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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.

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    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).2011-06-16
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    @Akhil: Should you perhaps add some categories related tag (more than the very scarce topos-theory, that is)?2011-06-16
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    @Theo: Thanks for the reference! I'll take a look and post back here.2011-06-16
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    @Asaf: I just did that.2011-06-16
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    @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.2011-06-17
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    Thank you for this information, this is about as much as I could understand as well. In fact, I think this would be a perfectly valid question for MO, where there are quite a few experts on topos theory who could certainly give a valuable answer. While it is maybe not exactly a research level question in a strict sense, I think the lack of experts on topoi here would amply justify a post on MO.2011-06-17
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    @Theo: Sure. I think I'll give it a couple of days to be safe, and then start a MO thread.2011-06-17
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    I understand. May I ask you to notify me when you do so? I'd be interested in reading some further explanations.2011-06-18
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    @Theo: Of course.2011-06-18
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    @Theo: I've just posted it to MO, see http://mathoverflow.net/questions/68335/what-do-coherent-topoi-have-to-do-with-completeness2011-06-21
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    @Akhil: Thanks a lot! The first vote is mine.2011-06-21

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There is now an accepted answer on MO. (I'm CWing and accepting this to make it clear that this question was answered.)