Georg Cantor

Biography

Childhood and studies

Georg Cantor was born on March 3rd, 1845 with Saint-Pétersbourg. His/her father is Georg Waldemar Cantor, a business man Danish and broker with the purse of St Pétersbourg; it is a enthusiastic Luthérien. His/her mother is Maria Anna Böhm, a woman of Austrian nationality , resulting from a family of musicians. Catholic of birth, it converts with Protestantism at the time of its marriage.

Georg Cantor, was high in the faith Lutheran, faith which it preserved all its life. Violin remarkable ist, it had inherited the artistic and musical talent of its maternal family. When the father of Cantor fell sick, the family sought winters less icy than in Saint Pétersbourg. She went to settle in Germany in 1856, initially with Wiesbaden, then with Frankfurt. In 1860, Cantor obtained a Diplôme with distinction with the Realschule of Darmstadt, where one noticed his exceptional performances in Mathématiques, in particular in Trigonométrie. In 1862, according to the wish of his/her father, Cantor integrated the federal Polytechnic school of Zurich and started higher learning in mathematics.

In 1863, with died of his/her father, Cantor preferred to continue his studies with the university of Berlin. It followed the courses of Weierstrass, Kummer and Kronecker. It bound friendship with Hermann Schwarz, then student. It spent the summer to the university of Göttingen, which became thereafter a great mathematical research center. In 1867, Berlin granted to him the title of Philosophiæ doctor for a thesis relating to the Théorie of the numbers, Of aequationibus secundi gradus indeterminatis .

Beginning of career

After having taught during one year in a school of girls with Berlin, Cantor settled in 1870 by the university of Market, where it made all its career. It obtained the necessary Habilitation thanks to its thesis. Cantor was promoted part-time lecturer in 1872.

In 1874, Cantor married Vally Guttmann. They will have six children, the last having incipient in 1886. Cantor was able to provide for the needs for the family in spite of modest academic wages thanks to the heritage of his/her father.

Heine had raised the question of the unicity of the writing of a periodic function of a real variable like sum series of goniometrical functions. Interressé by this problem, Cantor obtained unicity for the continuous functions. In 1872, Cantor gave an importance to the whole of the points of discontinuity of the function, which presupposes to handle infinite units. Thus Cantor started to wonder about the infinite one.

In 1874, Cantor published its first work on the subject in the Journal of Crelle. It gave the first proof that the whole of realities is not countable.

Hostilities between Cantor and Kroneker

In 1877, Cantor subjected its last article to the Newspaper of Crelle, in which it showed that a surface is in bijection with a real line. Kronecker, considered mathematician, was in disagreement with what founded work of Cantor in set theory. Kronecker, perceived today like a pioneer of the constructivism, did not think that one can consider an infinite whole like an entity: God created the integers; the remainder is the work of the man . Kronecker also thought that a proof of existence of a mathematical object satisfying certain properties was to give an explicit definition of such an object.

The thought of Kronecker is not contrary with the position of the mathematicians of its time.

In 1879, Cantor obtained a pulpit at the university of the Market. To reach more the high ranking at age the 34 years was a notable performance, but Cantor would have preferred to have a pulpit in a more prestigious university, in particular in Berlin where the best German university was. However, Kronecker was with the head of the sector of mathematics in Berlin until its death in 1891 and it did not wish to have Cantor as colleague.

In 1881, Edouard Heine, a colleague of Cantor of the university of Market dies, leaving an unoccupied pulpit. Market accepted the proposal of Cantor, according to which the pulpit could be allotted successively to Dedekind, Heinrich Weber, and Franz Mertens, but each one declined the pulpit. The lack of interest on behalf of Dedekind is surprising, since he taught in a school of engineer of low level and carried a heavy administrative load. This episode is revealing lack of reputation of the German department of mathematics of Market. Albert Wangerin was finally named, but never approached Cantor.

Depression

In 1884, Cantor suffered from its first access of depression. According to Eric Beautiful Temple, his crisis would come from a feeling of insecurity coming from a conflict freudien with his/her father. According to Joseph Dauben, it is more probable than this crisis is caused by the attacks of Kronecker.

This emotional crisis carried out it to give courses of Philosophie, rather than of Mathématiques. Each of the 52 letters that Cantor wrote with Mittag-Leffler during this year attacked Kronecker. Cantor recovered quickly, but a passage of the one of its letters reveals a loss of confidence in itself:

“… I do not know when I can turn over to the continuation of my scientific work. For the moment, I can absolutely nothing make for that, and that limits to me to give bare essential my course; O how much would like I to be active in sciences, so only I had mental promptness necessary. ”

Although it produced some work of value after 1884, it did not find the high level of its remarkable productions between 1874 and 1884. He proposed a reconciliation with Kronecker, which accepted without reserves. Despite everything, the philosophical dissension and the difficulties which divided them persisted. It is said sometimes that the recurring depressive accesses of Cantor were started by the opposition of Kronecker to its work. The relational difficulties of Cantor, the disorders of its mathematical production, were certainly exacerbated by its depression, but one can doubt that they are the cause.

In 1888, it published its correspondences with several philosophers about the philosophical implications of its Set theory. Edmund Husserl was one of his/her colleagues with Halle and a friend, between 1886 and 1901. The reputation of Husserl was done in philosophy, but at the time it prepared a doctorate of mathematics directed by Leo Königsberger, a student of Weierstrass. Cantor also wrote on the theological implications of its work in mathematics; it would have identified the “infinite absolute”, the infinite one of a clean class like that of all the cardinal or all the ordinal , with God.

