Georg Cantor（格奥尔格·康托尔，1845.3.3-1918.1.6)，生于俄国的德国数学家，集合论之父
Biography
富裕商人之家，艺术天赋，出色的小提琴手
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 and he was a man with a deep love of culture and the arts. Georg's mother, Maria Anna Böhm, was Russian and very musical. Certainly Georg inherited considerable musical and 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 home from a private tutor, Cantor attended primary school in St Petersburg, then in 1856 when he was eleven years old the family moved to Germany. However, Cantor:
... 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 and seemingly never wrote in the Russian language, which he must have known.
Cantor's father had poor health and the move to Germany was to 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 to 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 send him to the Höheren Gewerbeschule was that he wanted Cantor to become:-
... a shining star in the engineering firmament.
1862年进入苏黎世联邦理工学院，一年后转入柏林大学学习数学，与施瓦兹（Schwarz）同学，师从魏尔斯特拉斯（Weierstrass），库默尔（Kummer）和克罗内克（Kronecker），主要研究数论，1867年获得博士学位。
However, in 1862 Cantor had sought his father's permission to study mathematics at university and 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 complete his dissertation on number theory De aequationibus secundi gradus indeterminatis 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 to Halle in 1869, he presented his thesis, again on number theory, and received his habilitation.
1869年起执教于哈雷大学。受同事海涅（Heine）的影响，研究方向从数论转到分析。海涅用一个公开的难题挑战康托尔，即证明函数表示成三角级数的唯一性问题。康托尔于1870年4月解决此问题。随后两年康托尔发表更多关于三角级数的论文，这些工作体现出魏尔斯特拉斯对他的影响。
At Halle the direction of Cantor's research turned away from number theory and towards analysis. This was due to Heine, one of his senior colleagues at Halle, who challenged Cantor to prove the open 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 and 1872 dealing with trigonometric series and these all show the influence of Weierstrass's teaching.
Cantor was promoted to Extraordinary Professor at Halle in 1872 and 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 and in this paper Dedekind refers to Cantor's 1872 paper which Cantor had sent him.
1872年，康托尔结识戴德金（Dedekind）。康托尔发表关于三角级数的论文，用有理数的柯西序列来定义无理数。同年，戴德金也发表论文，用“戴德金分割”来定义实数。
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 to decide whether the real numbers were countable proved harder. He had proved that the real numbers were not countable by December 1873 and 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.
1873年，康托尔证明了代数数是可数的。
1874年，康托尔证明了实数是不可数的。论文中首次隐含地出现了“一一对应”的思想。
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.
1851年，刘维尔（Liouville）证明超越数是存在的。20年后，康托尔表明“几乎所有”实数都是超越数。
按：因为实数是不可数的，而代数数是可数的，实数除了代数数就是超越数，所以超越数一定是不可数的，由此得出结论：超越数比代数数多得多！
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 1-1 correspondence of points on each. In a letter to Dedekind dated 5 January 1874 he wrote [1]:-
Can a surface (say a square that includes the boundary) be uniquely referred to a line (say a straight line segment that includes the end 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 to be "no" that proof appears almost unnecessary.
The year 1874 was an important one in Cantor's personal life. He became engaged to Vally Guttmann, a friend of his sister, in the spring of that year. They married on 9 August 1874 and spent their honeymoon in Interlaken in Switzerland where Cantor spent much time in mathematical discussions with Dedekind.
Cantor continued to correspond with Dedekind, sharing his ideas and seeking Dedekind's opinions, and he wrote to Dedekind in 1877 proving that there was a 1-1 correspondence of points on the interval [0, 1] and points in p-dimensional space. Cantor was surprised at his own discovery and wrote:-
I see it, but I don't believe it!
1877年，康托尔证明了闭区间[0,1]上的点与p维空间的点是一一对应的。康托尔对此深感震惊，表示“我看到了，但是我不相信！”
Of course this had implications for geometry and 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, and only published after Dedekind intervened on Cantor's behalf. Cantor greatly resented Kronecker's opposition to his work and never submitted any further papers to Crelle's Journal.
