# Principle of Recursive Definition for Well-Ordered Sets

## Theorem

Let $J$ be a well-ordered set.

Let $C$ be any set.

Let $\mathcal F$ be the set of all functions that map initial segments $S_a$ of $J$ into $C$.

Then for any function of the form:

$\rho: \mathcal F \to C$

there exists a unique function:

$h: J \to C$

satisfying:

$\forall \alpha \in J: h\left({\alpha}\right) = \rho\left({ h {\restriction_{S_\alpha}} }\right)$

where ${\restriction}$ denotes the restriction of a mapping.