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Is there a quick way to prove the completeness theorem (every consistant theory has a model) from the compactness theorem (a theory has a model iff every finite subtheory of it has a model)? Usually the compactness theorem is a very easy result of the completeness theorem, but it can also be proved in other ways (e.g. using Tychonoff's theorem) and I wonder if this provides a "shortcut" to the completeness theorem.

I'm only asking about propositional calculus, but if the same holds for first order logic I'll be happy to hear it as well.

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    In the FOL versions both are equivalent to the Ultrafilter Lemma, as well to the Boolean Prime Ideal Theorem. In turn this is exactly Tychonoff's theorem for Hausdorff compact spaces.2012-04-09
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    This is all very nice but without a reference/proof sketch it doesn't help me too much.2012-04-09
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    And that is why I used a comment and not an answer... When I had to teach the compactness theorem for propositional calculus earlier this year I was knee-deep in references. I cannot find any of them right now. I'll post an answer if I have something good.2012-04-09

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