The strength of Martin-Löf type theory with a superuniverse. Part I

Abstract.

Universes of types were introduced into constructive type theory by Martin-Löf [12]. The idea of forming universes in type theory is to introduce a universe as a set closed under a certain specified ensemble of set constructors, say $\mathcal{C}$. The universe then “reflects”$\mathcal{C}$.

This is the first part of a paper which addresses the exact logical strength of a particular such universe construction, the so-called superuniverse due to Palmgren (cf. [16, 18, 19]).

It is proved that Martin-Löf type theory with a superuniverse, termed MLS, is a system whose proof-theoretic ordinal resides strictly above the Feferman-Schütte ordinal $\Gamma_0$ but well below the Bachmann-Howard ordinal. Not many theories of strength between $\Gamma_0$ and the Bachmann-Howard ordinal have arisen. MLS provides a natural example for such a theory.