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.