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Born 3 March 1845 St Petersburg, Russia Died 6 January 1918 Halle, Germany Summary Georg Cantor was a Russian-born mathematician who can be considered as the founder of phối theory và introduced the concept of infinite numbers with his discovery of cardinal numbers. He also advanced the study of trigonometric series.
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Georg Cantor"s father, Georg Waldemar Cantor, was a successful merchant, working as a wholesaling agent in St Petersburg, then later as a broker in the St Petersburg Stock Exchange. Georg Waldemar Cantor was born in Denmark và he was a man with a deep love of culture và the arts. Georg"s mother, Maria Anna Böhm, was Russian and very musical. Certainly Georg inherited considerable musical và artistic talents from his parents being an outstanding violinist. Georg was brought up a Protestant, this being the religion of his father, while Georg"s mother was a Roman Catholic.After early education at trang chủ from a private tutor, Cantor attended primary school in St Petersburg, then in 1856 when he was eleven years old the family moved lớn Germany. However, Cantor Ann. Of Sci.

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27 (1971), 345-391." href="../../Biographies/Cantor/#reference-21">21>:-... Remembered his early years in Russia with great nostalgia and never felt at ease in Germany, although he lived there for the rest of his life và seemingly never wrote in the Russian language, which he must have known.Cantor"s father had poor health và the move to Germany was to lớn find a warmer climate than the harsh winters of St Petersburg. At first they lived in Wiesbaden, where Cantor attended the Gymnasium, then they moved lớn Frankfurt. Cantor studied at the Realschule in Darmstadt where he lived as a boarder. He graduated in 1860 with an outstanding report, which mentioned in particular his exceptional skills in mathematics, in particular trigonometry. After attending the Höhere Gewerbeschule in Darmstadt from 1860 he entered the Polytechnic of Zürich in 1862. The reason Cantor"s father chose to lớn send him to the Höheren Gewerbeschule was that he wanted Cantor to lớn become:-... A shining star in the engineering firmament.However, in 1862 Cantor had sought his father"s permission lớn study mathematics at university & he was overjoyed when eventually his father consented. His studies at Zürich, however, were cut short by the death of his father in June 1863. Cantor moved to the University of Berlin where he became friends with Hermann Schwarz who was a fellow student. Cantor attended lectures by Weierstrass, Kummer and Kronecker. He spent the summer term of 1866 at the University of Göttingen, returning to Berlin to lớn complete his dissertation on number theory De aequationibus secundi gradus indeterminatis Indeterminate equations of the second degree">Ⓣ(Indeterminate equations of the second degree) in 1867.While at Berlin Cantor became much involved with a student Mathematical Society, being president of the Society during 1864-65. He was also part of a small group of young mathematicians who met weekly in a wine house. After receiving his doctorate in 1867, Cantor taught at a girl"s school in Berlin. Then, in 1868, he joined the Schellbach Seminar for mathematics teachers. During this time he worked on his habilitation and, immediately after being appointed khổng lồ Halle in 1869, he presented his thesis, again on number theory, & received his habilitation.At Halle the direction of Cantor"s research turned away from number theory & towards analysis. This was due lớn Heine, one of his senior colleagues at Halle, who challenged Cantor to lớn prove the mở cửa problem on the uniqueness of representation of a function as a trigonometric series. This was a difficult problem which had been unsuccessfully attacked by many mathematicians, including Heine himself as well as Dirichlet, Lipschitz and Riemann. Cantor solved the problem proving uniqueness of the representation by April 1870. He published further papers between 1870 & 1872 dealing with trigonometric series and these all show the influence of Weierstrass"s teaching.Cantor was promoted khổng lồ Extraordinary Professor at Halle in 1872 và in that year he began a friendship with Dedekind whom he had met while on holiday in Switzerland. Cantor published a paper on trigonometric series in 1872 in which he defined irrational numbers in terms of convergent sequences of rational numbers. Dedekind published his definition of the real numbers by "Dedekind cuts" also in 1872 và in this paper Dedekind refers to lớn Cantor"s 1872 paper which Cantor had sent him.In 1873 Cantor proved the rational numbers countable, i.e. They may be placed in one-one correspondence with the natural numbers. He also showed that the algebraic numbers, i.e. The numbers which are roots of polynomial equations with integer coefficients, were countable. However his attempts khổng lồ decide whether the real numbers were countable