I don't think this is quite analogous. For one, first-order arithmetic doesn't seem a uniquely human invention. Instead, it seems like it is mathematically univ

I don't think this is quite analogous. For one, first-order arithmetic doesn't seem a uniquely human invention. Instead, it seems like it is mathematically universal in a certain way—induction is a thing that appears across all sorts of different logics, and is usually tied to a marked increase in logic strength. And, there are plenty of logics that are complete, like presberger arithmetic; these logics just happen to be weaker than first-order arithmetic.