Lately I became very interested in the theory of computability and a fundamental early result you learn is the Recursion Theorem also known as the Fixed point theorem. At first sight you can see it's immediate usefulness in that it allows you to define computable functions using their own idices. A nice and simple application of it is the fact that we can compute infinitely many fixed points for any computable function. I know however that there are many interesting and clever applications of it and my question is
>What are some of your favorite uses of this theorem?