Quotients of Strongly Proper Forcings and Guessing Models

Abstract

We prove that a wide class of strongly proper forcing posets have quotients with strong properties. Specifically, we prove that quotients of forcing posets which have simple universal strongly generic conditions on a stationary set of models by certain nice regular suborders satisfy the ${\omega }_1$-approximation property. We prove that the existence of stationarily many ${\omega }_1$-guessing models in $P_{{\omega }_2}(H(\theta ))$, for sufficiently large cardinals $\theta$, is consistent with the continuum being arbitrarily large, solving a problem of Viale and Weiss.