This article provides the proof-theoretic analysis of the transfinitely iterated fixed point theories $\widehat{ID}_\alpha and \widehat{ID}_{ the exact proof-theoretic ordinals of these systems are presented.

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-Löf'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 $|{\rm MLM}|=\psi_{\Omega_1}(\Omega_{{\rm M}+\omega})$. (shrink)

We present well-ordering proofs for Martin-Löf's type theory with W-type and one universe. These proofs, together with an embedding of the type theory in a set theoretical system as carried out in Setzer show that the proof theoretical strength of the type theory is precisely ψΩ1Ω1 + ω, which is slightly more than the strength of Feferman's theory T0, classical set theory KPI and the subsystem of analysis + . The strength of intensional and extensional version, of the version à (...) la Tarski and à la Russell are shown to be the same. (shrink)

Induction–recursion is a powerful definition method in intuitionistic type theory. It extends inductive definitions and allows us to define all standard sets of Martin-Löf type theory as well as a large collection of commonly occurring inductive data structures. It also includes a variety of universes which are constructive analogues of inaccessibles and other large cardinals below the first Mahlo cardinal. In this article we give a new compact formalization of inductive–recursive definitions by modeling them as initial algebras in slice categories. (...) We give generic formation, introduction, elimination, and equality rules generalizing the usual rules of type theory. Moreover, we prove that the elimination and equality rules are equivalent to the principle of the existence of initial algebras for certain endofunctors. We also show the equivalence of the current formulation with the formulation of induction–recursion as a reflection principle given in Dybjer and Setzer 93). Finally, we discuss two type-theoretic analogues of Mahlo cardinals in set theory: an external Mahlo universe which is defined by induction–recursion and captured by our formalization, and an internal Mahlo universe, which goes beyond induction–recursion. We show that the external Mahlo universe, and therefore also the theory of inductive–recursive definitions, have proof-theoretical strength of at least Rathjen's theory KPM. (shrink)