"Are there finitely or infinitely many Fermat primes?": decidable? 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?
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.


http://en.wikipedia.org/wiki/Arithmetical_hierarchy
Currently no such proof exists. I doubt that one ever will.

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