Russell's Paradox

[Mike Strand]

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!

[Mike Strand]

Maybe lack of interest in this paradox indicates the same feeling I get about it. This paradox, even stated in terms of the barber, seems to pose a ridiculous situation, and so who cares if it leads to contradiction?

This paradox may primarily be of academic interest, but I was hoping there might be readers here that could point out any practical consequences.

[chaz wyman]

Is a set of all sets a set of itself?

Sets are not a natural category. It is an arbitrary act to create a set, and therefore an arbitrary act to form the rules of sets.
All Russell needed to do was to state the rule one way or another and there is no paradox.

As there is absolutely no practical problem as a result of this so-called paradox then we need not give it any mind.

[Mike Strand]

Thanks, chaz! The aspect of "arbitrariness" you present here appeals to me, and I'll try to think about it a little more.

I have the feeling that there isn't any practical problem raised by this paradox, but apparently some mathematicians are concerned with it. I was wondering if there were any practical applications of math that would thus be affected by these concerns.

[chaz wyman]

Thanks, chaz! The aspect of "arbitrariness" you present here appeals to me, and I'll try to think about it a little more.

I have the feeling that there isn't any practical problem raised by this paradox, but apparently some mathematicians are concerned with it. I was wondering if there were any practical applications of math that would thus be affected by these concerns.

I'm not sure, but would be wiling to think about any you can find.

[MGL]

The barber paradox is actually not the same as Russell's paradox, although they share the self-referencing feature common to many paradoxes.

The barber paradox is simply a contradiction and only seems like a paradox because it seems to mean something. If you put it into predicate logic the contradiction will become clearer. Basically a barber can not logically shave everyone who does not shave themselves if the domain of "everyone" embraces himself as well.

Russell's paradox refered to the class of classes that are not members of themselves. The practical significance of this paradox in mathematics is that it undermines Cantor's proof that the power class of any class has more members than that class. ( A power class is the class of sub-classes in a class). Russell's solution to the paradox was to prohibit the possibility of classes of belonging to themselves, by suggesting classes formed a hierarchy of types. However, it is not clear to me whether this preserved Cantor's proof which I think depends on the assumption that classes can be members of themselves.

A good explanation of this can be found in the book Paradoxes by R.M. Sainsbury.

[Mike Strand]

Thanks, MGL! I had come to the same view of the barber paradox, and only knew of a connection between Cantor and Russell's paradox, but not sure what the connection was. So your explanation is of great interest to me. I studied real analysis years ago in graduate school, and my classes covered Cantor and the ideas of cardinality and power class. Now I want to take my old math books off the shelf!

And your reference to Sainsbury is appreciated.

[Mark Question]

do generalizations, signs, symbols, words, sentences, ideas, rational structures, languages, thinkings, cultures have any practical significance? can those be more specific, clarified if we have problems when using them? barber problem no problemo if two specified classes: customer and barber. man as an barber can shave itself as an customer. two different roles. like a mother, a child and a lover can be one and the same person. the class of classes that are not members of themselves should have no problemos too if read it more specifically, more detail or carefully(sorry my english) what it says. it says: the class of classes that are not members of themselves. yes? so, is it a class that is not member of itself, or - is it the class of classes that are not members of themselves? can we same way generalize easily that a mother of president obama was black(or arab-american) too? or do we categorize people like "son of a bitch"? and is a set of all sets a set of itself, also more precisely, gibberish poetry or exactly what? unnatural or arbitrary ideas from natural and practical human brains?

[chaz wyman]

do generalizations, signs, symbols, words, sentences, ideas, rational structures, languages, thinkings, cultures have any practical significance?

practical significance? What is on your mind?


can those be more specific, clarified if we have problems when using them? barber problem no problemo if two specified classes: customer and barber. man as an barber can shave itself as an customer. two different roles. like a mother, a child and a lover can be one and the same person. the class of classes that are not members of themselves should have no problemos too if read it more specifically, more detail or carefully(sorry my english) what it says. it says: the class of classes that are not members of themselves. yes? so, is it a class that is not member of itself, or - is it the class of classes that are not members of themselves? can we same way generalize easily that a mother of president obama was black(or arab-american) too? or do we categorize people like "son of a bitch"? and is a set of all sets a set of itself, also more precisely, gibberish poetry or exactly what? unnatural or arbitrary ideas from natural and practical human brains?

Try a little grammar!

[Notvacka]

Discussions about paradoxes and self-reference usually prompts med to recommend Gödel, Escher, Bach: An Eternal Golden Braid by Douglas Hofstadter. Few books are so enlightening, entertaining and generally intellectually stimulating: http://www.amazon.com/G%C3%B6del-Escher ... 0465026567

[chaz wyman]

Discussions about paradoxes and self-reference usually prompts med to recommend Gödel, Escher, Bach: An Eternal Golden Braid by Douglas Hofstadter. Few books are so enlightening, entertaining and generally intellectually stimulating: http://www.amazon.com/G%C3%B6del-Escher ... 0465026567

Looks more like psychology.

[Notvacka]

Discussions about paradoxes and self-reference usually prompts med to recommend Gödel, Escher, Bach: An Eternal Golden Braid by Douglas Hofstadter. Few books are so enlightening, entertaining and generally intellectually stimulating: http://www.amazon.com/G%C3%B6del-Escher ... 0465026567

Looks more like psychology.

That must be due to some stupid sales pitch. The market for psychology books is bigger than the market for books about logic. Let me put it this way: If you appreciate Lewis Carrol, then you'll love this one. Just get your hands on a copy and thank me later for the tip.

Or you can read the Wikipedia entry first: http://en.wikipedia.org/wiki/G%C3%B6del,_Escher,_Bach

[Mark Question]

practical significance? What is on your mind?

what is on your mind? "practical" "thinking"? words? why? why we think? what about philosophy? no good? does not benefit us? sorry my english.

Try a little grammar!

try a little thinking.
and thank you for participating.

[chaz wyman]

practical significance? What is on your mind?

what is on your mind? "practical" "thinking"? words? why? why we think? what about philosophy? no good? does not benefit us? sorry my english.

Try a little grammar!

try a little thinking.
and thank you for participating.

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?

[Mark Question]

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.