Generic Functions in Scott Domains

Abstract

In this thesis we will apply forcing to domain theory. When a Scott domain represents a function space, each function will be a filter in the basis of the domain. By using the partially ordered basis as the forcing relation, each generic filter G yields a model of ZFC in which G is a function, given some other model of ZFC containing this basis. Such generic functions are the main concern of this thesis. By case studies and general abstractions of these, we will investigate whether G is a total function or not. We will specifically consider function spaces where R and (N -> N) are domains and N and R codomains. In the cases where the domain of G is sigma compact, G is total. For (X -> R) where X is a separable complete metric space, the main result is that G is total if and only if X is sigma compact, given some rather weak additional condition on X. When G is not total, we will explicitly construct some x for which G is not defined.