Russells Paradox is a problem in set theory. Suppose we have a statement about some elements of a set This 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 of for 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 object will be an element of if and only if But now suppose we ask whether or not is an element of itself. Plugging in for in the definition of we come to the conclusion that if and only if This is obviously impossible and we have a contradiction.
To avoid the paradox, mathematicians use a restricted definition of a set. If is a set, we define In this definition, only elements of are considered for membership in the set being defined. Among elements of only those that make the statement “... x ...” come out true are elements of the set. Obviously cannot 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. If is a set whose elements are sets, then is the intersection of all of the sets in Thus, for any  if and only if but if then the statement would be true no matter what x is, and therefore would be a set containing everything. Since Russell's Paradox shows that there can be no such set, it follows that is 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, then is the intersection of and all of the elements of In other words,  