3.6: Counting Past Infinity

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As children, we all learned to answer the question “How many?” by counting with numbers: 1, 2, 3, 4, .... But the question of “How many?” was asked and answered long before the abstract concept of number was invented. The answer can be given in terms of “as many as.” How many cousins do you have? As many cousins as I have fingers on both hands. How many sheep do you own? As many sheep as there are notches on this stick. How many baskets of wheat must I pay in taxes? As many baskets as there are stones in this box. The question of how many things are in one collection of objects is answered by exhibiting another, more convenient, collection of objects that has just as many members.

In set theory, the idea of one set having just as many members as another set is ex- pressed in terms of one-to-one correspondence. A one-to-one correspondence between two sets A and B pairs each element of A with an element of B in such a way that every element of B is paired with one and only one element of A. The process of counting, as it is learned by children, establishes a one-to-one correspondence between a set of n ob- jects and the set of numbers from 1 to n. The rules of counting are the rules of one-to-one correspondence: Make sure you count every object, make sure you don’t count the same object more than once. That is, make sure that each object corresponds to one and only one number. Earlier in this chapter, we used the fancy name ‘bijective function’ to refer to this idea, but we can now see it as as an old, intuitive way of answering the question

“How many?”