456 Mr. A. E. Harward on the 



employed is that from an aggregate M o£ cardinal number H 

 we can take away in succession aggregates 



each of cardinal number D so as to leave at each stage a 

 remainder of cardinal number 9L Since when we have 

 performed v operations we can always perform a (v+l)th, 

 it necessarily follows that the series of operations which we 

 can perforin is a series of type not less than co. So far the 

 argument is sound, but Mr. Jourdain goes on to infer that 

 some or all of the terms of M can be arranged in a progression 

 of agaregates P. It is this step which appears to me to be 

 unsound. We have only proved that we can go on performing 

 a certain endless process *, and we are not entitled to assume 

 that this endless process is equivalent to the completed 

 operation whereby some or all of the terms of M are 

 arranged in a progression of aggregates each of cardinal 

 number 0. 



If the arrangement used by Mr. Jourdain be sound there 

 is no reason why we should stop at the conclusion 



a=a+tf a; 



making use of that result we can go on taking from M 

 aggregates 



each of cardinal number 6, where j3 may bejany ordinal of the 

 second class, at each stage we can leave a remainder of 

 cardinal number ft, so we can always continue the process. 

 The series of operations which we can perform is therefore a 

 series of type not less than a>i. So far the argument is 

 sound and the conclusion true f. If we infer from this that 

 we can arrange some or all of the elements of M in a series 

 of type coi of aggregates each of cardinal number 0, we can 

 deduce the conclusion that E = H + tti6, and from this point 

 we can go on and prove in the same way that ft^H + ^u? 

 and so on indefinitely. These results are obviously untrue if 



* A finite series of operations which can he successively performed is 

 itself an operation, hnt an infinite series of operations is not an operation 

 hut an endless process, and hefore we treat such a process as equivalent 

 to a completed operation we must justify our procedure. 



f The series of operations which we can perform is of type a> v even in 

 the case where a = kS % . In this case the series of operations is of course 

 merely an endless process, and is not equivalent to any completed 

 operation. 



