Axiom of Choice - Equivalents

Equivalents

There are important statements that, assuming the axioms of ZF but neither AC nor ¬AC, are equivalent to the axiom of choice. The most important among them are Zorn's lemma and the well-ordering theorem. In fact, Zermelo initially introduced the axiom of choice in order to formalize his proof of the well-ordering theorem.

• Set theory
  • Well-ordering theorem: Every set can be well-ordered. Consequently, every cardinal has an initial ordinal.
  • Tarski's theorem: For every infinite set A, there is a bijective map between the sets A and A×A.
  • Trichotomy: If two sets are given, then either they have the same cardinality, or one has a smaller cardinality than the other.
  • The Cartesian product of any family of nonempty sets is nonempty.
  • König's theorem: Colloquially, the sum of a sequence of cardinals is strictly less than the product of a sequence of larger cardinals. (The reason for the term "colloquially", is that the sum or product of a "sequence" of cardinals cannot be defined without some aspect of the axiom of choice.)
  • Every surjective function has a right inverse.

• Order theory
  • Zorn's lemma: Every non-empty partially ordered set in which every chain (i.e. totally ordered subset) has an upper bound contains at least one maximal element.
  • Hausdorff maximal principle: In any partially ordered set, every totally ordered subset is contained in a maximal totally ordered subset. The restricted principle "Every partially ordered set has a maximal totally ordered subset" is also equivalent to AC over ZF.
  • Tukey's lemma: Every non-empty collection of finite character has a maximal element with respect to inclusion.
  • Antichain principle: Every partially ordered set has a maximal antichain.

• Abstract algebra
  • Every vector space has a basis.
  • Every unital ring other than the trivial ring contains a maximal ideal.
  • For every non-empty set S there is a binary operation defined on S that makes it a group. (A cancellative binary operation is enough.)

• Functional analysis
  • The closed unit ball of the dual of a normed vector space over the reals has an extreme point.

• General topology
  • Tychonoff's theorem stating that every product of compact topological spaces is compact.
  • In the product topology, the closure of a product of subsets is equal to the product of the closures.

• Mathematical logic
  • If S is a set of sentences of first-order logic and B is a consistent subset of S, then B is included in a set that is maximal among consistent subsets of S. The special case where S is the set of all first-order sentences in a given signature is weaker, equivalent to the Boolean prime ideal theorem; see the section "Weaker forms" below.