I am probably asking a stupid question but what I gather from a layman explanation of Godel's incompleteness theorem is that it is completely possible that a true statement cannot be derived from theorem and axioms.
While all evidence points out that P != NP. Isn't it a possibility that it cannot be proved after all because the system itself is incomplete?
P.S.: All my understanding comes from first four chapters of GEB by Douglas Hofstadter.
Yes, at our current state of knowledge it is certainly conceivable that P=NP is independent of any given axiomatic foundation for mathematics (such as Peano Arithmetic or ZFC). It might even be "independent of the axioms, but not provably so".
In contrast to some other enigmatic sentences, knowing that P=NP is independent of the axioms would not in itself help us much in figuring out whether it is actually true or false. This might be contrasted to, for example, Goldbach's conjecture -- the only way for that to be independent is if it is actually true.
The survey paper by Araronson that @fgp found looks excellent.
