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Your example is faulty.

> I'd ask you if this set intersected with [0, 1] has a lower bound / 'inf' that's contained in it, and if so, did I just describe that lower bound with language?

The "indescribable numbers" are dense in [0,1], and so (if the set exists) the inf of the set of indescribable numbers which are between 0 and 1 is 0. Perfectly describable.

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My example is incomplete, not faulty. I left it as a question (does the inf belong to the set?). If the answer is yes, we reached a contradiction. If the answer is no, we have to continue further zooming in to this interval (or some other construction along those lines).

See, I claim that this set is ill-defined, so I can't know its properties like whether or not it's dense, open, closed, Borel-measurable, etc. etc.

You have to tell me what its properties are, and I will come up with a concrete proof that the set in question is ill-defined.

EDIT: After I RTFA'd, this is actually the paradox in section 2.3 of the linked article

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