3. Sets, Classes, and Russell's Paradox

We will want to know how to do is how to construct sets. Unfortunately, as was discovered about 100 years ago, if you are not careful you get into trouble.

A Tentative Definition of Set Theory:

Axiom 1: Two sets are the same if and only if (write "iff") they have the same members.Symbolically,

$\QTR{Large}{X=Y}$

Axiom 2: Let be a proposition about mathematical objects then true $\QTR{Large}{\}}$ is a set.

This sort of works! Try it. Look at the definitions of and the open unit interval $\QTR{Large}{(}$ 0,1 $\QTR{Large}{)}$ on Page 0.

Here is more interesting example. Let

x $\QTR{Large}{(}$ x y $\QTR{Large}{(\ }$ y x $\QTR{Large}{<}$ y $\QTR{Large}{))}$

xy $\QTR{Large}{(}$ x $\QTR{Large}{\in A}$ y $\QTR{Large}{<}$ x y

Let

Then true $\QTR{Large}{\}}$ is the set of Dedekind cuts.

Russell's Paradox (Discovered by Bertrand Russell in 1901):

Let be the proposition " $\QTR{Large}{X}$ is a set that does not contain itself as a member." That is, ,which can be written . For almost any set that one thinks of true. The set , of integers is not an integer. However, by the Tentative Definition above, the set of all sets is a set so it is a member of itself.

true $\QTR{Large}{\}}$

Now consider the set $\QTR{Large}{S}$ of all sets that do not contain themselves as a member, that is true $\QTR{Large}{\}}$ . The question is ? Of course, as in the case of the barber , if then and if then .

Mathematical Fallout:

In point of fact, in order to do mathematics we do want to be able to work with some version of 2. In particular, the "collection" of mathematical objects satisfying a particular list of axioms. For example, in this course we want to talk about "metric spaces" and we will need to do this in a way that avoids Russell's Paradox. As a first step in the next section we take a very brief look at Axiomatic Set Theory .