Indestructibility of some compactness principles over models of PFA*

We show that PFA (Proper Forcing Axiom) implies that adding any number of Cohen subsets of w will not add an w2-Aronszajn tree or a weak w1-Kurepa tree, and moreover no acentered forcing can add a weak w1-Kurepa tree (a tree of height and size w1 with at least w2 cofinal branches). This partially answers an open problem whether ccc forcings can add w2-Aronszajn trees or w1-Kurepa trees (with not sign O omega 1in the latter case).We actually prove more: We show that a consequence of PFA, namely the guessing model principle, GMP, which is equivalent to the ineffable slender tree property, ISP, is preserved by adding any number of Cohen subsets of w. And moreover, GMP implies that no a-centered forcing can add a weak w1-Kurepa tree (see Section 2.1 for definitions). For more generality, we study variations of the principle 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.(c) 2023 Elsevier B.V. All rights reserved.

Automated summary

This paper shows that the Proper Forcing Axiom (PFA) has implications on the addition of Cohen subsets to w, in that it does not add specific types of trees (w2-Aronszajn trees and weak w1-Kurepa trees), and acentered forcing cannot add a weak w1-Kurepa tree. Furthermore, the paper studies variations of the principle GMP at higher cardinals and the indestructibility consequences they entail, addressing a question on weakly but not strongly inaccessible cardinals and proving the absence of weak aleph omega +1-Kurepa trees and aleph omega+2-Aronszajn trees in a model.