### 数学家——康托尔（Georg Ferdinand Ludwig Philipp Cantor）

<FONT color=#ff0000> </FONT><H1><FONT color=#ff0000>Georg Ferdinand Ludwig Philipp Cantor</FONT></H1>

<H3>Born:<FONT color=green> 3 March 1845 in St Petersburg, Russia

</FONT>Died:<FONT color=purple> 6 Jan 1918 in Halle, Germany</FONT></H3>

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<P>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. </P>

<P>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 :- </P>

<P>... 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. </P>

<P>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 Zurich in 1862. The reason Cantor's father chose to send him to the Höheren Gewerbeschule was that he wanted Cantor to became:- </P>

<P>... a shining star in the engineering firmament. </P>

<P>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 Zurich, 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 Herman 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. </P>

<P>While at Berlin Cantor became much involved with the 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. </P>

<P>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. </P>

<P>Cantor was promoted to Extraordinary Professor at Halle in 1872 and in that year he began a friendship with Dedekind who 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. </P>

<P>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. </P>

<P>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. </P>

<P>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 :- </P>

<P>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. </P>

<P>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. </P>

<P>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 and points in p-dimensional space. Cantor was surprised at his own discovery and wrote:- </P>

<P>I see it, but I don't believe it! </P>

<P>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. </P>

<P>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 and the unit square was not a continuous map. </P>

<P>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. </P>

<P>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. </P>

<P>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 :- </P>

<P>The major achievement of the Grundlagen was its presentation of the transfinite numbers as an autonomous and systematic extension of the natural numbers. </P>

<P>Cantor himself states quite clearly in the paper that he realises the strength of the opposition to his ideas:- </P>

<P>... 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. </P>

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</P> <P>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 :- </P><P>... 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. </P><P>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 (see for example and ). After this mental illness of 1884 :- </P><P>... 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. </P><P>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. </P><P>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:- </P><P>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. </P><P>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. </P><P>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. </P><P>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. </P><P>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. </P><P>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. </P><P>In 1897 Cantor attended the first International Congress of Mathematicians in Zurich. In their lectures at the Congress :- </P><P>... 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. </P><P>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. </P><P>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. </P><P>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. </P><P>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 :- </P><P>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. </P><P>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. </P><P>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 70 th 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. </P><P>Hilbert described Cantor's work as:- </P><P>...the finest product of mathematical genius and one of the supreme achievements of purely intellectual human activity. </P><P>

