Passive induction and a solution to a Paris–Wilkie open question

Abstract

In 1981, Paris and Wilkie raised the open question about whether and to what extent the axiom system I Σ 0 did satisfy the Second Incompleteness Theorem under Semantic Tableaux deduction. Our prior work showed that the semantic tableaux version of the Second Incompleteness Theorem did generalize for the most common definition of I Σ 0 appearing in the standard textbooks. However, there was an alternate interesting definition of this axiom system in the Wilkie–Paris article in the Annals of Pure and Applied Logic 35 (1987), pp. 261–302 which we did not examine in our year-2002 article in the Journal of Symbolic Logic. Our first goal is to show that the incompleteness results of our prior paper can generalize in this alternate context. We will also develop a formal analysis, using a new technique called Passive Induction, that is simpler than the formalism we had used before. A further reason our results are of interest is that we have shown in a companion paper published in Electronic Notes in Theoretical Computer Science 165 (2006), pp. 213–226 that some very unorthodox axiomizations for I Σ 0 are anti-thresholds for the Herbrandized version of the Second Incompleteness Theorem. Thus, different axiomizations for I Σ 0 have nearly fully opposite incompleteness properties. This paper is self-contained. It will not require a knowledge of our earlier results.