Proof Theory of Constructive Systems: Inductive Types and Univalence

Chapter
Part of the Outstanding Contributions to Logic book series (OCTR, volume 13)

Abstract

In Feferman’s work, explicit mathematics and theories of generalized inductive definitions play a central role. One objective of this article is to describe the connections with Martin–Löf type theory and constructive Zermelo–Fraenkel set theory. Proof theory has contributed to a deeper grasp of the relationship between different frameworks for constructive mathematics. Some of the reductions are known only through ordinal-theoretic characterizations. The paper also addresses the strength of Voevodsky’s univalence axiom. A further goal is to investigate the strength of intuitionistic theories of generalized inductive definitions in the framework of intuitionistic explicit mathematics that lie beyond the reach of Martin–Löf type theory.

Keywords

Explicit mathematics Constructive Zermelo–Fraenkel set theory Martin–Löf type theory Univalence axiom Proof-theoretic strength 

MSC

03F30 03F50 03C62 

Notes

Acknowledgements

Part of the material is based upon research supported by the EPSRC of the UK through grant No. EP/K023128/1. This research was also supported by a Leverhulme Research Fellowship and a Marie Curie International Research Staff Exchange Scheme Fellowship within the 7th European Community Framework Programme. This publication was also made possible through the support of a grant from the John Templeton Foundation.

Thanks are also due for the invitation to speak at the American Annual Meeting of the Association for Symbolic Logic (University of Connecticut, Storrs, 23 May, 2016) where the material of the first seven sections was presented.

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Authors and Affiliations

  1. 1.Department of Pure MathematicsUniversity of LeedsLeedsUK

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