Subtle cardinal

In mathematics, subtle cardinals and ethereal cardinals are closely related kinds of large cardinal number.

A cardinal ${\displaystyle \kappa }$is called subtle if for every closed and unbounded ${\displaystyle C\subset \kappa }$and for every sequence ${\displaystyle (A_{\delta })_{\delta <\kappa }}$of length ${\displaystyle \kappa }$such that ${\displaystyle A_{\delta }\subset \delta }$for all ${\displaystyle \delta <\kappa }$(where ${\displaystyle A_{\delta }}$is the ${\displaystyle \delta }$th element), there exist ${\displaystyle \alpha ,\beta }$, belonging to ${\displaystyle C}$, with ${\displaystyle \alpha <\beta }$, such that ${\displaystyle A_{\alpha }=A_{\beta }\cap \alpha }$.

A cardinal ${\displaystyle \kappa }$is called ethereal if for every closed and unbounded ${\displaystyle C\subset \kappa }$and for every sequence ${\displaystyle (A_{\delta })_{\delta <\kappa }}$of length ${\displaystyle \kappa }$such that ${\displaystyle A_{\delta }\subset \delta }$and ${\displaystyle A_{\delta }}$has the same cardinality as ${\displaystyle \delta }$for arbitrary ${\displaystyle \delta <\kappa }$, there exist ${\displaystyle \alpha ,\beta }$, belonging to ${\displaystyle C}$, with ${\displaystyle \alpha <\beta }$, such that ${\displaystyle {\textrm {card}}(\alpha )=\mathrm {card} (A_{\beta }\cup A_{\alpha })}$.

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}

Some equivalent properties to subtlety are known.

Subtle cardinals are equivalent to a weak form of Vopěnka cardinals. Namely, an inaccessible cardinal ${\displaystyle \kappa }$is subtle if and only if in ${\displaystyle V_{\kappa +1}}$, 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 ${\displaystyle \leq \kappa }$if and only if every transitive set ${\displaystyle S}$of cardinality ${\displaystyle \kappa }$contains ${\displaystyle x}$and ${\displaystyle y}$such that ${\displaystyle x}$is a proper subset of ${\displaystyle y}$and ${\displaystyle x\neq \varnothing }$and ${\displaystyle x\neq \{\varnothing \}}$.^{Corollary 2.6} If a cardinal ${\displaystyle \lambda }$is subtle, then for every ${\displaystyle \alpha <\lambda }$, every transitive set ${\displaystyle S}$of cardinality ${\displaystyle \lambda }$includes a chain (under inclusion) of order type ${\displaystyle \alpha }$.^{Theorem 2.2}

A hypersubtle cardinal is a subtle cardinal which has a stationary set of subtle cardinals below it.^p.1014

• List of large cardinal properties