960 



SCIENCE 



[N. S. Vol. XXX. No. 783 



posed of some but not all of the terms of 

 a and having to a a biuniform relation. 

 In respect to the finite cardinals, they 

 may be defined as follows, presenting them 

 in what, once order is defined, will be 

 called a series, 0, 1, 2, . . . Let zero (0) be 

 defined as above ; let the cardinal next after 

 the cardinal n be defined to be the cardinal 

 n -\-l; let N, the class of finite cardinals, 

 be defined to be the class of cardinals that 

 are contained in every class that contains 

 and contains n -\- 1 if it contains n. It 

 remains then to show that the two defini- 

 tions of finite cardinals are equivalent, and 

 that can be done. 



Cardinals, we have seen, are classes. 

 The ordinary rational numbers, or frac- 

 tions, are not classes, but are, as we shall 

 see, relations of finite cardinals. Let a 

 be any given finite cardinal, and let x and 

 y be any finite cardinals such that xa ■= y. 

 Denote by A the relation such that xAy is 

 equivalent to xa^y. Similarly, to any 

 finite cardinal n there corresponds a rela- 

 tion N whose domain and codomain are 

 respectively composed of all the finite car- 

 dinals X and y such that xn = y. If 

 ab^p and cd = p, that is, if ab = cd, 

 then aBp and cDp, whence pDc, so that 

 aBDc. The relation BD, the relative prod- 

 uct of B and the converse of D, is named 

 rational number, or fraction, and denoted 

 by b/d. If ab = cd, it readily follows 

 that b/d = a/c. The rational n/1 is com- 

 monly denoted by n, but the rational n 

 and the cardinal n are radically different, 

 the former being a relation while the latter 

 is a class. 



The cardinals and rationals are signless. 

 Like the rationals, positive and negative 

 integers and fractions are relations but 

 they are relations of a different type. Sup- 

 pose the finite cardinals arranged as by 

 their second definition above given. Let 

 R be such that xBy, x and y being finite 



cardinals, means that, in the mentioned 

 arrangement, y is the immediate successor 

 of x; then xBy means that y is the immedi- 

 ate predecessor of x. It is readily proved 

 that B" is the converse of (E)p or, what is 

 the same, of B". The relations B'p and R" 

 {p being a finite cardinal) are defined to 

 be the positive and negative integers fa- 

 miliarly denoted by -j- p and — p respec- 

 tively. Thus to each finite cardinal p 

 there corresponds a positive integer, -f- p, 

 and a negative integer, — p. If x, y and 

 p are finite cardinals, the propositions, 

 xB'fy and x-{-p = y, are equivalent; so, 

 too, are xB^'y and y -\- p = x or x — p = y- 

 Similarly if a; be a rational number, and 

 if y and z stand for any two rational num- 

 bers so related that y-\-x = z, the relation 

 in question is denoted by -|- x; but if y 

 and z are so related that y — x^z, the 

 relation is denoted by — x. 



Before speaking of the ordinal number, 

 it is necessary to tell what is meant by say- 

 ing of a class that it is ordered or that its 

 terms are arranged in a series. This, 

 which is one of Russell's most brilliant 

 achievements, was accomplished as follows. 

 I here but indicate the method and state 

 the result. The method was precisely that 

 of research in natural science, namely, he 

 collected together the various kinds of rela- 

 tion by which what is called order, whatever 

 order in its essence should turn out to be, 

 is generated. These relations, which he 

 found to belong to one or another of six 

 distinct types, turned out, upon penetra- 

 ting analysis, to be reducible to a single 

 type, namely, that of relations at once 

 transitive and asymmetric, an asymmetric 

 relation B being such that, if xBy, then 

 not yBx. The conclusion may be stated to 

 be that, a class being given, if there exist a 

 transitive asymmetric relation B such that, 

 X and y being any two whatever of its 

 terms, either xBy or else yBx, the class is 



