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.