Axiom:Axiom of Choice

Formulation 1
For every set of nonempty sets, we can provide a mechanism for choosing one element of each element of the set.


 * $\forall s:\left({ \varnothing \notin s \implies \exists \left({ f: s \to \bigcup s }\right): \forall t \in s: f(t) \in t }\right)$

That is, one can always create a choice function for selecting one element from each member of the set.

Formulation 2
Let $\left \langle {X_i} \right \rangle_{i \in I}$ be a family of sets all of which are non-empty, indexed by $I$ which is also non-empty.

Then there exists a family $\left \langle {x_i} \right \rangle_{i \in I}$ such that:
 * $\forall i \in I: x_i \in X_i$

That is, the Cartesian product of a non-empty family of sets which are non-empty is itself non-empty.

Formulation 3
Let $\mathcal S$ be a set of nonempty disjoint sets.

Then there is a set $C$ such that for all $S \in \mathcal S$, $C \cap S$ has exactly one element.

Symbolically:
 * $\forall s: \left({ (\varnothing \notin s \land \forall t,u \in s: t = u \lor t \cap u = \varnothing) \implies \exists c: \forall t \in s: \exists x: t \cap c = \{x\} }\right)$

Comment
Although it seems intuitively obvious ("surely you can just pick an element?"), when it comes to infinite sets of sets this axiom leads to non-intuitive results, notably the famous Banach-Tarski Paradox.

For this reason, the Axiom of Choice (often abbreviated AoC or AC) is often treated separately from the rest of the Zermelo-Fraenkel Axioms.

Set theory based on the Zermelo-Fraenkel axioms is referred to ZF, while that based on the Z-F axioms including the AoC is referred to as ZFC.

Additional forms
The following are equivalent, in ZF, to the Axiom of Choice:
 * Zorn's Lemma
 * Kuratowski's Lemma
 * Hausdorff Maximal Principle


 * Tukey's Lemma


 * Tychonoff's Theorem
 * Kelley's Theorem


 * Vector Space has Basis

Also see

 * Equivalence of Versions of Axiom of Choice