Godel's Completeness Theorem and Deligne's Theorem

摘要

These notes were written for a presentation given at the university Paris VIIin 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 showhow this theorem is equivalent to G\"odel's completeness theorem for firstorder logic. Because it was not possible to cover everything in only threehours, the focus was on Barr's and Deligne's theorems. This explains why thecorresponding sections have been given more attention. Section 1 and theAppendix were added in an attempt to make this document self-contained andunderstandable by a reader with a good knowledge of topos theory. A coherent topos is a topos which is equivalent to a Grothendieck topos thatadmits a site $(C,J)$ such that $C$ has finite limits and there exists a baseof $J$ with finite covering families. In order to prove that any coherent toposhas enough points, it is enough to show that for any coherent topos$\mathcal{E}$ there is a surjective geometric morphism from a topos with enoughpoints to $\mathcal{E}$. As sheaf topoi over topological spaces have enoughpoints, any surjective geometric morphism from some $Sh(X)$, with $X$ atopological space, to $\mathcal{E}$ is sufficient. These are provided by Barr'stheorem. This is the approach that will be taken here (following MacLane andMoerdijk, Sheaves in Geometry and Logic).