I was wondering more about mathematical uses for the kind proof theory that gets dirty with the syntax of proofs, normalisation results and so forth. It seems to me that most mathematical uses of logic don’t have much concern for the syntax of proofs. Once you have logical soundness and completeness, and you know the proof system is recursively enumerable, you cease to care about its syntax, or what its proofs look like. Infact, once you’ve proved Goedel’s results, it seems you may aswell forget about the proof theory altogether, and concern yourself only with the semantics of the language, as in model theory.

But, I wonder if there are any examples that would prove this suspicion wrong.

To continue the debate on that other post, things like this (the Goodstein result) demonstrate why mathematicians should care about foundations – if you’re struggling with a hard problem (P != NP for example) you’d rather like to know if it’s actually independent from the axioms you’re working with. It saves you wasting time trying to prove something from insufficient assumptions.