Classical Namba forcing can have the weak countable approximation property

We show that it is consistent from an inaccessible cardinal that classical Namba forcing has the weak ω 1 ω _1 -approximation property. In fact, this is the case if ℵ 1 _1 -preserving forcings do not add cofinal branches to ℵ 1 _1 -sized trees. The exact statement we obtain is similar to Hamkins’ Key Lemma. It follows as a corollary that Martin’s Maximum ( M M MM ) implies that there are stationarily many indestructibly weakly ω 1 ω _1 -guessing models that are not internally unbounded. This answers a question of Cox and Krueger and partially answers another. Our result on M M 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.