**December 11**

**Dima Sinapova**,
University of Chicago

**Iteration, reflection, and singular cardinals**

There is an inherent tension between stationary reflection and the failure of the singular cardinal hypothesis (SCH). The former is a compactness type principle that follows from large cardinals. 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.

Two classical results of Magidor are:

(1) from large cardinals it is consistent to have reflection at $\aleph_{\omega+1}$, and

(2) from large cardinals it is consistent to have the failure of SCH at $\aleph_\omega$.

As these principles are at odds with each other, the natural question is whether we can have both. We show the answer is yes.

We describe a Prikry style iteration, and use it to force stationary reflection in the presence of not SCH. Then we obtain this situation at $\aleph_\omega$ by interleaving collapses. This is joint work with Alejandro Poveda and Assaf Rinot.