I found some applications of the transfinite recursion theorem within set theory. For example, to prove the following theorem:
A set $A$ is infinite if and only if there exists a one-to-one function $f: A\rightarrow A$ that is not onto $A$.
However, all of such applications I've learnt so far are set-theoretic. So, are there some direct or almost direct applications of this theorem in other fields of (applied) mathematics such as in calculus, algebra, computer science, etc.? By direct or almost direct, I mean the application is not based on a chain of other results that required the theorem.
