Volume 1 / Part 4 — *153. THE RELATION-NUMBERS , AND .
Reading edition
*153. THE RELATION-NUMBERS , AND .
Summary of *153.
The relation-numbers and have already been defined (in *56), though it remains for the present number to show that they are relation-numbers. They are the ordinal 0 and 2 respectively, i.e. they are the ordinal numbers of well-ordered series of no terms and series of two terms respectively. But there is no means of introducing an ordinal 1 which shall be analogous to the cardinal 1 as completely as and are analogous to 0 and 2. The only relations whose fields are unit classes are relations of the form . We therefore put
*153·01.
The above definition gives the nearest possible approach to an ordinal 1. so defined is a relation-number, and is the relation-number corresponding to 1 in the sense that it is the relation-number of all such relations as have a field consisting of one term. But is not what is called an "ordinal number," because this term is confined by usage to the relation-numbers of well-ordered series, and is not a serial relation. It is essential to a serial relation to be contained in diversity; and if, by definition, we include among series, we introduce more exceptions than we avoid. Moreover does not have the kind of properties which we wish 1 to have; e.g. is not .
We do not use , because we shall at a later stage define as the class of those series whose fields have terms, so that , while and have the values and , as already defined. On account of this general definition of , we choose a different symbol for the relation-number 1, and has the merit of being as like as possible.
To illustrate, by anticipation, the way in which differs from proper ordinal numbers, we may point out that if is added to , we do not obtain . We shall define as the class of series which consist of three terms, i.e. the class of relations of the form where . We shall define the sum of two ordinal numbers as the ordinal number of the sum of two relations having these ordinal numbers (cf. *180), and it will appear that if and are relations whose fields have no members in common, then has a relation-number which is the sum of those of and . Suppose now and , where . Then This is not a member of , because of the additional term . Thus the addition of one term to a series does not give the same number as results from the addition of to . Hence the addition of 1 to an ordinal number has to be separately treated[12].
We prove in this number that (*153·11), that (*153·24; observe that we have to take a couple of classes (or relations) in order to be sure of the existence of two different objects of the class in question), and that ) (*153·32). We prove (*153·18), (*153·212), and (*153·36). We have also (not proved) and (*153·301). But we do not have ; e.g. if , but . We have (*153·12) and (*153·34), but from our primitive propositions we cannot deduce unless we rise above the lowest type of relations. The case is exactly analogous to that of (cf. *101); we have
*153·26·262.
But if, as monists aver, there is only one individual, we shall not have in the type of relations of individuals to individuals. Our primitive propositions do not suffice to disprove this supposition.
*153·01.
*153·1.
*153·101.
Dem.
*153·11.
*153·111.
*153·12.
*153·13.
*153·14.
Dem.
*153·15.
Dem.
*153·16.
Dem.
*153·17.
*153·18.
Dem.
*153·2.
*153·201.
*153·202.
*153·203.
Dem.
*153·21.
*153·211.
*153·212.
*153·22.
*153·23.
This proposition illustrates the reasons for not putting We want the inductive ordinals, like the inductive cardinals, to form a series in order of magnitude; but, as the above proposition illustrates, the relation-number of such relations as is not in the same series with and . The above proposition should be contrasted with *51·411.
*153·24.
*153·25.
*153·251.
Dem.
*153·26.
*153·261.
*153·262.
*153·27.
*153·28.
Dem.
*153·281.
The above proposition is used in the theory of series (*204·48).
*153·3.
*153·301.
*153·31.
Dem.
*153·311.
Dem.
*153·32.
*153·33.
*153·34.
Dem.
*153·341.
*153·35.
Dem.
*153·36.
Dem.