In mathematics, subtle cardinals and ethereal cardinals are closely related kinds of large cardinal number. A cardinal κ {\displaystyle \kappa } is called subtle if for every closed and unbounded C ⊂ κ {\displaystyle C\subset \kappa } and for every sequence ( A δ ) δ < κ {\displaystyle (A_{\delta })_{\delta <\kappa }} of length κ {\displaystyle \kappa } such that A δ ⊂ δ {\displaystyle A_{\delta }\subset \delta } for all δ < κ {\displaystyle \delta <\kappa } (where A δ {\displaystyle A_{\delta }} is the δ {\displaystyle \delta } th element), there exist α , β {\displaystyle \alpha ,\beta } , belonging to C {\displaystyle C} , with α < β {\displaystyle \alpha <\beta } , such that A α = A β ∩ α {\displaystyle A_{\alpha }=A_{\beta }\cap \alpha } . A cardinal κ {\displaystyle \kappa } is called ethereal if for every closed and unbounded C ⊂ κ {\displaystyle C\subset \kappa } and for every sequence ( A δ ) δ < κ {\displaystyle (A_{\delta })_{\delta <\kappa }} of length κ {\displaystyle \kappa } such that A δ ⊂ δ {\displaystyle A_{\delta }\subset \delta } and A δ {\displaystyle A_{\delta }} has the same cardinality as δ {\displaystyle \delta } for arbitrary δ < κ {\displaystyle \delta <\kappa } , there exist α , β {\displaystyle \alpha ,\beta } , belonging to C {\displaystyle C} , with α < β {\displaystyle \alpha <\beta } , such that card ( α ) = c a r d ( A β ∪ A α ) {\displaystyle {\textrm {card}}(\alpha )=\mathrm {card} (A_{\beta }\cup A_{\alpha })} . Subtle cardinals were introduced by Jensen & Kunen (1969). Ethereal cardinals were introduced by Ketonen (1974). 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 κ {\displaystyle \kappa } is subtle if and only if in V κ + 1 {\displaystyle V_{\kappa +1}} , 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 There is a subtle cardinal ≤ κ {\displaystyle \leq \kappa } if and only if every transitive set S {\displaystyle S} of cardinality κ {\displaystyle \kappa } contains x {\displaystyle x} and y {\displaystyle y} such that x {\displaystyle x} is a proper subset of y {\displaystyle y} and x ≠ ∅ {\displaystyle x\neq \varnothing } and x ≠ { ∅ } {\displaystyle x\neq \{\varnothing \}} .Corollary 2.6 If a cardinal λ {\displaystyle \lambda } is subtle, then for every α < λ {\displaystyle \alpha <\lambda } , every transitive set S {\displaystyle S} of cardinality λ {\displaystyle \lambda } includes a chain (under inclusion) of order type α {\displaystyle \alpha } .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
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