Forcing with matrices of countable elementary submodels

TL;DR

This paper studies a forcing notion based on matrices of countable elementary submodels, showing it adds a Kurepa tree with specific properties, including being almost Souslin under certain conditions.

Contribution

It introduces a new forcing construction using matrices of models and analyzes the properties of the resulting Kurepa trees, including their Souslin characteristics.

Findings

Forcing with matrices adds a Kurepa tree.

Suborder with continuous matrices yields an almost Souslin tree.

The constructed trees have specific stationary set properties.

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 $\omega_1$.