Mathematics
Well-Ordering and the Well-Ordering Theorem
Quick fact
The well-ordering theorem says that every set can be well-ordered, a statement equivalent to the axiom of choice. In fact, this theorem is so non-constructive that we cannot usually describe the well-ordering of a set like the real numbers—it just exists.
Why this is interesting
You’ve known since grade school that the natural numbers are ordered: 1, 2, 3, and so on. But what if every set—even the infinite ones—could be organized in a similar way, so that every non-empty part has a smallest element?