One caveat to this quote below is that Godel’s first incompleteness theorem relies on the assumption of the formal system being recursively enumerable, and if we drop this requirement, then we can get a consistent and complete description of say, first order arithmetic.
I strongly disagree with this: diagonalization arguments often cannot be avoided at all, not matter how you change the setup. This is what vexed logicians in the early 20th century: no matter how you change your formal system, you won’t be able to avoid Godel’s incompleteness theorems.
One caveat to this quote below is that Godel’s first incompleteness theorem relies on the assumption of the formal system being recursively enumerable, and if we drop this requirement, then we can get a consistent and complete description of say, first order arithmetic.
More here:
https://en.wikipedia.org/wiki/Gödel’s_incompleteness_theorems#Effective_axiomatization