Classical truth in higher types

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 the truth in the full set-theoretic model transfers to truth in other models. As a corollary we obtain that classical logic plus the axiom of choice is not conservative over Gödel’s system T for (closed) sentences of the form ∃F .r = 0. We hope that this study will contribute to a clearer delineation and perception of constructive mathematics from a classical perspective.