Zermelo’s Axiom and Axiomatization in Transition (1908–1918)

  • Gregory H. Moore
Part of the Studies in the History of Mathematics and Physical Sciences book series (HISTORY, volume 8)


As early as 1896, even before the discovery of set-theoretic paradoxes, a few mathematicians had suggested that set theory ought to be axiomatized. Yet interest in formulating such an axiomatization remained very faint even in 1903, when Russell restated Burali-Forti’s result of 1897 as a paradox, and published his own paradox as well. Hilbert, for example, viewed Russell’s paradox as revealing that contemporary logic failed to meet the demands of set theory.1 Russell asserted further that a solution to the paradoxes would result only from a reappraisal of the assumptions used in logic, rather than from technical mathematics [1906, 37]. Unperturbed by the paradoxes, Zermelo concentrated on axiomatizing set theory within mathematics rather than on revising the underlying logical assumptions.


Choice Function Propositional Function Transfinite Induction Hamel Basis Vicious Circle Principle 
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  1. 1.
    Letter of 7 November 1903 to Frege, in Frege 1980, 51–52.Google Scholar
  2. 1.
    Zermelo 1908, 108, 117. For a given set M and a given choice function f on the family of all non-empty subsets of M, T is a θ-chain if (1) T is a set of subsets of M; (2) M is in T; (3) if A is in T, then A - {f(A)} is in T; (4) if S is a subset of T, then ⋂ S is in T.Google Scholar
  3. 2.
    Zermelo corresponded with Jourdain in 1907, but apparently their letters focused on a generalization of König’s theorem (2.1.2); see Jourdain 1908, 512.Google Scholar
  4. 3.
    DMV 14 (1905), 61.Google Scholar
  5. 4.
    This was a jibe at Peano’s admiration for the Scholastics.Google Scholar
  6. 5.
    Zermelo 1908, 116. During 1906 he corresponded with Poincaré regarding his proof and his axiomatization of set theory. One of Poincaré’s letters, unfortunately undated, mentioned that in his article [1906, 315] he had intended to include a passage expressing reservations about deducing the Equivalence Theorem from the Well-Ordering Theorem. Then he added: “This does not mean that I have sided with the critics of your proof. I would be rather inclined to dismiss their objections, but I ask you to allow me the time to reflect further on this question before I adopt a definitive solution.” This letter, as well as three others from Poincaré, is kept in Zermelo’s Nachlass at the University of Freiburg im Breisgau.Google Scholar
  7. 6.
    In other words, Zermelo interpreted past events so that they naturally culminated in his Axiom. For a general discussion of this style of historical writing, see Butterfield 1965.Google Scholar
  8. 7.
    Zermelo had previously sent this example to Poincaré, who replied in a letter of 16 June 1906 that the impredicative definition in Cauchy’s proof could be replaced by “an entirely determinate process not admitting any ambiguity or any vicious circle.... It is not the same, on the contrary, for your Well-Ordering Theorem. There my doubts remain because I cannot conceive of an analogous process.Google Scholar
  9. 8.
    Zermelo 1908, 118–120; cf. 2.6.Google Scholar
  10. 9.
    As Zermelo later discovered when he edited Cantor’s works [1932], Cantor did regard it as a proof, but retained reservations which kept him from publishing it; see 1.6.Google Scholar
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    Zermelo 1908, 120–121; cf. Russell’s letter of 12 April 1904 to Jourdain in Grattan-Guinness 1977, 29.Google Scholar
  12. 11.
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  13. 12.
    Ibid. It is not known if (2.5.1) is weaker than the Well-Ordering Theorem.Google Scholar
  14. 1.
    Hilbert 1899, 1–50.Google Scholar
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    Cantor 1932, 447–448.Google Scholar
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    Harward 1905, 459; see Moore 1976. This definition was never published.Google Scholar
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    Zermelo 1908a, 262–267. These names for the axioms were his own.Google Scholar
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    Rubin and Rubin 1963, 75–77. Zermelo discovered (3.2.1) between August 1904 and early in 1906, for it occurs on a left-hand page in the notebook discussed in footnote 15 below.Google Scholar
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    Whitehead, writing to Russell on 25 April 1904, proposed for inclusion in Principia Mathematica a postulate that is essentially the Axiom of Extensionality. This letter is kept in the Russell Archives.Google Scholar
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    Baire et alii 1905, 264. On the relationship between Schröder’s and Frege’s views of the empty set, see Frege 1895, 437.Google Scholar
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    See 3.3. In the letter to Dedekind mentioned above, Cantor had stated non-axiomatically that any submultitude (Teilvielheit) of a set is a set, but did not specify how to determine a sub-multitude; see Cantor 1932, 444. In 1905 Lebesgue stated a rudimentary version of Separation, viewing it (incorrectly) as Cantor’s definition of a set; see Lebesgue in Baire et alii 1905, 265.Google Scholar
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    Burali-Forti 1897, 162n. Pieri [1906, 207] cited Burali-Forti [1896] when adopting the existence of an infinite set as a postulate.Google Scholar
  28. 15.
