An equiconsistency proof for $\mathrm{CZF} + V = L$

TL;DR

This paper proves that adding the axiom V = L to the constructive set theory CZF does not increase its consistency strength, using complex realizability models and recursion theory.

Contribution

It provides an equiconsistency proof for CZF + V = L, which is non-trivial due to the intuitionistic nature of CZF and the failure of L to be an inner model.

Findings

CZF + V = L is equiconsistent with CZF

The proof involves advanced realizability models

Recursion-theoretic arguments are employed

Abstract

In many axiomatic set theories, G\"odel's constructible universe $L$ is known as an inner model, that is, a definable class satisfying the same axioms (and containing the same ordinals). This gives a trivial proof that adding the axiom $V = L$ does not increase the consistency strength of the theory. In this paper, we shall look at a system of intuitionistic set theory known as $\mathrm{CZF}$, where $L$ fails to exhibit such nice properties. We will demonstrate that, here, the theory $\mathrm{CZF} + V = L$ is still equiconsistent with $\mathrm{CZF}$, but the proof will involve a much more complicated realisability model and a recursion-theoretic argument.