Interlude: An Excursion into the Number Concept

Abstract

To appreciate Cantor’s revolutionary ideas about the infinite, we must first make a brief excursion into the history of the number concept. The simplest type of numbers are, of course, the counting numbers 1, 2, 3,…. Mathematicians prefer to call them the natural numbers, or again the positive integers. Simple though they are, these numbers have been the subject of research and speculation since the dawn of recorded history, and many civilizations have assigned to them various mystical properties. An important branch of modern mathematics, number theory, deals exclusively with the natural numbers, and some of the most fundamental questions about them—for example, questions relating to the prime numbers—are without answer to this day. But without reservation, the single most important property of the natural numbers is this—there are infinitely many of them. The fact that there is no last counting number seems so obvious to us that we hardly bother to reflect upon its consequences. The entire system of calculations with numbers—our familiar rules of arithmetic—would have colapsed like a house of cards if there were a last number beyond which nothing else existed. Suppose for a moment that such a number did exist, say 1,000.