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-