Russell's paradox

Paradox. Let $R=\{S:S\notin S\}$ (the set of all sets that are not elements of themselves). Then:
- If $R\in R$, then $R\notin R$ (by definition of $R$).
- If $R\notin R$, then $R\in R$ (since $R$ would be a set not containing itself).

Contradiction. Both cases lead to contradiction.

Resolution. Naive set theory (any property defines a set) is inconsistent. ZFC restricts comprehension to elements of an existing set: $\{x\in A:P(x)\}$.

Conclusion. Russell's paradox motivates axiomatic set theory.