Constructibility, computational complexity and P versus NP

If an algorithm is to be counted as a practically working solution to a decision problem, then the algorithm must must verifiable in some constructed and ``trusted'' theory such as PA or ZF. In this paper, a class of decision problems related to inconsistency proofs for a general class of formal theories is used to demonstrate that under this constructibility restriction, an unformalizable yet arguably convincing argument can be given for the existence of decision problems for which there exists an explicitly constructible algorithm recognizing correct solutions in polynomial time, but for which there exists no explicitly constructible solution algorithm. While these results do not solve the P versus NP problem in the classical sense of supplying a proof one way or the other in a ``trusted'' formal theory, arguably they resolve the natural constructive version of the problem.