Unfortunately, the set of computable numbers, while countable, is more difficult to work with than the much more conceptually reasonable set of reals. Luckily, we have pretty great theoretical tools for dealing with uncountability, so I don't think the fact that the vast, vast majority of real numbers are unidentifiable is really that big of a problem. That, and the continuum is a really useful concept, even if it very well might not have any basis in physical reality.