In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, 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 .
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 ≠ {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.