Axiom of choice

Statement

Let $X$ be a set of nonempty sets. Then there is a choice function on $X$, that is, a function $c \colon X \to \bigcup X$ such that $c(S) \in S$ for each $S \in X$.

Remarks

An equivalent form: for each family $(A_i)_{i \in I}$ of nonempty sets, the product $\prod_{i \in I} A_i$ is nonempty. Over ZF, it is equivalent to Zorn's lemma, to the well-ordering theorem and to the comparability of cardinals. It is independent of ZF, if ZF is consistent. Kurt Gödel showed in 1938 that ZF does not refute it, and Paul Cohen showed in 1963 that ZF does not prove it.

History

Ernst Zermelo formulated the axiom in 1904 for his proof of the well-ordering theorem.

Explanations

L1 human

For finitely many nonempty boxes, you can pick one object from each box, one box at a time. For infinitely many boxes without a rule to pick, the axiom of choice says that a selection still exists. From infinitely many pairs of shoes, you can take the left shoe of each pair. From infinitely many pairs of identical socks, there is no such rule, and you need the axiom.