On Friday, August 24, 2012 12:58:24 PM UTC-4, TS742 wrote: > Are some hypotheses unprovable? Or do they all have a proof that is > > just not found yet? The Riemann hypothesis comes to mind.
Yes. There exits. That was demonstrated by Goedel in 1931. "Within any sistem of axioms of arithmetic there will be, ever, undecidable propositions ." Example: Before Pascal did introduce the Axiom of Mathematical Induction, the sum of the first n integers ( S = n(n+1)/2) was a hypothesis unprovable."
Perhaps the hypothesis of the infintude of twin primes (And others),needs the following new axiom: "The pairing of two Dirichlet Arithmetic Progresions produce infinitely many coincidences of two primes." ( D.A.P is a progression a.n + b such that a , b are coprimes.)
For the twin primes it suffices to pair the progressions 6n - 1 and 6n + 1