The Axiom System IΣ0 Manages to Simultaneously Obey and Evade the Herbrandized Version of the Second Incompleteness Theorem

In 1981, Paris and Wilkie [J. Paris and A. Wilkie, “Δ0 Sets and Induction”, 1981 Jadswin Conference Proceedings, (Leeds University Press) pp. 237-248] indicated it was an open question whether IΣ0 would satisfy the Second Incompleteness Theorem for Herbrand deduction. We will show that IΣ0 will both obey and defy the Herbrandized version of the Second Incompleteness Theorem, depending on which of several equivalent definitions of IΣ0 one examines.