Second Principle of Recursive Definition

Theorem
Let $\N$ be the natural numbers.

Let $T$ be a set.

Let $a \in T$.

For each $n \in \N_{>0}$, let $G_n: T^n \to T$ be a mapping.

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


 * $\forall x \in \N: f(x) = \begin{cases}

a & : x = 0\\ G_n \left({f(0), \ldots, f(n)}\right) & : x = n + 1 \end{cases}$

Also known as
Some authors go through considerable effort to define the sequence $(G_n)_n$ as a single mapping $G$.

The domain of such a mapping is then for example given as one of the following:


 * $\operatorname{dom} G := \left\{{f: \N_{0} }\right\} = \displaystyle \bigcup_{n \mathop\in \N} [\N_{0} } T^n$

At we deem the presentation with separate $G_n$ to be more enlightening.

Also see

 * Principle of Recursive Definition
 * Transfinite Recursion