Sigh .. you have to love our science media who like to beat up myths and legends rather than study things.
Let me first start with an aside that it is worth looking at Alan Turing and whether he was playing a private joke on dumb people when he published his halting theorem (
https://en.wikipedia.org/wiki/Halting_problem). I think he may well have been poking fun at the other papers in the area. The irony would be that it is the paper most associate him with, yet I think he was well aware of the liar paradox his paper enshrines.
Okay so lets cut to the chase and you sort of got to the problem that no theory can describe something that doesn't or can't be described
That is the basis of the liar paradox where a "liar declares he is lying". The whole paradox roles around a falsehood because if we impose that the liar must always lie then the liar can't make that statement. Analysis post that point is stupid and yet people over the years have attempted it and come to an understanding that it says "something profound".
Read carefully the authors description of what he has found
What I have found is how profoundly stupid intelligent people can sometimes be.
Again it's sort of like the girl appearing on the barstool joke which rolls on the fact that everything in the universe is in a state of flux. However no-one really believes it is possible because now you are demanding trillions of specific atoms pop into a precise location at the same moment, something that is forbidden. The joke works because layman think they are saying something about how crazy scientists are and the scientists take the deeper meaning of just how little layman really understand anything and the jokes on them.
This paper is a version of that, and goes into a pile of analysis on a question that can't happen. It was a second rate paper and why it got this much attention was beyond me but I guess in the mathematics world there isn't much news around at the moment.
