__Russell's Paradox__

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### 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 *y*Î*y*? Then by the equality
of sets

### * yÎy*
if and only if *yÎ*{*x*|*x*Ï*x*}.

### Thus, *yÎy *
iff *y*Ï*y*. This is the
paradox.

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