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Russell's paradox deals with the set of all sets that do not contain themselves as a member of that set. It ask whether that set of all sets is also the member of the set or not.

The barber paradox states :Consider a collection of barbers in a city who shaves only those men who do not shave themselves.And the questions arises whether the barber shaves himself or not.And the answer to which comes as barber should shaves himself and also should not shave himself. I understood the barber paradox.

But how should we relate it to Russell's paradox, what is the set here to be considered that does not contain themselves.

thanks in advance

Rob Arthan
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math student
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    Russel paradox says that ${x\mid x\notin x}$ is not a set. So, see $x$ as a person, and $x\notin x$ doesn't shave himself. – Surb Aug 25 '22 at 11:53
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    The Barber paradox "is a puzzle derived from Russell's paradox. It was used by Bertrand Russell as an illustration of the paradox- The puzzle shows that an apparently plausible scenario is logically impossible. Specifically, it describes a barber who is defined such that he both shaves himself and does not shave himself, which implies that no barber exists." – Mauro ALLEGRANZA Aug 25 '22 at 11:53

2 Answers2

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Russell forms the set $S=\{x \ |\ x\notin x\}$ and then proves that both $S\in S$ and $S\notin S$ are contradictory.

In the barber's paradox, take $S$ to be the set of men who are shaved by the (male) barber and then we see that the barber both must and cannot be in the set.

I'm not quite sure your description of the barber's paradox matches my understanding. To me, there is exactly one barber, who is male, and any number of other male people in the town. The barber shaves precisely the men who do not shave themselves.

If you want to be extremely literal about the mapping, let each man in the town be represented by a set, $x$, that contains precisely the people within the town that he shaves. (For uniqueness of each set, you might also want to add a unique identifying marker, like a number.) Let's call the barber's set representation $S$.

We are told that the barber shaves precisely the people who do not shave themselves i.e. the people who are not contained in their own set, so his set $S$ is precisely equal to $\{x\ |\ x\notin x\}$. Then the question "does the barber shave himself?" is precisely the question "does $S\in S$ hold?"

A.M.
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  • Ok I want to know what u mean by barber's set representation? – math student Aug 25 '22 at 13:25
  • @mathstudent I said that each individual in the town is represented by a set. The barber is an individual in the town, so call their set $S$. Its elements are precisely the sets that represent the people who the barber shaves. – A.M. Aug 25 '22 at 22:38
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Perhaps the OP's formulation is unnecessarily complicated. Take one male barber in the town, "the "one who shaves all those, and those only, who do not shave themselves". Then the set of men is non-empty. Now let $S$ simply be the set of all men in the town that the barber shaves. The barber can not belong to $S$, neither can he not belong to $S$. So $S$ does not exist. That is, there is no such barber.

Matematleta
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