Another take, from this essay by Martin Davis. His argument, in essence, is the following, quoted from the referenced essay.

There is a computable function f whose range K is not computable.

[...]

Consider propositions of the form "n does not belong to K", where n is a fixed natural number. We can suppose that, in a particular formal system these propositions are each represented by a corresponding string of symbols we may write as Pn.

[...]

Let F be a sound formal system. Then there is a number n0 such that n0 does not belong to K, but it is not the case that Pn0 is provable in F.

Let's consider another formal system F'. We can also prove that there is a number n1 such that n1 does not belong to K, but it is not the case that P'n1 is provable in F'.

Should we merge both numbers and think of a single one for all F? The problem is that the proof that Pn0 is not provable in F is based on an enumeration of theorems in F, and not in any F, but in the particular F that we trying to prove that Pn0 is not provable in.

So in principle, there is no reason to consider them equal. And, if we try to build it (them), it will be a construction, like Gödel's, that depends on the particular F, so it will be most surely different if we use different F.