A model of the Axiom of Determinacy in which every set of reals is universally Baire
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 c...
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| Format: | Article |
| Language: | English |
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Cambridge University Press
2025-01-01
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| Series: | Forum of Mathematics, Sigma |
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| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509425100534/type/journal_article |
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| author | Paul B. Larson Grigor Sargsyan Trevor Wilson |
| author_facet | Paul B. Larson Grigor Sargsyan Trevor Wilson |
| author_sort | Paul B. Larson |
| collection | DOAJ |
| description | 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
$\lambda $
that is a limit of Woodin cardinals and of
$\mathord {<}\lambda $
-strong cardinals’. The
$\Sigma ^2_1$
reflection property of our model is proved using genericity iterations as in Neeman [18] and Steel [22]. |
| format | Article |
| id | doaj-art-0a547800fd4f4010b8ca51cc2b109e86 |
| institution | OA Journals |
| issn | 2050-5094 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj-art-0a547800fd4f4010b8ca51cc2b109e862025-08-20T02:35:37ZengCambridge University PressForum of Mathematics, Sigma2050-50942025-01-011310.1017/fms.2025.10053A model of the Axiom of Determinacy in which every set of reals is universally BairePaul B. Larson0https://orcid.org/0000-0001-5931-7051Grigor Sargsyan1https://orcid.org/0009-0009-7639-3940Trevor Wilson2https://orcid.org/0000-0001-9513-3612Department of Mathematics, https://ror.org/05nbqxr67 Miami University , Oxford, Ohio, 45056, USAInstitute of Mathematics, https://ror.org/01dr6c206 Polish Academy of Sciences , 8 Śniadeckich Street, Warsaw, Poland; E-mail:Department of Mathematics, https://ror.org/05nbqxr67 Miami University , Oxford, Ohio, 45056, USA; E-mail: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 $\lambda $ that is a limit of Woodin cardinals and of $\mathord {<}\lambda $ -strong cardinals’. The $\Sigma ^2_1$ reflection property of our model is proved using genericity iterations as in Neeman [18] and Steel [22].https://www.cambridge.org/core/product/identifier/S2050509425100534/type/journal_article35E6003E1503E55 |
| spellingShingle | Paul B. Larson Grigor Sargsyan Trevor Wilson A model of the Axiom of Determinacy in which every set of reals is universally Baire Forum of Mathematics, Sigma 35E60 03E15 03E55 |
| title | A model of the Axiom of Determinacy in which every set of reals is universally Baire |
| title_full | A model of the Axiom of Determinacy in which every set of reals is universally Baire |
| title_fullStr | A model of the Axiom of Determinacy in which every set of reals is universally Baire |
| title_full_unstemmed | A model of the Axiom of Determinacy in which every set of reals is universally Baire |
| title_short | A model of the Axiom of Determinacy in which every set of reals is universally Baire |
| title_sort | model of the axiom of determinacy in which every set of reals is universally baire |
| topic | 35E60 03E15 03E55 |
| url | https://www.cambridge.org/core/product/identifier/S2050509425100534/type/journal_article |
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