Arch. Math. Logic (2000) 39: 155–181 c  Springer-Verlag 2000 Extending Martin-L ¨ of Type Theory by one Mahlo-universe Anton Setzer Department of Mathematics, Uppsala University, P.O. Box 480, S-751 06 Uppsala, Sweden (e-mail: setzer@math.uu.se) Received: 8 February 1996 Abstract. We define a type theory MLM, which has proof theoretical strength slightly greater then Rathjen’s theory KPM. This is achieved by replacing the universe in Martin-L¨ of’s Type Theory by a new universe V having the property that for every function f , mapping families of sets in V to families of sets in V, there exists a universe inside V closed under f . We show that the proof theoretical strength of MLM is ≥ ψ Ω 1 Ω M+ω . This is slightly greater than |KPM|, and shows that V can be considered to be a Mahlo-universe. Together with [Se96a] it follows |MLM| = ψ Ω 1 (Ω M+ω ). 1. Introduction An ordinal M is recursively Mahlo if M is admissible and every M-recursive closed unbounded subset of M contains an admissible ordinal. Equivalently, this is the case if and only if M is admissible and for all ∆ 0 formulas φ(x,y, z ), and all  z ∈ L M such that ∀x ∈ L M .∃y ∈ L M .φ(x,y, z ) there exists an admissible ordinal β< M such that ∀x ∈ L β ∃y ∈ L β .φ(x,y, z ) holds. One can easily see that M is inaccessible and that β can always be chosen to be inaccessible. On the basis of this definition Rathjen has in [Ra91] developed a formu- lation of Kripke Platek set theory with one recursively Mahlo ordinal called KPM and has analyzed it proof theoretically. ([Ra90, Ra91,Ra94]). This was a major break-through in proof theory after the treatment of inaccessibles. A universe in Martin-L¨ of Type Theory can be interpreted as the least fixed point of a certain operator. In the presence of the W-type, the fixed point is obtained by iterating the operator up to the first recursively inaccessible: an admissible is necessary for closure under the inductive definition of U,