He was exceptionally good at mathematics, particularly trigonometry. Georg Ferdinand Ludwig Philipp Cantor was born on month day 1845, at birth place, to Georg Waldemar Cantor and Maria Anna Cantor (born Böhm). Georg Cantor founded the set theory, the branch of mathematical logic that studies sets. He proved to be a brilliant student and graduated with distinction from the Realschule in Darmstadt in 1860. H Meschkowski, Zwei unveröffentlichte Briefe Georg Cantors. [35], In one of his earliest papers,[36] 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 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. It contained Cantor's reply to his critics and showed how the transfinite numbers were a systematic extension of the natural numbers. He is also known for inventing the Cantor set, which is now a fundamental theory in mathematics. [12] Cantor suffered from chronic depression for the rest of his life, for which he was excused from teaching on several occasions and repeatedly confined in various sanatoria. At the same time, there was growing opposition to Cantor's ideas, led by Leopold Kronecker, who admitted mathematical concepts only if they could be constructed in a finite number of steps from the natural numbers, which he took as intuitively given. Through this work he proved that real numbers are not countable. [74] Secondly, the notion of infinity as an expression of reality is itself disallowed in intuitionism, since the human mind cannot intuitively construct an infinite set. Cantor was promoted to extraordinary professor in 1872 and made full professor in 1879. [19][18] 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. In 1856, when Cantor was 11 years old, his family moved to Germany, although Cantor was never at ease in this country. From 1905, Cantor corresponded with his British admirer and translator Philip Jourdain on the history of set theory and on Cantor's religious ideas. [61], Cantor avoided paradoxes by recognizing that there are two types of multiplicities. Finally, Wittgenstein's attacks were finitist: he believed that Cantor's diagonal argument conflated the intension of a set of cardinal or real numbers with its extension, thus conflating the concept of rules for generating a set with an actual set. In an 1877 letter to Richard Dedekind, Cantor proved a far stronger result: for any positive integer n, there exists a 1-to-1 correspondence between the points on the unit line segment and all of the points in an n-dimensional space. [63], In 1908, Zermelo published his axiom system for set theory. In 1904, the Royal Society awarded Cantor its Sylvester Medal, the highest honor it can confer for work in mathematics. Moore devotes a chapter to this criticism: "Zermelo and His Critics (1904–1908)". Apart from the cardinal numbers, Georg Cantor is also credited for having furthered the study of trigonometry and transfinite numbers. He was awarded the Sylvester Medal, which is a highly prestigious award in mathematics. 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 (or equivalently, the cardinality of the reals is exactly aleph-one, rather than just at least aleph-one). I. For example, an ordinal α is transfinite because it can be increased to α + 1. I Grattan-Guinness, Missing materials concerning the life and work of Georg Cantor. He was born in Copenhagen, sometime 1810-1815 from parents of "israelitischen Eltern", i.e. Georg Ferdinand Ludwig Philipp Cantor (/ˈkæntɔːr/ KAN-tor, German: [ˈɡeːɔʁk ˈfɛʁdinant ˈluːtvɪç ˈfɪlɪp ˈkantɔʁ]; March 3 [O.S. This notation is still in use today. He then proceeded to spend a semester at the University of Göttingen in 1866 and wrote his doctoral thesis in 1867. I realize that in this undertaking I place myself in a certain opposition to views widely held concerning the mathematical infinite and to opinions frequently defended on the nature of numbers. In Cantor's published works and also in his Nachlass there are no statements by himself which relate to a Jewish origin of his ancestors. In August 1862, he then graduated from the "Höhere Gewerbeschule Darmstadt", now the Technische Universität Darmstadt. He then began looking for a 1-to-1 correspondence between the points of the unit square and the points of a unit line segment. He had two motivations for developing the axiom system: eliminating the paradoxes and securing his proof of the well-ordering theorem. He also gave a new method of constructing transcendental numbers in 1874 which were first constructed by Joseph Liouville in 1844. He received his education from the University of Berlin where he specialized in physics, philosophy, and mathematics before embarking on a successful academic career. Georg's mother, Maria Anna Böhm, was Russian and very musical. In an extensive attempt to persuade other Christian thinkers and authorities to adopt his views, Cantor had corresponded with Christian philosophers such as Tilman Pesch and Joseph Hontheim,[83] as well as theologians such as Cardinal Johann Baptist Franzelin, who once replied by equating the theory of transfinite numbers with pantheism. [62] Russell named paradoxes after Cesare Burali-Forti and Cantor even though neither of them believed that they had found paradoxes. Given a trigonometric series f(x) with S as its set of zeros, Cantor had discovered a procedure that produced another trigonometric series that had S1 as its set of zeros, where S1 is the set of limit points of S. If Sk+1 is the set of limit points of Sk, then he could construct a trigonometric series whose zeros are Sk+1. P Dugac, Georg Cantor and Henri Poincaré. Born in: Saint Petersburg, Russian Empire, place of death: Halle, Province of Saxony, German Empire, Diseases & Disabilities: Bipolar Disorder, discoveries/inventions: Eponymous Paradox, education: Humboldt University of Berlin, Realschule, ETH Zurich, Quotes By Georg Cantor |