> Especially since there are well-established math proofs of irrational numbers.

The logic is circular, simply because mathematicians are the ones who invented irrationals. Of course they have proofs on them. They also have proofs on lots of things that don't exist in this universe.

And as I pointed out elsewhere, many analysis textbooks define a real number to be "a (converging) sequence of rationals". The notion of convergence is defined before reals even enter into the picture, and a real number is merely the identifier for a given converging sequence of rationals.