Logic Seminar
Matthew Foreman
University of California Irvine
Games on weakly compact cardinals
Abstract: Weakly compact cardinals are equivalent to the statement that every $\kappa$-complete filter on a Boolean algebra $\mathcal{B}$ of size $\kappa$ can be extended to a $\kappa$-complete ultrafilter on $\mathcal{B}$. One can continue this finitely many times. Can it be continued transfinitely?
Fix a cardinal $\kappa$ and consider the following game $\mathcal G_\gamma$ of ordinal length $\gamma$: Player I plays a a sequence of collections $\mathcal S_\alpha\subseteq P(\kappa)$ of size $\kappa$ and player II plays an increasing sequence of $\kappa$-complete ultrafilters $U_\alpha$ on $\bigcup_{\beta\le \alpha}\mathcal S_\beta$. Player II wins if she can continue playing until stage $\gamma$.
Clearly if $\kappa$ is measurable then II wins the game of any length. Welch asked whether the property that ``II has a winning strategy in $\mathcal G_\gamma$" can hold at a non-measurable cardinal.
The main result in this talk is that if II wins $\mathcal G_{\omega_1}$ then there is a precipitous ideal on $\kappa$ whose quotient has a countably closed dense subset. Hence the answer to Welch's question, at least for $\gamma\ge \omega_1$, is no.
In joint work with Magidor, we prove that it is consistent at a non-measurable cardinal for II to have a winning strategy in $\mathcal G_{\omega_1}$, hence the theorem is not vacuous.
Tuesday April 30, 2019 at 3:30 PM in 427 SEO