Recursion theorem on the natural numbers ($f(0)=a,f(n+1)=H(n,f(n))$)

Given an initial value $a$ and a specified successor rule on a set, there is a unique function on the natural numbers with that initial value and successor rule. Finite compatible approximations give existence; induction gives agreement of any two approximations and uniqueness. Recursive definitions of addition and multiplication apply this theorem with the other argument fixed.