20.2 Russell’s paradox
Suppose that we have a set such that
That is, \say is the set of objects which are not elements of themselves.
However, is ? Well, if is in , then (by definition of ) is not in . If is not in , then (by definition of ) is in . This is a paradox - it cannot be true.
The solution to this paradox is to be very careful when defining sets - we cannot define sets based on arbitrary criteria; we must build them out of other, well-defined and pre-existing sets!