example, the set of positive rational numbers can easily be one-to-one mapped to the set of natural number pairs (2-tuples) because maps to . Since the set of natural number pairs is one-to-one mapped (actually one-to-one correspondence or bijection) to the set of natural numbers as shown above, the positive rational number set is proved as countable.

With the foresight of knowing that there are uncountable sets, we can wonder whether or not this last result can be pushed any further. The answer