Namba forcing, weak approximation, and guessing

Abstract

We prove a variation of Easton's lemma for strongly proper forcings, and use it to prove that, unlike the stronger principle $\text{IGMP}$, $\text{GMP}$ together with $2^{\omega }\le {\omega }_2$ is consistent with the existence of an ${\omega }_1$-distributive nowhere c.c.c. forcing poset of size ${\omega }_1$. We introduce the idea of a weakly guessing model, and prove that many of the strong consequences of the principle $\text{GMP}$ follow from the existence of stationarily many weakly guessing models. Using Namba forcing, we construct a model in which there are stationarily many indestructibly weakly guessing models which have a bounded countable subset not covered by any countable set in the model.