A sequence in which every element appears and has only finitely many predecessors establishes a one-to-one correspondence with the natural numbers.

We can reorder the integers, alternating between positive and negative: 0, -1, 1, -2, 2, -3, 3, …. This ordering has the same order type as the natural numbers, and thus enables a one-to-one correspondence between the natural numbers and the integers. One of Cantor’s most striking early observations is that the same is possible with the positive rational numbers. Every positive rational number can be written uniquely in lowest terms as some fraction \(p/q\), where \(p\) and \(q\) are positive in