I am amateur in the field of mathematical logic, so sorry for any confusing parts of this question.
It is well known that Gödel's incompleteness theorem shows there are great limits to what first-order logic can do. Is it possible to base our formal reasoning in some other system (different kind of logic, Type theory(?), something based on category theory) in such a way that Gödel's incompleteness theorem doesn't apply?
Has anyone tried to invent whole new system as foundation of mathematics to prevent this? Maybe one not based on formulae and proofs, but on something entirely different?