 The best-known set-theoretic paradoxes are Russell’s paradox and Cantor’s paradox. Russell’s paradox arises from considering the Russell set \(R\) of all sets that are not members of themselves, that is, the set defined defined by \(R = \{ x \mid  x \not\in x \}\). The contradiction is then derived by asking whether \(R\) is a member of itself, that is, whether \(R \in R\) holds. If \(R \in R\) then \(R\) is a member of itself, and thus \(R \not\in R\), by definition of \(R\). If, on the other hand, \(R \not\in R\) then \(R\) is not a member of itself, and thus \(R \in R\), again by definition of \(R\).