The Russell's Paradox, credited to Bertrand Russell, was one of those which forced the axiomatization of set theory.
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.