Zorn's Lemma implies Hausdorff's Maximal Principle

Theorem

Let Zorn's Lemma be accepted as true.

Then Hausdorff's Maximal Principle holds.

Proof

Recall Zorn's Lemma:

Let $\struct {S, \preceq}, S \ne \O$ be a non-empty ordered set.

Let $T \subseteq \powerset S$ be the set of subsets of $S$ that are totally ordered by $\preceq$.

Then every element of $T$ is a subset of a maximal element of $T$ under the subset relation.

$\Box$

Recall Hausdorff's Maximal Principle:

Let $A$ be a non-empty set of sets.

Let $S$ be the set of all chain of sets of $A$ (ordered under the subset relation).

Then every element of $S$ is a subset of a maximal element of $S$ under the subset relation.

$\Box$

It is seen directly that Hausdorff's Maximal Principle is a special case of Zorn's Lemma where the ordering is the subset relation.

$\blacksquare$

Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $4$: Superinduction, Well Ordering and Choice: Part $\text {II}$ -- Maximal principles: $\S 5$ Maximal principles: Proposition $5.9$