Understanding Mathematics by Peter Alfeld, Department of Mathematics, University of Utah

The set of real numbers is not denumerable.

   
   1     A.a1a2a3a4 ...
   2     B.b1b2b3b4 ...
   3     C.c1c2c3c4 ...
   4     D.d1d2d3d4 ...
   5     E.e1e2e3e4 ...
   6     F.f1f2f3f4 ...

So the first number has an integer part A and the decimal digits are a1, a2, a3, and so on. (Of course, it does not matter that we run out of letters, the notation is just due to the limitations of the html language).

We'll now construct a number z that is not in the above list. z is obtained as follows:

The integer part of z is zero. If a1=2 we make the first digit of z equal to 3, otherwise we make it equal to 2. Similarly, we make the second digit of z equal to 3 if b2=2 and 2 otherwise. In general, the i -th digit of z is 3 if the i -th digit of the i -th number in the list is 2 and 3 otherwise.

Now, z differs from every number in the list (in fact, it differs from the i -th number in the i -th digit), and so it cannot be in the list. The set of real numbers is larger than the set of rational numbers.

Note 1: There's a small technical difficulty in that the decimal representation of a real number is not necessarily unique. For example,

However, this problem can be overcome easily (by picking that representation of a number that ends with an infinite string of zeros rather than nines).

Note 2: Cantor showed that the set of real numbers is equivalent to the powerset of the natural numbers.

Fine print, your comments, more links, Peter Alfeld, PA1UM

[16-Aug-1996]