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In mathematics, an Erdős cardinal, also called a partition cardinal is a certain kind of large cardinal number introduced by . The Erdős cardinal is defined to be the least cardinal such that for every function there is a set of order type that is homogeneous for (if such a cardinal exists). In the notation of the partition calculus, the Erdős cardinal is the smallest cardinal such that : Existence of zero sharp implies that the constructible universe satisfies "for every countable ordinal , there is an -Erdős cardinal". In fact, for every indiscernible satisfies "for every ordinal , there is an -Erdős cardinal in (the Levy collapse to make countable)". However, existence of an -Erdős cardinal implies existence of zero sharp. If is the satisfaction relation for (using ordinal parameters), then existence of zero sharp is equivalent to there being an -Erdős ordinal with respect to . And this in turn, the zero sharp implies the falsity of axiom of constructibility, of Kurt Gödel. If κ is -Erdős, then it is -Erdős in every transitive model satisfying " is countable". ==References== * * * * 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Erdős cardinal」の詳細全文を読む スポンサード リンク
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