Zermelo's Well-Ordering Theorem/Converse

Theorem

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

Then the Axiom of Choice holds.

Proof 1

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.

$\blacksquare$

Proof 2

Let $\FF$ be an arbitrary collection of sets.

By assumption all sets can be well-ordered.

Hence the set $\bigcup \FF$ of all elements of sets contained in $\FF$ is well-ordered by some ordering $\preceq$.

By definition then, every subset of $\ds \bigcup \FF$ has a smallest element under $\preceq$.

Also, note that each set in $\FF$ is a subset of $\bigcup \FF$.

Thus, we may define the choice function $C$:

$C: \FF \to \bigcup \FF, \map C X = \min X$

where $\min X$ is the smallest element of $X$ under $\preceq$.

$\blacksquare$

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.