[al2o3cr]

A key thing that's missing here: the incompleteness theorem is a statement about provability within a specified system of axioms that apply to statements about natural numbers.

It's totally possible to prove a Gödel sentence for one model of arithmetic in another that's "more powerful"; at a minimum, one can construct an expanded system that explicitly includes the sentence as an axiom. That system, however, has its own Gödel sentence.

The natural numbers part is equally important - the statement "the set of all proofs is countable" is trivially false for a system based on the reals, for instance. (Construct all statements of the form "exists x such that x > y" for y in R, a one-to-one correspondence between true statements and an uncountable set)

Applying results like the incompleteness theorem to statements that are entirely outside of its intended domain of discourse isn't doing mathematics, it's just bullshitting.