarz were not agreeable to having Cantor as a colleague. Worse yet, Kronecker, who was peerless among German mathematicians while he was alive, fundamentally disagreed with the thrust of Cantor's work. Kronecker, now seen as one of the founders of the constructive viewpoint in mathematics, disliked much of Cantor's set theory because it asserted the existence of sets satisfying certain properties, without giving specific examples of sets whose members did indeed satisfy those properties. Cantor came to believe that Kronecker's stance would make it impossible for Cantor to ever leave Halle.

In 1881, Cantor's Halle colleague Eduard Heine died, creating a vacant chair. Halle accepted Cantor's suggestion that it be offered to Dedekind, Heinrich Weber, and Franz Mertens, in that order, but each declined the chair after being offered it. This episode is revealing of Halle's lack of standing among German mathematics departments. Wangerin was eventually appointed, but he was never close to Cantor.

In 1884, Cantor suffered his first known bout of depression. This emotional crisis led him to apply to lecture on philosophy rather than on mathematics. Every one of the 52 letters Cantor wrote to Gösta Mittag-Leffler that year attacked Kronecker. Cantor soon recovered, but a passage from one of these letters is revealing of the damage to his self-confidence:

"... I don't know when I shall return to the continuation of my scientific work. At the moment I can do absolutely nothing with it, and limit myself to the most necessary duty of my lectures; how much happier I would be to be scientifically active, if only I had the necessary mental freshness."

Although he performed some valuable work after 1884, he never attained again the high level of his remarkable papers of 1874-84. He eventually sought a reconciliation with Kronecker, which Kronecker graciously accepted. Nevertheless, the philosophical disagreements and difficulties dividing them persisted. It was once thought that Cantor's recurring bouts of depression were triggered by the opposition his work met at the hands of Kronecker. While Cantor's mathematical worries and his difficulties dealing with certain people were greatly magnified by his depression, it is doubtful whether they were its cause, which was probably bipolar disorder.

In 1888, he published his correspondence with several philosophers on the philosophical implications of his set theory. Edmund Husserl was his Halle colleague and friend from 1886 to 1901. While Husserl later made his reputation in philosophy, his doctorate was in mathematics and supervised by Weierstrass' student Leo Königsberger. On Cantor, Husserl, and Frege, see Hill and Rosado Haddock (2000). Cantor also wrote on the theological implications of his mathematical work; for instance, he identified the Absolute Infinite with God.

Cantor believed that Francis Bacon wrote the plays attributed to Shakespeare. During his 1884 illness, he began an intense study of Elizabethan literature in an attempt to prove his Bacon authorship thesis. He eventually published two pamphlets, in 1896 and 1897, setting out his thinking about Bacon and Shakespeare.

In 1890, Cantor was instrumental in founding the Deutsche Mathematiker-Vereinigung, chaired its first meeting in Halle in 1891, and was elected its first president. This is strong evidence that Kronecker's attitude had not been fatal to his reputation. Setting aside the animosity he felt towards Kronecker, Cantor invited him to address the meeting; Kronecker was unable to do so because his spouse was dying at the time.

After the 1899 death of his youngest son, Cantor suffered from chronic depression for the rest of his life, for which he was excused from teaching on several occasions and repeatedly confined in various sanatoria. He did not abandon mathematics completely, lecturing on the paradoxes of set theory (eponymously attributed to Burali-Forti, Russell, and Cantor himself) to a meeting of the Deutsche Mathematiker-Vereinigung in 1903, and attending the International Congress of Mathematicians at Heidelberg in 1904.

In 1911, Cantor was one of the distinguished foreign scholars invited to attend the 500th anniversary of the founding of the University of St. Andrews in Scotland. Cantor attended, hoping to meet Bertrand Russell, whose newly published Principia Mathematica repeatedly cited Cantor's work, but this did not come about. The following year, St. Andrews awarded Cantor an honorary doctorate, but illness precluded his receiving the degree in person.

Cantor retired in 1913, and suffered from poverty, even hunger, during World War I. The public celebration of his 70th birthday was cancelled because of the war. He died in the sanatorium where he had spent the final year of his life.

