\begin{definition}[Definition:Recursively Defined Mapping/Minimal Infinite Successor Set]
Let $\omega$ be the minimal infinite successor set.
Let $T$ be a set.
Let $a \in T$.
Let $g: T \to T$ be a mapping.
Let $f: \omega \to T$ be the mapping defined as:
:$\forall x \in \omega: \map f x = \begin{cases}
a & : x = 0 \\
\map g {\map f n} & : x = n^+
\end{cases}$
where $n^+$ is the successor set of $n$.
Then $f$ is said to be '''recursively defined''' on $\omega$.
\end{definition}