Recursion in Kolmogorov's R-operator and the ordinal σ3

Abstract

The theory of objects recursive in functions of higher types was developed originally by Kleene [Kl 1,2]. The sets of objects of finite types are defined by Kleene as follows:T0 = ω = the set of nonnegative integers,and, for all i ∈ ω,Ti+1 = Tiω = the set of total functions from Ti to ω.In order to study and classify the functions recursive in Kolmogorov's R-operator, it is necessary to discuss recursion in partial functions of higher types. Accordingly, we define the partial objects of higher finite types byand, for all i ∈ ω, = the set of partial functions from Ti to ω.In addition to considering the partial functions from Ti to ω, the “hereditary partial” functions also have to be considered. For example, Hinman has studied objects of type 2 which act on partial functions from ω to ω. In case of dealing with a function F which has partial functions in its domain, we usually require that F be consistent, i.e. if F(f) is defined and g is an object of the same type as f, defined at least at all points where f is defined (in short, g extends f—notation: g ⊇ f), then F(g) is defined and is equal to F(f). Now we can define the higher type partial objects which are consistent byand, for all i ∈ ω, = the set of consistent partial functions from to ω.