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Home , Ernst Zermelo

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 , 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 The best Heinz-Dieter Ebbinghaus in cooperation with parts of the book are the previously unpublished Volker Peckhaus letters.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, , 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 of Choice father died shortly before Zermelo finished high in 1904 and the theorem, based on the , that every 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 —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 , and (a combination that was Ludwig Boltzmann over the Poincaré Recurrence later shared by 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 but Cantor’s work had no particular influence on before that of Emile Borel and ; him at the time. His interest in set theory was his conflict with over whether stimulated later, circa 1900, by . axiomatic set theory should be formulated in Zermelo’s 1894 doctoral dissertation under first-order or second-order (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 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 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 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 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 λ, then every 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. Blumenthal informed Borel he is a first- teacher for the that König himself was the first to realize that his more sophisticated students, for proof was not valid, followed (independently of whom it is important to penetrate each other) by Cantor, Bernstein, and Zermelo [[1], the depths of science. They, like 74]. all the younger lecturers here with In 1907 Zermelo had been teaching as a Pri- whom he is a close friend, appreci- vatdozent at Göttingen for eight years, and his ate him extraordinarily. However, grant for doing so was finished. He either had to he is not a teacher for beginners…. find a salaried position somewhere or abandon (p. 107) academic life altogether. So Hilbert turned to the Zermelo did not obtain the professorship at Ministry of Cultural Affairs in Berlin (responsi- Würzburg in 1906, nor again in 1909. In 1910, ble for all universities in Germany) and made a when Bernstein was being considered at Göttingen case for establishing a lectureship in mathematical for an extraordinary professorship in actuar- logic at Göttingen and for Zermelo to hold this ial mathematics, the mathematics department lectureship. Hilbert persuaded Friedrich Althoff, urged the Minister to offer an extraordinary the all-powerful director of the First Education- professorship there to Zermelo as well. In the al Department at the ministry, to establish this end it was not offered because of news from lectureship—the first in in Ger- Switzerland—in 1910 Zermelo obtained a full many or, to the best of my knowledge, anywhere professorship at the University of Zurich. At that else. However, lecture courses in mathematical time, Zermelo’s friend was still logic had previously been given in Germany by professor at Zurich and was on the committee Ernst Schröder and Gottlob Frege, and in England to select his replacement. Schmidt vigorously by Russell. supported Zermelo’s application, which included Zermelo held his first course in mathematical a very strong recommendation from Hilbert. logic in 1908, but almost all that we know of this Zermelo held this professorship at Zurich only course comes from lecture notes taken by Kurt until 1916. It was ended at the request of the Grelling. Zermelo’s own notes for the course were university because of his tuberculosis. He had un- in Gabelsberger, a form of shorthand that Bernays dergone surgery for the disease and had missed and Gödel also used frequently. Almost no one several terms of teaching due to ill health. However knows Gabelsberger now, since it was superseded Fraenkel, who began working on axiomatic set the- in Germany by a different form of shorthand. Fur- ory around 1919 and whose relations with Zermelo ther, Ebbinghaus does not inform the reader that were strained, published in his autobiography a Zermelo’s lecture notes are in Gabelsberger and claim that Zermelo lost his professorship because, hence inaccessible even to . By contrast, when a guest at a hotel in Germany he wrote the Gödel Editorial Project trained someone in under his nationality “Thank God, not Swiss!”, a Gabelsberger in order to transcribe unpublished comment unfortunately read in the hotel register manuscripts by Gödel into German. Unfortunately, by the head of Zurich’s education department. the manuscripts of Zermelo and of Bernays have Ebbinghaus made thorough use of documents not been transcribed. from Zermelo’s official file at the University of Early in 1905 Zermelo fell seriously ill with an Zurich to show that his illness, by making him inflammation of the lungs, and had to abandon his unable to teach for several semesters, was the rea- teaching for several months. A year later, he again son for his involuntary retirement. He was given a had an illness in his lungs, finally diagnosed as generous pension. tuberculosis. Once again, he had to stop teaching. In 1921 Zermelo left Switzerland permanent- In 1906 Minkowski, who was asked to recommend ly and settled at Freiburg in Germany, where he candidates for a position at the University of remained until his death in 1953. In Freiburg

