Recursion theorem on the natural numbers
= Recursion theorem on the natural numbers
{title2=$f(0)=a,\qquad f(n+1)=H(n,f(n))$}
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.