Archive for Mathematical Logic

, Volume 40, Issue 3, pp 207–233 | Cite as

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

  • Michael Rathjen


Universes of types were introduced into constructive type theory by Martin-Löf [3]. 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 ?. The universe then “reflects”?.

This is the second part of a paper which addresses the exact logical strength of a particular such universe construction, the so-called superuniverse due to Palmgren (cf.[4–6]).

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 Γ0 but well below the Bachmann-Howard ordinal. Not many theories of strength between Γ0 and the Bachmann-Howard ordinal have arisen. MLS provides a natural example for such a theory. In this second part of the paper the concern is with the with upper bounds.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Michael Rathjen
    • 1
  1. 1.School of Mathematics, University of Leeds, Leeds LS2 9JT, UK. e-mail:

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