Hausdorff's Maximal Principle

Theorem

Formulation 1

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

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

Formulation 2

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.

Hausdorff's Maximal Principle and Axiom of Choice

Axiom of Choice implies Hausdorff's Maximal Principle

Let the Axiom of Choice be accepted.

Then Hausdorff's Maximal Principle holds.

Hausdorff's Maximal Principle implies Axiom of Choice

Let Hausdorff's Maximal Principle be accepted.

Then the Axiom of Choice holds.

Also known as

Hausdorff's Maximal Principle is also known as the Hausdorff Maximal Principle.

Some sources call it the Hausdorff Maximality Principle or the Hausdorff Maximality Theorem.

Also see

• Results about Hausdorff's maximal principle can be found here.

Source of Name

This entry was named for Felix Hausdorff.