Abstract.
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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Received: 8 December 1998 / Published online: 21 March 2001
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Rathjen, M. The strength of Martin-Löf type theory with a superuniverse. Part II. Arch. Math. Logic 40, 207–233 (2001). https://doi.org/10.1007/s001530000051
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DOI: https://doi.org/10.1007/s001530000051