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The Universal Computer, 3rd Edition by Martin Davis

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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:

1 + 1 = 2,1 + 2 = 3, ,1 + 99 = 100, 

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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