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

This book gives a detailed treatment of functional interpretations of arithmetic, analysis, and set theory. The subject goes back to Goedel'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, Goedel's incompleteness theorems.