# 2019 seminar talk: Capturing by normal ultrapowers

Talk held by Miha Habič (Czech Technical University in Prague, Czech Republic and Charles University, Prague, Czech Republic) at the KGRC seminar on 2019-03-28.

### Abstract

Title: If $\kappa$ is measurable and GCH holds, then any ultrapower by a normal measure on $\kappa$ will be missing some subset of $\kappa^+$. On the other hand, Cummings showed that, starting from a $(\kappa+2)$-strong $\kappa$, one can force to a model (without collapsing cardinals) where $\kappa$ carries a normal measure whose ultrapower captures the entire powerset of $\kappa^+$. Moreover, the large cardinal hypothesis is optimal. I will present an improvement of Cummings' result and show that this capturing property can consistently hold at the least measurable cardinal.

This is joint work with Radek Honzík.

There is a video recording of this talk available on YouTube.

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