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.