Subtle cardinal
In mathematics, subtle cardinals and ethereal cardinals are closely related kinds of large cardinal number.
A cardinal <math>\kappa</math> is called subtle if for every closed and unbounded <math>C\subset\kappa</math> and for every sequence <math>(A_\delta)_{\delta<\kappa}</math> of length <math>\kappa</math> such that <math>A_\delta\subset\delta</math> for all <math>\delta<\kappa</math> (where <math>A_\delta</math> is the <math>\delta</math>th element), there exist <math>\alpha,\beta</math>, belonging to <math>C</math>, with <math>\alpha<\beta</math>, such that <math>A_\alpha=A_\beta\cap\alpha</math>.
A cardinal <math>\kappa</math> is called ethereal if for every closed and unbounded <math>C\subset\kappa</math> and for every sequence <math>(A_\delta)_{\delta<\kappa}</math> of length <math>\kappa</math> such that <math>A_\delta\subset\delta</math> and <math>A_\delta</math> has the same cardinality as <math>\delta</math> for arbitrary <math>\delta<\kappa</math>, there exist <math>\alpha,\beta</math>, belonging to <math>C</math>, with <math>\alpha<\beta</math>, such that <math>\textrm{card}(\alpha)=\mathrm{card}(A_\beta\cup A_\alpha)</math>.<ref name="Ketonen74">Template:Citation</ref>
Subtle cardinals were introduced by Template:Harvtxt. Ethereal cardinals were introduced by Template:Harvtxt. Any subtle cardinal is ethereal,<ref name="Ketonen74" />p. 388 and any strongly inaccessible ethereal cardinal is subtle.<ref name="Ketonen74" />p. 391
CharacterizationsEdit
Some equivalent properties to subtlety are known.
Relationship to Vopěnka's PrincipleEdit
Subtle cardinals are equivalent to a weak form of Vopěnka cardinals. Namely, an inaccessible cardinal <math>\kappa</math> is subtle if and only if in <math>V_{\kappa+1}</math>, any logic has stationarily many weak compactness cardinals.<ref>W. Boney, S. Dimopoulos, V. Gitman, M. Magidor "Model Theoretic Characterizations of Large Cardinals Revisited" (2023).</ref>
Vopenka's principle itself may be stated as the existence of a strong compactness cardinal for each logic.
Chains in transitive setsEdit
There is a subtle cardinal <math>\leq\kappa</math> if and only if every transitive set <math>S</math> of cardinality <math>\kappa</math> contains <math>x</math> and <math>y</math> such that <math>x</math> is a proper subset of <math>y</math> and <math>x\neq\varnothing</math> and <math>x\neq\{\varnothing\}</math>.<ref name="Friedman02">H. Friedman, "Primitive Independence Results" (2002). Accessed 18 April 2024.</ref>Corollary 2.6 If a cardinal <math>\lambda</math> is subtle, then for every <math>\alpha<\lambda</math>, every transitive set <math>S</math> of cardinality <math>\lambda</math> includes a chain (under inclusion) of order type <math>\alpha</math>.<ref name="Friedman02" />Theorem 2.2
ExtensionsEdit
A hypersubtle cardinal is a subtle cardinal which has a stationary set of subtle cardinals below it.<ref name="Henrion87">C. Henrion, "Properties of Subtle Cardinals. Journal of Symbolic Logic, vol. 52, no. 4 (1987), pp.1005--1019."</ref>p.1014