Subtle cardinal

In mathematics, a cardinal κ is subtle iff for every closed and unbounded C ⊂ κ and for every sequence A of length κ for which element number δ (for an arbirary δ), A_δ ⊂ δ there are α, β belonging to C such that A_α=A_β∩α.

Theorem: There is a subtle cardinal ≤κ iff 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 &ne {Ø}. An infinite ordinal κ is subtle iff for every λ<κ, every transitive set S of cardinality κ includes a chain (under inclusion) of order type λ.

Subtle cardinals are a type of large cardinal.