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, Aδ ⊂ δ there are α, β, belonging to C, with α < β, such that Aα = Aβ ∩ 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, Aδ ⊂ δ and Aδ has the same cardinal as δ, there are α, β, belonging to C, with α < β, such that card = card.
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 ≠. An infinite ordinal κ is subtle if and only if for every λ < κ, every transitive set S of cardinality κ includes a chain of order type λ.
