December 31, 1909] 



SCIENCE 



961 



thus arranged in a series; and that order 

 otherwise generable is geuerable by such a 

 relation. The result is of coiirse subject to 

 such doubt as must always attend the 

 method employed, but its correctness seems 

 highly probable. It can be easily proved 

 that, given any three terms x, y, z of an 

 open series, M'e have xBy and yBz, or yHz 

 and zBx or zBx and xBy, that is, one of 

 the three terms is between the other two; 

 and if the series be closed, like that of the 

 points of a circle, it can be rendered open 

 by cutting it— that is, by regarding it as 

 beginning (or ending) with some (any) 

 definite term. 



We are now prepared to present the no- 

 tion of ordinal number. If, given two 

 series s, and So, there subsist between them, 

 regarded as classes, a biuniform relation B 

 such that, Oj and &i being any two terms 

 of Si and a, and h^ their respective corre- 

 spondents (through B) in S„, a, precedes 

 or follows &i according as a„ precedes or 

 follows 1)„, then the series s^ and s, are said 

 to be like. Plainly likeness is a transitive 

 and symmetric relation. Two like series 

 are said to have the same ordinal number 

 or the same order-type. Herewith ordinal 

 number, or order-type, of a series is yet not 

 defined. The definition is: the ordinal 

 number, or order- type, of a series s is the 

 class of all series like it. Or, defining like 

 relations to be such as generate like series, 

 we can define ordinal number, or order- 

 type, of a series-generating relation to be 

 the class (a relation by primitive propo- 

 sition) of series like it. The definition does 

 not distinguish finite and infinite and so ap- 

 plies to both. In case the terms of a series 

 constitute a finite class, the cardinal num- 

 ber of the class and the ordinal number of 

 the series obey the same laws and are com- 

 monly denoted by the same name and sym- 

 bol. Yet they are radically different no- 

 tions. For example, the cardinal three 



includes the class composed of a, b and c, 

 but not the series a, b and c as such, while 

 the ordinal three includes the series but 

 not the class. On transition to infinites 

 the distinction is forced upon us, for in- 

 finite cardinals obey, for example, the law 

 of commutation, while the infinite ordinals 

 do not. 



I have time for but a single indication 

 pointing the way to the concept and theory 

 of real numbers. Consider, for example, 

 the two familiar classes: A^ the class of 

 rationals less than 2; B, the class of ra- 

 tionals whose squares are less than 2. Each 

 of these classes possesses the properties : 

 (1) it does not contain all the rational 

 numbers; (2) it contains all the rational 

 numbers less than any one of its numbers ; 

 (3) every number in it is less than some 

 other number in it. Any class of rationals 

 that has the three properties is named seg- 

 ment (of rationals). Given a segment s, 

 the class of rationals not belonging to s 

 may be called the cosegment of s. It is 

 found that the cla.ss of all segments admits 

 of a theory precisely isomorphic with that 

 of the real numbers as usually defined. 

 Hence the segments are named real num- 

 bers. Segments faU into two classes ac- 

 cording as their cosegments have or have 

 not a smallest rational. In the former case 

 the segment is called a rational real num- 

 ber. Thus segment A is the rational real 

 two or 2. In the other case, the segment 

 is called an irrational real niimber. Thus 

 segment B is the irrational real commonly 

 denoted by \/2. It is obvious that seg- 

 ments and reals might just as well be 

 defined by the relation greater than instead 

 of less than. The decisive advantage of the 

 foregoing definition, which makes no ap- 

 peal to the (as yet) undefined notion of 

 limit, is that it avoids the necessity of 

 assuming a limit where there is none, as 

 in case of class B. 



