Revisiting the conservativity of fixpoints over intuitionistic arithmetic

Abstract

This paper presents a novel proof of the conservativity of the intuitionistic theory of strictly positive fixpoints, \(\widehat{{\textrm{ID}}}{}_{1}^{{\textrm{i}}}{}\), over Heyting arithmetic (\({\textrm{HA}}\)), originally proved in full generality by Arai (Ann Pure Appl Log 162:807–815, 2011. https://doi.org/10.1016/j.apal.2011.03.002). The proof embeds \(\widehat{{\textrm{ID}}}{}_{1}^{{\textrm{i}}}{}\) into the corresponding theory over Beeson’s logic of partial terms and then uses two consecutive interpretations, a realizability interpretation of this theory into the subtheory generated by almost negative fixpoints, and a direct interpretation into Heyting arithmetic with partial terms using a hierarchy of satisfaction predicates for almost negative formulae. It concludes by applying van den Berg and van Slooten’s result (Indag Math 29:260–275, 2018. https://doi.org/10.1016/j.indag.2017.07.009) that Heyting arithmetic with partial terms plus the schema of self realizability for arithmetic formulae is conservative over \({\textrm{HA}}\).