THE POSITIVE THEORY OF INFINITY 191 



row, and one more, namely o ; thus the number 

 of terms in the top row is obtained by adding one to 

 the number of the bottom row. So long, therefore, 

 as it was supposed that a number must be increased by 

 adding 1 to it, this state of things constituted a con- 

 tradiction, and led to the denial that there are infinite 

 numbers. 



The following example is even more surprising. 

 Write the natural numbers 1, 2, 3, 4, ... in the top 

 row, and the even numbers 2, 4, 6, 8, . . . in the 

 bottom row, so that under each number in the top row 

 stands its double in the bottom row. Then, as before, 

 the number of numbers in the two rows is the same, yet 

 the second row results from taking away all the odd 

 numbers an infinite collection from the top row. 

 This example is given by Leibniz to prove that there 

 can be no infinite numbers. He believed in infinite 

 collections, but, since he thought that a number must 

 always be increased when it is added to and diminished 

 when it is subtracted from, he maintained that infinite 

 collections do not have numbers. " The number of 

 all numbers," he says, " implies a contradiction, which 

 I show thus : To any number there is a corresponding 

 number equal to its double. Therefore the number 

 of all numbers is not greater than the number of 

 even numbers, i.e. the whole is not greater than its 

 part." l In dealing with this argument, we ought 

 to substitute " the number of all finite numbers ' 

 for "the number of all numbers"; we then obtain 

 exactly the illustration given by our two rows, one 

 containing all the finite numbers, the other only the even 

 finite numbers. It will be seen that Leibniz regards it as 

 self-contradictory to maintain that the whole is not 



1 Phil. Werke, Gerhardt's edition, vol. i. p. 338. 



