# Logic Seminar: Stationary reflection and singular cardinals

**Author: Lona** | Image: Lona

**Author: Lona** | Image: Lona

Speaker: Dima Sinapova (University of Illinois at Chicago)

Abstract: Two classical results of Magidor are:

- from large cardinals it is consistent to have reflection at $_{ω+}$, and
- from large cardinals it is consistent to have the failure of the singular cardinal hypothesis (SCH) at $_{ω}$.

These principles are at odds with each other. The former is a compactness type principle. (Compactness is the phenomenon where if a certain property holds for every smaller substructure of an object, then it holds for the entire object.) In contrast, failure of SCH is an instance of incompactness. The natural question is whether we can have both of these simultaneously. We show the answer is yes.

This is joint work with Alejandro Poveda and Assaf Rinot.