Gödel’s theorems don’t imply inconsistency for all large systems (unless “large” is taken to mean something strange), just for systems which are both not super-weak in what they can say, and complete (or if they have their own consistency as a theorem).

I don’t think Gödel’s theorems particularly support the claim you’re making.

In fact, here is an argument that a consistent rule-set (either can be extended to something consistent and complete, or ) can be extended to be made arbitrarily large and consistent:

take a ruleset which is consistent, but for which there is something for which it has no prescription one way or the other (neither explicitly nor implied collectively by other rules) (I.e. “not complete”). Then, add a rule specifying that thing and nothing else which isn’t implied by that thing. This will be consistent, as if it were not, then the negation of the rule added would have already been an implication.

This will either yield a larger ruleset of the same kind (consistent and incomplete), or it will yield one which is consistent and complete. Gödel’s theorems show that if the ruleset is an axiom system which is sufficiently expressive (e.g. contains Peano arithmetic) then the latter cannot be the result. So in this case, there are arbitrarily large extensions of the rule-set.

If it isn’t an axiom system, or is one for a rather weak system, then the “the result is a consistent and complete system” option, well, why would you want it to be larger?

I think parent might have been referring to the inconsistency that Gödel noticed in the US Constitution when applying for citizenship.

This is less of a problem in legal systems as the legal system self admits to resting on unproven axioms.