# What do coherent topoi have to do with completeness?

There is a theorem of Deligne that a "coherent" topos (e.g. one on a site where all objects are quasi-compact and quasi-separated) has enough points (i.e. isomorphisms can be detected via geometric morphisms to the topos of sets). I've heard it said that this is a form of Goedel's completeness theorem for first-order logic.

Why is that?

I'm sorry for not providing more motivation, but I don't know enough about this connection to do so!

This is now posted on MO as well.

-
Mac Lane and Moerdijk make this assertion after stating Deligne's theorem in Ch. IX.11 (Cor 3 on p. 521), I quote: "With classifying topoi based on Gentzen's rules as suggested at the end of §X.5, [Deligne's theorem] is essentially equivalent to Gödel's Completeness Theorem for first-order logic." They also elaborate a bit on that in §X.7 (see the Corollary 2 on p.569 and the remarks following it). – t.b. Jun 16 '11 at 2:09
@Akhil: Should you perhaps add some categories related tag (more than the very scarce topos-theory, that is)? – Asaf Karagila Jun 16 '11 at 8:42
@Theo: Thanks for the reference! I'll take a look and post back here. – Akhil Mathew Jun 16 '11 at 16:03
@Asaf: I just did that. – Akhil Mathew Jun 16 '11 at 16:03
@Theo: Thanks again for the reference. At this point I actually don't have much more to say, though! I think I get the loose idea (formulas hold in any topos iff they hold in sets in a "geometric theory" because there are enough points) but will have to get further in M-M before I can say anything intelligent here. – Akhil Mathew Jun 17 '11 at 5:51