[crypto5]

> If the supply of axioms and inference rules was infinite, with no finite alternate encoding, it would mean Godel numbering would fail because the process of assigning numbers to each axiom and rule would never complete.

I strongly disagree with that. There is no evidence of that.

> And if L1 happens to be strong enough to encode Peano arithmetic, you can construct a statement a Godel statement G1 which refers to logic L1.

As I said before, Godel proved that your G1 is unprovable in specific framework of Principia Mathematics, which is first/second/higher theory/logic (consist on quantors, functions, predicate, variables, rules).

I don't see evidence that there can be no other framework even with Peano arithmetic inside which can't deliver consistent theory. You started speculating about framework with infinite set of axiom, and I disagreed with that, but there can be other type of infinite framework, say when you allow proofs of infinite length, then Godel numbering may be impossible, and it can be example of such framework.