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Symbolically, if the cardinality of is denoted as , the cardinality of the continuum is c = 2 ℵ 0 > ℵ 0 . {\displaystyle {\mathfrak {c}}=2^{\aleph _{0}}>\aleph _{0}.} This was proven by Georg Cantor in his uncountability proof of 1874, part of his groundbreaking study of different infinities.
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.
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}} .
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 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.
When two sets, and , have the same cardinality, it is usually written as | | = | |; however, if referring to the cardinal number of an individual set , it is simply denoted | |, with a vertical bar on each side; [3] this is the same notation as absolute value, and the meaning depends on context.
Thus X has cardinality at least . If X is a separable , complete metric space with no isolated points, the cardinality of X is exactly 2 ℵ 0 {\displaystyle 2^{\aleph _{0}}} . If X is a locally compact Hausdorff space with no isolated points, there is an injective function (not necessarily continuous) from Cantor space to X , and so X has ...
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 ...