When it is necessary to speak of the numbers of classes in a way which makes it impossible to confuse them with relation-numbers, we shall call them "cardinal numbers." Thus cardinal numbers are the numbers appropriate to classes. These include the ordinary integers of daily life, and also certain infinite numbers, of which we shall speak later. When we speak of "numbers" without qualification, we are to be understood as meaning cardinal numbers. The definition of a cardinal number, it will be remembered, is as follows:—
The "cardinal number" of a given class is the set of all those classes that are similar to the given class.
The most obvious application of relation-numbers is to series. Two series may be regarded as equally long when they have the same relation-number. Two finite series will have the same relation-number when their fields have the same cardinal number of terms, and only then—i.e. a series of (say) 15 terms will have the same relation-number as any other series of fifteen terms, but will not have the same relation-number as a series of 14 or 16 terms, nor, of course, the same relation-number as a relation which is not serial. Thus, in the quite special case of finite series, there is parallelism between cardinal and relation-numbers. The relation-numbers applicable to series may be called "serial numbers" (what are commonly called "ordinal numbers" are a sub-class of these); thus a finite serial number is determinate when we know the cardinal number of terms in the field of a series having the serial number in question. If
is a finite cardinal number, the relation-number of a series which has
terms is called the "ordinal" number
. (There are also infinite ordinal numbers, but of them we shall speak in a later chapter.) When the cardinal number of terms in the field of a series is infinite, the relation-number of the series is not determined merely by the cardinal number, indeed an infinite number of relation-numbers exist for one infinite cardinal number, as we shall see when we come to consider infinite series. When a series is infinite, what we may call its "length," i.e. its relation-number, may vary without change in the cardinal number; but when a series is finite, this cannot happen.
We can define addition and multiplication for relation-numbers as well as for cardinal numbers, and a whole arithmetic of relation-numbers can be developed. The manner in which this is to be done is easily seen by considering the case of series. Suppose, for example, that we wish to define the sum of two non-overlapping series in such a way that the relation-number of the sum shall be capable of being defined as the sum of the relation-numbers of the two series. In the first place, it is clear that there is an order involved as between the two series: one of them must be placed before the other. Thus if