Does Gödel sentence depend on numbering?

Question

If we change Gödel's numbering definition of the $Prov$ predicate will change as well, but the meaning won't. How is that going to affect $G$? It seems to me like it will change as it is actually $\neg Prov([G])$ and $Prov$ is totally different. Also is it possible to make a numbering, such that $G$ is $Con(F)$ or any other sentence independent of the system $F$?

Answer 1

Yes; the "specific" sentence $G$ depends on the theory $F$ and on the coding mechanism used: see Gödel Numbering.

But, for every coding mechanism, we can find the corresponding sentence $G$, provided that the theory $F$ satisfies the condition of Gödel's Theorem.