Forcing Axioms and the Definabilty of the Nonstationary Ideal on $ω_1$
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2022
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| _version_ | 1866915342058520576 |
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| author | Hoffelner, Stefan Larson, Paul Schindler, Ralf Wu, Liuzhen |
| author_facet | Hoffelner, Stefan Larson, Paul Schindler, Ralf Wu, Liuzhen |
| contents | We show that under $\BMM$ and "there exists a Woodin cardinal$"$, the nonstationary ideal on $ω_1$ can not be defined by a $Σ_1$ formula with parameter $A \subset ω_1$. We show that the same conclusion holds under the assumption of Woodin's $(\ast)$-axiom. We further show that there are universes where $\BPFA$ holds and $\NS$ is $Σ_1(ω_1)$-definable. Last we show that if the canonical inner model with one Woodin cardinal $M_1$ exists, there is a universe where $\NS$ is saturated, $Σ_1(ω_1)$-definable and $\MA$ holds. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2208_05288 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Forcing Axioms and the Definabilty of the Nonstationary Ideal on $ω_1$ Hoffelner, Stefan Larson, Paul Schindler, Ralf Wu, Liuzhen Logic 03E35, 03E45, 03E47, 03E55, 03E57 We show that under $\BMM$ and "there exists a Woodin cardinal$"$, the nonstationary ideal on $ω_1$ can not be defined by a $Σ_1$ formula with parameter $A \subset ω_1$. We show that the same conclusion holds under the assumption of Woodin's $(\ast)$-axiom. We further show that there are universes where $\BPFA$ holds and $\NS$ is $Σ_1(ω_1)$-definable. Last we show that if the canonical inner model with one Woodin cardinal $M_1$ exists, there is a universe where $\NS$ is saturated, $Σ_1(ω_1)$-definable and $\MA$ holds. |
| title | Forcing Axioms and the Definabilty of the Nonstationary Ideal on $ω_1$ |
| topic | Logic 03E35, 03E45, 03E47, 03E55, 03E57 |
| url | https://arxiv.org/abs/2208.05288 |