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There is a man in a town that shaves all and only those that do not shave themselves.

That is, this barber shaves x if and only if x does not shave x, for all x.

Does the barber shave himself?

No, he cannot shave himself!

Where x and y are existent individuals and S is the relation 'shaves'...

For any x [Ay(xSy <-> ~(ySy)) -> (xSx <-> ~(xSx))], is the paradox.

The assumption that there is an x such that: Ay(xSy <-> ~(ySy)),

leads to the contradiction (xSx <-> ~(xSx)), therefore,

There is no x such that: Ay(xSy <-> ~(ySy)).

ie. Ax~Ay(xSy <-> ~(ySy)), ie. ~ExAy(xSy <-> ~(ySy)) is a theorem.

An x such that Ay(xSy <-> ~(ySy)) cannot exist in classical logic.

The x such that Ay(xSy <-> ~(ySy)) cannot exist either.

There is no existent individual that satisfies the description [an x such

that: Ay(xSy <-> ~(ySy)].

The 'barber' does not exist. He can't shave himself nor can he be shaven.

All primary predications of 'the barber' are false.

We cannot say what it is, but, we can say what it is not.

The barber is not among the members of: those that do shave themselves,

or, those that do not shave themselves.

The barber cannot be: a man, a woman, a robot, etc.

There is no entity that satisfies the description of 'the barber'.

Also..

There is no existent individual class that satisfies the description [an x

such that Ay(y e x <-> ~(y e y))].

The 'Russell Class' does not exist.

~ExAy(yRx <-> ~(yRy)) and ~ExAy(xRy <-> ~(yRy)) are theorems for all relations R.

Including:

1. ~ExAy(x(Shaves)y <-> ~(y(Shaves)y)).

2. ~ExAy(x=y <-> ~(y=y)).

3. ~ExAy(y e x <-> ~(y e y)).

etc.

The answer to Russell's question "Is the class of those classes that are

not members of themselves, a member of itself?" is No, because it does not

exist.

It is neither a member of any class nor is anything a member of it!

(Contrary to NBG, von Neumann-Bernays-Godel set theory, which claims that the Russell class is a proper class.)

The barber paradox, and the Russell Paradox, are resolved without the need for a theory of types or a special category of 'proper classes'.