Kurt Gödel kind of proved that it doesn't make much sense to build math based on axioms. This is what I consider the most important result of his work.
He didn't prove it doesn't make sense, he proved it's impossible to prove everything.
This is not.. really the right takeaway from his work. That not every true statement within a system is provable from its axioms, or that the consistency of a system is not provable within that system, does not mean that axioms are not useful. Indeed, what math is not grounded in the end by axioms?
in my mind, i compressed it to: there are always root assumption(s), that are just that, assumptions.
It doesn't make much sense... except for the fact that it works unreasonably well.