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 arbitrary (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 .[1]
Subtle cardinals were introduced by Jensen & Kunen (1969). Ethereal cardinals were introduced by Ketonen (1974). Any subtle cardinal is ethereal,[1]p. 388 and any strongly inaccessible ethereal cardinal is subtle.[1]p. 391
^ abcKetonen, Jussi (1974), "Some combinatorial principles" (PDF), Transactions of the American Mathematical Society, 188, Transactions of the American Mathematical Society, Vol. 188: 387–394, doi:10.2307/1996785, ISSN 0002-9947, JSTOR 1996785, MR 0332481
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