Upward Löwenheim-Skolem-Tarski numbers for abstract logics

Galeotti, Khomskii and Väänänen recently introduced the notion of the upward Löwenheim-Skolem-Tarski number for a logic, strengthening the classical notion of a Hanf number. A cardinal κ is the upward Löwenheim-Skolem-Tarski number (ULST number) of a logic $L$ if it is the least cardinal with the property that whenever M is a model of size at least κ satisfying a sentence φ in $L$, then there are arbitrarily large models satisfying φ and having M as a substructure. The substructure requirement is what differentiates the ULST number from the Hanf number and gives the notion large cardinal strength. While it is a theorem of ZFC that every logic has a Hanf number, Galeotti, Khomskii and Väänänen showed that the existence of the ULST number for second-order logic implies the existence of a partially extendible cardinal. We answer positively their conjecture that the ULST number for second-order logic is the least extendible cardinal.

We define the strong ULST number by strengthening the substructure requirement to elementary substructure. We investigate the ULST and strong ULST numbers for several classical strong logics: infinitary logics, the equicardinality logic, logic with the well-foundedness quantifier, second-order logic, and sort logics. We show that the ULST and the strong ULST numbers are characterized in some cases by classical large cardinals and in some cases by natural new large cardinal notions that they give rise to. We show that for some logics the notions of the ULST number, strong ULST number and least strong compactness cardinal coincide, while for others, it is consistent that they can be separated. Finally, we introduce a natural large cardinal notion characterizing strong compactness cardinals for the equicardinality logic.