Set theory MOC

Axiom of Choice

The Axiom of Choice is a controversial axiom of set theory. In addition to those of ZF it forms the final axiom of ZFC. Some equivalent formulations are zfc

1. For any set of inhabited sets, there exists a choice function .

2. Let be functions and be a Relation set. If is left-total, i.e. relates every with at least one , then there exists a choice function that selects such a for each , i.e.

3. The cartesian product of an arbitrary collection of inhabited sets is itself inhabited.

4. Every surjection in Category of sets is split epic. This structuralist formulation is an example of the External Axiom of Choice.

Proof of equivalence over ZF

proof

Other equivalences

• Set-theoretic
  • Well ordering principle
  • Cardinal comparability hypothesis
  • Maximal chain principle
  • Zorn’s lemma
• Topological
  • Tikhonov’s theorem

Relationship to other axioms

Weakenings

Over ZF

• Boolean prime ideal theorem
• Axiom of Dependent Choice Countable Axiom of Choice

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