On sets of terms with a given intersection type
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We are interested in how much of the structure of a strongly normalizable lambda term is captured by its intersection types and how much all the terms of a given type have in common. In this note we consider the theory BCD (Barendregt, Coppo, and Dezani) of intersection types without the top element. We show: for each strongly normalizable lambda term M, with beta-eta normal form N, there exists an intersection type A such that, in BCD, N is the unique beta-eta normal term of type A. A similar result holds for finite sets of strongly normalizable terms for each intersection type A if the set of all closed terms M such that, in BCD, M has type A, is infinite then, when closed under beta-eta conversion, this set forms an adaquate numeral system for untyped lambda calculus. A number of related results are also proved.
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