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

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.