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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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    @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.)