Counting infinite sets

In Chapter 10 a non-empty set X was defined to have cardinality n , where n is a positive integer, when there is a bijection ℕ n → X from the standard set ℕ n = {1,2,…, n }. Such sets, together with the empty set, were defined as finite and all others as infinite . We saw (Exercise 10.1) that, given a finite set X , another set Y is also finite of the same cardinality if and only if there is a bijection X → Y . In this case we say that X and Y are equipotent . A bijection X → Y guarantees that X and Y have the same cardinality. These ideas may be extended to infinite sets. We can think of any two sets as having the same cardinality if they are equipotent so that there is a bijection between them. We can then describe this cardinality by introducing additional standard sets. For example, when we embark on counting a set X we assign a distinct element of the set to each positive integer in turn. If we exhaust the elements on reaching the integer n then we have constructed a bijection ℕ n → X and so the set is finite with cardinality n . If we never exhaust the elements we may nevertheless reach each element of the set eventually so that we have a bijection ℤ + → X . In this case the set is equipotent to the standard set ℤ + and we say that the set, although infinite, is denumerable .