Principle of Recursive Definition for Minimally Inductive Set

Theorem
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.

Then there exists exactly one mapping $f: \omega \to T$ such that:


 * $\forall x \in \omega: \map f x = \begin {cases}

a & : x = \O \\ \map g {\map f n} & : x = n^+ \end {cases}$

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

Proof
Take the function $F$ generated in the second principle of transfinite recursion.

Set $f = F {\restriction_\omega}$.

Therefore, such a function exists.

Now, suppose there are two functions $f$ and $f'$ that satisfy this:


 * $\map f \O = \map {f'} \O$

Then:

This completes the proof.