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,<sup>p. 388</sup> and any strongly inaccessible ethereal cardinal is subtle.<sup>p. 391</sup>
Some equivalent properties to subtlety are known.
Subtle cardinals are equivalent to a weak form of VopÃÂnka 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.
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 .<sup>Corollary 2.6</sup> If a cardinal is subtle, then for every , every transitive set of cardinality includes a chain (under inclusion) of order type .<sup>Theorem 2.2</sup>
A hypersubtle cardinal is a subtle cardinal which has a stationary set of subtle cardinals below it.<sup>p.1014</sup>