The integers don't have the same cardinality as the reals, full stop. It comes from the definition of real numbers, and if you don't accept it, then you're not

The integers don't have the same cardinality as the reals, full stop. It comes from the definition of real numbers, and if you don't accept it, then you're not talking about the same real numbers that everybody else is. What this person is actually saying is "I don't believe real numbers exist". Okay, and? Does the number 4 really exist? Like, really exist? Who cares!? It's useful! Most mathematicians actually don't care whether something "exists" in some absolute-truth sense, and certainly not whether it physically exists in our universe or not. They use the word "exist" a lot, but in a slightly different way; the use it to mean something like "can this set of constraints ever be satisfied?" All models are wrong, but some models are useful. Whether those models exist in any real sense is pointless philosophical masturbation.