Work

Cantor was the originator of set theory, 1874-84. He was the first to see that infinite sets come in different sizes, as follows. He first showed that given any set A, the set of all possible subsets of A, called the power set of A, exists. He then proved that the power set of an infinite set A has a size greater than the size of A (this fact is now known as Cantor's theorem). Thus there is an infinite hierarchy of sizes of infinite sets, from which springs the transfinite cardinal and ordinal numbers, and their peculiar arithmetic. His notation for the cardinal numbers was the Hebrew letter aleph (א) with a natural number subscript; for the ordinals he employed the Greek letter omega.

Cantor was the first to appreciate the value of one-to-one correspondences (hereinafter denoted "1-to-1") for set theory. He defined finite and infinite sets, breaking down the latter into denumerable and nondenumerable sets. There exists a 1-to-1 correspondence between any denumerable set and the set of all natural numbers; all other infinite sets are nondenumerable. He proved that the set of all rational numbers is denumerable, but that the set of all real numbers is not and hence is strictly bigger. The cardinality of the natural numbers is aleph-null; that of the reals is larger, and is at least aleph-one (the latter being the next smallest cardinal after aleph-null).

Cantor's first 10 papers were on number theory, his thesis topic. At the suggestion of Eduard Heine, the Professor at Halle, Cantor turned to analysis. Heine proposed that Cantor solve an open problem that had eluded Dirichlet, Lipschitz, Bernhard Riemann, and Eduard Heine himself: the uniqueness of the representation of a function by trigonometric series. Cantor solved this difficult problem in 1869. Between 1870 and 1872, Cantor published more papers on trigonometric series, including one defining irrational numbers as convergent sequences of rational numbers. Dedekind, whom Cantor befriended in 1872, cited this paper later that year, in the paper where he first set out his celebrated definition of real numbers by Dedekind cuts.

Cantor's 1874 paper, "On a Characteristic Property of All Real Algebraic Numbers", marks the birth of set theory. It was published in Crelle's Journal, despite Kronecker's opposition, thanks to Dedekind's support. Previously, all infinite collections had been (silently) assumed to be of "the same size"; Cantor was the first to show that there was more than one kind of infinity. In doing so, he became the first to invoke the notion of a 1-to-1 correspondence, albeit not calling it such. He then proved that the real numbers were not denumerable, employing a proof more complex than the remarkably elegant and justly celebrated diagonal argument he first set out in 1891.

The 1874 paper also showed that the algebraic numbers, i.e., the roots of polynomial equations with integer coefficients, were denumerable. Real numbers that are not algebraic are transcendental. Liouville had established the existence of transcendental numbers in 1851. Since Cantor had just shown that the real numbers were not denumerable and that the union of two denumerable sets must be denumerable, it logically follows from the fact that a real number is either algebraic or transcendental that the transcendentals must be nondenumerable. The transcendentals have the same "power" (see below) as the reals, and "almost all" real numbers must be transcendental. Cantor remarked that he had effectively reproved a theorem, due to Liouville, to the effect that there are infinitely many transcendental numbers in each interval.

In 1874, Cantor began looking for a 1-to-1 correspondence between the points of the unit square and the points of a unit line segment. In an 1877 letter to Dedekind, Cantor proved a far stronger result: there exists a 1-to-1 correspondence between the points on the unit line segment and all of the points in a p-dimensional space. About this discovery Cantor wrote famously (and in French) "I see it, but I don't believe it!" This astonishing result has implications for geometry and the notion of dimension.

In 1878, Cantor submitted another paper to Crelle's Journal, which again displeased Kronecker. Cantor wanted to withdraw the paper, but Dedekind persuaded him not to do so; moreover, Weierstrass supported its publication. Nevertheless, Cantor never again submitted anything to Crelle.

This paper made precise the notion of a 1-to-1 correspondence, and defined denumerable sets as sets which can be put into a 1-to-1 correspondence with the natural numbers. Cantor introduces the notion of "power" (a term he took from Jakob Steiner) or "equivalence" of sets; two sets are equivalent (have the same power) if there exists a 1-to-1 correspondence between them. He then proves that the rational numbers have the smallest infinite power, and that R has the same power as R. Moreover, countably many copies of R have the same power as R. While he made free use of countable as a concept, he did not write the word "countable" until 1883. Cantor also discussed his thinking about dimension, stressing that his mapping between the unit interval and the unit square was not a continuous one.

