Transfinite Numbers. 441 



These classes or types o£ aggregates are the u cardinal 

 numbers." An aggregate is said to be infinite when it is 

 similar to a proper part* of itself. The cardinal number of 

 an infinite aggregate is said to be transfinite. If an aggre- 

 gate M is similar to part of an aggregate N. and if N is not 

 similar to the whole or to any part of M, then the cardinal 

 number of M is said to be " less than " the cardinal number 

 ofN. 



Before we can assert that of two different cardinal numbers 

 one must be less than the other, we require to prove that of 

 two dissimilar aggregates one must be similar to a part of 

 the other, and that it is not possible for each to be similar to 

 a part of the other. 



3. The addition and multiplication of cardinals are best 

 defined in the mariner explained in Russell's l Principles of 

 Mathematics,' ch. xii. The sum of any aggregate of cardinal 

 numbers is the cardinal number of the logical sum of an 

 aggregate of aggregates, of which no two have any common 

 element, and which have those cardinal numbers. 



The multiplicative class of an aggregate of aggregates, no 

 two of which have any common element, is the class each of 

 whose terms is an aggregate formed by taking one and only 

 one element from each of those aggregates. 



I provisionally assumef as an axiom that the multiplier - 

 tive class of an aggregate of aggregates is itself an aggregate. 

 The product of an aggregate of cardinal numbers is the 

 cardinal number of the multiplicative class of an aggregate 

 or aggregates which have those cardinal numbers. 



The expression d^ is defined as the product of an aggregate 

 of cardinal number D of cardinal numbers each equal to H. 



From the above axiom it follows that the class of sub-classes 

 of any aggregate is itself an aggregate, for in forming a 

 sub-class we have as regards each element two alternatives 

 (to include it, or to exclude it) , therefore if 91 be the cardinal 

 of the aggregate, then (including none and all as sub-classes) 

 the cardinal of the aggregate of subclasses is 2*, and it can 

 be shown that its cardinal number is greater than that of the 

 original aggregate. For if we suppose the sub-classes to be 

 correlated one to one with the terms of the aggregate, it can 

 be proved that there is at least one sub-class omitted from the 

 correlation, namely, that which is formed by rejecting or 

 including each element of the aggregate according as it 

 occurs or does not occur in the sub-class correlated with it. 



* I. e. a part which is not the whole. 



\ The necessity for assuming some axiom arises from the informal 

 character of our definition of the term aggregate, vide Note B. 



