In the case of set theory, you can exclude sets of sets and still have an extremely rich and deep mathematical theory of sets. (But it is definitely still incom

In the case of set theory, you can exclude sets of sets and still have an extremely rich and deep mathematical theory of sets. (But it is definitely still incomplete - for example ZFC can't prove the continuum hypothesis, and if you take the continuum hypothesis or its negation as an axiom, there are still infinitely more unprovable statements in that mathematical theory) In the case of Godel's incompleteness theorem, excluding the amount of math you need to in order to avoid unprovable sentences leaves you with only very simple axiom systems. If your mathematics is complete enough to even do simple arithmetic (i.e. Peano arithmetic), then there are unprovable statements. It's not as easy a fix as Russell's set theory paradox, which is why it's an important and foundational aspect of nearly all mathematics even today.