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In set theory, the cardinality of the continuum is the cardinality or "size" of the set of real numbers ... the set of all continuous functions from ...
The cardinality of the continuum is the size of the set of real numbers. The continuum hypothesis is sometimes stated by saying that no cardinality lies between that of the continuum and that of the natural numbers , ℵ 0 {\displaystyle \aleph _{0}} , or alternatively, that c = ℵ 1 {\displaystyle {\mathfrak {c}}=\aleph _{1}} .
As is standard in set theory, we denote by the least infinite ordinal, which has cardinality ; it may be identified with the set of natural numbers.. A number of cardinal characteristics naturally arise as cardinal invariants for ideals which are closely connected with the structure of the reals, such as the ideal of Lebesgue null sets and the ideal of meagre sets.
The independence proof just described shows that CH is independent of ZFC. Further research has shown that CH is independent of all known large cardinal axioms in the context of ZFC. [8] Moreover, it has been shown that the cardinality of the continuum can be any cardinal consistent with König's theorem.
The cardinality of is often called the cardinality of the continuum, and denoted by , or , or . The Cantor set is an uncountable subset of R {\displaystyle \mathbb {R} } . The Cantor set is a fractal and has Hausdorff dimension greater than zero but less than one ( R {\displaystyle \mathbb {R} } has dimension one).
The continuum hypothesis posits that the cardinality of the set of the real numbers is ; i.e. the smallest infinite cardinal number after , the cardinality of the integers. Paul Cohen proved in 1963 that it is an axiom independent of the other axioms of set theory; that is: one may choose either the continuum hypothesis or its negation as an ...
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A standard Borel space is characterized up to isomorphism by its cardinality, [3] and any uncountable standard Borel space has the cardinality of the continuum. For subsets of Polish spaces, Borel sets can be characterized as those sets that are the ranges of continuous injective maps defined on Polish spaces.