381 Dr. Dorothy Wrinch and Dr. H. Jeffreys on Certain 



Any set of finite numbers from this sequence has an 

 immediate successor. If the set is a finite one, the suc- 

 cessor is the finite number that follows the last of the set. 

 If it is an infinite one, the immediate successor is co ; for 

 in the first place co succeeds all terms of the set, and in the 

 second place any term before co must be finite and therefore 

 can only have a finite number of terms before it. Hence 

 co has the properties of being after all members of the set, 

 and of being the first ordinal that follows all members of 

 the set. These are the distinguishing features of the 

 immediate successor of a set. 



It is not always clear, even to physicists, that non- 

 well-ordered sequences exist. The whole of the modern 

 theory of functions is concerned only with well-ordered 

 series, and indeed with the simplest possible kind of 

 infinite ordinal type, namely co itself. The series of 

 rational proper fractions in order of magnitude offers the 

 simplest example of a non-well-ordered series. If we take 

 from this the set J, f , J, j, . . . , we easily see that this set 

 has an immediate successor in the series, namely \. On 

 the other hand, the set consisting of all terms Jess than 

 or equal to J has no immediate successor, for between any 

 successor and -J there is another successor of the set, so 

 that the suggested successor can always be shown not to 

 be immediate. This series can, however, be re-arranged so 

 as to be well-ordered, as follows : 



i i j, .1 2 J 1 2 3 (C\ 



1) 25 3> 4? 35 5' 6' .^' 4) V V J 



In this we have written down first those terms whose 

 denominators and numerators added together come to 2, 

 then to 3, and so on, putting first in each case those with 

 the smaller numerator. The sequence of all numbers, 

 rational and irrational, resembles the last in being not well- 

 ordered when arranged in order of magnitude ; but in this 

 case it is not known to be possible, and suspected to be 

 impossible, to rearrange it so that it will be well-ordered. 



We have suggested that the functional relations that 

 occur in physics can be arranged in a well-ordered series. 

 If we make the further assumption indicated by our previous 

 principles, that any of the relations we are in search of 

 can be reached in a finite number of steps, the number 

 or admissible functions cannot exceed the number of finite 

 integers, namely tf - This at once imposes a severe re- 

 striction on the classes of functions occurring in physics. 



