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Full Description
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.
Contents
Arithmetic: Primitive Recursive Functionals; Λ- (Diller - Nahm) Interpretation of Heyting Arithmetic in Finite Types; The Dialectica Interpretation and Equality Functionals; Simultaneous Recursions in Linear Types; Computability, Consistency, Continuity; Modified Realization and its Hybrids; Hybrids of the Λ-Interpretation; N-Interpretations; Interpretations of Classical Arithmetic; Extensionality and Majorizability; Analysis: Bar Recursive Functionals; Λ- and Dialectica Interpretation of Bar Induction by Bar Recursion; Functional Interpretations of Classical Analysis; Computability of Bar Recursive Functionals; Set Theory: Constructive Set Functionals; Kripke - Platek Set Theory and Its Functional Interpretations; Constructive Set Theory and Its Λ-Interpretation; Modified Realizations of Constructive Set Theory; The Q-Hybrid of the Λ-Interpretation of Constructive Set Theory in Finite Types; Majorizability of Constructive Set Functionals.
