Countable Set is Well-Orderable

Theorem

Let $S$ be a countable set.

Then $S$ is well-orderable.

Proof

By the Well-Ordering Principle, the set of natural numbers $\N$ under the usual ordering $\le$ forms a well-ordered set.

By definition of countable set, there exists an injection:

$f: S \to \N$

Let $V$ be a basic universe.

By definition of basic universe:

$S \in V$

and:

$\N \in V$

By the Axiom of Transitivity, both $S$ and $\N$ are classes.

From Class which has Injection to Subclass of Well-Orderable Class is Well-Orderable, it follows that $S$ is well-orderable.

Hence the result.

$\blacksquare$

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: $\S 1$ Introduction to well ordering: Discussion