Why is the well-ordering theorem so important in the set theory?
Every set can be well-ordered.
Mathematicians think the above theorem is very important but the below theorem is not so important.
Of course I know that the above theorem is stronger than the below theorem.
But why is the above theorem so important?
Every set can be totally-ordered.
Best Answer
Most simply, well-orderings let us do things that arbitrary total orders don't - namely, they support definition by (transfinite) recursion and proof by (transfinite) induction.
The difference is most clearly seen if we look at "constructions" of weird sets of reals: $\mathsf{ZF}$ obviously proves that $\mathbb{R}$ can be totally ordered, but a mere total ordering of $\mathbb{R}$ doesn't help us build a Vitali set, a Bernstein set, an undetermined game on the naturals, or etc. - for any of those, we need a well-ordering.