Countable Set has Choice Function

Theorem

Let $S$ be a countable set.

Let $\mathbb S = \powerset S \setminus \set \O$ be the power set of $S$ excluding the empty set $\O$.

Then there exists a choice function $C$ for $S$:

$\forall x \in \mathbb S: \map C x \in x$

Proof

From Countable Set is Well-Orderable, we have that $S$ is a well-orderable set.

The result follows from Well-Orderable Set has Choice Function.

$\blacksquare$

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