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.

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 ≠ {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.

References

• Friedman, Harvey (2001), "Subtle Cardinals and Linear Orderings", Annals of Pure and Applied Logic, 107 (1–3): 1–34, doi:10.1016/S0168-0072(00)00019-1 
• Jensen, R. B.; Kunen, K. (1969), Some Combinatorial Properties of L and V, Unpublished manuscript 
• Ketonen, Jussi (1974), "Some combinatorial principles", Transactions of the American Mathematical Society, Transactions of the American Mathematical Society, Vol. 188, 188: 387–394, doi:10.2307/1996785, ISSN 0002-9947, JSTOR 1996785, MR 0332481