Well-Ordering Theorem

The Well-Ordering Theorem was proved by Zermelos in 1904, and it states:

> Every set can be well-ordered.

This result surprised mathematicians everywhere. The Well-Ordering Theorem is equivalent of the Axiom of Choice, and no well-ordering relation has ever been explicitly constructed for uncountable sets. Thus, the mathematicians who reject the Axiom of Choice also reject this theorem.