# Principle of Recursive Definition for Well-Ordered Sets

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## Contents

## 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.

## Proof

## Also see

## Sources

- 2000: James R. Munkres:
*Topology*(2nd ed.): Supplementary Exercise $1.1$