Zermelo's Well-Ordering Theorem/Converse/Proof 1

Theorem

Let it be supposed that every set is well-orderable.

Then the Axiom of Choice holds.

Proof

Let $S$ be an arbitrary set.

By assumption $S$ is well-orderable.

From Well-Orderable Set has Choice Function, $S$ has a choice function.

As $S$ is arbitrary, the result follows.

$\blacksquare$

Also known as

Zermelo's Well-Ordering Theorem is also known just as the well-ordering theorem.

Some sources omit the hyphen: (Zermelo's) well ordering theorem.

It is also known just as Zermelo's Theorem.

Under this name it can often be seen worded:

Every set of cardinals is well-ordered with respect to $\le$.

This is called by some authors the Trichotomy Problem.

It is also referred to as the well-ordering principle, but this causes confusion with the result that states that the natural numbers are well-ordered.

Source of Name

This entry was named for Ernst Friedrich Ferdinand Zermelo.

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 $4.9$