A model of the Axiom of Determinacy in which every set of reals is universally Baire
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866909651171278848 |
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| author | Larson, Paul B. Sargsyan, Grigor Wilson, Trevor |
| author_facet | Larson, Paul B. Sargsyan, Grigor Wilson, Trevor |
| contents | The consistency of the theory $\mathsf{ZF} + \mathsf{AD}_{\mathbb{R}} + {}$``every set of reals is universally Baire'' is proved relative to $\mathsf{ZFC} + {}$``there is a cardinal that is a limit of Woodin cardinals and of strong cardinals.'' The proof is based on the derived model construction, which was used by Woodin to show that the theory $\mathsf{ZF} + \mathsf{AD}_{\mathbb{R}} + {}$``every set of reals is Suslin'' is consistent relative to $\mathsf{ZFC} + {}$``there is a cardinal $λ$ that is a limit of Woodin cardinals and of $\mathord{<}λ$-strong cardinals.'' The $Σ^2_1$ reflection property of our model is proved using genericity iterations as used by Neeman and Steel. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_20510 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A model of the Axiom of Determinacy in which every set of reals is universally Baire Larson, Paul B. Sargsyan, Grigor Wilson, Trevor Logic 03E60 The consistency of the theory $\mathsf{ZF} + \mathsf{AD}_{\mathbb{R}} + {}$``every set of reals is universally Baire'' is proved relative to $\mathsf{ZFC} + {}$``there is a cardinal that is a limit of Woodin cardinals and of strong cardinals.'' The proof is based on the derived model construction, which was used by Woodin to show that the theory $\mathsf{ZF} + \mathsf{AD}_{\mathbb{R}} + {}$``every set of reals is Suslin'' is consistent relative to $\mathsf{ZFC} + {}$``there is a cardinal $λ$ that is a limit of Woodin cardinals and of $\mathord{<}λ$-strong cardinals.'' The $Σ^2_1$ reflection property of our model is proved using genericity iterations as used by Neeman and Steel. |
| title | A model of the Axiom of Determinacy in which every set of reals is universally Baire |
| topic | Logic 03E60 |
| url | https://arxiv.org/abs/2502.20510 |