Axioms and (counter) examples in synthetic domain theory

Introduction The central idea of Synthetic Domain Theory (SDT) is that, if one formalizes abstract properties that a category of domains should have (relative to an intended range of applications), one might nd full subcategories of the category of sets enjoying these properties. Domains, then, would just be (special) sets, maps of domains would be arbitrary set-theoretic functions, and various constructions on domains would (ideally) be set-theoretic constructions. Unfortunately, one soon observes that this idea runs into trouble with classical set theory. For example, there are precious few sets with the property that any endofunction on them has a xed point. Remarkably however, as Dana Scott observed in 25], such inconsistencies do not arise if intuitionistic set theory is used instead. For this reason, Scott proposed that intuitionistic set theory might provide an intuitive and powerful framework for deriving domain-theoretic structure as set-theoretic structure. This proposal has since been vindicated both axiomatically in (many versions of) intuitionistic set theory, and semantically in models of intuitionistic set theory (especially toposes). To carry out an axiomatic treatment, one rst needs to x on a version of intuitionistic set theory. In this paper we adopt the most popular choice, the internal logic of an elementary topos (with nno), also chosen, e.g., in 23, 8, 26]. The principal beneets are that models of the logic (toposes) are ubiquitous, and the methods for constructing and analysing them are very well-established. For the purposes of the axiomatic part of this paper, we believe that it would also be Both authors acknowledge the PIONIER project \The Geometry of Logic", led by Professor I. Moerdijk, which employs the rst author, and enabled the second author to visit Utrecht in the period February{June 1998. 1 possible to use an (impredicative) intuitionistic type theory, as in 22], or even intuitionistic Zermelo-Fraenkel set theory ((27]), without changing the nature of the mathematics (only the metamathematics). An interesting challenge would be to attempt an axiomatic development in a predicative type theory. It seems that the best way of isolating a full subcategory of sets is to identify a category of predomains, carriers of computational values, not necessarily including any speciic \undeened" value (In classical domain-theoretic terms, predomains are cpo's without the requirement of a least element). One aims to place axioms that guarantee that the category is closed under important set-theoretic constructions (e.g. function spaces), and allows a treatment of domain-theoretic phenomena, …