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

David Gonzalez; Dino Rossegger; Hongyu Zhu; Steffen Lempp; Uri Andrews

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

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arxiv 2402.10029 v2 pith:TDRDD6AG submitted 2024-02-15 math.LO

Uri Andrews , David Gonzalez , Steffen Lempp , Dino Rossegger , Hongyu Zhu This is my paper

classification math.LO

keywords modelstheoriescompleteclasscomplexityfirst-ordergiveomega

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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 $\pmb\Pi_\omega^0$-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.

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