August 2009 Notices of the AMS 825 Zermelo became friends with Reinhold Baer and first-order sentence is provable) was published Arnold Scholz (both algebraists), who were succes- only after this dispute. Even now, many mathe- sively assistants there. In 1926 the two professors maticians do not understand what is meant by of mathematics at Freiburg, Lothar Heffter and “first-order logic” and, consequently, the import Alfred Loewy, applied to the Ministry of Education of the Löwenheim-Skolem Theorem and the Com- to make Zermelo a full honorary professor. This pactness Theorem for this logic. Because the chief permitted Zermelo to teach (which he did for sev- question at issue between Zermelo and Skolem en years without salary). During this period his was the latter’s formulation of set theory with- mathematical research and publications resumed, in first-order logic, we must be clear (as almost both in the foundations of mathematics and in all those at the time were not) about what was applied mathematics. Then, early in 1935, he was involved: the difference between first-order and dismissed from his position because he refused second-order logic. to give the Hitler salute. In 1946 he wrote to the The essential difference between these two log- rector of the university, asking to be rehabilitated: ics (relative to axiomatizing set theory) is in how “As an honorary professor I gave regular lectures to interpret a quantifier ranging over the in pure and applied mathematics for a number of of a given set within a given model A. In second- years until I was forced under the Hitler govern- order logic, the expression “for every subset of a ment by political intrigues to give up this activity. given set x” is interpreted in the way natural to Circumstances having now changed,…I therefore mathematicians who are not logicians, i.e., that ex- request the University…to look favorably on my pression means what it says: “every subset” means reappointment as an honorary professor” (p. 251). “every subset”, whether or not a given subset hap- His request was granted. pens to be in A. But in first-order logic, “for every Even after his dismissal, Zermelo maintained subset of a given set x” in a model A means only some scientific contacts. In 1941 Scholz organized those subsets of x that are members of A. Thus, a colloquium at Göttingen for Zermelo’s seventi- within second-order logic, “the set S of all subsets eth birthday, and among the speakers was Bartel of N”, where N is the set of all natural numbers, van der Waerden. Zermelo gave three talks at the is uncountable, whether looked at from inside a colloquium, all on applied mathematics. model A or from outside A. By contrast, in first- As the review has mentioned, this biography order logic, “the set S of all subsets of N” can be has many positive features. However, there are uncountable if looked at from inside the model A certain matters where a more critical approach but countable if looked at from outside the model is necessary. We leave to one side various minor A. From outside, the apparent uncountability of errors, due to its author writing in English, which is S is seen to be an artifact caused by the lack of not his mother tongue: “inconstructive character a certain inside A, a bijection between of the axiom of choice” (p. vii) rather than “non- S and N that is present outside A. This is the constructive character”, “nth derivation” (p. 12) so-called “Skolem paradox”, which was precisely for “nth derivative”, etc. Nor do we wish to empha- the point at issue between Zermelo and Skolem. size his misleading claim (p. 40) that Frege took a The extremely limited expressive power of first- logicist position on the foundations of geometry, order logic can be made clearer as follows: First- i.e., that geometrical objects are built on the basis order logic does not permit a theory of finite of logic alone (p. 40). This was not Frege’s view at groups. That is, there is no first-order axiomati- all, since he was very much a traditionalist in re- zation whose models are all and only the finite gard to geometry (unlike his treatment of the real groups. This is not accidental but essential since, numbers, where he was a logicist). Likewise, we by the Compactness Theorem, any collection of pass over Ebbinghaus’s failure (pp. 136–8), when first-order sentences that has arbitrarily large finite discussing at length the Axiom of Replacement models also has an infinite model. In this sense, the and its invention by Fraenkel and Skolem, to point models of first-order logic cannot distinguish the out that this axiom had actually been published finite from the infinite. Nor can first-order logic years before them by Dmitri Mirimanoff in 1917 distinguish between different infinite cardinali- [[5], 262]. ties; for any first-order theory that has countably But on one matter Ebbinghaus leaves a serious many primitive symbols (and these were the only misimpression that must be corrected: namely first-order theories considered before 1936), if the Zermelo’s 1929 dispute with Skolem on the foun- theory has an infinite model, then it has a model of dations of set theory and the role of first-order every infinite cardinality. The Löwenheim-Skolem (or second-order) logic in set theory. Ebbinghaus Theorem is the special case that if a first-order implies that Skolem was right, and Zermelo wrong theory has an infinite model, it has a model in the (and wrongheaded), in this dispute, since first- set of natural numbers. Thus, in first-order logic, order logic (with its Completeness Theorem) is the “set” of all real numbers has a model that is now the dominant form of mathematical log- a subset of the natural numbers, and the set of ic. Yet that Completeness Theorem (every valid all natural numbers also has uncountable models.

