Recursive models for constructive set theories

Abstract

We define recursive models of Martin-Löf's (type or) set theories. These models are a sort of recursive realizability; in fact, we show that for implication-free formulae of HA^ω, satisfaction in the model coincides with mr-HEO realizability. Using an idea of Aczel, we extend the model to a recursive model of the constructive set theories of Myhill and Friedman. Our models can be described without presupposing any knowledge of Martin-Löf's theories, and may make them seem less mysterious. We use our models to obtain several metamathematical results, for example consistency and independence results concerning continuity of functions on compact metric spaces. On the other hand, Martin-Löfs (latest) theories refute continuity of functions from N^N to N, as well as Church's thesis, although a show that all provably well-defined functions are continuous.