Well-Ordering, Proofs for Martin-Löf Type Theory

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 [Set93] show that the proof theoretical strength of the type theory is precisely ψΩ1ΩI+ω, which is slightly more than the strength of Feferman’s theory T0, classical set theory KPI and the subsystem of analysis (∆2−CA)+(BI). The strength of intensional and extensional version, of the version à la Tarski and à la Russell are shown to be the same.