440 Mr. A. E. Harward on the 



tively as a whole is an " unlimited class," other classes are 

 " limited classes." That there are unlimited classes is easily 

 shown, for the class of all classes is a class, and as such is a 

 member of itself. If all classes could be thought of col- 

 lectively as a whole the whole would be identical with one 

 among many individuals comprised in itself, which is a 

 contradiction in terms. 



Strictly speaking a distinction can be drawn between a 

 limited class qua class, and the aggregate of individuals 

 belonging to it, but as the distinction is not important for 

 my present purpose I shall, to avoid circumlocution, speak 

 of limited classes as " being " aggregates, and vice versa. 



It is useful to have a synonym for " class " which can be 

 used when it is necessary to make it evident that the word 

 has no collective import, and for this purpose I use the word 

 " type/' By " type " I mean merely " class " (qua " class," 

 and not qua " aggregate ") and not some mystical entity 

 which the members of the class are supposed to resemble. 



The following propositions follow immediately from the 

 meaning of the terms: — 



(a) Every class which has an unlimited sub-class is itself 



unlimited ; and conversely every sub-class of an 

 aggregate is an aggregate. 



(b) The logical sum* of an aggregate of aggregates is itself 



an aggregate, for in arranging all the members of 

 a class in an aggregate of aggregates we do thereby 

 exhibit them collectively as a whole f. 



(c) Any class of which the individuals can be correlated 



one to one with the elements of an aggregate is 

 itself an aggregate. 



(d) Any class of which the individuals can be correlated 



one to one with the members of an unlimited class 

 is itself unlimited. 

 2. Two aggregates or classes are said to be "similar" 

 when their elements can be correlated one to one so that one 

 term of either corresponds to one and only one term of the 

 other. The relation of similarity is symmetrical J and tran- 

 sitive to all the aggregates similar to a given aggregate from 

 a class such that any two members of the class are similar to 

 one another. By this relation of similarity all aggregates 

 are classified in exclusive § classes of similar aggregates. 



* Vide Russell's ' Principles of Mathematics,' p. 117. 

 t This argument does not amount to formal proof. It is rather an 

 appeal to intuition, vide Note B. 



X Vide ' Principles of Mathematics,' p. 218. 



§ Classes are said to be exclusive when no two have a common member 



