 The specification of \(\alpha(i,x,y)\) just given does not itself have the form of a recursive definition. But it is possible to define similar functions in a manner which generalizes the format of the scheme (\ref{gprimrec}). One means of doing so is to first use recursion on the type \(\mathbb{N} \rightarrow \mathbb{N}\)—a simple form of recursion at higher types as envisioned by Skolem and  Hilbert—to define an iteration functional as follows: 