How does the set of all sets that are not members of themselves create Russell’s paradox?

Russell’s paradox arises from the comprehension principle used in Gottlob Frege’s early axiomatic system of set theory. This principle allowed for the formation of a set based on any definable condition, including the set $R$ of all sets that are not members of themselves: $R=\{x\mid x\notin x\}$.12

The paradox occurs when evaluating 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, which is that it is not a member of itself ($R\notin R$).
• If $R$ is not a member of itself ($R\notin R$), it meets the defining condition of the set, meaning it must be a member of itself ($R\in R$).

This logical contradiction demonstrated that an unrestricted comprehension principle leads to an impossible situation, revealing a fundamental flaw in Frege’s system. Modern set theory avoids this by imposing restrictions on how sets are constructed, ensuring that no set can be as large as the "universal set" of all sets.12