Abstract
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 (1993) show that the proof theoretical strength of the type theory is precisely ψ Ω 1 Ω 1 + ω , which is slightly more than the strength of Feferman's theory T 0, classical set theory KPI and the subsystem of analysis ( Δ 1 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.
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