Alternative Formulation of Russell's Paradox

Question

Someone recently told me about a very interesting variant of Russell's Paradox they once saw that goes like this: $$S=\{x : \neg \exists y [ x \in y \ \& \ y \in x]\}.$$ Does anyone have any references that explain or mention this variant of Russell's Paradox, or that could perhaps explain this variant of the paradox themselves?

Answer 1

Suppose that $S$ is a set. Now we ask, is there some $x\in S$ such that $S\in x$?

If the answer is positive, then $x\in S$ and $S\in x$, and therefore $x\notin S$, by the very definition of $S$. Therefore the answer must be negative, but then $S\in S$, so by taking $x=S$ we get $S\in S$ and $S\in S$.

In either case, we run into a contradiction, and therefore $S$ is not a set.