Motivation: In Razborov and Rudichs article "Natural proofs" they define a class of proofs they call "natural proofs" and show that under certain assumptions you can't prove that $P\neq NP$ using a "natural proof". I know that this kind of results is common in complexity theory, but I don't know any good examples from other fields. This is why I ask:
Question: Can you give an example of a statement S that isn't known to be unprovable (it could be an unsolved problem or but it could also be a theorem), a promising-looking class of proofs and a proof that a proof from this class can't prove S.
I'm interested in both famous unsolved problems and in elementary examples, that can be used to explain this kind of thinking to, say, freshmen.
Best Answer
The fact that Fermat's Last Theorem is false over the $p$-adics shows that it cannot be proved using arguments using congruences.
The fact that the Steiner-Lehmus Theorem is false over the complexes shows that it cannot be proved using what John Conway calls "equality-chasing" arguments.
This one is probably not explainable at the freshman level but the fact that the Paris-Harrington theorem and the Robertson-Seymour graph minor theorem are not provable in first-order Peano arithmetic shows that some kind of "infinitary" reasoning or sophisticated induction is needed to prove them.