Volume 1 / Part 4 — SECTION B. ADDITION OF RELATIONS, AND THE PRODUCT OF TWO RELATIONS.

Summary of Section B.

In the present section, we have to consider the kind of addition of relations which is required in ordinal arithmetic. In cardinal arithmetic, if is a class of mutually exclusive classes, has the properties required of their sum, and thus we do not require a new kind of logical addition before dealing with arithmetical addition. But in ordinal arithmetic this is not so. Suppose and are the generating relations of two series, and we wish to add the -series at the end of the -series. Then we wish every term of the -series to precede every term of the -series; thus is not the generating relation of the new series, since gives no relation between the terms of the -series and the terms of the -series. The relation we want is since this makes every term of the -series precede every term of the -series. Hence we put It will be seen that is in general different from .

If and have no common terms, the sum of the relation-numbers of and is the relation-number of (cf. *180).

The addition of a single term to a series requires a new definition, and cannot be dealt with as a particular case of the addition of two relations. It might be thought that, just as gives the result of adding the one term to the class , so would give the result of adding the one term to the series . But this is not the case, since, when we add a term to a series, we do not want this term to precede itself, whereas is a relation which has to itself. What we want is a relation which every member of has to but which does not have to itself; thus we take as our relation, and put This definition defines the generating relation of the series obtained by adding at the end of the -series; similarly for adding at the beginning we put If is not a member of , the relation-number of is the sum of the relation-number of and the ordinal 1, which we represent by . (The ordinal 1 has no meaning by itself, but only as a summand.)

The sum of a series of series is defined in the same way as the sum of two series was defined. Let be a serial relation whose field consists of serial relations. Then the sum of all the series generated by members of , when these series are taken in the order generated by , must be a relation which holds between and whenever either (1) and both belong to the field of one of the series, and precedes in this series, or (2) belongs to the field of an earlier series than that to which belongs. In the first case, we have , i.e. . In the second case, we have , i.e. , i.e. . Hence the generating relation of the sum of all the series is . Hence we put The relation has all the properties which we should expect of the sum of a series of series.

If a series is to result from the addition of a series of series, it is necessary that no two of the series should have any common terms. For if we have we shall also have .

Hence instead of a series, we shall have cycles; for it is essential to a series that no term should precede itself. (What seem to be series in which there is repetition are always the result of a one-many correlation with series in which there is no repetition, so that a term can be counted once as the correlate of one term, and again as the correlate of a later term.) For this reason, as well as for many others, it is important to consider relations between mutually exclusive relations, i.e. between relations whose fields have no common terms. We put Then has much the same utility in relation-arithmetic as has in cardinal arithmetic. We have which is analogous to the proposition (*84·14) It will be found that in relation-arithmetic the relation often appears where appears in the analogous proposition of cardinal arithmetic.

Analogous to "" is the relation of double ordinal similarity. This holds between two relations and when they are ordinally similar relations between ordinally similar relations with known correlators, i.e. when, if is an ordinal correlator of and , so that , then if is a member of , and is the corresponding member of , so that , we shall have , and shall be able to specify a member of . But as in cardinals, so here, we have to frame our definition of double ordinal similarity in such a way as to minimize the use of the multiplicative axiom. We therefore take as our definition the following: and are said to have double ordinal similarity when there is a one-one relation which has for its converse domain, and is such that . A relation which has these properties is called a double correlator of and , i.e. we put a definition which, as will be perceived, is closely analogous to that of in *111. Two relations have double similarity when they have a double correlator, i.e. is a double correlator of and when is a correlator of and and is a correlator of and . This might be taken as the definition of a double correlator, since it is equivalent to the above definition.

If we assume the multiplicative axiom, we can prove that double similarity holds between similar relations of mutually exclusive similar relations, i.e. between two relations of mutually exclusive relations and which have a correlator such that, if , then and are always similar. In this case, . Thus if we assume the multiplicative axiom we have, if , , In the particular case in which the fields of and consist of well-ordered relations (i.e. relations generating well-ordered series), this equivalence can be proved without the use of the multiplicative axiom, because two similar well-ordered relations have only one correlator, so that the difficulty of selecting among correlators does not arise.

Double ordinal correlators have the same importance in proving the formal laws of relation-arithmetic that double cardinal correlators have in cardinal arithmetic. The construction of double correlators in various cases constitutes a large part of relation-arithmetic.

In defining the ordinal product of two relation-numbers, and in defining exponentiation, we use a relation which has properties analogous to those of . This relation is , of which the structure is as follows: Let , be two terms having the relation ; then form the two relations , . The relation holds between two couples and whenever ; thus it arranges couples whose referents are members of , and whose relata are , in an order similar to . The relations and are (by *150·03) the same as and . Thus arranges such relations as in an order similar to . Thus is similar to , and every member of its field is similar to . Thus the relation-number of is , and every member of its field has the relation-number . Moreover , as it is easy to see, is a relation of mutually exclusive relations. Hence it is suitable for defining the product of and , and we put In the next section, after we have defined the product of a relation of relations, we shall use the same relation for the definition of exponentiation, putting These two definitions should be compared with those in *113 and *116.

In virtue of the definition of , the relation holds between terms which either have one of the relations of the form , or belong respectively to the fields of two relations , , where . Thus the relation holds between and whenever and , and also between and whenever . Thus if, for the sake of illustration, and generate finite series, so that their fields are then the field of will consist of the couples and their order as arranged by is that in which they are written above. Thus the above couples in the above order constitute the series , and it is evident that this series has terms.

When the factors of a product are not enumerated, but are given as the field of a relation, a new definition of multiplication is required. This definition, which has the advantage of being applicable to infinite products, will be dealt with in the following section.