A Side Condition Principle With Applications

Abstract

We introduce a combinatorial principle $\textsf{SC}(κ)$, for any regular cardinal $κ\ge ω_2$, which asserts the existence of a family $\mathcal{Y} \subseteq [H(κ)]^ω$ of elementary substructures of $H(κ)$ which is closed under finite intersections and satisfies that for stationarily many $N \in \mathcal{Y}$, if $M \in \mathcal{Y}$ and $M \cap ω_1 < N \cap ω_1$, then $M \cap N \cap κ\in N$. We explore the idea of defining forcings with side conditions consisting of finite subsets of such a family $\mathcal{Y}$. If $\textsf{SC}(ω_2)$ holds, then there exists a thin stationary subset of $[ω_2]^ω$, and consequently $\textsf{SC}(ω_2)$ is independent of ZFC. Assuming that $μ$ is a Mahlo cardinal, the Lévy collapse $\mathrm{Col}(ω,<μ)$ forces that $\textsf{SC}(κ)$ holds for all regular cardinals $κ\ge ω_2$. Local instances of $\textsf{SC}$ can be forced using matrices of models or derived as consequences of square principles. We apply the principle $\textsf{SC}$ to define two kinds of forcings: a streamlined version of a strongly proper collapse, and a forcing which is strongly proper on a stationary set and adds arbitrarily many Baumgartner clubs. Finally, we prove two consistency results using forcings derived from $\textsf{SC}$ related to Aronszajn trees, the guessing model principle, strong failures of club guessing, and the higher dominating number.