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In mathematics, subtle cardinals and ethereal cardinals are closely related kinds of large cardinal number. A cardinal κ is called subtle if for every closed and unbounded ''C'' ⊂ κ and for every sequence ''A'' of length κ for which element number δ (for an arbitrary δ), Aδ ⊂ δ there are α, β, belonging to ''C'', with α<β, such that ''A''α=''A''β∩α. A cardinal κ is called ethereal if for every closed and unbounded ''C'' ⊂ κ and for every sequence ''A'' of length κ for which element number δ (for an arbitrary δ), Aδ ⊂ δ and Aδ has the same cardinal as δ, there are α, β, belonging to ''C'', with α<β, such that card(α)=card(''A''β∩''A''α). Subtle cardinals were introduced by . Ethereal cardinals were introduced by . Any subtle cardinal is ethereal, and any strongly inaccessible ethereal cardinal is subtle. == Theorem == There is a subtle cardinal ≤κ if and only if every transitive set ''S'' of cardinality κ contains ''x'' and ''y'' such that ''x'' is a proper subset of ''y'' and ''x'' ≠ Ø and ''x'' ≠ . An infinite ordinal κ is subtle if and only if for every λ<κ, every transitive set ''S'' of cardinality κ includes a chain (under inclusion) of order type λ. 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「subtle cardinal」の詳細全文を読む スポンサード リンク
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