Indestructibility of some compactness principles over models of PFA

Abstract

We show that $\mathrm{PFA}$ (Proper Forcing Axiom) implies that adding any number of Cohen subsets of ω will not add an ${\omega }_2$-Aronszajn tree or a weak ${\omega }_1$-Kurepa tree, and moreover no σ-centered forcing can add a weak ${\omega }_1$-Kurepa tree (a tree of height and size ${\omega }_1$ with at least ${\omega }_2$ cofinal branches). This partially answers an open problem whether ccc forcings can add ${\omega }_2$-Aronszajn trees or ${\omega }_1$-Kurepa trees (with $\neg {\square }_{{\omega }_1}$ in the latter case).

We actually prove more: We show that a consequence of $\mathrm{PFA}$, namely the guessing model principle, $\mathrm{GMP}$, which is equivalent to the ineffable slender tree property, $\mathrm{ISP}$, is preserved by adding any number of Cohen subsets of ω. And moreover, $\mathrm{GMP}$ implies that no σ-centered forcing can add a weak ${\omega }_1$-Kurepa tree (see Section 2.1 for definitions).

For more generality, we study variations of the principle $\mathrm{GMP}$ at higher cardinals and the indestructibility consequences they entail, and as applications we answer a question of Mohammadpour about guessing models at weakly but not strongly inaccessible cardinals and show that there is a model in which there are no weak ${\aleph }_{\omega +1}$-Kurepa trees and no ${\aleph }_{\omega +2}$-Aronszajn trees.