= Russell's paradox
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{title2=$R\in R\Longleftrightarrow R\notin R$}
{wiki}
Unrestricted formation of a <set> $R=\{x:x\notin x\}$ contradicts its own membership condition: substituting $R$ for $x$ gives the displayed equivalence. Thus a predicate cannot always define a <set> of all objects satisfying it. Restricted <set comprehension> forms a <subset> of a previously given <set>, avoiding this argument. The contradiction does not assert that a <set> whose elements happen not to contain themselves is impossible; it rules out the universal collection with exactly the specified membership condition.
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