Barber's paradox

**Edgardo** · Sep20-05, 03:26 PM

There was once a barber. Some say that he lived in Seville. Wherever he lived, all of the men in this town either shaved themselves or were shaved by the barber. And the barber only shaved the men who did not shave themselves.

That is a nice story. But it raises the question: Did the barber shave himself?

**croxbearer** · Sep21-05, 08:26 AM

Yes. Because i suppose he is the only barber in town. And to satisfy the third sentence, he must also shave. The barber is not included in "the men"(third person) as referred in the last sentence.

**DaveC426913** · Sep21-05, 11:23 AM

Part II:
A little known fact: There were actually TWO barbers in Seville (and the next nearest barber is a hundred mile train ride away in Quadalquivir).

The barber mentioned by Edguardo runs a very tight ship; his shop is spotlessly clean, as is his attire. He has perfectly groomed nails, teeth and hair - the very model of a clean, well-groomed gentleman. The other barber, on the other side of the tracks has a dirty shop, with hair on the floor. He's dressed in old clothes, with grimy nails, yellow teeth, B.O. and a terrible haircut - he's a slob.

You've just blown into town for a convention and absolutely must get a haircut. Who do you go to?

**croxbearer** · Sep21-05, 04:11 PM

>>>
I will go to the second barber - the slob one. Since, there are only TWO barbers in town, and it was mentioned that he has a terrible haircut. That means, the well-groomed barber did his haircut. And also maybe, the reason why his shop was dirty is because he has lot of customers, which means he is really a good a barber, and probably he did also the haircut of the first barber.

**Jeff Ford** · Sep21-05, 04:34 PM

Originally Posted by Edgardo

There was once a barber. Some say that he lived in Seville. Wherever he lived, all of the men in this town either shaved themselves or were shaved by the barber. And the barber only shaved the men who did not shave themselves.

That is a nice story. But it raises the question: Did the barber shave himself?

Since the barber only shaves the men who did not shave themselves, he could not shave himself. Therefore the barber had a beard.

**Edgardo** · Sep22-05, 01:51 PM

First of all,

a) the barber lives in Seville too. Therefore he should be included in the "men" mentioned in the orginal text.

Just a clue: examine the two cases with
(i) he shaves himself. What follows?
(ii) he does not shave himself. What follows?
The conclusion should be strange.

Remember, the barber is also among the "men".

I think Jeff understood the paradoxon. But to add something, the men definitely get shaved. So what's the problem with my story?

Secondly

b) Huh?!?!

What's that thing about Part II?!?!?! LoL, you're changing my original question, Dave. But it's good anyway, hehe.

**quark** · Sep22-05, 11:28 PM

Never mind it. The barber is low on hormones.

**Lisa!** · Sep23-05, 04:37 AM

Originally Posted by Edgardo

First of all,

a) the barber lives in Seville too. Therefore he should be included in the "men" mentioned in the orginal text.

Just a clue: examine the two cases with
(i) he shaves himself. What follows?

He just shaved men who did not shave themselves!

(ii) he does not shave himself. What follows?
The conclusion should be strange.

All of the men in this town were shaved anyway whether by themselves or by the barber!

But this story doesn't make sense to me:
1.Why someone can't/don't shave himself if he's able to shave others?(as you say there are some people who even aren't barber but can shave themselves!)

(2.If there are some people who can shave themselves, why they can't shave the barber?

3.Supppose the barber can't shave himself because of some reasons, so perhaps we can find someone else who's the same! I mean he can't shave himself but he can shave the barber!

4. It's important where he exactly lives! Perhaps he went to another city for shaving!

)

Lol! I'm not a man, so I have no information about men's shaving!

**LarrrSDonald** · Sep23-05, 03:52 PM

This is kind of a more commonly (and thus badly) worded version of Russell's Antinomy, i.e.

$LaTeX Code: R= \\{x:x \\notin x \\}$
$LaTeX Code: R \\in R?$

By some (common) definitions it's a paradox (similar to the liar paradox mentioned in another thread). Both $LaTeX Code: R \\in R$ and $LaTeX Code: R \\notin R$ violates the definitions. It was a rather embarresing problem at the time (or so I understand, I was too young to care). I have no idea what resolutions were offered or if it was just determined mostly irrelevant for general appliactions - I imagine it's mostly a philosophical problem rather then a common issue with definitions.

