Well-Orderable Set has Choice Function
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Theorem
Let $S$ be a well-orderable set.
Then $S$ has a choice function.
Proof
Let $\preccurlyeq$ be a well-ordering on $S$.
Let $T \subseteq S$ be an arbitrary non-empty subset of $S$.
As $S$ is a well-ordered set, $T$ has a unique smallest element by $\preccurlyeq$.
Thus, we may define the choice function $C: \powerset S \setminus \set \O \to S$ as:
- $\forall T \in \powerset S \setminus \set \O: \map C T$ is the smallest element of $T$ under $\preccurlyeq$
This is the choice function we require.
$\blacksquare$
Sources
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $4$: Superinduction, Well Ordering and Choice: Part $\text I$ -- Superinduction and Well Ordering: $\S 4$ Well ordering and choice: Theorem $4.9$