Russel’s Paradox

What is Russell’s Paradox?  What is the solution to it?

Background (you will need to know to understand the answer):

Russell’s Paradox arises when sets get “too big” and sets contain sets.  It is possible for sets to contain sets, but not all such combinations are acceptable.  We will now explain how the idea of a “set containing a set” can be problematic.

To start, first notice that many sets, even if they are infinite in size, can generally be expressed in English sentences containing a few words.  For example, “The set of all even numbers” is a sentence describing an infinite set—the even numbers—and the description should be very clear to anyone, even someone in the second grade.  Yet this description only takes six words!  (Side point:  In general mathematicians like to be very terse, and the brevity of mathematical statements can confuse people who are not used to paying attention to the detail and exact meaning of every word—an absolute necessity in mathematics.)  In spite of the tendency towards terseness, there are some sets that might take many words to describe, for example, consider this set:  “Let K = the set of all people who (1) were born in New York, (2) have red hair, (3) moved to California after age 10, (4) drive a Ford, (5) did not graduate college, (6) like to follow football games on television, (7) like Big Macs, (8) own a computer, etc. “  If we were to list 100 different conditions, even using terse wording, the description would certainly be over 100 words long.  Of course, it could be there is no person on earth who meets all 100 conditions, especially if some of them are rare, such as (over 6 feet 5 finches tall).  While that one condition might be met by some people, the number of people meeting this condition might be relatively small, so it could be there is no one who meets all 100 conditions.  If this is the case, then K, the set described is the empty set, but we would not know that without a survey unless we created a ridiculous condition, such as the person must be over 10 feet tall.  In short, this set could very well not be the empty set, but the description could be very long—over 100 words.

Now let H be the set of all sets that can be described in an English sentence containing less than 100 words.  By definition, the elements of H are themselves sets.  Clearly, the set K described above is not inside of H, because its description is too long.  However, all other sets mentioned above are sets that are inside of H—as well as many other sets, such as the set of real numbers, the set of rational numbers, the set of all people on earth, etc.  These sets are infinite, but their descriptions are short, so they are all inside of H.  In fact, the set H is also inside of H because the description of H above contained 17 words (which is certainly less than 100).  Thus we have something like the following H ={H, N, Z, Q, S, T, ….}  H is in fact a member of H!  This concept is counter-intuitive, because if we think a set as a collection, a collection cannot obviously contain itself.  So while it is nice to think of a set as a collection, the analogy breaks down when we deal with more complicated examples, as is so in this case.

The Answer as it Should Appear on a Test:

Now, here is Russell’s Paradox.  All the sets we have above, or “standard” sets, such as whole numbers, natural numbers, integers, etc. we will call “ordinary” sets.  However, any set, such as H, which contains itself, we will call an “extra-ordinary” set.

Let Obe the set of all ordinary sets.  Question:  Is O  ordinary or extra-ordinary?  We will find we have a problem (contradiction) or paradox, either way we look at it.  If O  is ordinary, then since O  contains all ordinary sets (by its very definition) then it must contain itself (since it is ordinary)—and by containing itself it would then become extra-ordinary—a contradiction!  Again, if O is extra-ordinary, then O  must contain itself, by the definition of the word “extra-ordinary.”  But this fact contradicts the definition of O , since O is supposed to contain only ordinary sets, not an extra-ordinary ones!  So as a result, we cannot decide whether O is ordinary or extra-ordinary.

What is the solution to Russell’s Paradox?

The solution to this paradox, is to redefine or rethink what we mean by sets.  They are not simply any collections, but they must be limited so they do not grow too big.  There are several ways to achieve this re-definition—but most are too sophisticated to explain here.  We will explain, briefly (not in detail) the simplest method here.

We start by defining some set, which we call U, the universal set.  U could be the real numbers, natural numbers, or any other set you want it to be.  However, we cannot just create “new” sets at random.  They must be formed by certain rules—for example, by taking elements from U, by taking subsets from U, or by taking unions or intersections, or complements of these sets.  So while we can make an infinite number of sets, and we can even create a set of subsets, we can’t create “higher levels” of sets containing sets, so we stay clear of this paradox.

Questions:  Let S be the set of all sets.  Does S exist?  

Answer:  No.  It is too big.  The solution to Russell’s Paradox sets limits as to how big a set can get—and S violates those limits.  We need to have some “upper” set such as the real numbers, and all other sets are subsets of this upper set or universal set.