Barber's Paradox!

Anthony.R.Brown · 2007-03-20 04:55:05

Suppose you walk past a barber's shop one day, and see a sign that says

"Do you shave yourself? If not, come in and I'll shave you! I shave anyone who does not shave himself, and noone else."

This seems fair enough, and fairly simple, until, a little later, the following question occurs to you - does the barber shave himself? If he does, then he mustn't, because he doesn't shave men who shave themselves, but then he doesn't, so he must, because he shaves every man who doesn't shave himself... and so on. Both possibilities lead to a contradiction.

A.R.B

The Barbers Wife shaves the Barber!! the first time! and then every other time! after the Barber has shaved himself once! in a row Only!!

Dross · 2007-03-20 07:05:28

But if the Barber's wife goes to shave the Barber, then the barber is not shaving himself.

He must therefore, according to his sign, stop his wife from performing said shaving act and then take it upon himself to shave. But then he's shaving himself, making him someone he will not shave.

Then the wife of the Barber must shave him. But if the Barber's wive goes to shave the Barber...

Ricky · 2007-03-20 07:24:48

The barber shaves everyone who doesn't shave themselves. Does this include people who don't shave? What about women? They don't shave. Does the barber shave them?

Anthony.R.Brown · 2007-03-21 01:15:51

The Original " Barber's Paradox! " is a Question involving a 50% Contradiction!

So I,m giving an Answer! that satisfies Both Situations! 50%

an unequal answer! for an unequal Question!

mathsyperson · 2007-03-21 01:49:32

The way I interpret this, the barber doesn't have to shave people who don't shave themselves. It's just that those are the only people that he will shave. Therefore, if someone else shaves him then we won't get a contradiction.

Alternatively, if you say that the barber shaves everyone who don't shave themselves and no one else, then you could get out of the paradox by extending on Ricky's thinking and saying that the barber is a woman.

Maelwys · 2007-03-21 01:56:19

mathsyperson wrote:

The way I interpret this, the barber doesn't have to shave people who don't shave themselves. It's just that those are the only people that he will shave. Therefore, if someone else shaves him then we won't get a contradiction.
Alternatively, if you say that the barber shaves everyone who don't shave themselves and no one else, then you could get out of the paradox by extending on Ricky's thinking and saying that the barber is a woman.

I agree with this thinking. It says "I shave anyone" and not "I must shave everyone", so I don't think he's forcing people to let him shave them. So that would mean that anybody else is free to shave anybody they want (including the barber himself), and anybody (including the barber) is free to not get shaved at all (if they're female, or want to grow a beard). So the barber has 3 choices that don't contradict the sign, either he has somebody else shave him, he lets his beard grow in, or she doesn't need to be shaved.

Alternatively, if the sign implies "I must shave everyone who doesn't shave themselves" then I'd never want to live in this town, because I like my beard, my wife wouldn't want some psycho forcing a razer at her, and neither would my son ;-)

Anthony.R.Brown · 2007-03-21 02:01:28

The problem with some of this is! that some woman! do have Beards! I can only imagine they do sometimes shave!

My answer was only to Split the task into two! which is half doing and half not! I think the Original " Barber's Paradox! " should have had a Clause! about Others! ie Woman!

Last edited by Anthony.R.Brown (2007-03-21 02:03:29)

Anthony.R.Brown · 2007-03-22 02:55:39

Another interesting point is! if the Barber had never Shaved once! ( Male or Female ) and just let their Beard grow! they could in fact Shave themselves! once! because they would have no History of being a person that Shaved themselves!

Anthony.R.Brown · 2007-03-23 02:28:41

More Info!

A paradox is an apparently sound statement or proposition which leads to a logically unacceptable conclusion. One such statement, known as the barber's paradox, goes thus:

In a village there is a barber. The barber cuts the hair of everybody in the village who doesn't cut their own hair. In other words: If someone cuts their own hair then the barber doesn't cut their hair. If the barber cuts their hair then they don't cut their own hair.

The paradox arrises when you consider who cuts the barbers hair. If the barber cuts his own hair then the barber doesn't cut his hair. If the barber doesn't cut his own hair then then the barber cuts it.

Russell's paradox, named after its discoverer Bertrand Russell, is a mathematical paradox based on set theory. Consider a set called S which is the set of sets that do not contain themselves. If S is not an element of S, then by definition it must be an element of S. If S is an element of S then it cannot be an element of S.

n.b. the following does not display correctly in Microsoft Internet Explorer (the boxes should show the "not an element of" symbol):

S[x | x ∉ x]
S ∈ S → S ∉ S
S ∉ S → S ∈ S

This effectively means that true = false and false = true. This could be of great usefulness in maths exams because it means that if you can prove that you have got a question wrong then you can also mathematicaly prove that it is also correct.

Ricky · 2007-03-23 03:34:35

This effectively means that true = false and false = true. This could be of great usefulness in maths exams because it means that if you can prove that you have got a question wrong then you can also mathematicaly prove that it is also correct.

