|
In mathematics, a weakly compact cardinal is a certain kind of cardinal number introduced by ; weakly compact cardinals are large cardinals, meaning that their existence cannot be proven from the standard axioms of set theory. Formally, a cardinal κ is defined to be weakly compact if it is uncountable and for every function ''f'': () 2 → there is a set of cardinality κ that is homogeneous for ''f''. In this context, () 2 means the set of 2-element subsets of κ, and a subset ''S'' of κ is homogeneous for ''f'' if and only if either all of ()2 maps to 0 or all of it maps to 1. The name "weakly compact" refers to the fact that if a cardinal is weakly compact then a certain related infinitary language satisfies a version of the compactness theorem; see below. Every weakly compact cardinal is a reflecting cardinal, and is also a limit of reflecting cardinals. This means also that weakly compact cardinals are Mahlo cardinals, and the set of Mahlo cardinals less than a given weakly compact cardinal is stationary. Some authors use a weaker definition of weakly compact cardinals, such as one of the conditions below with the condition of inaccessibility dropped. == Equivalent formulations == The following are equivalent for any uncountable cardinal κ: # κ is weakly compact. # for every λ<κ, natural number n ≥ 2, and function f: ()n → λ, there is a set of cardinality κ that is homogeneous for f. # κ is inaccessible and has the tree property, that is, every tree of height κ has either a level of size κ or a branch of size κ. # Every linear order of cardinality κ has an ascending or a descending sequence of order type κ. # κ is -indescribable. # κ has the extension property. In other words, for all ''U'' ⊂ ''V''κ there exists a transitive set ''X'' with κ ∈ ''X'', and a subset ''S'' ⊂ ''X'', such that (''V''κ, ∈, ''U'') is an elementary substructure of (''X'', ∈, ''S''). Here, ''U'' and ''S'' are regarded as unary predicates. # For every set S of cardinality κ of subsets of κ, there is a non-trivial κ-complete filter that decides S. # κ is κ-unfoldable. # κ is inaccessible and the infinitary language ''L''κ,κ satisfies the weak compactness theorem. # κ is inaccessible and the infinitary language ''L''κ,ω satisfies the weak compactness theorem. A language ''L''κ,κ is said to satisfy the weak compactness theorem if whenever Σ is a set of sentences of cardinality at most κ and every subset with less than κ elements has a model, then Σ has a model. Strongly compact cardinals are defined in a similar way without the restriction on the cardinality of the set of sentences. 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「Weakly compact cardinal」の詳細全文を読む スポンサード リンク
|