Mathematics
The Well-Ordering Principle and Induction Axioms as Foundations
Quick fact
The well-ordering principle and the principle of mathematical induction are logically equivalent: in the presence of the other Peano axioms, assuming one allows you to derive the other. This means the entire structure of natural number arithmetic can be built on the simple idea that every nonempty set of natural numbers has a least element.
Why this is interesting
You've probably used induction to prove formulas like 1+2+...+n = n(n+1)/2, but have you ever wondered why that proof method is actually valid? It turns out that the simple fact that every set of natural numbers has a smallest element is the hidden engine behind the entire technique.