There are two ways to prove the compactness theorem for propositional logic - either using the completeness theorem and going from semantic entailment to syntactic proof, or by a topological argument in Stone spaces. The latter, I feel, is an unexpected way of doing it - but I don't know the history of the subject so I'm probably not qualified to comment whether it was fundamentally new or not. Certainly in light of Stone's representation theorem, it seems unsurprising that there could be a topological proof of a theorem in logic, and as I understand it this connection is further investigated in topos theory?
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