1 Introduction Induction-recursion is a powerful definition method in intuitionistic type theory in the sense of Scott ("Constructive Validity") [31] and Martin-L"of [17, 18, 19]. The first occurrence of formal induction-recursion is Martin-L"of's definition of a universe `a la Tarski [19], which consists of a set U

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We define a type theory MLM, which has proof theoretical strength slightly greater then Rathjens theory KPM. This is achieved by replacing the universe in Martin-Lof's Type Theory by a new universe V, which has the property that for every function f , mapping families of sets in V to families of sets in V, there exists a universe closed under f . We show that the proof theoretical strength of MLM is /\Omega 1\Omega M+! . Therefore we reach a strength slightly greater than jKPMj and V can be considered as a Mahlo-universe. Together with [Se96a] it follows jMLMj = /\Omega 1(\Omega M+! ). 1 Introduction An ordinal M is recursively Mahlo iff M is admissible and every M-recursive closed unbounded subset of M contains an admissible ordinal. Equivalently, this is the case iff M is admissible and for all \Delta 0 formulas OE(x; y; ~z), and all ~z 2 LM such that 8x 2 LM :9y 2 LM :OE(x; y; ~z) there exists an admissible ordinal fi ! M such that 8x 2 L fi 9y 2 L fi :OE(x; y; ~z) holds. ...

We present a type theory T T M, extending Martin-Löf Type Theory by adding one Mahlo universe V, a universe being the type theoretic analogue of one recursive Mahlo ordinal. A model, formulated in a Kripke-Platek style set theory KP M +, is given and we show that the proof theoretical strength of T T M is ≤ |KP M + | = ψΩ1 (ΩM+ω). By [Se96a], this bound is sharp. 1