Impossible barber paradox

The barber paradox is a puzzle derived from Russell's paradox.

It is about a barber who is defined such that he both shaves himself and doesn't shave himself.

Let's have an explanation :

Our barber is very active and he

• shaves those and only those who don't shave themselves.

• doesn't shave those who shave themselves.

Now the question is: Does the barber shave himself?

While answering you'll face a contradiction.

As we have seen a barber is the one who shaves those who don't shave themselves. So-

• if he shaves himself, he can't be a barber. Because the barber only shaves them who don't shave themselves.

• if he doesn't shave, he falls in the category of those who do not shave themselves, and so, cannot be a barber.

In 1901, mathematician and philosopher Bertrand Russell was investigating set theory. That time a central idea was every property can be defined with set. For example :

A set of green things, a set of numbers without 9 etc. Moreover, a set can contains many other sets. This very set is called universal set.

Now say, a universal set have exactly two elements. The problem comes when pondering the possibility of a universal set that do not contain themselves — like the barber, this seems to be impossible.

The puzzle of impossible barber shows that an apparently plausible scenario is logically impossible.

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