Russel's paradox: what is the contradiction with $R \not\in R$? Let the Russel's Set be: $$R = \\{S | S \notin S\\}$$ Where $S$ is a set 1. Suppose $R \in R$, but by definition $R \not\in R$, contradiction. 2. Suppose $R \not\in R$... (I am not sure what should be the contradiction here) My guess: then $\\{S | S \not\in S\\} \not\in R$, then $R$ is not the set of all sets such that $S \not\in S$ OR $R$ is empty, but clearly it cannot be empty (?) Can someone point out the contradiction in the second one?

Suppose $R\
ot \in R$. Then $R$ is a set that does not have itself as an element.

Since $R$ satisfies this condition, we conclude $R\in R$.... woops.