Functional interpretation of Aczel's constructive set theory

Abstract

In the present paper we give a functional interpretation of Aczel's constructive set theories CZF − and CZF in systems T ∈ and T ∈ + of constructive set functionals of finite types. This interpretation is obtained by a translation ×, a refinement of the ∧-translation introduced by Diller and Nahm (Arch. Math. Logik Grundlagenforsch. 16 (1974) 49–66) which again is an extension of Gödel's Dialectica translation. The interpretation theorem gives characterizations of the definable set functions of CZF − and CZF in terms of constructive set functionals. In a second part we introduce constructive set theories in all finite types. We expand the interpretation to these theories and give a characterization of the translation ×. We further show that the simplest non-trivial axiom of extensionality (for type 2) is not interpretable by functionals of T ∈ and T ∈ + . We obtain this result by adapting Howard's notion of hereditarily majorizable functionals to set functionals. Subject of the last section is the translation ∨ that is defined in Burr (Arch. Math. Logic, to appear) for an interpretation of Kripke-Platek set theory with infinity ( KP ω).