Between 1879 and 1884, Cantor published a series of six articles in Mathematische Annalen that together formed an introduction to his set theory. By agreeing to publish these articles, the editor displayed courage, because of the growing opposition to Cantor's ideas, led by Kronecker. Kronecker admitted mathematical concepts only if they could be constructed in a finite number of steps from the natural numbers, which he took as intuitively given. For Kronecker, Cantor's hierarchy of infinities was inadmissible.

The fifth paper in this series, "Foundations of a General Theory of Aggregates", published in 1883, was the most important of the six and was also published as a separate monograph. It contained Cantor's reply to his critics and showed how the transfinite numbers were a systematic extension of the natural numbers. It begins by defining well-ordered sets. Ordinal numbers are then introduced as the order types of well-ordered sets. Cantor then defines the addition and multiplication of the cardinal and ordinal numbers. In 1885, Cantor extended his theory of order types so that the ordinal numbers simply became a special case of order types.

Cantor's 1883 paper reveals that he was well aware of the opposition his ideas were encountering:

"... 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."

Hence he devotes much space to justifying his earlier work, asserting that mathematical concepts may be freely introduced as long as they are free of contradiction and defined in terms of previously accepted concepts. He also cites Aristotle, Descartes, Berkeley, Leibniz, and Bolzano on infinity.

Cantor was the first to formulate what later came to be known as the continuum hypothesis or CH: there exists no set whose power is greater than that of the naturals and less than that of the reals (or equivalently, the cardinality of the reals is exactly aleph-one, rather than just at least aleph-one). His inability to prove the continuum hypothesis caused Cantor considerable anxiety but, with the benefit of hindsight, is entirely understandable: a 1940 result by Gödel and a 1963 one by Paul Cohen together imply that the continuum hypothesis can neither be proved nor disproved using standard Zermelo-Fraenkel set theory plus the axiom of choice (the combination referred to as "ZFC").

In 1882, the rich mathematical correspondence between Cantor and Dedekind came to an end. Cantor also began another important correspondence, with Gösta Mittag-Leffler in Sweden, and soon began to publish in Mittag-Leffler's journal Acta Mathematica. But in 1885,[Mittag-Leffler asked Cantor to withdraw a paper from Acta while it was in proof, writing that it was "... about one hundred years too soon." Cantor complied, but wrote to a third party:

"Had Mittag-Leffler had his way, I should have to wait until the year 1984, which to me seemed too great a demand! ... But of course I never want to know anything again about Acta Mathematica."

Thus ended his correspondence with Mittag-Leffler, as did Cantor's brilliant development of set theory over the previous 12 years. Mittag-Leffler had meant well, but this incident reveals how even Cantor's most brilliant contemporaries often failed to appreciate his work.

In 1895 and 1897, Cantor published a two-part paper in Mathematische Annalen under Felix Klein's editorship; these were his last significant papers on set theory. (The English translation is Cantor 1955.) The first paper begins by defining set, subset, etc., in ways that would be largely acceptable now. The cardinal and ordinal arithmetic are reviewed. Cantor wanted the second paper to include a proof of the continuum hypothesis, but had to settle for expositing his theory of well-ordered sets and ordinal numbers. Cantor attempts to prove that if A and B are sets with A equivalent to a subset of B and B equivalent to a subset of A, then A and B are equivalent. Ernst Schroeder had stated this theorem a bit earlier, but his proof, as well as Cantor's, was flawed. Felix Bernstein supplied a correct proof in his 1898 Ph.D. thesis; hence the name Cantor-Schroeder-Bernstein theorem.

Around this time, the set-theoretic paradoxes began to rear their heads. In an 1897 paper on an unrelated topic, Cesare Burali-Forti set out the first such paradox, the Burali-Forti paradox: the ordinal number of the set of all ordinals must be an ordinal and this leads to a contradiction. Cantor discovered this paradox in 1895, and described it in an 1896 letter to Hilbert. Curiously, Cantor was highly critical of Burali-Forti's paper.

