Volume 1 / Part 3 — *112. THE ARITHMETICAL SUM OF A CLASS OF CLASSES.
Reading edition
*112. THE ARITHMETICAL SUM OF A CLASS OF CLASSES.
Summary of *112.
In this number, we return to the arithmetical operations. The definition of addition in *110 was only applicable to a finite number of summands, because the summands had to be enumerated. In the present number, we define the arithmetical sum of a class of classes, so that the summands are given as the members of a class, and do not require to be enumerated. Hence the definition in this number is as applicable to an infinite number of summands as to a finite number.
If is a class of mutually exclusive classes, the number of will be the sum of the numbers of members of ; i.e. if we write "" for the sum of the numbers of members of , But when the members of are not mutually exclusive, a term which is a member of two members (say and ) of has to be counted twice over in obtaining the arithmetical sum of , whereas in the logical sum is only counted once. Thus we need a construction which shall duplicate , taking it first as a member of , and then as a member of . This is effected if we replace first by , and then by . In fact, has the kind of arithmetical properties which we mean to secure when we speak of " considered as a member of "—a phrase which, as it stands, does not serve our purpose, for is simply however we may choose to consider it. Thus we replace by and by and so on; i.e. (using *85·5), we replace by and by and so on. These new classes are similar to and and so on, and are mutually exclusive. Hence their logical sum has the number of terms which is wanted for the arithmetical sum of the members of . Thus we put
With regard to the second of these definitions, it is to be observed that is not a function of , unless no two members of are similar; for cannot contain the same number twice over. For the same reason, if is a class of cardinals, and we define "," we do not get what is wanted for arithmetical addition, because our definition will not enable us to deal with summations in which there are numbers that are repeated. We could, if it were worth while, define "" as follows: Take a class of classes , consisting of one class having each number which is a member of , i.e. let be a selection from ; then will have the required number of terms. i.e. we might put But since this definition is only available for sums in which no number is repeated, it is not worth while to introduce it.
In this number we prove the following propositions among others.
*112·15.
This is an extension of *110·32.
*112·17.
The chief point in the above proposition is that it does not require , .
*112·2—·24 are concerned with the use of the multiplicative axiom and the propositions of *111 in which it appears as hypothesis. We have
*112·22.
whence we derive the proposition
*112·24.
I.e. assuming the multiplicative axiom, two classes which each consist of classes of terms each have the same number of terms in their sum. This number would naturally be defined as multiplied by , but owing to the necessity of the multiplicative axiom in this proposition, we have selected a different definition of multiplication (*113) which does not depend upon the multiplicative axiom. The reader should observe that the similarity of two classes, each of which consists of mutually exclusive sets of terms, cannot be proved in general without the multiplicative axiom.
The remaining propositions of this number give properties of in special cases. We prove that (*112·3), that (*112·321), that (*112·34), which connects the definition of addition in this number with that in *110. Finally we prove the general associative law for addition, in the following two forms:
*112·41.
*112·43.
*112·01.
*112·02.
*112·1.
*112·101.
*112·102.
Dem.
*112·103.
*112·11.
*112·12.
*112·13.
*112·14.
Dem.
*112·15.
*112·151.
Dem.
The following proposition is a lemma for *112·153, which is required for *112·16. *112·16 in turn is used in *112·17, which is a fundamental proposition in the theory of addition.
*112·152.
Dem.
In the following proposition, we have a double correlator of a sort which will frequently occur in cardinal arithmetic, namely with its converse domain limited, where is a given double correlator (or single correlator, on other occasions). As appears from the propositions used in the above proof of *112·152, if is a correlator whose converse domain includes and has as a member, . Thus is an operation which, when operating on suitable relations of individuals to classes (including selectors), turns the individuals into their correlates and the classes into the classes of their members' correlates. This is why it is a useful relation.
*112·153.
Dem.
*112·16.
*112·17.
Dem.
*112·18.
Dem.
*112·2.
Dem.
*112·21.
*112·22.
*112·23.
Dem.
*112·231.
Dem.
*112·24.
Dem.
*112·3.
*112·301.
Dem.
*112·302.
Dem.
Thus if is a member of a class of classes, it does not affect the value of their arithmetical sum.
*112·303.
Dem.
*112·304.
Dem.
*112·31.
Dem.
*112·311.
Dem.
*112·32.
Dem.
*112·321.
*112·33.
*112·331.
*112·34.
Dem.
This proposition establishes the agreement of the two definitions of addition, namely that in *110 and that in *112. It will be seen that the definition of *112 is inapplicable to the addition of a class to itself, if this is to give the double of the class, instead of (like logical addition) simply reproducing the class. Hence the need of the condition in the above proposition.
*112·341.
Dem.
*112·35.
Dem.
Similar propositions can obviously be proved for any finite number of summands.
*112·4.
Dem.
*112·41.
Dem.
*112·42.
Dem.
*112·43.
Dem.
The above is the associative law for arithmetical addition.