Classical Namba forcing can have the weak countable approximation property

Abstract

We show that it is consistent from an inaccessible cardinal that classical Namba forcing has the weak $\ddot{I}{‰}_1$-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 $\ddot{I}{‰}_1$-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.

Revised version: The statements are the same as the submitted version, plus corrections from a helpful referee report