i 9 o SCIENTIFIC METHOD IN PHILOSOPHY 



There are two respects in which the infinite numbers 

 that are known differ from finite numbers : first, infinite 

 numbers have, while finite numbers have not, a property 

 which I shall call reflexiveness ; secondly, finite numbers 

 have, while infinite numbers have not, a property which 

 I shall call inductiveness. Let us consider these two 

 properties successively. 



(i) Reflexiveness. A number is said to be reflexive 

 when it is not increased by adding i to it. It follows 

 at once that any finite number can be added to a reflexive 

 number without increasing it. This property of infinite 

 numbers was always thought, until recently, to be self- 

 contradictory ; but through the work of Georg Cantor 

 it has come to be recognised that, though at first 

 astonishing, it is no more self-contradictory than the 

 fact that people at the antipodes do not tumble ofF. 

 In virtue of this property, given any infinite collection 

 of objects, any finite number of objects can be added 

 or taken away without increasing or diminishing the 

 number of the collection. Even an infinite number of 

 objects may, under certain conditions, be added or taken 

 away without altering the number. This may be made 

 clearer by the help of some examples. 



Imagine ail the natural numbers o, i, 2, 3, ... to be 

 written down in a row, and immediately beneath them 



1, 2, 3, 4, . . . n-\- 1 . . . 

 write down the numbers 1, 2, 3, 4, . . ., so that 1 is 

 under o, 2 is under 1, and so on. Then every number 

 in the top row has a number directly under it in 

 the bottom row, and no number occurs twice in either 

 row. It follows that the number of numbers in the 

 two rows must be the same. But all the numbers 

 that occur in the bottom row also occur in the top 



