Let R be the set of all sets that are not members of themselves.

If R qualifies as a member of itself, it would contradict its own definition as a set containing all sets that are not members of themselves.

On the other hand, if such a set is not a member of itself, it would qualify as a member of itself by the same definition

let R = {x | xx }, then RR <=> RR

A informal example is the barber paradox:

A barber who shaves all men who do not shave themselves and only men who do not shave themselves, so does the barber shave himself?