Are there any interesting theorems outside of set theory that use ordinals in their proofs? The only example I know of is Goodstein's theorem, and I haven't been able to find anything else. In other (more vague) words, what is the use of ordinals? (Other than Goodstein.) Theorems that use the w...

I know what transfinite induction is, but not sure how it is used to prove something. Can anyone show how transfinite induction is used to prove something? A simple case is OK.