Forcing with Symmetric Systems of Models of Two Types

The purpose of this paper is to present a general method for forcing on $\omega_2$ and $\omega_3$ with finite conditions, while preserving all cardinals and some fragments of $\mathrm{GCH}$. This method is based on the technique of forcing with finite symmetric systems of elementary submodels, and improves earlier versions of this forcing by including models of two types. We will present several applications of the pure side condition forcing and variants thereof, by adding a Kurepa tree on $\omega_2$, a club subset of $\omega_2$ that avoids infinite sets from the ground model, a function bounding every canonical function below $\omega_3$ on a club, and a simplified $(\omega_2,1)$-morass.