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 $\omega_2 \cap \textrm{cof}(\omega)$, and adding an $\omega_1$-Kurepa tree.