Tired of waiting for getting my beard trimmed ,I decided to dive into my mathematical thoughts:-

It was a fine evening and I was tired of my schedule that day .I was sweaty and I was like what my mom would call as " எண்ணை  கு டம் ".My beard had grown, it started to itch me .I was irritated and decided to get it trimmed. The lazy me decided not to do it myself and so I went to the barber.

As usual someone was already up there and the barber said to wait for 20 min. I was tired already and to cool down I sat and thought "What hidden math could be in this place ?" I had the answer "Barber's Paradox"

Originally Bertrand Russell's Paradox but to popularize it ,he stated it as,

A city of people has one and only one barber  and he has been defined as

A barber is a person who shaves people and only those people who do not shave themselves.

Now the question is Should the barber shave himself or not?

If he shaves himself,he can no longer be called a barber.If he doesn't then how should he get shaved?

Russell's Paradox:- Does the set of all those sets which do not contain themselves contain itself?

This paradox is important as it couldn't be explained by George Cantor's theory of "What is a Set?" and it clearly showed that a strong Foundation of Set theory is needed.

Resolving the paradox:-

If one would think to resolve it , he or she would arrive at the conclusion that such a barber doesn't exist or The barber must be a women .etc.

Mathematically the problem boils down to ,

Let B = { x ε A : x ∉ x } .....(*) be a set

(Constructed using Axioms of Specification and Extension) .

Now the question is does Bε A ?

Solution:

Suppose B ε A 

Either B∈B or B∉B

then by (*) ,

i)If BεB we must have B ∉ B which is a contradiction so B ∉ A. 

ii)if B ∉ B then as B ε A ⇒ Bε B  which is a contradiction  B ∉ A.

So definitely B ∉ A. So we cannot conjure up a set by saying magic words like " Set of all the sets.."  or in other words 

 nothing can contain "Everything"

As I completed my analysis by thinking deep ,20 mins were completed and  it was my turn to get trimmed. A barber mentioned above may not exist but I was happy to have a barber who could shave me and save my time.

"As for everything else, so for mathematical theory: beauty can be perceived but not explained."

                                         —Arthur Cayley.