Friday, October 11, 2002

Experience Mathematics # 17 -- If it is, then it is not

A set can be thought of as a collection of objects. But what is it, really? The above sentence does not say: A set is a collection of objects. So is a set a collection of objects, or can it only be thought of as a collection of objects?

Sets can be of two types: those that contain themselves, and those that do not. For example, consider the set $F$ of fruits in your home. This set is not a fruit, so cannot contain itself. Now consider the set $A$. The set $A$ contains all sets that can be described in less than sixteen words. The above sentence has only $15$ words and describes $A$, so $A$ must be a member of itself.

Now consider the set $R$ of all sets that do not contain themselves as a member. In particular, $F$ is a member of $R$. The question is: Is $R$ a member of itself?

Well, if it is, then by definition $R$ consists of sets that do not contain themselves as a member. So $R$ is not a member of $R$. In short, if it is, then it is not.

Conversely, suppose $R$ is not a member of itself. Then since $R$ contains all sets that are not members of themselves, $R$ must be an element of $R$. Thus, if it is not, it is!

This paradox—pointed out the famous philosopher, Bertrand Russell—led to the formalization of set theory. Formally speaking, a ‘set’ and the relation ‘is an element of’ are undefined notions that satisfy certain axioms. However, we can continue to think of a set as a collection of objects. Just make sure that we consider only well defined sets—where we can decide whether any given object is an element of the set or not. That saves us from all Russellian disasters.

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