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Has anyone ever proven that there exists a proof or disproof that there are finitely many Fermat primes. I know that it's an unsolved problem whether there are finitely or infinitely many Fermat primes but my question is only whether it has been proven to be possible to prove or disprove it. If so, how can I access such a proof?

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It's a $\Large{\Pi_2}$ statement of an incomplete theory (Peano arithmetic) so we can't a priori say "there exists a proof or disproof of this theorem".

If it was $\Large \Sigma_1$ statement then there would certainly exist a proof of it were true - but not necessarily if it were false. If it was a statement from a complete theory we could say for sure there was a proof without having one.

Personal opinion: On the other hand it's clearly true that there are only 5 Fermat primes, and proving must be possible (since it doesn't do any horrible self-reference or encoding into it) but by ideas that have not been discovered.

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Currently no such proof exists. I doubt that one ever will.

A simple search for "Fermat Prime" will tell you a lot.

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Just to make sure we're all on the same page here, I think OP is asking whether it has been proved that the existence of infinitely many Fermat primes is a decidable question. Will a search for "Fermat prime" tell us anything about that? – Gerry Myerson Oct 24 '12 at 4:47
If not, what search would? – marty cohen Oct 27 '12 at 6:16
@martycohen, "I doubt that one ever will." why not? – user58512 Jan 31 '13 at 12:49

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