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