Specializing Trees with Small Approximations I

Abstract

Assuming $\mathrm{PFA}$, we shall use internally club ${\omega }_1$-guessing models as side conditions to show that for every tree $T$ of height ${\omega }_2$ without cofinal branches, there is a proper and ${\aleph }_2$-preserving forcing notion with finite conditions which specialises $T$. Moreover, the forcing has the ${\omega }_1$-approximation property.

References in corpus (1)

• Squares, ascent paths, and chain conditions