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 $$A$$ of length $$\kappa$$ such that $$A_\delta\subset\delta$$ for arbitrary $$\delta<\kappa$$ (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 $$A$$ 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 $$\textrm{card}(\alpha)=\mathrm{card}(A_\beta\cup A_\alpha)$$.

Subtle cardinals were introduced by. Ethereal cardinals were introduced by. 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 $$\leq\kappa$$ 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 $$x\neq\varnothing$$ and $$x\neq\{\varnothing\}$$. Corollary 2.6 An infinite ordinal $$\kappa$$ is subtle if and only if for every $$\lambda<\kappa$$, every transitive set $$S$$ of cardinality $$\kappa$$ includes a chain (under inclusion) of order type $$\lambda$$.

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