# Georg Cantor

### Our next story about great Petersburger is rather unexpected by non-specialist audiences. Did you know that great founder of set theory was born in St.Petersburg and was Russian citizen for whole his life^{*}? Honestly, we also were surprised to learn it.

### *According to Prof. Sergey Vostokov (in Russian)

Georg Cantor was born in 1845 in the western merchant colony of Saint Petersburg, Russia, and brought up in the city until he was eleven. Cantor’s paternal grandparents were from Copenhagen and fled to Russia from the disruption of the Napoleonic Wars. Cantor’s father had been a member of the Saint Petersburg stock exchange. Georg, the oldest of six children, was regarded as an outstanding violinist. His mother, Maria Anna Böhm, was an Austro-Hungarian born in Saint Petersburg. His grandfather Franz Böhm (1788–1846) was a well-known musician and soloist in a Russian imperial orchestra.

In 1853 George started his education at Saint Peter’s School. When his father became ill, the family moved to Germany in 1856, first to Wiesbaden, then to Frankfurt, seeking milder winters than those of Saint Petersburg.

In 1860, Cantor graduated with distinction from the Realschule in Darmstadt; his exceptional skills in mathematics, trigonometry in particular, were noted. In August 1862, he then graduated from the “Höhere Gewerbeschule Darmstadt”, now the Technische Universität Darmstadt. In 1862, Cantor entered the Swiss Federal Polytechnic. After his father’s death in June 1863, Cantor shifted his studies to the University of Berlin, attending lectures by Leopold Kronecker, Karl Weierstrass and Ernst Kummer. He spent the summer of 1866 at the University of Göttingen, then and later a center for mathematical research. Cantor submitted his dissertation on number theory at the University of Berlin in 1867. After teaching briefly in a Berlin girls’ school, Cantor took up a position at the University of Halle, where he spent his entire career. He was awarded the requisite habilitation for his thesis on number theory.

In 1872 Cantor met Richard Dedekind, who later became his close friend. Many Cantor’s ideas were discussed in correspondence with Dedekind.

Cantor was promoted to extraordinary professor in 1872 and made full professor in 1879. To attain the latter rank at the age of 34 was a notable accomplishment, but Cantor desired a chair at a more prestigious university, in particular at Berlin, at that time the leading German university. However, his work encountered too much opposition for that to be possible.

In 1882, the mathematical correspondence between Cantor and Dedekind came to an end. Cantor also began another important correspondence, with Gösta Mittag-Leffler in Sweden, and soon began to publish in Mittag-Leffler’s journal Acta Mathematica. But in 1885, Mittag-Leffler was concerned about the philosophical nature and new terminology in a paper Cantor had submitted to Acta. He asked Cantor to withdraw the paper from Acta while it was in proof, writing that it was “… about one hundred years too soon.” Cantor complied.

Cantor suffered his first known bout of depression in May 1884 and was hospitalized in sanatorium. Criticism of his work weighed on his mind. This crisis led him to apply to lecture on philosophy rather than mathematics. But Cantor recovered soon thereafter, and subsequently made further important contributions, including his diagonal argument and theorem.

In 1889, Cantor was instrumental in founding the German Mathematical Society and chaired its first meeting in Halle in 1891, where he first introduced his diagonal argument; his reputation was strong enough to ensure he was elected as the first president of this society. Georg Cantor was also instrumental in the establishment of the first International Congress of Mathematicians, which was held in Zürich, Switzerland, in 1897.

Cantor suffered from chronic depression for all of his later years, for which he was excused from teaching on several occasions and repeatedly confined in various sanatoria. After 1904 there was a series of hospitalizations at intervals of two or three years. He did not abandon mathematics completely, however, lecturing on the paradoxes of set theory to a meeting of the Deutsche Mathematiker-Vereinigung in 1903, and attending the International Congress of Mathematicians at Heidelberg in 1904. In the same year, the Royal Society awarded Cantor its Sylvester Medal, the highest honor it can confer for work in mathematics.

In 1911, Cantor was one of the distinguished foreign scholars who attended the 500th anniversary of the founding of the University of St. Andrews in Scotland. The following year, St. Andrews awarded Cantor an honorary doctorate, but illness precluded his receiving the degree in person.

Cantor retired in 1913, living in poverty and suffering from malnourishment during World War I. Georg Cantor had a fatal heart attack on January 6, 1918, in the sanatorium where he had spent the last year of his life.

**Mathematical work**

Cantor’s work between 1874 and 1884 is the origin of set theory. Prior to this work, the concept of a set was a rather elementary one that had been used implicitly since the beginning of mathematics, dating back to the ideas of Aristotle. No one had realized that set theory had any nontrivial content. Before Cantor, there were only finite sets (which are easy to understand) and “the infinite” (which was considered a topic for philosophical, rather than mathematical, discussion). By proving that there are (infinitely) many possible sizes for infinite sets, Cantor established that set theory was not trivial, and it needed to be studied. Set theory has come to play the role of a foundational theory in modern mathematics, in the sense that it interprets propositions about mathematical objects (for example, numbers and functions) from all the traditional areas of mathematics (such as algebra, analysis and topology) in a single theory, and provides a standard set of axioms to prove or disprove them. The basic concepts of set theory are now used throughout mathematics.

In one of his earliest papers, Cantor proved that the set of real numbers is “more numerous” than the set of natural numbers; this showed, for the first time, that there exist infinite sets of different sizes. He was also the first to appreciate the importance of one-to-one correspondences in set theory. He used this concept to define finite and infinite sets, subdividing the latter into denumerable (or countably infinite) sets and nondenumerable sets (uncountably infinite sets).

Cantor introduced fundamental constructions in set theory, such as the power set of a set A, which is the set of all possible subsets of A. He later proved that the size of the power set of A is strictly larger than the size of A, even when A is an infinite set; this result soon became known as Cantor’s theorem. Cantor developed an entire theory and arithmetic of infinite sets, called cardinals and ordinals, which extended the arithmetic of the natural numbers.

The Continuum hypothesis, introduced by Cantor, was presented by David Hilbert as the first of his twenty-three open problems in his address at the 1900 International Congress of Mathematicians in Paris.

Memorial board on Vasilievskiy Island in St. Petersburg, saying “Georg Cantor, the great mathematician and the founder of set theory was born in this house and lived from 1845 until 1854”