# Definition:Recursively Defined Mapping/Minimal Infinite Successor Set

## Definition

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 $\omega$.