RUSSELL’S PARADOX

 

I) THE PARADOX:

 

The paradox can be simply stated as follows:

 

R is a set of all sets, which are not members in themselves.

 

The statement itself may appear quite paradoxical even if one has no idea about what it is talking about. We will give a brief account of set theory to readers who are either unaware of it or in need of refreshing their memories.

Text Box: An elementary introduction to Set Theory A set is a well-defined collection of objects- the objects can be physical objects or even objects of our thought or intuition (as per Cantor who is the father of set theory). For example, if we take a school, we can think of the inmates as boys, girls, teachers – three distinct sets; or as high school students and middle school students; or students with height more than 4 feet tall and less than 4 feet tall etc. All arithmetic operations can be done – addition, subtraction etc- on sets. Sets need not contain only physical objects. For all positive integers, we can think of a set of even numbers and a set of odd numbers. Now coming to the above paradox: a) What are sets, which are members of themselves? and b) What are sets, which are not members of themselves? Supposing the set itself has the property, which defines the membership to the set, then it can be a member of itself. For example, if we think of two sets – a set containing all chairs in the world and a set containing all items, which are not chairs. By definition, the second set can be a member of itself as it is not a chair. Now with this brief explanation of sets, let us go back to the main article.

 

 

 

The paradox is whether R is a member of itself or not? If it is a not a member, by the very definition of this set, it should be a member. But then if it is a member, then it cannot be a member, as it will not satisfy the qualifying property to be a member.

 

This type of paradox was known to ancient Greeks, who reveled in such paradoxical logical stalemates. Supposing I say, “I am a liar!” am I a liar? If I am a liar, then I am lying and hence I am not a liar. But then, if I am not a liar then I am telling the truth and hence I am a liar after all. This circular logic will never stop.

 

 

 

 

 

Many people think that this type of paradox is on account of  ‘self reference statements’. It need not always be a self-referred statement. Let us take two people, say Tim and Tom, who always like to play pranks. Tim says, “ Tom always tells the truth.” Tom says, “ Tim always likes to lie!” What do you believe? You can start with Tom and go on a ‘logical merry go around ’ or with Tim and go on an anticlockwise ‘ logical merry go around ’.

 
 
 

 

 

 

 


II) WHY THE PARADOX IS SO IMPORTANT?

 

Had this paradox been posted in a website like this, it probably would have been taken as a prank on set theory and would have died a natural death once the novelty had worn off.

But Bertrand Russell wanted the Mathematical fraternity of his day to take this paradox seriously.

 

If you pause for a moment, you will realize why this paradox needed to be squarely addressed. One of the three logical methods used by Mathematicians, Reductio Ad Absurdum (the other two being proof by deduction and proof by induction) is based on the firm belief that if any logical argument based on an assumption leads to an absurd statement then the assumption needs to be thrown out of the window.

 

Hence if the set theory leads to absurd conclusions, then the theory itself may be flawed and not based on solid foundations. Hence what was at stake was one of the new and promising branches of Mathematics.

 

III) EXAMPLES

 

Normally, the different mathematicians who explain this paradox give two popular examples to clear the obscurity from the riddle. In order not to deviate much from our brethren, let us also stick to the same examples.

 

a)      A puzzled barber

 

Let us take a village, which has a lone barber who shaves all those people who do not shave themselves. Thus by definition, he only shaves only those who do not shave themselves.

 

It may look quite innocuous and simple to people like us who are simple minded. Now the point is whether the barber should shave himself. If he does, he is not a barber. If he doesn’t he should.

 

b) A perplexed librarian

 

Let us look at another example. Imagine a huge library, which stocks books on various subjects. We can broadly classify them as fiction and nonfiction. The books under fiction fall under further subcategories like Historical novels, Murder mysteries, Comics, Horror, Humor etc. The books under nonfiction also can be further categorized as physics, chemistry, biology, computer science, history, geography, mathematics etc.

 

In the library there are catalogues separately for each subcategory of books. The catalogues under the fiction category also are listed in the catalogues themselves, which represent them whereas the catalogues under the nonfiction category are not listed in the catalogues themselves.

 

Now the librarian wants to prepare two consolidated ‘catalogues of catalogues’ which will list these catalogues- (1) one for those catalogues which list themselves and (2) another one which do not list themselves. When the conscientious librarian is about to complete the job he is plagued by a doubt – Should he list the 2nd catalogue in that catalogue itself?

 

You see, this catalogue (2nd one) is supposed to contain all catalogues, which do not list themselves. Hence if he decides not to list it, it qualifies to be included and be listed. If he agrees and decides to include it, then the mere inclusion disqualifies it from being included as it is supposed to contain only those catalogues, which do not list themselves.

 

IV) RESOLUTION OF THE PARADOX

 

Well, then. How was the paradox resolved? There were several methods and approaches by which this paradox was laid to rest. We will confine ourselves to a very brief redefinition of the set by which a set was no longer allowed to be a member of itself.

 

Russell’s response to the paradox is contained in his so-called theory of types. His basic idea is that we can avoid reference to S (the set of all sets that are not members of themselves) by arranging all sentences into a hierarchy. This hierarchy will consist of sentences (at the lowest level) about objects, sentences (at the next lowest level) about sets of objects, sentences (at the next lowest level) about sets of sets of objects, etc. It is then possible to refer to all objects for which a given condition (or predicate) holds only if they are all at the same level or of the same "type".

 

What it means in layman’s words is as follows. When we say that set A is a subset of B, the earlier definition of Cantor allowed the set A along with the members of A to be members of set B. You can see that this way we are effectively duplicating or doubling the set A when we consider it as a subset of B. Actually, when we say set A is a subset of B, we mean that all the members of A are also members of B and set B has some more distinct members who are not members of A .Set A itself is not a member of set B but its members are.

 

Thus by redefining the definition of set, which excluded the set to be a member of itself, the paradox itself was avoided.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

BACK TO HOME

Hosted by www.Geocities.ws

1