Uncountable Limits and the lambda Calculus
In this paper we address the problem of solving recursive domain equations using uncountable limits of domains. These arise for instance, when dealing with the ω1-continuous function-space constructor and are used in the denotational semantics of programming languages which feature unbounded choice constructs. Surprisingly, the category of cpo's and ω1-continuous embeddings is not ω0-cocomplete. Hence the standard technique for solving reflexive domain equations fails. We give two alternative methods. We discuss also the issue of completeness of the λβηcalculus w.r.t reflexive domain models. We show that among the reflexive domain models in the category of cpo's and ω0-continuous functions there is one which has a minimal theory. We give a reflexive domain model in the category of cpo's and ω1-continuous functions whose theory is precisely the λβη theory. So ω1-continuous λ-models are complete for the λβη-calculus.