80 Facts About Georg Cantor

Georg Cantor played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics.

Georg Cantor defined the cardinal and ordinal numbers and their arithmetic.

Georg Cantor's recurring bouts of depression from 1884 to the end of his life have been blamed on the hostile attitude of many of his contemporaries, though some have explained these episodes as probable manifestations of a bipolar disorder.

Georg Cantor, born in 1845 in Saint Petersburg, Russia, was brought up in that city until the age of eleven.

Georg Cantor's grandfather Franz Bohm was a well-known musician and soloist in a Russian imperial orchestra.

Georg Cantor's father had been a member of the Saint Petersburg stock exchange; when he 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, Georg Cantor graduated with distinction from the Realschule in Darmstadt; his exceptional skills in mathematics, trigonometry in particular, were noted.

In 1862 Georg Cantor entered the Swiss Federal Polytechnic in Zurich.

Georg Cantor spent the summer of 1866 at the University of Gottingen, then and later a center for mathematical research.

Georg Cantor was a good student, and he received his doctoral degree in 1867.

Georg Cantor submitted his dissertation on number theory at the University of Berlin in 1867.

Georg Cantor was awarded the requisite habilitation for his thesis, on number theory, which he presented in 1869 upon his appointment at Halle University.

Georg Cantor was able to support a family despite his modest academic pay, thanks to his inheritance from his father.

Georg Cantor was promoted to extraordinary professor in 1872 and made full professor in 1879.

Kronecker, now seen as one of the founders of the constructive viewpoint in mathematics, disliked much of Georg Cantor's set theory because it asserted the existence of sets satisfying certain properties, without giving specific examples of sets whose members did indeed satisfy those properties.

Whenever Georg Cantor applied for a post in Berlin, he was declined, and the process usually involved Kronecker, so Georg Cantor came to believe that Kronecker's stance would make it impossible for him ever to leave Halle.

In 1882, the mathematical correspondence between Georg Cantor and Dedekind came to an end, apparently as a result of Dedekind's declining the chair at Halle.

Georg Cantor began another important correspondence, with Gosta Mittag-Leffler in Sweden, and soon began to publish in Mittag-Leffler's journal Acta Mathematica.

Georg Cantor suffered his first known bout of depression in May 1884.

Georg Cantor began an intense study of Elizabethan literature, thinking there might be evidence that Francis Bacon wrote the plays attributed to William Shakespeare ; this ultimately resulted in two pamphlets, published in 1896 and 1897.

Georg Cantor recovered soon thereafter, and subsequently made further important contributions, including his diagonal argument and theorem.

Georg Cantor eventually sought, and achieved, a reconciliation with Kronecker.

In 1889, Georg Cantor was instrumental in founding the German Mathematical Society, and he chaired its first meeting in Halle in 1891, where he first introduced his diagonal argument; his reputation was strong enough, despite Kronecker's opposition to his work, to ensure he was elected as the first president of this society.

Georg Cantor was instrumental in the establishment of the first International Congress of Mathematicians, which took place in Zurich, Switzerland, in 1897.

Since the paper had been read in front of his daughters and colleagues, Georg Cantor perceived himself as having been publicly humiliated.

Georg Cantor did not abandon mathematics completely 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 1911, Georg Cantor was one of the distinguished foreign scholars invited to the 500th anniversary of the founding of the University of St Andrews in Scotland.

Georg Cantor attended, hoping to meet Bertrand Russell, whose newly published Principia Mathematica repeatedly cited Georg Cantor's work, but the encounter did not come about.

Georg Cantor had a fatal heart attack on January 6,1918, in the sanatorium where he had spent the last year of his life.

In one of his earliest papers, Georg 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.

Georg Cantor was the first to appreciate the importance of one-to-one correspondences in set theory.

Georg Cantor used this concept to define finite and infinite sets, subdividing the latter into denumerable sets and nondenumerable sets.

Georg Cantor developed important concepts in topology and their relation to cardinality.

For example, he showed that the Georg Cantor set, discovered by Henry John Stephen Smith in 1875, ishere dense, but has the same cardinality as the set of all real numbers, whereas the rationals are everywhere dense, but countable.

Georg Cantor showed that all countable dense linear orders without end points are order-isomorphic to the rational numbers.

Georg 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 Georg 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.

The US philosopher Charles Sanders Peirce praised Georg Cantor's set theory and, following public lectures delivered by Georg Cantor at the first International Congress of Mathematicians, held in Zurich in 1897, Adolf Hurwitz and Jacques Hadamard both expressed their admiration.

At that Congress, Georg Cantor renewed his friendship and correspondence with Dedekind.

From 1905, Georg Cantor corresponded with his British admirer and translator Philip Jourdain on the history of set theory and on Georg Cantor's religious ideas.

At the suggestion of Eduard Heine, the Professor at Halle, Georg Cantor turned to analysis.

