Quotients of Strongly Proper Forcings and Guessing Models

TL;DR

This paper demonstrates that many strongly proper forcing posets have quotients with desirable properties, and shows the consistency of stationarily many -guessing models with arbitrarily large continuum, addressing a problem by Viale and Weiss.

Contribution

It establishes that quotients of strongly proper forcings can have strong properties and proves the consistency of many -guessing models with large continuum.

Findings

Quotients of certain strongly proper forcings satisfy the -approximation property.

Existence of stationarily many -guessing models is consistent with arbitrarily large continuum.

Addresses a problem posed by Viale and Weiss.

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.