**March 17**

Jonathan Osinski,
University of Hamburg

**Model-Theoretic Characterizations of Weak Vopěnka's Principle**

It has been known since the 1980s that Vopěnka's Principle (VP) is equivalent to certain statements about logics, e.g. to the schema 'Every logic has a compactness cardinal.' On the other hand, it was only recently shown by Trevor Wilson that a related statement statement called Weak Vopěnka's Principle (WVP) is strictly weaker than VP. In fact, Joan Bagaria and Wilson showed that WVP is equivalent to the existence of $\Pi_n$-strong cardinals for all natural numbers $n$. We generalize logical characterizations of strong cardinals to achieve a characterization of $\Pi_n$-strong cardinals and therefore of WVP in terms of properties of strong logics. This is partly joint work with Will Boney and partly with Trevor Wilson.