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| '''Georg Ferdinand Ludwig Philipp Cantor''' (], ], ] – ], ], ]) was a ] ]. He is best known as the creator of ]. Cantor established the importance of ] between sets, defined ] and ], and proved that the ]s are "more numerous" than the ]s. In fact, ] implies the existence of an "infinity of infinities." He defined the ] and ] numbers, and their arithmetic. Cantor's work is of great philosophical interest, a fact of which he was well aware. | |||
| Cantor's work encountered ] from mathematical contemporaries such as ] and ], and later from ] and ]. ] raised ]. His recurring bouts of ] from ] 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 ]. | |||
| Today, the vast majority of mathematicians who are neither ] nor ] accept Cantor's work on transfinite sets and arithmetic, recognizing it as a major ]. In the words of ]: "No one shall expel us from the Paradise that Cantor has created." | |||
| ==Life== | |||
| Cantor was born in ] in ], ], and brought up in a Lutheran ] mission in St. Petersburg. Cantor's father, Georg Woldemar Cantor, was a ] man of ] religion born in either ] or ] and a ] on the St Petersburg Stock Exchange. His mother, Maria Anna Böhm, was born in St. Petersburg and came from an ]n ] family. She had converted to ] upon marriage. His ethnic background has been disputed for a long time among biographers<ref> See note number 6 at </ref> but nowadays it is very well accepted that he was of ] descent<ref>About the past denial of Gerog Cantor Jewishness,partly for an Anti-semitic reasons, see at:Redner, Harry., | |||
| 2002. ''Philosophers and Anti-semitism'', Modern Judaism - Volume 22, Number 2, pp. 115-141. Oxford Uni. Press.</ref>. | |||
| 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 ]ist, having inherited his parents' considerable musical and artistic talents. | |||
| When Cantor's father became ill, the family moved to ] in ], first to ] then to ], seeking winters milder than those of ]. In 1860, Cantor graduated with distinction from the Realschule in ]; his exceptional skills in mathematics, ] in particular, were noted. In 1862, following his father's wishes, Cantor entered the ] in ], today the ] and began studying mathematics. | |||
| After his father's death in 1863, Cantor shifted his studies to the ], attending lectures by ], ], and ], and befriending his fellow student ]. 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 ] for a ] on ], ''De aequationibus secundi gradus indeterminatis''. After teaching one year in a Berlin girls' school, Cantor took up a position at the ], where he spent his entire career. He was awarded the requisite ] 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 ], Cantor spent much time in mathematical discussions with ], 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, ], who headed mathematics at Berlin until his death in 1891, and his colleague ] 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 ], 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 ] died, creating a vacant chair. Halle accepted Cantor's suggestion that it be offered to ], ], and ], 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. ] 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 ] rather than on mathematics. Every one of the 52 letters Cantor wrote to ] that year attacked Kronecker. Cantor soon recovered, but a passage from one of these letters is revealing of the damage to his self-confidence: <blockquote>"... 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." </blockquote> 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. ] 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 ]' student ]. On Cantor, Husserl, and ], see Hill and Rosado Haddock (2000). Cantor also wrote on the theological implications of his mathematical work; for instance, he identified the ] with ]. | |||
| Cantor believed that ] wrote the plays attributed to ]. During his 1884 illness, he began an intense study of ] 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 '']'', 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 ]. He did not abandon mathematics completely, lecturing on the paradoxes of set theory (eponymously attributed to ], ], and ] 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 ] in ]. Cantor attended, hoping to meet ], whose newly published '']'' 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 ]. 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 ], 1874-84. He was the first to see that ] come in different sizes, as follows. He first showed that given any set ''A'', the set of all possible subsets of ''A'', called the ] of ''A'', exists. He then proved that the ] of an infinite set ''A'' has a size greater than the size of ''A'' (this fact is now known as ]). Thus there is an infinite hierarchy of sizes of infinite sets, from which springs the ] ] and ]s, and their peculiar arithmetic. His notation for the cardinal numbers was the Hebrew letter ] ('''א''') with a natural number subscript; for the ordinals he employed the Greek letter omega. | |||
| Cantor was the first to appreciate the value of ]s (hereinafter denoted "1-to-1") for set theory. He defined ] and ]s, breaking down the latter into ] and ]s. There exists a 1-to-1 correspondence between any denumerable set and the set of all ]s; all other infinite sets are nondenumerable. He proved that the set of all ]s is denumerable, but that the set of all ]s is not and hence is strictly bigger. The ] of the natural numbers is ]; that of the reals is larger, and is at least ] (the latter being the next smallest cardinal after aleph-null). | |||
| Cantor's first 10 papers were on ], his thesis topic. At the suggestion of ], the Professor at Halle, Cantor turned to ]. Heine proposed that Cantor solve an open problem that had eluded ], ], ], and ] himself: the uniqueness of the representation of a ] by ]. Cantor solved this difficult problem in 1869. Between 1870 and 1872, Cantor published more papers on ], including one defining ]s as convergent sequences of ]s. 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 ]. | |||
| Cantor's 1874 paper, "On a Characteristic Property of All Real Algebraic Numbers", marks the birth of set theory. It was published in ], despite ]'s opposition, thanks to ]'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 ] he first set out in 1891. | |||
