Classical Namba forcing can have the weak countable approximation property

TL;DR

This paper demonstrates that classical Namba forcing can have the weak -approximation property under certain set-theoretic assumptions, with implications for models of  and stationary sets.

Contribution

It shows the consistency of Namba forcing having the weak -approximation property from an inaccessible cardinal and explores implications for -preserving forcings and -guessing models.

Findings

Namba forcing can have the weak -approximation property under specific conditions.

-preserving forcings do not add cofinal branches to -sized trees.

-Miller forcing implies the existence of certain indestructibly weakly -guessing models.

Abstract

We show that it is consistent from an inaccessible cardinal that classical Namba forcing has the weak $\omega_1$-approximation property. In fact, this is the case if $\aleph_1$-preserving forcings do not add cofinal branches to $\aleph_1$-sized trees. The exact statement we obtain is similar to Hamkins' Key Lemma. It follows as a corollary that $\mathsf{MM}$ implies that there are stationarily many indestructibly weakly $\omega_1$-guessing models that are not internally unbounded. This answers a question of Cox and Krueger and partially answers another. Our result on $\mathsf{MM}$ gives a short proof of a weakening of Cox and Krueger's main result by removing their use of higher Namba forcings, but we find another application of their ideas by answering a question of Adolf, Apter, and Koepke on preservation of successive cardinals by singularizing forcings.