A Syntactic and Categorical Derivation of G\"{o}del's Completeness Theorem

TL;DR

This paper provides a purely syntactic construction of the syntactic category for a first-order theory, demonstrating its properties and connection to models, thereby offering a categorical perspective on G"odel's Completeness Theorem.

Contribution

It introduces a fully syntactic construction of the syntactic category for first-order theories and links it to models via coherent categories and Deligne's theorem.

Findings

The syntactic category $ ext{Syn}(T)$ is a consistent coherent category.

Morphisms from $ ext{Syn}(T)$ to Set correspond to models of $T$.

The approach offers a categorical proof of G"odel's Completeness Theorem.

Abstract

Considering classical first-order logic with equality, we give a "fully syntactic" construction of the (weak) syntactic category $\text{Syn}(T)$ associated to a consistent theory $T$; we show it is a consistent coherent category; and we show that a morphism of coherent categories $\text{Syn}(T)\to \bf{Set}$ gives rise to a model $\mathcal{M}$ of $T$ in the usual sense. We then invoke Deligne's theorem on small consistent coherent categories.