Zermelo's Well-Ordering Theorem/Proof 1

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Theorem

Let the Axiom of Choice be accepted.

Then every set is well-orderable.


Proof

Let $S$ be a set.

Let $\powerset S$ be the power set of $S$.

By the Axiom of Choice, there exists a choice function on $S$.

The result follows from Set with Choice Function is Well-Orderable.

$\blacksquare$


Axiom of Choice

This theorem depends on the Axiom of Choice.

Because of some of its bewilderingly paradoxical implications, the Axiom of Choice is considered in some mathematical circles to be controversial.

Most mathematicians are convinced of its truth and insist that it should nowadays be generally accepted.

However, others consider its implications so counter-intuitive and nonsensical that they adopt the philosophical position that it cannot be true.


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