| The 1874 paper also showed that the ]s, i.e., the ]s of ] equations with ] ]s, were denumerable. Real numbers that are not algebraic are ]. ] had established the existence of transcendental numbers in 1851. Since Cantor had just shown that the ]s 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 ], 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 ], Cantor submitted another paper to ], which again displeased Kronecker. Cantor wanted to withdraw the paper, but Dedekind persuaded him not to do so; moreover, ] supported its publication. Nevertheless, Cantor never again submitted anything to ]. | |||
| This paper made precise the notion of a 1-to-1 correspondence, and defined ]s as sets which can be put into a 1-to-1 correspondence with the ]. Cantor introduces the notion of "power" (a term he took from ]) 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'''<sup>''n''</sup> has the same power as '''R'''. Moreover, countably many copies of '''R''' have the same power as '''R'''. While he made free use of ] as a concept, he did not write the word "countable" until 1883. Cantor also discussed his thinking about ], stressing that his ] between the unit interval and the unit square was not a continuous one. | |||
| Between ] and 1884, Cantor published a series of six articles in '']'' 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 ] 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 ]s were a systematic extension of the natural numbers. It begins by defining ]ed sets. ] are then introduced as the order types of ]ed sets. Cantor then defines the addition and multiplication of the ] and ]s. 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: <blockquote>"... 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."</blockquote> 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 ] and defined in terms of previously accepted concepts. He also cites ], ], ], ], and ] on infinity. | |||
| Cantor was the first to formulate what later came to be known as the ] 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 ] result by ] and a ] one by ] together imply that the continuum hypothesis can neither be proved nor disproved using standard ] plus the ] (the combination referred to as "ZFC").<ref>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 ].</ref> | |||
| In ], the rich mathematical correspondence between Cantor and Dedekind came to an end. Cantor also began another important correspondence, with ] 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:<blockquote>"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''."</blockquote> 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 ] and ], Cantor published a two-part paper in '']'' under ]'s editorship; these were his last significant papers on set theory. (The English translation is Cantor ].) The first paper begins by defining set, ], etc., in ways that would be largely acceptable now. The ] and ] 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 ]s and ]s. 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. ] had stated this theorem a bit earlier, but his proof, as well as Cantor's, was flawed. ] supplied a correct proof in his ] Ph.D. thesis; hence the name ]. | |||
| Around this time, the set-theoretic ]es began to rear their heads. In an 1897 paper on an unrelated topic, ] set out the first such paradox, the ]: the ] 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 ]. Curiously, Cantor was highly critical of Burali-Forti's paper. | |||
| In ], Cantor discovered his eponymous ]: 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'' (] again). This paradox, together with Burali-Forti's, led Cantor to formulate his concept of ], <sup>'']''</sup> 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 ]es. | |||
| One common view among mathematicians is that these paradoxes, together with ], demonstrate that it is not possible to take a "]", or non-axiomatic, approach to set theory without risking contradiction, and it is certain that they were among the motivations for ] and others to produce ] of set theory. Others note, however, that the paradoxes do not obtain in an informal view motivated by the ], 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.{{Fact|date=January 2007}} | |||
| Cantor's work did attract favorable notice beyond Hilbert's celebrated encomium. In public lectures delivered at the first ], held in Zurich in 1897, ] and ] both expressed their admiration for Cantor's set theory. At that Congress, Cantor also renewed his friendship and correspondence with Dedekind. ] in America also praised Cantor's set theory. In ], Cantor began a correspondence, later published, with his ] admirer and translator ], on the history of ] and on Cantor's religious ideas. | |||
| ==Notes== | |||
| <references /> | |||
| == See also == | |||
| * ] | |||
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| * ] - award by the ] in honor of Georg Cantor. | |||
| * ] which Cantor subscribed to | |||
| == Bibliography == | |||
| Primary literature in English: | |||
| * Cantor, Georg, 1955 (1915). ''Contributions to the Founding of the Theory of Transfinite Numbers''. ], ed. and trans. Dover. | |||
| * Ewald, William B., ed., 1996. ''From ] to ]: 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: | |||
| * Cantor, Georg, 1932. . -88mb! , ed. by ]. Almost everything that Cantor wrote. | |||
| 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. | |||
| * ], 2000. ''The Search for Mathematical Roots: 1870-1940''. Princeton Uni. Press. | |||
| *Hallett, Michael, 1984. ''Cantorian set theory and limitation of size''. Oxford Uni. Press. | |||
| * ], 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. | |||
| *], 2004. ''The Road to Reality''. Alfred A. Knopf. Chpt. 16 reveals how Cantorian thinking intrigues a leading contemporary theoretical physicist. | |||
| * ], 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. | |||
| == External links == | |||
| * O'Connor, J. J., and Robertson, E.F. MacTutor archive. The following are the source for much of this entry: | |||
| ** {{MacTutor Biography|id=Cantor}} | |||
| ** '''' Mainly devoted to Cantor's accomplishment. | |||
| * {{MathGenealogy |id=29561}} | |||
| * | |||
| * | |||
| * Stanford Encyclopedia of Philosophy: by Thomas Jech. | |||
| *''Encyclopedia Britannica'': | |||
| * Grammar school Georg-Cantor Halle(Saale): | |||
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