The Complexity of the Set of Validities of a Theory

TL;DR

This paper investigates the complexity of the set of all logical schemata valid in a given theory, providing a model-theoretic characterization and addressing a longstanding question about the common validities of decidable theories.

Contribution

It offers a complete model-theoretic analysis of the Turing degree of validities for decidable theories and answers Vaught's 1960 question on their shared complexity.

Findings

Characterizes the Turing degree of validities of decidable theories

Shows the set of validities differs from standard first-order valid formulas

Provides insights into the common validities among all decidable theories

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.