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And we can map the powerset of into the Cantor set, a subset of the real numbers. So statements about H ℵ 1 {\displaystyle H_{\aleph _{1}}} can be converted into statements about the reals. Therefore, H ℵ 1 ⊂ L ( R ) {\displaystyle H_{\aleph _{1}}\subset L(R)} , where L ( R ) is the smallest transitive inner model of ZF containing all the ...
All subsets of a set S (all possible choices of its elements) form the power set P(S). Georg Cantor proved that the power set is always larger than the set, i.e., |P(S)| > |S|. A special case of Cantor's theorem is that the set of all real numbers R cannot be enumerated by natural numbers, that is, R is uncountable: |R| > |N|.
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}} .
the power set of the set of real numbers, so it is the number of subsets of the real line, or the number of sets of real numbers the power set of the power set of the set of natural numbers the set of all functions from R {\displaystyle \mathbb {R} } to R {\displaystyle \mathbb {R} } ( R R {\displaystyle \mathbb {R} ^{\mathbb {R} }} )
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 cardinality of the set of real numbers (cardinality of the continuum) is 2 ℵ 0. It cannot be determined from ZFC (Zermelo–Fraenkel set theory augmented with the axiom of choice) where this number fits exactly in the aleph number hierarchy, but it follows from ZFC that the continuum hypothesis (CH) is equivalent to the identity 2 ℵ 0 ...
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 goal of a cardinal assignment is to assign to every set A a specific, unique set that is only dependent on the cardinality of A. This is in accordance with Cantor 's original vision of cardinals: to take a set and abstract its elements into canonical "units" and collect these units into another set, such that the only thing special about ...