Model theory of class-sized logics
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866908989697032192 |
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| author | Osinski, Jonathan Wilson, Trevor |
| author_facet | Osinski, Jonathan Wilson, Trevor |
| contents | We study compactness and Löwenheim-Skolem properties of fragments of the class-sized logic $\mathcal{L}_{\infty \infty}$ and of class-sized versions of second-order and sort logics. In these fragments, certain combinations of infinitary quantifiers and boolean connectives are banned. While model-theoretic properties fail for unrestricted class logics, this drastically changes in our more restricted setting. We show that model-theoretic properties of class logics characterise a wide array of large cardinals, and that some of them can even be obtained in ZFC. In particular, we give a characterisation of Weak Vopěnka's Principle and Ord is Woodin by downwards Löwenheim-Skolem properties, and a characterisation of Shelah cardinals by a compactness property of class-sized logics. We further strengthen many known results about properties of set-sized logics by studying how they transfer to class-sized extensions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_21678 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Model theory of class-sized logics Osinski, Jonathan Wilson, Trevor Logic 03E55, 03C95, 03C75, 03C55 We study compactness and Löwenheim-Skolem properties of fragments of the class-sized logic $\mathcal{L}_{\infty \infty}$ and of class-sized versions of second-order and sort logics. In these fragments, certain combinations of infinitary quantifiers and boolean connectives are banned. While model-theoretic properties fail for unrestricted class logics, this drastically changes in our more restricted setting. We show that model-theoretic properties of class logics characterise a wide array of large cardinals, and that some of them can even be obtained in ZFC. In particular, we give a characterisation of Weak Vopěnka's Principle and Ord is Woodin by downwards Löwenheim-Skolem properties, and a characterisation of Shelah cardinals by a compactness property of class-sized logics. We further strengthen many known results about properties of set-sized logics by studying how they transfer to class-sized extensions. |
| title | Model theory of class-sized logics |
| topic | Logic 03E55, 03C95, 03C75, 03C55 |
| url | https://arxiv.org/abs/2604.21678 |