Subtle cardinal

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), A_δ ⊂ δ, there exist 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), A_δ ⊂ δ and A_δ has the same cardinal as δ, there exist α, β, belonging to C, with α < β, such that card(α) = card(A_β ∩ A_α).

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.

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