**Tide** · Sep29-05, 03:57 AM

The logical inconsistency that follows constitutes a proof that such a barber cannot exist. Brand the person who provided the question a liar (a Cretan??) and move on. :)

**Curious3141** · Oct7-05, 11:06 PM

Originally Posted by LarrrSDonald

This is kind of a more commonly (and thus badly) worded version of Russell's Antinomy, i.e.

$LaTeX Code: R= \\{x:x \\notin x \\}$
$LaTeX Code: R \\in R?$

By some (common) definitions it's a paradox (similar to the liar paradox mentioned in another thread). Both $LaTeX Code: R \\in R$ and $LaTeX Code: R \\notin R$ violates the definitions. It was a rather embarresing problem at the time (or so I understand, I was too young to care). I have no idea what resolutions were offered or if it was just determined mostly irrelevant for general appliactions - I imagine it's mostly a philosophical problem rather then a common issue with definitions.

One can read more about Russell's Paradox here.

Russell himself attempted to resolve the paradox with his Theory of Types. Modern set theory - Zermelo Fraenkel (ZF) - also resolves this paradox.

**mezarashi** · Oct7-05, 11:13 PM

Originally Posted by Edgardo

There was once a barber. Some say that he lived in Seville. Wherever he lived, all of the men in this town either shaved themselves or were shaved by the barber. And the barber only shaved the men who did not shave themselves.

That is a nice story. But it raises the question: Did the barber shave himself?

The answer is that no such barber exists. The answer is clearly indeterminate through ad absurdum. You logically reduce it to adsurdity, in which the statement will have no meaning.

1. Assume the barber does not shave.
2. Then by the said rules, he will shave himself.
3. If he shaves himself, that means that he will not shave himself, for he does not shave people who shave themselves.
4. If he does not shave himself, then by said rules he will shave himself. (this loops back to #2)

There's nothing too fancy about this. Here, I have a "paradox" of my own. Have fun.

The sentence below is true.
The sentence above is false.

**amcavoy** · Oct8-05, 07:15 PM

Originally Posted by mezarashi

The answer is that no such barber exists. The answer is clearly indeterminate through ad absurdum. You logically reduce it to adsurdity, in which the statement will have no meaning.

1. Assume the barber does not shave.
2. Then by the said rules, he will shave himself.
3. If he shaves himself, that means that he will not shave himself, for he does not shave people who shave themselves.
4. If he does not shave himself, then by said rules he will shave himself. (this loops back to #2)

There's nothing too fancy about this. Here, I have a "paradox" of my own. Have fun.

The sentence below is true.
The sentence above is false.

This one goes around and around.

Alex

**nnnnnnnn** · Oct12-05, 02:27 PM

Originally Posted by mezarashi

The answer is that no such barber exists. The answer is clearly indeterminate through ad absurdum. You logically reduce it to adsurdity, in which the statement will have no meaning.
1. Assume the barber does not shave.
2. Then by the said rules, he will shave himself.
3. If he shaves himself, that means that he will not shave himself, for he does not shave people who shave themselves.
4. If he does not shave himself, then by said rules he will shave himself. (this loops back to #2)
There's nothing too fancy about this. Here, I have a "paradox" of my own. Have fun.
The sentence below is true.
The sentence above is false.

Well if he doesnt exist then no, he doesn't shave himself.

**Lisa!** · Oct12-05, 03:00 PM

This thread is still alive? Where is Edgardo?

**Edgardo** · Oct14-05, 03:24 PM

Hello, I am still here

As mentioned by some posters here, the paradox was discovered by the famous mathematician Bertrand Russel.

Here are some websites that explain what the problem is about (WARNING, SPOILER!):
http://plus.maths.org/issue20/xfile/
http://www.jimloy.com/logic/russell.htm

I personally found the paradox quite interesting and thought about it about an hour until I fully understood it.

Have fun!