m'~]p~d..~_..,,,-- David E. R o w e , EditorJ

The Double Life of Felix Hausdorff/ Paul Mongr WALTER PURKERT Mathematical Institute Bonn University Beringstr. 7 D53115, Bonn, Germany e-marl: [email protected]~-bonnde

EDITOR'S NOTE Nearly every mathematician is familiar with the name Felix Hausdorff, and yet until fairly recently little was known about his life. Despite an extraordinarily productive career, perhaps unparalleled in intellectual and creative breadth, this multitalented figure only received scant attention from biographers or historians of mathematics. The circumstances of his life and tragic death, even the broader scope of his mathematical work, were forgotten, cast into the shadows. That situation changed dramatically, however, in the 1990s when a team of scholars began work on a unique endeavor in the annals of mathematical publishing: to produce a Hausdorff edition containing his Gesammelte Werke in nine volumes. Its projected structure and contents are: Band I: Biographie. Hausdorff als akademischer Lehrer. Arbeiten Ober geordnete Mengen Band II: Grundzu'ge der Mengenlehre (1914) Band III: Mengenlehre (1927, 1935). Arbeiten zur deskriptiven Mengenlehre und Topologie Band IV: Analysis, Algebra und Zahlentheorie Band V: Astronomie, Optik und Wahrscheinlichkeitstheorie Band VI: Geometrie, Raum und Zeit Band VII: Philosophisches Werk (Sant' Ilario (1897). Das Chaos in kosmischer Auslese (1898). Essays zu Nietzsche) Band VIII: Literarisches Werk (Ekstasen, Der Arzt seiner Ehre, Essays) Band IX: Korrespondenz

Send submisstons to David E. Rowe, Fachbereich 0 8 ~ l n s t l t u t fLir Mathematik, Johannes Gutenberg Umversity, D55099 Mainz, Germany.

This edition will ultimately contain reprints of all of Hausdorff's published astronomical and mathematical works along with detailed commentary. The volumes will also present for the first time a considerable number of carefully edited unpublished texts from a vast manuscript collection now housed at the Bonn University Library. Yet this is

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not all; as evident from the list of volumes, the edition will also contain Hausdorff's literary and philsophical works, written under a pseudonym, many of which enjoyed high critical acclaim in their day. The initial plans for this project grew out of activities in Bonn that took place in 1992, the fiftieth anniversary of Hausdorff's death. To mark that occasion Professor Egbert Brieskorn organized a special commemorative exhibit devoted to Hausdorff's life and work (Brieskorn 1992). Alongside this he also arranged a colloquium out of which emerged the essays in Brieskorn (1996). Brieskorn has continued to collect and assemble documents related to Hausdorff's biography for many years, and these documents will eventually appear in volume 1 of the edition. He was also instrumental in raising funds to finance the cataloguing of Hausdorffs papers, work carried out by Walter Purkert from 1993 to 1995. This catalogue (Findbuch) is now available online at www.aic.uni-wuppertal.de/fb7/hausdorff/findbuch.asp. It can also be used for searches, for example to determine whether, and if so in which documents, a person or concept of interest happens to appear. In order to pursue the broader objectives of this editorial project, it was necessary to bring together scholars representing a broad spectrum of expertise. In fact, the novelty of this venture is reflected by the backgrounds of the contributing editors who have been or are still associated with this project: 16 mathematicians, four historians of mathematics, two literary scholars, one philosopher, and one astronomer. Their nationalities are also diverse, representing Germany, Switzerland, Russia, the Czech Republic, and Austria. Since January 2002 the Hausdorff edition has been taken on as an official project of the Nordrhein-Westf~lischen Akademie. Editorial responsibility for the edition as a whole lies with Egbert Brieskom, supported by Friedrich Hirzebruch, Reinhold Remmert, Walter Purkert, and Erhard Scholz. Springer-

Verlag has assumed responsibility for publishing the Hausdorff edition, and five o f its nine volumes h a v e already a p p e a r e d : volumes IV (2001), II (2002), VII (2004), V (2005), a n d III (2008). T h e s e h a n d s o m e books, w h i c h ought

to be f o u n d in every major mathematics library, might even lead s o m e to brash u p their G e r m a n and try to savor the mathematical and literary delights they contain. As an a d d e d inducement, Walter Purkert, w h o has

m a n a g e d the meticulous daily w o r k of this difficult u n d e r t a k i n g from the beginning, offers the following c o m pelling portrait of Hausdorff's career and his extraordinary intellectual accomplishments. D.E.I1.

The Double Life of Felix Hausdorff/ Paul Mongr6 by Walter Purkert Translated by Hilde Rowe and David E. Rowe

n a 1921 review of Hausdorff's principal work, Grundzgige der Mengenlehre (1914), the American mathematician Henry Blumberg wrote: It w o u l d be difficult to n a m e a volu m e in any field of mathematics, e v e n in the u n c l o u d e d d o m a i n of n u m b e r theory, that surpasses the Grundztige in clearness a n d precision. (Blumberg 1921, 116) C o m p a r e that statement with another r e m a r k about Hausdorff in a letter from Paul Lauterbach, writer a n d translator, written to the musician a n d Nietzsche scholar, Heinrich K6selitz ( p s e u d o n y m : Peter Gast): A Dionysian mathematician! That s o u n d s incredible; but let him send s o m e t h i n g to you a n d w e will wager that there is s o m e t h i n g about him to be experienced. 1 "Dionysian" refers to Dionysius, the G r e e k g o d of wine, fertility, b u t also of ecstasy and the intoxicating, irrational, ecstatic elements necessary for experiencing the world or the creative process. An individual w h o writes b o o k s of mathematics of such unsurp a s s e d clarity and precision, o n the one hand, and is c o n s i d e r e d to be Dionysian, on the other, surely must l e a d a remarkable d o u b l e e x i s t e n c e - a n d Hausdorff was just such a man. As Felix Hausdorff he was an important mathematician w h o s e w o r k has rem a i n e d relevant and influential up to the p r e s e n t day; as Paul Mongrd he was a m a n of letters, a p h i l o s o p h e r , a n d a

critical essayist, a figure w h o m the journalist Paul Fechter recalled in 1948 in his autobiography, Menschen u n d Zeiten, as "one of the most r e m a r k a b l e individuals to a p p e a r in the first decades of the twentieth century" a n d w h o "has wrongfully b e e n forgotten b y the y o u n g e r generation." (Fechter 1948, 156) Naturally, in this d o u b l e life m a n y visible a n d invisible threads b e c a m e in-

tertangled, a n d these must b e retraced to u n d e r s t a n d p r o p e r l y the man and his work. Felix Hausdorff was b o r n in Breslau on N o v e m b e r 8, 1868. His father, a Jewish b u s i n e s s m a n n a m e d Louis Hausdorff (1843-1896), m o v e d in the Fall of 1870 with his y o u n g family to Leipzig, w h e r e he m a n a g e d various companies including linen and cotton

Figure I. Felix Hausdorff working in his home study in Bonn during the early 1920s.

1Paul Lauterbach. Letter to Heinrich K6sehtz from 30 December, 1893. Goethe- und Sch~llerarch~v Weimar. 102/417,

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shops. He was an e d u c a t e d man w h o at age 13 had o b t a i n e d the title Morenu. 2 He wrote several papers, including a long o n e o n the Aramaic translation of the Bible from the perspective of Talmudic Law, which a p p e a r e d in the Monatsschrifi f~r

Geschichte und Wissenschafi des Judenthums. For m a n y y e a r s Louis Hausdorff was involved with the "DeutschIsraelitischen G e m e i n d e b u n d , "3 and was m a d e a m e m b e r o f the executive committee b e c a u s e its presiding officer thought it w o u l d b e desirable to have the d e c i d e d l y conservative position, for w h i c h Mr. Louis Hausdorff was known, to b e represented on the committee. 4 In an 1896 obituary o f Louis Hausdorff, the G e m e i n d e b u n d recalled: His great and n o b l e heart beat warmly for the affairs of his fellow believers. At the s a m e time, he was a devoted, self-sacrificing father in the true Jewish sense; in the same manner, his bountiful acts of charity c o r r e s p o n d e d to the most beautiful tradition of our p e o p l e . (Ibid., Nr. 44, 1896.) Felix Hausdorff's mother, Hedwig (1848-1902; she was called J o h a n n a in various documents), w a s a m e m b e r of the w i d e l y d i s p e r s e d Jewish family Tietz. From one b r a n c h of this family came Hermann Tietz, the founder of the first d e p a r t m e n t store and later the principal o w n e r of a chain of department stores "Hermann Tietz." During the p e r i o d of the National-Socialist dictatorship, the firm was "aryanized" und e r the n a m e HERTIE. We d o not k n o w h o w Felix Hausdorff was r e a r e d as a child, but it seems likely he h a d a strict religious upbringing. In a report to the executive committee of the "Deutsch-Israelitischen G e m e i n d e b u n d " his father said, The center of J u d a i s m is not found in the sermon, n o r in the religious services. Its true focus is much more to b e f o u n d in the religious life of the family.

H o w Felix Hausdorff reacted to his upbringing is also unclear, but evidence points to sharp differences with his father's views. In one of his aphorisms, he later wrote: W h o e v e r invented the fable of the h a p p i n e s s of c h i l d h o o d forgot three things: religion, upbringing, and the early phases of sexuality. (Hausdorff 1897a, 254) In a n o t h e r aphorism, he writes, in reg a r d to the rearing of c h i l d r e n in his day, But the m e t h o d is still the s a m e today: exterminate, hinder, cut off, deny, restrict, p r o h i b i t - - i t w a s a fundamentally negative, privatistic, prohibitive m e t h o d or rearing, improving, punishing---eradicating instead o f creating, amputating instead of healing. (Hausdorff 1897a, 62) The results of Felix Hausdorff's religious training were the o p p o s i t e of w h a t his father w a n t e d to achieve: H a u s d o r f f gave up practicing the Jewish faith. He b e c a m e an agnostic w h o critically disputed the tenets of Jewish religion just as he did the Christian. Still, he w a s never baptized, a religious rite that w o u l d have offered him considerable advantages. Hausdorff's educational b a c k g r o u n d was in m a n y ways typical for a child from a middle-class family with high aspirations. For three years he attended the former s e c o n d Btirgerschule in Leipzig; afterward, b e g i n n i n g in 1878, he w e n t to the Nicolai Gymnasium. This school had an excellent reputation as a humanistic educational institution. Hausdorff was an outstanding pupil, the b e s t in his class over m a n y years, a n d h e often was given the h o n o r of r e a d i n g the p o e m s he h a d c o m p o s e d in Latin or G e r m a n during school vacations. In his graduating class of 1887 he w a s the only pupil to receive the cumulative grade of "I." The focus of the g y m n a s i u m education w a s on classical languages, which c o m p r i s e d app r o x i m a t e l y 45% of the o b l i g a t o r y curriculum. Hausdorff was required, for

example, in the final examination for graduation to write a Latin essay on the theme: "Cupidius q u a m verius Cicero dicit res u r b a n a s bellicis rebus a n t e p o n e n d a s esse'" (freely translated: "it corr e s p o n d s m o r e to Cicero's interests than the truth w h e n he states that matters of public welfare have priority o v e r those of warfare"). (Jahresbericht des" Nicolai-Gymnasiums ffir das Jahr 1887, X-XI) The c h o i c e of field for his university studies may well have b e e n a difficult o n e for the muhitalented Felix Hausdorff. Magda Dierkesmann, a student in B o n n from 1926-1932 w h o w a s often a guest in Hausdorff's home, reported m a n y years later: His versatile musical talent was so great that it was only due to the urging of his father that he gave u p his plans to study music and b e c o m e a composer. 5 By the time h e graduated, the d e c i s i o n had b e e n r e a c h e d (though w e d o not k n o w w h a t p r o m p t e d it): in the a n n u a l report of the Nicolai G y m n a s i u m for 1887, next to the list of graduates, o n e finds a c o l u m n giving the "future field

Figure 2. Hausdorff as he appeared during his tenure in Greifswald, 1913-1921.