Article by: J J O'Connor and E F Robertson</P> Georg Cantor put forth the modern theory on infinite sets that revolutionized almost every mathematics field. However, his new ideas also created many dissenters and made him one of the most assailed mathematicians in history. <P>Georg Ferdinand Ludwig Philipp Cantor was born in St. Petersburg, Russia, on March 3, 1845. Georg's background was very diverse. His father was a Danish Jewish merchant that had converted to Protestantism while his mother was a Danish Roman Catholic. The family stayed in Russia for eleven years until the father's ailing health forced them to move to the more acceptable environment of Frankfurt, Germany, the country Georg would call home for the rest of his life. <P>All the Cantor children displayed an early artistic talent with Georg excelling in mathematics. His father, the eternal pragmatic, saw this gift and tried to push his son into the more profitable but less challenging field of engineering. In one of his letters, he pressed upon his son that his entire family and God Himself were expecting him to become a "shining star" as an engineer. Georg was not at all happy about this idea but he lacked the assertiveness to stand up to his father and relented. However, after several years of training, he became so fed up with the idea that he mustered up the courage to beg his father to become a mathematician. Finally, just before entering college, his father let Georg study mathematics. The son accepted his decision with the same submission that he had before, thanking his father for the fact that he would not "displease him." <P>In 1862, Georg Cantor entered the University of Zurich only to transfer the next year to the University of Berlin after his father's death. At Berlin he studied mathematics, philosophy and physics. There he studied under some of the greatest mathematicians of the day including Kronecker and <a href="http://www.shu.edu/projects/reals/history/weierstr.html" target="_blank" >Weierstrass</A>. After receiving his doctorate in 1867, he was unable to find good employment and was forced to accept a position as an unpaid lecturer and later as an assistant professor at the backwater University of Halle. In 1874, he married and eventually had six children. <P>It was in that same year of 1874 that Cantor published his first paper on the theory of sets. While studying a problem in analysis, he had dug deeply into its "foundations," especially sets and infinite sets. What he found flabbergasted him so much that he wrote to a friend: "I see it but I don't believe it.". In a series of papers from 1874 to 1897, he was able to prove among other things that the set of integers had an equal number of members as the set of even numbers, squares, cubes, and roots to equations; that the number of points in a line segment is equal to the number of points in an infinite line, a plane and all mathematical space; and that the number of transcendental numbers, values such as <img src="http://www.shu.edu/projects/reals/symbols/pi.gif">and <I>e</I> that can never be the solution to any algebraic equation, were much larger than the number of integers. Interestingly, the Jesuits also used his theory to "prove" the existence of God and the Holy Trinity. However, Cantor, who was also an excellent theologian, quickly distanced himself away from such "proofs." <P>Before in mathematics, infinity had been a taboo subject. Previously, Gauss had stated that infinity should only be used as "a way of speaking" and not as a mathematical value. Most mathematicians followed his advice and stayed away. However, Cantor would not leave it alone. He considered infinite sets not as merely going on forever but as completed entities, that is having an actual though infinite number of members. He called these actual infinite numbers transfinite numbers. By considering the infinite sets with a transfinite number of members, Cantor was able to come up his amazing discoveries. For his work, he was promoted to full professorship in 1879. <P>However, his new ideas also gained him numerous enemies. Many mathematicians just would not accept his groundbreaking ideas that shattered their safe world of mathematics. One great mathematician, Henri Poincare expressed his disapproval, stating that Cantor's set theory would be considered by future generations as "a disease from which one has recovered." However, he was kinder than another critic, Leopold Kronecker. Kronecker was a firm believer that the only numbers were integers and that negatives, fractions, imaginary and especially irrational numbers had no business in mathematics. He simply could not handle "actual infinity." Using his prestige as a professor at the University of Berlin, he did all he could to suppress Cantor's ideas and ruin his life. Among other things, he delayed or suppressed completely Cantor's and his followers' publications, raged both written and verbal personal attacks against him, belittled his ideas in front of his students and blocked Cantor's life ambition of gaining a position at the prestigious University of Berlin. <P>Not all mathematicians were antagonistic to Cantor's ideas. Some greats such as Mittag-Leffler, <a href="http://www.shu.edu/projects/reals/history/weierstr.html" target="_blank" >Karl Weierstrass</A>, and long-time friend Richard Dedekind supported his ideas and attacked Kronecker's actions. However, it was not enough. Like with his father before, Cantor simply could not handle it. Stuck in a third-rate institution, stripped of well-deserved recognition for his work and under constant attack by Kronecker, he suffered the first of many nervous breakdowns in 1884. The rest of his life was spent in and out of mental institutions and his work nearly ceased completely. Much too late for him to really enjoy it, his theory finally began to gain recognition by the turn of the century. In 1904, he was awarded a medal by the Royal Society of London and was made a member of both the London Mathematical Society and the Society of Sciences in Gottingen. He died in a mental institution on January 6, 1918. <P>Today, Cantor's work is widely accepted by the mathematical community. His theory on infinite sets reset the foundation of nearly every mathematical field and brought mathematics to its modern form. In addition, his work has helped to explain <a href="http://www.shu.edu/projects/reals/history/zeno.html" target="_blank" >Zeno's</A> paradoxes that plagued mathematics for 2500 years. However, his theory also has led to many new questions, especially about set theory, that should keep mathematicians busy for centuries. </P><P><B>Sources</B> <P><B></B><UL><LI>Bell, E.T. <I>Men of Mathematics.</I> New York: Simon and Schuster, Inc., 1937. <LI>Muir, Jane. <I>Of Men and Numbers. </I>New York: Dodd, Mead & Company, 1962. <LI>Wilson, John H. "Cantor." <I>Encyclopedia of World Biography.</I> New York: McGraw-Hill, Inc., 1973. vol. 2, pp. 356-357. </LI></UL><P><FONT size=-1>Historical information compiled by <B>Paul Golba</B></FONT> </P> <P>康 托 尔：集合论的创始人

康托尔(1845—1918)，德国著名的数学家，集合论的创始人。他的数学理论对19世纪末至20世纪初的世界数学产生了重大影响。

康托尔1845年3月3日出生在俄国彼得堡的一个丹麦犹太血统的富商家庭中。1856年康托尔随父母迁居德国法兰克福。宗教是康托尔家庭的重要组成部分，康托尔的父亲原是犹太教徒，后来皈依了新教，而他母亲则生来就是罗马天主教徒。由于家庭中这种混合的宗教信仰，使童年时代的康托尔对神学产生了一种终生的兴趣，特别是那些与无穷性质有关的神学问题对康托尔研究数学产生了很大的影响。从幼年时代，康托尔就表现出了对数学的强烈兴趣和卓越天才，他自己也决心成为一名数学家。康托尔的家庭还显示了明显的艺术素质。在他的家庭中，音乐受到特别的尊崇。康托尔本人是一个很不错的素描画家，他留给后人一些很能表现他天赋的铅笔画。实际上康托尔也具备了艺术家的才能。

