Georg Cantor (photo below) is a German mathematician who created set theory and introduced the concept of transfinite numbers, infinitely large, but different from each other. He also gave a definition of ordinal and cardinal numbers and created their arithmetic.
Georg Cantor: A Short Biography
Born in St. Petersburg 03.03.1845. His father was the Dane of the Protestant faith Georg-Waldemar Kantor, who was engaged in trade, including on the stock exchange. His mother, Maria Bem, was Catholic and came from a family of prominent musicians. When Georg's father fell ill in 1856, the family moved first to Wiesbaden and then to Frankfurt in search of a milder climate. The boy's mathematical talents appeared even before his 15th birthday while studying at private schools and gymnasiums in Darmstadt and Wiesbaden. In the end, George Cantor convinced his father that he was determined to become a mathematician, not an engineer.
After a short training at the University of Zurich in 1863, Kantor transferred to Berlin University to study physics, philosophy and mathematics. There he was taught:
- Karl Theodor Weierstrass, whose specialization in analysis, probably had the greatest impact on George;
- Ernst Eduard Kummer, who taught higher arithmetic;
- Leopold Kronecker, a number theory specialist who subsequently opposed Cantor.
After spending one semester at the University of Göttingen in 1866, the following year, George wrote a doctoral dissertation entitled “In mathematics, the art of asking questions is more valuable than problem solving”, concerning a problem that Karl Friedrich Gauss left unresolved in his Disquisitiones Arithmeticae (1801) . After brief teaching at the Berlin School for Girls, Kantor began working at Halle University, where he remained until the end of his life, first as a teacher, from 1872 as an assistant professor and from 1879 as a professor.
Research
At the beginning of a series of 10 works from 1869 to 1873, George Cantor examined number theory. The work reflected the enthusiasm for the subject, his research by Gauss and the influence of Kronecker. At the suggestion of Heinrich Eduard Heine, a colleague of Cantor in Halle, who recognized his mathematical talent, he turned to the theory of trigonometric series, in which he expanded the concept of real numbers.
Based on the work on the function of the complex variable of the German mathematician Bernhard Riemann in 1854, in 1870, Kantor showed that such a function can be represented in only one way - trigonometric series. Consideration of a totality of numbers (points) that would not contradict such a representation led him, first, in 1872 to determine irrational numbers in terms of convergent sequences of rational numbers (fractions of integers) and then to the beginning of work on the work of his whole life, set theory and the concept of transfinite numbers.
Set theory
Georg Cantor, whose set theory originated in correspondence with the mathematician of the Braunschweig Technical Institute Richard Dedekind, was friends with him since childhood. They came to the conclusion that sets, finite or infinite, are a collection of elements (for example, numbers, {0, ± 1, ± 2...}) That have a certain property, while preserving their individuality. But when Georg Cantor applied a one-to-one correspondence to study their characteristics (for example, {A, B, C} to {1, 2, 3}), he quickly realized that they differ in the degree of their belonging, even if they were infinite sets , i.e., sets whose part or subset includes as many objects as it itself. His method soon yielded amazing results.
In 1873, George Cantor (mathematician) showed that rational numbers, although infinite, are countable, because they can be put into one-to-one correspondence with natural numbers (i.e., 1, 2, 3, etc.). He showed that the set of real numbers, consisting of irrational and rational, is infinite and uncountable. More paradoxically, Kantor proved that the set of all algebraic numbers contains as many elements as the set of all integers, and that transcendental numbers that are not algebraic, which are a subset of irrational numbers, are uncountable and, therefore, their number is greater than integers , and should be considered infinite.
Opponents and Supporters
But Cantor's work, in which he first put forward these results, was not published in the Krell journal, as one of the reviewers, Kronecker, was categorically opposed. But after Dedekind’s intervention, it was published in 1874 under the title "On the characteristic properties of all real algebraic numbers."
Science and personal life
In the same year, during a honeymoon with his wife, Valley Gutman in Interlaken, Switzerland, Cantor met Dedekind, who spoke favorably about his new theory. George's salary was small, but with the money of his father, who died in 1863, he built a house for his wife and five children. Many of his works were published in Sweden in the new journal Acta Mathematica, whose editor and founder was Gesta Mittag-Leffler, who was among the first to recognize the talent of German mathematician.
The connection with metaphysics
Cantor's theory has become a completely new subject of research concerning the mathematics of the infinite (for example, series 1, 2, 3, etc., and more complex sets), which largely depended on a one-to-one correspondence. Kantor’s development of new methods for raising questions about continuity and infinity gave his research an ambiguous character.
When he argued that infinite numbers really exist, he turned to ancient and medieval philosophy regarding actual and potential infinity, as well as to the early religious education his parents gave him. In 1883, in his book, Fundamentals of the General Theory of Sets, Cantor combined his concept with the metaphysics of Plato.
Kronecker, who claimed that only integers exist (“God created integers, the rest is the work of man”), for many years vehemently rejected his reasoning and prevented his appointment at the University of Berlin.
Transfinite Numbers
In the years 1895-97. Georg Cantor fully formed his concept of continuity and infinity, including infinite ordinal and cardinal numbers, in his most famous work, published under the title “Contribution to the creation of the theory of transfinite numbers” (1915). This composition contains his concept, to which he was led by a demonstration that an infinite set can be brought into one-to-one correspondence with one of its subsets.
By the smallest transfinite cardinal number, he meant the power of any set that can be put into one-to-one correspondence with natural numbers. Cantor called him Aleph Zero. Large transfinite sets are denoted by alef-one, alef-two , etc. Then he developed the arithmetic of transfinite numbers, which was similar to finite arithmetic. Thus, he enriched the concept of infinity.
The opposition he faced and the time it took for his ideas to be fully accepted are explained by the difficulty of re-evaluating the ancient question of what the number is. Cantor showed that many points on the line have a higher power than Aleph-Zero. This led to the well-known problem of the continuum hypothesis - there are no cardinal numbers between the Aleph zero and the power of the points on the line. This problem in the first and second half of the 20th century aroused great interest and was studied by many mathematicians, including Kurt Godel and Paul Cohen.
Depression
The biography of George Cantor since 1884 was overshadowed by the mental illness that had begun in him, but he continued to work actively. In 1897, he helped hold the first international mathematical congress in Zurich. Partly because Kronecker opposed him, he often sympathized with young novice mathematicians and sought to find a way to rid them of harassment from teachers who feel threatened by new ideas.
Confession
At the turn of the century, his work was fully recognized as the basis for function theory, analysis, and topology. In addition, the books of Cantor Georg served as an impetus for the further development of intuitive and formalistic schools of the logical foundations of mathematics. This has significantly changed the teaching system and is often associated with “new mathematics”.
In 1911, Cantor was among those invited to celebrate the 500th anniversary of the University of St Andrews in Scotland. He went there hoping to meet with Bertrand Russell, who in his recently published work Principia Mathematica repeatedly referred to German mathematician, but this did not happen. The university awarded Kantor an honorary degree, but due to his illness, he could not accept the award personally.
Cantor retired in 1913, lived in poverty and starved during the First World War. The celebrations in honor of his 70th birthday in 1915 were canceled due to the war, but a small ceremony took place at his home. He died on January 6, 1918 in Galle, in a psychiatric hospital, where he spent the last years of his life.
Georg Cantor: biography. A family
On August 9, 1874, a German mathematician married Valley Gutman. The spouses had 4 sons and 2 daughters. The last child was born in 1886 in a new house acquired by Cantor. He was helped by his father’s inheritance to support his family. Cantor's health condition was greatly affected by the death of his youngest son in 1899 - since then depression has not left him.