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A proof that every aggregate can be well-ordered

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Acta Mathematica

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Literatur

  1. Math. Ann., Vol. XXI, p. 550; orGrundlagen einer allgemeinen Mannigfaltisgkeitslehre, Leipzig 1883, p. 6. Cf. Cantor’sContributions to the Founding of the Theory of Transfinite numbers, Chicago and London, 1915, pp. 62, 62, 66.

  2. Cf. a remark due toE. Zermelo inMath. Ann., Vol. LXV, 1908, p. 125.

  3. For example,Hardy explicitly used it in the paper to be mentioned below. In my paper of 1904 also mentioned below I first relied on Hardy’s result, but afterwards (Math. Ann., Vol. LX, 1905, p. 68) made use explicity of the notion of an infinity of arbitrary selections (cf.Rev. de Math., Vol. VIII, 1906, p. 9, note I).

  4. Borel (Math. Ann., Vol. LX, 1905, p. 195, andLeçons sur la théorie des fonctions, Second ed., Paris, 1914, pp. 135–181). Any reasons Borel may have had for his rejection of a series of arbitrary choices are not given, and it seems that he passed over an important logical point involved, since he admitted any enumerable infinity of choices and rejected a non-enumerable infinity of choices (cf.Hobson,The Theory of Functions of a Real Variable and the Theory of Fourier’s Series, Cambridge, 1907, p. 210, note; cf. pp. 196–197). Borel made use of an enumerable infinity of choices in the aboveLeçons, for example, pp. 12–13.

  5. H. Weber,Jahresber. der D. M. V., Vol. II, 1891–2, pp. 20;Maths. Ann., Vol. XLIII, 1893, p. 15. See alsoHobson,op cit., The Theory of Functions of a Real Variable and the Theory of Fourier’s Series, Cambridge, 1907, pp. 196–197. Cf.Schoenflies,Encycl. der. math. Wiss., Vol. I, Part I, p. 188,Die Entwicklung der Lehre von den Punktmannigfaltigkeiten, Leipzig, 1900, p. 5, andEntwicklung der Mengenlehre und ihrer Anwendungen, Leipzig and Berlin, 1913, pp. 6–7.

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  6. Math. Ann., Vol. XLVI, 1895, p. 493;Contributions, p. 105. On p. 205 of theContributions, I wrongly assumed that Cantor, likeRussell (The Principles of Mathematics, Cambridge, 1903, pp. 122–123) selected eacht arbitrarily.

  7. Math. Ann., Vol. XLVI, 1895, p. 485;Contributions, p. 92.

  8. Math. Ann., Vol. XLVI, 1895, p. 487;Contributions, p. 95.Schoenflies drew attention to the fact, which must have been the one that ledCantor to his definition, that multiplication could be defined for an infinity of cardinal numbers. The idea was worked out in the symbols of «mathematical logic» byA. N. Whitehead (Amer. Journ. of Math., Vol. XXIV, 1902, pp. 383–385), who did not however mention that the essential idea is due toSchoenflies; although elsewhere (ibid. Amer. Journ. of Math., p. 367) he mentioned Schoenflies’s book. The fact of an axiom being required here and in many other cases was certainly not noticed by either Whitehead or Russell before 1903. (cf.ibid., Amer. Journ. of Math., pp. 368, 380;Russell,op cit., The Principles of Mathematics, Cambridge, 1903, pp. 122–123) selected eacht arbitrarily pp. 122–123; and my paper inQuart. Journ. Math., 1907, p. 364), and was not pointed out by them in print until after Zermelo’s discovery was generally known. In many cases it was Zermelo or others who also were not «mathematical logicians» who first pointed out that the principle of arbitrary selection is tacily used in much mathematical reasoning (cf.Math. Ann., Vol. LIX, 1904, p. 516; Vol. LXV, 1908, pp. 113–115). From this and from the historical remark in my above-cited paper (pp. 360–366), it must, I think, be concluded that «mathematical logic» has not been of help in perceiving the logical difficulties that beset an infinite series of arbitrary choices. It has not been of any help in solving these difficulties.

  9. Cf.Contributions, p. 205. The first to publish a remark that the principle of selection was used in this place seems to have been Zermelo (Math. Ann., Vol. LXV, 1908, p. 114, fifth paragraph).

