An intrinsic treatment of ubiquity phenomena in higher-order computability models

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 the same ‘hereditarily total’ functionals. This results in a cleaner, more ‘intrinsic’ formulation of the latter theorems, which we are moreover able to generalize to cover certain computability models of a non-deterministic character. Within our axiomatic framework, we also investigate extensions of the ubiquity results to a much wider class of types, arising from simple types via subset and quotient constructions. This enables us to cover many types of interest that arise naturally in mathematical practice (e.g. in real or complex analysis), and establishes the existence of robust computability notions at such types. In the present Part I, we introduce our axioms and the conceptual framework, and carry through the required proofs for firstand second-order types. The focus here is on the Kreisel-Lacombe-Shoenfield theorem, which we adapt to our axiomatic setting in various ways. In Part II, we will extend our results to type 3 and above, using the construction of Normann which we exploited in our earlier work on ubiquity.