proved harder. He had proved that the real numbers were not countable by December 1873 & published this in a paper in 1874. It is in this paper that the idea of a one-one correspondence appears for the first time, but it is only implicit in this work.A transcendental number is an irrational number that is not a root of any polynomial equation with integer coefficients. Liouville established in 1851 that transcendental numbers exist. Twenty years later, in this 1874 work, Cantor showed that in a certain sense "almost all" numbers are transcendental by proving that the real numbers were not countable while he had proved that the algebraic numbers were countable.Cantor pressed forward, exchanging letters throughout with Dedekind. The next question he asked himself, in January 1874, was whether the unit square could be mapped into a line of unit length with a đối kháng correspondence of points on each. In a letter to lớn Dedekind dated 5 January 1874 he wrote Dictionary of Scientific Biography (New York 1970-1990). See THIS LINK." href="../../Biographies/Cantor/#reference-1">1>:-Can a surface (say a square that includes the boundary) be uniquely referred to lớn a line (say a straight line segment that includes the over points) so that for every point on the surface there is a corresponding point of the line and, conversely, for every point of the line there is a corresponding point of the surface? I think that answering this question would be no easy job, despite the fact that the answer seems so clearly khổng lồ be "no" that proof appears almost unnecessary.The year 1874 was an important one in Cantor"s personal life. He became engaged lớn Vally Guttmann, a friend of his sister, in the spring of that year. They married on 9 August 1874 & spent their honeymoon in Interlaken in Switzerland where Cantor spent much time in mathematical discussions with Dedekind.Cantor continued lớn correspond with Dedekind, sharing his ideas và seeking Dedekind"s opinions, & he wrote lớn Dedekind in 1877 proving that there was a đơn correspondence of points on the interval <0, 1> and points in ppp-dimensional space. Cantor was surprised at his own discovery & wrote:-I see it, but I don"t believe it!Of course this had implications for geometry & the notion of dimension of a space. A major paper on dimension which Cantor submitted to Crelle"s Journal in 1877 was treated with suspicion by Kronecker, & only published after Dedekind intervened on Cantor"s behalf. Cantor greatly resented Kronecker"s opposition to his work & never submitted any further papers to Crelle"s Journal.The paper on dimension which appeared in Crelle"s Journal in 1878 makes the concepts of solo correspondence precise. The paper discusses denumerable sets, i.e. Those which are in 1-1 correspondence with the natural numbers. It studies sets of equal power, i.e. Those sets which are in 1-1 correspondence with each other. Cantor also discussed the concept of dimension & stressed the fact that his correspondence between the interval <0, 1> & the unit square was not a continuous map.Between 1879 and 1884 Cantor published a series of six papers in Mathematische Annalen designed to lớn provide a basic introduction khổng lồ set theory. Klein may have had a major influence in having Mathematische Annalen published them. However there were a number of problems which occurred during these years which proved difficult for Cantor. Although he had been promoted to lớn a full professor in 1879 on Heine"s recommendation, Cantor had been hoping for a chair at a more prestigious university. His long standing correspondence with Schwarz ended in 1880 as opposition to lớn Cantor"s ideas continued to lớn grow và Schwarz no longer supported the direction that Cantor"s work was going. Then in October 1881 Heine died & a replacement was needed to lớn fill the chair at Halle.Cantor drew up a danh mục of three mathematicians to lớn fill Heine"s chair and the list was approved. It placed Dedekind in first place, followed by Heinrich Weber & finally Mertens. It was certainly a severe blow lớn Cantor when Dedekind declined the offer in the early 1882, và the blow was only made worse by Heinrich Weber and then Mertens declining too. After a new các mục had been drawn up, Wangerin was appointed but he never formed a close relationship with Cantor. The rich mathematical correspondence between Cantor và Dedekind ended later in 1882.Almost the same time as the Cantor-Dedekind correspondence ended, Cantor began another important correspondence with Mittag-Leffler. Soon Cantor was publishing in Mittag-Leffler"s journal Acta Mathematica but his important series of six papers in Mathematische Annalen also continued to lớn appear. The fifth paper in this series Grundlagen einer allgemeinen Mannigfaltigkeitslehre Foundations of a general theory of manifolds">Ⓣ(Foundations of a general theory of manifolds) was also published as a separate monograph and was especially important for a number of reasons. Firstly Cantor realised that his theory of sets was not finding