Good Scales and Non-Compactness of Squares
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866915866084376576 |
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| author | Levine, Maxwell Mildenberger, Heike |
| author_facet | Levine, Maxwell Mildenberger, Heike |
| contents | Cummings, Foreman, and Magidor investigated the extent to which square principles are compact at singular cardinals. The first author proved that if $κ$ is a singular strong limit of uncountable cofinality, all scales on $κ$ are good, and $\square^*_δ$ holds for all $δ<κ$, then $\square_κ^*$ holds. In this paper we will present a strongly contrasting result for $\aleph_ω$. We construct a model in which $\square_{\aleph_n}$ holds for all $n<ω$, all scales on $\aleph_ω$ are good, but in which $\square_{\aleph_ω}^*$ fails and some weak forms of internal approachability for $[H(\aleph_{ω+1})]^{\aleph_1}$ fail. This requires an extensive analysis of the dominating and approximation properties of a version of Namba forcing. We also prove some supporting results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_16071 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Good Scales and Non-Compactness of Squares Levine, Maxwell Mildenberger, Heike Logic 03E04, 03E35, 03E55 Cummings, Foreman, and Magidor investigated the extent to which square principles are compact at singular cardinals. The first author proved that if $κ$ is a singular strong limit of uncountable cofinality, all scales on $κ$ are good, and $\square^*_δ$ holds for all $δ<κ$, then $\square_κ^*$ holds. In this paper we will present a strongly contrasting result for $\aleph_ω$. We construct a model in which $\square_{\aleph_n}$ holds for all $n<ω$, all scales on $\aleph_ω$ are good, but in which $\square_{\aleph_ω}^*$ fails and some weak forms of internal approachability for $[H(\aleph_{ω+1})]^{\aleph_1}$ fail. This requires an extensive analysis of the dominating and approximation properties of a version of Namba forcing. We also prove some supporting results. |
| title | Good Scales and Non-Compactness of Squares |
| topic | Logic 03E04, 03E35, 03E55 |
| url | https://arxiv.org/abs/2412.16071 |