Note that Gödel's incompleteness theorems do not apply to just any system of logic: they are about particular formal systems that can prove certain facts about

Note that Gödel's incompleteness theorems do not apply to just any system of logic: they are about particular formal systems that can prove certain facts about the arithmetics of integers. So, for them to fail, it doesn't even take a non-mathematical formal system, just something that has nothing to do with natural numbers, for example, Euclidean geometry, which happens to be fully decidable.