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.