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)