Maximal Principles

Theorem

The maximal principles are a collection of theorems which can be considered as forms of Zorn's Lemma.

Kuratowski's Lemma

Let $S$ be a set of sets which is closed under chain unions.

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

Tukey's Lemma

Let $S$ be a non-empty set of finite character.

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

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.

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.

Also known as

The maximal principles are also known as:

• Kuratowski Principles (for Kazimierz Kuratowski)
• Hausdorff Principles (for Felix Hausdorff)

Also see

• Results about the maximal principles can be found here.

Historical Note

The Maximal Principles were discovered independently by Felix Hausdorff in $1914$, Kazimierz Kuratowski in $1922$, and Max August Zorn in $1935$.

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