Volume 1 / Part 4 — SECTION D. ARITHMETIC OF RELATION-NUMBERS.

Summary of Section D.

In the present section, we shall be concerned with the arithmetical operations on relation-numbers. Their purely logical properties have been dealt with in Section A; in the present section, it is their arithmetical properties that are to be established. These properties result immediately from the arithmetical properties of relations which have been established in Sections B and C. The subjects treated of in the present section are analogous to those treated of in Section B of Part III. with the exception of such as have already had their analogues discussed in Sections B and C of Part IV. The analogy is sufficiently close to render it often unnecessary to give proofs, since these are often step by step analogous to the proofs of corresponding propositions in Part III, Section B.

The two chief requisites in defining the arithmetical operations with relation-numbers are (1) to take due account of types, (2) to construct what may be called separated relations, i.e. relations of mutually exclusive relations derived from and ordinally similar to given relations. Each of these points calls for some preliminary explanations.

The sum of two relation-numbers , will be denoted by "," in order to distinguish this kind of addition from (the arithmetical addition of classes) and (the addition of cardinals). In defining , we have to take account of the following considerations.

Suppose and are two relations which are of the same type, and have mutually exclusive fields. Then obviously we shall want to frame our definition of the sum of two relation-numbers in such a way that the sum of and shall be . But if and are not of the same type, is meaningless; and if and overlap, may be too small to have as its relation-number the sum of the relation-numbers of and . Both these difficulties can be met by observing that, if and , we must make such definitions as to have Hence, in defining the sum of the relation-numbers of and , we may replace and by any two relations and which are respectively like and . Therefore what we require for our definition is to find two relations and which (1) are respectively like and , (2) are of the same type, (3) have mutually exclusive fields. All these three requisites are satisfied if we put We then define as meaning , and we define the sum of the relation-numbers of and as the relation-number of . This procedure is exactly analogous to that of *110; in fact, we have

In defining the sum of the relation-numbers of a field, we do not have to consider types, because the members of a field are necessarily all of the same type. But we do have to consider the question of overlapping. If a term occurs both in and in where , we want a method of counting x twice over in forming the arithmetical sum. Thus cannot be taken as the sum of the relation-numbers of members of , unless . Suppose, for instance, we have three series These each have three terms; and we want the sum of their relation-numbers to be the relation-number of a series of nine terms. But if we put and if we further put so that places the above three series in the above order, we have which is not a series, and does not have the relation-number which we require as the sum of the relation-numbers of , , .

What is wanted is a method of distinguishing the various occurrences of and and . For this reason, when occurs as a member of the field of , we replace it by ; when as a member of the field of , by ; and when as a member of the field of , by . Thus the series is replaced by ; is replaced by ; and is replaced by . The sum of these three series then has the relation-number which is required as the sum of the relation-numbers of , , .

The above process is symbolized as follows. The generating relation of the series is ; thus the three relations whose sum is to be taken are , , , i.e. using the notation of *182, according to which we put , our three relations are , , . But the generating relation of the series is , since . Thus is the relation required for defining the sum of the relation-numbers of members of the field of ; i.e. we put We will call the separated relation corresponding to . is constructed, as above, by replacing every member of , where , by ; so that if belongs both to and to , it is duplicated by being transformed once into , and once again into .

For the treatment of products, we do not require , because has been so defined as to effect the requisite separation. We might, however, by the use of , have dispensed with as a fundamental notion, and contented ourselves with ; for we have Thus we might have taken as the fundamental notion, and defined by means of it.

The addition of unity to a relation-number has to be treated separately from the addition of two relation-numbers, for the same reasons which necessitate the treatment of and separately from . There is no ordinal number 1, but we can define the addition of one to a relation-number. If and , we must have where we write "" for unity as an addendum. We do not write "," because we shall, at a later stage, give a general definition of , in virtue of which, if is an inductive cardinal, will be the corresponding ordinal. This definition entails , and therefore we use a different symbol "" for 1 as addendum. The symbol is only defined in its uses, and has no significance except in a use which has been specially defined.

We define the product as the relation-number of , when and . The product so defined obeys the associative law, and obeys the distributive law in the form but not, in general, in the form The latter form holds when , , are finite ordinals, as we shall prove at a later stage (*262). The commutative law also does not hold in general for ordinal addition and multiplication, but holds where finite ordinals are concerned.

The product of the numbers of the members of , in the order generated by , is defined as being , and is denoted by . It will be seen that is not a function of , since the value of a product depends upon the order of the factors; it is also not a function of , unless no two members of have the same relation-number. The properties of result from *172 and *174.

" to the th power" is denoted by "" and is defined as the relation-number of , where and . Its properties result from the propositions of *176 and *177.