Forcing closed unbounded subsets of $\aleph_{\omega_{1}+1}$

Abstract

Using square sequences, a stationary subset $S_T$ of $\aleph_{\omega_{1}+1}$ is constructed from a tree $T$ of height $\omega_1$, uniformly in $T$. Under suitable hypotheses, adding a closed unbounded subset to $S_T$ requires adding a cofinal branch to $T$ or collapsing at least one of $\omega_1$, $\aleph_{\omega_{1}}$, and $\aleph_{\omega_1+1}$. An application is that in ZFC there is no parameter free definition of the family of subsets of $\aleph_{\omega_1+1}$ that have a closed unbounded subset in some $\omega_1$, $\aleph_{\omega_{1}}$, and $\aleph_{\omega_1+1}$ preserving outer model.