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Ernst Zermelo: an Approach to His Life and Work Reviewed by Gregory H Book Review Ernst Zermelo: An Approach to His Life and Work Reviewed by Gregory H. Moore half a century), and on work by earlier historians Ernst Zermelo: An Approach to His Life and Work of mathematics, including the reviewer. The best Heinz-Dieter Ebbinghaus in cooperation with parts of the book are the previously unpublished Volker Peckhaus letters. Yet relatively few of them from Zermelo Springer, Berlin, 2007 appear in the book, even when they were available, US$64.95, xiv + 356 pages such as his letters to Hilbert. Thanks largely to his ISBN-13: 978-3-540-49551-2 widow, this book contains more photographs of Zermelo than any other source. The biography begins with Zermelo’s ancestors, Ernst Zermelo is familiar to mathematicians as then turns to his youth spent in Berlin, where his the creator of the controversial Axiom of Choice father died shortly before Zermelo finished high in 1904 and the theorem, based on the Axiom of Choice, that every set can be well ordered. school and where Zermelo already experienced Many will be aware that in 1908 he axiomatized the poor health that would plague him through set theory—in a form later modified by Abra- much of his life. Next the book discusses his uni- ham Fraenkel (1922) and then by Zermelo himself versity studies, which focused on mathematics, (1930). Some will know of Zermelo’s conflict with physics, and philosophy (a combination that was Ludwig Boltzmann over the Poincaré Recurrence later shared by Paul Bernays and Kurt Gödel). Like Theorem and its role in understanding the Second many German students of the time, he studied at Law of Thermodynamics. several universities—in his case Berlin, Freiburg, Fewer will know of the following: Zermelo’s and Halle. At Halle he attended lectures by Georg work in the calculus of variations at the start of Cantor on elliptic functions and number theory, his career; his pioneering work in game theory but Cantor’s work had no particular influence on before that of Emile Borel and John von Neumann; him at the time. His interest in set theory was his conflict with Thoralf Skolem over whether stimulated later, circa 1900, by David Hilbert. axiomatic set theory should be formulated in Zermelo’s 1894 doctoral dissertation under first-order or second-order logic (and the result- H. A. Schwarz was on the calculus of variations. ing conflict over the Löwenheim–Skolem Theorem Schwarz, who had been Weierstrass’s student at and the existence of a countable model of set Berlin, succeeded him there in 1892, and Zermelo theory); his contributions to infinitary logic; or was Schwarz’s first doctoral student. After his finally his conflict with Gödel over the latter’s In- dissertation, Zermelo worked for three years as completeness Theorems and Zermelo’s attempts assistant to Max Planck at the Berlin Institute for to circumvent them. Theoretical Physics. It was during this period that The book under review is the first full-length Zermelo’s polemical dispute with Boltzmann over biography of Zermelo. It is based on a great the Second Law of Thermodynamics occurred. deal of archival research, on the cooperation of In 1897 Zermelo wrote to Felix Klein at Göt- Zermelo’s widow Gertrud (who outlived him by tingen, expressing the wish to come there to Gregory H. Moore is professor of mathematics and prepare a Habilitation in theoretical physics or statistics at McMaster University. His email address is mechanics. Although Klein’s reply is not extant, [email protected]. apparently he encouraged Zermelo, for Zermelo August 2009 Notices of the AMS 823 came, and his 1899 Habilitationsschrift at Göt- Weierstrassian calculus of varia- tingen was in theoretical hydrodynamics. (In the tions, and so I would miss him German academic system a Habilitationsschrift, here most of all. (pp. 35–36) which was like a second doctoral dissertation, Zermelo, as he wrote three decades later (in a was required before one was allowed to give a passage first published in 1980 by the reviewer), lecture course at a university.) At Göttingen, Zer- was much influenced by Hilbert in starting to do melo then returned to researching the calculus research on set theory: of variations, on which he had many conversa- tions with Constantin Carathéodory, who became Thirty years ago, when I was a a lifelong friend. In fact, they undertook a joint Privatdozent in Göttingen, I began, book on the calculus of variations, which, as Her- under the influence of D. Hilbert, mann Minkowski wrote to Wilhelm Wien in 1906, to whom I owe more than anyone “promises to become the best in this field” (p. 33). else for my scientific development, Unfortunately the book was never finished. to concern myself with questions One of the finest parts of the biography con- about the