Definition.

The initial functions are the following:

a.     zero function, Z(x) = 0 for all x Î N,

b.     successor function. S(x) = x+1 for all x Î N,

c.     projection functions, U_iⁿ(x₁, …, x_n) = x_i for all x₁, ..., x_n x Î N; one function for each n and i where n = 1, 2, 3, ... and 1 £ i £ n.

d.     the integers, k for each k Î N.  (These are numbers, not functions, but for our purposes it is convenient to include them as "initial functions".)

We can now define the class of recursive functions: a function is recursive if and only if it can be obtained from the initial functions by a finite number of applications of the operations of substitution, recursion and m-operator. This can be expressed more formally.

Definition.

The function f is a recursive function if there is a finite sequence f_i, f₂, f₃, …, f_n of functions such that f_n = f and for each i (1 £ i £ n) either f_i is an initial function or f_i is obtained from entries earlier in the sequence by substitution, recursion or the m-operator.