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

Introduction Mathematics is usually developed on the basis of set theory. When trying to use type theory as a new basis for mathematics, most of mathematics has to be reformulated. This is of great use, because then the step to programs is direct and one can expect to get the best programs. However, it seems that most mathematicians will continue to work in set theory. Even when changing to type theory for the formalisation, usually the proofs will be developed first having classical set theory in the background. Therefore methods for transferring directly set theoretical arguments to type theory could make the step from traditional mathematics to type theory and therefore to computer science far easier. The reason why set theory is used in mathematics is its high flexibility and that it allows to write down expressions without having to care about the type of the object. Therefore, if set theoretical proofs can be transferred to type theory, one could use set theory as a progr