Forcing with Adequate Sets of Models as Side Conditions

We present a general framework for forcing on ${\omega }_2$ with finite conditions using countable models as side conditions. This framework is based on a method of comparing countable models as being membership related up to a large initial segment. We give several examples of this type of forcing, including adding a function on ${\omega }_2$, adding a nonreflecting stationary subset of ${\omega }_2\cap \text{cof}(\omega )$, and adding an ${\omega }_1$-Kurepa tree.