康托尔投给克雷尔杂志的论文受到该杂志的编辑克罗内克的阻挠。后来在戴德金的干预下，康托尔的论文才得以发表。康托尔从此与克罗内克交恶(wù)，决定不再向克雷尔杂志投递论文。
按：克罗内克是一个数学直觉主义者，他认为算术与数学分析都必须以整数为基础，而研究无理数和无穷集合是没有意义的，因为它们根本不存在！而康托尔的思想属于数学逻辑主义，这两派的思想观念可谓针锋相对，势同水火。因此，克罗内克终生都极力反对和攻击康托尔的工作。鉴于当时克罗内克在数学界的地位，其他数学家对康托尔的工作也总是持怀疑态度，这对康托尔的工作造成很大的负面影响。
The paper on dimension which appeared in Crelle's Journal in 1878 makes the concepts of 1-1 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 and stressed the fact that his correspondence between the interval [0, 1] and the unit square was not a continuous map.
在1878年的论文中，康托尔给出了“一一对应”概念的精确表述，并研究了等势集和维度，强调闭区间[0, 1]上的点虽然与单位平方的上的点一一对应，但是对应的函数是不连续的。
Between 1879 and 1884 Cantor published a series of six papers in Mathematische Annalen designed to provide a basic introduction to 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 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 Cantor's ideas continued to grow and Schwarz no longer supported the direction that Cantor's work was going. Then in October 1881 Heine died and a replacement was needed to fill the chair at Halle.
反对康托尔思想的声音不断增长。1880年，施瓦兹不再支持康托尔的数学研究方向，他们的通信因此结束。
Cantor drew up a list of three mathematicians to fill Heine's chair and the list was approved. It placed Dedekind in first place, followed by Heinrich Weber and finally Mertens. It was certainly a severe blow to Cantor when Dedekind declined the offer in the early 1882, and the blow was only made worse by Heinrich Weber and then Mertens declining too. After a new list had been drawn up, Wangerin was appointed but he never formed a close relationship with Cantor. The rich mathematical correspondence between Cantor and Dedekind ended later in 1882.
1882年，康托尔与戴德金富有成果的数学通信中止。与此同时，康托尔开始与另一位数学家米塔-列夫勒（Mittag-Leffler）建立通信。
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 appear. The fifth paper in this series Grundlagen einer allgemeinen Mannigfaltigkeitslehre 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 and the Grundlagen was designed to reply to the criticisms. Secondly:-
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 to his ideas:-
... I realise 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.
1884年，康托尔有记录的首次抑郁症发作。
At the end of May 1884 Cantor had the first recorded attack of depression. He recovered after a few weeks but now seemed less confident. He wrote to Mittag-Leffler at the end of June :-
... 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.
At one time it was thought that his depression was caused by mathematical worries and 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. After this mental illness of 1884:-
... he took a holiday in his favourite Harz mountains and for some reason decided to try to reconcile himself with Kronecker. Kronecker accepted the gesture, but it must have been difficult for both of them to forget their enmities and the philosophical disagreements between them remained unaffected.
曾经认为，康托尔的抑郁症是由于对数学的忧虑引起的，特别是他与克罗内克的敌对关系。然而，对精神病的最新研究意味着我们现在可以肯定，康托尔的数学忧虑和糟糕的人际关系被他的抑郁症大大增强了，而不是他得抑郁症的原因。
Mathematical worries began to trouble Cantor at this time, in particular he began to 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 to quickly find his error.
康托尔开始担心自己证明不了连续统假设（continuum hypothesis）。
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 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.
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 to Cantor's rapid development of set theory over about 12 years seems to have almost stopped.
米塔-列夫勒说服康托尔撤回了一篇已到达证明阶段的论文。此举虽然出于善意，但也体现出米塔-列夫勒对康托尔的工作不够欣赏。康托尔随即断绝了与米塔-列夫勒的信件往来，引领康托尔在集合论领域突飞猛进达12年之久的新思想的涌流似乎已经停止了。
In 1886 Cantor bought a fine new house on Händelstrasse, a street named after the German composer Handel. Before the end of the year a son was born, completing his family of six children. He turned from the mathematical development of set theory towards two new directions, firstly discussing the philosophical aspects of his theory with many philosophers (he published these letters in 1888) and 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, and despite the bitter antagonism between himself and 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 and died shortly afterwards. Cantor was elected president of the Deutsche Mathematiker-Vereinigung at the first meeting and held this post until 1893. He helped to organise the meeting of the Association held in Munich in September 1893, but he took ill again before the meeting and 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 to 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, and 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 to include a proof of the continuum hypothesis in the second part. However, it was not to be, but the second paper describes his theory of well-ordered sets and ordinal numbers.
