On the intuitionistic strength of monotone inductive definitions

Abstract

We prove here that the intuitionistic theory T₀↾+UMID_N, or even EETJ↾+UMID_N, of Explicit Mathematics has the strength of Π₂¹-CA₀. In Section 1 we give a double-negation translation for the classical second-order μ-calculus, which was shown in [Moe02] to have the strength of Π₂¹-CA₀. In Section 2 we interpret the intuitionistic μ-calculus in the theory EETJ↾+UMID_N. The question about the strength of monotone inductive definitions in T₀ was asked by S. Feferman in 1982, and — assuming classical logic — was addressed by M. Rathjen.