Abstract

[41], with editorial changes and some minor corrections. Part 2 presents what happened next, together with some further development of the material. The first part begins with an elementary set-theoretical model of the λβ-calculus. Functions are modelled in a similar way to that normally employed in set theory, by their graphs; difficulties are caused in this enterprise by the axiom of foundation. Next, based on that model, a model of the λβη-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 λβη-calculus which are continuous complete lattices. In the second part we begin with a brief discussion of models of the λ-calculus in set theories with anti-foundation axioms. Next we review the model of the λβ-calculus of Part 1 and also the closely related—but different!—models of Scott [51, 52] and of Engeler [19, 20]. Then we discuss general frameworks in which elementary constructions of models can be given. Following Longo [36], one can employ certain Scott-Engeler algebras.

Citations