In mathematics, subtle cardinals and ethereal cardinals are closely related kinds of large cardinal number. A cardinal \kappa is called subtle if for every closed and unbounded and for every sequence of length \kappa such that for all (where A_\delta is the \deltath element), there exist, belonging to C, with , such that. A cardinal \kappa is called ethereal if for every closed and unbounded and for every sequence of length \kappa such that and A_\delta has the same cardinality as \delta for arbitrary, there exist , belonging to C, 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 \kappa 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 \leq\kappa if and only if every transitive set S of cardinality \kappa contains x and y such that x is a proper subset of y and and. Corollary 2.6 An infinite ordinal \kappa is subtle if and only if for every, every transitive set S of cardinality \kappa includes a chain (under inclusion) of order type \lambda.

Extensions

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

Citations