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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2509.18991 |
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Table of 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 $\aleph_ω$.