If one tries to "add the missing axioms" to avoid the incompleteness of the system, then one has to add either "p" or "not "p"" as axioms. But then the definition of "being a Gödel number of a proof" of a statement changes. which means that the formula Bew("x") is now different. Thus when we apply the diagonal lemma to this new Bew, we obtain a new statement "p", different from the previous one, which will be undecidable in the new system if it is ω-consistent.