826 Notices of the AMS Volume 56, Number 7 Moreover, the set of all first-order sentences true proposed by A. Tarski and other of the natural numbers has a countable model logicians presupposes the general that is not isomorphic with the set of all natural notion of set. [Skolem in [7], 13] numbers. All of these aspects of first-order log- In the discussion following Skolem’s lecture, Tarski ic are common knowledge to logicians, although responded to this criticism: usually unfamiliar to mathematicians who are not [I] object to the desire shown by Mr. logicians. Skolem to reduce every theory to Given all of the above, why would logicians a denumerable model…. Because formulate set theory within first-order logic? (The of a well-known generalization of dominance of first-order logic only began in the the Löwenheim-Skolem Theorem, 1950s when Alfred Tarski developed model theo- every formal system that has an ry.) In December 1938 at a conference in Zurich infinite model has a model whose on the foundations of mathematics—a conference power is any transfinite cardinal to which Zermelo regretted not being invited— given in advance. From this, one Skolem returned again to the existence of count- can just as well argue for excluding able models for set theory and to the Skolem denumerable models from consid- paradox [[2], 37]. On this occasion he chose to eration in favor of uncountable emphasize the relativism, not only of set theory, models. [Tarski in [2], 17] but of mathematics as a whole. The discussion that Skolem aimed to cripple set theory. But Tarski’s followed Skolem’s lecture revealed both interest view, allowing models of all infinite in and ambivalence about the Löwenheim-Skolem within first-order logic, has dominated later devel- Theorem. Bernays commented at length on this opments and is a partial vindication of Zermelo’s matter: views. It is only partial, since today almost all set The axiomatic restriction of the theorists formulate set theory within first-order notion of set [to first-order logic] logic with its rich theory of models. does not prevent one from ob- taining all the usual theorems…of References Cantorian set theory…. Neverthe- [1] Pierre Dugac, Sur la correspondance de Borel less,…this way of making the no- et le théorème de Dirichlet-Heine-Weierstrass-Borel- tion of set (or that of predicate) Schoenflies-Lebesgue, Archives internationales d’histoire precise has a consequence of an- des sciences 39 (1989), 69–110. other kind: the interpretation of [2] Ferdinand Gonseth (ed.), Les Entretiens de the system is no longer necessar- Zurich sur les Fondements et la Méthode des Sciences ily unique…. It is to be observed Mathématiques. 6–9 Décembre 1938, Leemann, Zurich, 1941. that the impossibility of character- [3] Ivor Grattan-Guinness, The Search for izing the finite with respect to the Mathematical Roots: 1870–1940. , Set Theories, and infinite comes from the restrictive- the Foundations of Mathematics from Cantor through ness of the [first-order] formalism. Russell to Gödel, Princeton University Press, Princeton, The impossibility of characteriz- 2000. ing the countable with respect to [4] Gerhard Kowalewski, Bestand und Wandel. the uncountable in a sense that is Meine Lebenserinnerungen zugleich ein Beitrag zur in some way unconditional—does neueren Geschichte der Mathematik, Oldenbourg, Mu- this reveal, one might wonder, a nich, 1950. [5] Gregory H. Moore, Zermelo’s Axiom of Choice: Its certain inadequacy of the method Origins, Development and Influence, Springer, New York, under discussion here [first-order 1982. logic] for making axiomatizations [6] , Towards a History of Cantor’s Continuum precise? [Bernays in [2], 49–50] Problem, in The History of Modern Mathematics, volume Skolem objected vigorously to Bernays’ suggestion 1, pp. 78–121, as edited by David E. Rowe and John Mc- and insisted that a first-order axiomatization is Cleary, Academic Press, Boston, 1989. Thoralf Skolem surely the most appropriate. [7] , Une relativisation des no- tions mathématiques fondamentales, Colloques Interna- In 1958, at a colloquium held in Paris, Skolem tionaux du Centre National de la Recherche Scientifique, reiterated his views on the relativism of fundamen- Paris, 1958, pp. 13–18. tal mathematical notions and criticized Tarski’s contributions: It is self-evident that the dubi- ous character of the notion of set renders other notions dubious as well. For example, the semantic definition of mathematical truth

August 2009 Notices of the AMS 827