Some consequences from Proper Forcing Axiom together with large continuum and the negation of Martin's Axiom

Abstract

Recently, David Asperó and Miguel Angel Mota discovered a new method of iterated forcing using models as side conditions. The side condition method with models was introduced by Stevo Todorčević in the 1980s. The Asperó–Mota iteration enables us to force some $\Pi_2$-statements over $H(\aleph_2)$ with the continuum greater than $\aleph_2$. In this article, by using the Asperó–Mota iteration, we prove that it is consistent that $\mho$ fails, there are no weak club guessing ladder systems, $\mathfrak{p}= {\mathrm{add}}(\mathcal{N}) = 2^{\aleph_0}>\aleph_2$ and ${MA}_{\aleph_1}$ fails.