the incompleteness theorems are a part of this broader point as well, since they basically say we can’t choose a perfect system that has everything we want. but still, the incompleteness theorems themselves require making assumptions. you still need to assume some axioms for them to apply, in addition to picking a set of logical rules to follow. and those logical rules aren’t set in stone either. some mathematicians don’t subscribe to the law or the excluded middle, and it makes for some interesting mathematics. for example, it lets you define an infinitesimal as something that’s basically “not not zero”, while still being different from zero.