Well the issue I see is the very nature of Proof itself. Axioms
   and proofs. I can't remember off the top of my head completely,
   but I believe that set theory, which I *think* is at the head of
   the families of mathematics generally, has the paradox of the
   complete and incomplete set and their relationship. I don't
   remember the verbage: I can find it though. But here's the
   resolution of the paradox: You ultimately have to move outside
   of the system. Because, here's the thing: It's a system. There
   are other systems. You reach the limits of one then you have to
   move to another. It's unsatisfying of course but it's how you
   get around paradoxes. Of course the simplest way to get around a
   paradox? "I don't know." Accepting that as The Answer. Not "It
   can't be known" - that's a step too far. Rather it's "I don't
   know". This eliminates the possibility that maybe somebody ELSE
   knows or that someday it might be known. Of course, there's
   currently no room for the Subjective "I" in the system itself.
   It's not a flaw of the system, just a limitation of it. In other
   systems, unique, personal subjectivity is welcome.