An incompleteness theorem is one to the effect that there is no recursive definition of a given statement-class.
The first such theorem was that there is no recursive definition of the class of truths of the form ‘x is a real number.’ This was proved by George Cantor in the 1870’s, and the proof involved a technique known as ‘diagonalization.’
Fifty years later, Gödel tried to come up with a recursive definition of the class of arithmetical truths. He couldn’t and then decided to prove that no such definition was possible.
He succeeded. His proof involved an exotic form of diagonalization.