Russell's Paradox

Russells Paradox is a problem in set theory. Suppose we have a statement about some elementsof a setThis statement will be true for some values of x and false for others. It is tempting to think that we could form the set of all values offor which the statement is true. In other words, it is tempting to think that the expression

should be accepted as a definition of a set. This can lead to a contradiction.

Suppose that all expressions of the type displayed above name sets. Consider the following definition of a set

According to this definition, an objectwill be an element ofif and only ifBut now suppose we ask whether or not is an element of itself. Plugging inforin the definition ofwe come to the conclusion thatif and only ifThis is obviously impossible and we have a contradiction.
To avoid the paradox, mathematicians use a restricted definition of a set. Ifis a set, we define

In this definition, only elements ofare considered for membership in the set being defined. Among elements ofonly those that make the statement “... x ...” come out true are elements of the set. Obviouslycannot be the Universal set since we would have Russell's Paradox again so that Russell's Paradox can be thought of as a proof by contradiction that there can be no set that contains absolutely everything.

Russell's Paradox also explains why there is a restriction on intersections of families of sets. Ifis a set whose elements are sets, thenis the intersection of all of the sets inThus, for anyif and only ifbut ifthen the statementwould be true no matter what x is, and thereforewould be a set containing everything. Since Russell's Paradox shows that there can be no such set, it follows thatis not a set. For this reason, there must be at least one set in the intersection. If U is a set and F is a family of sets, thenis the intersection ofand all of the elements ofIn other words,