Forcing with Adequate Sets of Models as Side Conditions

TL;DR

This paper introduces a versatile framework for forcing on 92 using finite conditions and countable models as side conditions, enabling various set-theoretic constructions.

Contribution

It develops a general method for forcing with models as side conditions based on membership comparison up to an initial segment, with multiple applications.

Findings

Framework for forcing on 92 with finite conditions and models as side conditions

Examples include adding functions, stationary sets, and 91-Kurepa trees

Provides a unified approach for various forcing constructions

Abstract

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.