the acceptance that he had hoped và the Grundlagen was designed to reply to the criticisms. Secondly Georg Cantor: His Mathematics và Philosophy of the Infinite (Cambridge, Mass, 1979; reprinted 1990)." href="../../Biographies/Cantor/#reference-3">3>:-The major achievement of the Grundlagen was its presentation of the transfinite numbers as an autonomous and systematic extension of the natural numbers.Cantor himself states quite clearly in the paper that he realises the strength of the opposition khổng lồ his ideas:-... I realise that in this undertaking I place myself in a certain opposition lớn views widely held concerning the mathematical infinite and to opinions frequently defended on the nature of numbers.At the kết thúc of May 1884 Cantor had the first recorded attack of depression. He recovered after a few weeks but now seemed less confident. He wrote khổng lồ Mittag-Leffler at the kết thúc of June Georg Cantor: His Mathematics và Philosophy of the Infinite (Cambridge, Mass, 1979; reprinted 1990)." href="../../Biographies/Cantor/#reference-3">3>:-... 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 khổng lồ the most necessary duty of my lectures; how much happier I would be khổng lồ be scientifically active, if only I had the necessary mental freshness.At one time it was thought that his depression was caused by mathematical worries & as a result of difficulties of his relationship with Kronecker in particular. Recently, however, a better understanding of mental illness has meant that we can now be certain that Cantor"s mathematical worries and his difficult relationships were greatly magnified by his depression but were not its cause (see for example Georg Cantor: His Mathematics & Philosophy of the Infinite (Cambridge, Mass, 1979; reprinted 1990)." href="../../Biographies/Cantor/#reference-3">3> & Ann. Of Sci. 27 (1971), 345-391." href="../../Biographies/Cantor/#reference-21">21>). After this mental illness of 1884 Georg Cantor: His Mathematics and Philosophy of the Infinite (Cambridge, Mass, 1979; reprinted 1990)." href="../../Biographies/Cantor/#reference-3">3>:-... He took a holiday in his favourite Harz mountains và for some reason decided lớn try khổng lồ reconcile himself with Kronecker. Kronecker accepted the gesture, but it must have been difficult for both of them lớn forget their enmities & the philosophical disagreements between them remained unaffected.Mathematical worries began khổng lồ trouble Cantor at this time, in particular he began khổng lồ worry that he could not prove the continuum hypothesis, namely that the order of infinity of the real numbers was the next after that of the natural numbers. In fact he thought he had proved it false, then the next day found his mistake. Again he thought he had proved it true only again lớn quickly find his error.All was not going well in other ways too, for in 1885 Mittag-Leffler persuaded Cantor to withdraw one of his papers from Acta Mathematica when it had reached the proof stage because he thought it "... About one hundred years too soon". Cantor joked about it but was clearly hurt:-Had Mittag-Leffler had his way, I should have khổng lồ wait until the year 1984, which to lớn me seemed too great a demand! ... But of course I never want lớn know anything again about Acta Mathematica.Mittag-Leffler meant this as a kindness but it does show a lack of appreciation of the importance of Cantor"s work. The correspondence between Mittag-Leffler and Cantor all but stopped shortly after this event and the flood of new ideas which had led khổng lồ Cantor"s rapid development of set theory over about 12 years seems lớn have almost stopped.In 1886 Cantor bought a fine new house on Händelstrasse, a street named after the German composer Handel. Before the kết thúc of the year a son was born, completing his family of six children. He turned from the mathematical development of mix theory towards two new directions, firstly discussing the philosophical aspects of his theory with many philosophers (he published these letters in 1888) và secondly taking over after Clebsch"s death his idea of founding the Deutsche Mathematiker-Vereinigung which he achieved in 1890. Cantor chaired the first meeting of the Association in Halle in September 1891, và despite the bitter antagonism between himself & Kronecker, Cantor invited Kronecker to address the first meeting.Kronecker never addressed the meeting, however, since his wife was seriously injured in a climbing accident in the late summer và died shortly afterwards. Cantor was elected president of the Deutsche Mathematiker-Vereinigung at the first meeting và held this post until 1893. He helped khổng lồ organise the meeting of the Association held in Munich in September 1893, but he took ill again before the meeting & could not attend.Cantor published a rather strange paper in 1894 which listed the way that all even numbers up to 1000 could be written as the sum of two primes. Since a verification of Goldbach"s conjecture up khổng lồ 10000 had been done 40 years before, it is likely that this strange paper says more about Cantor"s state of mind than it does about Goldbach"s conjecture.His last major papers on set theory appeared in 1895 and 1897, again in Mathematische Annalen under Klein"s editorship, & are fine surveys of transfinite arithmetic. The rather long gap between the two papers is due to the fact that although Cantor finished writing the second part six months after the first part was published, he hoped lớn include a proof of the continuum hypothesis in the second part. However, it was not to lớn be, but the second paper describes his theory of well-ordered sets and ordinal numbers.In 1897 Cantor attended the first International Congress of Mathematicians in Zürich. In their lectures at the Congress

