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

TL;DR

This work characterizes the Borel (descriptive set-theoretic) complexity of the space of models of a first-order theory via Wadge reductions. It shows a sharp dichotomy: a complete theory has a $\boldsymbol\Pi^0_\omega$-complete set of models precisely when it is not boundedly axiomatizable, and for any finite $n$ a theory is $\boldsymbol\Pi^0_n$-complete exactly when it has a $\forall_n$-axiomatization (with $\boldsymbol\Sigma^0_n$-hardness otherwise). The results build on Knight’s and Solovay’s framework and the uniform $\alpha$-system approach to produce reductions from arbitrary $\boldsymbol\Pi^0_\omega$ sets to Mod$(T)$; they connect infinitary definability in $L_{\omega_1\omega}$ to the Borel classification of models. Consequences include $\Pi^0_\omega$-completeness for complete extensions of PA and for sequential foundational theories, and a precise correspondence between quantifier complexity and Wadge degrees of Mod$(T)$. The work advances understanding of how logical axiomatization interacts with descriptive set-theoretic complexity and has implications for the interpretability of model-theoretic properties within a Borel hierarchy.

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 $\pmbΠ_ω^0$-complete set of models. In particular, any sequential theory (a class of foundational theories isolated by Pudlák) has a $\pmbΠ_ω^0$-complete set of models. We also give sharp conditions for theories to have a $\pmbΠ^0_n$-complete set of models.

Theorems & Definitions (43)

• Definition 3.1
• Theorem 3.2
• Theorem 3.3
• Example 3.4
• Corollary 3.5
• Corollary 3.6
• proof
• Definition 3.7
• Lemma 4.1
• Definition 4.2