THE POSITIVE THEORY OF INFINITY 195 



greater and less can be perfectly well applied to them. 

 The whole of Simplicius's difficulty comes, as is evident, 

 from his belief that, if greater and less can be applied, a 

 part of an infinite collection must have fewer terms than 

 the whole ; and when this is denied, all contradictions 

 disappear. As regards greater and less lengths of lines, 

 which is the problem from which the above discussion 

 starts, that involves a meaning of greater and less which is 

 not arithmetical. The number of points is the same in 

 a long line and in a short one, being in fact the same as 

 the number of points in all space. The greater and less 

 of metrical geometry involves the new metrical concep- 

 tion of congruence, which cannot be developed out of 

 arithmetical considerations alone. But this question has 

 not the fundamental importance which belongs to the 

 arithmetical theory of infinity. 



(2) Non-inductiveness. The second property by which 

 infinite numbers are distinguished from finite numbers 

 is the property of non-inductiveness. This will be best 

 explained by defining the positive property of inductive- 

 ness which characterises the finite numbers, and which 

 is named after the method of proof known as " mathe- 

 matical induction." 



Let us first consider what is meant by calling a property 

 " hereditary ' in a given series. Take such a property 

 as being named Jones. If a man is named Jones, so is 

 his son ; we will therefore call the property of being 

 called Jones hereditary with respect to the relation of 

 father and son. If a man is called Jones, all his de- 

 scendants in the direct male line are called Jones ; this 

 follows from the fact that the property is hereditary. 

 Now, instead of the relation of father and son, consider 

 the relation of a finite number to its immediate successor, 

 that is, the relation which holds between o and 1, between 



