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arXiv · 2506.08901

The Complexity of the Set of Validities of a Theory

Denis R. Hirschfeldt·Henry Towsner·Scott Weinstein
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Abstract

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.

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Denis R. Hirschfeldt, Henry Towsner, Scott Weinstein. 2026-05-23. The Complexity of the Set of Validities of a Theory. https://arxiv.org/abs/2506.08901

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