alt Hacker News> “incompleteness theorems” established that no formal system of mathematics — no finite set of rules, or axioms, from which everything is supposed to follow — can ever be complete.'
There is usually a 'not sufficiently complex' clause in that definition. Presburger arithmetic is complete: https://en.wikipedia.org/wiki/Presburger_arithmetic
Right, you need to be able to construct numbers for Gödel's proof to apply.
Hilbert's incidence geometry, for instance, is consistent and complete. It's just rather small.