Transfinite Numbers. 459 



In precisely the same way, assuming the proposition 

 ^ = tf y to be true for all values of y < k, we can prove 



that the aggregate of couples of ordinals less than w K forms a 

 series of type co K when the couples are arranged according to 

 the order of magnitude of their greater components, couples 

 which have the same greater component being arranged inter se 

 according to the order of magnitude of their lesser components. 



Note B. 



The definition given in this article of the distinction be- 

 tween limited and unlimited classes must be regarded as 

 provisional. It is not a formal definition of the kind which 

 we require for mathematical purposes. With this definition 

 we cannot formally prove ab initio that any class containing a 

 multiplicity of terms is a limited class ; it is only by an appeal 

 to intuition that wo can justify the statement that finite and 

 enumerable classes are limited. We can prove that any 

 transfinite cardinals which exist must form a well-ordered 

 class ; but we cannot determine how far the class extends 

 without the aid of some assumption. 



The formal definition proposed by Mr. Jourdain (Phil. 

 Mag. Jan 1904, p. 67) cannot be accepted as satisfactory, 

 because we cannot define the class of ordinals (or any class 

 ordinally similar thereto) in a manner free from contradiction 

 without first making the very distinction which we seek to 

 define. Before we can determine the proper mode of formally 

 defining the distinction in question, it is necessary to investi- 

 gate the subject from the standpoint of the informal definition^ 

 by assuming as the basis of formal reasoning general principles 

 suggested by intuition. This is what I have endeavoured to 

 do in the present paper. I may remark that when this paper 

 was written I had not succeeded in framing any suitable 

 formal definition. In a paper of later date *, I have discussed 

 the logical relations between the various axioms which we 

 may assume, and have arrived at a formal definition which 

 appears to me to be satisfactory and which obviates the 

 necessity for assuming any axiom. I need only remark here 

 that the axiom regarding the multiplicative class which I 

 have assumed in the present paper, includes a disputable pro- 

 position (viz., that ftfcis a cardinal number when b is transfinite) 

 which it is not necessary for our purpose to assume. 

 As the axiom is only used to prove that there is no greatest 

 cardinal number, any other axiom from which that can be 



* Despatched to the Editors of the Phil. Mag. in March before this 

 paper came back to me for revision. 



