Model-theoretic characterizations of large cardinals (Re)${}^2$visited
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_ | 1866916749551599616 |
|---|---|
| author | Boney, Will Osinski, Jonathan |
| author_facet | Boney, Will Osinski, Jonathan |
| contents | We characterize several large cardinal notions by model-theoretic properties of extensions of first-order logic. We show that $Π_n$-strong cardinals, and, as a corollary, ``Ord is Woodin" and weak Vopěnka's Principle, are characterized by compactness properties involving Henkin models for sort logic. This provides a model-theoretic analogy between Vopěnka's Principle and weak Vopěnka's Principle. We also characterize huge cardinals by compactness for type omission properties of the well-foundedness logic $\mathbb L(Q^{\text{WF}})$, and show that the compactness number of the Härtig quantifier logic $\mathbb L(I)$ can consistently be larger than the first supercompact cardinal. Finally, we show that the upward Löwenheim-Skolem-Tarski number of second-order logic $\mathbb L^2$ and the sort logic $\mathbb L^{s,n}$ are given by the first extendible and $C^{(n)}$-extendible cardinal, respectively. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_15574 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Model-theoretic characterizations of large cardinals (Re)${}^2$visited Boney, Will Osinski, Jonathan Logic 03E55, 03C85, 03C95 We characterize several large cardinal notions by model-theoretic properties of extensions of first-order logic. We show that $Π_n$-strong cardinals, and, as a corollary, ``Ord is Woodin" and weak Vopěnka's Principle, are characterized by compactness properties involving Henkin models for sort logic. This provides a model-theoretic analogy between Vopěnka's Principle and weak Vopěnka's Principle. We also characterize huge cardinals by compactness for type omission properties of the well-foundedness logic $\mathbb L(Q^{\text{WF}})$, and show that the compactness number of the Härtig quantifier logic $\mathbb L(I)$ can consistently be larger than the first supercompact cardinal. Finally, we show that the upward Löwenheim-Skolem-Tarski number of second-order logic $\mathbb L^2$ and the sort logic $\mathbb L^{s,n}$ are given by the first extendible and $C^{(n)}$-extendible cardinal, respectively. |
| title | Model-theoretic characterizations of large cardinals (Re)${}^2$visited |
| topic | Logic 03E55, 03C85, 03C95 |
| url | https://arxiv.org/abs/2505.15574 |