Volume 1 / Part 4 — *172. THE PRODUCT OF THE RELATIONS OF A FIELD.
Reading edition
*172. THE PRODUCT OF THE RELATIONS OF A FIELD.
Summary of *172.
In this number we have to consider the form of product which is applicable to any relation of relations, whether mutually exclusive or not. If our relation were a , we could take , and order selected classes from by first differences. This would give us a relation whose field would be . But if any two fields overlap, this method fails. We might substitute for , and order the members of by first differences; but this method will not give what we want if two or more members of have the same field. In order to avoid any confusion due to repetition, we must, if and , consider in connection with , not merely with . That is, the relations in the field of the product of must be such as concern themselves with the ordered couple , not merely with . The simplest way of effecting this is to consider . A member of , say , is a relation which picks out a representative of from the field of every which is a member of ; that is, whenever , . Since we have , not , two relations may have the same field and yet we can distinguish the occurrence of a given term as the representative of the one from its occurrence as the representative of the other. Thus no degree of overlapping will cause confusion.
The relations which compose are to be ordered by first differences, but in order to distinguish different occurrences of a given term, we must give a slightly different form to the principle of first differences from that employed in *170 or *171. The new form of the principle is as follows: Consider two relations and which are members of . Let be a member of in which chooses a representative which precedes that of , i.e. in which ; and let all earlier relations than , i.e. all relations such that and , have . Then we say that precedes . This principle may also be stated as follows: We may divide the members of into four classes, not in general mutually exclusive, namely:
(1) those in which , i.e. in which the -representative precedes the -representative;
(2) those in which ,
(3) those in which ,
(4) those in which no one of the above three relations of and occurs.
Then we shall say that precedes if there is a member of class (1) whose predecessors all belong to class (3).
In case all the members of are serial, the fourth of the above classes is null, and the other three are mutually exclusive. If, further, is well-ordered, any two different members of must be such that one precedes the other in the above-defined order. Thus in this case the product of a series of series is a series (cf. *251).
The definition of the product is Owing to the complication of this definition, the proofs of propositions of the present number are apt to be long.
Various other definitions might be adopted for , but we have found the above definition on the whole the best.
We might, for example, drop the condition in the definition; we could then write our definition in the simpler form: which, with our definition, is only available when . But if we adopt this simplification, we no longer have which is a very useful proposition, required in the proofs of *183·13, *185·21 and other important propositions.
On the other hand, we might frame our definition on the analogy of rather than, as above, on the analogy of . The definition would then be:
This definition does not assume that there is a first relation for which the -representative precedes the -representative. Thus it might be thought that it would give better results in cases where is not well-ordered. But in fact this is not the case. If is not well-ordered, it may happen that every for which is preceded by one for which , and vice versa; in this case, we shall have neither nor . Thus our suggested new definition does not secure that shall be a series whenever and all the members of are series, and therefore has no substantial advantage over the simpler definition which we have adopted, and has the disadvantage of greater complication.
In the present number, we first prove that (*172·13) and that (*172·14), so that a product is null if any one of its factors is null. We then proceed to propositions about , , etc. We have
*172·162.
*172·17.
Hence we derive propositions as to the existence of . We have
*172·181.
Thus assuming the multiplicative axiom, a product which has factors none of which are null is not null.
We then consider , and where . We have
*172·2.
which is a useful proposition, and
*172·23.
which connects the two definitions of multiplication, showing that they lead to equivalent results for any finite number of factors, i.e. whenever the definition of *166 is applicable.
We next consider and , proving
*172·32.
with a similar proposition for (*172·321), and
*172·35.
which is a form of the associative law using both kinds of multiplication. The kind which uses only will be proved in *174.
We have next the proof (with its immediate consequences) that if and have double likeness, . We prove
*172·43.
This proposition should be compared with *114·51, which is its cardinal analogue. It will be seen that the correlator only differs by the substitution of for . From *172·43 we obtain
*172·44.
whence
*172·45.
Other propositions about will be given in *174.
*172·01.
*172·1.
*172·11.
Dem.
*172·12.
Dem.
*172·13.
Dem.
*172·14.
Dem.
*172·141.
The following propositions are concerned with , , etc. *172·15·151·16·161 are lemmas for *172·162·17.
*172·15.
Dem.
*172·151.
*172·16.
Dem.
*172·161.
Dem.
The following proposition is important. It shows that, if consists of series, if any member of has no first term, has no first term, but if every member of has a first term, the selection of all these first terms is the first term of .
*172·162.
Dem.
The following proposition is much used.
*172·17.
Dem.
*172·171.
*172·18.
*172·181.
Dem.
*172·182.
*172·19.
Note that we cannot proceed to , because is meaningless, owing to the fact that the field of consists of non-homogeneous relations.
*172·191.
Dem.
*172·192.
Dem.
The following proposition is sometimes useful. (It is used in *173·22. *182·2.*185·21.)
*172·2.
Dem.
The following propositions are concerned with the nature of the connection between and . The connection is such as might be desired, except when , in which case, as shown above, is like , and is therefore not like .
*172·21.
Dem.
*172·22.
Dem.
*172·23.
The following propositions are lemmas for *172·32.
*172·3.
Dem.
*172·31.
Dem.
*172·32.
Dem.
*172·321.
The following proposition is a lemma for *172·34, which is required in proving *172·35 (as well as *176·4).
*172·33.
Dem.
*172·34.
Dem.
*172·35.
The above proposition is important, being a form of the associative law.
The following propositions are extensions of *172·23. It is obvious that they may be extended to any finite number of factors.
*172·36.
Dem.
*172·361.
*172·37.
Dem.
The following propositions are concerned with the construction of a correlator of with when we are given a double correlator of with . If the double correlator is or , the correlator of with is
*172·4.
Dem.
*172·401.
Dem.
*172·402.
Dem.
*172·403.
Dem.
*172·404.
Dem.
*172·41.
Dem.
The following proposition is important, since it gives the required correlator of with .
*172·42.
Dem.
The following proposition is a lemma for *172·43.
*172·421.
Dem.
*172·43.
*172·44.
*172·45.
The following proposition shows that if two relations have the same field, and if the parts of them that are contained in diversity are the same, they have the same product. Thus e.g. , in virtue of *91·541.
*172·5.
Dem.
The following proposition is used in *182·42.
*172·51.
*172·52.
Dem.
Thus we shall always have unless there are members of which have no referent except themselves.