A GENERALIZATION OF BETH MODEL TO FUNCTIONALS OF HIGH TYPES

Abstract

Beth model is one of the tools in intuitionistic proof theory. Van Dalen constructed a Beth model for intuitionistic analysis with choice sequences. Bernini and Wendel investigated intuitionistic theories with many types of choice sequences, n-functionals, and their relation with classical type theory. In this paper we construct a Beth model for a basic intuitionistic theory with many types of functionals using recursive approach to define nodes in the Beth model. This model is a tool for justifying consistency of intuitionistic principles for functionals of high types. The principles that hold in our Beth model can be added to the initial basic theory in order to develop it into a relatively strong intuitionistic theory, where a significant part of classical type theory can be interpreted, with the purpose of contributing to the programme of justifying classical mathematics from the intuitionistic point of view. In this paper we show that Kripke’s schema for n-functionals and standard axioms for lawless n-functionals hold in our Beth model.