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On Cantor’s continuum problem and well ordering: what really happened at the 1904 International Congress of Mathematicians in Heidelberg. (English) Zbl 1334.01018

Rowe, David E. (ed.) et al., A delicate balance: global perspectives on innovation and tradition in the history of mathematics. A Festschrift in honor of Joseph W. Dauben. Cham: Birkhäuser/Springer (ISBN 978-3-319-12029-4/hbk; 978-3-319-36594-7/pbk; 978-3-319-12030-0/ebook). Trends in the History of Science, 3-24 (2015).
The author is well known for his historical analysis of Georg Cantor’s early understanding of the antinomies of set theory [Zbl 0627.01008] and for his Cantor biography of 1987 [Zbl 0617.01018]. Currently, he is putting the finishing touches to a multi-volume edition of Felix Hausdorff’s “Works” (Bonn).
The paper under review is basically a translation of a German publication of the author from 2004 [Zbl 1081.01020]. In its first part, the author shows the origins of Cantor’s set theory in investigations of trigonometric series around 1870. Then, he describes Hausdorff’s turn toward set theory in the 1890s which was partly influenced by his philosophical interests and which culminated in his “Grundzüge der Mengenlehre” [Leipzig: Veit & Comp. (1914; JFM 45.0123.01)]. During the ICM at Heidelberg in August 1904, the Hungarian J. König gave a talk [Verh. d. 3. intern. Math. Kongr. Heidelb., 144–147 (1905; JFM 36.0096.01)] which was meant to refute Cantor’s famous continuum hypothesis. The author, as partly I. Grattan-Guinness (2000) before him, makes it clear that it was F. Hausdorff, in a small internal “post-congress” in Wengen in the Swiss Alps (with Cantor, Hensel, Hilbert, and Schönflies attending) and in a subsequent publication, who convinced the other mathematicians and König himself that the latter’s talk was based on an erroneous assumption. The error was in a formula from the cardinal arithmetic of alephs in F. Bernstein’s dissertation [Untersuchungen aus der Mengenlehre. Göttingen (1901; JFM 32.0073.02)], which turned out not to be generally valid. In the publication under review, the author adds material from a publication by H.-D. Ebbinghaus [Hist. Math. 34, No. 4, 428–432 (2007; Zbl 1135.01014)]. A postcard written by Ernst Zermelo to Max Dehn on 27 October 1904 makes it very likely that Zermelo too, independently of Hausdorff, had found the mistake in Bernstein’s dissertation very soon. The paper under review discounts, however, the colorful, but unreliable report [Bestand und Wandel. München: Verlag R. Oldenbourg (1950; Zbl 0037.00203)] by G. Kowalewski on the Heidelberg Congress (which the latter apparently not even attended) and on its aftermath, which exaggerates the role of Zermelo in finding the mistake and ignores Hausdorff altogether. In a final section “Some subsequent developments”, the author shows the positive side of König’s inequality on sequences of cardinal numbers, which was later generalized by the Russian I. I. Zhegalkin [JFM 38.0094.04] and by Zermelo [JFM 39.0097.03]. The paper under review thus presents and analyzes important new findings for the history of set theory and, in particular, the continuum hypothesis.
For the entire collection see [Zbl 1320.01005].

MSC:

01A55 History of mathematics in the 19th century
01A60 History of mathematics in the 20th century
01A70 Biographies, obituaries, personalia, bibliographies
03-03 History of mathematical logic and foundations
03E50 Continuum hypothesis and Martin’s axiom
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