In this thesis I will relate standard computability to computability of reals|and related spaces|and link these notions with denotational semantics of computer languages. The framework for the proposed study was worked out in collaboration with Steve Awodey, Lars Birkedal and Dana Scott. The rst section gives a motivation for our setup and indicates how standard computability ts in it. The subs...

We introduce some simple conditions on typed partial combinatory algebras (viewed as models of higher-order computability) which suffice for an axiomatic development of some non-trivial computability theory, including various forms of the Kreisel-LacombeShoenfield theorem, along with the author’s earlier ‘ubiquity’ theorems which say that a wide range of computability models all give rise to th...