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 ${\displaystyle {\textrm {card}}(\alpha )=\mathrm {card} (A_{\beta }\cup A_{\alpha })}$.¹

Subtle cardinals were introduced by Jensen & Kunen (1969). Ethereal cardinals were introduced by Ketonen (1974). Any subtle cardinal is ethereal,^{1p. 388} and any strongly inaccessible ethereal cardinal is subtle.^{1p. 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.²

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 ${\displaystyle x\neq \{\varnothing \}}$.^{3Corollary 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 ${\displaystyle \alpha }$.^{3Theorem 2.2}

A hypersubtle cardinal is a subtle cardinal which has a stationary set of subtle cardinals below it.^4p.1014