2Morenu being Hebrew for "our teacher"; this title was conferred on those who quahfled to teach as rabbas. 3This organization was founded after the creation of the German Re~ch to represent the rnterests of German Jews w~thfn the new state 4Mittheilungen des Deutsch-lsraelltischen Gemeindebundes, Nr. 5 (1878). The conservatives maintained a strict line with respect to convenhonal rel@ous practices. They fought, for example, to have Jewish pupils freed from attending the Gymanslum on the Sabbath so that they could attend the Synagogue, or at a minimum that they be freed from wnting tests on these days. 5(Dierkesmann 1967, 51-52) In a conversation with Egbert Brieskorn, Frau D~erkesmann assured him that she was told this d~rectly by Hausdorff

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Figure 3. The elderly Hausdorff remained active mathematically up until his suicide in 1942.

of study," which for Felix Hausdorff w a s "natural sciences." (Jahresbericht

des Nicolai-Gymnasiums f ~ r das Jahr 1887, XVl) B e t w e e n 1887 and 1891, Hausdorff s t u d i e d mathematics a n d a s t r o n o m y in Leipzig, though with interruptions of o n e semester each to study in Freiburg (SS 1888) and in Berlin (WS 1888/1889). H e had exceptionally b r o a d interests a n d t o o k courses in mathematics, astronomy, physics, chemistry, and geography. He also a t t e n d e d lecture courses in p h i l o s o p h y a n d history of philosophy, languages a n d literatures, a n d on the history of socialism a n d the l a b o r movement. O n e of his electives w a s a course on the scientific foundations of belief in a p e r s o n a l God, a n d a n o t h e r dealt with the relationship bet w e e n mental disorders a n d crime. He also a t t e n d e d lectures b y the Leipzig musicologist Paul on the history of music. Hausdorff's early love for music rem a i n e d with him all his life. Already as a student in Leipzig he h a d a special affinity for and excellent k n o w l e d g e of the music of Richard Wagner. Later, he often invited friends to his h o m e for musical evenings and r e g a l e d them at the piano.

During his last semesters as a student in Leipzig, Hausdorff w o r k e d closely with Heinrich Bruns (1848-1919), professor of a s t r o n o m y and director of the astronomical observatory. Bruns, a student of Karl Weierstrass, was k n o w n a b o v e all for his w o r k on the threeb o d y p r o b l e m and on optics (Bruns's Eikonal). He gave Hausdorff a dissertation topic on the refraction of light in the a t m o s p h e r e (Hausdorff 1891). This w o r k w a s f o l l o w e d b y two further p u b lications o n the same subject, leading up to Hausdorff's Habilitation for w h i c h he s u b m i t t e d a study on the extinction of light in the a t m o s p h e r e (Hausdorff 1895). T h e s e early astronomical w o r k s by Hausdorff w e r e - - t h e i r excellent mathematical presentation notwiths t a n d i n g - o f no further consequence. As it t u r n e d out, Bruns's principal i d e a was u n w o r k a b l e (astronomical observations of refraction near the h o r i z o n were required, which, as Julius Bauschinger thereafter showed, w e r e impossible to obtain with the necessary exactitude). Beyond this, n e w technology o p e n e d the possibility of obtaining atmospheric data directly b y m e a n s of test balloons, thereby obviating the n e e d for difficult calculations of these data b a s e d on refraction observations. With the Habilitation, Hausdorff could b e g i n his academic career as a Privatdozent in Leipzig. He offered a wide range of courses, but alongside his teaching and research he also continued to pursue literary a n d philosophical interests. This brought him into contact with a circle of n o t e w o r thy writers, artists, and publishers that included H e r m a n n Conradi, Richard Dehmel, Otto Erich Hartleben, Gustav Kirstein, Max Klinger, Max Reger, a n d Frank W e d e k i n d . During the p e r i o d from 1897 to 1904--the high p o i n t of his o w n literary and philosophical crea t i v i t y - h e p u b l i s h e d eighteen of the twenty-two w o r k s that a p p e a r e d u n d e r his p s e u d o n y m , including a v o l u m e of p o e m s , a play, a b o o k o n e p i s t e m o l ogy, a n d a v o l u m e of aphorisms. The b o o k of aphorisms was the first a m o n g Hausdorff's works to a p p e a r u n d e r the p e n n a m e Paul Mongr~. He entitled it Sant' Ilario. Gedanken aus

der Landschafi Zarathustras (Hausdorff 1897a). The choice of p s e u d o n y m already suggests his orientation: ft m o n g r e - - a f t e r my o w n taste. This reflected an individuality, spiritual autonomy, and a rejection of prejudices and conformity in political, social, religious, or other spheres o f h u m a n affairs. The subtitle, " G e d a n k e n aus der Landschaft Zarathustras" stems from the circumstance in w h i c h Hausdorff c o m p l e t e d his b o o k while recuperating on the Ligurian coast n e a r Genoa, the same locale w h e r e Friedrich Nietzsche wrote the first two parts of Also sprach Zarathustra; the subtitle also naturally suggests a strong spiritual affinity to Nietzsche. In a p r e v i e w of Sant'Ilario that a p p e a r e d in the w e e k l y magazine Die Zukunft, Hausdorff explicitly acknowle d g e d his debt: O n this blissful coast [- 9 -] I followed the lonely paths of Zarathustra's creator-wonderful, narrow paths along b a n k s a n d cliffs that have no r o o m for m o v i n g an army. If o n e should thus wish to count me a m o n g Nietzsche's followers, then let this serve as m y o w n confession. 6 Hausdorff did not attempt to c o p y Nietzsche let a l o n e surpass him; as one reviewer put it, there is "not a trace of mimicking Nietzsche." Hausdorff positions himself, so to speak, next to Nietzsche in an effort to release his individual thoughts a n d to gain the freed o m n e e d e d to question conventional norms. He also m a i n t a i n e d a critical distance to Nietzsche's later works. In his essay on Nietzsche's Der Wille z u r Macht, a b o o k c o m p i l e d from various fragments in the Nietzsche-Archiv, he wrote: Nietzsche g l o w s like a fanatic. If his moral order b a s e d on breeding w e r e to be established drawing on our m o d e r n k n o w l e d g e of biology and physiology, that could lead to a world historical scandal c o m p a r e d to which the Inquisition and witch trials w o u l d a p p e a r like merely harmless confusions. (Hausdorff 1902, 1336) Yet Hausdorff t o o k his critical standard from the y o u n g e r Nietzsche, from the gracious, moderate, under-

6(Hausdorff 1897b, 361). For details on Hausdorff's relationshtp to Nletzsche, see Stegma~er (2002) as well as the h~stoncal introductton to Hausdorff (2004)

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Figure 4. The International Conference Center, University of Greifswald, is named for Felix Hausdorff. Photo by Stan Sherer. standing, free spirit Nietzsche and from the cool, dogma-free, systemless skeptic Nietzsche [. 9.] (Ibid, 1338) Any attempt to d e s c r i b e the contents of a v o l u m e of a p h o r i s m s w o u l d b e senseless, but in o r d e r to say at least something a b o u t it, o n e can point to two ideas that Hausdorff takes up over and again: first, he e x p r e s s e s a d e e p skepticism with regard to all forms of teleology and, even more, ideologies or theories for improving the world that claim to k n o w the true meaning and p u r p o s e of humanity. As exemplars of this, consider these t w o excerpts from the first and third Aphorisms: The w o r l d is so full of outrageous nonsense, cracks, fragmentation, chaos, "free will"; I envy those w h o s e good, synthesizing eyes are able to see the w o r l d as the unfolding of an "idea," a single idea. (Hausdorff 1897a, 4) If not truth itself, t h e n surely the belief in holding truth is, to a dangerous degree, antagonistic to life and m u r d e r o u s for the future. Not one of those w h o d e l u d e d themselves that they w e r e b l e s s e d with the truth hesitated for a m o m e n t to p r o n o u c e the grand finale, or the great day, or some other e n d point, turning point, or climax for humanity, and every time this m e a n t that all future humanity was to b e m o l d e d by their image, their stamp, a n d their narrowness. (Hausdorff 1897a, 6)

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This raises the question of the relationship b e t w e e n individuals a n d society. For Hausdorff, as for Nietzsche, the individual is no m e r e figure within an historical process that subordinates his individuality to a higher order. O n the contrary, individuals, especially those w h o are creative, should b e p l a c e d in the c e n t e r and their rights s h o u l d be d e f e n d e d . From Aphorism 35: Fruitful is a n y o n e w h o calls something his own, w h e t h e r m a k i n g or enjoying, in s p e e c h or gesture, in

longing or posessing, in science or culture; fruitful is everything that occurs less than twice, every tree g r o w i n g in its soil and reaching u p to its sky, every smile that b e l o n g s to only o n e face, every thought that is only o n c e right, every e x p e r i e n c e that b r e a t h e s forth the heartstrengthening smell of the individual! (Hausdorff 1897a, 37) The year 1898 saw the a p p e a r a n c e of Hausdorffs critical epistemological s t u d y - - a g a i n under the p s e u d o n y m Paul M o n g r ~ - - D a s Chaos in kosmischer Auslese (Chaos in cosmic selection). Its critique of metaphysics resulted from Hausdorff's effort to come to terms with Nietzsche's idea of eternal recurrence. His aim is nothing less than to destroy permanently every type of metaphysics. Regarding the world in itself, a transcendental world core as Hausdorff calls it, w e k n o w nothing and can k n o w nothing. W e must take "the world in itself' to be u n d e t e r m i n e d and indeterminant, a m e r e chaos. Our world of experience, o u r cosmos, is a result of selection, w h i c h w e have always involuntarily u n d e r t a k e n and continue to undertake according to our possiblilities of knowledge. Starting from that chaos, there are any n u m b e r of other orders, other cosmoi, that could be conceived, but from the w o r l d of our cosmos there is no possibility of drawing conclusions regarding a transcendental world.

