Founder of set theory - Canto

Georg Cantor, a Russian-born mathematician who can be considered the founder of set theory, introduced the concept of infinite numbers when he discovered cardinality. He also advanced the study of triangular series.

Born March 3, 1845, St. Petersburg, Russia

Died January 6, 1918 in Halle, Germany

Georg Cantor's father, Georg Waldemar Cantor, was a successful businessman who worked as a wholesale agent in St. Petersburg and later as a broker on the St. Petersburg Stock Exchange. Georg Waldemar Cantor was born in Denmark and is a lover of culture and art. Georg's mother, Maria Anna Böhm, is Russian and loves music very much. Of course, George inherited a considerable amount of musical and artistic talent from his parents, who were brilliant violinists. George grew up protestant, which was the religion of his father, while George's mother was Roman Catholic.

After receiving early education as a governess, Cantor attended elementary school in St. Petersburg, before moving to Germany in 1856 when he was 11 years old. However, Cantor[21]:-

... Nostalgic for his early days in Russia, never felt at ease in Germany, although he had spent the rest of his life there and never seemed to have written in Russian, which he must have known.

Cantor's father was in poor health and moved to Germany to find a warmer climate than the harsh winters in St. Petersburg. At first they lived in Wiesbaden, where Canto went to the gymnasium, and then they moved to Frankfurt. Cantor studied at Realschule in Darmstadt and boarded there. Upon his graduation in 1860 he published an excellent report in which he specifically mentioned his special skills in mathematics, especially trigonometry. After joining Höhere Gewerbeschule in Darmstadt in 1860, he entered the Zurich Institute of Technology in 1862. Cantor's father chose to send him to Höheren Gewerbeschule because he wanted Cantor to be:-

...... A shining star in the engineering world.

However, in 1862, Cantor sought his father's permission to study mathematics at university, and when his father eventually agreed, he was ecstatic. However, in June 1863 his father died and his studies in Zurich were interrupted. Cantor moved to the University of Berlin, where he became friends with classmate Hermann Schwartz. Cantor attended lectures by Weierstrass, Kummer, and Kronecker. He spent the summer of 1866 at the University of Göttingen and returned to Berlin to complete his thesis on number theory, De aequationibus secundi gradus indeterminatis (T) in 1867.

Although in Berlin Canto more participated in the Student Mathematical Society, became president of the association in 1864-65. He was also part of a small group of young mathematicians who gathered weekly at the tavern. After receiving his doctorate in 1867, Cantor taught at a girls' school in Berlin. Then, in 1868, he joined the Schellbach Seminar for Mathematics Teachers. During this time, he worked on his professional qualifications and submitted his thesis, also on number theory, immediately after his appointment to Halley in 1869, and obtained his professional qualification.

At Halley, Cantor shifted his research direction from number theory to analysis. This is due to Heine, a senior colleague at Halle, who challenged Cantor to demonstrate the open-ended question about the uniqueness of functions represented as trigonometric series. It was a conundrum that had been overcome unsuccessfully by many mathematicians, including Heine himself, as well as Dilikré, Lippchitz, and Riemann. Cantor solved the problem of proving uniqueness in April 1870. He published more papers dealing with trigonometric progressions between 1870 and 1872, all of which show the influence of Weierstrass's teaching.

Cantor was promoted to Distinguished Professor at Harley in 1872 and in the same year formed a friendship with Deider king, whom he had met on vacation in Switzerland. Cantor published a paper on trigonometric series in 1872 in which he defined irrational numbers based on convergent sequences of rational numbers. Dedekind also published his definition of real numbers in 1872 through the "Dedekind cut", in which Dedekind referred to the paper that Cantor sent him in 1872.

In 1873 Canto proved that rational numbers can be counted, i.e., they can be placed on a one-to-one correspondence with the natural numbers. He also proved that algebraic numbers, i.e., numbers that are the roots of polynomial equations with integer coefficients, are countable. However, his attempts to determine whether real numbers are countable proved more difficult. By December 1873, he had proved that real numbers were uncountable, and in 1874 he published a paper. The idea of a one-to-one correspondence first appears in this article, but is only implicit in this article.

A transcendent number is an irrational number that is not the root of any polynomial equation with integer coefficients. Lloville established the existence of transcendent numbers in 1851. Twenty years later, in this work of 1874, Cantor proved that in a sense " almost all" numbers are transcendent, he proved that real numbers are uncountable, and he proved that algebraic numbers are countable.

Canto moved forward, exchanging letters with Dedekin throughout. The next question he asked himself in January 1874 was whether it was possible to map a unit square to a line of unit length, where 1 - 1 corresponded to each point. In a letter dated 5 January 1874 to Deiderkin, he wrote [1]:-

Is it possible to uniquely refer to a surface, such as a square containing a boundary, as a line (for example, a straight line segment containing an endpoint) so that each point on the surface has a corresponding point of the line, and vice versa, for each point of the line, there is a corresponding point on the surface? I don't think it's easy to answer this question, even though the answer seems so obviously "no" that there is little need to provide evidence.

