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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}} .
In simple terms, the Continuum Hypothesis (CH) states that the set of real numbers has minimal possible cardinality which is greater than the cardinality of the set of integers. That is, every set S ⊆ R {\displaystyle S\subseteq \mathbb {R} } of real numbers can either be mapped one-to-one into the integers or the real numbers can be mapped ...
Cantor's diagonal argument shows that is strictly greater than , but it does not specify whether it is the least cardinal greater than (that is, ).Indeed the assumption that = is the well-known Continuum Hypothesis, which was shown to be consistent with the standard ZFC axioms for set theory by Kurt Gödel and to be independent of it by Paul Cohen.
An infinite set may have the same cardinality as a proper subset of itself, as the depicted bijection f(x)=2x from the natural to the even numbers demonstrates. Nevertheless, infinite sets of different cardinalities exist, as Cantor's diagonal argument shows.
The cardinality of the set () of such filter bases is ... The set of all real-valued continuous functions on a separable space has a cardinality equal to ...
The cardinality of the natural numbers is denoted aleph-null (), while the cardinality of the real numbers is denoted by "" (a lowercase fraktur script "c"), and is also referred to as the cardinality of the continuum.
Continuous probability distribution: Sometimes this term is used to mean a probability distribution whose cumulative distribution function (c.d.f.) is (simply) continuous. Sometimes it has a less inclusive meaning: a distribution whose c.d.f. is absolutely continuous with respect to Lebesgue measure. This less inclusive sense is equivalent to ...