The Universal Computer, 3rd Edition

CHAPTER 4

Cantor: Detour through Infinity

The sequence of numbers 1, 2, 3, …, the so-called natural or counting numbers, goes on forever. No matter how large a number you start with, you can always get a larger number by adding 1. One may conceive of the natural numbers as generated by a process, beginning with 1 and successively adding 1:

$\begin{array}{ccc}\text{1}\text{+}\text{1}\text{=}\text{2,}& \text{1}\text{+}2\text{=}3\text{,}\dots \text{,}& 1+99=100,\dots \end{array}$

Such a process, continuing beyond any finite bound, was characterized by Aristotle as a “potential infinity.” However, Aristotle was not willing to accept as legitimate the culmination of this process: the infinite set of all natural numbers. This would be a “completed” or “actual” infinity, and Aristotle declared that such were illegitimate.¹ Aristotle’s views heavily ...