What contradiction is produced by the idea of a set containing every set?

The idea of a set containing every set—often called the universal set—leads to logical contradictions when paired with an unrestricted comprehension principle, which allows for the formation of a set from any definable condition.12

Two primary contradictions arise from this premise:

• Russell’s Paradox: If one defines a set $R$ as the collection of all sets that are not members of themselves ($R=\{x\mid x\notin x\}$), a contradiction emerges: if $R$ is a member of itself, it must satisfy the condition of not being a member of itself; if it is not a member of itself, it satisfies the condition to be a member of itself.13
• Cantor’s Paradox: By Cantor’s theorem, the power set $P(U)$ of any set $U$ must have a strictly greater cardinality than $U$ itself. If $U$ is the set of all sets, then $P(U)$ must be a subset of $U$, implying the cardinality of $P(U)$ cannot exceed that of $U$, which directly contradicts Cantor's theorem.24

To resolve these, modern set theories like Zermelo-Fraenkel (ZFC) restrict the comprehension principle, effectively prohibiting the existence of a universal set.14