Georg Kantor (photo is given later in the article) is a German mathematician who created set theory and introduced the concept of transfinite numbers, infinitely large, but different from each other. He also defined ordinal and cardinal numbers and created their arithmetic.
Georg Kantor: short biography
Born in St. Petersburg on 1845-03-03. His father was a Dane of the Protestant faith, Georg-Valdemar Kantor, who was engaged in trade, including on the stock exchange. His mother Maria Bem was a 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 showed up even before his 15th birthday while studying at private schools and gymnasiums in Darmstadt and Wiesbaden. In the end, Georg Cantor convinced his father of his firm intention to become a mathematician, not an engineer.
After a brief study at the University of Zurich, in 1863 Kantor transferred to the University of Berlin to study physics, philosophy and mathematics. There himtaught:
- Karl Theodor Weierstrass, whose specialization in analysis probably had the greatest influence on Georg;
- Ernst Eduard Kummer, who taught higher arithmetic;
- Leopold Kronecker, number theorist who later opposed Cantor.
After spending one semester at the University of Göttingen in 1866, the following year Georg wrote his doctoral dissertation en titled "In mathematics the art of asking questions is more valuable than solving problems", concerning a problem that Carl Friedrich Gauss had left unsolved in his Disquisitiones Arithmeticae (1801). After briefly teaching at the Berlin School for Girls, Kantor began working at the University of Halle, 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 papers from 1869 to 1873, Georg Cantor considered number theory. The work reflected his passion for the subject, his studies of Gauss and the influence of Kronecker. At the suggestion of Heinrich Eduard Heine, Cantor's colleague 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 a complex variable by the German mathematician Bernhard Riemann in 1854, in 1870 Kantor showed that such a function can be represented in only one way - by trigonometric series. Consideration of a set of numbers (points) thatwould not contradict such a view, led him, firstly, in 1872 to the definition of irrational numbers in terms of convergent sequences of rational numbers (fractions of integers) and further to the beginning of work on his life's work, set theory and the concept of transfinite numbers.
Set Theory
Georg Kantor, whose set theory originated in correspondence with the mathematician of the Technical Institute of Braunschweig Richard Dedekind, was a friend of him since childhood. They concluded that sets, whether finite or infinite, are collections of elements (eg numbers, {0, ±1, ±2…}) that have a certain property while retaining their individuality. But when Georg Cantor used a one-to-one correspondence (for example, {A, B, C} to {1, 2, 3}) to study their characteristics, he quickly realized that they differ in their degree of membership, even if they were infinite sets., i.e. sets, a part or subset of which includes as many objects as it itself. His method soon gave amazing results.
In 1873, Georg Cantor (mathematician) showed that rational numbers, although infinite, are countable because they can be put in one-to-one correspondence with natural numbers (i.e. 1, 2, 3, etc.). d.). He showed that the set of real numbers, consisting of irrational and rational ones, is infinite and uncountable. More paradoxically, Cantor proved that the set of all algebraic numbers contains as many elements ashow many are the set of all integers, and that transcendental numbers, which are not algebraic, which are a subset of irrational numbers, are uncountable and, therefore, their number is greater than integers, and should be considered as infinite.
Opponents and supporters
But Kantor's paper, in which he first put forward these results, was not published in Krell, as one of the reviewers, Kronecker, was vehemently opposed. But after the intervention of Dedekind, it was published in 1874 under the title "On the characteristic properties of all real algebraic numbers."
Science and private life
The same year, while on his honeymoon with his wife Wally Gutman in Interlaken, Switzerland, Kantor met Dedekind, who spoke favorably of 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 papers were published in Sweden in the new journal Acta Mathematica, edited and founded by Gesta Mittag-Leffler, who was among the first to recognize the talent of the German mathematician.
Connection with metaphysics
Cantor's theory became a completely new subject of study concerning the mathematics of the infinite (eg series 1, 2, 3, etc., and more complex sets), which depended heavily on one-to-one correspondence. Kantor's development of new staging methodsquestions concerning 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 that his parents gave him. In 1883, in his book Foundations of General Set Theory, Kantor combined his concept with Plato's metaphysics.
Kronecker, who claimed that only integers “exist” (“God created the 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 1895-97. Georg Cantor fully formed his notion of continuity and infinity, including infinite ordinal and cardinal numbers, in his most famous work, published as Contributions to the Establishment of the Theory of Transfinite Numbers (1915). This essay contains his concept, to which he was led by demonstrating that an infinite set can be put in a one-to-one correspondence with one of its subsets.
Under the least transfinite cardinal number, he meant the cardinality of any set that can be put in one-to-one correspondence with natural numbers. Cantor called it aleph-null. Large transfinite sets are denoted aleph-one, aleph-two, etc. He further developed the arithmetic of transfinite numbers, which was analogous to finite arithmetic. so, heenriched the concept of infinity.
The opposition he faced and the time it took for his ideas to be fully accepted is due to the difficulty of re-evaluating the ancient question of what a number is. Cantor showed that the set of points on a line has a higher cardinality than aleph-zero. This led to the well-known problem of the continuum hypothesis - there are no cardinal numbers between aleph-zero and the cardinality of 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 Gödel and Paul Cohen.
Depression
The biography of Georg Kantor since 1884 was overshadowed by his mental illness, but he continued to work actively. In 1897 he helped hold the first international mathematical congress in Zurich. Partly because he was opposed by Kronecker, he often sympathized with young aspiring mathematicians and sought to find a way to save them from the harassment of teachers who felt threatened by new ideas.
Recognition
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 the intuitionist and formalist schools of the logical foundations of mathematics. This significantly changed the teaching system and is often associated with the "new mathematics".
In 1911, Kantor was among those invited tocelebration of the 500th anniversary of the University of St. Andrews in Scotland. He went there in the hope of meeting Bertrand Russell, who, in his recently published work Principia Mathematica, repeatedly referred to the German mathematician, but this did not happen. The university awarded Kantor an honorary degree, but due to illness he was unable to accept the award in person.
Kantor retired in 1913, lived in poverty and starved during the First World War. 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 1918-06-01 in Halle, in a psychiatric hospital, where he spent the last years of his life.
Georg Kantor: biography. Family
August 9, 1874, a German mathematician married Wally Gutman. The couple had 4 sons and 2 daughters. The last child was born in 1886 in a new house purchased by Kantor. His father's inheritance helped him support his family. Kantor's he alth was greatly affected by the death of his youngest son in 1899, and depression has not left him since.
