Dialectica Principles via G\"odel Doctrines

TL;DR

This paper develops a categorical framework for G"odel's Dialectica interpretation, introducing G"odel doctrines and fibrations, and demonstrates their logical soundness and correspondence with classical principles.

Contribution

It introduces G"odel doctrines as a categorical model of the Dialectica interpretation, extending previous work with a new intrinsic categorical presentation.

Findings

G"odel fibrations are equivalent to Hofstra's Dialectica fibrations.

The categorical model satisfies Markov and Independence of Premise principles.

Characterization of categories from the tripos-to-topos construction applied to G"odel doctrines.

Abstract

G\"odel's Dialectica interpretation was conceived as a tool to obtain the consistency of Peano arithmetic via a proof of consistency of Heyting arithmetic in the 40s. In recent years, several proof-theoretic transformations, based on G\"odel's Dialectica interpretation, have been used systematically to extract new content from classical proofs, following a suggestion of Kreisel. Thus, the interpretation has found new relevant applications in several areas of mathematics and computer science. Several authors have explained the Dialectica interpretation in categorical terms. In our previous work, we introduced an intrinsic categorical presentation of the Dialectica construction via a generalisation of Hofstra's work, using the notion of G\"odel fibration and its proof-irrelevant version, a G\"odel doctrine. The key idea is that G\"odel fibrations can be thought of as fibrations generated by some basic elements playing the role of quantifier-free elements. This categorification of quantifier-free elements is crucial not only to show that our notion of G\"odel fibration is equivalent to Hofstra's Dialectica fibration in the appropriate way, but also to show how G\"odel doctrines embody the main logical features of the Dialectica Interpretation. To show that, we derive the soundness of the interpretation of the implication connective, as expounded by Troelstra, in the categorical model. This requires extra logical principles, going beyond intuitionistic logic, namely Markov Principle and the Independence of Premise principle, as well as some choice. We show how these principles are satisfied in the categorical setting, establishing a tight correspondence between the logical system and the categorical framework. Finally, to complete our analysis, we characterise categories obtained as results of the tripos-to-topos of Hyland, Johnstone and Pitts applied to G\"odel doctrines.