Volume 1 / Part 3 — *113. ON THE ARITHMETICAL PRODUCT OF TWO CLASSES OR OF TWO CARDINALS.
Reading edition
*113. ON THE ARITHMETICAL PRODUCT OF TWO CLASSES OR OF TWO CARDINALS.
Summary of *113.
In this number, we give a definition of multiplication which can be extended to any finite number of factors, but not to an infinite number of factors. We define first the arithmetical class-product of two classes and , and thence the product of two cardinals and as the number of terms in the product of and when has terms and has terms. In *114, we shall give a definition of multiplication which is not restricted to a finite number of factors. The advantages of the definition to be given in this number are, that it does not require the factors to be of the same type, and that it enables us to multiply a class by itself without (as in logical addition and multiplication) simply reproducing the class in question. The disadvantage of the definition in this number is the impossibility of extending it to an infinite number of factors.
The arithmetical class-product of two classes and , which we denote by [5], is the class of all ordinal couples which take their referent from and their relatum from , i.e. it is the class of all such relations as , where and . For a given , the class of couples we obtain is , which is similar to ; and the number of such classes, for varying , is . Thus we have classes of couples, and is the logical sum of these classes of couples. The class of such classes as , where , is important again in connection with exponentiation; we have , whence the class of such classes, when is varied among the 's, is , and which we take as the definition of .
We represent the arithmetical product of and by . This, as well as , is defined in terms of exactly as, in *110, the sum was defined in terms of .
The present number contains many propositions which belong to the theory of rather than (specially) of ; and many propositions are rather logical than arithmetical in their nature, i.e. they might have been given in *55. The line is, however, so hard to draw that it has seemed better to deal simultaneously with all propositions on or on its sum, which is . Thus in the present number, the early propositions, down to *113·118, deal mainly with logical properties of and ; the following propositions, down to *113·13, deal mainly with arithmetical properties of ; the propositions *113·14—·191 are concerned mainly with arithmetical properties of ; *113·2—·27 deal with the simpler properties of ; *113·3—·34 give propositions involving the multiplicative axiom, and exhibiting the connection (assuming this axiom) of addition and multiplication; *113·4—·491 are concerned with various forms of the distributive law; *113·5—*113·541 deal with the associative law of multiplication, and the remaining propositions deal with multiplication by 0 or 1 or 2.
The most important propositions in the present number are the following:
*113·101.
This merely embodies the definition of .
*113·105.
This proposition is especially useful in dealing with exponentiation (*116).
*113·114.
It is in virtue of this proposition that a product of a finite number of factors only vanishes when one of its factors vanishes.
*113·118.
This proposition is chiefly useful in the analogous theory of ordinal products (*165, *166), where it enables us to apply *74·773. Unless , we have , and unless (*113·116).
*113·12.
I.e. unless is null, consists of mutually exclusive classes each having members.
*113·127.
This is an important proposition, since it gives a double correlator of with whenever simple correlators of with and of with are given. It leads at once to
*113·13.
This proposition is fundamental in the theory of multiplication, since it shows that the number of members of depends only upon the numbers of members of and . It is also fundamental in the theory of exponentiation, as will appear in *116.
*113·141.
This is the source of the commutative law of multiplication (*113·27).
*113·146.
This connects our present theory of multiplication with the theory of selections.
We come next to propositions concerning . We have
*113·204.
The use of this proposition, like that of *110·4, is for avoiding trivial exceptions.
*113·23.
*113·25.
This proposition enables us to infer propositions on products of cardinals from propositions on products of classes, and is therefore constantly used.
*113·27.
This is the commutative law of cardinal multiplication.
The chief proposition using the multiplicative axiom is
*113·31.
I.e. assuming the multiplicative axiom, the sum of the numbers of members in classes of terms is . If we had taken this sum as defining , almost all propositions on multiplication would have required the multiplicative axiom. The advantage of is that, given and , we can construct a double correlator of with , without using the multiplicative axiom. This is proved in *113·127 (mentioned above).
The distributive law, which is next considered, has various forms. We have, to begin with,
*113·4.
whence, using also the commutative law, we easily deduce
*113·43.
But the distributive law also holds when, instead of enumerated summands , or , , the summands are given as the members of a class , which may be infinite. We have
*113·48.
whence, using the definitions of *112, we find
*113·491.
