In mathematics, subtle cardinals and ethereal cardinals are closely related kinds of large cardinal number.

is called subtle if for every closed and unbounded.

.

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, any logic has stationarily many weak compactness cardinals.

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

Chains in transitive sets

.Corollary 2.6.Theorem 2.2

Extensions

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

See also

• List of large cardinal properties

References

Citations