The year 1874 was an important year in Cantor's personal life. That same spring, he became engaged to his sister's friend, Wally Gutman. They married on 9 August 1874 and honeymooned in Interlaken, Switzerland, where Canto spent much time with Deidkin on mathematical discussions. Cantor continued to correspond with Dedekin, sharing his ideas and consulting Dedeggin, and in 1877 wrote to Dedekin to prove the existence of 1-1 corresponding points on the interval [ 0 , 1 ].

And points out phosphorus fiber space. Cantor was surprised by his discovery and wrote:-

I saw it, but I didn't believe it!

Of course, this has implications for geometry and the concept of spatial dimensions. In terms of size, its Canto submitted to the main paper-making Kreller's magazine was treated with suspicion in 1877, only after its publication Dade intervened on behalf of Cantor. Cantor hated Kronecker's opposition to his work and never submitted any further papers to Claire's journal.

This emerged about the size of the paper Kreiler's magazine in 1878 so that the concepts 1-1 correspond accurately. This article discusses countable sets, i.e. those that correspond to natural numbers at points 1 - 1. It studies sets of equal powers, i.e. those that correspond to each other 1 - 1. Canto also discusses the concept of size and emphasizes the fact that the correspondence between his intervals [0,1] and the unit square is not a continuous map. 1879 to 1884

Cantor published a series of six papers in Mathematische Annalen aimed at providing a basic introduction to set theory. Klein may have had a significant impact on Mathematische Annalen's publication of them. However, many of the problems that have occurred over the years have been difficult for Canto. Although he was promoted to full professor in 1879 at heyne's recommendation, Cantor always wanted to be a professor at a more prestigious university. His long correspondence with Schwartz ended in 1880 as opposition to Canto's ideas continued to grow, and Schwartz no longer supported the direction of Canto's work. Then in October 1881 Heine died, and Harley's chair needed a replacement to fill.

Cantor drafted a list of three mathematicians to fill Heine's seat, which was approved. It ranked The Dedekind first, followed by Heinrich Weber, and finally Mertens. In early 1882, Dedekin rejected Cantor's offer, which undoubtedly dealt a heavy blow to Cantor, and Heinrich Weber and Mertens also refused, which made the blow worse. Wangerin was appointed after the new list was drawn up, but he never had a close relationship with Cantor. The rich mathematical correspondence between Cantor and Dedekin ended later in 1882.

Almost at the same time as the Conto-Deidkin correspondence ended, Conto began another important communication with Mitag Leffler. Soon, Cantor published in Mitta-Leffler's journal Acta Mathematica, but six of his major series of papers in Mathematische Annalen also continued to appear. The fifth paper in this series, Grundlagen Einer allgemeinen Mannigfaltigkeitslehre (T), was also published as a separate monograph and was particularly important for a number of reasons. First, Cantor realized that his set theory was not getting the acceptance he had hoped, and Grundlagen aimed to respond to criticism. Second [ 3 ]:-

Grundlagen's main achievement is the autonomy and system extension of expressing out-of-limit numbers as natural numbers.

Canto himself makes it very clear in the paper that he is aware of the power to oppose his ideas:-

...... I realized that in this work I placed myself in some sort of opposition to the widely held view of mathematical infinity and the often justified nature of numbers.

In late May 1884, Cantor first recorded an episode of depression. He recovered a few weeks later, but now seems less confident. He wrote to Mittag-Leffler at the end of June [3 ]:-

...... I don't know when I'll be able to get back to continuing my scientific work. At the moment I am completely powerless against it, limiting myself to the most necessary duties of the speech; if I had the necessary spiritual freshness, I would be happier to engage in scientific activity.

Once upon a time, it was thought that his depression was caused by mathematical problems, especially as a result of his difficult relationship with Kronecker. However, a better understanding of mental illness recently means that we can now be sure that Cantor's mathematical problems and his difficult relationship were greatly amplified by his depression, but not the cause of it (see, for example, [3] and [21]). After mental illness [3] in 1884:-

... He was on vacation in his favorite Hartz Mountains and for some reason decided to try to reconcile with Kronecker. Kronecker accepted the gesture, but it must have been hard for the two of them to forget their hostility, and the philosophical differences between them remained unaffected.

The troubles of mathematics began to trouble Cantor at this time, especially when he began to worry that he would not be able to prove the continuum hypothesis that the infinite order of real numbers was second to the order of natural numbers. In fact, he thought he had proven it to be fake, and then discovered his mistake the next day. He again thought he had proven it, but soon discovered his mistake.

In other respects, things did not go well, as in 1885 Mitag-Leffler persuaded Cantor to withdraw one of his papers from the Acta Mathematica when he reached the stage of proof, because he believed that "... About a hundred years too early." Cantor joked, but was clearly hurt :-

If Mitta-Leffler had his way, I would have waited until 1984, which seemed like too much of a need for me! ...... But of course, I don't want to know anything about Acta Mathematica anymore.

