Subtle cardinal

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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 Aα=Aβ∩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(AβAα).

Subtle cardinals were introduced by Jensen & Kunen (1969). Ethereal cardinals were introduced by Ketonen (1974). Any subtle cardinal is ethereal, and any strongly inaccessible ethereal cardinal is subtle.

Theorem[edit]

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 ≠ {Ø}. 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.

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