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 (where is the th element), 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.W. Boney, S. Dimopoulos, V. Gitman, M. Magidor "Model Theoretic Characterizations of Large Cardinals Revisited" (2023).
Vopěnka'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 .H. Friedman, "Primitive Independence Results" (2002). Accessed 18 April 2024.Corollary 2.6 If a cardinal is subtle, then for every , every transitive set of cardinality includes a chain (under inclusion) of order type .Theorem 2.2
Extensions A hypersubtle cardinal is a subtle cardinal which has a stationary set of subtle cardinals below it.C. Henrion, "Properties of Subtle Cardinals. Journal of Symbolic Logic, vol. 52, no. 4 (1987), pp.1005--1019."p.1014
See also List of large cardinal properties
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