Incompleteness and Inconsistency

Graham Priest's In Contradiction (Dordrecht: Marinus Nijhoff Publishers, 1987, chapter 3) contains an argument concerning the intuitive, or 'naive' notion of (arithmetic) proof, or provability. He argues that the intuitively provable arithmetic sentences constitute a recursively enumerable set, which has a Godel sentence which is itself intuitively provable. The incompleteness theorem does not apply, since the set of provable arithmetic sentences is not consistent. The purpose of this article is to sharpen Priest's argument, avoiding reference to informal notions, consensus, or Church's thesis. We add Priest's dialetheic semantics to ordinary Peano arithmetic PA, to produce a recursively axiomatized formal system PA* that contains its own truth predicate. Whether one is a dialetheist or not, PA* is a legitimate, rigorously defined formal system, and one can explore its proof-theoretic properties. The system is inconsistent (but presumably non-trivial), and it proves its own Godel sentence as well as its own soundness. Although this much is perhaps welcome to the dialetheist, it has some untoward consequences. There are purely arithmetic (indeed, Pi (0)) sentences that are both provable and refutable in PA*. So if the dialetheist maintains that PA* is sound, then he must hold that there are true contradictions in the most elementary language of arithmetic. Moreover, the thorough dialetheist must hold that there is a number g which both is and is not the code of a derivation of the indicated Godel sentence of PA*. For the thorough dialetheist, it follows ordinary PA and even Robinson arithmetic are themselves inconsistent theories, I argue that this is a bitter pill for the dialetheist to swallow.