3 typo

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 know 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.

2 "the" -> "a" and checked "community wiki"; [made Community Wiki]

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# Examples of statements that provably can't be proved using the promising looking method

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 know 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.