9.4: Russell’s Paradox
Russell's Paradox is a well-known logical paradox involving self-reference. It is a little tricky, so you may want to read this carefully and slowly. If you have a list of lists that do not list themselves, then that list must list itself, because it doesn't contain itself. However, if it lists itself, it then contains itself, meaning it cannot list itself. This makes logical usages of lists of lists that don't contain themselves somewhat difficult. Did you get it? Probably not. Let me explain this using a couple of examples below.
Since this introduction defines Russell’s Paradox with lists, let’s begin by making up a few lists.
List 1: Apples, California, James, Canada, sun, basketball
List 2: Adam, computer, beef, List 2, dachshund, washing machine
List 3: Boys, tables, anger, List 1
Note that List 2 has itself (“List 2”) as one of the items on the list. List 3 has “List 1” as an item, but not itself (“List 3”). Good so far?
OK, now, let’s create a new list, List L, which consists of all lists that do not list themselves. List 2 lists itself as an item, so List 2 is excluded from List L. The other two are included since they do not include themselves as an item. So, we have
List L: List 1, List 3, ….
Here is Russell’s question: is List L an item under itself? In other words, among all the items in List L (such as List 1 and List 2), do you expect List L to appear?
If List L appears as an item under itself, then List L DOES include itself, but remember what List L was? It was supposed to be the list of all lists that do NOT list themselves. So clearly this cannot happen. Therefore, List L cannot appear as an item under itself. Right?
Well, then suppose List L does not appear as an item under itself. Then, again by the very definition of List L (it is supposed to be the list of all lists that do NOT list themselves), so List L must appear as an item under itself.
So here is a quick summary so far: If List L appears as an item under itself, then it cannot appear as an item under itself. If, on the other hand, List L does NOT appear as an item under itself, then by definition it must appear as an item under itself.
That, my friend, is a paradox. This is Russell’s Paradox.
There is another version of this paradox which may be a bit easier to understand. This is often called “The barber’s dilemma.” Suppose there is a large group of men, one of whom is a barber. This particular barber was ordered to shave those men who do not shave themselves and ONLY those men who do not shave themselves. Got it? Think of dividing the group into two subgroups: Group A, the “shavers” and Group B, the “non-shavers.” The barber shaves only those in Group B. So if you are one of the men (not the barber), and if you do not shave yourself (you would be in Group B), the barber will shave you. If you shave yourself (you would be in Group A), the barber will not shave you. Easy, right?
The question is this: Does the barber shave himself or not? If he does shave himself, he would be in Group A, so he does not shave himself. If he does not shave himself, he would be in Group B, so he must shave himself. Yes, this guy has a serious dilemma.
You probably noticed the similarity between the two explanations above. This is a paradox generated by self-referencing.
There are many other forms of this paradox, and much has been written about it. See if you can come up with your own story or example that illustrates Russell’s Paradox.
This paradox was originally devised in 1901 by Bertrand Russell, probably the only mathematician who won the Nobel Prize in Literature. Interested readers can find many online resources on the exciting life of Russell, a philosopher, mathematician, author, and one of the most brilliant people who ever lived.
Contributors and Attributions
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Saburo Matsumoto
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