Functional Interpretations: From the Dialectica Interpretation to Functional Interpretations of Analysis and Set Theory eBook

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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.

This is a digital product.

Functional Interpretations: From the Dialectica Interpretation to Functional Interpretations of Analysis and Set Theory is written by Justus Diller and published by World Scientific. The Digital and eTextbook ISBNs for Functional Interpretations: From the Dialectica Interpretation to Functional Interpretations of Analysis and Set Theory are 9789814551410, 9814551414 and the print ISBNs are 9789814551397, 9814551392. Additional ISBNs for this eTextbook include 9789814551403.