Well, typically, I'd say, "A proposition P(x) of first order logic describes a particular number iff, for all numbers x and y such that P(x) and P(y) are true,

Well, typically, I'd say, "A proposition P(x) of first order logic describes a particular number iff, for all numbers x and y such that P(x) and P(y) are true, x = y." Then, your system of description is exactly as powerful as your system of logical axioms. But I think you're correct in that you can construct a number where it's undecidable whether it has such a description. That doesn't mean that the set of undescribable numbers is ill-defined (relative to your axiom system): it just means that you can't always answer whether it contains that number.