This is an extension of the distributive law to the case where the number of summands may be infinite.
The associative law
*113·54.
is proved without any difficulty.
We prove next that when, and only when, or , , being existent cardinals (*113·602); that a cardinal is unchanged when it is multiplied by 1 (*113·62·621); that (*113·66) and that (*113·671).
*113·02.
*113·03.
*113·04.
*113·05.
In relation to types, *113·03·04·05 call for similar remarks to those made in *110 for addition.
*113·1.
*113·101.
*113·102.
Dem.
*113·103.
*113·104.
*113·105.
Dem.
*113·106.
*113·107.
*113·11.
Dem.
*113·111.
*113·112.
Dem.
*113·113.
*113·114.
*113·115.
Dem.
*113·116.
*113·117.
*113·118.
*113·12.
*113·121.
*113·122.
*113·123.
*113·124.
Dem.
*113·125.
*113·126.
Dem.
*113·127.
*113·128.
*113·13.
*113·14.
*113·141.
*113·142.
Dem.
*113·143.
Dem.
*113·144.
Dem.
Note to *113·144. In virtue of *113·143 and *55·61 we have
At a later stage (in *150) we shall put
Thus we shall have, anticipating this notation,
Hence we have
*113·145.
*113·146.
*113·147.
Dem.
The advantage of this proposition is that it exhibits the correlator of and as a function of .
*113·148.
Dem.
*113·15.
Dem.
*113·151.
*113·152.
Dem.
The following proposition is only significant when and are classes of relations. It is used in relation-arithmetic (*172·34).
*113·153.
Dem.
*113·16.
Dem.
*113·17.
Dem.
*113·171.
Dem.
Note that the hypothesis is only significant when and are of the same type.
*113·172.
Dem.
*113·18.
Dem.
*113·181.
Dem.
*113·182.
*113·183.
Dem.
*113·19.
Dem.
*113·191.
Dem.
*113·2.
*113·201.
*113·202.
Dem.
*113·203.
*113·204.
*113·205.
*113·21.
*113·22.
Dem.
*113·221.
*113·222.
Dem.
*113·23.
Dem.
*113·24.
*113·25.
This proposition constitutes part of the reason for our definitions. It is obvious that such definitions ought, if possible, to be chosen as will yield this proposition.
*113·251.
*113·26.
Dem.
*113·261.
Here "." includes all ascending derivatives of . We shall only prove the result for and , since it is proved in just the same way for the other cases. or or etc. will serve equally well; i.e. it is not necessary to take the same derivative of as of .
Dem.
As appears in the above proof, if and are any derivatives of and , the above proposition holds provided we have
Thus it holds for all ascending derivatives, but not always for descending derivatives.
*113·27.
Dem.
Note that this proposition is not confined to the case in which and are cardinals. When either or both are not cardinals,
*113·3.
Dem.
*113·31.
*113·32.
*113·33.
*113·34.
The above propositions give the connection of addition and multiplication.
The following propositions are concerned with various forms of the distributive law.
*113·4.
Dem.
*113·401.
*113·41.
Dem.
*113·42.
*113·421.
*113·43.
Dem.
The following propositions are concerned with various forms of the distributive law, when the summands are not enumerated, but given as the members of a class.
The first of them (*113·44) gives the distributive law with regard to arithmetical class-multiplication and logical addition of classes.
*113·44.
Dem.
*113·45.
Dem.
*113·46.
Dem.
*113·47.
This is the distributive law for arithmetical multiplication and arithmetical addition of the kind defined in *112.
*113·48.
Dem.
*113·49.
Dem.
*113·491.
The following propositions are concerned with the associative law for arithmetical multiplication.
*113·5.
Dem.
*113·51.
Dem.
*113·511.
*113·52.
*113·53.
Dem.
*113·531.
*113·54.
Dem.
*113·541.
*113·6.
Dem.
*113·601.
Dem.
*113·602.
Dem.
The following propositions are concerned with multiplication by a unit class or by 1 or 2.
*113·61.
Dem.
*113·611.
*113·612.
*113·62.
Dem.
*113·621.
Dem.
Observe that if is a typically definite cardinal, is the "same" cardinal rendered typically ambiguous; while if is typically ambiguous, in every type.
*113·63.
Dem.
*113·64.
Dem.
*113·65.
Dem.
*113·66.
Dem.
*113·67.
Dem.
*113·671.