Set-Theoretical and Other Elementary Models of the lambda-Calculus

Part 1 of this paper is the previously unpublished 1972 memorandum [43], with editorial changes and some minor corrections. Part 2 presents what happened next, together with some further development of the material. The rst part begins with an elementary set-theoretical model of the -calculus. Functions are modeled in a similar way to that normally employed in set theory, by their graphs; di culties are caused in this enterprise by the axiom of foundation. Next, based on that model, a model of the -calculus is constructed by means of a natural deduction method. Finally, a theorem is proved giving some general properties of those non-trivial models of the -calculus which are continuous complete lattices. The second part begins with a brief discussion of models of the -calculus in set theories with anti-foundation axioms. Next the model of the calculus of Part 1 and also the closely related|but di erent!|models of Scott [53, 54] and of Engeler [21, 22] are reviewed. Then general frameworks in which elementary constructions of models can be given are discussed. Following Longo [38], one can employ certain Scott-Engeler algebras. Following Coppo, Dezani-Ciancaglini, Honsell and Longo [13], one can obtain lter models from their Extended Applicative Type Structures. An extended discussion is given of various ways of constructing models of the -calculus, and the connections between them. Finally an extension of the theorem to complete partial orders is given. The theme of the paper is the consideration of means of constructing models. There is hardly any analysis of their properties; there is no discussion of their application. Part 1: A Set-theoretical de nition of application