Universes in type theory part I—Inaccessibles and Mahlo

Abstract

We give an overview over universes in Martin-Lof type theory and consider the following universe constructions: a simple universe, E. Palmgren’s super universe and the Mahlo universe. We then introduce models for these theories in extensions of Kripke-Platek set theory having the same proof theoretic strength. The extensions of Kripke-Platek set theory used formalise the existence of a recursively inaccessible ordinal, a recursively hyper-inaccessible ordinal, and a recursively Mahlo ordinal. Using these models we determine upper bounds for the proof theoretic strength of the theories in questions. In case of simple universes and the Mahlo universe, these bounds have been shown by the author to be sharp. This article is an overview over the main techniques in developing these models, full details will be presented in a series of future articles.