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It would at least require some careful setup to be coherent. Like you won't be able to define "the least positive integer that can't be uniquely described in fewer than fifteen words". Basically your language won't be consistent if it allows you to talk about things that can / can't be uniquely described in the language itself.

My disagreement about this post is: I don't see what it has to do with the axiom of choice. The only "choice" used here is that this nonempty set of non-describable numbers (if indeed we can define it at all) contains at least one element. That's a tautology and doesn't require choice.

Also, the author may be interested in Skolem's paradox: in particular, if the theory of the real numbers is consistent, then there must be a construction with only countably many reals, and so if I understand correctly, they would all be describable.