Forcing with Adequate Sets of Models as Side Conditions

We present a general framework for forcing on $ω_2$ 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 $ω_2$, adding a nonreflecting stationary subset of $ω_2 \cap \textrm{cof}(ω)$, and adding an $ω_1$-Kurepa tree.

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