Russell's Paradox

[chaz wyman]

Words, words, words. All those words and still you have said nothing.
You cast a long list of things and ask if they have practical significance? Are you stupid?
(generalizations, signs, symbols, words, sentences, ideas, rational structures, languages, thinkings, cultures)
eg. Does a language have practical significance?
What are you asking about?

To whom, to what end?

maybe i have said nothing - to you?
does stupid recognize a rhetorical question when he see one?
thank you for trying.

A rhetorical question has to have impact and meaning otherwise its just a bad question.

[Mark Question]

A rhetorical question has to have impact and meaning otherwise its just a bad question.

you are right! there was no "impact and meaning" - to you.

[chaz wyman]

A rhetorical question has to have impact and meaning otherwise its just a bad question.

you are right! there was no "impact and meaning" - to you.

ANyone else out there???

[Owen]

Bertrand Russell and before him Ernst Zermelo discovered this one, but Russell was the first to bring it to public attention, I think. This paradox has a popular version, called the "Barber Paradox":

Suppose a male barber in his town shaves all men in the town who do not shave themselves and only men who do not shave themselves. Does this barber shave himself?

If I answer yes, then the barber shaves himself, which violates the premise that he only shaves men who don't shave themselves. If I answer no, then the premise that he shaves all men in the town who don't shave themselves is violated.

Can anyone explain what's going on here?

I'm tempted to just say that the premisses are ridiculous -- a more realistic assumption is that he shaves all clients of his who decide to pay him for a shave, either for a change or by custom. And the barber may or may not shave himself -- maybe he goes to a barber in another town on occasion. But given the assumptions, the paradox mystifies me.

Apparently many of the academic discussions of this paradox center on set theory, a bastion of mathematics. Also, Ludwig Wittgenstein claimed to dispose of the paradox as follows:

"The reason why a function cannot be its own argument is that the sign for a function already contains the prototype of its argument, and it cannot contain itself. For let us suppose that the function F(fx) could be its own argument: in that case there would be a proposition 'F(F(fx))', in which the outer function F and the inner function F must have different meanings, since the inner one has the form O(f(x)) and the outer one has the form Y(O(fx)). Only the letter 'F' is common to the two functions, but the letter by itself signifies nothing. This immediately becomes clear if instead of 'F(Fu)' we write '(do) : F(Ou) . Ou = Fu'. That disposes of Russell's paradox. (Tractatus Logico-Philosophicus, 3.333)"

Good luck!

Russell's paradox and the Barber paradox are resolved by a theorem of first order predicate logic, without set theory or Wittgenstein's remark.

The theorem is: ~(some y)(all x)(yRx <-> ~(xRx)) or its equivalent ~(some y)(all x)(xRy <-> ~(xRx)).

The Barber Paradox is an instance of (some y)(all x)((yRx) <-> ~(xRx)), when R=shaves.

When x=y (all x)((y shaves x) <-> ~(x shaves x)), is the contradiction (all x)((x shaves x) <-> ~(x shaves x)).
p <-> ~p, is contradictory. That is, (all x)(yRx <-> ~(xRx)) is false for every y.

(all y)~(all x)((y shaves x) <-> ~(x shaves x)).
therefore,
~(some y)(all x)((y shaves x) <-> ~(x shaves x)).

That is to say, there is no such barber, the description fails to refer.


Russell's Paradox is an instance of (some y)(all x)(xRy <-> ~(xRx)), when R=is a member of.

When x=y, (all x)((x is a member of y) <-> ~(x is a member of x)) is the contradiction (all x)((x is a member of x) <-> ~(x is a member of x)).
That is, (all x)((x is a member of y) <-> ~(x is a member of x)) is false for all y.

(all y)~(all x)((x is a member of y) <-> (x is a member of x)).
therefore,
~(some y)(all x)((x is a member of y) <-> (x is a member of x)).

That is, the class of those classes that are not members of themselves does not exist.