Godel's Completeness Theorem and Deligne's Theorem

TL;DR

This paper explains the proof of Deligne's theorem on coherent topoi having enough points and shows its equivalence to G"odel's completeness theorem for first-order logic, connecting topos theory and logic.

Contribution

It provides a detailed explanation of Deligne's theorem and its equivalence to G"odel's completeness theorem, emphasizing Barr's theorem and topos-theoretic methods.

Findings

Coherent topoi have enough points.

Deligne's theorem is equivalent to G"odel's completeness theorem.

Surjective geometric morphisms from sheaf topoi over spaces prove the theorem.

Abstract

These notes were written for a presentation given at the university Paris VII in January 2012. The goal was to explain a proof of a famous theorem by P. Deligne about coherent topoi (coherent topoi have enough points) and to show how this theorem is equivalent to G\"odel's completeness theorem for first order logic. Because it was not possible to cover everything in only three hours, the focus was on Barr's and Deligne's theorems. This explains why the corresponding sections have been given more attention. Section 1 and the Appendix were added in an attempt to make this document self-contained and understandable by a reader with a good knowledge of topos theory. A coherent topos is a topos which is equivalent to a Grothendieck topos that admits a site $(C,J)$ such that $C$ has finite limits and there exists a base of $J$ with finite covering families. In order to prove that any coherent topos has enough points, it is enough to show that for any coherent topos $\mathcal{E}$ there is a surjective geometric morphism from a topos with enough points to $\mathcal{E}$. As sheaf topoi over topological spaces have enough points, any surjective geometric morphism from some $Sh(X)$, with $X$ a topological space, to $\mathcal{E}$ is sufficient. These are provided by Barr's theorem. This is the approach that will be taken here (following MacLane and Moerdijk, Sheaves in Geometry and Logic).