logic · paradox
Russell's paradox
The set of all sets that do not contain themselves.
A bag of apples holds apples, not the bag. This bag holds every bag that is not inside itself. Can this bag go in?
A bag holds things. "Inside itself" means the bag is one of the things in the bag.
The apple bag fails that test on purpose. It holds apples, so it stays out of the special bag.
The special bag holds every bag that is not inside itself, and it holds nothing else.
Put the special bag in. Then it is inside itself, so it broke the rule that got it in.
Leave it out. Then it is not inside itself, so the rule says it belongs in.
The same trap is a barber who shaves every man who does not shave himself, and nobody else.
If the barber shaves himself, he is a man who shaves himself, so he must not. If he does not, he must.