Axiom of Choice
The Axiom of Choice states that for any set A of sets, there exists a choice function which picks a single element of each x in A. While intuitive, it has some surprising consequences, such as the Banach-Tarski Paradox. It is logically independent of the Zermelo-Frankl axioms, so one may choose whether or not to include it.
This concept has the prerequisites:
- Zermelo-Frankl axioms (The Axiom of Choice is often added to the Z-F axioms.)
- Know the statement of the Axiom of Choice
- Be aware that it is controversial (e.g. because of the Banach-Tarski paradox), and it is often noted specifically when a theorem depends on it
- Be aware that it is independent of the Zermelo-Frankl axioms
Core resources (read/watch one of the following)
Location: Article "Axiom of Choice"
→ Elements of Set Theory
An introductory textbook on axiomatic set theory.
Location: Chapter 6, "Cardinal numbers and the Axiom of Choice," subsection "Axiom of Choice," pages 151-157
- The material about cardinal numbers is optional (and has additional prerequisites).
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