# 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:

## 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