Russell's Paradox

Shortly after the turn of the 19th century, Bertrand Russell demonstrated a hole in mathematical logic of set theory at the time.  A set can be member of itself. For sets \(R\) and \(S\)

\(R = \{S | R \notin S\}\)

The set \(R\) contains all sets that do not have themselves as members.

However, is \(R\) a member of itself?

Clearly not, since by definition \(R\) is the set of all sets that do not have themselves as member.

But then, if \(R\) does not have itself as a member then it must be a member of the set \(R\)

At that point in time, it created a huge stir among the mathematical community since most of what they do are based upon the foundation of sets.