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In set theory, the cardinality of the continuum is the cardinality or "size" of the set of ... the set of all continuous functions from ... additional terms may apply.
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.
Hence, the set {banana, apple, pear} has the same cardinality as {yellow, red, green}. With infinite sets such as the set of integers or rational numbers, the existence of a bijection between two sets becomes more difficult to demonstrate. The rational numbers seemingly form a counterexample to the continuum hypothesis: the integers form a ...
In mathematics, the terms continuity, continuous, ... The latter cardinality is equal to the cardinality of the set of all subsets of a countably infinite set.
There are two ways to define the "cardinality of a set": 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 cardinality of the set is the first uncountable cardinal number, . The ordinal ω 1 {\displaystyle \omega _{1}} is thus the initial ordinal of ℵ 1 {\displaystyle \aleph _{1}} . Under the continuum hypothesis , the cardinality of ω 1 {\displaystyle \omega _{1}} is ℶ 1 {\displaystyle \beth _{1}} , the same as that of R {\displaystyle ...
In set theory, the concept of cardinality is significantly developable without recourse to actually defining cardinal numbers as objects in the theory itself (this is in fact a viewpoint taken by Frege; Frege cardinals are basically equivalence classes on the entire universe of sets, by equinumerosity).