    This and the following axioms occur in the notebook, part of Zermelo’s Nachlass, where he outlined the lectures that he gave on set theory at Göttingen in 1901. The lectures are in shorthand on the right-hand pages, while the axioms and other notes on the left-hand pages were written later. They appear to date from the period 1904–1906.Google Scholar
  29. 16.
    In German, Zermelo’s four axioms read: “I. Eine wohldefinirte Menge enthält niemals sich selbst als Element. MM. IL Ein einziges Element m 0 definirt eine Menge {m 0} ≠ m 0; ist M eine wohldefinirte Menge und m′ ein beliebiges weiteres Element, das in M nicht vorkommt (z.B. M selbst!), so bildet auch (M, m′) eine wohldefinirte Menge. III. Ist M eine wohldefinirte Menge und E irgend eine Eigenschaft, die einem Element m von M zukommen oder nicht zukommen kann, ohne das noch eine Willkür möglich ist, so bilden die Elemente m′ welche die Eigenschaft E haben eine wohldefinirte Menge, eine Teilmenge M′ von M, sowie der Komplementive M″. IV. Auch die Gesamtheit aller Teilmengen von M bildet selbst eine wohldef. Menge, ebenso alle diejenige Teilmengen M′ welche eine wohldefinirte Eigenschaft besitzen.”Google Scholar
  30. 17.
    DMV 15 (1906), 406–407.Google Scholar
  31. 18.
    Zermelo’s lecture notebook of 1906 is kept in his Nachlass at the University of Freiburg im Breisgau.Google Scholar
  32. 19.
    See, for example, Beth 1959, 494; Bourbaki 1969, 47–48; Kline 1972, 1185; Quine 1966, 17; and van Heijenoort 1967, 199.Google Scholar
  33. 20.
    Russell’s letter of 15 March 1908 in Grattan-Guinness 1977, 109.Google Scholar
  34. 21.
    Hausdorff 1908, 435–437.Google Scholar
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    Zermelo 1908a, 261; cf. 1908, 115–116.Google Scholar
  36. 23.
    Zermelo 1908, 107, 110, 125.Google Scholar
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  38. 25.
    Zermelo 1908, 115, 118–127.Google Scholar
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    Zermelo 1908a, 261; cf. 1908, 124. He elaborated this idea two decades later in his cumulative type theory; see 4.9.Google Scholar
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    Zermelo 1908, 118–119. Rang and Thomas [1981] have analyzed what is known about Zermelo’s discovery.Google Scholar
  41. 28.
    See, for example, the description of Zermelo’s character in Fraenkel 1967, 149.Google Scholar
  42. 1.
    Russell in Grattan-Guinness 1977, 109.Google Scholar
  43. 2.
    This letter can be found in Zermelo’s Nachlass at the University of Freiburg; the two articles are probably Russell 1906 and 1906a. The Nachlass also contains a brief card from Dedekind, dated 18 December 1907, and a similar card from Frege, dated 29 December 1907, both thanking Zermelo for sending his articles on set theory—presumably [1908] and [1908a]. Dedekind and Frege made no substantive comments, and no other correspondence from them exists in the Nachlass. It appears that these two articles were in print by December 1907, unless Zermelo sent page proofs.Google Scholar
  44. 3.
    Zermelo in Meschkowski 1967, 267.Google Scholar
  45. 4.
    Russell and Whitehead 1910, vii. Zermelo’s Nachlass contains notes by Kurt Grelling of the course on mathematical logic that Zermelo gave at Göttingen during the summer of 1908. In the first lecture Zermelo asserted that the chief question was to what extent mathematics is “a logical science.” He carefully took the middle ground between those, from Leibniz to Peano and Russell, who affirmed that arithmetic is part of logic, and those, from Kant to Poincaré, who claimed that it is not. In conclusion, Zermelo argued that logic should be developed in terms of Hilbert’s axiomatic method.Google Scholar
  46. 5.
    Zermelo 1908a, 262.Google Scholar
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    Poincaré 1909a, 473.Google Scholar
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    Ibid., 474–475.Google Scholar
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  50. 9.
    Zermelo 1909, 192. During 1907 he had corresponded with Poincaré about submitting this article to the Revue de Métaphysique et de Morale. In his reply of 19 June 1907, Poincaré found the paper too mathematical for readers of the Revue but added that he had proposed it to Mittag-Leffler for publication in Acta Mathematica. This is where the paper appeared two years later.Google Scholar
  51. 10.
