Separating Maximality Principles
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911116935823360 |
|---|---|
| author | Gappo, Takehiko Lietz, Andreas |
| author_facet | Gappo, Takehiko Lietz, Andreas |
| contents | We investigate fragments of generic absoluteness principles known as Maximality Principles. We determine the consistency strength of $Σ_n$-$\mathsf{MP}(\mathbb R)$ and $Π_n$-$\mathsf{MP}(\mathbb R)$, the boldface Maximality Principle restricted respectively to $Σ_n$- and $Π_n$-formulas. Further, we show that no implication between $Σ_n$-$\mathsf{MP}(\mathbb R)$ and $Π_n$-$\mathsf{MP}(\mathbb R)$ is provable in $\mathsf{ZFC}$. We also establish the consistency, relative to a Woodin cardinal, of the Maximality Principle for $ω_1$-preserving posets with countable ordinal parameters and prove its consistency strength is bounded below by a Ramsey cardinal. Finally, we resolve questions of Ikegami-Trang and Goodman by separating the Maximality Principle for stationary set preserving posets restricted to $Σ_2$-formulas from $\mathsf{MM}^{++}$ in the presence of large cardinals. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_16506 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Separating Maximality Principles Gappo, Takehiko Lietz, Andreas Logic 03E57, 03E35, 03E55, 03E45 We investigate fragments of generic absoluteness principles known as Maximality Principles. We determine the consistency strength of $Σ_n$-$\mathsf{MP}(\mathbb R)$ and $Π_n$-$\mathsf{MP}(\mathbb R)$, the boldface Maximality Principle restricted respectively to $Σ_n$- and $Π_n$-formulas. Further, we show that no implication between $Σ_n$-$\mathsf{MP}(\mathbb R)$ and $Π_n$-$\mathsf{MP}(\mathbb R)$ is provable in $\mathsf{ZFC}$. We also establish the consistency, relative to a Woodin cardinal, of the Maximality Principle for $ω_1$-preserving posets with countable ordinal parameters and prove its consistency strength is bounded below by a Ramsey cardinal. Finally, we resolve questions of Ikegami-Trang and Goodman by separating the Maximality Principle for stationary set preserving posets restricted to $Σ_2$-formulas from $\mathsf{MM}^{++}$ in the presence of large cardinals. |
| title | Separating Maximality Principles |
| topic | Logic 03E57, 03E35, 03E55, 03E45 |
| url | https://arxiv.org/abs/2508.16506 |