Functional Interpretations: From The Dialectica Interpretation To Functional Interpretations Of Analysis And Set Theory

Diller Justus

ISBN 10: 9814551392 / ISBN 13: 9789814551397

Published by World Scientific Publishing Company, Incorporated, 2020

About this Item

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Bibliographic Details

• Title: Functional Interpretations: From The Dialectica Interpretation To Functional Interpretations Of Analysis And Set Theory
• Publisher: World Scientific Publishing Company, Incorporated
• Publication Date: 2020
• Language: English
• ISBN 10: 9814551392
• ISBN 13: 9789814551397
• Binding: Hardcover
• Condition: New

About this title

Synopsis

This book gives a detailed treatment of functional interpretations of arithmetic, analysis, and set theory. The subject goes back to Gödel's Dialectica interpretation of Heyting arithmetic which replaces nested quantification by higher type operations and thus reduces the consistency problem for arithmetic to the problem of computability of primitive recursive functionals of finite types. Regular functional interpretations, in particular the Dialectica interpretation and its generalization to finite types, the Diller-Nahm interpretation, are studied on Heyting as well as Peano arithmetic in finite types and extended to functional interpretations of constructive as well as classical systems of analysis and set theory. Kreisel's modified realization and Troelstra's hybrids of it are presented as interpretations of Heyting arithmetic and extended to constructive set theory, both in finite types. They serve as background for the construction of hybrids of the Diller-Nahm interpretation of Heyting arithmetic and constructive set theory, again in finite types. All these functional interpretations yield relative consistency results and closure under relevant rules of the theories in question as well as axiomatic characterizations of the functional translations.

From the Back Cover

This book gives a detailed treatment of functional interpretations of arithmetic, analysis, and set theory. The subject goes back to G del's Dialectica interpretation of Heyting arithmetic which replaces nested quantification by higher type operations and thus reduces the consistency problem for arithmetic to the problem of computability of primitive recursive functionals of finite types. Regular functional interpretations, i.e. Dialectica and Diller Nahm interpretation as well as Kreisel's modified realization, together with their Troelstra-style hybrids, are applied to constructive as well as classical systems of arithmetic, analysis, and set theory. They yield relative consistency and conservativity results and closure under relevant rules of the theories in question as well as axiomatic characterizations of the functional translations.

Prerequisites are: familiarity with classical and intuitionistic predicate logic, basics of computability theory, G del's incompleteness theorems.