Russell's Paradox
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 P: (x does not belong to x). We define C to be the collection of all x with the property P. Now comes the question: does C have the property P? 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.
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