this paper will work for any of the different formulations of Martin-Lof's type theory. 2 The construction of the interpretation

this paper show that the exact formulation of the rules of type theory is very important for the power of the subset type; it actually turns out that there are propositions involving subsets which are trivially true in naive set theory but which cannot be proved in type theory. We will look at the provability of propositions of the form

this paper is to present a new paradox for Type Theory, which is a type-theoretic refinement of Reynolds' result [24] that there is no set-theoretic model of polymorphism. We discuss then one application of this paradox, which shows unexpected connections between the principle of excluded middle and the axiom of description in impredicative Type Theories. 1 Minimal and Polymorphic Higher-Order Logic

Introduction Type theory may be used as a programming language with an integrated programming logic. An implementation of the theory should then give an environment allowing you to specify, construct and verify programs interactively. We have used NUPRL [10], which is such an environment for type theory, to develop some examples. NUPRL's logic is based on Martin-Lof's type theory [7] and provides you with a proof system, a function which extracts programs from proofs, a program evaluator, a library system and a possibility to define your own syntax and proof tactics for (partial) automatization of proofs. The usual way to specify a program in type theory is by identifying the specification with a type and a program satisfying the specification is then an element of this type. For instance a specification of a function which, given a non-empty list, returns any member of that list may be given by the proposition: (#x #<F

The theory we will be concerned with in this paper is Martin-Löf's polymorphic type theory with intensional equality and the universe of small sets. We will give a different proof of normalization for this theory.