Can all undecidable statements on natural numbers be given by Godel's numbering

Question

In the standard proof for Godel incompleteness theorem he code our language with Godel's numbering and of course there are is a lot of freedom in our coding. The proof then provide us with an undecidable statement about the natural numbers. I wonder if it's known whether or not all undecidable statements (maybe up to some isomorphism) can be achieved in this way using a suitable coding for each such statement.