Upward Löwenheim-Skolem numbers for abstract logics

This is a talk at the CUNY Logic Workshop, CUNY Graduate Center, November 10, 2023.
Slides

Galeotti, Khomskii and Väänänen recently introduced the notion of the upward Löwenheim Skolem (ULS) number for an abstract logic. A cardinal $\kappa$ is the upward Lowenheim Skolem number for a logic $\mathcal L$ if it is the least cardinal with the property that whenever $M$ is a model of size at least $\kappa$ satisfying a sentence $\varphi$ in $\mathcal L$, then there are arbitrarily large models $N$ satisfying $\varphi$ and having $M$ as a substructure (not necessarily elementary). If we remove the requirement that $M$ has to be a substructure of $N$, we get the classic notion of a Hanf number. While $\rm ZFC$ proves that every logic has a Hanf number, having a ULS number often turns out to have large cardinal strength. In a joint work with Jonathan Osinski, we study the ULS numbers for several classical logics. We introduce a strengthening of the ULS number, the strong upward Löwenheim Skolem number SULS which strengthens the requirement that $M$ is a substructure to full elementarity in the logic $\mathcal L$. It is easy to see that both the ULS and the SULS number for a logic $\mathcal L$ are bounded by the least strong compactness cardinal for $\mathcal L$, if it exists.