George Cantor

Georg Ferdinand Ludwig Philipp Cantor (3 March 1845, Saint Petersburg - 6 January 1918, Halle) was a Russian-born mathematician of Austrian and Jewish descent. He was the inventor with Dedekind of set theory, which is the basis of modern mathematics. Thanks to his daring research on infinite sets he was the first able to formalize the notion of infinity in the form of transfinite numbers (cardinal and ordinal).

He lived through bouts of depression, originally attributed to criticism and his failed attempts to prove the continuum hypothesis, although it is now believed that he suffered from some form of "manic-depressive illness". He died of a heart attack in the Halle psychiatric clinic.

Biography

Cantor towards 1870

He was the son of the merchant Georg Waldemar Cantor and Marie Böhm. His father was born in Copenhagen (Denmark), in a Jewish family, but emigrated to Saint Petersburg and was baptized as a Lutheran. There his son was born and they lived until in 1856 a lung disease prompted the father to move his family to Frankfurt am Main (Germany). All these events caused different nations to claim Georg Cantor as their own, after his death.

Georg Cantor's primary education was initially entrusted to a private teacher, then he went to elementary school in St. Petersburg. When the family moved to Germany, Cantor attended private schools in Frankfurt and Darmstadt until he entered the Wiesbaden Institute at the age of fifteen. He graduated in 1860 with an extraordinary report in which special mention was made of his great talent in mathematics, in particular trigonometry. During his childhood he studied music and was recognized as a prodigy violinist, but eventually gave up music to study mathematics..

Georg Cantor's university studies began in 1862 in Zurich, but the following year, after the death of his father, he went to the University of Berlin where he specialized in mathematics, philosophy and physics, although the young man's interest focused on the first two. There he became friends with Hermann Schwarz, who was his partner. He had as teachers in the field of mathematics Ernst Kummer, Karl Weierstrass and Leopold Kronecker.

While in Berlin, Cantor was part of a small group of young mathematicians who met weekly in a wine bar. After obtaining his doctorate in 1867, Cantor was a teacher at a girls' school in Berlin. Later, in 1868, he joined the Schellbach Seminary for teachers of mathematics. During this stage, he worked on his habilitation and immediately after he obtained a place at the University of Halle in 1869, he presented his work, again on number theory, and received his habilitation.

In Halle he changed the direction of Cantor's research from number theory to mathematical analysis. This was due to Eduard Heine, one of his older colleagues at Halle, who challenged Cantor to prove the open problem on the uniqueness of the representation of a function as a trigonometric series. This was a difficult problem that had been attacked by many mathematicians, including Heine himself as well as Dirichlet, Lipschitz, and Riemann. Cantor solved the problem by proving the uniqueness of the representation in April 1870. Between 1870 and 1872 he published several articles dealing with trigonometric series, which showed Weierstrass's teachings.

In 1872, when he was twenty-seven years old, he became Extraordinary Professor at the University of Halle, beginning his main investigations.

University of Halle, where Georg Cantor worked almost his entire life

His early work with Joseph Fourier's series led to the development of a theory of irrational numbers and in 1874 his first work on set theory appeared. At this time he had an extremely interesting correspondence with Dedekind, in which they discussed Cantor's new ideas and proofs.However, relations between Dedekind and Cantor encountered problems and had great ups and downs.

In 1873 Cantor proved that the rational numbers are countable, that is, they can be put in one-to-one correspondence with the natural numbers. He also proved that algebraic numbers are countable. However, his attempts to decide whether real numbers are countable proved more difficult. In December 1873 he managed to prove that the set of real numbers was not countable and in 1874 he published it in an article. It is in this article that the idea of a one-to-one correspondence appears for the first time, although it only remains implicit in the work.

In his 1874 paper, Cantor proved that in a sense 'almost all' numbers are transcendental, by proving that real numbers are not countable, while algebraic numbers are.

