Axioms and (counter)examples in synthetic domain theory

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 settheoretic functions and various constructions on domains would ideally be settheoretic 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 poin