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Jourdain’s 1904 Generalization of Grundlagen

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Proofs of the Cantor-Bernstein Theorem

Part of the book series: Science Networks. Historical Studies ((SNHS,volume 45))

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Abstract

In 1904, Jourdain (see Grattan-Guinness 1977 prologue) published two papers. The first, in January (1904), was titled “On the transfinite cardinal numbers of well-ordered aggregates”; the second, in March (1904a), was titled “On the transfinite cardinal numbers of number-classes in general”. The papers are remarkable because they matched Cantor’s theory of inconsistent sets and offered a general construction of Cantor’s scale of number-classes, both unpublished at the time (see ‎Chap. 4).

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Notes

  1. 1.

    Jourdain usually translated Cantor’s term Menge as ‘aggregate’. For example in Cantor 1915. He made an exception in this paper and used ‘manifold’ for ‘consistent aggregates’ (see below). In this chapter, we will use both ‘aggregate’ and ‘set’, as well as ‘class’, interchangeably. ‘Manifold’ we will use when we wish to stress Jourdain’s terminology. We might also use ‘collection’ for inconsistent sets and ‘subset’ for Jourdain’s ‘part’. We will use ‘similar’ interchangeably with Cantor’s ‘equivalent’ and also with Russell’s ‘ordinally similar’ hoping that the context will clarify our intention.

  2. 2.

    Grattan-Guinness did not bring Jourdain’s letter. From Cantor’s answer it seems that Jourdain sent him only the content of his 1904a, not that of 1904b.

  3. 3.

    We agree with Moore (1982 p 51) that the communication to Hilbert is the letter of June 28, 1899, with the attachment to Schoenflies, and the communication to Dedekind is the one of August 3, 1899.

  4. 4.

    Certain constructs in set theory did become unnecessary in view of AC. Jourdain raised such an argument against Bernstein’s concept of ‘multiple similarity’ (Jourdain 1907b p 363 footnote *).

  5. 5.

    Jourdain (1904b p 295 footnote *, 300) rejects Schoenflies view (1900 p 48) that new principles are necessary to generate limit ordinals for longer than ω successions.

  6. 6.

    ‘Segment’ in the sense of an initial segment as in Cantor’s 1897 Beiträge §13, to which Jourdain refers.

  7. 7.

    Though Jourdain speaks here of ‘cardinal numbers’ he means the alephs.

  8. 8.

    For initial numbers Jourdain introduced the notation ω γ (1904b p 295 footnote †) which he attributes to Russell (1903 p 322). In fact, Cantor used the same notation in his August 3, 1899, letter to Dedekind (see Chap. ‎3).

  9. 9.

    The final quote is first in the original.

  10. 10.

    There he attributed this definition to several mathematicians, including Bernstein, Whitehead, Kőnig and Peano. Jourdain gave no references. We suggest the following: Bernstein 1905 p 127ff or Bernstein 1904 p 559, Whitehead 1902 p 380 *3.13 (it is the second *3.12 in the original), Peano 1906 p 360. With regard to Kőnig, probably J. Kőnig, we have not located a reference. Cf. Kuratowski-Mostowski 1968 p 185.

  11. 11.

    Cf. Sect. 13.1 for Zermelo’s Theorem IV, which is a version of CBT using ≥. Note that using the Cantorian meaning of ≥ and Dedekind’s definition of ‘infinite’, the following formulation of CBT is trivial (Jourdain 1907b p 356): 𝔞≥𝔟 and 𝔞 ≤ 𝔟 entails 𝔞 = 𝔟, because under our assumptions, 𝔞 > 𝔟, 𝔟 > 𝔞, 𝔞 = 𝔟 are exclusive of each other (Mańka-Wojciechowska 1984 p 191).

  12. 12.

    Russell was aware of the problem (Grattan-Guinness 1977 p 49).

  13. 13.

    Harward (1905 p 459) pointed at the circularity of Jourdain’s formal definition of W. Jourdain retorted (1907b p 356 footnote * last paragraph) that he did not make this “glaringly vicious circle”. He may have meant his covert reference to 𝔅 (see the text).

  14. 14.

    𝔅 was apparently mentioned in Jourdain’s letter to Cantor for Cantor related to it. On 𝔅 cf. Jourdain 1905b.

  15. 15.

    Jourdain references Schröder 1898 without giving a page number and we could not locate where Schröder’s view is expressed.

  16. 16.

    In the same footnote Jourdain compares this procedure to transfinite induction, “ordinal induction” he calls it or “the conclusion from {ν} to ω”. He references Schoenflies 1900 (pp 45, 52, 60, 67) for examples where this procedure is used. Jourdain’s differentiation between cardinal and ordinal induction seems superficial.

  17. 17.

    The counterexample against the general validity of the procedure applied is entirely superfluous, as it comes out of the next paragraphs.

  18. 18.

    Incidentally, this was the month in which Whitehead’s 1902 paper was published, with an erroneous proof of CBT by Russell (see Chap. 15). We can speculate that Jourdain found the argument he mentioned when he tried to proof-process Russell’s CBT proof. Jourdain clearly studied Whitehead 1902 carefully because he attempted to prove two open problems stated in that paper (see the next section).

  19. 19.

    In fact, the proof requires the natural numbers and properties of ℵ0 (not the type ω + 1).

  20. 20.

    Prior to his 1908a paper Jourdain did communicate with Zermelo (see below).

  21. 21.

    This result, for 𝔞 an aleph, follows from the Sum Lemma (see Sect. 2.2).

  22. 22.

    Russell used the mentioned inequality in his proof of CBT (see Sect. 15.1).

  23. 23.

    This result, for a an aleph, follows from the Union Lemma (see Sect. 2.2).

  24. 24.

    Without loss of generality we can take 𝔟 = 𝔞 (1904a p 73).

  25. 25.

    Jourdain explicitly referenced the pages mentioned here from Whitehead 1902.

  26. 26.

    Similarly pointless observations, but intuitively necessary and naively satisfying, regarding the numbers in general number-classes, Jourdain brought in §3 of 1904b.

  27. 27.

    Cantor too had not stressed that his proof of the Sequent Lemma, required CBT.

  28. 28.

    𝔲 is here a dummy that fixes the order of α, β. Apparently, ordered-pairs were not yet invented in 1904. Cantor used indexed dummy with the same purpose in letters to Dedekind of the late 1873 (see ‎Sect. 7.3). In Jourdain’s 1908a, ordered-pairs are already used.

  29. 29.

    In terms borrowed from Lakatos’ theory of research programs (Lakatos 1978b) it may seem that the inconsistent sets emerged as a protective belt to Cantor’s naive set theory while the consistent sets belong to the core of the theory. However, Cantor embraced the inconsistent sets as an essential part of his theory, not as a protective maneuver.

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  1. The Cohn Institute for the History and Philosophy of Science and Ideas, Tel Aviv University, Tel Aviv, Israel

    Arie Hinkis

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Hinkis, A. (2013). Jourdain’s 1904 Generalization of Grundlagen . In: Proofs of the Cantor-Bernstein Theorem. Science Networks. Historical Studies, vol 45. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0224-6_17

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