THE PROBLEM OF INFINITY 181 



one little step would plunge us into the infinite. If it is 

 assumed that the first infinite number is reached by a 

 succession of small steps, it is easy to show that it is self- 

 contradictory. The first infinite number is, in fact, 

 beyond the whole unending series of finite numbers. 

 " But," it will be said, " there cannot be anything beyond 

 the whole of an unending series." This, we may point 

 out, is the very principle upon which Zeno relies in the 

 arguments of the race-course and the Achilles. Take the 

 race-course : there is the moment when the runner still 

 has half his distance to run, then the moment when he 

 still has a quarter, then when he still has an eighth, and 

 so on in a strictly unending series. Beyond the whole 

 of this series is the moment when he reaches the goal. 

 Thus there certainly can be something beyond the whole 

 of an unending series. But it remains to show that this 

 fact is only what might have been expected. 



The difficulty, like most of the vaguer difficulties 

 besetting the mathematical infinite, is derived, 1 think, 

 from the more or less unconscious operation of the idea 

 of counting. If you set to work to count the terms in an 

 infinite collection, you will never have completed your 

 task. Thus, in the case of the runner, if half, three- 

 quarters, seven-eighths, and so on of the course were 

 marked, and the runner was not allowed to pass any of 

 the marks until the umpire said " Now," then Zeno's 

 conclusion would be true in practice, and he would never 

 reach the goal. 



But it is not essential to the existence of a collection, 

 or even to knowledge and reasoning concerning it, that we 

 should be able to pass its terms in review one by one. 

 This may be seen in the case of finite collections ; we can 

 speak of " mankind " or " the human race," though many 

 of the individuals in this collection are not personally 



