Correct me if I am wrong here.‘∀x¬Gdl(x,y)' simply states that There does not exist godel number for a given number y, right? So if we say that there exist a diagonalization of ‘∀x¬Gdl(x,y)', then we want to prove that there does exist a godel number for a given y, which leads to contradiction.
So how does it mean that diagonalization of ‘∀x¬Gdl(x,y)' is unprovable?
What is the definition of the relation $Gdl(x,y)$ ?
There is no "godel number" for a given number $y$. Usually, we associate godel numbers to expressions of the language; thus, a relation between godel numbers must express a relation between the corresponding expressions of the language, like in :
where we express the fact that no godel number $x$ is the godel number of a derivation (that is a sequence of formulae) of the formula with godel number $y$.
See Arithmetization of the Formal Language into Gödel's Incompleteness Theorems.