On Indestructible Strongly Guessing Models

Abstract

In \cite{MV} we defined and proved the consistency of the principle ${\rm GM}^+(ω_3,ω_1)$ which implies that many consequences of strong forcing axioms hold simultaneously at $ω_2$ and $ω_3$. In this paper we formulate a strengthening of ${\rm GM}^+(ω_3,ω_1)$ that we call ${\rm SGM}^+(ω_3,ω_1)$. We also prove, modulo the consistency of two supercompact cardinals, that ${\rm SGM}^+(ω_3,ω_1)$ is consistent with ZFC. In addition to all the consequences of ${\rm GM}^+(ω_3,ω_1)$, the principle ${\rm SGM}^+(ω_3,ω_1)$, together with some mild cardinal arithmetic assumptions that hold in our model, implies that any forcing that adds a new subset of $ω_2$ either adds a real or collapses some cardinal. This gives a partial answer to a question of Abraham \cite{AvrahamPhD} and extends a previous result of Todorčević \cite{Todorcevic82} in this direction.