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,^{[1]p. 388} and any strongly inaccessible ethereal cardinal is subtle.^{[1]p. 391}