Pathologies in satisfaction classes
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866913382844596224 |
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| author | Abdul-Quader, Athar Łełyk, Mateusz |
| author_facet | Abdul-Quader, Athar Łełyk, Mateusz |
| contents | We study subsets of countable recursively saturated models of $\mathsf{PA}$ which can be defined using pathologies in satisfaction classes. More precisely, we characterize those subsets $X$ such that there is a satisfaction class $S$ where $S$ behaves correctly on an idempotent disjunction of length $c$ if and only if $c \in X$. We generalize this result to characterize several types of pathologies including double negations, blocks of extraneous quantifiers, and binary disjunctions and conjunctions. We find a surprising relationship between the cuts which can be defined in this way and arithmetic saturation: namely, a countable nonstandard model is arithmetically saturated if and only if every cut can be the "idempotent disjunctively correct cut" in some satisfaction class. We describe the relationship between types of pathologies and the closure properties of the cuts defined by these pathologies. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_18069 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Pathologies in satisfaction classes Abdul-Quader, Athar Łełyk, Mateusz Logic 03C62, 03H15 We study subsets of countable recursively saturated models of $\mathsf{PA}$ which can be defined using pathologies in satisfaction classes. More precisely, we characterize those subsets $X$ such that there is a satisfaction class $S$ where $S$ behaves correctly on an idempotent disjunction of length $c$ if and only if $c \in X$. We generalize this result to characterize several types of pathologies including double negations, blocks of extraneous quantifiers, and binary disjunctions and conjunctions. We find a surprising relationship between the cuts which can be defined in this way and arithmetic saturation: namely, a countable nonstandard model is arithmetically saturated if and only if every cut can be the "idempotent disjunctively correct cut" in some satisfaction class. We describe the relationship between types of pathologies and the closure properties of the cuts defined by these pathologies. |
| title | Pathologies in satisfaction classes |
| topic | Logic 03C62, 03H15 |
| url | https://arxiv.org/abs/2303.18069 |