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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… (More)

The purpose of this note is to observe a generalization of the concept " computable in.. . " to arbitrary partial combinatory algebras. For every partial combinatory algebra (pca) A and every partial endofunc-tion on A, a pca A[f ] is constructed such that in A[f ], the function f is representable by an element; a universal property of the construction is… (More)

Introduction The purpose of this short paper is to sketch the development of a few basic topics in the history of Realizability. The number of topics is quite limited and reeects very much my own personal taste, biases and prejudices. Realizability has, over the past 60 years, developed into a subject of such dimensions that a comprehensive overview would… (More)

We study a notion of realizability with a local operator J which was first considered by A.M. Pitts in his thesis [7]. Using the Suslin-Kleene theorem, we show that the representable functions for this realizability are exactly the hyperarithmetical (∆ 1 1) functions. We show that there is a realizability interpretation of nonstandard arithmetic, which,… (More)