On the ubiquity of certain total type structures

Abstract

It is an empirical observation arising from the study of higher type computability that a wide range of approaches to defining a class of (hereditarily) total functionals over$\nat$leads in practice to a relatively small handful of distinct type structures. Among these are the type structure C of Kleene–Kreiselcontinuous functionals, its effective substructure Ceffand the type structure HEO of thehereditarily effective operations. However, the proofs of the relevant equivalences are often non-trivial, and it is not immediately clear why these particular type structures should arise so ubiquitously.In this paper we present some new results that go some way towards explaining this phenomenon. Our results show that a large class ofextensional collapseconstructions always give rise to C, Ceffor HEO (as appropriate). We obtain versions of our results for both the ‘standard’ and ‘modified’ extensional collapse constructions. The proofs make essential use of a technique due to Normann.Many new results, as well as some previously known ones, can be obtained as instances of our theorems, but more importantly, the proofs apply uniformly to a whole family of constructions, and provide strong evidence that the three type structures under consideration are highly canonical mathematical objects.