The provably total functions of basic arithmetic and its extensions

Abstract

We study Basic Arithmetic, \(\textsf{BA}\) introduced by Ruitenburg (Notre Dame J Formal Logic 39:18–46, 1998). \(\textsf{BA}\) is an arithmetical theory based on basic logic which is weaker than intuitionistic logic. We show that the class of the provably total recursive functions of \(\textsf{BA}\) is a proper sub-class of the primitive recursive functions. Three extensions of \(\textsf{BA}\), called \(\textsf{BA}+\mathsf U\), \(\mathsf {BA_{\mathrm c}}\) and \(\textsf{EBA}\) are investigated with relation to their provably total recursive functions. It is shown that the provably total recursive functions of these three extensions of \(\textsf{BA}\) are exactly the primitive recursive functions. Moreover, among other things, it is shown that the well-known MRDP theorem does not hold in \(\textsf{BA}\), \(\textsf{BA}+\mathsf U\), \(\mathsf {BA_{\mathrm c}}\), but holds in \(\textsf{EBA}\).