Specializing Trees with Small Approximations I

TL;DR

Under the assumption of PFA, the paper introduces a forcing method that specializes certain trees of height ω₂ without cofinal branches, preserving cardinalities and possessing the ω₁-approximation property.

Contribution

It develops a new forcing technique using internally club ω₁-guessing models to specialize trees of height ω₂ under PFA, with finite conditions and preservation properties.

Findings

For every tree of height ω₂ without cofinal branches, a proper, ω₂-preserving forcing specializes it.

The forcing has the ω₁-approximation property, ensuring certain structural preservation.

The method relies on side conditions involving club models under PFA.

Abstract

Assuming $\rm PFA$, 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.