88 MYSTICISM AND LOGIC 



less. This method can be applied generally. If there is 

 some relation which, like marriage, connects the things 

 in one collection each with one of the things in another 

 collection, and vice versa, then the two collections have 

 the same number of terms. This was the way in which 

 we found that there are as many even numbers as there 

 are numbers. Every number can be doubled, and every 

 even number can be halved, and each process gives just 

 one number corresponding to the one that is doubled or 

 halved. And in this way we can find any number of 

 collections each of which has just as many terms as there 

 are finite numbers. If every term of a collection can be 

 hooked on to a number, and all the finite numbers are 

 used once, and only once, in the process, then our 

 collection must have just as many terms as there are 

 finite numbers. This is the general method by which the 

 numbers of infinite collections are defined. 



But it must not be supposed that all infinite numbers 

 are equal. On the contrary, there are infinitely more 

 infinite numbers than finite ones. There are more ways 

 of arranging the finite numbers in different types of 

 series than there are finite numbers. There are probably 

 more points in space and more moments in time than 

 there are finite numbers. There are exactly as many 

 fractions as whole numbers, although there are an infinite 

 number of fractions between any two whole numbers. 

 But there are more irrational numbers than there are 

 whole numbers or fractions. There are probably exactly 

 as many points in space as there are irrational numbers, 

 and exactly as many points on a line a millionth of an 

 inch long as in the whole of infinite space. There is a 

 greatest of all infinite numbers, which is the number of 

 things altogether, of every sort and kind. It is obvious 

 that there cannot be a greater number than this, because, 