foundations of mathe- cerns Hilbert’s repeated attempts to help Zerme- matics, especially with the funda- lo’s career. In 1901 Zermelo, who as a Privatdozent mental problems of Cantorian set (lecturer) at Göttingen had to subsist solely on fees theory, which only came to my from students attending his lectures, applied for full consciousness in the produc- a grant intended to help such young lecturers. His tive cooperation among Göttingen application was supported by Hilbert, who wrote: mathematicians. (p. 28) Dr. Zermelo is a gifted scholar Since 1981 it has been known that Zermelo dis- with a sharp judgment and a quick covered Russell’s Paradox before Bertrand Russell intellectual grasp. He shows a live- did and that Zermelo’s version of the paradox was ly interest and open understanding discussed by mathematicians at Göttingen, includ- of the questions of our [mathemat- ing Hilbert, before Russell published his paradox ical] science and, moreover, has in 1903. comprehensive theoretical knowl- But Zermelo’s first immortal and yet controver- edge in the domain of mathemati- sial achievement in set theory was his proof, in a cal physics. I am in continual sci- letter to Hilbert of September 1904, that every set entific exchange with him. (p. 34) can be well ordered. A month earlier, at the Interna- Thanks to Hilbert’s support, Zermelo’s application tional Congress of Mathematicians in Heidelberg, was successful. Julius König had given a lecture purporting to Unfortunately, Hilbert’s support did not get Zer- disprove Cantor’s Continuum Hypothesis and to melo a position in 1903 at the University of Breslau. show that the set of all real numbers cannot be well That university asked Hilbert to rate candidates ordered. König’s argument relied essentially on a for the position there (among whom Zermelo was “theorem” in Felix Bernstein’s doctoral disserta- not listed). Hilbert replied to Breslau in May 1903 tion. Cantor, Hilbert, and Schoenflies contributed by rating them, and then added: to the official discussion following the lecture. Now, concerning further names, What happened next has been the subject of I immediately start with the one dispute. Ebbinghaus tends to rely on the dubious whom I consider the real candidate account of Gerhard Kowalewski, written almost for the Breslau Faculty, namely a half-century after the event. The reviewer pub- Zermelo. lished a letter from Felix Hausdorff to Hilbert of Zermelo is a modern mathe- matician who combines versatility September 24, 1904, in Leipzig, where Hausdorff with depth in a rare way. He is an ex- had checked in the library soon after his return pert in the calculus of variations…. home: “After the Continuum Problem had plagued I regard the calculus of variations me at Wengen almost like a monomania, naturally as a branch of mathematics which I looked first here at Bernstein’s dissertation” [[6], will belong to the most important 108]. Hausdorff informed Hilbert that the error lay ones in the future…. where it had been suspected—in Bernstein’s argu- You must not presume that I ment that if κ is an infinite cardinal with κ < λ and intend to praise Zermelo into leav- if B is a set with cardinality λ, then every subset ing [Göttingen]. Before Minkowski A of B (where A has cardinality κ) lies in an initial came here and before Blumenthal segment of λ. This is false when κ is ℵ0 and λ is matured, Zermelo was my main ℵω—precisely the case that König needed for his mathematical company. I have argument. The seminal concept hidden here, and learned a lot from him, e.g., the waiting for Hausdorff to discover it, was cofinality. 824 Notices of the AMS Volume 56, Number 7 Kowalewski [[4], 202] had claimed that Zermelo Würzburg, gave a strong recommendation to Zer- found the error the day after König’s talk. Grattan- melo, whom he described as “a mathematician of Guinness [[3], 334] asserted that Zermelo had not the highest qualities, of the broadest knowledge, found the error. Ebbinghaus (p. 52) discovered a of quick and penetrating grasp, of rare critical letter of October 27, 1904, from Zermelo to Max gift” (p. 106). But Minkowski also commented on Dehn, which showed that Zermelo was one of his personality: those to suspect the error lay in Bernstein’s “the- Above all, his conspicuous lack of orem”, but was able to verify this only when, after good luck stems from his outer ap- the holidays, he returned to Göttingen and could pearance, his nervous haste which visit the library. Consequently, Grattan-Guinness shows in his speaking and conduct. was mistaken. However, the clearest light on what Only very recently it is giving way happened is shed by a letter (not mentioned by to a more calm, serene nature. Be- Ebbinghaus) from Otto Blumenthal to Emile Borel cause of the clarity of his intellect on December 1, 1904.
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