In defining the arithmetical operations, the only correct procedure is to construct an actual class (or relation, in the case of relation-numbers) having the required number of terms. This sometimes demands a certain amount of ingenuity, but it is essential in order to prove the existence of the number defined. Take, as the simplest example, the case of addition. Suppose we are given a cardinal number , and a class which has terms. How shall we define ? For this purpose we must have two classes having terms, and they must not overlap. We can construct such classes from in various ways, of which the following is perhaps the simplest: Form first all the ordered couples whose first term is a class consisting of a single member of , and whose second term is the null-class; then, secondly, form all the ordered couples whose first term is [Pg 117] the null-class and whose second term is a class consisting of a single member of . These two classes of couples have no member in common, and the logical sum of the two classes will have terms. Exactly analogously we can define , given that is the number of some class and is the number of some class . Such definitions, as a rule, are merely a question of a suitable technical device. But in the case of multiplication, where the number of factors may be infinite, important problems arise out of the definition. Multiplication when the number of factors is finite offers no difficulty. Given two classes and , of which the first has terms and the second terms, we can define as the number of ordered couples that can be formed by choosing the first term out of and the second out of . It will be seen that this definition does not require that and should not overlap; it even remains adequate when and are identical. For example, let be the class whose members are , , . Then the class which is used to define the product is the class of couples: This definition remains applicable when or or both are infinite, and it can be extended step by step to three or four or any finite number of factors. No difficulty arises as regards this definition, except that it cannot be extended to an infinite number of factors. The problem of multiplication when the number of factors may be infinite arises in this way: Suppose we have a class consisting of classes; suppose the number of terms in each of these classes is given. How shall we define the product of all these numbers? If we can frame our definition generally, it will be applicable whether is finite or infinite. It is to be observed that the problem is to be able to deal with the case when is infinite, not with the case when its members are. If [Pg 118] is not infinite, the method defined above is just as applicable when its members are infinite as when they are finite. It is the case when is infinite, even though its members may be finite, that we have to find a way of dealing with.