The doctrinal G\"odel's completeness theorem and the type space functor

Marco Abbadini; Francesca Guffanti

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arXiv · 2607.21115

The doctrinal G\"odel's completeness theorem and the type space functor

Marco Abbadini·Francesca Guffanti

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Abstract

We give a self-contained proof of G\"odel's completeness theorem entirely within the formalism of first-order Boolean doctrines (an algebraic approach to classical many-sorted first-order logic). Moreover, we show that G\"odel's completeness theorem entails that the fiberwise Stone dual of a first-order Boolean doctrine is its type space functor; roughly speaking, this means that the Stone dual of the Boolean algebra of formulas in context $X$ is the Stone space of $X$-pointed models modulo elementary equivalence.

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Marco Abbadini, Francesca Guffanti. 2026-07-23. The doctrinal G\"odel's completeness theorem and the type space functor. https://arxiv.org/abs/2607.21115

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