*42. MISCELLANEOUS PROPOSITIONS.
Project Gutenberg EPUB3 edition
Whitehead, Alfred North and Russell, Bertrand
Summary of *42.
The present number contains various propositions concerning products and sums of classes. They are concerned chiefly with classes of classes of classes, or with relations of relations of relations. These are required respectively in cardinal and in ordinal arithmetic. Thus *42·1 is used in *112 and *113, which are concerned with cardinal addition and multiplication, while *42·12·2 are used in *160 and *162, which are concerned with ordinal addition. *42·22, though not explicitly referred to, is useful in facilitating the comprehension of propositions on series of series of series, or rather on relations between relations between relations, which are required in connection with the associative law of multiplication in relation-arithmetic.
*42·1.
Here must, for significance, be a class of classes of classes. The proposition states that if we take each member, , of , and form , and then form the sum of all the classes so obtained, the result is the same as if we form the sum of the sum of . This is the associative law for , and is (as will appear later) the source of the associative law of addition in cardinal arithmetic. The way in which this proposition comes to be the associative law for may be seen as follows: Suppose consists of two classes, and ; suppose in turn consists of the two classes and , and of the two classes and . Then . (This will be proved later.) Thus has two members, one of which is , while the other is . Thus But has four members, namely , , , . Thus .
Thus our proposition leads to which is obviously a case of the associative law.
Our proposition states the associative law generally, including the case where the number of brackets, or of summands in any bracket, is infinite. The proof is as follows.
Dem.
*42·11.
Dem.
This is the associative law for products. Supposing again, for illustration, that consists of the two classes , , while consists of the two classes , and \beta of the two classes , , then consists of the two classes and , so that , while . Thus our proposition becomes
A descriptive function whose arguments are classes or classes of classes may be said to obey the associative law provided
This equation may be interpreted as follows: Given a class , divide it into any number of subordinate classes, so that no member is left out, though one member may belong to two or more classes. Let the classes into which is divided make up the class , so that is a class of classes, and . Then the above equation asserts that if we first form the of the various sub-classes of , and then the of the resulting class, the result is the same as if we formed the of directly.
In some cases—for example, that of arithmetical addition of cardinals—the above equation holds only when no two members of have a common term, i.e. when the parts into which is divided are mutually exclusive.
For a descriptive function whose arguments are relations of relations, we shall find another form for the associative law; this form plays in ordinal arithmetic a part analogous to that played by the above form in cardinal arithmetic.
*42·12.
Dem.
*42·13.
Dem.
*42·2.
This proposition assumes that is a relation between relations. For example, suppose we have a series of series, whose generating relations are ordered by the relation . Then is the class of these generating relations; is the relation "one or other of the generating relations which compose ," and is the class of all the terms occurring in any of the series. is the fields of the various series, and is again all the terms occurring in any of the series. is all the terms belonging to fields of series which are members of , and is all members of fields of members of the field of ; each of these again is all the terms occurring in any of the series. The proof is as follows:
Dem.
The following propositions apply to a relation of relations of relations. These propositions are useful for proving associative laws in ordinal arithmetic, since these laws deal with series of series of series, and series of series of series are most simply constituted by supposing the generating relations of the constituent series to be ordered by relations which are themselves ordered by a relation .
*42·21.
Dem.
*42·22.
If , in the above proposition, is a relation which generates a series of series of series, the above gives various forms for the class of ultimate terms of these series. Thus suppose ; then is a relation between generating relations of series. If now , is the generating relation of a series which we may regard as composed of individuals. The class of individuals so obtainable may be expressed in any of the above forms, as well as in others which are not given above.
*42·3.
Dem.
*42·31.