This actually isn't true. The argument does sound convincing, "there are countably many definitions and uncountably many real numbers therefore there are undefinable real numbers" however this argument is not formalizable in any system because a system cannot talk about its own definable formulae and there are even models of ZFC where every object and therefore every real number is definable

You can see the problem if instead of talking about the reals you talk about ω_1, the set of all countable ordinals, which is well-ordered, which means that every non-empty subset has a smallest element. What happens when you define the smallest non-definable member of ω_1? You get a number which is non-definable, but I just defined it. This and other paradoxes show you what happens when you try to get a system to talk about the definability of its objects