The Borel complexity of the class of models of first-order theories

TL;DR

This paper analyzes the descriptive set-theoretic complexity of the class of models of first-order theories, providing conditions for when these classes are complete at various levels of the Borel hierarchy.

Contribution

It establishes precise criteria for the complexity of model classes of first-order theories, especially for complete and sequential theories, in the Borel hierarchy.

Findings

Complete theories can have $oldsymbol ext{Pi}_ ext{omega}^0$-complete sets of models.

Sequential theories also have $oldsymbol ext{Pi}_ ext{omega}^0$-complete sets of models.

Conditions for $oldsymbol ext{Pi}_n^0$-completeness of models are characterized.

Abstract

We investigate the descriptive complexity of the set of models of first-order theories. Using classical results of Knight and Solovay, we give a sharp condition for complete theories to have a $\pmb\Pi_\omega^0$-complete set of models. In particular, any sequential theory (a class of foundational theories isolated by Pudl\'ak) has a $\pmb\Pi_\omega^0$-complete set of models. We also give sharp conditions for theories to have a $\pmb\Pi^0_n$-complete set of models.