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:

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

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 ...

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