Principle of Recursive Definition for Well-Ordered Sets
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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
Sources
- 2000: James R. Munkres: Topology (2nd ed.): Supplementary Exercise $1.1$