 A class \(\mathbf{X}\) of recursively defined functions may be specified by giving a class of initial functions \(I_{\mathbf{X}}\) which is then closed under one or more functionals from a class \(Op_{\mathbf{X}}\). It is in general possible to define a class in this manner on an arbitrary set of initial functions. However, all of the function classes considered in this entry will determine functions of type \(\mathbb{N}^k \rightarrow \mathbb{N}\)—i.e., they will take \(k\)-tuples of natural numbers as inputs and (if defined) return a single natural number as output.