Figure 5. Hausdorff's home at Am Graben 5, Greifswald, .with mathematical tourists. Photo by Stan Sherer.

W e will have to s h o w the full diversity of both w o r l d s a n d the untenability of any r e a s o n i n g from empirical consequences to transcendental premises [" 9 -] a n d to do so in a comprehensive generality that also goes b e y o n d Kant's result in a practical way [. 9 .] (Hausdorff 1898, 4) As methodology, he p r o p o s e s , [. 9 .] w e have [- 9 -] simply to determ i n e those transcendental variations that leave a given empirical phen o m e n o n unchanged. (Hausdorff 1898, 9) In Chaos in kosmischer Auslese he att e m p t e d to carry out this p r o g r a m for the categories of time and space. Consider the following passage from his Leipzig inaugural lecture Das Raumproblem. By studying a m a p one can never determine the form of the original space without knowing the method of projection used to obtain it. Thus [- 9 -] our empirical s p a c e is just such a physical map, an i m a g e of the absolute space [absolute in the sense of transcendental]; but [ ' - ' ] w e d o n o t k n o w the m e t h o d of projection a n d so w e cannot k n o w the original. The two spaces are related by m e a n s of an u n k n o w n a n d undetermined c o r r e s p o n d e n c e , a completely arbitrary point transformation. Still, the empirical space maintains its value as a m e a n s of orientation; w e are able to find our w a y with this m a p a n d w e c a n c o m m u nicate with those w h o also possess this map; the distortion n e v e r enters o u r consciousness b e c a u s e not only the objects but w e ourselves and our measuring instruments are uniformly affected by this. [. 9 .] If this v i e w p o i n t is correct, then it must be possible for the p r e i m a g e to u n d e r g o an arbitrary transformation without changing the image:

[...]7

The simplest such transformation w o u l d be a uniform shrinking or exp a n d i n g of the transcendental space by a constant factor. But Hausdorff was

c o n c e r n e d with arbitrary transformations, w h i c h means that the transcendental s p a c e must remain c o m p l e t e l y u n d e t e r m i n e d and i n d e t e r m i n a n t - such a s p a c e is thus a senseless concept, scientifically speaking. Hausdorff w o r k e d intensively on the s p a c e p r o b lem for m a n y years; in the winter semester 1903 to 1904 he offered a lecture c o u r s e in Leipzig on Zeit u n d R a u m (Time a n d Space) (NL Hausdorff: Kapsel 24: Fasz. 71), in which he s p o k e of his p a s s i o n for this p r o b l e m . The fundamental concept of a topological space, w h i c h he later created, was conceived in o r d e r to a c c o m o d a t e practically every situation in which "spatiality," in the topological sense, plays a role. This c o n c e p t was p r o b a b l y influe n c e d b y his philosophical reflections on the s p a c e problem. It is especially striking that in Chaos in kosmischer Auslese, a p h i l o s o p h i c a l study, Hausdorff brought in e l e m e n t s from the very newest mathematics, set theory. This unique but also p r o b l e m atic a s p e c t surely m a d e the w o r k ' s reception m o r e difficult. In 1904 the periodical Die n e u e R u n d s c h a u published Hausdorffs o n e act p l a y D e r A r z t seiner Ehre (The Surgeon o f his Honour). This earthy satire dealt with duelling and the c o n v e n tional c o d e of h o n o r of aristocrats a n d the Prussian officers' corps. By Hausdorff's day, such forms of chivalry w e r e b e c o m i n g o u t m o d e d in bourgeois society. The H a m b u r g e r Echo on 15 Nov e m b e r 1904 n o t e d in a review, Mongre has the courage to s h o w duelling in the light that it deserves. He treats it as comedy, a b o u t w h i c h one can agree over a glass of w i n e so long as one is not c h a i n e d like a vain fool to the fashion d e m o n of "honor". Der A r z t seiner Ehre was Hausdorff's greatest literary success. B e t w e e n 1904 and 1912 it was p e r f o r m e d over 300 times on stages in Berlin, Brunswick, Bremen, Breslau, Bromberg, Budapest, Dusseldorf, Dortmund, Elberfeld, El-

bing, Frankfurt, Furth, Graz, Hamburg, Hannover, Kassel, Cologne, Koenigsberg, Krefeld, Leipzig, Magdeburg, Muhlhausen, Munich, Nuremberg, Prague, Riga, Strassburg, Stuttgart, Wien, W i e s b a d e n and Zurich. 8 Hausdorff was anything but an obscure playwright. In June 1912 he attended a banquet at the Hotel Esplanade in Berlin held in h o n o r o f Frank W e d e k i n d , arriving in the c o m p a n y of Max Reinhardt, Felix Holl~inder, a n d Arthur Kahane, the cr~me de la cr~me o f the Berlin theatrical scene. 9 W e must content ourselves with these few glimpses of Hausdorff's literary and philosophical works without touching on his p o e t r y volume Ekstasen (1900) or his essays, true pearls in this literary genre, a b o u t which see Vollhardt 2000. Most of the essays app e a r e d in the periodical Neue Deutsche Rundschau (Freie Btihne) (later ren a m e d Die n e u e Rundschau (Freie Btihne)), the then leading literary journal about w h i c h was said: "Remember, that y o u r life will pass, even if y o u m a d e it into the Neue Rundschau (Gedenke, Mensch, dass Du vergehst, auch w e n n Du in d e n Neuen Rundschau stehst)." After the S e c o n d W o r l d War, Hausdorff's philosophical writings were totally forgotten, as w e r e his literary works. One might conjecture that antiSemitism and the cultural barbarism of the Nazi distatorship contributed to this neglect. Up until the Nazi era there was still a public a w a r e n e s s of Hausdorff as a p h i l o s o p h e r a n d writer, as this entry in the 1931 edition of the Grolgen Brockhaus s h o w s (dates and localities of his life have b e e n omitted): Hausdorff, Felix, Mathematician a n d Writer. He is the a u t h o r of: G r u n d z ~ g e der Mengenlehre (1914), a n d u n d e r the p s e u d o n y m Paul Mongre the epistemological study "Das Chaos in kosmischer Auslese" (1898), the works "Sant' Ilario. G e d a n k e n aus d e r Landschaft Zarathustras" (1897) and "Ekstasen"

7(Hausdorff 1903, 15) How such.transformattons might affect physical properties remains open here In a posthumous (unfortunately undated) fragment "Transformationsprincip" Hausdorff wrote about thts: "That the physical content also m~ght take part in the transformation needs to be considered more carefully. That is perhaps not so simple Perhaps ~n this respect the principle is even objectionable--an idea I find attractive now that I've noticed that others (Po~ncar6) have taken up th~s pnnctpleW' (NL Hausdorff. Kapsel 49: Fasz. 1079, BI 3 ) 8For the review cited above and the information about these performances I thank Udo Roth, Munich. 9Frank Wedekind. Gesammelte Bnefe Hrsg von Fritz Strich. Bd. 2, M0nchen 1924, p 269, for th~s reference I thank Ariane Martin, Mainz

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(1900). He has close affinities t o the fundamental ideas o f Nietzsche; he rejects all metaphysics and regards the w o r l d of e x p e r i e n c e as a segment d r a w n by consciousness out of a lawless chaos. Regarding his philosophical contributions, W. Stegmaier w r o t e in the preface to volume VII of Hausdorff's G e s a m melte Werke,

The m o r e I delve into Felix Hausdorff's writings, the m o r e they comm a n d m y respect: for their clarity, their honesty, their n o b l e modesty, their intellectual i n d e p e n d e n c e , and above all for their astonishing present-day relevance. Perhaps n o w the time has come, after one h u n d r e d years, that they can l e a d to fruitful philosophical orientation as they so deserve. (Hausdorff 2004, VII.) In 1899 Hausdorff married Charlotte Goldschmidt, the d a u g h t e r of a Jewish physician, Siegismund Goldschmidt, from Bad Reichenhall. This man's stepmother, incidentally, w a s the famous feminist and p r e s c h o o l p e d a g o g u e , Henriette Goldschmidt. In 1900 the Hausdorffs' daughter Lenore (Nora), their only child, was born; she survived through the Nazi era a n d died in 1991 in Bonn. In D e c e m b e r of 1901 Hausdorff was a p p o i n t e d as an unofficial associate professor (aulgerplanm~i6iger Extraordinarius) at Leipzig University. In submitting the faculty's p r o p o s a l for this appointment, which c o n t a i n e d a very favorable assessment given by his colleagues and c o m p o s e d b y Heinrich Bruns, the Dean a d d e d the following remark:

Figure 6. The plaque outside Hausdorff's home in Greifswald. Photo by Stan Sherer.

The faculty considers itself, however, duty b o u n d to inform the Royal Ministry that the present p r o p o s a l w a s not a p p r o v e d by all m e m b e r s in the meeting on the 2nd of Nov e m b e r this year, but rather b y a v o t e o f 22 to 7. The minority w h o v o t e d against Dr. Hausdorff did so b e c a u s e he is of the Jewish faith. 1~ This a m e n d a t o r y remark illuminates at a g l a n c e the o p e n anti-Semitism that was especially on the rise across the entire G e r m a n Empire after the financial crash (GrQnderkrach) that followed its f o u n d i n g in 1871. Leipzig w a s at the center of the anti-Semitic m o v e m e n t , in w h i c h students p l a y e d a large role. This m a y w e l l have b e e n one r e a s o n w h y Hausdorff never felt particularly comfortable teaching there; a n o t h e r reason was the strong sense of hierarchy a m o n g the full professors (Ordinarien), w h o t e n d e d to disregard their junior colleagues. Later in Bonn Hausdorff c o m m e n t e d retrospectively in a letter to Friedrich Engel: In B o n n one has the feeling, even as a junior faculty m e m b e r (NichtOrdinarius), of being formally acc e p t e d , a sense I could n e v e r bring m y s e l f to feel in Leipzig [an der Pleisse] (Letter from 21 February 1911. NL Engel, UB Gielgen, Handschriftenabteilung) Hausdorffs mathematical research covered unusually broad terrain: he wrote p a p e r s on such diverse topics as optics (Hausdorff 1896), non-Euclidean g e o m e t r y (Hausdorff 1899), hypercomplex n u m b e r systems (Hausdorff 1900b), insurance mathematics (Hausdorff 1897c), and probability t h e o r y (Hausdorff 1901b). The last two w o r k s contain several noteworthy results that w e r e not without influence. In Hausdorff (1897c) he introduced the varia n c e o f an insurer's losses as a measure o f risk. T o d a y theories of individual risk have given w a y to collective risk theories, but variance of loss nevertheless remains a fundamental quantity for the evaluation o f insurance plans with fixed coverages a n d premiums. (In this p a p e r Hausdorff also pres e n t e d a first correct p r o o f of the Theo r e m of Hattendorff.) For various types

of life insurance he calculated the variance of loss, a n d these results w e r e taken up i m m e d i a t e l y afterward in the textbook literature. In Hausdorff (1901b) he called special attention to the concept of conditional probability, a notion of fundamental importance that had only b e e n used implicitly u p until then. He also introduced the n e w terminology ("relative probability") along with a suitable notation for it. 11 In this s a m e p a p e r (and i n d e p e n d e n t of Thiele) he dealt with semi-invariants and gave highly simplified derivations of the Gram-Charlier series of T y p e A. His e x a m p l e o f a s e q u e n c e of independent, identically distributed r a n d o m variables X1, X2, . . with density q~(x) = 89e -jx], for which Zn =

