Zorn's Lemma

• categories: Set theory, Theorem, Real Analysis

Statement: Let $S$ be a partially ordered set in which every chain (i.e., totally ordered subset) has an upper bound in $S$. Then, $S$ contains at least one maximal element.

A maximal element $m\in S$ is an element such that there is no other element in $S$ strictly greater than $m$ with respect to the partial order.

Intuition

Zorn’s Lemma ensures that under certain conditions, a set with a partial ordering contains elements that cannot be “extended” further within the set. It’s particularly useful in proofs where we want to show the existence of a maximal or “largest” element without needing to construct it explicitly.

Key Concepts

• Partial Order: A relation $\le$ on $S$ that is reflexive, antisymmetric, and transitive.
• Chain: A subset $C\subset S$ in which every pair of elements is comparable, i.e., for any $x,y\in C$, either $x\le y$ or $y\le x$.
• Upper Bound: An element $u\in S$ is an upper bound of a chain $C\subset S$ if $x\le u$ for all $x\in C$.

Applications

Zorn’s Lemma is crucial in many areas of mathematics, particularly in situations where constructive methods are difficult or impossible:

• Existence of Bases: It is used to prove that every vector space has a basis (Hamel basis), even for infinite-dimensional spaces.
• Existence of Maximal Ideals: In ring theory, Zorn’s Lemma guarantees the existence of maximal ideals in any nontrivial ring with unity.
• Extension of Linear Operators: It helps extend linear functionals from subspaces to entire vector spaces.

Relation to the Axiom of Choice

Zorn’s Lemma is equivalent to the Axiom of Choice and Hausdorff’s Maximal Principle in standard set theory (Zermelo-Fraenkel set theory, ZF). This means any one of these statements can be derived from the others, and they are often used interchangeably to establish existence results in mathematics.