Subtle cardinal

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

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

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

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 $\kappa$ is subtle if and only if in $V_{\kappa +1}$, 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.

Chains in transitive sets

There is a subtle cardinal $\le \kappa$ if and only if every transitive set $S$ of cardinality $\kappa$ contains $x$ and $y$ such that $x$ is a proper subset of $y$ and $x\ne \varnothing$ and $x\ne \{\varnothing \}$.^{Corollary 2.6} If a cardinal $\lambda$ is subtle, then for every $\alpha <\lambda$, every transitive set $S$ of cardinality $\lambda$ includes a chain (under inclusion) of order type $\alpha$.^{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