Indestructible Guessing Models and the Continuum

TL;DR

The paper introduces indestructibly -guessing models and the IGMP principle, which captures many consequences of PFA and is consistent with large continuum, advancing understanding of set-theoretic models.

Contribution

It defines indestructibly -guessing models, introduces the IGMP principle, and shows its consistency with arbitrarily large continuum, extending the scope of PFA consequences.

Findings

IGMP captures many PFA consequences such as the Suslin hypothesis.

IGMP is consistent with the continuum being arbitrarily large.

Indestructibly -guessing models strengthen previous guessing models.

Abstract

We introduce a stronger version of an $\omega_1$-guessing model, which we call an indestructibly $\omega_1$-guessing model. The principle IGMP states that there are stationarily many indestructibly $\omega_1$-guessing models. This principle, which follows from PFA, captures many of the consequences of PFA, including the Suslin hypothesis and the singular cardinal hypothesis. We prove that IGMP is consistent with the continuum being arbitrarily large.