Hausdorff Maximal Principle

Theorem

Each chain in the poset $\left({\mathcal P, \preceq}\right)$ is included in a maximal chain in $\mathcal P$.

Proof

Let $\preceq$ be a partial order on the set $\mathcal P$ and let $X$ be a chain in $\left({\mathcal P, \preceq}\right)$.

By definition, a maximal chain in $\mathcal P$ that includes $X$ is a chain $Y$ in $\mathcal P$ such that $X \subseteq Y$ and there is no chain $Z$ in $\mathcal P$ with $X \subseteq Z$ and $Y \subsetneq Z$.

Let us define $\mathcal C$ as:

$\mathcal C = \left\{{Y : Y}\right.$ is a chain in $\mathcal P$ and $\left.{X \subseteq Y}\right\}$

Then:

• a maximal chain in $\mathcal P$ that includes $X$ and
• a maximal element of $\mathcal C$ under the partial order induced on $\mathcal C$ by inclusion

are one and the same.

It thus suffices to show that $\mathcal C$ contains such a maximal element.

According to Zorn's Lemma, $\mathcal C$ contains a maximal element if each chain in $\mathcal C$ has an upper bound in $\mathcal C$.

Let $W$ be a chain in $\mathcal C$, and let $Z = \bigcup W$.

Let $a, b \in Z$.

Then $\exists A, B \in W: a \in A, b \in B$.

Since $W$ is a chain in $\mathcal C$, one of $A$ and $B$ includes the other.

Suppose WLOG $A \subseteq B$.

Then $a, b \in B$.

Since $B$ is a chain in $\mathcal P$, either $a \preceq b$ or $b \preceq a$.

Therefore $Z$ is a chain in $\mathcal P$.

By definition of $Z$, we have that $\forall A \in W: A \subseteq Z$.

Thus $Z$ is an upper bound for $W$ under inclusion.

Finally we have that:

$\forall A \in W: X \subseteq A$

and so $X \subseteq Z$.

Thus $Z \in \mathcal C$

$\blacksquare$

Axiom of Choice

This proof depends on the Axiom of Choice, by way of Zorn's Lemma.

Because of some of its bewilderingly paradoxical implications, the Axiom of Choice is considered 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.

Source of Name

This entry was named for Felix Hausdorff.