A history of set theory (Boston, Mass., 1972)." href="../../Biographies/Cantor/#reference-4">4>:-... Hurwitz openly expressed his great admiration of Cantor & proclaimed him as one by whom the theory of functions has been enriched. Jacques Hadamard expressed his opinion that the notions of the theory of sets were known và indispensable instruments.At the Congress Cantor met Dedekind and they renewed their friendship. By the time of the Congress, however, Cantor had discovered the first of the paradoxes in the theory of sets. He discovered the paradoxes while working on his survey papers of 1895 and 1897 & he wrote lớn Hilbert in 1896 explaining the paradox khổng lồ him. Burali-Forti discovered the paradox independently và published it in 1897. Cantor began a correspondence with Dedekind lớn try to lớn understand how lớn solve the problems but recurring bouts of his mental illness forced him to lớn stop writing to Dedekind in 1899.Whenever Cantor suffered from periods of depression he tended khổng lồ turn away from mathematics and turn towards philosophy & his big literary interest which was a belief that Francis Bacon wrote Shakespeare"s plays. For example in his illness of 1884 he had requested that he be allowed to lecture on philosophy instead of mathematics và he had begun his intense study of Elizabethan literature in attempting khổng lồ prove his Bacon-Shakespeare theory. He began lớn publish pamphlets on the literary question in 1896 and 1897. Extra bức xúc was put on Cantor with the death of his mother in October 1896 and the death of his younger brother in January 1899.In October 1899 Cantor applied for, & was granted, leave from teaching for the winter semester of 1899-1900. Then on 16 December 1899 Cantor"s youngest son died. From this time on until the end of his life he fought against the mental illness of depression. He did continue to lớn teach but also had lớn take leave from his teaching for a number of winter semesters, those of 1902-03, 1904-05 và 1907-08. Cantor also spent some time in sanatoria, at the times of the worst attacks of his mental illness, from 1899 onwards. He did continue to lớn work and publish on his Bacon-Shakespeare theory and certainly did not give up mathematics completely. He lectured on the paradoxes of phối theory lớn a meeting of the Deutsche Mathematiker-Vereinigung in September 1903 and he attended the International Congress of Mathematicians at Heidelberg in August 1904.In 1905 Cantor wrote a religious work after returning trang chủ from a spell in hospital. He also corresponded with Jourdain on the history of set theory and his religious tract. After taking leave for much of 1909 on the grounds of his ill health he carried out his university duties for 1910 and 1911. It was in that year that he was delighted to receive an invitation from the University of St Andrews in Scotland khổng lồ attend the 500th anniversary of the founding of the University as a distinguished foreign scholar. The celebrations were 12-15 September 1911 but Ann. Of Sci. 27 (1971), 345-391." href="../../Biographies/Cantor/#reference-21">21>:-During the visit he apparently began to lớn behave eccentrically, talking at great length on the Bacon-Shakespeare question; then he travelled down khổng lồ London for a few days.Cantor had hoped to meet with Russell who had just published the Principia Mathematica. However ill health and the news that his son had taken ill made Cantor return to Germany without seeing Russell. The following year Cantor was awarded the honorary degree of Doctor of Laws by the University of St Andrews but he was too ill to lớn receive the degree in person.Cantor retired in 1913 & spent his final years ill with little food because of the war conditions in Germany. A major sự kiện planned in Halle khổng lồ mark Cantor"s 70th birthday in 1915 had khổng lồ be cancelled because of the war, but a smaller event was held in his home. In June 1917 he entered a sanatorium for the last time and continually wrote khổng lồ his wife asking to be allowed to lớn go home.

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He died of a heart attack.Hilbert described Cantor"s work as:-...the finest product of mathematical genius và one of the supreme achievements of purely intellectual human activity.