George Cantor, the father of modern set theory, in effect proved in 1874 that if, as is customarily assumed, there are only as many names as there are natural numbers, then there is no way of naming all the real numbers. Since one wants to say that real numbers exist and yet one cannot name each of them, it is not unreasonable to relinquish the connection between naming an object and making an existence claim about it. However, we can still use the predicate 'is a real number' embedded in a quantified sentence to talk of real numbers, for example, ... '(x)(If x is a real number then ----)'. The reference and the ontological commitment are accomplished by the semantic relation of predication. In other words, we can apply 'is a real number' to each of the real numbers without naming each one of them individually. Variables stand in the same position as names and, in cases like the above, the reference cannot be made by names but only by variables. Variables and predication therefore can be used to register our ontological commitments where names cannot. (W. V. Quine, p. 27)
In essence, the claim is that we cannot name the real numbers because there is a shortage of names. That is, we run out of names. This is surely confused. If numerals are treated as names of numbers, then there is a name for each natural number (e.g., '1' is the name of 1, '2' of 2, '3' of 3, ad infinitum), and there are plenty of other names besides. In fact, there is such an abundance of names that the natural numbers could each have a second (e.g., 'I', 'II', 'III', ad infinitum), or even a third (e.g., '.', '. .', '. . .' ad infinitum). Upon reflection, it seems that each natural number could have an infinity of names. In addition, the surplus of names includes the signs for each real number, not to mention names like 'John' and 'Jane'.
It's a good thing, too. I was afraid I'd have to fight with a number for one of the available names.
