# Zorn's Lemma

## Summary

Zorn's Lemma states that if every chain in a partially ordered set S has an upper bound, then S has a maximal element. Zorn's Lemma is equivalent to the Axiom of Choice.

## Context

This concept has the prerequisites:

- functions and relations as sets (Proving Zorn's Lemma requires viewing relations as sets.)
- Axiom of Choice (Zorn's Lemma is equivalent to the Axiom of Choice.)
- bases (A canonical application of Zorn's Lemma is to show that every vector space has a basis.)
- order relations (Zorn's Lemma is stated in terms of partial orderings.)

## Goals

- Know the statement of Zorn's Lemma

- Be aware that it is equivalent to the Axiom of Choice

- Prove Zorn's Lemma, assuming the Axiom of Choice

- Prove the Axiom of Choice using Zorn's Lemma

- Use Zorn's Lemma to show that every vector space has a basis

## Core resources (read/watch one of the following)

## -Free-

→ Notes on Set Theory (2013)

Lecture notes for a course on axiomatic set theory.

## -Paid-

→ Elements of Set Theory

An introductory textbook on axiomatic set theory.

- Part of Chapter 6, "Cardinal numbers and the Axiom of Choice," subsection "Axiom of Choice," pages 151-154
- Chapter 7, "Orderings and ordinals," subsection "Debts paid," pages 195-199

Additional dependencies:

- cardinality

## See also

-No Additional Notes-