*54. CARDINAL COUPLES.
Project Gutenberg EPUB3 edition
Whitehead, Alfred North and Russell, Bertrand
Summary of *54.
Couples are of two kinds, namely (1) , in which there is no order as between and , and (2) , in which there is an order. We may distinguish these two kinds of couples as cardinal and ordinal respectively, since (as will be shown hereafter) the class of all couples of the form (where ) is the cardinal number 2, while the class of all couples of the form (where ) is the ordinal number 2, to which, for the sake of distinction, we assign the symbol "," where the suffix "" stands for "relational," because the ordinal 2 is a class of relations. In the present and the following numbers, we shall define 2 and as the classes of cardinal and ordinal couples respectively, leaving it to a later stage to show that 2 and , so defined, are respectively a cardinal and an ordinal number. An ordinal couple will also be called an ordered couple or a couple with sense. Thus a couple with sense is a couple of which one comes first and the other second.
We introduce here the cardinal number 0, defined as . That 0 so defined is a cardinal number, will be proved at a later stage; for the present, we postpone the proof that 0 so defined has the arithmetical properties of zero.
Cardinal couples are much less important, even in cardinal arithmetic, than ordinal couples, which will be considered in the two following numbers (*55 and *56). It is necessary, however, to prove some of the properties of cardinal couples, and this will be done in the present number. Some properties of cardinal couples which have been already proved are here repeated for convenience of reference. The definitions of 0 and 2 are:
*54·01.
*54·02.
Most of the propositions of the present number, except those that merely embody the definitions (*54·1·101·102) are used very seldom. The following are among the most important.
*54·26.
*54·3.
*54·4.
*54·53.
*54·56.
*54·01.
*54·02.
*54·1.
*54·101.
*54·102.
The two following propositions have already occurred in *51, but are here repeated, because they belong to the subject of the present number.
*54·21.
*54·22.
*54·25.
Dem.
*54·26.
Dem.
*54·27.
*54·271.
Dem.
*54·3.
Dem.
*54·4.
Dem.
This proposition shows that a class contained in a couple is either the null-class or a unit class or the couple itself, whence it will follow that 0 and 1 are the only numbers which are less than 2.
*54.41.
Dem.
*54·411.
*54·42.
Dem.
*54·43.
Dem.
From this proposition it will follow, when arithmetical addition has been defined, that 1 + 1 = 2.