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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 ...
The best known example of an uncountable set is the set of all real numbers; Cantor's diagonal argument shows that this set is uncountable. The diagonalization proof technique can also be used to show that several other sets are uncountable, such as the set of all infinite sequences of natural numbers (see: (sequence A102288 in the OEIS)), and the set of all subsets of the set ...
Cantor defined cardinality in terms of bijective functions: two sets have the same cardinality if, and only if, there exists a bijective function between them. Between any two real numbers a < b , no matter how close they are to each other, there are always infinitely many other real numbers, and Cantor showed that they are as many as those ...
By defining the notion of the size of a set in terms of its cardinality, the issue can be settled. Since there is a bijection between the two sets involved, this follows in fact directly from the definition of the cardinality of a set. Hilbert's paradox of the Grand Hotel illustrates more paradoxes of enumeration.
As a consequence, the cardinality of the real numbers, which is the same as that of the power set of the integers, is strictly larger than the cardinality of the integers; see Cardinality of the continuum for details. The theorem is named for Georg Cantor, who first stated and proved it at the end of the 19th century.
The cardinality of a set A is defined as its equivalence class under equinumerosity. A representative set is designated for each equivalence class. The most common choice is the initial ordinal in that class. This is usually taken as the definition of cardinal number in axiomatic set theory.
The proof of Cantor's second theorem came from Dedekind. However, it omits Dedekind's explanation of why the limits a ∞ and b ∞ exist. [45] Cantor restricted his first theorem to the set of real algebraic numbers. The proof he was using demonstrates the countability of the set of all algebraic numbers. [20]
The above definition of the Baire space generalizes to one where the elements of the countably infinite sequence (,,) are chosen from a set () of cardinality. Such a space is called a Baire space of weight κ {\displaystyle \kappa } and can be denoted as B ( κ ) {\displaystyle B(\kappa )} . [ 1 ]