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 \omega _1 -approximation property. In fact, this is the case if ℵ 1 \aleph _1 -preserving forcings do not add cofinal branches to ℵ 1 \aleph _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 \mathsf {MM} ) implies that there are stationarily many indestructibly weakly ω 1 \omega _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 \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.
Fields of Study & Subjects
Advanced Topology and Set Theory
Economic theories and models
Mathematical and Theoretical Analysis
Publication details
- Journal / source
- Proceedings of the American Mathematical Society
- Publisher
- American Mathematical Society (AMS)
- Volume
- 154
- Issue
- 4
- Pages
- 1787–1796
- Publication date
- 5 February 2026
- Language
- EN