Classical truth in higher types

Abstract

We study, from a classical point of view, how the truth of a statement about higher type functionals depends on the underlying model. The models considered are the classical set-theoretic finite type hierarchy and the constructively more meaningful models of Continuous Functionals, Hereditarily Effective Operations, as well as the closed term model of Gödel's system T . The main results are characterisations of prenex classes for which truth in the full set-theoretic model transfers to truth in the other models. As a corollary we obtain that the axiom of choice is not conservative over Gödel's system T with classical logic for closed ∃ 2 -formulas. We hope that this study will contribute to a clearer delineation and perception of constructive mathematics from a classical perspective.