ABSTRACT
We are so accustomed to thinking of finite collections that our intuitions become disturbed when we first consider infinite sets like the set of positive integers. Consider first how we count finite sets. Suppose that there are two piles, one of red balls, the other of blue, and you wish to determine which, if either, is more numerous. The most straightforward way would be to pair off the red balls with the blue ones and see whether either colour was exhausted
before the other: if you could pair them off one-to-one without remainder then you could conclude that there were just as many red balls as blue ones, that the piles were equinumerous. Obviously if there were any unmatched blue balls then you would conclude that there were more blue balls than red. That two sets have the same number when the members of one correspond one-to-one with the members of the other is an idea which is fundamental to our practice of counting and our notion of number. David Hume (1710-76) knew this: ‘When two numbers are so combin’d, as that the one has always an unit corresponding to every unit of the other, we pronounce them equal.’ Counting the balls in each pile, and comparing the results, implicitly involves such one-to-one matching. You match each of the red balls with a different one of the first 423 whole numbers, say, and if you can do the same with the blue balls without remainder you conclude the piles are equinumerous. This is justified because if the red balls pair off with the first 423 integers, and the first 423 integers pair off with the blue balls, the red balls must pair off with the blue ones. Of course, you would normally use the integers in their natural ascending order, starting with 1 and ending with 423, especially if you did not know the number of balls in each pile to start with. But it would be sufficient for establishing equinumerosity that you could pair off the red and blue balls respectively with the first 423 integers taken in any order.
