My point was - mathematicians still argue about whether the axiom of choice is a good idea or not. They probably always will.

So mathematical axioms are in flux.

Axioms are not in flux, set of axioms are in flux. Axioms are fixed and finite. This is how software works.

Gödel's incompleteness theorems is built around this. By saying axioms change then one could create an universal math theorem. Since axioms are finite in definition and applied in a finite space, functions, there cannot be an universal math.

Example: Fibonacci sequence only works with positive integers and zero. It will work with negative integers wrapped in absolute value. The axioms of Fibonacci did not change while wrapped in absolute value. Rational and irrational numbers can work with Fibonacci too by applying another function bound to absolute value.