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 $ω_1$-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 $ω_1$-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.

Theorems & Definitions (18)

• proposition 1
• theorem 3
• corollary 4
• theorem 5
• proof : Proof of \ref{['maintheorem']}
• claim 7
• proof
• claim 8
• proof
• claim 9