Classical Namba forcing can have the weak countable approximation property
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866917963826724864 |
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| author | Levine, Maxwell |
| author_facet | Levine, Maxwell |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2312_14083 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Classical Namba forcing can have the weak countable approximation property Levine, Maxwell Logic 03E35, 03E55 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. |
| title | Classical Namba forcing can have the weak countable approximation property |
| topic | Logic 03E35, 03E55 |
| url | https://arxiv.org/abs/2312.14083 |