A relation is serial when it is an aliorelative, transitive, and connected; or, what is equivalent, when it is asymmetrical, transitive, and connected. A series is the same thing as a serial relation. It might have been thought that a series should be the field of a serial relation, not the serial relation itself. But this would be an error. For example, are six different series which all have the same field. If the field were the series, there could only be one series with a given field. What distinguishes the above six series is simply the different ordering relations in the six cases. Given the ordering relation, the field and the order are both determinate. Thus the ordering relation may be taken to be the series, but the field cannot be so taken. Given any serial relation, say , we shall say that, in respect of this relation, "precedes" if has the relation to , which we shall write "" for short. The three characteristics which must have in order to be serial are: (1) We must never have , i.e. no term must precede itself. (2) must imply , i.e. if precedes and precedes , must precede . (3) If and are two different terms in the field of , we shall have or , i.e. one of the two must precede the other. The reader can easily convince himself that, where these three properties are found in an ordering relation, the characteristics we expect of series will also be found, and vice versa. We are therefore justified in taking the above as a definition of order [Pg 34] or series. And it will be observed that the definition is effected in purely logical terms. Although a transitive asymmetrical connected relation always exists wherever there is a series, it is not always the relation which would most naturally be regarded as generating the series. The natural-number series may serve as an illustration. The relation we assumed in considering the natural numbers was the relation of immediate succession, i.e. the relation between consecutive integers. This relation is asymmetrical, but not transitive or connected. We can, however, derive from it, by the method of mathematical induction, the "ancestral" relation which we considered in the preceding chapter. This relation will be the same as "less than or equal to" among inductive integers. For purposes of generating the series of natural numbers, we want the relation "less than," excluding "equal to." This is the relation of to when is an ancestor of but not identical with , or (what comes to the same thing) when the successor of is an ancestor of in the sense in which a number is its own ancestor. That is to say, we shall lay down the following definition:— An inductive number is said to be less than another number when possesses every hereditary property possessed by the successor of .