The Principle of Recursive Definition. Let $A$ be a set, $a_{0}\in A$ , $F_{n}$ be the set of all functions $f:\{1,...,n\}\rightarrow A$ , and $\rho:\cup_{n\in\mathbb{Z}_{+}}F_{n}\rightarrow A$ be a recursive rule. Then there exists a unique function (recursion formula) $h:\mathbb{Z}_{+}\rightarrow A$ such that $h(1)=a_{0}$ and for $i>1$ , $h(i)=\rho(h|\{1,...,i-1\})$ .