Weak compactness cardinals for strong logics and subtlety properties of the class of ordinals
Philipp Lücke
TL;DR
The paper develops an ordinal-combinatorial framework linking abstract logics to large-cardinal phenomena, showing that weak compactness for abstract logics can hold without forcing strongly inaccessible cardinals. It identifies precise equivalences between various ordinal-subtlety notions and the existence of large cardinals within the $C^{(n)}$-hierarchy, notably connecting ${ m Ord}$ being essentially subtle to $C^{(n)}$-strongly unfoldable cardinals, and ${ m Ord}$ being essentially faint to stationary classes of $C^{(n)}$-weakly shrewd cardinals. It further demonstrates that the schemes governing weak compactness and LST properties for all abstract logics can be separated from, and are not provably equivalent to, single-sentence axiomatizations, with inner-model reflections in ${ m HOD}$ playing a central role. The results culminate in equiconsistency and separation theorems that illuminate the interplay between class-axioms, logics, and inner-model theory, and raise open questions about the SCH and eventual axiomatizability of subtlety properties. Overall, the work advances a robust framework connecting abstract logics, ordinal combinatorics, and large-cardinal phenomena, with implications for the limits of axiomatizability and the structure of the inner universe.
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 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 HOD of all hereditarily ordinal definable sets.