The result of Godel's theorems was that we knew for sure that a formal axiomatic system wasn't capable to derive all of mathematics. The math derived under the system cannot be consistent and complete.
What I'm going to write here isn't a widely accepted position, but it has no unanswered criticisms.
Mathematical knowledge, like all other knowledge, is not derived from anything. Any argument uses premises and rules of inference from which a conclusion supposedly follows - if the premises and rules are correct then so is the conclusion. But we have no way of guaranteeing the truth of premises or rules so arguments can't be used to prove conclusions.
Maths isn't guaranteed to be complete and consistent for many reasons, including Godel's theorem. Another reason is that all our reasoning is conducted using physical systems, such as pen and paper and human brains. Those systems make errors, so mathematics can't be guaranteed to be error-free.
Mathematical knowledge is created by guessing mathematical ideas as solutions to problems and criticising the guesses, not by deriving them from conclusions. The same is true for other knowledge.
See the books by Popper listed here for more on the epistemology described above:
and "The Fabric of Reality" by David Deutsch, especially chapter 10 which is specifically about maths.