The year 1874 was an important one in Cantor's personal life. He became engaged to Vally Guttmann, a friend of his sister's, in the spring of that year. They were married on August 9, 1874, and honeymooned in Interlaken, Switzerland, where Cantor spent much time in mathematical discussions with Dedekind.

An important article that Cantor sent to the Journal de Crelle in 1877 was treated with suspicion by Kronecker, and was only published after Dedekind intervened on Cantor's behalf. Cantor deeply resented Kronecker's opposition to his work and never submitted another article to the Journal de Crelle.

As for the study of infinite sets, which were considered by his teacher Kronecker to be mathematical madness, Cantor discovered that they do not always have the same size, that is, the same cardinal pattern: for example, the set of rationals is numerable, that is, of the same size as the set of natural numbers, while that of the real numbers is not: there are, therefore, various infinities, one larger than the other.

This fact was a challenge for a spirit as religious as that of Georg Cantor. And the accusations of blasphemy by certain envious colleagues or those who did not understand his discoveries did not help him. In late May 1884 Cantor had his first recorded bout of depression. He recovered after a few weeks but felt more insecure. He suffered from depression, and was repeatedly admitted to psychiatric hospitals.

His depression was once thought to be caused by mathematical concerns such as various paradoxes of set theory, which seemed to invalidate his entire theory (make it inconsistent or contradictory > in the sense that a certain property could be both true and false) and as a result of his relationship with Kronecker, in particular. Recently, however, a better understanding of mental illness has led to claims that Cantor's mathematical concerns and difficult relationships were greatly exaggerated by his depression, but not the main cause.

In addition, he tried for many years to prove the continuum hypothesis, which is known today to be impossible, and which has to be accepted (or rejected) as an additional axiom of the theory. Constructivism will deny this axiom, among other things, by developing a whole alternative mathematical theory to modern mathematics.

In about 1888 he adopted the idea of founding the Deutsche Mathematiker-Vereinigung (German Association of Mathematicians) which he achieved in 1890. Cantor chaired the first meeting of the Association in Halle in September 1891 and Despite his bitter antagonism with Kronecker, Cantor invited him to give a lecture at the first meeting. However, Kronecker never spoke at the meeting, as his wife was in an accident and died shortly after. Cantor was elected president of the Deutsche Mathematiker-Vereinigung at the first meeting, a post he held until 1893.

He began to equate the concept of absolute infinity (which is not conceivable by the human mind) with God, and wrote religious articles on the subject.

He systematized the set ℝ of real numbers and used the concept of an open set. A promoter of research in Russia, along the lines of Euler, he is the author of the "Principle of embedded intervals", creator of certain sets in topology and measure theory.

His last important papers on set theory appeared in 1895 and 1897, again in the Mathematische Annalen now edited by Klein, and they are beautiful accounts of transfinite arithmetic.

Whenever Cantor suffered from periods of depression, he tended to move away from mathematics and turn towards philosophy and his great literary interest, believing that it was Francis Bacon who wrote Shakespeare's plays (see Authorship of Shakespeare's plays). Shakespeare). He began publishing pamphlets on literary issues in 1896 and 1897. The death of his mother in October 1896 and his younger brother in 1899 placed further pressure on Cantor's health.

In October 1899, Cantor applied for and obtained a leave of absence from teaching for the winter semester of 1899-1900. Later, on December 16, 1899, the youngest of his children died. From this moment until the end of his days he fought against his mental illness of depression. He continued to teach, but was absent from teaching for several winter semesters. Cantor spent some seasons in sanitariums, when he suffered the worst attacks of his disease, from 1899 onwards. He continued to work and publish on his Bacon-Shakespeare theory and certainly did not abandon mathematics entirely. He lectured on the paradoxes of set theory at a meeting of the Deutsche Mathematiker-Vereinigung in September 1903 and attended the International Congress of Mathematicians in Heidelberg in August 1904.

Cantor retired in 1913 and spent his last years sick and short on food because of the war in Germany. A major gathering planned in Halle to celebrate Cantor's seventieth birthday in 1915 had to be canceled because of the war, but a smaller celebration was held at his home. In June 1917, he entered a sanitarium for the last time, and continually wrote to his wife, asking her to be allowed to return home.

