A Solovay-like model at $\aleph_ω$
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866909018823327744 |
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| author | Poveda, Alejandro Thei, Sebastiano |
| author_facet | Poveda, Alejandro Thei, Sebastiano |
| contents | Assuming the consistency of ZFC with appropriate large cardinal axioms we produce a model of ZFC where $\aleph_ω$ is a strong limit cardinal and the inner model $L(\mathcal{P}(\aleph_ω))$ satisfies the following properties: (1) Every set $A\subseteq (\aleph_ω)^ω$ has the $\aleph_ω$-PSP. (2) There is no scale at $\aleph_ω$. (3) The Singular Cardinal Hypothesis (SCH) fails at $\aleph_ω$. (4) Shelah's Approachability property (AP) fails at $\aleph_ω$. (5) The Tree Property (TP) holds at $\aleph_{ω+1}$. The above provides the first example of a Solovay-type model at the level of the first singular cardinal, $\aleph_ω$. Our model also answers, in the context of ZF+$\mathrm{DC}_{\aleph_ω}$, a well-known question by Woodin on the relationship between the SCH and the AP at $\aleph_ω$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_18991 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Solovay-like model at $\aleph_ω$ Poveda, Alejandro Thei, Sebastiano Logic Assuming the consistency of ZFC with appropriate large cardinal axioms we produce a model of ZFC where $\aleph_ω$ is a strong limit cardinal and the inner model $L(\mathcal{P}(\aleph_ω))$ satisfies the following properties: (1) Every set $A\subseteq (\aleph_ω)^ω$ has the $\aleph_ω$-PSP. (2) There is no scale at $\aleph_ω$. (3) The Singular Cardinal Hypothesis (SCH) fails at $\aleph_ω$. (4) Shelah's Approachability property (AP) fails at $\aleph_ω$. (5) The Tree Property (TP) holds at $\aleph_{ω+1}$. The above provides the first example of a Solovay-type model at the level of the first singular cardinal, $\aleph_ω$. Our model also answers, in the context of ZF+$\mathrm{DC}_{\aleph_ω}$, a well-known question by Woodin on the relationship between the SCH and the AP at $\aleph_ω$. |
| title | A Solovay-like model at $\aleph_ω$ |
| topic | Logic |
| url | https://arxiv.org/abs/2509.18991 |