There are compelling benefits to using foundational type theory as a framework for programming language semantics. I give a semantics of an expressive programming calculus in the foundational type theory of Nuprl. Previous typetheoretic semantics have used less expressive type theories, or have sacrificed important programming constructs such as recursion and modules. The primary mechanisms of this semantics are partial types, for typing recursion, set types, for encoding power and singleton kinds, which are used for subtyping and module programming, and very dependent function types, for encoding signatures. Keywords Semantics, program verification, type theory, functional programming 1 INTRODUCTION Type theory has become a popular framework for formal reasoning in computer science and has formed the basis for a number of automated deduction systems, including Automath, Nuprl, HOL and Coq, among others. In addition to formalizing mathematics, these systems are widely used for the a...