Hausdorff Maximal Principle/Proof 2

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Theorem

Let $\struct {\PP, \preceq}$ be a partially ordered set.

Then there exists a maximal chain in $\PP$.


Proof

Let $\preceq$ be an ordering on the set $\PP$.

Let $X$ be a chain in $\left({\PP, \preceq}\right)$.

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


Let us define $\CC$ as:

$\CC = \leftset {Y : Y}$ is a chain in $\PP$ and $\rightset {X \subseteq Y}$

Then:

a maximal chain in $\PP$ that includes $X$

and:

a maximal element of $\CC$ under the partial order induced on $\CC$ by inclusion

are one and the same.

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


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

Let $W$ be an arbitrary chain in $\CC$.

Let $Z = \bigcup W$.

It will be shown that $Z$ is an upper bound for $W$ in $\CC$.


Let $a, b \in Z$.

Then:

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

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

Without loss of generality, suppose $A \subseteq B$.

Then $a, b \in B$.

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

Therefore $Z$ is a chain in $\PP$.

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:

$Z \in \CC$

We have shown that for an arbitrary chain $W$ in $\CC$, $W$ has an upper bound $Z$ in $\CC$.

The result follows.

$\blacksquare$


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 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.


Source of Name

This entry was named for Felix Hausdorff.