Russell's Paradox
This paradox, due to Bertrand Russell, was one of those which forced the axiomatization of set theory. We start with the property : (
does not belong to
). We define
to be the collection of all
with the property
. Now comes the question: does
have the property
? Assuming it does, it cannot be in itself, in spite of satisfying its own membership criterion, a contradiction. Assuming it doesn't, it must be in itself, in spite of not satisfying its own membership criterion. This is the paradox.