## Logic Seminar

Dima Sinapova

UIC

Ordinal definable subsets of singular cardinals

**Abstract:**A remarkable theorem of Shelah states that if $\kappa$ is a singular strong limit cardinal of uncountable cofinality, then there is a subset $x$ of $\kappa$, such that $HOD_x$ contains the powerset of $\kappa$. We show that in general this is not the case for countable cofinality. Using a version of diagonal supercompact extender Prikry forcing, we construct a generic extension in which there is a singular cardinal $\kappa$ with countable cofinality, such that $\kappa^+$ is supercompact in $HOD_x$ for all $x\subset\kappa$. This result was obtained during a SQuaRE meeting at AIM and is joint with Cummings, Friedman, Magidor, and Rinot.

Tuesday September 27, 2016 at 4:00 PM in SEO 427