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.


Type Theory Constructive Type Universe Construction Logical Strength Constructive Type Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

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:

Personalised recommendations