Functional Interpretations (eBook)
by justus diller (Author)

 60,699 Words
 248 Pages
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 DillerNahm 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 DillerNahm 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:
 Theories of Primitive Recursive Functionals
 Heyting Arithmetic in All Finite Types and Its ΛInterpretation
 Dialectica Interpretation and Equality Functionals
 Simultaneous Recursions in Linear Types
 Computability, Consistency, Continuity
 Modified Realization and Its Hybrids
 Hybrids of the ΛInterpretation
 Between ΛInterpretation 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 CZF^{ω–} and Its ΛInterpretation
 Modified Realizations of Constructive Set Theory
 Hybrids of the ΛInterpretation of CZF^{ω–}
 TypeExtensionality and Majorizability of Constructive Set Functionals
Readership: Researchers in mathematical logic, in particular in proof theory of constructive systems, and set theory.Dialectica Interpretation;Dillerâ€“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
 Released: November 18, 2019
 Categories: Education
 Language: English
 Publisher: World Scientific Publishing Company
 ISBN10: 9814551414
 ISBN13: 9789814551410