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), Aδ ⊂ δ, there exist 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 (for an arbitrary δ), Aδ ⊂ δ and Aδ has the same cardinal as δ, there exist α, β, belonging to C, with α < β, such that card(α) = card(Aβ ∩ Aα).
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.
Subtle cardinals are equivalent to a weak form of Vopěnka cardinals. Namely, an inaccessible cardinal is subtle if and only if in , any logic has stationarily many weak compactness cardinals.[1]
Vopenka's principle itself may be stated as the existence of a strong compactness cardinal for each logic.
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}.[2]Corollary 2.6 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.
A hypersubtle cardinal is a subtle cardinal which has a stationary set of subtle cardinals below it.[3]p.1014