Forcing "$\mathrm{NS}_{ω_1}$ is $ω_1$-dense" From Large Cardinals
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917613853999104 |
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| author | Lietz, Andreas |
| author_facet | Lietz, Andreas |
| contents | We answer a question of Woodin by showing that assuming an inaccessible cardinal $κ$ which is a limit of ${<}κ$-supercompact cardinals exists, there is a stationary set preserving forcing $\mathbb{P}$ so that $V^{\mathbb P}\models``\mathrm{NS}_{ω_1}\text{ is }ω_1\text{-dense}"$. We also introduce a new forcing axiom $\mathrm{QM}$, show it is consistent assuming a supercompact limit of supercompact cardinals and prove that it implies $\mathbb{Q}_{\mathrm{max}}\text{-}(*)$. Consequently, $\mathrm{QM}$ implies ``$\mathrm{NS}_{ω_1}$ is $ω_1$-dense". |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_09020 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Forcing "$\mathrm{NS}_{ω_1}$ is $ω_1$-dense" From Large Cardinals Lietz, Andreas Logic 03E57 03E55 03E50 03E35 We answer a question of Woodin by showing that assuming an inaccessible cardinal $κ$ which is a limit of ${<}κ$-supercompact cardinals exists, there is a stationary set preserving forcing $\mathbb{P}$ so that $V^{\mathbb P}\models``\mathrm{NS}_{ω_1}\text{ is }ω_1\text{-dense}"$. We also introduce a new forcing axiom $\mathrm{QM}$, show it is consistent assuming a supercompact limit of supercompact cardinals and prove that it implies $\mathbb{Q}_{\mathrm{max}}\text{-}(*)$. Consequently, $\mathrm{QM}$ implies ``$\mathrm{NS}_{ω_1}$ is $ω_1$-dense". |
| title | Forcing "$\mathrm{NS}_{ω_1}$ is $ω_1$-dense" From Large Cardinals |
| topic | Logic 03E57 03E55 03E50 03E35 |
| url | https://arxiv.org/abs/2403.09020 |