In order to make matters clear, it will be best to begin with examples. Let us first consider various different kinds of series which can be made out of the inductive numbers arranged on various plans. We start with the series which, as we have already seen, represents the smallest of infinite serial numbers, the sort that Cantor calls . Let us proceed to thin out this series by repeatedly performing the [Pg 89] operation of removing to the end the first even number that occurs. We thus obtain in succession the various series: and so on. If we imagine this process carried on as long as possible, we finally reach the series in which we have first all the odd numbers and then all the even numbers. The serial numbers of these various series are . Each of these numbers is "greater" than any of its predecessors, in the following sense:— One serial number is said to be "greater" than another if any series having the first number contains a part having the second number, but no series having the second number contains a part having the first number. If we compare the two series we see that the first is similar to the part of the second which omits the last term, namely, the number 2, but the second is not similar to any part of the first. (This is obvious, but is easily demonstrated.) Thus the second series has a greater serial number than the first, according to the definition—i.e. is greater than . But if we add a term at the beginning of a progression instead of the end, we still have a progression. Thus . Thus is not equal to . This is characteristic of relation-arithmetic generally: if and are two relation-numbers, the general rule is that is not equal to . The case of finite ordinals, in which there is equality, is quite exceptional. The series we finally reached just now consisted of first all the odd numbers and then all the even numbers, and its serial [Pg 90] number is . This number is greater than or , where is finite. It is to be observed that, in accordance with the general definition of order, each of these arrangements of integers is to be regarded as resulting from some definite relation. E.g. the one which merely removes 2 to the end will be defined by the following relation: " and are finite integers, and either is 2 and is not 2, or neither is 2 and is less than ." The one which puts first all the odd numbers and then all the even ones will be defined by: " and are finite integers, and either is odd and is even or is less than and both are odd or both are even." We shall not trouble, as a rule, to give these formulæ in future; but the fact that they could be given is essential. The number which we have called , namely, the number of a series consisting of two progressions, is sometimes called . Multiplication, like addition, depends upon the order of the factors: a progression of couples gives a series such as