442 Mr. A. E. Harward on the 



From this it follows that there cannot be any greatest 

 cardinal number. 



4. At this stage it is necessary to introduce ordinal notions*. 

 A simply-ordered series is any aggregate of elements such 

 that there is a certain transitive asymmetrical relation R 

 such that if a and b be any two elements of the aggregate, 

 then either a has the relation R to b, or b has the relation R 

 to a. In one of these cases (it does not matter which we 

 choose) we say that a is before b, in the other we say that 

 a is after b. 



It should be observed that the term " series " is by this 

 definition restricted to aggregates. 



It may happen that the terms of an unlimited class are 

 related inter se in the manner described above, in that case 

 it is called a u simply-ordered serial class," but must not be 

 called a series. 



In what follows I use the word series to mean simply- 

 ordered series. 



Two series are said to be ordinally similar when their 

 terms can be correlated one to one, so that earlier terms 

 correspond to earlier and later to later |. 



Hereafter, whenever I speak of two series as similar it is 

 to be understood that I mean ordinally similar. 



The relation of ordinal similarity is symmetrical and tran- 

 sitive, so all the series ordinally similar to a given series form 

 a class such that any two members of the class are ordinally 

 similar to each other. 



In this way all series are classified in exclusive classes of 

 similar series. These classes or types of series may be de- 

 noted by symbols, and these symbols can be combined by 

 addition and multiplication. The symbol a + (3 means the 

 type of the series formed by a series of type a followed by a 

 series of type ft. 



The symbol u(3 means the type of the series formed by 

 an aggregate of series of type a arranged in a series of 



type/31. 



By a part of a series is meant any aggregate of terms 

 belonging to a series. By a segment of a series is meant 



* The idea that any progress can be made in the study of cardinals 

 without introducing ordinal notions appears to me to he chimerical. It 

 is true that the notion " aggregate " is logically prior to the notion 

 " series," but it is only by arranging the elements of aggregates in com- 

 parable series of some kind that we can determine whether they are 

 similar or dissimilar. 



t I. e., if x correspond to x' and y to y', then if x be before y, x' must 

 be before y', and vice versa. 



X a 4-/3 must be distinguished from (3+ a, and a/3 from /3a. 



