On a Theorem by Bezboruah & Shepherdson

Abstract

We discuss an incompleteness result proven by Bezboruah and Shepherdson. This result tells us that the weak theory ${\mathsf{PA}}^{-}$ does not prove the consistency of any theory (under certain assumptions explained in the paper). Kreisel argued that such a result is not meaningful. We discuss Kreisel's objection and conclude that his argument does not hold water. We compare Pudlák's extension of the Second Incompleteness Theorem with the Bezboruah-Sheperdson Theorem. Finally, we reprove the Bezboruah-Sheperdson Theorem for a sequence coding based on an insight of Nielsen and Markov.

A mistake towards the end of the paper was repaired. The correct result is now given in Theorem 4.6