Constructive Models of Set Theory

Abstract

In this chapter we will present model constructions, or otherwise expressed, “concrete” realizability interpretations, for constructive set theories (sub-theories of intuitionistic ZF), patterned after the models of EM ₀ and ML ₁ given in preceding chapters. These models are of interest for two distinct reasons:

• (i) They sharpen our conception of the notion of “set” described in constructive set theory by giving specific and concrete examples of the kind of universe to which these theories can apply.

• (ii) They (or rather their formalized versions) provide interpretations of constructive set theory in various other theories, such as EM0, ML1, or subsystems of analysis, which help to clarify the relations between these different theories, both in terms of the notion of set described and in terms of the formal proof-theoretic strength of the theories.