Abstract
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.
Many mathematicians still stand opposed to the Axiom of Choice. With the increasing recognition that there are questions in mathematics which cannot be decided without this axiom, the resistance to it must increasingly disappear.
Ernst Steinitz [1910, 170–171]
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References
Letter of 7 November 1903 to Frege, in Frege 1980, 51–52.
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.
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.
DMV 14 (1905), 61.
This was a jibe at Peano’s admiration for the Scholastics.
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.
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.
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.”
Zermelo 1908, 118–120; cf. 2.6.
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.
Zermelo 1908, 120–121; cf. Russell’s letter of 12 April 1904 to Jourdain in Grattan-Guinness 1977, 29.
Zermelo 1908, 124–125.
Ibid. It is not known if (2.5.1) is weaker than the Well-Ordering Theorem.
Hilbert 1899, 1–50.
Burali-Forti 1896, 46; 1896a, 236.
Cantor 1932, 447–448.
Ibid., 444.
Harward 1905, 459; see Moore 1976. This definition was never published.
Russell 1906, 37–40, 45–47.
Russell’s letter of 10 September 1906 to Jourdain in Grattan-Guinness 1977, 91.
Zermelo 1908a, 262–267. These names for the axioms were his own.
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.
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.
Cantor 1932, 444. There is some reason to believe, however, that Zermelo may have read the similar letter of 1896 or 1897 from Cantor to Hilbert; see also footnote 1 of 1.6.
Baire et alii 1905, 264. On the relationship between Schröder’s and Frege’s views of the empty set, see Frege 1895, 437.
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.
Burali-Forti 1897, 162n. Pieri [1906, 207] cited Burali-Forti [1896] when adopting the existence of an infinite set as a postulate.
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.
In German, Zermelo’s four axioms read: “I. Eine wohldefinirte Menge enthält niemals sich selbst als Element. M ∉ M. 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.”
DMV 15 (1906), 406–407.
Zermelo’s lecture notebook of 1906 is kept in his Nachlass at the University of Freiburg im Breisgau.
See, for example, Beth 1959, 494; Bourbaki 1969, 47–48; Kline 1972, 1185; Quine 1966, 17; and van Heijenoort 1967, 199.
Russell’s letter of 15 March 1908 in Grattan-Guinness 1977, 109.
Hausdorff 1908, 435–437.
Zermelo 1908a, 261; cf. 1908, 115–116.
Zermelo 1908, 107, 110, 125.
Ibid., 110, 115–116, 124; cf. 1908a, 266.
Zermelo 1908, 115, 118–127.
Zermelo 1908a, 261; cf. 1908, 124. He elaborated this idea two decades later in his cumulative type theory; see 4.9.
Zermelo 1908, 118–119. Rang and Thomas [1981] have analyzed what is known about Zermelo’s discovery.
See, for example, the description of Zermelo’s character in Fraenkel 1967, 149.
Russell in Grattan-Guinness 1977, 109.
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.
Zermelo in Meschkowski 1967, 267.
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.
Zermelo 1908a, 262.
Poincaré 1909a, 473.
Ibid., 474–475.
Ibid., 416–477.
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.
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.
Hessberg 1909, 82, 86–90, 130–133, 103.
Grelling 1910, 9, 21.
Schoenflies 1911, 227, 241.
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.
Ibid., 244, 229, 254, 222.
Fraenkel 1919, 134–135.
Ibid., 136–137, 125–128.
For a list of equivalents, see Rubin and Rubin [1963, 111–124] as well as Appendix 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.
Whitehead 1902, 383; see (1.7.14).
Hausdorff 1907, 117–118; see 2.5.
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.
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.
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.
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.
Kuratowski 1922, 88–89.
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.
See, for example, Rubin and Rubin 1963, 6.
Schoenflies 1908, 36.
An algebraically closed field is a field in which every polynomial in one indeterminate can be decomposed into linear factors. The algebraic closure of a field F is the least algebraically closed field which includes F.
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.
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.
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.
Ibid., 288–289, 299.
Noether 1916, 103–106, 123.
Zermelo 1914, 442.
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.
Ibid., 461–463.
Wilson 1908, 440–442.
Russell 1911, 31; this lecture is translated in Grattan-Guinness 1977, 161–174.
Russell and Whitehead 1910, 503, and 1912, 190, 228, 278.
Poincaré 1909a, 472–478.
Poincaré 1909, 196.
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.
Hadamard in Borei 1914a, 72–73.
Lebesgue 1922, 61. Correspondence with his heirs has failed to uncover any surviving manuscript of this exchange.
In the second edition [1934] of his book, de la Vallée-Poussin attempted to eliminate all his earlier uses of the Axiom.
Sierpiński 1918, 124–125; see (4.1.13) and (4.1.20).
Lebesgue 1918, 238–239.
Cipolla 1913, 1–2.
Ibid., 7.
Kennedy 1980, 191. Tonelli had just obtained his doctorate from the University of Bologna.
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.
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].
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.
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.
See the forthcoming doctoral thesis of Juris Steprans at the University of Toronto.
Schoenflies and Hahn 1913, 170.
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].
Dénes König in Julius Konig 1914, v.
J. König 1914, 1–2, 148.
Likewise, in his first study of large cardinals, the German mathematician Paul Mahlo [1911, 187] briefly expressed reservations about the Axiom.
Dingier 1911, 13–14, 8.
Fraenkel 1919, 125–128, 146–148, 143.
Hausdorff 1914, 133–138.
Hausdorff 1914a, 428.
Ibid., 429.
Stefan Banach [1923] established that there is a solution for the line and the plane; see 4.11.
Hausdorff 1914a, 430–433.
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.
Jourdain 1906a, 16.
The last of these was Jourdain 1910. See also Jourdain 1906a, 1906b, 1907, 1908, 1908a, discussed briefly in 2.7.
Jourdain 1918b, 388. The same lemma occurs in his articles 1918d, 985; 1918e, 303; and 1921, 240.
Jourdain in Grattan-Guinness 1977, 146.
Jourdain in Grattan-Guinness 1977, 149.
Littlewood in Grattan-Guinness 1977, 152.
Laura Jourdain in Grattan-Guinness 1977, 153.
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.
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Moore, G.H. (1982). Zermelo’s Axiom and Axiomatization in Transition (1908–1918). In: Zermelo’s Axiom of Choice. Studies in the History of Mathematics and Physical Sciences, vol 8. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-9478-5_4
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