Cantor believed that Francis Bacon had written the parts allotted to Shakespeare. For its period of disease, in 1884, it started a thorough study of the literature élisabéthaine, with an aim of proving this thesis. It published finally two lampoons, in 1896 and 1897, which exposed its sights.

In 1890, Cantor takes part in the foundation of the Deutsche Mathematiker-Vereinigung . It organizes of it the first meeting with Halle in 1891 and president is elected by it. That shows clearly that the attitude of Kronecker was not fatal with its reputation. In spite of the animosity which it tested for Kronecker, Cantor invited it to speak at this meeting; Kronecker was unable, because his wife was at this time with the article of death.

After the younger death of sound wire, in 1899, Cantor suffers from a chronic depression, which followed it until the end of its life, and for which it was exempted of teaching and was on several occasions locked up in a repetitive way in sanatorium. It did not give up mathematics completely: it gives courses on the paradoxes of the Set theory (allotted in manner éponyme to Burali-Forti, Russell, and Cantor itself) at the time of a conference of the Deutsche Mathematiker-Vereinigung , in 1903, and attends the international Congress of the Mathematicians with Heidelberg in 1904.

In 1903, he is prize winner of the Médaille Sylvester of the Royal Society.

Cantor took its retirement in 1913; he suffered from poverty and even from hunger during the First World War. The public celebration of its 70 years was cancelled because of the war. He died in the hospital where he had spent the last year of his life.

Work

Cantor was the initiator of the Set theory, starting from 1874. Some, as Galileo had already noticed that an infinite unit, like the squares of the integers, could be put in correspondence with an infinite unit the container strictly, in fact all the entireties. There is in a certain way “as much” of squares of integers that integers. Cantor is the first to give a precise direction to this remark, using the concept of Bijection which it introduces (under another name) on the occasion, then to systematize it. For example Cantor shows that there is as many rational numbers, those represented by fractions, that integers. Cantor goes further and discovers that there are several infinite, with the direction where they cannot be put in correspondence between them by a Bijection: it shows into 1874 which the real line contains more irrational numbers (“much more”) that rational numbers, but also that of algebraic numbers (solutions of polynomial equations to whole coefficients).

Cantor introduces the overall Dénombrable or infinite concept Dénombrable: a unit which can be put in bijection with the integers, i.e. one can, in a certain way, to number all its elements by entireties (without repetition but it is not essential). It shows that the whole of the relative integers, the number rational, and the algebraic numbers is all countable, but the whole of the real numbers is not it.

It gives a proof elegant and very short of this last result in 1891, where it uses what is known maintaining like the diagonal argument of Cantor, and which since was very much used, in particular in Logique mathematics and Théorie of the calculability. It uses this argument to show that the set of all the subsets of a unit has , called Ensemble of the parts of has , has strictly more elements than has , even if has is infinite, i.e. these two units cannot be put in bijection. This proposal is today called Théorème of Cantor. It has as a consequence, the existence of a strict hierarchy of infinite units.

To study the infinite one, Cantor introduces two concepts of numbers and their arithmetic particular (nap, produced…). The first is that of Cardinal number, which characterizes a class of units being able to be put in bijection. More the infinite cardinal small number is that of the natural entireties, the countable one. The cardinal of the real numbers, or way equivalent of the whole of the subsets of the natural entireties, is the Puissance of continuous the. Cantor introduces the Hebraic letter א (Aleph) to appoint the cardinals, notation always of use today. Thus the cardinal of the whole of the natural entireties is noted ℵ₀ (to read aleph zero). The power of continuous is a cardinal inevitably higher or equal to the cardinal following the countable one immediately, that one notes ℵ₁. Cantor supposed that it was ℵ₁, it is the Hypothèse of continuous the.

The second is that of ordinal Nombre, which generalizes the entireties as they are ordered. It uses for that the concept of Bon order, which it introduces in 1883. Cantor notes the ordinal ones with Greek letters, smallest ordinal infinite, that of the whole of the natural entireties, is noted ω₀ (today simply ω). For the cardinal numbers it uses in fact ordinal in index of the letter ℵ.

The first ten productions of Cantor related to the Théorie of the numbers, the subject of its thesis. According to the suggestion of the professor Edouard Heine, Cantor is directed towards the Analyze. Heine proposes in Cantor to solve a problem from which the solution escaped Dirichlet, Lipschitz, Bernhard Riemann and Edouard Heine itself: the unicity of the representation of a function by a Fourier series. Cantor résou this difficult problem in 1869. Between 1870 and 1872, Cantor publishes other work on the trigonometrical series, including a definition of the irrational numbers like convergent continuations of rational numbers. It is one of both usual constructions of the real numbers. Dedekind, with which Cantor bound of friendship in 1872, quotes this work in the publication containing its own construction of the real numbers, from what one now calls the cuts of Dedekind.

Publication of Cantor of 1874, " On an index property of all realities algébriques" , which is that marked the birth of its Set theory. It was published in the Journal of Crelle, in spite of the opposition of Kronecker and thanks to the support of Dedekind. It is in this one which it proves that the real numbers are not countable, by using a proof more complex than the remarkably elegant diagonal argument, celebrates rightly, than it establishes in 1891.

Publication of 1874 watch whereas the algebraic numbers, i.e. the polynomial roots of equation S with whole coefficients, are countable. The real numbers which are not algebraic are transcendent. Liouville had established the existence of transcendent numbers in 1851. Cantor having shown that the whole of the real numbers is not countable, and that the union of two countable units must be countable, and as a real number is either algebraic, or transcendent, the whole of the transcendent numbers is not countable. There are in fact as many transcendent numbers of real numbers. One from of deduced a very simple proof of a theorem, which had in Liouville, according to which there is an infinity of transcendent numbers in each interval.

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