Abstract

Introduction Domain Theory, type theory (both in the style of Martin-Lof [40, 41] and in the polymorphic style of Girard/Reynolds [23, 56]), and topos theory (both in the topological/sheaf-theoretic treatments and in the realizability approach going back to the early work of Kleene) have attempted to improve on set theory by providing a large suite of closure conditions on domains/types/objects as well as a far-reaching logic of properties emphasizing the computable/constructive aspects of the definitions and qualities of functions. Scott's domain theory, (and the many variations proposed and studied; see [2] and [75] for recent introductions with references) has been especially successful in allowing for recursive definitions of types (i.e., solutions to domain equations) but at the expense of introducing a complex structure of "partial elements" in order to have solutions to fixed-point equations in the domains. Moreover, the topological and e

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