Volume 1 / Part 4 — SUMMARY OF PART IV.
Reading edition
The subject to be treated in this Part is a general kind of arithmetic of which ordinal arithmetic is a particular application. The form of arithmetic to be treated in this Part is applicable to all relations, though its chief importance is in regard to such relations as generate series. The analogy with cardinal arithmetic is very close, and the reader will find that what follows is much facilitated by bearing the analogy in mind.
The outlines of relation-arithmetic are as follows. We first define a relation between relations, which we shall call ordinal similarity or likeness, and which plays the same part for relations as similarity plays for classes. Likeness between and is constituted by the fact that the fields of and can be so correlated by a one-one relation that if any two terms have the relation , their correlates have the relation , and vice versa. If and generate series, we may express this by saying that and are like if their fields can be correlated without change of order. Having defined likeness, our next step is to define the relation-number of a relation as the class of relations which are like , just as the cardinal number of a class is the class of classes which are similar to . We then proceed to addition. The ordinal sum of two relations and is defined as the relation which holds between and when and have the relation or the relation , or when is a member of and is a member of . If and generate series, it will be seen that this defines the sum of and as the series resulting from adding the -series after the end of the -series. The sum is thus not commutative. The sum of the relation-numbers of and is of course the relation-number of their sum, provided and have no common terms.
The ordinal product of two relations and is the relation between two couples , , when , belong to and , belong to and either or . Thus, for example, if the field of consists of , , , and the field of consists of , , the relation will hold from any earlier to any later term of the following series: It is plain that, denoting the ordinal product of and by , we have where the second "" as standing between classes has the meaning defined in *113·01.
Infinite ordinal sums and products will also be defined, but the definitions are somewhat complicated.
The arithmetic which results from the above definitions satisfies all those of the formal laws which are satisfied in ordinal arithmetic, when this is not confined to finite ordinals; that is to say, relation-numbers satisfy the associative law for addition and for multiplication[10], they satisfy the distributive law in the shape (where the + and are those appropriate to relation-numbers) and they satisfy the exponential laws They do not in general satisfy the commutative law either in addition or in multiplication, nor do they satisfy the distributive law in the form nor the exponential law But in the particular case in which the relations concerned are finite serial relations, the corresponding relation-numbers do satisfy these additional formal laws; hence the arithmetic of finite ordinals is exactly analogous to that of inductive cardinals (cf. Part V, Section E).
If the relations concerned are limited to well-ordered relations, relation-arithmetic becomes ordinal arithmetic as developed by Cantor; but many of Cantor's propositions, as we shall see in this Part, do not require the limitation to well-ordered relations.