Interpretability suprema in Peano Arithmetic

Interpretability suprema in Peano Arithmetic This paper develops the philosophy and technology needed for adding a supremum operator to the interpretability logic $$\mathsf {ILM}$$ ILM of Peano Arithmetic ( $$\mathsf {PA}$$ PA ). It is well-known that any theories extending $$\mathsf {PA}$$ PA have a supremum in the interpretability ordering. While provable in $$\mathsf {PA}$$ PA , this fact is not reflected in the theorems of the modal system $$\mathsf {ILM}$$ ILM , due to limited expressive power. Our goal is to enrich the language of $$\mathsf {ILM}$$ 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 $$\mathsf {GL}$$ GL . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Archive for Mathematical Logic Springer Journals

Interpretability suprema in Peano Arithmetic

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Publisher
Springer Berlin Heidelberg
Copyright
Copyright © 2017 by The Author(s)
Subject
Mathematics; Mathematical Logic and Foundations; Mathematics, general; Algebra
ISSN
0933-5846
eISSN
1432-0665
D.O.I.
10.1007/s00153-017-0557-4
Publisher site
See Article on Publisher Site

Abstract

This paper develops the philosophy and technology needed for adding a supremum operator to the interpretability logic $$\mathsf {ILM}$$ ILM of Peano Arithmetic ( $$\mathsf {PA}$$ PA ). It is well-known that any theories extending $$\mathsf {PA}$$ PA have a supremum in the interpretability ordering. While provable in $$\mathsf {PA}$$ PA , this fact is not reflected in the theorems of the modal system $$\mathsf {ILM}$$ ILM , due to limited expressive power. Our goal is to enrich the language of $$\mathsf {ILM}$$ 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 $$\mathsf {GL}$$ GL .

Journal

Archive for Mathematical LogicSpringer Journals

Published: May 26, 2017

References

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