Namba forcing, weak approximation, and guessing

TL;DR

This paper explores the relationships between various forcing principles, introduces weakly guessing models, and uses Namba forcing to construct models with specific combinatorial properties related to stationarily many weakly guessing models.

Contribution

It introduces the concept of weakly guessing models and demonstrates their significance in deriving consequences of the principle GMP, using Namba forcing to construct models with particular stationarity properties.

Findings

GMP with $2^ ext{omega} \,\le\, \omega_2$ is consistent with certain weakly guessing models.

Many strong consequences of GMP follow from stationarily many weakly guessing models.

Constructed models with stationarily many indestructibly weakly guessing models using Namba forcing.

Abstract

We prove a variation of Easton's lemma for strongly proper forcings, and use it to prove that, unlike the stronger principle $\textsf{IGMP}$, $\textsf{GMP}$ 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 $\textsf{GMP}$ 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.