Generic Absoluteness Revisited
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909817072779264 |
|---|---|
| author | Fuchino, Sakaé Gappo, Takehiko Parente, Francesco |
| author_facet | Fuchino, Sakaé Gappo, Takehiko Parente, Francesco |
| contents | The present paper is concerned with the relation between recurrence axioms and Laver-generic large cardinal axioms in light of principles of generic absoluteness and the Ground Axiom.
M. Viale proved that Martin's Maximum$^{++}$ together with the assumption that there are class many Woodin cardinals implies $\mathcal{H}(\aleph_2)^{\mathsf{V}}\prec_{Σ_2}\mathcal{H}(\aleph_2)^{\mathsf{V}[\mathbb{G}]}$ for a generic $\mathbb{G}$ on any stationary preserving $\mathbb{P}$ which also preserves Bounded Martin's Maximum. We show that a similar but more general conclusion follows from each of $(\mathcal{P},\mathcal{H}(κ))_{Σ_2}$-${\sf RcA}^+$ (which is a fragment of a reformulation of the Maximality Principle for $\mathcal{P}$ and $\mathcal{H}(κ)$), and the existence of the tightly $\mathcal{P}$-Laver-generically huge cardinal.
While under "$\mathcal{P}=$ all stationary preserving posets", our results are not very much more than Viale's Theorem, for other classes of posets, "$\mathcal{P}=$ all proper posets" or "$\mathcal{P}=$ all ccc posets", for example, our theorems are not at all covered by his theorem.
The assumptions (and hence also the conclusion) of Viale's Theorem are compatible with the Ground Axiom. In contrast, we show that the assumptions of our theorems (for most of the common settings of $\mathcal{P}$ and with a modification of the large cardinal property involved) imply the negation of the Ground Axiom. This fact is used to show that fragments of Recurrence Axiom $(\mathcal{P},\mathcal{H}(κ))_Γ$-${\sf RcA}^+$ can be different from the corresponding fragments of Maximality Principle ${\sf MP}(\mathcal{P},\mathcal{H}(κ))_Γ$ for $Γ=Π_2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_15384 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Generic Absoluteness Revisited Fuchino, Sakaé Gappo, Takehiko Parente, Francesco Logic 03E45, 03E50, 03E55, 03E57, 03E65 The present paper is concerned with the relation between recurrence axioms and Laver-generic large cardinal axioms in light of principles of generic absoluteness and the Ground Axiom. M. Viale proved that Martin's Maximum$^{++}$ together with the assumption that there are class many Woodin cardinals implies $\mathcal{H}(\aleph_2)^{\mathsf{V}}\prec_{Σ_2}\mathcal{H}(\aleph_2)^{\mathsf{V}[\mathbb{G}]}$ for a generic $\mathbb{G}$ on any stationary preserving $\mathbb{P}$ which also preserves Bounded Martin's Maximum. We show that a similar but more general conclusion follows from each of $(\mathcal{P},\mathcal{H}(κ))_{Σ_2}$-${\sf RcA}^+$ (which is a fragment of a reformulation of the Maximality Principle for $\mathcal{P}$ and $\mathcal{H}(κ)$), and the existence of the tightly $\mathcal{P}$-Laver-generically huge cardinal. While under "$\mathcal{P}=$ all stationary preserving posets", our results are not very much more than Viale's Theorem, for other classes of posets, "$\mathcal{P}=$ all proper posets" or "$\mathcal{P}=$ all ccc posets", for example, our theorems are not at all covered by his theorem. The assumptions (and hence also the conclusion) of Viale's Theorem are compatible with the Ground Axiom. In contrast, we show that the assumptions of our theorems (for most of the common settings of $\mathcal{P}$ and with a modification of the large cardinal property involved) imply the negation of the Ground Axiom. This fact is used to show that fragments of Recurrence Axiom $(\mathcal{P},\mathcal{H}(κ))_Γ$-${\sf RcA}^+$ can be different from the corresponding fragments of Maximality Principle ${\sf MP}(\mathcal{P},\mathcal{H}(κ))_Γ$ for $Γ=Π_2$. |
| title | Generic Absoluteness Revisited |
| topic | Logic 03E45, 03E50, 03E55, 03E57, 03E65 |
| url | https://arxiv.org/abs/2410.15384 |