Archive for Mathematical Logic

, Volume 56, Issue 5–6, pp 555–584 | Cite as

Interpretability suprema in Peano Arithmetic

  • Paula HenkEmail author
  • Albert Visser
Open Access


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 \(\mathsf {GL}\).


Provability logic Interpretability Peano Arithmetic 

Mathematics Subject Classification

03F45 03F25 03F30 


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Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Institute for Logic, Language, and InformationUniversity of AmsterdamAmsterdamThe Netherlands
  2. 2.Philosophy, Faculty of HumanitiesUtrecht UniversityUtrechtThe Netherlands

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