TABLE OF CONTENTS As mathematicians refined the idea of a limit, they also came to a better understanding of continuity. They found that a big challenge for a successful theory of continuity was to put the real numbers into one-to-one correspondence with the points of the number line so that the numbers would form a continuum.
Many mathematicians, ancient and modern, worked hard to meet this challenge. Eudoxus, Dedekind, and Cantor all developed theoretical concepts that they hoped would fill the gaps in the ordered set of rational numbers. They wanted a geometric interpretation, one that produced a straight line that is continuous or unbroken. Their efforts were successful, and we may now speak of a "real number continuum."
But it was the discovery of irrational numbers over 2000 years ago by the Pythagoreans that first caused the need for an exact definition of the continuum. That produced a division in mathematical thinking that has lasted to the present. One school of thought has concentrated on working out the implication of Pythagoras' original belief that the framework of the universe was to be found in the natural numbers 1, 2, 3, 4, . This sequence and the set of integers (including the negative versions of the natural numbers), ordered according to size, are the discrete numbers. We apply the term "discrete" to any sequence and to any ordered set in which every term (except the first, if any) has a unique predecessor, and every term (except...
