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.