I was not refering to your definition of P and NP (which I've just reread and I think it's not good; you want "accept", not "recognize"). I was refering to the fact that you do not make the difference between "true" and "provable", which can be found in any standard introductory book on logic; or, if you prefer a novel, Hofstadter's Goedel, Escher, Bach: an Eternal Golden Braid.