Russell's Paradox
From AoPSWiki
The Russell's Paradox, credited to Bertrand Russell, was one of those which forced the axiomatization of set theory.
Paradox
We start with the property 
: (
does not belong to ). We define 
to be the collection of all with the property . Now comes the question: does have the property ? Assuming it does, it cannot be in itself, in spite of satisfying its own membership criterion, a contradiction. Assuming it doesn't, it must be in itself, in spite of not satisfying its own membership criterion. This is the paradox.