    Hessenberg 1909. He had already spoken approvingly of Zermelo’s system in his previous paper [1908, 147] which criticized Schoenflies, 1908 report on set theory.Google Scholar
  52. 11.
    Hessberg 1909, 82, 86–90, 130–133, 103.Google Scholar
  53. 12.
    Grelling 1910, 9, 21.Google Scholar
  54. 13.
    Schoenflies 1911, 227, 241.Google Scholar
  55. 14.
    Ibid., 231–232, 251. A serious difficulty with Schoenflies’ dictum, but one which he did not realize, was the following: Two sets A and B might each be consistent with a given system S of set theory; but if both A and B were adjoined to S, the resulting system could be inconsistent.Google Scholar
  56. 15.
    Ibid., 244, 229, 254, 222.Google Scholar
  57. 16.
    Fraenkel 1919, 134–135.Google Scholar
  58. 17.
    Ibid., 136–137, 125–128.Google Scholar
  59. 1.
    For a list of equivalents, see Rubin and Rubin [1963, 111–124] as well as Appendix 2.Google Scholar
  60. 2.
    Although the term maximal principles has become standard in the recent mathematical literature, a more appropriate designation would be extremal principles. Indeed, for many maximal principles there is a corresponding minimal principle asserting the existence of a minimal element, i.e., an element such that no other element is strictly smaller in the given partial order. For a history of maximal principles whose interpretation differs somewhat from the present book, see Campbell 1978.Google Scholar
  61. 3.
    Whitehead 1902, 383; see (1.7.14).Google Scholar
  62. 4.
    Hausdorff 1907, 117–118; see 2.5.Google Scholar
  63. 5.
    Contrary to the claim of Grattan-Guinness [1977, 61, 159], Hausdorff did not use a maximal principle in order to avoid the Axiom of Choice and transfinite induction.Google Scholar
  64. 6.
    Zoretti 1909, 487. In a topological space, a continuum is a perfect set which is connected, i.e., not the union of two disjoint closed sets.Google Scholar
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    Mazurkiewicz 1910, 296–298. Nevertheless, his stated aim was not to eliminate the Axiom from the proof but rather to avoid the use of transfinite ordinals.Google Scholar
  66. 8.
    Brouwer 1911, 138. Consequently we must reject Campbell’s claim [1978, 78] that these two results were special cases of a maximal principle, as well as his claim [1978, 80] that Brouwer used the Well-Ordering Theorem in his proof. Such a use would have been illegitimate from Brouwer’s intuitionistic standpoint.Google Scholar
  67. 9.
    Kuratowski 1922, 88–89.Google Scholar
  68. 10.
    Russell and Whitehead 1910, 561–565. In an unpublished manuscript of 1906, Russell had conjectured that (3.4.4) and (3.4.5) were equivalent to the Axiom of Choice and the Multiplicative Axiom respectively; cf. (2.7.4) and (2.7.5). At that time he thought the latter to be weaker than the former.Google Scholar
  69. 11.
    See, for example, Rubin and Rubin 1963, 6.Google Scholar
  70. 12.
    Schoenflies 1908, 36.Google Scholar
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    Läuchli [1962, 5–7] used the Fraenkel-Mostowski method to provide a model of ZFU in which a certain field had no algebraic closure, while Pincus [1972, 722–723] obtained a model of ZF containing such a field.Google Scholar
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    Steinitz 1910, 251. A field is absolutely algebraic if all its elements are algebraic relative to its prime subfield. A prime field is a field which has no proper subfield.Google Scholar
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    Ibid., 171, 292–293. A field F 1 is a purely transcendental extension of a field F if F 1 is isomorphic to a field of quotients obtained by adjoining some number of indeterminates to F. F 2 is an algebraic extension of F 1 if every element of F 2 is algebraic with respect to F 1.Google Scholar
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    Ibid., 288–289, 299.Google Scholar
  76. 6.
    Noether 1916, 103–106, 123.Google Scholar
  77. 7.
    Zermelo 1914, 442.Google Scholar
  78. 8.
    D. König 1916, 461. A graph is of nth degree if each vertex is on exactly n edges. A pair graph is a graph such that any closed path contains an even number of vertices.Google Scholar
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    Ibid., 461–463.Google Scholar
  80. 1.
