Bertrand Russell's Paradox revealed a flaw in 1901 in early or naive set theory. (It is likely others saw this flaw before Bertrand Russell.) It is a form of the liar's paradox expressed in the terms of set theory.

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. In formal terms,

$S\in S \iff S\not\in S.$

Russel's paradox and others like it can be avoided only at the expense of giving up the idea that any criteria can be used to construct sets. For example in Zermelo-Fraenkel set theory the construction $S=\{T\mid T\not\in T\}$ would not qualify as well-founded. Some formulations of set theory would allow such a construction but then S would be a proper class rather than a set.