Weak compactness cardinals for strong logics and subtlety properties of the class of ordinals

Abstract

Motivated by recent work of Boney, Dimopoulos, Gitman, and Magidor, we characterize the existence of weak compactness cardinals for all abstract logics through combinatorial properties of the class of ordinals. This analysis is then used to show that, in contrast to the existence of strong compactness cardinals, the existence of weak compactness cardinals for abstract logics does not imply the existence of a strongly inaccessible cardinal. More precisely, it is proven that the existence of a proper class of subtle cardinals is consistent with the axioms of $\mathrm{ZFC}$ if and only if it is not possible to derive the existence of strongly inaccessible cardinals from the existence of weak compactness cardinals for all abstract logics. Complementing this result, it is shown that the existence of weak compactness cardinals for all abstract logics implies that unboundedly many ordinals are strongly inaccessible in the inner model $\mathrm{HOD}$ of all hereditarily ordinal definable sets.