Thus progressions as we have defined them have the five formal properties from which Peano deduces arithmetic. It is easy to show that two progessions are "similar" in the sense defined for similarity of relations in Chapter VI. We can, of course, derive a relation which is serial from the one-one relation by which we define a progression: the method used is that explained in Chapter IV., and the relation is that of a term to a member of its proper posterity with respect to the original one-one relation.
Two transitive asymmetrical relations which generate progressions are similar, for the same reasons for which the corresponding one-one relations are similar. The class of all such transitive generators of progressions is a "serial number" in the sense of Chapter VI.; it is in fact the smallest of infinite serial numbers, the number to which Cantor has given the name
, by which he has made it famous.
But we are concerned, for the moment, with cardinal numbers. Since two progressions are similar relations, it follows that their domains (or their fields, which are the same as their domains) are similar classes. The domains of progressions form a cardinal number, since every class which is similar to the domain of a progression is easily shown to be itself the domain of a progression. This cardinal number is the smallest of the infinite cardinal numbers; it is the one to which Cantor has appropriated the Hebrew Aleph with the suffix 0, to distinguish it from larger infinite cardinals, which have other suffixes. Thus the name of the smallest of infinite cardinals is
.
To say that a class has
terms is the same thing as to say that it is a member of