Unrestricted formation of a set contradicts its own membership condition: substituting for 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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Russell's paradox is a fundamental problem in set theory, discovered by the philosopher and logician Bertrand Russell in 1901. It arises within naive set theory, where sets can be defined by any property or condition. The paradox challenges the notion of a "set of all sets." To illustrate the paradox, consider the set \( R \) defined as the set of all sets that do not contain themselves as a member.