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| Natura: | Preprint |
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2025
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| Accesso online: | https://arxiv.org/abs/2505.01393 |
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| _version_ | 1866915278987722752 |
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| author | Goldberg, Gabriel Hathaway, Dan |
| author_facet | Goldberg, Gabriel Hathaway, Dan |
| contents | We put together Woodin's $Σ^2_1$ basis theorem of AD$^+$ and Vopěnka's theorem to conclude the following: If there is a proper class of Woodin cardinals, then every $(Σ^2_1)^{\mbox{uB}}$ statement that is true in $V$ is true in $\mbox{HOD}$. Moreover, this is true even if we allow a parameter $C \subseteq \mathbb{R}$ such that $C$ and its complement have scales that are $\mbox{OD}$ and universally Baire. We also investigate whether $(Σ^2_1)^{\mbox{uB}}$ statements are upwards absolute from $\mbox{HOD}$ to $V$ under large cardinal hypotheses, observing that this is true if $\mbox{HOD}$ has a proper class of Woodin cardinals. Finally, we discuss $(\forall^{\mathbb{R}})\, (Σ^2_1)^{\mbox{uB}}$ absoluteness and conclude that this much absoluteness between $\mbox{HOD}$ and $V$ cannot be implied by any large cardinal axiom consistent with the axiom ``$V =$ Ultimate $L$''. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_01393 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On $(Σ^2_1)^{uB}$ Absoluteness Between V and HOD Goldberg, Gabriel Hathaway, Dan Logic We put together Woodin's $Σ^2_1$ basis theorem of AD$^+$ and Vopěnka's theorem to conclude the following: If there is a proper class of Woodin cardinals, then every $(Σ^2_1)^{\mbox{uB}}$ statement that is true in $V$ is true in $\mbox{HOD}$. Moreover, this is true even if we allow a parameter $C \subseteq \mathbb{R}$ such that $C$ and its complement have scales that are $\mbox{OD}$ and universally Baire. We also investigate whether $(Σ^2_1)^{\mbox{uB}}$ statements are upwards absolute from $\mbox{HOD}$ to $V$ under large cardinal hypotheses, observing that this is true if $\mbox{HOD}$ has a proper class of Woodin cardinals. Finally, we discuss $(\forall^{\mathbb{R}})\, (Σ^2_1)^{\mbox{uB}}$ absoluteness and conclude that this much absoluteness between $\mbox{HOD}$ and $V$ cannot be implied by any large cardinal axiom consistent with the axiom ``$V =$ Ultimate $L$''. |
| title | On $(Σ^2_1)^{uB}$ Absoluteness Between V and HOD |
| topic | Logic |
| url | https://arxiv.org/abs/2505.01393 |