Georg Cantor died in Halle, Germany, on January 6, 1918 at the age of seventy-two of a heart attack. Today, his work is widely recognized and has been the recipient of several honors.

A bijective function, defined by Cantor in 1874, although he did not call it with that name.

Illustration of Cantor's diagonal argument for the existence of countless sets The sequence below cannot appear anywhere in the infinite list of sequences above.

Plaque in memory of Cantor in his house of St.Petersburg which prays (in Russian): "In this building was born and lived from 1845 to 1854 the great mathematician and creator of the theory of ensembles Georg Cantor".

Acknowledgments

• The lunar crater Cantor bears this name in his memory, honor shared with the historian of mathematics Stiven Cantor (1829–1920).
• The asteroid (16246) Cantor also commemorates its name.

Importance of Cantor

• Infinite numbers
• Cantor Set
• Argument of the Cantor diagonal
• Hotel Infinito
• Constructivism
• Cantor's thought: "I see this, but I don't believe it" by discovering that a segment and a square have equal points.

Cantor's Works

In English

• Cantor, Georg (1955). Philip Jourdain, ed. Contributions to the Founding of the Theory of Transfinite Numbers. New York: Dover. ISBN 978-0-486-60045-1..

In German

• Cantor, Georg (1874). "Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen". Journal für die Reine und Angewandte Mathematik 1874 (77): 258-262. S2CID 199545885. doi:10.1515/crll.1874.77.258.
• Cantor, Georg (1878). «Ein Beitrag zur Mannigfaltigkeitslehre». Journal für die Reine und Angewandte Mathematik 1878 (84): 242-258. doi:10.1515/crll.1878.84.242..
• Georg Cantor (1879). «Ueber unendliche, lineare Punktmannichfaltigkeiten (1)». Mathematische Annalen 15 (1): 1-7. S2CID 179177510. doi:10.1007/bf01444101.
• Georg Cantor (1880). «Ueber unendliche, lineare Punktmannichfaltigkeiten (2)». Mathematische Annalen 17 (3): 355-358. S2CID 179177438. doi:10.1007/bf01446232.
• Georg Cantor (1882). «Ueber unendliche, lineare Punktmannichfaltigkeiten (3)». Mathematische Annalen 20 (1): 113-121. S2CID 177809016. doi:10.1007/bf01443330.
• Georg Cantor (1883). «Ueber unendliche, lineare Punktmannichfaltigkeiten (4)». Mathematische Annalen 21 (1): 51-58. S2CID 179177480. doi:10.1007/bf01442612.
• Georg Cantor (1883). «Ueber unendliche, lineare Punktmannichfaltigkeiten (5)». Mathematische Annalen 21 (4): 545-591. S2CID 121930608. doi:10.1007/bf01446819. Published separately as: Grundlagen einer allgemeinen Mannigfaltigkeitslehre.
• Georg Cantor (1891). «Ueber eine elementare Frage der Mannigfaltigkeitslehre». Jahresbericht der Deutschen Mathematiker-Vereinigung 1: 75-78.
• Cantor, Georg (1895). «Beiträge zur Begründung der transfiniten Mengenlehre (1)». Mathematische Annalen 46 (4): 481-512. S2CID 177801164. doi:10.1007/bf02124929. Archived from the original on 23 April 2014.
• Cantor, Georg (1897). «Beiträge zur Begründung der transfiniten Mengenlehre (2)». Mathematische Annalen 49 (2): 207-246. S2CID 121665994. doi:10.1007/bf01444205.
• Cantor, Georg (1932). Ernst Zermelo, ed. «Gesammelte Abhandlungen mathematischen und philosophischen inhalts». Berlin: Springer. Archived from the original on February 3, 2014.. Almost everything that Cantor wrote. Includes excerpts of his correspondence with Dedekind (p. 443–451) and Fraenkel's Cantor biography (p. 452–483) in the appendix.