What contradiction did Bertrand Russell discover in naive set theory?

In the context of the philosophy of mathematics, Bertrand Russell discovered a fundamental flaw in naive set theory known as Russell’s paradox in 1901.12

The paradox arises from the "unrestricted comprehension principle," which allowed for the formation of a set from any condition expressible by a formula. Russell observed that this principle permits the construction of the set $R$ of all sets that are not members of themselves:1
$$R=\{x\mid x\notin x\}$$

The contradiction emerges when one asks whether $R$ is a member of itself:
* If $R$ is a member of itself ($R\in R$), it must satisfy the condition of the set, meaning it is not a member of itself ($R\notin R$).12
* If $R$ is not a member of itself ($R\notin R$), it satisfies the condition for membership in $R$, meaning it must be a member of itself ($R\in R$).12

This "vicious circle" demonstrated that naive set theory was inconsistent, necessitating the development of axiomatic systems—such as Zermelo-Fraenkel set theory—that impose restrictions on how sets can be constructed to avoid such contradictions.13