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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