Notre Dame Journal of Formal Logic

Unifying Functional Interpretations

Paulo Oliva

Abstract

This article presents a parametrized functional interpretation. Depending on the choice of two parameters one obtains well-known functional interpretations such as Gödel's Dialectica interpretation, Diller-Nahm's variant of the Dialectica interpretation, Kohlenbach's monotone interpretations, Kreisel's modified realizability, and Stein's family of functional interpretations. A functional interpretation consists of a formula interpretation and a soundness proof. I show that all these interpretations differ only on two design choices: first, on the number of counterexamples for A which became witnesses for ¬A when defining the formula interpretation and, second, the inductive information about the witnesses of A which is considered in the proof of soundness. Sufficient conditions on the parameters are also given which ensure the soundness of the resulting functional interpretation. The relation between the parametrized interpretation and the recent bounded functional interpretation is also discussed.

Article information

Source
Notre Dame J. Formal Logic, Volume 47, Number 2 (2006), 263-290.

Dates
First available in Project Euclid: 25 July 2006

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1153858651

Digital Object Identifier
doi:10.1305/ndjfl/1153858651

Mathematical Reviews number (MathSciNet)
MR2240624

Zentralblatt MATH identifier
1113.03052

Subjects
Primary: 03F07: Structure of proofs
Secondary: 03F10: Functionals in proof theory

Keywords
functional interpretations dialectica interpretation modified realizability monotone functional interpretations majorizability proof mining

Citation

Oliva, Paulo. Unifying Functional Interpretations. Notre Dame J. Formal Logic 47 (2006), no. 2, 263--290. doi:10.1305/ndjfl/1153858651. https://projecteuclid.org/euclid.ndjfl/1153858651


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