Functional Interpretations

Functional Interpretations

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

Justus Diller


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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.

  • Arithmetic:
    • Theories of Primitive Recursive Functionals
    • Heyting Arithmetic in All Finite Types and Its Λ-Interpretation
    • Dialectica Interpretation and Equality Functionals
    • Simultaneous Recursions in Linear Types
    • Computability, Consistency, Continuity
    • Modified Realization and Its Hybrids
    • Hybrids of the Λ-Interpretation
    • Between Λ-Interpretation and Modified Realization
    • Interpretations of Classical Arithmetic
    • Extensionality and Majorizability
  • Analysis:
    • Bar Recursive Functionals
    • Intuitionistic Analysis and Its Functional Interpretations
    • Classical Analysis and Its Functional Interpretations
    • Strong Computability of Bar Recursive
  • Set Theory:
    • Constructive Set Functionals
    • Kripke–Platek Set Theory and Its Functional Interpretations
    • Constructive Set Theory in Finite Types CZFω– and Its Λ-Interpretation
    • Modified Realizations of Constructive Set Theory
    • Hybrids of the Λ-Interpretation of CZFω–
    • Type-Extensionality and Majorizability of Constructive Set Functionals

Readership: Researchers in mathematical logic, in particular in proof theory of constructive systems, and set theory.Dialectica Interpretation;Diller–Nahm Interpretation;Modified Realization;Functionals of Finite Type;Functional Interpretations of Arithmetic, of Analysis, and of Set Theory0Key Features:
  • It may become a standard reference in the field of functional interpretations
  • It should be present in libraries covering mathematical logic