Passive induction and a solution to a Paris-Wilkie open question
The paper is devoted to the question formulated in 1981 by \textit{J. Paris} and \textit{A. Wilkie} in the paper ``\(\Delta_{0}\) sets and induction'' [Jadwisin Conf. Proc. 237--248 (1981)]: whether and to what extent the axiom system I\(\Sigma_{0}\) does satisfy the Second Incompleteness Theorem under Semantic Tableaux deduction. In the paper ``How to extend the semantic tableaux and cut-free versions of the second incompleteness theorem almost to Robinson's arithmetic Q'' [J. Symb. Log. 67, 465--496 (2002; Zbl 1004.03050)], the author showed that the semantic tableaux version of the Second Incompleteness Theorem does generalize for the most common definition of I\(\Sigma_{0}\) appearing in standard textbooks. In the paper under review it is shown that the incompleteness results of the mentioned paper can be generalized in the context of an alternate definition of the axiom system I\(\Sigma_{0}\) given also in the paper by Paris and Wilkie [loc. cit.]. Further, a formal analysis using a new technique called Passive Induction is developed. It is simpler than the formalism used by the author earlier in the paper from 2002.
scientific article; zbMATH DE number 5156495
scientific article