Numbers of the Eicponential Form. 51 



can easily convince ourselves, apply the extended method of 

 induction, provided that exponentiation with this fcransfinite 

 number leave- some Aleph unaltered. 



This extended principle of induction is very closely con- 

 nected (through Cantor's "third principle of generation ") 

 with the question a< to whether the ordinal number of the 

 series of all the ordinal numbers defined by Cantor can he 

 defined without contradiction, and hence with the argument 

 of Burali-Forti *. I have returned, then, in the following 

 section, to the considerations which I have advanced in the 

 January (1904) number of this Magazine. 



In defining an aggregate which should serve as a criterion 

 whether any given aggregate is ''consistent" or "incon- 

 sistent "+, I haveused the conception, mentioned by SehonmVs. 

 of the (well-ordered) series ftftj J such that every well-ordered 

 series is similar to it or to a segment of it. 



This series (TJl was, now, stated by Schonfiies § to be 

 similar to the series (W) of all the ordinal numbers, as 

 defined by Cantor by the help of his three generating 

 principles ||. 



This statement appears to me to be incorrect , in fact, I 

 shall now show that we must agree to regard the series of 

 these " Cantor's ordinal numbers " as similar to a segment 



* Bid. p. 64. 



f Ibid. p. 67, Hue 18. The wording in the definition of W is to be 

 replaced by the slightly different wording given above. 



\ We consider in the criterion the aggregate which is the field of the 

 ^enerating-relation of the series £2U. 



§ " Die Entwickelung ...."' p. 41. 



|| The purpose of the third principle of generation is sometimes 

 misunderstood. For example, in the in some respects excellent fourth 

 Xote ('" Sur la theorie des ensembles et des nombres infinis ") on pp. 617 

 655 of Couturafs book " De l'infini mathematique," Paris, 1896 (see 

 egp. pp. 639-642), the object of this principle is taken to be to enable 

 one to surpass the second number-class, just as the second principle has 

 enabled one to surpass the first. This view seems to agree with that 

 of Schonfiies [op. cit. p. 48: cf. Phil. Mag. March 1904, p. 300); but 

 rather further on. a different, and self-contradictory, view of this object 

 i< taken. The third principle shows, namely, the occasion for using the 

 second principle to create a new number after (ill those generated by 

 ill.- application of the first two principles to a fundamental number 

 o). Q. . ■ • 



The true view was clearly stated by Cantor in his 'Grundlagen.' The 

 first two principle- create an infinite Beries of ordinal numbers, while 

 the third principle' enable- us to separate out various number-classes in 

 this series {cf. J§7, 8 



E •> 



