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arXiv · 2402.10029

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

Uri Andrews·David Gonzalez·Steffen Lempp·Dino Rossegger·Hongyu Zhu
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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.

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Uri Andrews, David Gonzalez, Steffen Lempp, Dino Rossegger, Hongyu Zhu. 2025-03-14. The Borel complexity of the class of models of first-order theories. https://arxiv.org/abs/2402.10029

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