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 𝑋 of inhabited sets, there exists a choice function 𝑓 :𝑋 ↣⋃𝑋.

(∀𝔐⁡𝑋)[(∀𝑥∈𝑋)(∃𝑦∈𝑥)⟹(∃𝑓:𝑋→⋃𝑋)(∀𝐴∈𝑋)[𝑓(𝐴)∈𝐴]]

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

(∀𝑥∈𝐴)(∃𝑦∈𝐵)𝑃(𝑥,𝑦)⟹(∃𝑓:𝐴→𝐵)(∀𝑥∈𝐴)𝑃(𝑥,𝑓(𝑥))

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

(∀𝛼∈𝐴)[𝑋𝛼≠∅]⟹∏𝛼∈𝐴𝑋𝛼≠∅

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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