Constructive Zermelo-Fraenkel set theory, CZF, can be interpreted in Martin-Löf type theory via the so-called propositionsas-types interpretation. However, this interpretation validates more than what is provable in CZF. We now ask ourselves: is there a reasonably simple axiomatization (by a few axiom schemata say) of the set-theoretic formulae validated in Martin-Löf type theory? The answer is yes for a large collection of statements called the mathematical formulae. The validated mathematical formulae can be axiomatized by suitable forms of the axiom of choice. The paper builds on a self-interpretation of CZF (developed in [M. Rathjen, The formulae-as-classes interpretation of constructive set theory, in: Proof Technology and Computation (Proceedings of the International Summer School Marktoberdorf 2003) IOS Press, Amsterdam, 2004 (in press)]) that provides an "inner" model of CZF which also validates the so-called -axiom of choice, ΠΣ-AC. The crucial technical step taken in the present paper is to investigate the absoluteness properties of this model under the hypothesis ΠΣ-AC.It is also shown that CZF plus the -axiom of choice possesses the disjunction property, the numerical existence property and the existence property for an important group of formulae.

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Abstract.We prove here that the intuitionistic theory T0↾ + UMIDN. or even EETJ↾ + UMIDN, of Explicit Mathematics has the strength of –CA0. In Section 1 we give a double-negation translation for the classical second-order μ-calculus, which was shown in [Mö02] to have the strength of –CA0. In Section 2 we interpret the intuitionistic μ-calculus in the theory EETJ↾ + UMIDN. 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.

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