On Indestructible Strongly Guessing Models

Rahman Mohammadpour, Boban Velickovic

Published 2023-03-30Version 1

In [14] 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 ${\rm GSM}^+(\omega_3, \omega_1)$, 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 [1] and extends a previous result of Todor\v{c}evi\'c [15] in this direction.