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The Princeton Companion to Mathematics by Imre Leader, June Barrow-Green, Timothy Gowers

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III.11 Countable and Uncountable Sets

Infinite sets arise all the time in mathematics: the natural numbers, the squares, the primes, the integers, the rationals, the reals, and so on. It is often natural to try to compare the sizes of these sets: intuitively, one feels that the set of natural numbers is “smaller” than the set of integers (as it contains just the positive ones), and much larger than the set of squares (since a typical large integer is unlikely to be a square). But can we make comparisons of size in a precise way?

An obvious method of attack is to build on our intuition about finite sets. If A and B are finite sets, there are two ways we might go about comparing their sizes. One is to count their elements: we obtain two nonnegative ...

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