I think if we take "description of a number" to mean "ZF formula that uniquely picks out that number", then that cannot be defined, because a formula picks out a number when it is true for that number and false for all others, but by Tarski[0], the truth predicate cannot be defined inside the logic itself. So "the set of all numbers which cannot be described" cannot be talked about using ZF.
However, there is a way around it, by taking as axiom that ZF is consistent, choosing some model M of ZF, and then talking about the set S of numbers inside M that cannot be described.
However, there is a way around it, by taking as axiom that ZF is consistent, choosing some model M of ZF, and then talking about the set S of numbers inside M that cannot be described.
[0] https://en.wikipedia.org/wiki/Tarski%27s_undefinability_theo...