Forcing with matrices of countable elementary submodels

Abstract

We analyze the forcing notion P of finite matrices whose rows consist of isomorphic countable elementary submodels of a given structure of the form Hθ. We show that forcing with this poset adds a Kurepa tree T. Moreover, if Pc is a suborder of 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 ω1.