The issue is one of representation and algorithms. For integers, you can provide a simple representation together with simple algorithms for addition, subtracti

The issue is one of representation and algorithms. For integers, you can provide a simple representation together with simple algorithms for addition, subtraction, multiplication, division with modulo, printing in decimal form, comparison which one is larger, etc. For real numbers, we can come up with precise representations as well. Basically ASTs of permitted operations like addition, root of this polynomial, integral of this function, etc. If your set of permitted operations is large enough, you can represent any number that human language can encode. But operations on those representations are non-trivial. Basically, for reals there is no representational space which is easy for "powerful" operations to navigate. For integers, there is the n-ary representation. But even that representation makes some operations non-trivial. What is the next larger prime number? What are a given integer's prime factors? Switching the representation to represent integers through their prime factors makes answering that question easier, but then suddenly other questions become non-trivial like addition of two numbers. None of the two issues has anything to do with cantor.