86 MYSTICISM AND LOGIC 



of another, the one which is a part has fewer terms than 

 the one of which it is a part. This maxim is true of finite 

 numbers. For example, Englishmen are only some among 

 Europeans, and there are fewer Englishmen than Euro- 

 peans. But when we come to infinite numbers, this is no 

 longer true. This breakdown of the maxim gives us the 

 precise definition of infinity. A collection of terms is 

 infinite when it contains as parts other collections which 

 have just as many terms as it has. If you can take away 

 some of the terms of a collection, without diminishing 

 the number of terms, then there are an infinite number 

 of terms in the collection. For example, there are just 

 as many even numbers as there are numbers altogether, 

 since every number can be doubled. This may be seen 

 by putting odd and even numbers together in one row, 

 and even numbers alone in a row below : 



1, 2, 3, 4, 5, ad infinitum. 



2, 4, 6, 8, 10, ad infinitum. 



There are obviously just as many numbers in the row 

 below as in the row above, because there is one below for 

 each one above. This property, which was formerly 

 thought to be a contradiction, is now transformed into a 

 harmless definition of infinity, and shows, in the above 

 case, that the number of finite numbers is infinite. 



But the uninitiated may wonder how it is possible to 

 deal with a number which cannot be counted. It is im- 

 possible to count up all the numbers, one by one, because, 

 however many we may count, there are always more to 

 follow. The fact is that counting is a very vulgar and 

 elementary way of finding out how many terms there 

 are in a collection. And in any case, counting gives us 

 what mathematicians call the ordinal number of our 

 terms ; that is to say, it arranges our terms in an order or 



