Subtle Cardinal

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