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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="omega 1"> <mml:semantics> <mml:msub> <mml:mi> ω </mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">\omega _1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -approximation property. In fact, this is the case if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal alef 1"> <mml:semantics> <mml:msub> <mml:mi mathvariant="normal"> ℵ </mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">\aleph _1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -preserving forcings do not add cofinal branches to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal alef 1"> <mml:semantics> <mml:msub> <mml:mi mathvariant="normal"> ℵ </mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">\aleph _1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -sized trees. The exact statement we obtain is similar to Hamkins’ Key Lemma. It follows as a corollary that Martin’s Maximum ( <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif upper M sans-serif upper M"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">M</mml:mi> <mml:mi mathvariant="sans-serif">M</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathsf {MM}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ) implies that there are stationarily many indestructibly weakly <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="omega 1"> <mml:semantics> <mml:msub> <mml:mi> ω </mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">\omega _1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -guessing models that are not internally unbounded. This answers a question of Cox and Krueger and partially answers another. Our result on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif upper M sans-serif upper M"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">M</mml:mi> <mml:mi mathvariant="sans-serif">M</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathsf {MM}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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.