\begin{definition}[Definition:Recursively Defined Mapping/Natural Numbers]
Let $p \in \N$ be a natural number.
Let $S = \set {x \in \N: p \le x}$.
Let $T$ be a set.
Let $g: T \to T$ be a mapping.
Let $f: S \to T$ be the mapping defined as:
:$\forall x \in S: \map f x = \begin{cases}
a & : x = p \\
\map g {\map f n} & : x = n + 1
\end{cases}$
for $a \in T$.
Then $f$ is said to be '''recursively defined''' on $S$.
\end{definition}