Abstract

This paper establishes a new, limitative relation between the polymorphic lambda calculus and the kind of higher-order type theory which is embodied in the logic of toposes. It is shown that any embedding in a topos of the cartesian closed category of (closed) types of a model of the polymorphic lambda calculus must place the polymorphic types well away from the powertypes oe !\Omega of the topos, in the sense that oe !\Omega is a subtype of a polymorphic type only in the case that oe is empty (and hence oe !\Omega is terminal) . As corollaries, we obtain strengthenings of Reynolds' result on the non-existence of settheoretic models of polymorphism. Introduction The results reported in this paper have their origin in Reynolds' discovery that the standard set-theoretic model of the simply typed lambda calulus cannot be extended to model the polymorphic, or second-order, typed lambda calculus. In [9] Reynolds speculated that there might be a model of polymorphism in which the typ...

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