Non-trivial Power Types Can't Be Subtypes of Polymorphic Types

This paper establishes a new limitative relation between the polymorphic lambda calculus and the kind of higher order type theory which is em bodied 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 poly morphic lambda calculus must place the poly morphic types well away from the powertypes of the topos in the sense that is a subtype of a polymorphic type only in the case that is empty and hence is ter minal As corollaries we obtain strengthenings of Reynolds result on the non existence of set theoretic models of polymorphism