Well-ordering theorem

Statement

Assume the axiom of choice. Then each set $A$ has a well-order.

Remarks

Another name is Zermelo's theorem. Over ZF, it is equivalent to the axiom of choice and to Zorn's lemma. A well-order of $\mathbb{R}$ is hard or impossible to visualize, because its construction needs the axiom of choice. With a well-order, each set allows proofs by transfinite induction.

History

Ernst Zermelo proved the theorem in 1904 with the axiom of choice, which he formulated for this purpose.

Explanations

L1 human

The theorem says that each set has an order with a first element, a next element, and so on, possibly beyond infinity. For $\mathbb{N}$ the usual order works. For $\mathbb{R}$ such an order exists, but it is very hard to picture.