A Simple Model Construction for the Calculus of Constructions

We present a model construction for the Calculus of Constructions (CC) where all dependencies are carried out in a set-theoretical setting. The Soundness Theorem is proved and as a consequence of it Strong Normalization for CC is obtained. Some other applications of our model constructions are: showing that CC 4Classical logic is consistent (by constructing a model for it} and showing that the Axiom of Choice is not derivable in CC (by constructing a model in which the type that represents the Axiom of Choice is empty). 1 I n t r o d u c t i o n In the literature there are many investigations on the semantics of polymorphic A-calculus with dependent types (see for example [12, 11, 10, 1, 5, 13]). Most of the existing models present a semantics for systems in which the inhabitants of the impredicative universe (~ypes) are "lifted" to inhabitants of the predicative universe (kinds) (see [16]). Such systems are convenient to be modeled by locally Cartesian-closed categories having small Cartesian-closed subcategories. A wellknown instance of these categorical models is the category of w-sets (or D-sets) and its subcategory of modest sets, which is isomorphic to the category of partial equivalence relations (PER). Then the types are interpreted as PERs and then "lifted" through an isomorphism to modest sets and hence to o J-sets. In practical applications, however, one prefers to use a different simple syntactical presentation of type systems the so-called Pure Type Sys tems (PTSs). A semantics of such a system is usually obtained by implicitly or explicitly encoding the system into the system with "lifted" types, so the types are interpreted in the same way. The resulting semantics, even the one presented by concrete models (see [12, 13]) is still complicated as it gives an indirect meaning of PTSs. Moreover, most concrete models of such type systems are extensional in the sense that the interpretation of a type is a set with an equivalence relation on it with the equivalence relation on the function space defined as the extensional equality * Part of this research was performed while the author was working at the University of Nijmegen, on the ESPRIT BRA project 'Types for Proofs and Programs'