Interpretability suprema in Peano Arithmetic

Abstract

This paper develops the philosophy and technology needed for adding a supremum operator to the interpretability logic mathsf {ILM} of Peano Arithmetic (mathsf {PA}). It is well-known that any theories extending mathsf {PA} have a supremum in the interpretability ordering. While provable in mathsf {PA}, this fact is not reflected in the theorems of the modal system mathsf {ILM}, due to limited expressive power. Our goal is to enrich the language of mathsf {ILM} by adding to it a new modality for the interpretability supremum. We explore different options for specifying the exact meaning of the new modality. Our final proposal involves a unary operator, the dual of which can be seen as a (nonstandard) provability predicate satisfying the axioms of the provability logic mathsf {GL}.