Russell's Paradox

Bertrand Russell's Paradox reveals a flaw in the early or naive set theory. (It is doubtful Bertrand Russell was the first to see this basic flaw.)

The defect is as follows. Let S be the set of all sets that do not contain themselves as members. In other words, set T is an element of S if, and only if, T is not an element of T:

$S=\{T\mid T\not \in T\}.$

Here is the flaw. Is Set S a member of itself? If S is a member of itself, then it cannot be a member of itself by the very definition of S. But if S is not a member of itself, then it must be a member of itself, again by its very definition. Hence there is a fundamental logical contradiction in this type of set, and in any theory that allows it.