# Talk: Level-by-level virtual large cardinals

I am giving an invited talk at the set theory seminar at the City University of New York, on Friday 15 February. I will be talking about virtual large cardinals, which grew out of my previous work on Ramsey-like cardinals. Here’s an abstract:

“A virtual large cardinal is (usually) the critical point of a generic elementary embedding from a rank-initial segment of the universe into a transitive $M\subseteq V$, as introduced by Gitman and Schindler (2018). A notable feature is that all virtual large cardinals are consistent with $V=L$, and they’ve proven useful in characterising several properties in descriptive set theory. We’ll work with the virtually $\theta$-measurable, $\theta$-strong and $\theta$-supercompact cardinals, where the $\theta$ in particular indicates that the generic embeddings have $H_\theta^V$ as domain, and investigate how these level-by-level virtual large cardinals relate both to each other and to the existence of winning strategies in certain games. This is work in progress and joint with Philipp Schlicht.”