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In fact, the cardinality of ℘ (), by definition , is equal to . This can be shown by providing one-to-one mappings in both directions between subsets of a countably infinite set and real numbers, and applying the Cantor–Bernstein–Schroeder theorem according to which two sets with one-to-one mappings in both directions have the same ...
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 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 ...
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 ...
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 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 cardinal number of X itself is often defined as the least ordinal a with |a| = |X|. [2] This is called the von Neumann cardinal assignment; for this definition to make sense, it must be proved that every set has the same cardinality as some ordinal; this statement is the well-ordering principle. It is however possible to discuss the ...
This follows from the fact that a continuous function is completely determined by its value on a dense subset of its domain. [2] Thus, the cardinality of the set of continuous real-valued functions on the reals is no greater than the cardinality of the set of real-valued functions of a rational variable. By cardinal arithmetic: