That's a common technique in math proofs, to disprove a statement by providing a counterexample. But there probably are improvable statements that are not self-referential, e.g. certain statements about infinite sets.
But the grander point, that Hofstadter pushes a lot in GEB, is that any sufficiently powerful symbolic system will be able to self-reference. For example, consider a programming language that is not Turing complete. You will basically need to disallow loops, because with those you can implement a virtual machine to break all your self-reference-banning rules.
But the grander point, that Hofstadter pushes a lot in GEB, is that any sufficiently powerful symbolic system will be able to self-reference. For example, consider a programming language that is not Turing complete. You will basically need to disallow loops, because with those you can implement a virtual machine to break all your self-reference-banning rules.