In mathematics, subtle cardinals and ethereal cardinals are closely related kinds of large cardinal number.
is called subtle if for every closed and unbounded
δ < κ
th element), there exist
α , β
α < β
is called ethereal if for every closed and unbounded
δ < κ
δ < κ
α , β
α < β
[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 Some equivalent properties to subtlety are known.
Subtle cardinals are equivalent to a weak form of Vopěnka cardinals.
Namely, an inaccessible cardinal
, any logic has stationarily many weak compactness cardinals.
[2] 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
is a proper subset of
[3]Corollary 2.6 An infinite ordinal
λ < κ
, every transitive set
includes a chain (under inclusion) of order type
A hypersubtle cardinal is a subtle cardinal which has a stationary set of subtle cardinals below it.
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