Changes

: <math>S=\{T\mid T\not\in T\}.</math>

: <math>S=\{T\mid T\not\in T\}.</math>

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.

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,

 

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 <math>S=\{T\mid T\not\in T\}</math> 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.

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 <math>S=\{T\mid T\not\in T\}</math> 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.