 However, Cantor also observed that different well-orderings of the same infinite sets will produce a different order type. For instance, we can define an ordering on the integers where every non-negative integer comes before every negative integer, with any two integers of the same sign being ordered by their absolute value. To approximately represent this ordering we could write it as \(0, 1, 2, 3, \dots, -1, -2, -3, \dots\). In this ordering, every non-empty subset still has a first element (the non-negative element of lowest absolute value if it contains any non-negative elements, and the negative element of lowest absolute value if it only contains negative elements). If we pair up the first element of this ordering with the first of the standard ordering on the natural numbers, and the second with the second, and the third with the third, and so on, then the negative integers would not be paired with any natural number. But we can put the natural numbers into the same order type by declaring the odd numbers to be before the even numbers, and ordering them by size within these two sets: \(1, 3, 5, \dots, 0, 2, 4, \dots\). A single infinite set can be given orderings of many different order types, and also different orderings of the same order type (for instance if we put the even numbers first and the odd numbers second).