The Complexity of the Set of Validities of a Theory
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866911711785648128 |
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| author | Hirschfeldt, Denis R. Towsner, Henry Weinstein, Scott |
| author_facet | Hirschfeldt, Denis R. Towsner, Henry Weinstein, Scott |
| contents | We study the collection of first-order logical schemata all of whose instances are theorems of a given theory $T$; we call these the validities of $T$ ($\mathsf{V}(T)$). It is easy to see that if $T$ is a decidable theory, then $\mathsf{V}(T)$ is distinct from the set of valid formulas of first-order logic as customarily understood. We provide a complete model-theoretic characterization of the complexity, in the sense of Turing degree, of $\mathsf{V}(T)$ for decidable theories $T$, and answer a question posed by Vaught in 1960 concerning the complexity of the collection of validities common to all decidable theories. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_08901 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Complexity of the Set of Validities of a Theory Hirschfeldt, Denis R. Towsner, Henry Weinstein, Scott Logic We study the collection of first-order logical schemata all of whose instances are theorems of a given theory $T$; we call these the validities of $T$ ($\mathsf{V}(T)$). It is easy to see that if $T$ is a decidable theory, then $\mathsf{V}(T)$ is distinct from the set of valid formulas of first-order logic as customarily understood. We provide a complete model-theoretic characterization of the complexity, in the sense of Turing degree, of $\mathsf{V}(T)$ for decidable theories $T$, and answer a question posed by Vaught in 1960 concerning the complexity of the collection of validities common to all decidable theories. |
| title | The Complexity of the Set of Validities of a Theory |
| topic | Logic |
| url | https://arxiv.org/abs/2506.08901 |