Infinity – The never-ending struggle

Abstract

Infinity has fascinated mankind since time immemorial. Zeno revealed that, whether we consider space and time to be infinitely divisible or consisting of tiny indivisible atoms, in both cases paradoxes appear. Despite this uncomfortable problem, practical mathematicians continued to use a range of infinitesimal and indivisible methods of calculation through to the 17th century development of the calculus and beyond. At the beginning of the 19 century, infinitesimal methods were still widely used. Dedekind’s construction of the real numbers suggested that the real line consists only of rationals and irrationals with no room for infinitesimals. He began with the set Q of rational numbers and proceeded to construct a set R of ‘cuts’ of the set Q which consist of two subsets A, B where every element of A is less than every element in B. He showed that these cuts were of two types. The first type corresponded to a rational number r with rational numbers less than r in A and rational numbers greater than r in B. (In this case the rational number r could be in either A or in B.) The second type did not have a rational number sitting between A and B. He showed that the set of cuts formed a system with elements of the first type corresponding to rational numbers and elements of the second type corresponding to irrational numbers. This construction ‘completed’ the real line by adding irrational numbers to ‘fill in the gaps’ between the rational numbers. In such a number line, there is ‘no room’ for infinitesimal quantities. The arithmetization of analysis by Riemann confirmed this view that no number a on the real line could be ‘arbitrarily small’, for if 0 < a < r for all positive real numbers r, then 2a is positive and even smaller than a. Infinitesimals therefore did not fit into the real number system. When Cantor constructed the concept of infinite cardinal and ordinal numbers, he developed a remarkable extension of counting finite sets to define the cardinal number of an infinite set with an operation of addition corresponding to the union of two sets and multiplication corresponding to the Cartesian product of two sets. Two infinite sets are said to have ‘the same cardinal number’ when they can be put in one-one correspondence. This was not without its difficulties. For instance, in the infinite case, a set and a proper subset could now have the same cardinal number, which contradicts finite experience and continues to cause confusion in those learning the theory today. The arithmetic of cardinals also has no use for infinitesimals because infinite cardinals do not have multiplicative inverses. By the beginning of the twentieth century, infinitesimal ideas were theoretically under attack, but they still continued to flourish in the practical world of engineering and science, often as a ‘facon de parler’, representing not a fixed infinitesimal quantity, but a variable that could become ‘arbitrarily small’.