Extending Martin-Löf Type Theory by one Mahlo-universe

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-Lof'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 \(\geq \psi_{\Omega_1}\Omega_{{\rm M}+\omega}\). This is slightly greater than \(|{\rm KPM}|\), and shows that V can be considered to be a Mahlo-universe. Together with [Se96a] it follows \(|{\rm MLM}|=\psi_{\Omega_1}(\Omega_{{\rm M}+\omega})\).