Ernst Zermelo, the later editor of Cantor's collected works, found the error already the next day.

The beginning of set theory as a branch of mathematics is often marked by the publication of Cantor's 1874 article, " Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen " (" On a Property of the Collection of All Real Algebraic Numbers ").

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

Cantor's construction of the real numbers is similar to the above construction ; the real numbers are the completion of the rational numbers using the ordinary absolute value to measure distances.

And in fact, Cantor's diagonal argument is constructive, in the sense that given a bijection between the real numbers and natural numbers, one constructs a real number which doesn't fit, and thereby proves a contradiction.

But Cantor's diagonal argument proves that the real numbers ( and therefore also the complex numbers ) are uncountable ; so the set of all transcendental numbers must also be uncountable.

* The role of the absolute infinite in Cantor's conception of set

Some believe that Georg Cantor's set theory was not actually implicated by these paradoxes ( see Frápolli 1991 ); one difficulty in determining this with certainty is that Cantor did not provide an axiomatization of his system.

Hilbert adopted and warmly defended Georg Cantor's set theory and transfinite numbers.

However, most real numbers are not definable: the set of all definable numbers is countably infinite ( because the set of all logical formulas is ) while the set of real numbers is uncountably infinite ( see Cantor's diagonal argument ).

According to Husserl, this view of logic and mathematics accounted for the objectivity of a series of mathematical developments of his time, such as n-dimensional manifolds ( both Euclidean and non-Euclidean ), Hermann Grassmann's theory of extensions, William Rowan Hamilton's Hamiltonians, Sophus Lie's theory of transformation groups, and Cantor's set theory.

Writing decades after Cantor's death, Wittgenstein lamented that mathematics is " ridden through and through with the pernicious idioms of set theory ," which he dismissed as " utter nonsense " that is " laughable " and " wrong ".

Kronecker, now seen as one of the founders of the constructive viewpoint in mathematics, disliked much of 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.

He did not abandon mathematics completely, however, lecturing on the paradoxes of set theory ( Burali-Forti paradox, Cantor's paradox, and Russell's paradox ) to a meeting of the Deutsche Mathematiker – Vereinigung in 1903, and attending the International Congress of Mathematicians at Heidelberg in 1904.

Cantor's work between 1874 and 1884 is the origin of set theory.

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.

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

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

An illustration of Cantor's diagonal argument for the existence of uncountable set s. The sequence at the bottom cannot occur anywhere in the infinite list of sequences above.

The best known example of an uncountable set is the set R of all real numbers ; Cantor's diagonal argument shows that this set is uncountable.

" Kronecker even objected to Cantor's proofs that the algebraic numbers are countable, and that the transcendental numbers are uncountable, results now included in a standard mathematics curriculum.

Cantor's work established the ubiquity of transcendental numbers.

The Absolute Infinite is mathematician Georg Cantor's concept of an " infinity " that transcended the transfinite numbers.

For example, if we can enumerate all such definable numbers by the Gödel numbers of their defining formulas then we can use Cantor's diagonal argument to find a particular real that is not first-order definable in the same language.

Cantor's theory of transfinite numbers was originally regarded as so counter-intuitive — even shocking — that it encountered resistance from mathematical contemporaries such as Leopold Kronecker and Henri Poincaré and later from Hermann Weyl and L. E. J. Brouwer, while Ludwig Wittgenstein raised philosophical objections.

Some Christian theologians ( particularly neo-Scholastics ) saw Cantor's work as a challenge to the uniqueness of the absolute infinity in the nature of God — on one occasion equating the theory of transfinite numbers with pantheism — a proposition which Cantor vigorously rejected.

And yet Cantor's diagonal argument shows that real numbers have higher cardinality.

In short, one who takes the view that real numbers are effectively computable interprets Cantor's result as showing that the real numbers are not recursively enumerable.

The original statement of the paradox, due to Richard ( 1905 ), has a relation to Cantor's diagonal argument on the uncountability of the set of real numbers.

This conflicts with Cantor's theorem that the power set of any set ( whether infinite or not ) always has strictly higher cardinality than the set itself.

The suggestion has therefore been made that Hume's Principle ought better be called " Cantor's Principle ".

Cantor has used HSX's moviestock prices to assist Cantor's gambling operations in the United Kingdom, in which bettors can place bets on how much money US films will gross.

Because of Cantor's theorem each set in the preceding sequence has cardinality strictly greater than the one preceding it.

Then ( in the von Neumann formulation of cardinality ) C is a set and therefore has a power set 2 < sup > C </ sup > which, by Cantor's theorem, has cardinality strictly larger than that of C. Demonstrating a cardinality ( namely that of 2 < sup > C </ sup >) larger than C, which was assumed to be the greatest cardinal number, falsifies the definition of C. This contradiction establishes that such a cardinal cannot exist.

For his part, Ernst Zermelo in his ( 1908 ) A new proof of the possibility of a well-ordering ( published at the same time he published " the first axiomatic set theory ") laid claim to prior discovery of the antinomy in Cantor's naive set theory.

Peano was motivated by Georg Cantor's earlier counterintuitive result that the infinite number of points in a unit interval is the same cardinality as the infinite number of points in any finite-dimensional manifold, such as the unit square.

Cantor had apparently discovered the same paradox in his ( Cantor's ) " naive " set theory and this become known as Cantor's paradox.

A generalized form of the diagonal argument was used by Cantor to prove Cantor's theorem: for every set S the power set of S, i. e., the set of all subsets of S ( here written as P ( S )), is larger than S itself.

This fact is a direct consequence of Cantor's theorem on the cardinality of the power set of a set.

For other examples, see proof that the square root of 2 is not rational and Cantor's diagonal argument.

For a more complete account of this proof, see Cantor's theorem.

For a proof, see Cantor's first uncountability proof.

Since the cardinal numbers are well-ordered by indexing with the ordinal numbers ( see Cardinal number, formal definition ), this also establishes that there is no greatest ordinal number ; conversely, the latter statement implies Cantor's paradox.