Principle of Recursive Definition for Well-Ordered Sets

Theorem

Let $J$ be a well-ordered set.

Let $C$ be any set.

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

Then for any function of the form:

$\rho: \FF \to C$

there exists a unique function:

$h: J \to C$

satisfying:

$\forall \alpha \in J: \map h \alpha = \map \rho {h \restriction_{S_\alpha} }$

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

Proof

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Also see

• Principle of Recursive Definition

Sources

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