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This theorem is proved using Cantor's first uncountability proof, which differs from the more familiar proof using his diagonal argument. The title of the article, " On a Property of the Collection of All Real Algebraic Numbers " ("Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen"), refers to its first theorem: the set ...
Georg Cantor published this proof in 1891, [1] [2]: 20– [3] but it was not his first proof of the uncountability of the real numbers, which appeared in 1874. [ 4 ] [ 5 ] However, it demonstrates a general technique that has since been used in a wide range of proofs, [ 6 ] including the first of Gödel's incompleteness theorems [ 2 ] and ...
Cantor's theorem and its proof are closely related to two paradoxes of set theory. Cantor's paradox is the name given to a contradiction following from Cantor's theorem together with the assumption that there is a set containing all sets, the universal set. In order to distinguish this paradox from the next one discussed below, it is important ...
Georg Ferdinand Ludwig Philipp Cantor (/ ˈ k æ n t ɔːr / KAN-tor; German: [ˈɡeːɔʁk ˈfɛʁdinant ˈluːtvɪç ˈfiːlɪp ˈkantoːɐ̯]; 3 March [O.S. 19 February] 1845 – 6 January 1918 [1]) was a mathematician who played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics.
The continuum hypothesis was advanced by Georg Cantor in 1878, [1] and establishing its truth or falsehood is the first of Hilbert's 23 problems presented in 1900. The answer to this problem is independent of ZFC, so that either the continuum hypothesis or its negation can be added as an axiom to ZFC set theory, with the resulting theory being ...
The modern study of set theory was initiated by the German mathematicians Richard Dedekind and Georg Cantor in the 1870s. In particular, Georg Cantor is commonly considered the founder of set theory. The non-formalized systems investigated during this early stage go under the name of naive set theory.
[20] The rejection of Cantor's infinitary ideas influenced the development of schools of mathematics such as constructivism and intuitionism. [citation needed] Wittgenstein did not object to mathematical formalism wholesale, but had a finitist view on what Cantor's proof meant.
According to Bernstein, Cantor had suggested the name equivalence theorem (Äquivalenzsatz). [2] Cantor's first statement of the theorem (1887) [3] 1887 Cantor publishes the theorem, however without proof. [3] [2] 1887 On July 11, Dedekind proves the theorem (not relying on the axiom of choice) [4] but neither publishes his proof nor tells ...