On Indestructible Strongly Guessing Models

In \cite{MV} we defined and proved the consistency of the principle \({\rm GM}^+(\omega_3,\omega_1)\) which implies that many consequences of strong forcing axioms hold simultaneously at \(\omega_2\) and \(\omega_3\). In this paper we formulate a strengthening of \({\rm GM}^+(\omega_3,\omega_1)\) that we call \({\rm SGM}^+(\omega_3,\omega_1)\). We also prove, modulo the consistency of two supercompact cardinals, that \({\rm SGM}^+(\omega_3,\omega_1)\) is consistent with ZFC. In addition to all the consequences of \({\rm GM}^+(\omega_3,\omega_1)\), the principle \({\rm SGM}^+(\omega_3,\omega_1)\), together with some mild cardinal arithmetic assumptions that hold in our model, implies that any forcing that adds a new subset of \(\omega_2\) 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 Todorčevi\'{c} \cite{Todorcevic82} in this direction.