Quick InfoBorn 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.
BiographyGeorg 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
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.