The Barber paradox is a paradox that relates to mathematical logic and set theory. The paradox considers a town with a male barber who shaves every man who does not shave himself, and no one else. Such a town cannot exist:
* If the barber does not shave himself, he must abide by the rule and shave himself.
* If he does shave himself, according to the rule he will not shave himself.
Thus the rule results in an impossible situation.
Whoever wrote that Wikipedia article or edited it to add that such a town cannot exist, deferred to practical reality, which does not allow those conditions, yet supposedly there is a paradox in logic there!!!
The conditions as given reduce to the assertion that a set includes and excludes itself, so there is a direct contradiction--not a paradox--but mathematicians and logicians teach it as a paradox, so they are not telling the truth.
To understand the reduction, just read the description that's in the Wikipedia article! The requirement that the barber shaves every man who does not shave himself in the town directly contradicts with the barber shaving himself or not shaving himself, so you get that reduction to a statement that the barber shaves and does not shave himself.
You might think it a trivial issue, but let's resolve it anyway by relying on the real world, where it makes sense if in the town the barber shaves every man who does not shave himself, and no one else, except himself, as that barber can exist!!!
So you have the idea of an exception--the barber.
Well now we get to consider THE set that only has sets that do not include themselves and has all such sets--except itself.
Necessarily, the set made up of all and only sets that do not include themselves, except itself, cannot include itself, as then you'd have a contradiction as it would both include and exclude itself.
There must, logically then, be a set of all sets that exclude themselves, except itself.
I call that set a superset, as there is only one set that only has sets that exclude themselves and has them all, except itself.
Let that sink in--there can be only one.
Super sets of that type are unknown in "modern" set theory as it is primitive.
You need the real world to understand logic.
If mathematicians and logicians focused in the real world versus being "pure" they could have resolved the supposed paradox by seeing the contradiction and hypothesized the existence of the superset with the easy proof of its existence.
Concentrate your mind on a set that includes all sets that exclude themselves, except itself.
To me that is an example of super-information as in any reality any sentient beings who came across this issue would, if smart enough, eventually hypothesize the exact same set. There can be only one across all realities, all means of existence.
That's kind of cool.