No, it means there is something faulty in the underlying premises. Specifically, that there exists such a set which contains sets that don't contain themselves. In Zermelo Fraenkel set theory, axioms are constructed in such a way that this set does not exist.

It is your assumption that it exists that is faulty.

Anthony.R.Brown · 2007-03-25 01:23:28

To Ricky!

the above is not a Quote: of mine! only Info!

egeniius · 2007-03-25 12:04:36

No, it means there is something faulty in the underlying premises. Specifically, that there exists such a set which contains sets that don't contain themselves. In Zermelo Fraenkel set theory, axioms are constructed in such a way that this set does not exist.

Wouldn't the fault in the premise be that a set which contains all sets that do not contain themselves can be constructed?

Anthony.R.Brown · 2007-06-03 00:59:45

To egeniius

Quote:" Pi is exactly three!" - Professor Frink "

A.R.B

Must have been the Biggest fool!.......................................................................................

LQ · 2007-06-03 03:09:59

I bet that since the barber is much like santa claus is on giving presents, very tired of shaving "the anointed" hence instead lightens the beard with a match. That or he doesn't have a beard. That would explain why he wants to shave everyone that doesn't shave themselves. "The shave man" we can call him

LQ · 2007-06-03 03:17:57

But that wouldn't work either ofcourse, since he even shaves the beardless I asume.

Anthony.R.Brown · 2007-06-04 01:33:59

To LQ

Quote:" But that wouldn't work either ofcourse, since he even shaves the beardless I asume. "

A.R.B

But the Barber can of course use IMAC hair removal cream on anyone!.....................................

Last edited by Anthony.R.Brown (2007-06-04 23:08:39)

LQ · 2007-06-04 02:03:08

Why don't we call him anti-santa by the way, that's much more fitting.

Anthony.R.Brown · 2007-06-04 23:10:09

To LQ

Quote:" Why don't we call him anti-santa by the way, that's much more fitting. "

A.R.B

Does/Did Santa Exist?

LQ · 2007-06-05 06:36:54

Not if the barber did. They can't exist simultaneously you see. Santa has a beard. You see now?

Last edited by LQ (2007-06-05 06:37:33)

Anthony.R.Brown · 2007-06-06 01:05:04

To LQ

Quote:" Not if the barber did. They can't exist simultaneously you see. Santa has a beard. You see now? "

A.R.B

Santa could of had a False beard! ( Not sure how that Affect's the Shaving Rules!? )

Identity · 2007-06-06 01:32:12

What if the barber did not have hair? What if the barber was a computer program which sent information to robots to cut client's hair?

Anthony.R.Brown · 2007-06-06 01:35:15

To Identity

Quote:" What if the barber did not have hair? What if the barber was a computer program which sent information to robots to cut client's hair? "

A.R.B

Bizarre Stuff! Keep taking the Medicine!....

LQ · 2007-06-06 07:44:42

Identity wrote:

What if the barber did not have hair? What if the barber was a computer program which sent information to robots to cut client's hair?

DESTWOY, DESTWOY, buzzzzzzzzzzzz

A fitting name for their sequels would be Cybarb. Travel through space to cut beard and conquer.

PS. I like you even though you are insane and i find the solution quite possible. Now make a similar for santa...

Last edited by LQ (2007-06-06 07:52:01)

Anthony.R.Brown · 2007-06-06 22:36:56

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Math Is Fun Forum

#1 2007-03-20 04:55:05

Barber's Paradox!

#2 2007-03-20 07:05:28

Re: Barber's Paradox!

#3 2007-03-20 07:24:48

Re: Barber's Paradox!

#4 2007-03-21 01:15:51

Re: Barber's Paradox!

#5 2007-03-21 01:49:32

Re: Barber's Paradox!

#6 2007-03-21 01:56:19

Re: Barber's Paradox!

#7 2007-03-21 02:01:28

Re: Barber's Paradox!

#8 2007-03-22 02:55:39

Re: Barber's Paradox!

#9 2007-03-23 02:28:41

Re: Barber's Paradox!

#10 2007-03-23 03:34:35

Re: Barber's Paradox!

#11 2007-03-25 01:23:28

Re: Barber's Paradox!

#12 2007-03-25 12:04:36

Re: Barber's Paradox!

#13 2007-06-03 00:59:45

Re: Barber's Paradox!

#14 2007-06-03 03:09:59

Re: Barber's Paradox!

#15 2007-06-03 03:17:57

Re: Barber's Paradox!

#16 2007-06-04 01:33:59

Re: Barber's Paradox!

#17 2007-06-04 02:03:08

Re: Barber's Paradox!

#18 2007-06-04 23:10:09

Re: Barber's Paradox!

#19 2007-06-05 06:36:54

Re: Barber's Paradox!

#20 2007-06-06 01:05:04

Re: Barber's Paradox!

#21 2007-06-06 01:32:12

Re: Barber's Paradox!

#22 2007-06-06 01:35:15

Re: Barber's Paradox!

#23 2007-06-06 07:44:42

Re: Barber's Paradox!

#24 2007-06-06 22:36:56

Re: Barber's Paradox!

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