54 Mr. Jourdain on Transfinite Cardinal 



The series of all ordinal numbers may, it seems to me, 

 properly be called an " absolutely " infinite series. For, if a 

 well-ordered series has a type, it is, in a certain sense. 

 completed ; while the above series W cannot, as is shown by 

 Burali-Forti's contradiction, have a type. 



This seems to be the most promising way of regarding 

 Burali-Forti's contradiction, and the words " absolutely 

 infinite " seem preferable to the equivalent word " incon- 

 sistent/'' which I, in common with Cantor, have used 

 hitherto ; because an u inconsistent " aggregate is not itself 

 contradictory (it exists, in the mathematical sense of the 

 word), but a cardinal number or type of it does not exist. 

 HoweA r er, I shall, in the next section, enter briefly into the 

 history of the use of this word in the theory of aggregates. 



9. 



The conception and name of an u inconsistent " aggregate 

 originated with Cantor"*, but the only published reference 

 to them occurs in two papers by Hilbert f. 



With regard to Hilbert' s statements, it does not seem to 

 follow that if the axioms of arithmetic (which are, according 

 to Hilbert, the laws of operation with real numbers and the 

 axiom of continuity) do not contradict one another, then 

 the real number-continuum is u consistent/'' For it does 

 not appear to be doubtful that the laws of operation witJi 

 ordinal numbers or Alephs form a system free from contra- 

 diction, and yet the aggregate of all ordinal number- or 

 Alephs is "inconsistent." 



Further, Hilbert states that a "similar'' method to that 

 pursued by him for the axioms of real numbers, when 

 applied to all Alephs, fails, so the totality of all Alephs is an 

 b ' inconsistent" aggregate (a mathematically non-existent 



* In a letter to me of January 6th, 1901, Professor Cantor said : — 

 u Ich unterscheide auf's strengste zwisclien unendlichen Mengen (con- 

 sistenten Vielheiten) einerseits imd den ihnen zukommenden abstracttu 

 unendlichen Zahlen andrerseits." There was no further explanation of 

 the term " consistency," and I confused it with Schroder's requirement 

 in the conception of a "common manifold" Q Yorlesuns'en iiber die 

 Algebra der Logik (exakte Logik),' Bd. i. ]890, pp. 147-148), On 

 finding that the aggregate of all ordinal numbers had no cardinal number, 

 I applied the name "inconsistent" used by Schroder (' Algebra und 

 Logik der Relative.' 1895, p. 4) to this a^qregate (Phil. Mag. Jan. 1904. 

 p. 67). 



t "Ueber den ZahlbegrifF," Jahresber. d. d. M.-V. Bd. viii. (1900) 

 pp. 180-184: " Matheinatische Probleme," Gott. XacJu: 1900, pp. 253- 

 297, see especially pp. 264-260. 



