# 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.