I think you have your argument backwards, we use math to describe reality. So whatever axiom system you choose to build your physical theory, there is then some

I think you have your argument backwards, we use math to describe reality. So whatever axiom system you choose to build your physical theory, there is then some correspondence between the objects of your axiom system and reality. And if you somehow manage to construct a physical theory with integers alone, then there will only be integers in your description. Interestingly: Indescribable numbers do not appear in physical theories. Proof: The number would need to be described by the theory in order to have a correspondence to nature. ( None of the above should be constructed as a statement about the existence of Platonic ideals.)