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a~X k

with ak =

1

(k +89

d o e s not c o n v e r g e to a normal distribution, p r o v i d e d the motivation for P. L6vy to formulate an interesting conjecture a b o u t the d e c o m p o s i t i o n of the normal distribution into two i n d e p e n dent c o m p o n e n t s . This conjecture from the early 1930s was p r o v e d in 1936 b y H. Cram~r (for details, see Hausdorff 2005, 579-583). Hausdorff's principal field of research, h o w e v e r , soon b e c a m e set theory, especially the theory of o r d e r e d sets. Initially it was his philosophical interests that l e d him to Cantor's ideas (see Hausdorff 2002, 3-5). In the summer semester of 1901, Hausdorff offered a lecture course on set theory; this was nearly a first in Germany, o n l y Ernst Zermelo's course in G6ttingen the previous semester p r e c e d e d it. (Cantor himself n e v e r offered lectures on set theory in Halle.) It was in the context of teaching this course that Hausdorff m a d e his first discovery in set theory: the type class T(t%) of all c o u n t a b l e o r d e r types has the p o w e r I~ of the continuum. He s o o n found, though, that this t h e o r e m h a d a p p e a r e d in Felix Bernstein's D i s s e r t a t i o n - - a n d carefully n o t e d in the margin of his manuscript: Presented o n 27 June 1901. Dissertation of F. Bernstein received o n 29 June 1901. (NL Hausdorff: Kapse103: Fasz. 12, Bk 37)

lOArchiv der Universit&t Leipzig, PA 547. The full report is reproduced in Beckert and Purkert (1987), 231-234. 11Kolmogoroff later adopted this notation (PB(A))in his book Grundbegriffe der Wahrscheinlichkeitsrechnung (1933).

42

2 k=l

Hausdorff u n d e r t o o k a t h o r o u g h study of o r d e r e d sets, motivated in large part by Cantor's continuum p r o b l e m , which p o s e s the question of finding the place o c c u p i e d by t~ = 2~0 in the s e q u e n c e of t h e N~.~2 In a letter to Hilbert dated 29 S e p t e m b e r 1904, he said this problem "had p l a g u e d me almost like an obsession" (NL Hilbert, Niedersfichsische Staats- u n d Universit~itsbibliothek zu G~Sttingen, Handschriftenabteilung, Nr. 136). He thought that the t h e o r e m card (T(t%)) = t~ offered a n e w strate g y for attacking the p r o b l e m . Cantor h a d long conjectured that 11 = ~1, but it h a d only b e e n p r o v e d that t~ -> ~ , w h e r e ~ represents the "number" of p o s s i b l e well-orderings of a countable set. It turned out that t~ is the "number" of all possible orderings of such a set, which naturally led to the study of special types of orderings that were m o r e general than well-orderings but less general than arbitrary orderings. This was precisely what Hausdorff did in his first set-theoretic publication (Hausdorff 1901a) in w h i c h he studied "graded sets" (gestufte Mengen). Today w e k n o w , from the results o f K. GOdel a n d P. Cohen, that this strategy for solving the continuum p r o b l e m h a d no m o r e chance of attaining its goal than Cantor's approach, which tried to generalize the Cantor-Bendixson t h e o r e m for c l o s e d sets to the case of arbitrary u n c o u n t a b l e point sets. In 1904 Hausdorff p u b l i s h e d the recursion formula that n o w carries his name: for every nonlimit ordinal tx

This formula, together with the concept of cofinality that he later introduced, s e r v e d as the foundation for all further results on the e x p o n e n t i a t i o n of alephs. Hausdorff's precise k n o w l e d g e of the problematics of recursion formulae of this t y p e e n a b l e d him to d e t e c t an error in Julius K6nig's lecture presentation at the 1904 International Congress of Mathematicians in Heidelberg. K~Snig c l a i m e d to have "proved" that the continuum cannot be well-ordered, which

dered sets, as Hausdorff showed. If, for example, the o r d e r e d d e c o m p o s i t i o n A = P + Q represents a g a p - t h a t is, P has n o greatest a n d Q no smallest elem e n t - t h e n there exist two uniquely determined regular initial numbers to~, ton such that P is cofinal with to~ a n d Q is coinitial with to,~. Hausdorff called the pair (to~,to,~) =: c~n the character of the gap. By this means, he o b t a i n e d from the decomposition A = P + {a} + Q a uniquely d e t e r m i n e d character for the element a, t h o u g h here one must allow for characters of the type (1,to~), (to~,l), or (1, 1). Thus, in the set o f rational numbers (with the natural orm-1 (0,1) is cofinal with { - - - 7 ] m e N a n d dering), all gaps a n d elements have the character Coo. If W is a given set of characters coinitial with { 1 }u~ N. These concepts (for elements a n d gaps), for e x a m p l e carry over to order types: for e x a m p l e , W = {Coo, Col, q0, c22}, the question the t y p e ,~ associated with the set of naturally arises w h e t h e r there exists an real n u m b e r s u n d e r its natural ordering o r d e r e d set having precisely W as its is cofinal with the type to of the natset of characters. A necessary condition ural numbers. for W is relatively easy to find, but An ordinal n u m b e r is called regular Hausdorff s u c c e e d e d in showing that if it is not cofinal with a smaller ordi- this condition is also sufficient. For this nal number, otherwise it is called sin- p u r p o s e one n e e d s a large reservoir of gular. Hausdorff called the smallest o r d e r e d sets, a n d this Hausdorff w a s n u m b e r in each of Cantor's n u m b e r able to create using his theory of genclasses an initial n u m b e r (An- eral o r d e r e d products a n d p o w e r s (see fangszahl) 1~: to = too, to> to2. . . . . ~o~o, Hausdorff 2002, 604-605.) In this resertoo,+], 9 9 9 All to~+] are regular, but voir o n e finds such interesting structures to,, = limn to,, is cofinal with to a n d thus as Hausdorffs rl~ non'nal types. Cantor an e x a m p l e of a singular initial numhad already considered the type 77 = r/0 ber. Hausdorff asked w h e t h e r there ex- of the rational numbers in their natural ist regular initial numbers with a limit ordering. He discovered that this type is n u m b e r as index, a query that s e r v e d universal with respect to the type class as the p o i n t of departure for the theT(N0) of all countable order types, that ory of inaccessible cardinal numbers. is, for every countable order type /~ He realized that, were such n u m b e r s to there exists a subset in r/ that has the exist, they w o u l d have to be of "exor- type /x. Hausdorff's r/~ type accombitant size. ''15 plishes the same thing for the type class In c o n n e c t i o n with this theory HausT(N~. The question w h e t h e r there exdorff p r o v e d the following f u n d a m e n ist rl~+l sets with the least possible cartal theorem: for every d e n s e o r d e r e d dinality t ~ + ] then leads to the question set A without b o u n d a r y there exist t w o whether 2~ = N~+I holds. It was in this uniquely d e t e r m i n e d regular initial context that Hausdofff raised the gennumbers to~, to,7 such that A is cofinal eralized continuum hypothesis for the with to~ a n d coinitial with to~ ( w h e r e * first time. His r/~ sets also served as the signifies the inverse ordering). This thepoint of departure for the notion of sato r e m offers a sensitive instrument for urated structures, which has since characterizing gaps and elements in or- played a major role in m o d e l theory. 16 w o u l d have implied that its cardinality is not an aleph, a result that e v o k e d c o n s i d e r a b l e interest. It was only recently d i s c o v e r e d that it was Hausdorff w h o u n c o v e r e d this error. 13 B e t w e e n 1906 and 1909 Hausdorff p u b l i s h e d his fundamental w o r k s o n o r d e r e d sets (Hausdorff 1906, 1907a, 1907b, 1908, 1909). For this theory, his concepts of cofinality and coinitiality were of k e y importance. If A is an ord e r e d set a n d M C A, then A is said to b e cofinal (coinitial) with M if for e v e r y a E A there exists an m E M such that m --> a ( m -- a). Thus, for example,

12Hausdorff's reflections on time as background for h~s study of order structures are taken up by E. Scholz in his article Loglsche Ordnungen im Chaos Hausdorffs frOhe Beltr2,ge zur Mengenlehre (in Brieskorn 1996, 107-134). 13Detailed ~nformat~on can be found ~n Hausdorff (2002, 9-12) and ~n Purkert (2004). Further ~mportant source material on this story can be found in Ebbinghaus (2007). ~4Today these are identified with the cardinal numbers: ~0, ~1, . ~,~, ~,~+~ . . . . ~5See Hausdorff (2002) and the commentary by U. Feigner, pp. 598-601. 16On this, see the essay by U. Feigner. D~e Hausdorffsche Theone der .q~-Mengen und ihre W~rkungsgeschlchte, In Hausdorff (2002, 645-674).

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Hausdorff's general products and p o w e r s also led him to the c o n c e p t of partially o r d e r e d sets. It turned out that the final gradations o f s e q u e n c e s and functions he had b e e n intensively studying w e r e partial orderings. His p r o o f of the existence o f (ah,aJ~) gaps in the maximal o r d e r e d subsets of these s e m i o r d e r e d sets is a m o n g his d e e p e s t results in set theory. Hausdorff s h o w e d that every o r d e r e d subset of a partially o r d e r e d set is c o n t a i n e d in a maximal o r d e r e d subset; this t h e o r e m is k n o w n t o d a y as Hausdorff's maximal chain theorem ("Maximalkettensatz'). Not only does the t h e o r e m follow from the well-ordering t h e o r e m (resp. the axiom of choice), it was later s h o w n to be equivalent to both o f them. 17 In 1908, Arthur Schoenflies had pointed out that the m o r e recent theory of ordered sets (that is, the extensions of this theory undertaken after Cantor) were almost exclusively due to Hausdorff (Schoenflies 1908, 40.) Indeed, developments within set theory immediately after Cantor have, in comparison with foundational issues, received comparatively little attention in the historical literature with the exception of the work of Zermelo. This applies in particular to the contributions of Hausdorff and Gerhard Hessenberg. In the summer semester of 1910, Hausdorff was a p p o i n t e d to a position as official associate professor (planm~i6iger Extraordinarius) at Bonn University. As mentioned previously, he found the academic atmosphere in Bonn far more to his liking than that in Leipzig. There he had not taught any courses in set theory since 1901, even though this was his primary field of research. After arriving in Bonn, he immediately gave a course on set theory, which he repeated in the summer semester of 1912 in a revised and e x p a n d e d form. It was during that summer that Hausdorff began work on his m a g n u m opus, Grundz~ge der Mengenlehre. He c o m p l e t e d it in Greifswald, where he b e g a n teaching as a full professor (Ordinarius) in the summer semester of 1913; his b o o k a p p e a r e d in print in April 1914.

