Russell's paradox, named after its discovery Bertrand Rusell, is a mathematical paradox based on set theory. Russell appears to have discovered his paradox in the late spring of 1901, while working on his principles of mathematics(1903). To understand Russell's paradox consider a real life example. In a town lives a barber who shaves exactly those who do not shave themselves. The question is whether the barber shaves himself. Answer : The barber shaves himself if and only if he does not shave himself! Here the set S of all those who do not shave themselves is represented by the barber and the question reduces to Russell's paradox.

     Let Ø(x) denote the property that x is not a member of itself. Then by the comprehension principle, we can construct a set y whose elements are those sets x which are not member of themselves. Symbolically we can write this as 

                        y:={x| Ø(x)}={x|x Ï x}.

Now we ask the question wheather yÎy? Then by the equality of sets

                        yÎy if and only if yÎ{x|xÏx}.

Thus, yÎ iff  yÏy. This is the paradox.

 

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