This paper deals with universes in explicit mathematics. After introducing some basic definitions, the limit axiom and possible ordering principles for universes are discussed. Later, we turn to least universes, strictness and name induction. Special emphasis is put on theories for explicit mathematics with universes which are proof-theoretically equivalent to Feferman's T 0 . 1 Introduction In some form or another, universes play an important role in many systems of set theory and higher order arithmetic, in various formalizations of constructive mathematics and in logics for computation. One aspect of universes is that they expand the set or type formation principles in a natural and perspicuous way and provide greater expressive power and proof-theoretic strength. The general idea behind universes is quite simple: suppose that we are given a formal system Th comprising certain set (or type) existence principles which are justified on specific philosophical grounds. Then it may be a...

Contents 1 Introduction 3 2 The theory T 0 5 3 The W-type 10 4 The theories FID(K ) 14 5 The model construction 17 6 Restricted inductions 26 7 Universes 27 1 Introduction In this thesis we are going to define an analogue of Martin-Lof's W-type for explicit mathematics and show that in the framework of explicit mathematics with elementary comprehension and join the W-type is equivalent to the principle of inductive generation. Furthermore we describe a new model construction for T 0 by a non-monotone inductive definition which we think is very natural. The construction will be formalized in Jager's theory FID(I). We will show how in this model restricted inductions can be treated in an adequate way. Then we extend our model construction to systems of explicit mathematics with universes. Systems of explicit mathematics were first presented by Feferma