康托尔关于集合论的最后两篇论文分别于1895和1897年发表于《数学年鉴》上，是对超限算术的详细考察。康托尔企图在第二篇论文中加入对连续统假设的证明，但没能做到，不过他描述了良序集和序数的理论。
In 1897 Cantor attended the first International Congress of Mathematicians in Zürich. In their lectures at the Congress:-
... Hurwitz openly expressed his great admiration of Cantor and 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 and 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 and he wrote to Hilbert in 1896 explaining the paradox to him. Burali-Forti discovered the paradox independently and published it in 1897. Cantor began a correspondence with Dedekind to try to understand how to solve the problems but recurring bouts of his mental illness forced him to stop writing to Dedekind in 1899.
1897年，康托尔出席了在苏黎世举行的第一届国际数学家大会。胡尔维茨（Hurwitz）和哈达马（Hadamard）公开表达对康托尔集合论的赞赏，康托尔的工作终于得到主流数学界的认可。康托尔与戴德金恢复友谊。康托尔首次发现集合论中的悖论，并试图与戴德金合作解决此悖论，然而反复发作的精神病迫使康托尔停止给戴德金写信。
Whenever Cantor suffered from periods of depression he tended to turn away from mathematics and turn towards philosophy and 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 and he had begun his intense study of Elizabethan literature in attempting to prove his Bacon-Shakespeare theory. He began to publish pamphlets on the literary question in 1896 and 1897. Extra stress was put on Cantor with the death of his mother in October 1896 and the death of his younger brother in January 1899.
每当康托尔罹患抑郁症时，他往往会远离数学，转向哲学和他最大的文学兴趣，即证明弗朗西斯·培根写了莎士比亚的戏剧。
In October 1899 Cantor applied for, and 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 teach but also had to take leave from his teaching for a number of winter semesters, those of 1902-03, 1904-05 and 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 work and publish on his Bacon-Shakespeare theory and certainly did not give up mathematics completely. He lectured on the paradoxes of set theory to a meeting of the Deutsche Mathematiker-Vereinigung in September 1903 and he attended the International Congress of Mathematicians at Heidelberg in August 1904.
1899年12月，康托尔的小儿子去世。从这个时候开始，直至生命的尽头，康托尔都在与抑郁症作持续的斗争。他继续教书，但经常请假，疾病严重时会在疗养院待一阵。他继续研究并发表他的培根-莎士比亚理论，也没有完全放弃数学，为听众讲授集合论的悖论，并于1904年8月出席了在海德堡举行的国际数学家大会。
In 1905 Cantor wrote a religious work after returning home 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 to attend the 500th anniversary of the founding of the University as a distinguished foreign scholar. The celebrations were 12-15 September 1911 but:-
During the visit he apparently began to behave eccentrically, talking at great length on the Bacon-Shakespeare question; then he travelled down to 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 receive the degree in person.
康托尔受邀参加苏格兰圣安德鲁斯大学于1911年9月举行的该大学成立500周年的庆祝活动。在访问期间，康托尔表现古怪，长篇大论地谈论培根-莎士比亚问题。然后他去了伦敦几天，希望与刚刚出版了《数学原理》的罗素会面。然而，健康状况不佳，以及他儿子生病的消息让康托尔赶回德国，没有看到罗素。第二年，康托尔被圣安德鲁斯大学授予荣誉法学博士学位，但他病得很重，无法亲自领取学位。
Cantor retired in 1913 and spent his final years ill with little food because of the war conditions in Germany. A major event planned in Halle to mark Cantor's 70th birthday in 1915 had to 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 to his wife asking to be allowed to go home. He died of a heart attack.
1913年康托尔退休。由于第一次世界大战所造成的食物短缺，康托尔的最后几年都是在疾病和饥饿中度过。
1917年6月，康托尔最后一次进入疗养院，并不断写信给妻子请求回家。
1918年1月6日，康托尔死于心脏病发作。
Hilbert described Cantor's work as:-
...the finest product of mathematical genius and one of the supreme achievements of purely intellectual human activity.
希尔伯特高度评价康托尔的工作：
...数学天才最优秀的作品，也是人类纯粹智力活动的最高成就之一。
本文传记内容来自https://mathshistory.st-andrews.ac.uk/Biographies/Cantor/
添加中文概要和注释，以供学习和参考。
