Failure of Approachability at the Successor of the first Singular for any Cofinality
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866914077018685440 |
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| author | Jakob, Hannes Levine, Maxwell |
| author_facet | Jakob, Hannes Levine, Maxwell |
| contents | We solve two long-standing open problems regarding the combinatorics of $\aleph_{ω+1}$. We answer a question of Shelah by showing that it is consistent for any $n\geq 1$ that $\mathsf{GCH}$ holds and there is a stationary set of points of cofinality $\aleph_n$ which is not in the approachability ideal. As a corollary, we obtain a model where the notions of goodness and approachability are distinct for stationarily many points of cofinality $\aleph_1$, answering an open question of Cummings, Foreman, and Magidor. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_18898 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Failure of Approachability at the Successor of the first Singular for any Cofinality Jakob, Hannes Levine, Maxwell Logic 03E04, 03E05, 03E35, 03E55 We solve two long-standing open problems regarding the combinatorics of $\aleph_{ω+1}$. We answer a question of Shelah by showing that it is consistent for any $n\geq 1$ that $\mathsf{GCH}$ holds and there is a stationary set of points of cofinality $\aleph_n$ which is not in the approachability ideal. As a corollary, we obtain a model where the notions of goodness and approachability are distinct for stationarily many points of cofinality $\aleph_1$, answering an open question of Cummings, Foreman, and Magidor. |
| title | Failure of Approachability at the Successor of the first Singular for any Cofinality |
| topic | Logic 03E04, 03E05, 03E35, 03E55 |
| url | https://arxiv.org/abs/2503.18898 |