Set theory, as this area of mathematics was understood at the time, included not just the general theory of sets but also point sets and the theories of content and measure. Hausdorff's w o r k was the first textbook that dealt systematically with all aspects of set theory in this comprehensive sense and which provided complete proofs in a masterful form. Moreover, it went well b e y o n d the presentation of k n o w n results: it contained a n u m b e r of significant original contributions by its author, which can only briefly b e described here. The first six chapters of the Grundzu'gedeal with general set theory. Hausdorff begins by setting out an algebra for sets that includes s o m e n e w concepts that w o u l d prove influential (Differenzenketten, rings and fields of sets, a n d go--systems). These introductory paragraphs on sets and their operations also contain the m o d e m set-theoretic concept of a function; here we encounter, so to speak, many of the ingredients that form the m o d e r n language of mathematics. There follows in chapters three to five the classical theory of cardinal numbers, order types, and ordinal numbers. In the sixth chapter on "Relations between ordered and wellordered sets (Beziehungen zwischen geordneten und wohlgeordneten Mengen)" Hausdorff presents, a m o n g other things, the most important results from his o w n researches on o r d e r e d sets. The chapters on "point sets"---one might better say topology---exude the spirit of a n e w era. Here Hausdorff presents for the first time, beginning with his axioms for neighborhoods, a systematic theory of topological spaces, to which he a d d e d the separation axiom k n o w n today by his name. This theory arose through a comprehensive synthesis involving the w o r k of other mathematicians as well as his o w n reflections on the space problem. The concepts and theorems from classical point set theory in ~u are e x t e n d e d - - s o far as this is poss i b l e - t o the general case, w h e r e they are s u b s u m e d into the newly created general or set-theoretic topology. Yet in the course of carrying out this "transla-

tion work," Hausdorff created a n u m b e r of fundamentally new constructions for topology, such as the interior and closure operations, while developing the fundamental concepts of o p e n set (which he called a "Gebiet") and compactness, a concept he took from Fr~chet. He also established and develo p e d the theory of connectedness, introducing in particular the notions of "components" and "quasi-components." He further specialized general topological spaces b y means of the first and second Hausdorff countability axioms. The metric spaces comprise a large class of spaces that satisfy the first countability axiom. These were introduced in 1906 by Fr~chet, w h o called them "classes (E)"; the terminology "metrischer Raum" is derived from Hausdorff. In his Grundzu'ge, he gave a systematic presentation of the theory of metric spaces, to which he a d d e d several n e w concepts (Hausdorff metric, completion, total boundedness, p-connectedness, reducible sets). Frechet's w o r k (Fr~chet 1906) had received little attention; it was through Hausdorff's Grundz~ge that metric spaces b e c a m e widely familiar to mathematicians.18 Both the chapter on mappings as well as the final chapter of the Grundz~ge on measure theory and integration are impressive for the generality of their approach a n d the originality of the presentation. Hausdorffs laconic remarks pointing to the significance of measure theory for probability w o u l d prove to be highly insightful. The final chapter also contains the first correct proof of Borel's strong law of large numbers. 19 Finally, the a p p e n d i x contains the single most spectacular result in the whole book, namely, Hausdorff's theorem that o n e cannot define a finitely additive measure, invariant u n d e r congruences, on all b o u n d e d subsets in ~'~ for n -> 3. Hausdorff's p r o o f is b y means of a famous paradoxical decomposition of the sphere, for which it is necessary to invoke the axiom of choice. 2~ In the c o u r s e of the twentieth century it b e c a m e standard practice to place mathematical theories on a set-

17Regard;ng this theorem and slm~lar results of C. Kuratowskt and M. Zorn, see the commentary by U Feigner in Hausdorff (2002, 602-604). 18Detatled commentaries on Hausdorff's contributions to general topology and the theory of metnc spaces can be found ~n Hausdorrf (2002, 675-787). 19S. D. Chatterjr g~ves commentary on measure theory and integration ~n the Grundzugein Hausdorff (2002, 788-800), see also Chatterji (2002). 2~ the historical tmpact of Hausdorff's sphere paradox, see Hausdorff (2001, 11-18); see also the article by P. SchreJber ~n Brieskorn (1996, 135-148), and the monograph by S. Wagon (1993)

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theoretic and axiomatic basis. The creation of axiomatically g r o u n d e d theories, as for example in general topology, served among other things to expose the structural elements c o m m o n to various concrete situations or special areas and to place these in an abstract theory that subsumed them as special cases. By so doing, there is a considerable gain in simplicity, unity, and ultimately in e c o n o m y of thought. Hausdorff placed special emphasis on this viewpoint irt his Grundzf2ge (Hausdorff 1914, 211). In this respect, the topological chapters in the Grundz~ige represent a pioneering achievement that p a v e d the w a y for the development of m o d e r n mathematics. This modern conception of the essence of mathematics, made manifest through this new methodological orientation, was conceived by Hausdorff m a n y years before he c o m p o s e d the Grundziige, indeed well before the appearance of the relevant works of Frechet (1906) and F. Riesz (1907, 1908). An important impulse in this direction surely came from the Grundlagen der Geometrie, which David Hilbert published in 1899. In Hausdorff's lecture course on Time and Space, held during the winter semester of 1903 to 1904, he remarked about mathematics: Mathematics stands completely apart not only from the actual meaning that one attributes to its concepts but also from the actual validity one ascribes to its propositions. Its undefinable concepts are arbitrarily chosen objects of thought, its axioms are also arbitrary, though c h o s e n so as to be free from contradiction. Mathematics is a science of pure thought, just as is formal logic (NL Hausdorff: Kapsel 24: Fasz. 71, B1. 4). And about space in particular, Thus: space is a logical construction, namely it includes all propositions that follow logically from the arbitrarily chosen axioms, whereas the concepts employed are arbitrarily chosen objects of thought (Ibid., B1. 31).

In light of these quotations, one might w o n d e r w h y Hausdorff did not undertake to secure the ultimate foundations, "das Fundament des Fundamentes" (Grundzage, 1) by developing set theory on an axiomatic basis. He was, of course, familiar with Zermelo's axiomatization. But he regarded this theory as only provisional: By stipulating suitable conditions E. Zermelo undertook the [. 9 -] necessary attempt to curtail the processes leading to a boundless construction of sets. However, these highly astute investigations cannot yet be regarded as complete, and since an introduction to set theory along this path w o u l d surely lead to great difficulties for beginners, we prefer here to admit the naive c o n c e p t of set, taking due account of the restrictions necessary in order to cut off the path leading to the paradoxes. (Hausdorff 1914, 2) It surely did not escape Hausdorff's notice that Zermelo's concept of "definite property" (definite Eigenschaft) lacked precision (see Feigner 1979, 3-8 and 49-91). In the remainder of the Grundzage, he avoided foundational questions. 21 The Grundzfz'ge der Mengenlehre appeared at the dawning of the First World War. When it broke out, in August 1914, scientific life in Europe was affected in the most dramatic ways. Under these circumstances, Hausdorff's b o o k had little impact for five to six years. After the war, a new generation of researchers began to take up the many suggestive impulses it contained, especially for topology, n o w a central field of interest. The reception of Hausdorff's ideas was enhanced by the founding in 1920, of a new journal in Poland, Fundamenta Mathematicae. This was the first mathematical journal specializing in the fields of set theory, topology, the theory of real functions, measure theory and integration, functional analysis, logic, and the foundations of mathematics. Within this spec-

tram of interests, general topology occupied a central place. Hausdorff's Grundz~ge was cited with great frequency beginning with the very first issue of Fundamenta Mathematicae. In the 558 articles (excluding the three written by Hausdorff himself) that appeared in the first twenty volumes between 1920 and 1933, no fewer than 88 referred to the Grundzf~ge. Here one must also take account that Hausdorff's concepts had b e c o m e so commonplace that one finds these in several articles in which he was not explicitly cited. Hausdorff's Grundzf~ge had a similar influence on the Russian topological school f o u n d e d by Paul Alexandroff and Paul Urysohn. This is evident from his correspondence with Alexandroff and Urysohn (after Urysohn's early death with Alexandroff alone) as well as from Urysohn's M6moire sur les multiplicit~s Cantoriennes (Urysohn 1925/ 1926), a work the size of a b o o k in which Urysohn set forth his theory of dimension, citing the Grundzf~ge no less than sixty times. The demand for Hausdorff's b o o k continued until well after the Second World War, as attested by the three Chelsea reprints that appeared in 1949, 1965, and 1978. In 1916, Hausdorff and Alexandroff solved (independently) the continuum problem for Borel sets22: Every Borel set in a complete separable metric space is either at most countable or has the p o w e r of the continuum. This result generalizes the theorem of CantorBendixson, which makes the same assertion for closed subsets in ~n. Earlier in 1903, W. H. Young extended this theorem to linear Ga-sets (Young 1903), and in 1914, Hausdorff proved it for Gala--sets in the Grundzt2gen. The t h e orem of Alexandroff and Hausdorff proved to be a powerful impulse for the further development of descriptive set theory. 23 Among Hausdorff's publications from his tenure in Greifswald, one in particular occupies a special place: his article on dimension and outer measure

21On these matters, see P Koepke' Metamathematlsche Aspekte der Hausdorffschen Mengenlehre, in Brreskom 1996, 71-106). There one finds an ~nteresting parallel between set-theoretic relativism and epistemological relatMsm in Chaos in kosmtscher Auslese. 22(Hausdorff 1916), (Alexandroff 1916) The not{on of a "Bore1 set" in the modern sense was introduced by Hausdorff ~n the GrundzOge. Schoenflies had used the term "Borel sets" merely for the case of Gs-sets 23(Alexandroff and Hopf 19351 20). For further tnformatton see the commentary of V Kanovel and P. Koepke in Hausdortf (2002, 779-782) and in Hausdorff (2008, 439-442).

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(Dimension und #ufleres Marl) (Hausdorff 1919a). This publication is still highly relevant; it has p r o b a b l y b e e n cited m o r e often in recent years than any other research p a p e r from the d e c a d e 1910 to 1920. Here a few technicalities are required. Let ~ be a system of b o u n d e d sets in Rq such that each set A C Rq can b e c o v e r e d b y the union of at most c o u n t a b l y m a n y sets UE og with diameters d(U) < E (E > 0 arbitrary). Let A(x) b e a continuous, strictly m o n o t o n e increasing nonnegative function o n [0, co). Then for A C Rq Hausdorff introduces

L~(A) = inf {n~>--IA(d(Un)): A C U and

Un' d(Un)

La(A)= lim L{ (A).