  10. Thus, in letters to me he expressed approval of the method of Hardy (1903) referred to below, and the discovery (1904) ofJulius König on the infinite products of certain cardinal numbers — which depends on the legitimacy of making an infinity of arbitrary selections.

  11. Cantor communicated this proof to me on November 4, 1903 because I had previously communicated to him (October 29, 1903) an almost identica proof which I had independently discovered.

  12. In his letter just mentioned, Cantor wrote: «Nimmt man nun irgend eine unendliche VielheitV und setzt voraus, dass ihrkein Aleph als Cardinalzahl zukommt, so betrachte ich es mit Ihnen als einclucb tend, dass in diesesV das SystemW hineinprojicirt gedacht werden kann; ...»

  13. References to this and some other papers to be dealt with below are given in my above-cited paper of 1907, pp. 363–365.

  14. Cf.Hobson,op. cit. The Theory of Functions of a Real Variable and the Theory of Fourier’s Series, Cambridge, 1907, pp. 191–194, 207–208, 210–211.

  15. Phil. Mag., January, 1904, Series 6, Vol. VII, pp. 61–75.

  16. Various aspects of this difference have been recognised by Hobson (Proc. Lond. Math. Soc. (2), Vol. III, 1905, pp. 171, 184–185, andop. cit., Hobson (Proc. Lond. Math. Soc. (2), Vol. III, 1905, pp. 195, 208–210) and Russell (Proc. Lond. Math. Soc. (2), Vol. IV, 1906, p. 29). However it appears from § X that the two difficulties cannot be separated so much as Russell thought, while Russellop. cit. Proc. Lond. Math. Soc. (2) Vol. IV, 1906, pp. 34–35, 43–44) failed to grasp that then my theory was that there is a class of ordinal numbers, but the series of all ordinal numbers has no type and no associated cardinal number (cf. my remark in ibid., p,. 282). In consequence of § IX below, it is necessary to admit that there is no such thing as a class of all ordinal numbers; and another point which my theory has led to be modified is due to the fact that there is a mistake inibid. pp. 271–272. It seems impossibles to avoid the theory that there are ordinal numbers beyond those indicated by Cantor, and from which. This theory has the avantage over the theory (held since 1905) of Russell, that it includes much more of the theory of the transfinite; while Russell’s very limited theory does not exlude false appearances of classes at all more effectively than my present theory.

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  17. Leçons sur l’intégration et la recherche des fonctions primitives, Paris, 1904, pp. 104–105; cf.Hardy,Cource of Pure Mathematics, Second ed., Cambridge, 1914, pp. 186–188; andSchoenflies,op. cit. Encycl. der math. Wiss, Vol. I, Parts I, 1900, pp. 51–52, Part II, Leipzig, 1908, pp. 76–80, andop. cit. Engycl. der math. Wiss. Vol. I, Part I, 1913, pp. 234–252.

  18. «Über das Problem der Wohlordnung»,Math. Ann., Vol. LXXVI, 1915, pp. 438–443.

  19. Cf. my paper of 1907 cited above, p. 366.

  20. It must be mentioned that, as is shown below in § X, this result depends on an axiom formulated by Zermelo in 1908, which is other than the principle of selection and which can be proved by using — &, it seems, only by using — the principle of selection or my rule given below.

  21. We may define a «chain», in a way which is, perhaps, preferable from a logical point of view, as follows. An «M-chain» is a class of couples(m, a), wherem is a member ofM anda is an ordinal number, and the couples are such that in each chain nom ora occurs more than once, and, ifa occurs, all ordinals less thana occur also. We will suppose that this chain is well-ordered by arranging the couples in the order of magnitude of the righ-hand members (a). We say that a chain «exhausts»M if the class of left-hand members (m) of the couples of the chain consists of all the members ofM. This new definition of the word «exhausts» obviously conforms closely to the usual sense if an «M-chain» is, as in the text, a part ofM. It may also be mentioned that, in the sense of this note, an «M-chain» is a one-valuedfunction where the argument consits of ordinal numbers. Such functions are considered byOswald Veblen (“Continuous Increasing Functions of Finite and Transfinite Ordinals’,Trans. Amer. Math. Soc., Vol. IX, 1908, pp. 280–292).

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  22. Se § I above.

