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On a proof that every aggregate can be well-ordered

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Literatur

  1. „Beweis, daß jede Menge wohlgeordnet werden kann”, Math. Ann. Bd. 59 (1904), pp. 514–516 [dated Sept. 24th, 1904].

  2. ‘On the transfinite Cardinal Numbers of Well-ordered Aggregates’, Phil. Mag., Jan. 1904, pp. 61–75 [dated Dec. 2nd, 1903]; ‘On the Transfinite Cardinal Numbers of Number-Classes in General’, ibid., March 1904, pp. 294–303 [dated Dec. 23nd, 1903]; ‘On Transfinite Cardinal Numbers of the Exponential Form’, ibid., Jan. 1905, pp. 42–56 [dated Sept. 6th, 1904].

  3. See Phil. Mag., Jan. 1904, pp. 74–75, and March 1904, pp. 295–300. Cf. the second note on p. 50 of the Phil. Mag. for Jan. 1905.

  4. See Phil. Mag., Jan. 1904, pp. 65–66.

  5. Ibid., See Phil. Mag., Jan. 1904, p. 64.

  6. Thelatter contradiction is the one given by me (ibid., Phil. Mag., Jan. 1904, pp. 64, 66); theformer was used by Cantor in a letter of Nov. 4th, 1903 (cf. ibid., Phil. Mag., Jan. 1904, p. 70, note).

  7. The following considerations appeared first in the Phil. Mag. for Jan. 1905 (see pp. 51–53).

  8. In this notation, ωγ is the first ordinal number of the (γ+1) th number class (see Phil. Mag., March 1904, p. 295, note).

  9. As in maintained by Russell (see his ‘Principles of Mathematics’, vol. 1, Cambridge 1903, pp. 111–116, 130–132, 242).

  10. See Phil. Mag., Jan. 1904, pp. 72–74, and March 1904, p. 301.

  11. Cf. Cantor, Math. Ann. Bd. 20 (1882), p. 115. If the cardinal number of a classu is, as Russell maintains (op. cit., p. 115), the class of all these classes which are equivalent tou, the number in question appears to be well-defined as soon asu is. It seems that aggregates likeW exist as many, but notas one (cf Russell, op. cit., pp. 104–105); so that if, and only if,u is a class as one, the cardinal number ofu exists, and the latter class can only be said to exist as many. This view seems to agree with that (held by Cantor) in Phil. Mag., Jan. 1904, p. 67.

  12. Cf. Russell, op. cit. Math. Ann. Bd. 20 (1882), p. 105.

  13. Cf. Phil. Mag., Jan. 1905, p. 54.

  14. Cf. Zermelo, loc. cit., pp. 515–516, paragraph V.

  15. Our very limited resources foractually building up such series (or the accompanying coverings) are examined in the Phil. Mag. for January 1905, pp. 42, 43–46.

  16. The apparently opposite statement in the Phil. Mag., Jan. 1904, p. 67, is due to the fact that I there explicityassumed that the continuum is a ‘consistent’ aggregate.

  17. Cf. Russell, op. cit. Math. Ann. Bd. 20 (1882), p. 105.

  18. „On the General Theory of Functions”, Journ. für Math., Bd. 128 (1905), pp. 169–210 (see esp. p. 171); Phil. Mag., Jan. 1905, pp. 45–46.

  19. „Über die Reihe der transfiniten Ordnungszahlen”, these Annalen Bd. 60 (1905), pp. 187–193.

  20. Loc. cit. „Über die Reihe der transfiniten Ordnungszahlen”, these Annalen Bd. 60 (1905), p. 188.

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  1. Broadwindsor, England

    Philip E. B. Jourdain

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Jourdain, P.E.B. On a proof that every aggregate can be well-ordered. Math. Ann. 60, 465–470 (1905). https://doi.org/10.1007/BF01561091

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