    Wilson 1908, 440–442.Google Scholar
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    Russell 1911, 31; this lecture is translated in Grattan-Guinness 1977, 161–174.Google Scholar
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    Russell and Whitehead 1910, 503, and 1912, 190, 228, 278.Google Scholar
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    Poincaré 1909a, 472–478.Google Scholar
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    Poincaré 1909, 196.Google Scholar
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    Poincaré 1912, 2–4; cf. 2.3. One notable difference between the two distinctions was that Poincaré regarded the Pragmatists as philosophical idealists, whereas Lebesgue branded the Empiricists’ opponents with that term.Google Scholar
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    Hadamard in Borei 1914a, 72–73.Google Scholar
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    Lebesgue 1922, 61. Correspondence with his heirs has failed to uncover any surviving manuscript of this exchange.Google Scholar
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    In the second edition [1934] of his book, de la Vallée-Poussin attempted to eliminate all his earlier uses of the Axiom.Google Scholar
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    Sierpiński 1918, 124–125; see (4.1.13) and (4.1.20).Google Scholar
  90. 11.
    Lebesgue 1918, 238–239.Google Scholar
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    Cipolla 1913, 1–2.Google Scholar
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    Ibid., 7.Google Scholar
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    Kennedy 1980, 191. Tonelli had just obtained his doctorate from the University of Bologna.Google Scholar
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    Tonelli 1913, 6. A function f is lower semi-continuous at a point x if f(x) equals the limit, as r approaches zero, of the greatest lower bound of f within an open ball of radius r. Google Scholar
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    Ibid., 7–13; cf. 1.8. It seems that the only Italian mathematician to accept the Axiom explicitly at this time was Giuseppe Bagnera of Palermo. In a calculus text Bagnera adopted the Denumerable Axiom for sets of real numbers [1915, iii].Google Scholar
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    While Luzin did not publish a proof of (3.6.7) in his article [1917], Luzin and Sierpiński did so in their joint article [1918, 44–47] on analytic sets. There they relied indirectly on the Axiom when they used the proposition that a countable union of measurable sets is measurable.Google Scholar
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    Luzin 1917, 93–94. Hausdorff[1916] had established that every uncountable Borei set in ℝ has a perfect subset, but here too the Axiom was used in an unavoidable way. Paul Alexandroff [1916], a student of Luzin, independently arrived at the same result as Hausdorff.Google Scholar
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    See the forthcoming doctoral thesis of Juris Steprans at the University of Toronto.Google Scholar
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    Schoenflies and Hahn 1913, 170.Google Scholar
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    Ibid., 180, 224. Perhaps the acceptance of Zermelo’s proofs in this report on set theory reflected the views of its coauthor Hans Hahn more than of Schoenflies. Certainly, in his later textbook on real functions, Hahn [1921, 25] embraced the proof without much ado. At that time, however, Schoenflies also accepted it within the context of Zermelo’s axiomatization [1922, 102].Google Scholar
  101. 22.
    Dénes König in Julius Konig 1914, v.Google Scholar
  102. 23.
    J. König 1914, 1–2, 148.Google Scholar
  103. 24.
    Likewise, in his first study of large cardinals, the German mathematician Paul Mahlo [1911, 187] briefly expressed reservations about the Axiom.Google Scholar
  104. 25.
    Dingier 1911, 13–14, 8.Google Scholar
  105. 26.
    Fraenkel 1919, 125–128, 146–148, 143.Google Scholar
  106. 1.
    Hausdorff 1914, 133–138.Google Scholar
  107. 2.
    Hausdorff 1914a, 428.Google Scholar
  108. 3.
    Ibid., 429.Google Scholar
  109. 4.
    Stefan Banach [1923] established that there is a solution for the line and the plane; see 4.11.Google Scholar
  110. 5.
    Hausdorff 1914a, 430–433.Google Scholar
  111. 6.
    Borei 1914, 255–256. Since Gödel’s work on constructible sets (see 4.10), it is known to be consistent with ZF that the non-measurable sets A, B, C given by Hausdorff are definable (by means of ordinals). In particular, they are definable if we assume the Axiom of Constructibility.Google Scholar
  112. 1.
    Jourdain 1906a, 16.Google Scholar
  113. 2.
    The last of these was Jourdain 1910. See also Jourdain 1906a, 1906b, 1907, 1908, 1908a, discussed briefly in 2.7.Google Scholar
  114. 3.
    Jourdain 1918b, 388. The same lemma occurs in his articles 1918d, 985; 1918e, 303; and 1921, 240.Google Scholar
  115. 4.
    Jourdain in Grattan-Guinness 1977, 146.Google Scholar
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    Jourdain in Grattan-Guinness 1977, 149.Google Scholar
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    Littlewood in Grattan-Guinness 1977, 152.Google Scholar
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    Laura Jourdain in Grattan-Guinness 1977, 153.Google Scholar
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    Mittag-Leffler in Jourdain 1921, 239. It may well be that Mittag-Leffler, who did not specify these new points of view, published the paper mainly out of respect for Jourdain.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1982

Authors and Affiliations

  • Gregory H. Moore
    • 1
  1. 1.Department of Mathematics and Institute for History and Philosophy of Science and TechnologyUniversity of TorontoTorontoCanada

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