ABSTRACT
The concept of cardinality when applied to infinite sets is much more interesting and at times may seem counter-intuitive. An excellent illustration of this is: If a set B is a proper subset of a finite set A, then it is obvious that A and B have different cardinalities. However, among infinite sets, it is perfectly possible for a set and a proper subset of itself to have the same cardinality! Time for an example or two. In 1891 the mathematician Georg Cantor published a lovely proof, known as the Cantor diagonalization argument, that there are infinite sets which are not countable, and the real numbers R are one of those. Such sets are called uncountable. The real numbers R are uncountable based on proof by Cantor Diagonalization. The proof is by contradiction. Just suppose that we are able to produce a list of all real numbers strictly between 0 and 1, written in their decimal representation.
