# Axiom of Choice

## Summary

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.

## Context

This concept has the prerequisites:

- Zermelo-Frankl axioms (The Axiom of Choice is often added to the Z-F axioms.)

## Goals

- 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)

## -Free-

→ Wikipedia

## -Paid-

→ 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

Other notes:

- The material about cardinal numbers is optional (and has additional prerequisites).

## See also

-No Additional Notes-