 Unfortunately, naive set theory is inconsistent. Russell gave the most convincing proof of this, although his was not the first paradox to be discovered: let \(P(x)\) be the property \(x \not\in x\). By the axiom of comprehension, there is a set \(R\) such that for any \(x, x \in R\) iff \(x \not\in x\). But it follows immediately that \(R \in R\) iff \(R \not\in R\), which is a contradiction.