~$0 This La(A) is t o d a y called the Hausdorff measure for the function A(x). Hausdorff assigned to a set A the dimension [A] if 0 < La(A) < 0o. A fundamental and difficult question n o w arises: for a given function A do there always exist sets A C Rq having dimension [A]? Hausdorff was able to s h o w that this is i n d e e d so for every strictly m o n o t o n e increasing, everyw h e r e concave continuous function A(x) : [0,~) --+ [0,0o) w i t h A(0) = 0 a n d l i m x ~ A ( x ) = m. In the case w h e r e A(x) = xP, p positive real, one obtains the usual concepts associated with Hausdorff measure a n d Hausdorff dimension. The Hausdorff dimension of a set A is then the n u m b e r a for which a := sup{p > 0: Lq2)(A) = 0o} = inf{p > 0: 13P~(a) = 0}, w h e r e LCp) = La a n d with a(x) = aP. Hausdorffs c o n c e p t of dimension is a finely-tuned instrument for characterizing and c o m p a r i n g sets that are "highly jagged." The concepts in Dimension und diuageres Marl have b e e n

a p p l i e d and further d e v e l o p e d in num e r o u s areas, for example, in the theory o f dynamical systems, geometric m e a s u r e theory, the theory of self-similar sets and fractals, the t h e o r y of stochastic processes, harmonic analysis, potential theory, a n d n u m b e r theory. 2a Unfortunately the b o o m of interest in "fractal theory" has often led to misunderstandings a n d misinterpretations a b o u t Hausdorff's conceptions. 25 The University of Greifswald was a small Prussian provincial university of m e r e l y local importance. Its mathematics institute was small, a n d in the summ e r semester of 1916 a n d the following winter semester, Hausdorff was the only mathematician teaching there! Thus his teaching activities w e r e almost c o m p l e t e l y d o m i n a t e d by elementary courses. His situation improved m a r k e d l y from a scientific standpoint w h e n he went to Bonn in 1921. Here he h a d the o p p o r t u n i t y to e x p a n d his t e a c h i n g to a w i d e n u m b e r of themes a n d to lecture over a n d again on his current research interests. Particularly noteworthy, for example, is the lecture course he offered in the s u m m e r sem e s t e r of 1923 on probability theory, 26 in w h i c h he p l a c e d this t h e o r y on axiomatic and measure-theoretic foundations, ten years before the publication o f A. N. Kolmogoroff's Grundbegriffe der Wahrscheinlichkeitsrechnung. In Bonn, Hausdorff f o u n d in Eduard Study a n d later Otto Toeplitz colleagues w h o w e r e not only outstanding mathematicians but w h o also b e c a m e g o o d friends. During this s e c o n d p e r i o d in Bonn, Hausdorff p r o d u c e d important w o r k in analysis. In Hausdorff (1921), he dev e l o p e d an entire class of s u m m a t i o n m e t h o d s for divergent series w h i c h tod a y are k n o w n as Hausdorff methods. 27 The classical m e t h o d s of H61der and Cesb.ro are special cases of these Hausdorff methods. Each such Hausdorff m e t h o d is given b y a s e q u e n c e of moments; in this context Hausdorff pres e n t e d an elegant solution of the prob-

lem of m o m e n t s for a finite interval that bypasses the theory of continued fractions. In Hausdorff (1923b) he dealt with a special m o m e n t p r o b l e m for a finite interval (subject to certain restrictions o n the generating density ~(x), for e x a m p l e that ~(x) E LP[0,1]). Hausdorff s p e n t many years w o r k i n g on criteria for the solvability a n d determination of m o m e n t problems, as e v i d e n c e d b y h u n d r e d s of pages left in his p o s t h u m o u s papers. 28 Hausdorff m a d e a major contribution to the e m e r g e n c e of functional analysis in the 1920s with his extension of the Fischer-Riesz t h e o r e m to L p spaces in Hausdorff (1923a). There he also p r o v e d the inequalities n a m e d after him a n d W. H. Young29: If an are the Fourier coefficients o f f E LV(0,2rr), q--<2, 1 + P

1 = 1 , then q

lan

--

~

LFIq

.

if s176 converges, then there exists an f E LP(0,2rr) having these a,, as its Fourier coefficients, and furthermore 1

1

2"n"

IZP

-<

la~l q

9

The Hausdorff-Young inequalities served as the point of departure for wide-ranging n e w d e v e l o p m e n t s (see the c o m m e n t a r y by S. D. Chatterji in Hausdorff (2001, 182-190). In 1927 Hausdorff p u b l i s h e d his b o o k Mengenlehre as the s e c o n d edition of the Grundzu'ge. In reality this was a totally n e w book. In o r d e r to app e a r in the G 6 s c h e n series, it was necessary to p r o v i d e a far m o r e restricted presentation than in the Grundz~ige. Thus large parts of the theory of ord e r e d sets a n d the sections on m e a s u r e theory a n d integration had to b e d r o p p e d . "Even more regrettable than these o m i s s i o n s ' - - a c c o r d i n g to Hausdorff in his p r e f a c e - was the n e e d to save further r o o m

24On the historical impact of Dimension und ~uBeres Ma& see the articles by Bandt/Haase and Bothe/Schmehng ~n Brieskorn (1996, 149-183 and 229-252) as well as the commentary by S D Chatterji in Hausdorff (2001, 44-54, and the hterature c~ted there~n) 25Aboutthis, see K. Steffen' Hausdorff-Dimens~on, regulare Mengen und total ~rregulare Mengen, in Brieskom (1996, 185-227). 26NL Hausdorff: Kapsel 21: Fasz 64, reprinted in its entirety with detailed commentary tn Hausdorff (2005), 595-756 271nHardy's classical study (Hardy 1949) he devotes an entire chapter to Hausdorff methods. 28On the entire complex of these published and unpublished works, see Hausdorff (2001, 105-171, 191-235, 255-267, and 339-373) 29Young had proved these for the special case p = 2n, n = 2, 3,

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in point set theory by sacrificing the topological standpoint, d e s p i t e its attractions for many readers of the first edition, and instead confining the discussion to the simpler theory of metric spaces, [" 9 "] (Hausdorff 1927, 5--6). Some reviewers of the w o r k expressly regretted this circumstance. As a form of compensation, however, Hausdorff offered an up-to-date presentation of the state of research in descriptive set theory. This insured that his n e w b o o k received almost as strong a reception as h a d the Grundziige, especially in Fundamenta Mathematicae. It b e c a m e a highly p o p u l a r t e x t b o o k a n d app e a r e d again in 1935 in an e x p a n d e d s e c o n d edition, which was r e p r o d u c e d b y D o v e r in 1944. An English translation was p u b l i s h e d in 1957 with n e w printings in 1962, 1978, a n d 1991. A Russian edition came out in 1937, alt h o u g h this is not really a true translation; parts of the b o o k w e r e r e w o r k e d b y Alexandroff and Kolmogoroff in order to bring the topological standpoint to the f o r e g r o u n d 9 In 1928 Hans Hahn w r o t e a review of the Mengenlehre (reprinted in Hausdorff 2008, 416-417). Possibly Hahn alr e a d y sensed the dangers of G e r m a n anti-Semitism. He e n d e d his review with these words: This in every respect masterful presentation of a difficult a n d haza r d o u s subject is a w o r k o f the type written b y those w h o have carried the fame of G e r m a n science a r o u n d the world, a w o r k of w h i c h the author as well as all G e r m a n mathematicians may be proud. (Hahn 1928, 58) Like most G e r m a n academics, Hausdorff never e n g a g e d directly in political activity. His views w e r e far more liberal, however, than most of his colleagues. After the First W o r l d War he j o i n e d the n e w l y f o u n d e d G e r m a n Democratic Party (DDP), w h i c h for a brief time represented a sizable leftistliberal constituency in the W e i m a r Republic. Several leading Jewish politicians and intellectuals w e r e d r a w n to the DDP, including Albert Einstein, but

its p o p u l a r i t y quickly e v a p o r a t e d during w h i c h time its more conservative wing t o o k control. Although never an active m e m b e r , Hausdorff d r o p p e d out of the DDP altogether in the mid-1920s; the party languished on during the years that followed, b e c o m i n g virtually irrelevant b y the time that National Socialists b e g a n their dramatic surge. O n e y e a r before they c a m e to power, Hausdorff e x p r e s s e d his o p i n ion of the Nazi m o v e m e n t in a letter to Hilbert. The occasion was the latter's seventieth birthday, so the c o m m e n t was p e r h a p s p h r a s e d in a m o r e ironic tone than it might have b e e n otherwise. Hausdorff h a d b e e n contemplating an a p p r o p r i a t e honorific title for Hilbert, similar to the one given to Gauss in the twilight o f his career: the Prince of Mathematicians. So he e x p l a i n e d to Hilbert that since the title princeps mathematicorum has already b e e n b e stowed, I w o u l d have suggested calling you the dux mathematiconLm w e r e it not that t o d a y the title duce Ff2hrer is so discredited b y those w h o offer to lead the G e r m a n p e o p l e on the basis of a license (F,2hrerschein) that t h e y have b e s t o w e d on themselves. 31 With the assumption of p o w e r b y the National Socialists, anti-Semitism b e c a m e an official state doctrine. Hausdorff was not directly affected in 1933 by the notorious "law to restore the civil service," b e c a u s e he had already b e e n a G e r m a n civil servant since before 1914. His teaching activity was, h o w ever, a p p a r e n t l y affected b y activities u n d e r t a k e n by Nazi student functionaries. In his manuscript for his lecture course Infinitesimalrechnung III held during the winter semester of 1934 to 1935 he n o t e d on p a g e 16: "Interrupted 20 November" (NL Hausdorff: Kapsel 19: Fasz. 59). Two days later, on 22 N o v e m b e r 1934, the "Westdeutsche Beobachter" r e p o r t e d in an article entitled "Party educates the Political Students" that "during these days" a w o r k i n g conference of the Nazi Student Union was taking place at Bonn University. The focus of their w o r k during this semester was the t h e m e of "race a n d folklore." These circum-

