In fact, the entire foundation of math -- its system of axioms -- has had to be fixed due to contradictions existing in previous iterations. The most well known perhaps being Russell's paradox in naive set theory: "Let X be the set of all sets that do not contain themselves. Does X contain itself?"

In fact, there have been many paradoxes that had to be resolved by the set theory we use today.

There are no correct axioms. You can change the axioms as you wish and make your own math2.0. And you will be able to apply it to things that follow thoose axioms but finding such things that follow them is the only hard part. We define 1+1=2 and that is true because we define it that way. If it does not hold true in any physical or something then it is that you are applying a correct math for a system which doesnt work with that math(i.e, you are the problem for assuming the same axiom is true for the real system)

Example, imaginary numbers weren’t a thing for a good while and some stuff didn’t work correctly

And here's Lewis Carroll to regale us with a tale that absolutely won't be misunderstood and taken at face value by later generations about how foolish these silly mathematicians are with their wonky numbers.

Axioms serve as a starting point.