**March 9**

**Brent Cody**,
Virginia Commonwealth University

**The weakly compact reflection principle and orders of weak compactness Part II**

This is a continuation of my talk from last semester. Many theorems regarding the nonstationary ideal can be generalized to the ideal of non--weakly compact sets. For example, Hellsten showed that under GCH, if $W\subseteq \kappa$ is a weakly compact set then there is a cofinality-preserving forcing extension in which there is a $1$-club $C\subseteq W$ and all weakly compact subsets of $W$ remain weakly compact. I will discuss some recent results in this direction related to the *weakly compact reflection principle*, which generalize work of Mekler and Shelah on the nonstationary ideal. One can easily observe that if the weakly compact reflection principle holds at $\kappa$ then $\kappa$ must be $\omega$-weakly compact. By developing a forcing to add a non-reflecting weakly compact set, I will prove that the converse can fail: if $\kappa$ is $(\alpha+1)$-weakly compact then there is a forcing extension in which $\kappa$ remains $\alpha$-weakly compact and the weakly compact reflection principle fails at $\kappa$. I will also discuss a proof of a result joint with Hiroshi Sakai: if the weakly compact reflection principle holds at $\kappa$ then there is a forcing extension preserving this in which $\kappa$ is the least $\omega$-weakly compact cardinal. Hence the weakly compact reflection principle at $\kappa$ need not imply that $\kappa$ is $(\omega+1)$-weakly compact.