stances make it likely that Hausdorff's decision to b r e a k off his lectures was c o n n e c t e d with this political activity. At no o t h e r time in his long career, except for the brief p e r i o d of the K a p p Putsch, did he ever cancel a lecture course. O n 31 March 1935, after some b a c k and forth, Hausdorff retired as an emeritus professor in Bonn. For his forty years of successful labor in G e r m a n higher education he received n o t a w o r d of thanks from the then r e s p o n sible authorities. He c o n t i n u e d to w o r k on indefatigably, publishing not only the n e w l y revised version of his b o o k Mengenlehre b u t also seven p a p e r s on t o p o l o g y and descriptive set theory, all of w h i c h a p p e a r e d in two Polish journals: o n e in Studia Mathematica, the others in Fundamenta Mathematicae. This w o r k is r e p r i n t e d in v o l u m e III of the Gesammelte Werke (Hausdorff 2008) with detailed commentary. In his final publication (Hausdorff 1938), Hausdorff s h o w e d that a continuous m a p p i n g from a closed subset F of a metric s p a c e E can be e x t e n d e d to all of E (allowing for the possibility that the image s p a c e can also be extended). In particular, a h o m e o m o r phism defined o n F can b e e x t e n d e d to a h o m e o m o r p h i s m o n all of E. This was a continuation of earlier investigations p u b l i s h e d in Hausdorff (1919b) and Hausdorff (1930). In Hausdorff (1919b) he gave a n e w p r o o f of the Tietze extension theorem, and in Hausdorff (1930) he s h o w e d that if E is a metric space a n d F C E closed, a n d if on F a n e w metric is given that leaves the original t o p o l o g y invariant, then this n e w metric can be e x t e n d e d to the entire space without altering its topology. In Hausdorff (1935b) he studied spaces that fulfill the Kuratowski closure axioms, e x c e p t for the axiom demanding that the closure o p e r a t i o n be idempotent. He called these "gestufte R~iume" (today they are usually k n o w n as closure spaces); he u s e d t h e m to study relations b e t w e e n Fr~chet's limit spaces and topological spaces. The u n p u b l i s h e d p a p e r s in Hausdorff's Nachlalg also s h o w h o w he con-

3~ complete text of the Mengenlehreis reprinted in Hausdorff (2008, 41-351) Background and reception to the work appear in an historical introduction (1-40), and the text ~tself receives detailed commentary (352-398) regarding mathematical as well as historical matters. 31Hausdorff to Hilbert, 21 January 1932, Cod Ms D HiIbert 452c, Nledersachsische Staats- und Universitatsblbliothek G6ttingen

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tinued not only to w o r k on but to follow the most recent d e v e l o p m e n t s in areas that interested him during these ever more difficult times. A major source of s u p p o r t c a m e from Erich Bessel-Hagen, w h o r e m a i n e d a faithful friend of the Hausdorff family throughout their ordeal. Bessel-Hagen brought b o o k s and journals from the mathematics library, w h i c h Hausdorff, as a Jew, was no longer a l l o w e d to enter. Several articles w o u l d not suffice to list all the perfidious laws, decrees, ordinances, a n d other legalistic machinations d e s i g n e d to discriminate and isolate the J e w s a n d to d e p r i v e them of their property a n d rights. Historians have c o u n t e d t h e m though: up to the N o v e m b e r 1938 p o g r o m there were more than 500 such proclamations. O n e w o n d e r s w h y Hausdorff, an internationally r e c o g n i z e d scholar living under such conditions, d i d not attempt to emigrate during the m i d 1930s. The answer can only remain conjectural: in Bonn he h a d his h o m e , his library a n d the possibility to work, some true friends, a n d although h e was always a skeptic, even he w o u l d not have considered it possible that the Nazi regime w o u l d destroy the e c o n o m i c foundations established b y elderly p e o p l e in the course of their l o n g lives, or that ultimately they w o u l d p a y with their lives. The N o v e m b e r pogrom, which c a m e to be k n o w n as the Night of the Broken Glass (Reichskristallnacht), with its o p e n brutality m a d e all this quite evident and clear. Hausdorff, n o w over 70, at last m a d e an attempt to emigrate. Richard Courant wrote to Herm a n n Weyl o n 10 F e b r u a r y 1939: Dear Weyl, I just r e c e i v e d the enclosed short a n d very touching letter from Professor Felix Hausdorff (which p l e a s e return), w h o is seventy years old a n d w h o s e wife is sixty-five years old. He certainly is a mathematician of very great merit and still quite active. He asks me w h e t h e r it w o u l d b e possible to find a research fellowship for h i m Y Weyl and John von N e u m a n n p r o v i d e d letters of r e c o m m e n d a t i o n that were

p r e s u m a b l y sent to American institutions a n d colleagues. Weyl e m p h a s i z e d Hausdorff's m a n y accomplishments a n d contributions to mathematics, calling him "A man with a universal intellectual outlook, a n d a p e r s o n of great culture and charm." These efforts of W e y l a n d von N e u m a n n were, however, evidently unsuccessful. F r o m several sources, in particular the letters of Bessel-Hagen, w e k n o w that H a u s d o r f f a n d his family w e r e f o r c e d to u n d e r g o a n u m b e r o f humiliations, especially after N o v e m b e r 1938. 33 In mid-1941 the Nazi governm e n t b e g a n to d e p o r t the J e w s in B o n n to the m o n a s t e r y "Zur e w i g e n A n b e t u n g " in Bonn-Endenich, from w h i c h the nuns had b e e n e x p e l l e d . F r o m there they w e r e then t r a n s p o r t e d to the e x t e r m i n a t i o n c a m p s in the east. In J a n u a r y 1942, Felix Hausdorff, his wife, a n d her sister Edith P a p p e n h e i m , w h o lived with them, w e r e o r d e r e d to resettle in the i n t e r n m e n t c a m p in Bonn-Endenich. O n 26 January, all t h r e e t o o k their o w n lives w i t h an o v e r d o s a g e of Veronal. Their last resting p l a c e is the c e m e t e r y in B o n n - P o p pelsdorf. Some of Bonn's Jewish citizens may still h a v e had illusions a b o u t the c a m p in Endenich; Hausdorff h a d none. Erwin N e u e n s c h w a n d e r f o u n d Hausdorff's farewell letter to the Jewish l a w y e r Hans Wollstein in the p a p e r s of Bessel-Hagen, 34 from w h i c h w e cite the b e g i n n i n g and end: Dear Friend Wollstein! By the time you receive this letter, w e three will have solved this problem in another w a y - - t h e w a y you always tried to dissuade us from. The feeling of safety that y o u pred i c t e d w o u l d b e ours o n c e the difficulties of moving h a d b e e n overc o m e has not c o m e a b o u t at all. On the contrary: Even Endenich Is p e r h a p s not yet the e n d (das Ende nich)! W h a t has h a p p e n e d to the J e w s in the last months a w a k e s justified anxiety in us that w e will no l o n g e r be

a l l o w e d to e x p e r i e n c e bearable conditions. After e x p r e s s i n g his gratitude to friends, and with great c o m p o s u r e in formulating his last wishes regarding his funeral and last will, Hausdorff wrote further: Excuse us for causing you troubles even after death; I a m c o n v i n c e d that y o u will d o w h a t you c a n ( a n d that is p e r h a p s not very much). Excuse us also for our desertion! W e h o p e that y o u and all our friends will e x p e r i e n c e better times. Your truly devoted, Felix Hausdorff This last wish of Hausdorff's was not fulfilled: the lawyer Wollstein was murd e r e d in Auschwitz. Hausdorff's library was sold b y his son-in-law a n d sole heir Arthur K6nig. His p o s t h u m o u s papers w e r e p r e s e r v e d b y a friend o f the family, the B o n n Egyptologist Hans Bonnet, w h o later wrote a b o u t their further fate (Bonnet 1967). H a u s d o r f f s papers, he wrote, w e r e not y e t saved, for in D e c e m ber 1944 a b o m b e x p l o s i o n destroyed m y h o u s e a n d the m a n u scripts w e r e mired in rubble from a c o l l a p s e d wall. I dug t h e m out without b e i n g able to p a y attention to their o r d e r and certainly without saving t h e m all. Then in J a n u a r y 1945 I h a d to leave Bonn [ . . - ] . W h e n I r e t u r n e d in the s u m m e r of 1946 almost all the furniture had disa p p e a r e d , b u t the p a p e r s of Hausdorff w e r e essentially intact. T h e y were worthless for treasure hunters. Nevertheless, they suffered losses and the remaining scattered p a g e s were m i x e d together m o r e than ever. The o n c e well-ordered cosmos h a d b e c o m e a chaos. (Bonnet 1967, 76 (152)) The late Professor GOnter Bergmann from MOnster p e r f o r m e d a great service b y carefully ordering the surviving 25,978 p a g e s of the Hausdorff Nachlass. In 1980 he transfered the n o w secure results o v e r to the Bonn University library. Bergmann also p u b l i s h e d some of the p r e s e r v e d p a p e r s in two facsimile v o l u m e s (Hausdorff 1969).

32Veblen Papers, Library of Congress, Container 31, folder Hausdorff We thank Reinhard Siegmund-Schultze, Kristiansand, for making a copy of th~s letter available. He was unable to find Hausdorff's original letter. 33Neuenschwander, E.: Fehx Hausdorffs letzte Lebensjahre nach Dokumenten aus dem BesseI-Hagen-Nachla8, in Bneskorn (1996, 253-270). 34BesseI-Hagen, Un~vers~t&tsarchiv Bonn Pnnted in Brieskorn (1996, 263-264 and in facsimile, 265-267).