In 1899, Cantor discovered his eponymous paradox: what is the cardinal number of the set of all sets? Clearly it must be the greatest possible cardinal. Yet for any sets A, the cardinal number of the power set of A > cardinal number of A (Cantor's theorem again). This paradox, together with Burali-Forti's, led Cantor to formulate his concept of limitation of size, according to which the collection of all ordinals, or of all sets, was an "inconsistent multiplicity" that was "too large" to be a set. Today they would be called proper classes.

One common view among mathematicians is that these paradoxes, together with Russell's paradox, demonstrate that it is not possible to take a "naive", or non-axiomatic, approach to set theory without risking contradiction, and it is certain that they were among the motivations for Zermelo and others to produce axiomatizations of set theory. Others note, however, that the paradoxes do not obtain in an informal view motivated by the iterative hierarchy, which can be seen as explaining the idea of limitation of size. Some also question whether the Fregean formulation of naive set theory (which was the system directly refuted by the Russell paradox) is really a faithful interpretation of the Cantorian conception.

Cantor's work did attract favorable notice beyond Hilbert's celebrated encomium. In public lectures delivered at the first International Congress of Mathematicians, held in Zurich in 1897, Hurwitz and Hadamard both expressed their admiration for Cantor's set theory. At that Congress, Cantor also renewed his friendship and correspondence with Dedekind. Charles Peirce in America also praised Cantor's set theory. In 1905, Cantor began a correspondence, later published, with his British admirer and translator Philip Jourdain, on the history of set theory and on Cantor's religious ideas.

Notes

  1. See note number 6 at this list of Jewish Mathematicians, citing a letter of Cantor's about his father's Jewish parentage, and other evidence. Biographers have varied between declaring Cantor Jewish on both sides and on neither.
  2. Some mathematicians consider these results to have settled the issue, and, at most, allow that it is possible to examine the formal consequences of CH or of its negation, or of axioms that imply one of those. Others continue to look for "natural" or "plausible" axioms that, when added to ZFC, will permit either a proof or refutation of CH, or even for direct evidence for or against CH itself; among the most prominent of these is W. Hugh Woodin. One of Goedel's last papers argues that the CH is false, and the continuum has cardinality Aleph-2.

See also

Bibliography

Primary literature in English:

  • Cantor, Georg, 1955 (1915). Contributions to the Founding of the Theory of Transfinite Numbers. Philip Jourdain, ed. and trans. Dover.
  • Ewald, William B., ed., 1996. From Kant to Hilbert: A Source Book in the Foundations of Mathematics, 2 vols. Oxford Uni. Press.
    • 1874. "On a property of the set of real algebraic numbers," 839-43.
    • 1883. "Foundations of a general theory of manifolds," 878-919.
    • 1891. "On an elementary question in the theory of manifolds," 920-22.
    • 1872-82, 1899. Correspondence with Dedekind, 843-77, 930-40.

Primary literature in German:

Secondary literature:

  • Aczel, Amir D., 2000. The mystery of the Aleph: Mathematics, the Kabbala, and the Human Mind. Four Walls Eight Windows. A popular treatment of infinity, in which Cantor is the key player.
  • Dauben, Joseph W., 1979. Georg Cantor : his mathematics and philosophy of the infinite. Harvard Uni. Press. The definitive biography to date.
  • Ivor Grattan-Guinness, 2000. The Search for Mathematical Roots: 1870-1940. Princeton Uni. Press.
  • Hallett, Michael, 1984. Cantorian set theory and limitation of size. Oxford Uni. Press.
  • Paul Halmos, 1998 (1960). Naive Set Theory. Springer.
  • Hill, C. O., and Rosado Haddock, G. E., 2000. Husserl or Frege? Meaning, Objectivity, and Mathematics. Chicago: Open Court. Three chpts. and 18 index entries on Cantor.
  • Roger Penrose, 2004. The Road to Reality. Alfred A. Knopf. Chpt. 16 reveals how Cantorian thinking intrigues a leading contemporary theoretical physicist.
  • Rudy Rucker, 2005 (1982). Infinity and the Mind. Princeton Uni. Press. Deeper than Aczel.
  • Suppes, Patrick, 1972 (1960). Axiomatic Set Theory. Dover. Although the presentation is axiomatic rather than naive, Suppes proves and discusses many of Cantor's results, thereby revealing Cantor's importance for the edifice of foundational mathematics.

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