Forcing with Symmetric Systems of Models of Two Types

Abstract: The main purpose of this note is to present a matrix version of Neeman's forcing with sequences of models of two types. In some sense, the forcing defined here can be seen as a generalization of Asper\'o-Mota's pure side condition forcing from arXiv:1203.1235 and arXiv:1206.6724. In our case, the forcing will consist of certain symmetric systems of countable and uncountable elementary submodels of some $H(\kappa)$, which will be closed by the relevant intersections, in Neeman's sense. The goal of this note is to show that this forcing preserves all cardinals by showing that it is strongly proper for the class of countable submodels of $H(\kappa)$ and for the class of countably closed submodels of $H(\kappa)$ of size $\aleph_1$, and that it has has the $\aleph_3$-cc. Moreover, we will show that it preserves $2^{\aleph_1}=\aleph_2$.