The Barber Paradox, the Implication & the Solution

In a village, the barber shaves everyone who does not shave himself, but no one else.

This statement appears to be innocent but when you ask the question,"Who shaves the barber?", it all starts falling apart.

Possible attempts at answering:

• Someone else shaves barber :- If someone else shaves the barber, by definition the barber shaves himself, violating the answer.
• Barber shaves himself :- If the barber shaves himself, by definition the barber does not shave himself.
• Barber does not shave :- Similar to the first one, if the barber does not shave himself, that means he shaves himself.
• Barber is a woman :- I know that cheeky answers seem to solve the problem but no, they don't. If barber does't shave him/herself, barber shaves him/herself, irrespective of whether or not there is anything to shave. Same goes for other answers that somehow imply that barber does not shave or does not need to shave. Eg - Barber trims his beard.
• Another Barber shaves him and he him :- Nope. The statement doesn't allow it.

This paradox was composed by Bertrand Russell to make non-mathematicians understand a paradox he had discovered in the old set theory. Set theory is a foundational concept of mathematics. Much of higher mathematics is based upon this theory. The old definition of a set was an intuitive one:

A set is a collection of objects, which are called elements. These elements can be whatever we would want them to be: numbers, people, functions or even other sets.

This naive definition was used by mathematicians for years.

Russell discovered a definition of set that was equivalent to the barber story above. He defined the set S to be the set of all sets that are not elements of themselves. Symbolically it is written as:

S = { A | A is not an element of A }

Now, if S does not contain itself, then it must be an element of itself by definition and if it contains itself, then it cannot be an element of itself. So, we are again reasoning in circles.

The resolution of the barber paradox is similar to the set paradox above. Just as there can be no such barber, there can be no such set. We can instead talk in terms of a class of sets to avoid Russell’s paradox. For further reading on this and other solutions proposed, visit IEP - Russell's Paradox.

Russell’s paradox showed that something as fundamental as a set needs to be carefully thought about. This is the reason I really like this paradox. It is the proof that one must not take anything for granted without giving sufficient thought to it, however fundamental it may seem to be, and especially if it is fundamental. It shows that fundamentals can be questioned and must be questioned. After all, they say prevention is better than cure and if the fundamentals aren’t right, there will be problems in the future. So, prevention is better than cure; it is important to make sure one’s foundation is sound before moving forward. And this applies to everything in life: from entering into a relationship, to choosing a life purpose; from learning a new skill to doing a mundane task.

References:

http://www.logicalparadoxes.info/barber/

http://statistics.about.com/od/ProbHelpandTutorials/a/What-Is-The-Barber-Paradox.htm

http://www.iep.utm.edu/par-russ/

P.S.: I saw another expression of the Russell's Paradox relating to properties:

"The other form is a contradiction involving properties. Some properties seem to apply to themselves, while others do not. The property of being a property is itself a property, while the property of being a cat is not itself a cat. Consider the property that something has just in case it is a property (like that of being a cat) that does not apply to itself. Does this property apply to itself? Once again, from either assumption, the opposite follows."

I was unable to understand it. Please explain it to me if you can.