Russell’s Paradox begins with a statement of "Let $R$ be the set of sets that are not members of themselves", i.e. $R=\{S\mid S\notin S\}$.
I'm a little bit confused with the statement, for example, let $S=\{1,\{2,3\}\}$, of course $S\notin S$ since $S$ doesn't have an element $\{1,\{2,3\}\}$, $S$ only have two elements which are $1$ and $\{2,3\}$, So $R$ only have one element i.e. $R=\{\{1,\{2,3\}\}\}$, and again, of course $R\notin R$, I must have something wrong here but I don't know where I go wrong.
Best Answer
Sets are collections of mathematical objects which are themselves mathematical objects. That means that in principle, a set can be a member of itself. We can therefore ask which sets are not members of themselves, and that is a valid question from a mathematical point of view.
Russell's paradox shows us that the collection of sets which are not members of themselves is not a set. It is a collection we can define, but it cannot be a set. And that was the main purpose of the paradox, to dispel the notion that every collection we can define forms a set.
Your example is not good, though, because the paradox applies to the entire mathematical universe. It encompasses all the sets out there. Not just that specific $S$.
Formally speaking, modern set theory has an axiom which prevents a set from being a member of itself. But it is possible to replace that axiom by one of several axioms which guarantee the existence of sets which are members of themselves (e.g. $x=\{x\}$, and more), and these new axioms do not introduce new contradictions to mathematics.