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.
This concept has the prerequisites:
- 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)
→ Notes on Set Theory (2013)
Lecture notes for a course on axiomatic set theory.
→ 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
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