Volume 1 / Part 3 — *115. MULTIPLICATIVE CLASSES AND ARITHMETICAL CLASSES.
Reading edition
*115. MULTIPLICATIVE CLASSES AND ARITHMETICAL CLASSES.
Summary of *115.
Whenever is a class of mutually exclusive classes, is similar to ; hence
Now is of the same type as ; and when is a class of mutually exclusive classes, consists of all classes formed by selecting one representative from each member of . It often happens that is easier to deal with than ; hence when possible (i.e. when , it is convenient to use , rather than , as the standard member of . We therefore put
We shall call the "multiplicative class" of .
The associative law, requires not merely , but also . The combination of these two hypotheses gives a completely disjointed class of classes of classes, i.e. a class of classes of classes which can be obtained by dividing a given class into mutually exclusive portions, and then dividing each of those portions into mutually exclusive portions. For example, take a square (a class of points) and divide it by horizontal lines, and then divide each of the resulting rectangles by vertical lines; then the resulting rows of little rectangles form such a class, each row of rectangles being one member of the class. Such a class we call an "arithmetical" class, and denote by "."
The present number is concerned with the properties of multiplicative classes and arithmetical classes. Some of these properties will be useful in dealing with exponentiation.
The present number begins with various propositions concerning which are merely repetitions of previous propositions of *83, *84, *85 or *113. Thus we have
*115·141. ,
*115·142. ,
*115·143. ,
*115·16. ,
and various other properties.
We then proceed to consider . We prove
*115·22.
and *115·23 gives a similar proposition substituting "" for .
After a few more propositions on , we proceed to the associative law for (*115·34), i.e.
(This proposition, *115·34, also states that, with the same hypothesis, . Hence we have
*115·35.
We have also
*115·42.
*115·44.
We have next to prove that if two classes of classes have double similarity, so have their multiplicative classes. The proof is simple, since the double correlator is the same as for the original classes, i.e.
*115·502.
whence
*115·51.
The number ends with some propositions which result from *114·64·65 and are analogous to them. One of these is used in the following number, in proving , namely,
*115·6.
The subject of this number will be useful in dealing with exponentiation, since we shall define by means of , where and .
*115·01.
*115·02.
*115·1.
*115·101.
*115·11.
Owing to this proposition, can be treated without any reference to whenever .
*115·12.
It is this proposition that makes the notation appropriate for the multiplicative class.
*115·13.
*115·131.
*115·14.
*115·141.
*115·142.
*115·143.
*115·144.
*115·145.
*115·15.
*115·151.
*115·152.
*115·153.
*115·154.
*115·16.
The following proposition is used in the theory of well-ordered series (*250·5).
*115·17.
Dem.
*115·18.
*115·2.
*115·21.
*115·211.
Dem.
*115·22.
Dem.
Observe that, although "" follows from the converse implication does not hold. If there were two different classes and having the same sum, we might have , i.e. , without having , in spite of "." In proofs, less use can be made of "" than of "." If or , the latter implies .
*115·23.
Dem.
*115·24.
*115·25.
*115·26.
In the above proposition, does not require the hypothesis , being true always. It is merely included here for convenience of reference.
*115·27.
We have now to prove the associative law for "," i.e.
In virtue of *115·12, we have only to prove (under the hypothesis) which, by *85·44, will follow from which, by *114·52, will follow from
Now
Thus the correlator which will give our proposition will be . We have only to prove that this is a , and the rest follows.
*115·3.
Dem.
*115·31.
Dem.
*115·32.
*115·33.
*115·34.
This proposition gives the associative law for "."
The following proposition embodies the last three propositions.
*115·35.
In connection with and , there remain two propositions of sufficient interest to deserve proof, namely and
Of these, the first is deduced from the second, while the second is proved by means of *114·51, putting for the which appears in that proposition, and for the of that proposition.
*115·4.
Dem.
*115·41.
*115·42.
Dem.
*115·43.
Dem.
*115·44.
The following proposition is a lemma for *115·46.
*115·45.
Dem.
*115·46.
Dem.
The above proposition is used in dealing with products in relation-arithmetic (*174·42).
*115·5.
*115·501.
*115·502.
Dem.
*115·51.
The above propositions show how, in certain respects, is more convenient than . We cannot have , because is a class of relations, not a class of classes; and the correlator of and is by no means so simple a function of the correlator of and as , which correlates and , in virtue of *115·502.
The following propositions are a continuation of those given in *114·601 ff.
*115·6.
Dem.
*115·601.
Dem.
*115·602.
*115·61.
*115·62.
*115·63.