Principle of Recursive Definition for Well-Ordered Sets
Jump to navigation
Jump to search
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 $1.1$