It will be useful to give a definition of the number which is that of the inductive cardinals. For this purpose we will first define the kind of series exemplified by the inductive cardinals in order of magnitude. The kind of series which is called a "progression" has already been considered in Chapter I. It is a series which can be generated by a relation of consecutiveness: [Pg 81] every member of the series is to have a successor, but there is to be just one which has no predecessor, and every member of the series is to be in the posterity of this term with respect to the relation "immediate predecessor." These characteristics may be summed up in the following definition:[20]— [20]Cf. Principia Mathematica, vol. II. * 123. A "progession" is a one-one relation such that there is just one term belonging to the domain but not to the converse domain, and the domain is identical with the posterity of this one term. It is easy to see that a progression, so defined, satisfies Peano's five axioms. The term belonging to the domain but not to the converse domain will be what he calls "0"; the term to which a term has the one-one relation will be the "successor" of the term; and the domain of the one-one relation will be what he calls "number." Taking his five axioms in turn, we have the following translations:— (1) "0 is a number" becomes: "The member of the domain which is not a member of the converse domain is a member of the domain." This is equivalent to the existence of such a member, which is given in our definition. We will call this member "the first term." (2) "The successor of any number is a number" becomes: "The term to which a given member of the domain has the relation in question is again a member of the domain." This is proved as follows: By the definition, every member of the domain is a member of the posterity of the first term; hence the successor of a member of the domain must be a member of the posterity of the first term (because the posterity of a term always contains its own successors, by the general definition of posterity), and therefore a member of the domain, because by the definition the posterity of the first term is the same as the domain. (3) "No two numbers have the same successor." This is only to say that the relation is one-many, which it is by definition (being one-one). [Pg 82] (4) "0 is not the successor of any number" becomes: "The first term is not a member of the converse domain," which is again an immediate result of the definition. (5) This is mathematical induction, and becomes: "Every member of the domain belongs to the posterity of the first term," which was part of our definition.