THE POSITIVE THEORY OF INFINITY 199 



of this proposition have been published by many writers, 

 including myself, but up to the present no valid proof 

 has been discovered. The infinite numbers actually 

 known, however, are all reflexive as well as non-induc- 

 tive ; thus, in mathematical practice, if not in theory, 

 the two properties are always associated. For our pur- 

 poses, therefore, it will be convenient to ignore the bare 

 possibility that there may be non-inductive non-reflexive 

 numbers, since all known numbers are either inductive 

 or reflexive. 



When infinite numbers are first introduced to people, 

 they are apt to refuse the name of numbers to them, 

 because their behaviour is so different from that of finite 

 numbers that it seems a wilful misuse of terms to call 

 them numbers at all. In order to meet this feeling, we 

 must now turn to the logical basis of arithmetic, and 

 consider the logical definition of numbers. 



The logical definition of numbers, though it seems an 

 essential support to the theory of infinite numbers, was 

 in fact discovered independently and by a different man. 

 The theory of infinite numbers that is to say, the arith- 

 metical as opposed to the logical part of the theory was 

 discovered by Georg Cantor, and published by him in 

 1 8 82-3. * The definition of number was discovered 

 about the same time by a man whose great genius has 

 not received the recognition it deserves I mean Gottlob 

 Frege of Jena. His first work, Begriffsschrifl, published 

 in 1879, contained the very important theory of hereditary 

 properties in a series to which I alluded in connection 

 with inductiveness. His definition of number is con- 

 tained in his second work, published in 1884, and entitled 

 Die Grundlagen der Arithmetik, eine logisch-mathematische 



1 In his Grundlagen einer allgemeinen Mannichfaltigkeitslehre and in. 

 articles in Acta Mathematical vol. ii. 



