Transfinite Numbers. 443 



any part which is such that i£ it includes any two terms it 

 also includes all the terms between those two. By an initial 

 segment is meant any part which is such that i£ it includes 

 any term it also includes all the terms before that term. 



A well-ordered series is one which is such that every part 

 of it has a first term"*. 



An unlimited serial class which is such that any aggregate 

 of terms selected from it has a first term is a " well-ordered 

 serial class."" 



It follows immediately from the definition of a well-ordered 

 series, that every term which has any terms after it has a 

 term which is next after it, and if any part of the series is such 

 that these are terms of the series after all the terms of the 

 part, then there is one term of the series which is the first 

 after all the terms of the part. 



The correlation of the terms of two similar well-ordered 

 series is unique, i. e., there is one and only one term of the one 

 series which can correspond to any given term of the other. 

 For the first term of the one must correspond to the first 

 term of the other, and the first term after any initial segment 

 of the one must correspond to the first term after the 

 corresponding initial segment of the other. 



If two well-ordered series are not similar then one of them 

 must be similar to some particular initial segment of the 

 other. 



For the first term of the one can be correlated with the 

 first term of the other and the next with the next, and 

 the first term after any initial segment of the one with the 

 first term after the corresponding initial segment of the other, 

 and so on until one or other series is exhausted. 



Conversely, if a well-ordered series is similar to an initial 

 segment of another well-ordered series, then the two series 

 are dissimilar. 



A progression is a well-ordered series which has no last 

 term and which is such that each term except the first has a 

 term next before it f. 



All well-ordered series can be classified in exclusive classes 

 such that any two of the same class are similar to each other. 



* It is not necessary to state that the series (or class) has a first term, 

 as this follows from the definition. In the case of the series this is 

 obvions ; in the case of the unlimited class we can proceed as follows. 

 Take any term if this he not the first, then take one before it, if this be 

 not the first then one before it and so on ; in this way we must in a 

 finite number of steps arrive at the first term of the class, otherwise we 

 should have an aggregate of terms having no first term. 



t It is here tacitly assumed that an enumerable class is a limited 

 class. 



