Subtle cardinal

In mathematics, subtle cardinals and ethereal cardinals are closely related kinds of large cardinal number.
A cardinal is called subtle if for every closed and unbounded and for every sequence of length such that for all , there exist, belonging to, with, such that.
A cardinal is called ethereal if for every closed and unbounded and for every sequence of length such that and has the same cardinality as for arbitrary, there exist, belonging to, with, such that.
Subtle cardinals were introduced by. Ethereal cardinals were introduced by. Any subtle cardinal is ethereal,^{p. 388} and any strongly inaccessible ethereal cardinal is subtle.^{p. 391}

Characterizations

Some equivalent properties to subtlety are known.

Relationship to Vopěnka's Principle

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.
Vopenka's principle itself may be stated as the existence of a strong compactness cardinal for each logic.

Chains in transitive sets

There is a subtle cardinal if and only if every transitive set of cardinality contains and such that is a proper subset of and and.^{Corollary 2.6} If a cardinal is subtle, then for every, every transitive set of cardinality includes a chain of order type.^{Theorem 2.2}

Extensions

A hypersubtle cardinal is a subtle cardinal which has a stationary set of subtle cardinals below it.^p.1014