Mittag-Leffler means that this is a form of goodwill, but it does indicate a lack of awareness of the importance of Cantor's work. Communication between Mitag Leffler and Cantor all but ceased shortly after this event, and the torrent of new ideas that led Tortol to develop set theory rapidly over a period of about 12 years seems to have all but ceased.

In 1886 Canto bought the beautiful new house 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 moved from the mathematical development of set theory to two new directions, first discussing the philosophical aspects of his theory with many philosophers (he published these letters in 1888), and secondly taking over the idea of founding the German Mathematical Society, which he realized in 1890, after Clebsch's death. Cantor presided over the first meeting of the Society in Halley in September 1891, and despite the fierce antagonism between him and Cronecker, Conto invited Cronecker to speak at the first meeting. However, Crohneck never spoke at the conference because his wife was seriously injured in a mountaineering accident in late summer and died shortly after. Cantor was elected president of the Deutsche Mathematiker-Vereinigung

At the first meeting and held the position until 1893. He helped organize the meeting of the association in Munich in September 1893, but before the meeting he fell ill again and was unable to attend.

Cantor published a rather strange paper in 1894, which listed how all even numbers up to 1000 can be written as the sum of two prime numbers. Since the goldbach conjecture had been validated 10,000 times 40 years earlier, this strange paper is likely to illustrate Cantor's state of mind better than the Goldbach conjecture. His last major paper on set theory was published in

In 1895 and 1897, he worked as an editor at Klein in the Mathematical Almanac and conducted elaborate surveys of over-limit arithmetic. The considerable gap between the two papers is because although Canto 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, this is not the case, but the second paper describes his theory of good ordinal sets and ordinal numbers.

In 1897 Canto attended the first international conference of mathematicians in Zurich. In their speeches at the conference [4]:-

...... Hurwitz openly expressed his high esteem for Cantor and declared that Cantor was a rich man of function theory. Jacques Hadamard said that the concept of set theory is a well-known and indispensable tool.

At the convention, Cantor met Dedekin, and they re-established their friendship. By the time of the Congress, however, Cantor had discovered the first paradox in set theory. He discovered these paradoxes in investigative papers in 1895 and 1897 and wrote to Hilbert in 1896 explaining them to him. Burali-Forti independently discovered the paradox and published it in 1897. Cantor began to correspond with Dedekin to try to understand how to solve the problem, but his recurring mental illness forced him to stop writing to Dedekin in 1899.

Whenever Canto suffered from periods of depression, he tended to turn away from mathematics and turn to philosophy and his keen interest in literature, believing that Francis Bacon had written Shakespeare's plays. For example, when he fell ill in 1884, he asked for permission to teach philosophy rather than mathematics, and he began to delve into Elizabethan literature in an attempt to prove his Bacon-Shakespearean theory. He began publishing pamphlets on literary issues in 1896 and 1897. The death of his mother in October 1896 and the death of his brother in January 1899 put additional pressure on Canto. 1899

In October, Cantor applied for and was granted a suspension for the winter semester of 1899–1900. Then in December 1899 the youngest son of the Cantor Company died. From then until the end of his life, he struggled with mental illness from depression. He continued to teach, but also had to take several winter semesters off from his teaching, those from 1902 - 03, 1904 - 05 and 1907 - 08. Cantor also spent some time in a nursing home from 1899 onwards, at the height of his mental illness. He did continue to work and publish his bacon-Shakespeare theory, and certainly did not abandon mathematics entirely. In September 1903 he lectured on the paradox of set theory at a conference in Deutsche Mathematiker-Vereinigung, and in August 1904 he attended the International Congress of Mathematicians in Heidelberg. In 1905 Canto wrote religious work after returning from a mantra in the hospital. He also corresponded with Jourdain on the history of set theory and his religious pamphlets. For most of 1909, after taking a sabbatical on the grounds of ill health, he fulfilled his university duties in 1910 and 1911.

. It was during this year that he was pleased to receive an invitation from the University of St Andrews Scotland to attend the 500 anniversaries of the University's founding as a distinguished foreign scholar. The celebration took place from 12 to 15 September 1911, but [21] :-

During the visit, he apparently began to act eccentric, gushing about the Bacon-Shakespeare issue. Then he went to London for a few days.

Cantor had hoped to meet Russell, who had just published Principia Mathematica. However, news of poor health and his son's illness led Cantor to return to Germany without seeing Russell. The following year, Cantor was awarded an honorary juris doctorate by the University of St Andrews, but he was too ill to receive his degree in person. Cantor retired in 1913 and spent his last years in illness due to the war conditions in Germany. A major event planned to commemorate Cantor in Harley on the 70th of 1915 had been cancelled due to war, but smaller events were held in his hometown. June 1917

He entered the nursing home for the last time and kept writing to his wife asking for permission to go home. He died of a heart attack.

Hilbert describes Canto's work as:-

... The best product of mathematical genius and one of the highest achievements of purely intellectual activity in human beings.