Kurt Gödel Research Center

2020 seminar talk: Structural reflection and shrewd cardinals

Talk held by Philipp Lücke (University of Barcelona, Spain) at the KGRC seminar on 2020-10-29.

Abstract

In my talk, I want to present work dealing with the interplay between extensions of the Downward Löwenheim–Skolem Theorem to strong logics, large cardinal axioms and set-theoretic reflection principles, focussing on the characterization of large cardinal notions through model- and set-theoretic reflection properties. The work of Bagaria and his collaborators shows that various important objects in the middle and upper reaches of the large cardinal hierarchy can be characterized through principles of structural reflection. I will discuss recent results dealing with possible characterizations of notions from the lower part of this hierarchy through the principle $\mathrm{SR}^-$, introduced by Bagaria and Väänänen. These results show that the principle $\mathrm{SR}^-$ is closely connected to the notion of shrewd cardinals, introduced by Rathjen in a proof-theoretic context, and embedding characterizations of these cardinals that resembles Magidor's classical characterization of supercompactness.

The slides for this talk are available.

Time and Place

Talk at 3:00pm via Zoom