  23. Principia Mathematica, Vol. II, Cambridge 1912, pp. 3, 187–190, 207–210, 278–288. Cf. § VI below.

  24. In this paper, is the first number of the (2+) theory of «number-classes» of Cantor.

  25. The problem indicated here is completely solved in § XIV below.

  26. This is merely the way of stating the axioms referred to in the test below, which exclude such aggregates asW.

  27. Math. Ann., Vol. LXXVI, 1915, pp. 438–440.

  28. Ibid. Math. Ann., Vol. LXXVI, 1915, p. 421.

  29. It might be as well to distinguish this property as «having being» from «existing» in the sense of having at least one member. Thus, the null-class will not «exist» but will «have being».

  30. Zermelo has concluded in this way in his paper «Sur les ensembles finis et le principe de l’induction complète»,Acta Math., Vol. XXXII, 1909, pp. 185–193.

  31. ThisK, it must be noted,also defines theK-class of which it is the member with the greatest type. Thus, if the type (ordinal number) ofK has no immediate predecessor, we must be careful to specify which one is meant of the twoK-classes which correspond toK.

  32. Of course the members of aK-class have no intrinsic order of their own.

  33. Note that this condition is not fulfilled for a series of type of chains of an enumerable aggregate.

  34. If γ istransfinite and has an immediate predecessor, it may be that there are chains of type γ−1 that exhaustM, and thus cannot be continued by anyM-chain. See the end of next section.

  35. Cf. §§ V and VI above.

  36. In fact, ifM-chains other thanK could continue as segment all the members ofk, K would not exhaustM.

  37. And also, rather more simply, by the argument of §

  38. Phil. Mag., January, 1904. See § I.

  39. Proc. Lond. Math. Soc. (2), Vol. IV, 1906, p. 29.

  40. It is important that Cantor seems to have been conscious that he assumed as axiomatic the principle of selection. I did not recognise that I had made any assumption until long afterwards (cf.Math. Ann., Vol. LX, 1905, p. 68).

  41. Math. Ann., Vol. LXV, 1908, p. 261.

  42. Cf.Journ. für Math., Vol. CXXXV, 1909, pp. 86–90;Math. Ann., Vol. LXXVI, 1915, pp. 438–439.

  43. This is the type of that chain (L) which may be said to «limit»M (cf. § VII), and was founded byHartogs (Math. Ann., Vol. LXXVI, p. 4–40) unnecessarily on a non-logical axiom.

  44. Russell andWhitehead, basing their attitudes on their theory of «logical types», hold that there is no reason to think that there is a series of type ωω. But, on the one hand, the extent to which Cantor’s ordinal numbers are preserved in this theory has been stated differently at different times and is not yet fixed (cf.Russell,Proc. Lond. Math. Soc. (2), Vol. IV, 1906, p. 46;Rev. de Metaphys., Vol. XIV, 1906, p. 639;Amer. Journ. Math., Vol. XXX, 1908, pp. 258, 261;Whitehead andRussell,Principia Mathematica., Vol. II, Cambridge, 1912, pp. 189–190; Vol. III, Cambridge, 1913, pp. 170, 173, and ond the other hand, it is not quite evident that there is not a ω th logical type in some sense analogous to that in which the number whichCantor denoted by ≫ωω≫ immediately follows those ordinal numbers obtained by exponentiating ω with ν, but is not ω exponentiated by ω.

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  45. I have brought forward this point of view inScience Progress, Vol. XIII, 1918.

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The undersigned does not accept the principal wiew on which is based the above paper of the regretted, highly esteemed mathematician Philip B. Jourdain, which paper seems to be the last one written by him. But it contains so many new points of wiew that I have thought I would do the mathematical Public a service by publishing it. At the same time, however, I wish to point out that this journal will not to any further extent be at the disposal for papers of the same kind.G. Mittag-Leffler.

(Cf.G. Mittag-Leffler, «Die Zahl, Enleitung zur Theorie der analytischen Functionen», The Tôhoku Mathematical Journal, Vol. 17, Nos 3, 4, May 1920. — «Talet, inledning till teorien för analytiska funktioner», Det Kgl. Danske Vid. Selsk., Math.-fys. Meddelelser, II, 5. 1920).

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Jourdain, P.E.B. A proof that every aggregate can be well-ordered. Acta Math. 43, 239–261 (1922). https://doi.org/10.1007/BF02401758

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