在法兰克福，康托尔曾在几家私立学校学习，15岁进入威斯巴登大学预科学校。1862年康托尔去苏黎世上大学，第二年转入柏林大学学习数学、哲学和物理学。1867年他在柏林大学获博士学位。1869年成为哈勒大学的讲师，1872—1905年一直任该校的教授。在上大学期间，这位敏感的年轻人特别擅长数学，他受当时几位数学大师的影响，从1862年他就做出了准备献身数学的决定。最初他父亲并不希望他从事纯粹科学研究，而是力促他学工。但是康托尔越来越多地受到数学的吸引。尽管他父亲对他的这一选择是否明智曾表示怀疑，但仍以极大的热情支持儿子的事业。由于康托尔一开始就具有献身数学的信念，这就为他创立集合论，取得数学史上的成就奠定了基础。

我们可以说全部数学都是由“无限”这一概念衍生而来的。从古希腊时代起，人们就开始探求无限的奥秘。而到19世纪末，真正对无限问题作出最深刻的数学研究的就是康托尔。他所创立的集合论被誉为20世纪最伟大的数学创造。它不仅影响了现代数学，也深深地影响了现代哲学和逻辑。在当时，他所从事的关于连续性和无穷的研究从根本上背离了数学中关于无穷的传统解释，因此引起了激烈的争论甚至是严厉的遣责。康托尔的老师克朗内克几乎从一开始就反对康托尔的思想，粗暴地攻击他达10年之久。康托尔不顾众多数学家和哲学家的反对坚定地捍卫了他的集合论观点。真理是不可战胜的，集合论最终获得了数学界的承认，集合论已成为整个数学大厦的基础。康托尔这种坚定的信念使他义无反顾地走向数学家之路，并因此成为世纪之交的最伟大的数学家之一。康托尔本人由于长期被人敌视反对，也由于研究数学问题的巨大压力，再加上生活上的失意，他患上了精神分裂症，并几度复发。康托尔生命的最后几年几乎是在疾病的折磨下度过的。他一直希望能进柏林大学任教，但由于世人对他的理论的非议，终究未能如愿。康托尔的一生是很困苦的。1918年1月6日，他在因精神病发作再次住院期间，不幸逝世。对于一位伟大的数学家来说，这真是一个令人悲痛的结局。

康托尔的集合论是现代数学的基础，他严密证明了“无穷”并不是铁板一块的不可分割的概念。他认为无穷也可以比较大小，他从伽利略的一个悖论(整数与偶数可以一一对应，从而认为他们同样多；但偶数又是整数的一部分，这样一来部分可以等于整体)中吸取了合理的内核，建立了一一对应的概念，从而提出了比较无穷集合大小的方法：如果在两个集合的元素之间可以建立某种一对一的对应关系，则这两个集合就定义为等势的(大小是一样的)。这是测量无穷集合的一把尺子，用这尺子康托尔证明了正偶数集合与全体正整数集合是等势的。因为这两个集合的元素间的配对可以永远进行下去而不漏掉任一元素。他用对角线的方法证明了全体有理数的集合与整数集合是可以配对的。这种集合叫做“可数集合”。而整数与直线上的点集(即实数)之间不可能存在这样一一对应的关系，实数集是不可数的。

康托尔用有理“基本序列”来定义无理数。他认为任何一个无理数都可用一个无穷有理数序列来表示。例如√2可以用1，1．4，1．414…这样一个无穷有理数序列来表示。康托尔是实数理论的创建人之一，实数理论的建立是分析学逻辑基础发展史上的重大事件。康托尔还提出了著名的“连续统假设”，定义了“超穷序数”的概念。

康托尔对数学的最主要贡献是创立了集合论。因此他也成为数学史上最富有想像力和最有争议的人物之一。尽管康托尔一生对集合论的研究始终面对重重困境，但他却从没有对自己工作的价值丧失信心。他在谈到争议很大的有关无穷的观点时写道：“我认为是唯一正确的这种观点，只有极少数人赞同，虽然我可能是历史上明确持有这种观点的第一人，但就其全部逻辑结果而言，我确信我将不是最后一人。”康托尔对数学的独特贡献是他以特殊提问的方式开辟了广阔的研究领域。他所提出的问题，一部分被他自己解决，一部分被他的后继者解决，一些尚未解决的问题则引导和支配着数学的某一发展方向。历史对康托尔的工作给出了公正的评价，集合论在本世纪已渗透到数学的各个分支，成为分析理论、测度论、拓扑学及数理科学中不可少的工具。作为集合论创始人的康托尔，他的名字连同他的不可磨灭的伟大功绩，将永远镌刻在人类科学发展的里程碑上。

</P><P>转载自中华一家人之人物传记</P>

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