Definition:Recursively Defined Mapping/Minimally Inductive 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: f \left({x}\right) = \begin{cases}

a & : x = 0 \\ g \left({f \left({n}\right)}\right) & : x = n^+ \end{cases}$

where $n^+$ is the successor set of $n$.

Then $f$ is said to be recursively defined on $\omega$.

Also see

 * Principle of Recursive Definition/Proof 2