Well, my definition does not imply that what is False is easy to decide, and that's OK. If N != NP, then the statement "X is factorable" would be hard to decide, even if it's False (or True for that matter).

Godel, in the proof of GIT, chose inconsistency (by concluding that G is True, even though it is also False).

So, by your own conclusions, I choose GIT to be false, and there are no contradictions.