Infinite terms and recursion in higher types

Systems of infinite terms defining functionals of finite type were first considered by Tait [10] and further developed by Feferman [3] initially in a proof-theoretic context. Later in unpublished notes Feferman introduced the system T of infinite o terms inductively generated from variables of all finite types and constants for the ordinary primitive recursive functions by application, abstraction and autonomous enumeration: if for each n, f(n) codes a term tn~ T o and f is itself defined by a term of T O then the term<tn>n~ N is in T o . This definition can be relativized to an arbitrary functional~and the resulting system of terms is denoted by To(~) . Feferman proved that if~is of type 2 then the functions definable in To(~) are precisely the functions recursive in~(This also follows from our results here together with [11]) . This immediately poses the problem of whether infinite terms can be used to characterize full Kleene recursion in higher types and more specifically whether, for~of type n+2, To(T) gives a characterization of the n+1 section of ~.