**October 14**

**Philipp Lücke**,
University of Barcelona

**Large cardinals, strong logics and reflection principles**

Various results establish deep connections between the existence of large cardinals, regularity properties of strong logics and the validity of set-theoretic reflection principles. In particular, several compactness properties of strong logics were proven to be equivalent to large cardinal axioms. An important example of such an equivalence is given by a theorem of Makowsky that shows that *Vopěnka's Principle* is equivalent to the existence of strong compactness cardinals for all abstract logics. Motivated by work of Boney, Dimopoulos, Gitman and Magidor, I recently proved an analogous combinatorial characterization of the existence of weak compactness cardinals for all abstract logics that is closely connected to the notion of *subtle cardinals*, introduced by Kunen and Jensen in their studies of strong diamond principles, and the concept of *shrewd cardinals*, defined by Rathjen in proof-theoretic work. In my talk, I want to first discuss the details of this characterization and then present connections to recent joint work with Joan Bagaria (Barcelona) on recurring patterns in the large cardinal hierarchy.