Forcing with matrices of countable elementary submodels

Abstract

We analyze the forcing notion $\mathcal{P}$ of finite matrices whose rows consists of isomorphic countable elementary submodels of a given structure of the form $H_{\theta }$. We show that forcing with this poset adds a Kurepa tree $T$. Moreover, if ${\mathcal{P}}_c$ is a suborder of $\mathcal{P}$ containing only continuous matrices, then the Kurepa tree $T$ is almost Souslin, i.e. the level set of any antichain in $T$ is not stationary in ${\omega }_1$.