Zermelo's Well-Ordering Theorem/Proof 3

From ProofWiki
Jump to navigation Jump to search


Let the Axiom of Choice be accepted.

Then every set is well-orderable.


Let $S$ be a non-empty set.

Let $C$ be a choice function for $S$.

Let $A$ be an arbitrary set.

Let the mapping $h$ be defined as:

$\map h A = \begin {cases}

\map C {S \setminus A} & : A \subsetneqq S \\ x & : \text {otherwise} \end {cases}$

where $x$ is an arbitrary element such that $x \notin S$.

The latter is known to exist from Exists Element Not in Set.

From the Transfinite Recursion Theorem: Formulation $5$, there exists a mapping $F$ on the class of all ordinals $\On$ such that:

$\forall \alpha \in \On: \map F \alpha = \map h {F \sqbrk \alpha}$

It remains to show the following:

$(1): \quad \exists \delta \in \On: \map F \delta \notin S$
$(2): \quad$ Let $\beta$ be the smallest ordinal such that $\map F \beta \notin S$. Then:
$F \sqbrk \beta = S$
$F \restriction \beta$ is injective
and so $\beta$ can be put into a one-to-one correspondence with $S$.

Hence $S$ can be well-ordered.

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.