Heine proposed that Georg Cantor solve an open problem that had eluded Peter Gustav Lejeune Dirichlet, Rudolf Lipschitz, Bernhard Riemann, and Heine himself: the uniqueness of the representation of a function by trigonometric series.

Between 1870 and 1872, Georg Cantor published more papers on trigonometric series, and a paper defining irrational numbers as convergent sequences of rational numbers.

Dedekind, whom Georg Cantor befriended in 1872, cited this paper later that year, in the paper where he first set out his celebrated definition of real numbers by Dedekind cuts.

Georg Cantor published an erroneous "proof" of the inconsistency of infinitesimals.

Georg Cantor proved that the collection of real numbers and the collection of positive integers are not equinumerous.

Georg Cantor's article contains a new method of constructing transcendental numbers.

Georg Cantor starts his second construction with any sequence of real numbers.

Between 1879 and 1884, Georg Cantor published a series of six articles in Mathematische Annalen that together formed an introduction to his set theory.

Georg Cantor then defines the addition and multiplication of the cardinal and ordinal numbers.

In 1885, Georg Cantor extended his theory of order types so that the ordinal numbers simply became a special case of order types.

Georg Cantor applied the same idea to prove Cantor's theorem: the cardinality of the power set of a set A is strictly larger than the cardinality of A This established the richness of the hierarchy of infinite sets, and of the cardinal and ordinal arithmetic that Cantor had defined.

Georg Cantor's argument is fundamental in the solution of the Halting problem and the proof of Godel's first incompleteness theorem.

In 1895 and 1897, Georg Cantor published a two-part paper in Mathematische Annalen under Felix Klein's editorship; these were his last significant papers on set theory.

Georg Cantor wanted the second paper to include a proof of the continuum hypothesis, but had to settle for expositing his theory of well-ordered sets and ordinal numbers.

Ernst Schroder had stated this theorem a bit earlier, but his proof, as well as Georg Cantor's, was flawed.

Georg Cantor then began looking for a 1-to-1 correspondence between the points of the unit square and the points of a unit line segment.

About this discovery Georg Cantor wrote to Dedekind: "" The result that he found so astonishing has implications for geometry and the notion of dimension.

In 1878, Georg Cantor submitted another paper to Crelle's Journal, in which he defined precisely the concept of a 1-to-1 correspondence and introduced the notion of "power" or "equivalence" of sets: two sets are equivalent if there exists a 1-to-1 correspondence between them.

Georg Cantor defined countable sets as sets which can be put into a 1-to-1 correspondence with the natural numbers, and proved that the rational numbers are denumerable.

Georg Cantor proved that n-dimensional Euclidean space R has the same power as the real numbers R, as does a countably infinite product of copies of R While he made free use of countability as a concept, he did not write the word "countable" until 1883.

Georg Cantor discussed his thinking about dimension, stressing that his mapping between the unit interval and the unit square was not a continuous one.

Georg Cantor was the first to formulate what later came to be known as the continuum hypothesis or CH: there exists no set whose power is greater than that of the naturals and less than that of the reals.

Georg Cantor believed the continuum hypothesis to be true and tried for many years to prove it, in vain.

In 1883, Georg Cantor divided the infinite into the transfinite and the absolute.

In 1883, Georg Cantor introduced the well-ordering principle "every set can be well-ordered" and stated that it is a "law of thought".

Georg Cantor extended his work on the absolute infinite by using it in a proof.

Georg Cantor used this inconsistent multiplicity to prove the aleph theorem.

Georg Cantor avoided paradoxes by recognizing that there are two types of multiplicities.

Georg Cantor had two motivations for developing the axiom system: eliminating the paradoxes and securing his proof of the well-ordering theorem.

Georg Cantor defined a set as a class that is a member of some class and stated the axiom: A class is not a set if and only if there is a one-to-one correspondence between it and the class of all sets.

Von Neumann used his axiom to prove the well-ordering theorem: Like Georg Cantor, he assumed that the ordinals form a set.

Georg Cantor directly addressed this intersection between these disciplines in the introduction to his Grundlagen einer allgemeinen Mannigfaltigkeitslehre, where he stressed the connection between his view of the infinite and the philosophical one.

Georg Cantor was a devout Lutheran whose explicit Christian beliefs shaped his philosophy of science.

In 1888, Georg Cantor published his correspondence with several philosophers on the philosophical implications of his set theory.

Georg Cantor even sent one letter directly to Pope Leo XIII himself, and addressed several pamphlets to him.

Meanwhile, Georg Cantor himself was fiercely opposed to infinitesimals, describing them as both an "abomination" and "the cholera bacillus of mathematics".

Georg Cantor cites Aristotle, Rene Descartes, George Berkeley, Gottfried Leibniz, and Bernard Bolzano on infinity.

Georg Cantor's mother, Maria Anna Bohm, was an Austro-Hungarian born in Saint Petersburg and baptized Roman Catholic; she converted to Protestantism upon marriage.

Understandably, Georg Cantor launched a thorough campaign to discredit Veronese's work in every way possible.