But this is a real one. Is it possible to describe an arbitrary real number? Almost all reals are not describable. But you cannot find a single such number.

The 'paradox' is that the search itself is self-failing - a broken strategy. Of course, now we have the language of sets and functions between them and cardinalities and we resolve this for us in a way that is meaningful. But still now you know the 'existence' of this thing? Can't be described.

It's interesting because of the property of creating with finite words universes of infiniteness.

This is the stuff of magic and folklore, and neatly resolved by Ursula K. LeGuin in _A Wizard of Earthsea_.

Where is it a proof that there are many unnamed things ? I could only see it as an argument that there are no "unnameable" things.