On Indestructible Strongly Guessing Models

Abstract

In \cite{MV} we defined and proved the consistency of the principle ${\mathrm{GM}}^{+}(\ddot{I}{‰}_3,\ddot{I}{‰}_1)$ which implies that many consequences of strong forcing axioms hold simultaneously at $\ddot{I}{‰}_2$ and $\ddot{I}{‰}_3$. In this paper we formulate a strengthening of ${\mathrm{GM}}^{+}(\ddot{I}{‰}_3,\ddot{I}{‰}_1)$ that we call ${\mathrm{SGM}}^{+}(\ddot{I}{‰}_3,\ddot{I}{‰}_1)$. We also prove, modulo the consistency of two supercompact cardinals, that ${\mathrm{SGM}}^{+}(\ddot{I}{‰}_3,\ddot{I}{‰}_1)$ is consistent with ZFC. In addition to all the consequences of ${\mathrm{GM}}^{+}(\ddot{I}{‰}_3,\ddot{I}{‰}_1)$, the principle ${\mathrm{SGM}}^{+}(\ddot{I}{‰}_3,\ddot{I}{‰}_1)$, together with some mild cardinal arithmetic assumptions that hold in our model, implies that any forcing that adds a new subset of $\ddot{I}{‰}_2$ 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.

Minor corrections and improvements. Accepted version by the JSL