Comment by ketzu

I wonder if the set of undescribable numbers is well defined at all. It seems to run into the same problem as the "minimum un-interesting number is interesting" as a function chosing a number from that set would be a finite description of that number, removing it from the set. (This already happens if you could select a finite subset of those numbers, as finite numbers could be ordered and you could just select the minimum.)

edit: Hm... thinking about it. The finitely-many-symbols argument also only holds if you consider finitely many interpretations of those symbols and if there actually are only finitely many symbols.