444 Mr. A. E. Harward on the 



These classes or types of well-ordered series are the ordinal 

 numbers *. 



Any two progressions are ordinally similar. The type of 

 well-ordered series to which a progression belongs is an 

 ordinal number denoted by the symbol w. 



If a and ft be two ordinal numbers and if a series of type a 

 be similar to an initial segment of a series of type ft, then a 

 is said to be less than ft. 



It necessarily follows that if a and ft be any two different 

 ordinals, then either a < ft or ft<a. 



The class of ordinals is therefore simply ordered. 



It is obvious that the initial segments of a well-ordered 

 series themselves form a well-ordered series, for each segment 

 can be correlated with the term next after if. Therefore the 

 initial segments form a series similar to the series obtained by 

 removing the first term of the original series. If the 

 original series be transfmite, the series so obtained is of 

 the same type as the original series. 



The ordinals less than any given ordinal ft can be correlated 

 in order of magnitude with the initial segments of a series of 

 type ft, and therefore they form a well-ordered series. 



If ft be transfmite the ordinals less than ft form a series 

 of type ft. If ft be finite the series has one term less than ft. 



The class of ordinals is therefore a well-ordered serial 

 class. 



There cannot be a greatest ordinal, for every w r ell-ordered 

 series can be increased by adding on terms at the end. It 

 follows that the ordinals are not a series f, for if they were a 

 series, the type of that series would be the greatest ordinal. 

 The class of ordinals is therefore an unlimited serial class. 

 In other words_, the ordinals cannot without contradiction be 

 thought of collectively as a whole, because if we attempt to 

 think of them in that manner we are at once led to the 

 conclusion that there are other ordinals not comprised in the 

 collection of which we are trj'ing to think. 



It immediately follows from this that the elements of 

 any aggregate can be arranged in a well-ordered series, for 

 we can take elements from the aggregate one by one and 

 correlate them with the ordinals in order of magnitude, and 

 this process must at some stage come to an end by exhaustion 

 of the aggregate, for if it did not, the whole or some part of 

 the aggregate would be similar to the whole class of ordinals 

 and this is not possible because that class is unlimited. 



* Hereafter letters of the Greek alphabet will loe used to denote 

 ordinal numbers exclusively. 



