[housecarpenter]

I think you misunderstood the parent comment. They're not talking about defining the reals by setting up a mapping where each real number corresponds to a unique natural number. They're talking about defining the reals by setting up a mapping where each real number corresponds to a unique mapping from the natural numbers to {0, 1} (i.e. a unique binary sequence). The set of all binary sequences is isomorphic to the power set of the natural numbers, which is uncountable.