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REFERENCES K6nigl. Sachs. Ges. der Wlss. zu Leipzig. Alexandroff, P. 1916. Sur la puissance des enMath.-phys. Classe 47, 401-482. sembles mesurables B Comptes rendus Hausdorff, F. 1896. Infinitesimale AbblldunAcad. Sci. Paris 162, 323-325. gender Optlk. Ber. 0ber die Verhandlungen Alexandroff, P., Hopf, H. 1935. Topeleg~e. der K6nigl. Sachs. Ges. der Wiss. zu Leipzig. Springer-Verlag, Berlin. Math.-phys. Classe 48, 79-130. Beckert, H., Purkert, W. (Hrsg.). 1987. Leipziger Hausdorff, F. (P. Mongre). 1897a. Sant' Ilario-mathemabsche Antrittsvorlesungen. TeubGedanken aus der Landschaft Zarathusner-Archiv zur Mathematik, Band 8, Leipzig. tras Verlag C. G. Naumann, Leipzig. VIII + Blumberg, H. 1921. Hausdorff's GrundzOge 379 S. Wiederabdruck des Gedichts "Der der Mengenlehre. Bull. AMS 27, 116-129. Dichter" und der Aphorismen 293,309, 313, See also: [Hausdorff 2002], 844-853. 324, 325, 337, 340, 346, 349 in Der Bonnet, H. 1967. Geleltwort. Jahresbericht der Zwiebelflsch 3 (1911), S. 80 u. 88-90. DMV 69, 75(151)-76(152). Hausdorff, F. (P. Mongre). 1897b. Sant' Ilario--Brieskorn, E. 1992. Fehx Hausdorff EleGedanken aus der Landschaft Zarathusmente einer Biographie. In: Katalog der tras Selbstanzeige, DieZukunft, 20.11. 361. Ausstellung "Felix Hausdorff--Paul Mon- Hausdorff, F. 1897c. Das Risico bei Zugr~ (1868-1942)," Mathematlsches Institut fallsspielen. Ber. uber die Verhandlungen der der UniversitAt Bonn, 24. Januar bts 28. K6ntgl. SAchs. Ges. der Wiss. zu Leipzig. Februar. Math.-phys. Classe 49, 497-548. Brieskom, E. (Hrsg.). 1996. Felix Hausdorffzum Hausdorff, F. (P. Mongre). 1898. Das Chaos in Ged~chtnis Aspekte seines Werkes kosmlscher Auslese---Ein erkenntnlskritlVleweg, Braunschweig/Wiesbaden. scher Versuch Verlag C. G. Naumann, Brieskorn, E. 1997. Gustav Landauer und der Leipzig. Vl und 213 S. Mathematiker Fehx Hausdorff. In: H. Delf; G. Hausdorff, F. 1899. Analyt~sche Beitr~ge zur Mattenklott (Eds.): Gustav Landauer im nlchteuklidischen Geometrie. Ber. uber die Gespr~ch--Sympos~um zum 125 GeburtsVerhandlungen der KTntgl. SAchs. Ges. der tag. TObtngen, 105-128. Wtss. zu Leipzig. Math.-phys. Classe 51, Chatterji, S. D. 2002. Hausdorff als MaStheo161-214. retlker. Math. Semesterberichte 49, 129- Hausdorff, F. (P. Mongre). 1900a. Ekstasen. 143. Gedlchtband. Verlag H. Seemann Nachf., Chatterjl, S. D. 2007. The Central Limit TheoLeipzig. 216 S. rem b la Hausdorff Expositiones Math. 25, Hausdorff, F. 1900b. Zur Theorie der Systeme 215-234. komplexer Zahlen. Ber. Qber die VerhandDierkesmann, M. 1967. Fehx Hausdorff Ein lungen der Konigl. Sachs. Ges. der Wiss. zu Lebensblld. Jahresbericht der DMV 69, Leipzig. Math.-phys. Classe 52, 43-61. 51 (127)-54(130). Hausdorff, F. 1901a. Uber elne gewisse Art Ebbinghaus, H.-D. 2007. Ernst Zermelo. An geordneter Mengen. Ber. 0ber die Verhandapproach to his fife and work Springer, lungen der KTnigl. SAchs. Ges. der Wiss. zu Berlin, etc. Leipzig. Math.-phys. Classe 53, 460-475. Eichhorn, E., Thiele, E.-J. 1994. Vorlesungen Hausdorff, F. 1901b. Beitr~ge zur Wahrzum Gedenken an Felix Hausdorff. Helderscheinhchkeitsrechnung. Ber. Qber die Vermann Verlag, Berlin. handlungen der KTnigl. SAchs. Ges. der Fechter, P 1948. Menschen und ZeJten. BerWiss. zu Leipzig. Math.-phys. Classe 53, telsmann, GQtersloh. 152-178. Feigner, U. (Hrsg.). 1979. Mengenlehre. Wiss. Hausdorff, F. (P. Mongr6). 1902. Der Wille zur Buchges., Darmstadt. Macht Neue Deutsche Rundschau (Freie Frechet, M. 1906. Sur quelques points du calB0hne) 13(12), 1334-1338. cul fonctionnel. Rendiconti del Circolo Mat. Hausdorff, F. 1903. Das Raumproblem di Palermo 22, 1-74. (Antrittsvorlesung an der UniversitAt Leipzig, Hausdorff, F. 1891. Zur Theone der asgehalten am 4.7.1903). Ostwalds Annalen tronomischen Strahlenbrechung (Dissertader Naturphilosophie 3, 1-23. tion). Ber. 0ber die Verhandlungen der Hausdorff, F. (P. Mongr6). 1904a. Der Arzt KTnigl. SAchs. Ges. der Wiss. zu Leipzig. seiner Ehre, Groteske Die neue Rundschau Math.-phys. Classe 43, 481-566. (Freie BQhne) 15(8), 989-1013. NeuherausHausdorff, F. 1895. Ober d~e Absorpbon des gabe als: Der Arzt selner Ehre Komddle in Lichtes in der Atmospt~re (Habilitationse/nem Akt mit e~nem Epilog Mit 7 Bildnisschrift). Ber. 0ber die Verhandlungen der sen, Holzschnitte von Hans Alexander MOiler

nach Zeichnungen von Waiter Tiemann, 10 BI., 71 S. FOnfteordentliche VerOffentlichung des Leipziger Bibhophilen-Abends, Leipzrg 1910. Neudruck: S. Fischer, Berlin 1912, 88 S. Hausdorff, F. 1904b. Der Potenzbegriff in der Mengenlehre Jahresbericht der DMV 13, 569-571. Hausdorff, F. 1906. Untersuchungen uber Ordnungstypen I, II, III Ber. 0ber die Verhandlungen der KSnigl. SAchs. Ges. der Wiss. zu Leipzig. Math.-phys. Klasse 58, 106-169. Hausdorff, F. 1907a. Untersuchungen dber Ordnungstypen IV, V. Ber. 0ber die Verhandlungen der K6nigl. SAchs. Ges. der Wiss. zu Leipzig. Math.-phys. Klasse 59, 84-159. Hausdorff, F. 1907b. Ober dlchte Ordnungstypen Jahresbericht der DMV 16, 541-546. Hausdorff, F. 1908. Grundzdge elner Theorie dergeordneten Mengen Math. Annalen 65, 435-505. Hausdorff, F. 1909. Die Graduierung nach dem Endverlauf Abhandlungen der KTnigl. Sachs. Ges. der Wiss. zu Leipzig. Math.phys. Klasse 31, 295-334. Hausdorff, F. 1914. Grundzdge der Mengenlehre. Verlag Veit & Co, Leipzig. 476 S. mit 53 Figuren. Nachdrucke: Chelsea Pub. Co., New York, 1949, 1965, 1978. Hausdorff, F. 1916. Die M#chtigkeit der Borelschen Mengen Math. Annalen 77, 430-437. Hausdorff, F. 1919a. D~mension und &uSeres Mal3. Math. Annalen 79, 157-179. Hausdorff, F. 1919b. Ober halbstetige Funktionen und deren Verallgemeinerung Math. Zeitschrift 5, 292-309. Hausdorff, F. 1921. Summatlonsmethoden und Momentfolgen I, II Math. Zeitschrift 9, I: 74-109, II: 280-299. Hausdorff, F. 1923a. Eine Ausdehnung des Parsevalschen Satzes dber Fourierreihen. Math. Zeitschrift 16, 163-169. Hausdorff, F. 1923b. Momentprobleme for eln endfiches Intervall Math. Zettschrift 16, 220-248. Hausdorff, F. 1927. Mengenlehre, zweite, neubearbeitete Auflage. Verlag Walter de Gruyter & Co., Berlin. 285 S. mit 12 Figuren. Hausdorff, F. 1930. Erwetterung einer HomOomorphie Fundamenta Mathemattcae 16, 353-360. Hausdorff, F. 1935a. Mengenlehre, dritte Auflage. Mit einem zusAtzlichen Kapitel und einpgen NachtrAgen. Veriag Walter de Gruyter & Co., Berlin. 307 S. mit 12 Rguren. Nachdruck: Dover Pub. New York, 1944. Englische Aus-

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gabe: Set theory. Qbersetzung aus dem Deutschen von J. R. Aumann, et al, Chelsea Pub. Co., New York 1957, 1962, 1978, 1991. Hausdorff, F. 1935b. GestuRe R~ume. Fundamenta Mathematlcae 25, 486-502. Hausdorff, F. 1938. Erweiterung e~nerstetlgen Abblldung. Fundamenta Mathematica 30, 40-47. Hausdorff, F. 1969. Nachgelassene Schriften. 2 B~inde. Ed.: G. Bergmann, Teubner, Stuttgart 1969. Band I enthalt aus dem Nachla6 die Faszikel 510-543, 545-559, 561-577, Band II die Faszikel 578-584, 598-658 (alle Faszikel sind im Faksimiledruck wiedergegeben). Hausdorff, F. 2001. Gesammelte Werke. Band IV. Analysis, Algebra und Zahlentheorie Springer-Verlag, Berlin, Heidelberg. Hausdorff, F. 2002. Gesammelte Werke. Band II. "GrundzOge der Mengenlehre "SpringerVerlag, Berhn, Heidelberg. Hausdorff, F. 2004. Gesammelte Werke. Band VII. Phflosophlsches Werk. Springer-Verlag, Berhn, Heidelberg. Hausdorff, F. 2005. Gesammelte Werke. Band V" Astronomle, Optik und Wahrscheinlichkeltstheone. Springer-Verlag, Berlin, Heidelberg.

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Hausdorff, F. 2008. Gesammelte Werke Band II1" Deskriptive Mengenlehre und Topolog~e. Springer-Verlag, Berlin, Heidelberg. Hahn, H. 1928. F Hausdorff, Mengenlehre Monatshefte for Mathematik und Physik 35, 56-58. Hardy, G. H. 1949. Divergent Series. Oxford Univ. Press, Oxford. Jahresbericht des Nicolai-Gymnaslums fdr das Jahr 1887. Stadtarchiv Leipzig, Bestand Nicolai-Gymnasium. Lorentz, G. G. 1967. Das mathematische Werk von Felix Hausdorff Jahresbericht der DMV 69, 54 (130)-62 (138). Purkert, W. 2004. Kontinuumproblem und Wohlordnung--dle spektakul#ren Erelgnisse auf dem Intemationalen Mathematikerkongre8 1904 ~nHeidelberg. In: Selslng, R., Folkerts, M., Hashagen, U. (Hrsg.): Form, Zahl, Ordnung, Boethius Bd. 48, Franz Steiner Verlag, Wiesbaden, S.223-241. Riesz, F. 1907 D~e Genesis des Raumbegriffs. Math. und Naturwiss. Berichte aus Ungarn 24, 309-353. In his Gesammelte Arbeiten I, 110-161. Rlesz, F. 1908. Stetigkeltsbegnff und abstrakte Mengenlehre Attl del Congr. Internaz. dei Mat., Roma 2, 18-24.

Schoenflies, A. 1908. Die Entwickelung der Lehre von den Punktmannigfaltigkeiten. Tell II. Jahresbericht der DMV, 2. Erganzungsband, Teubner, Leipzig. Stegmaier, W. 2002. Ein Mathematlker in der Landschaft Zarathustras Felix Hausdorff als Phllosoph Nietzsche-Studien 31, 195-240. Urysohn, P. 1925/1926. Memolre sur les multlpllcit~s Cantoriennes. Fundamenta Math. 7, 30-137; 8, 225-351. Vollhardt, F. 2000. Von der Sozlalgeschichte zur Kulturwissenschaft? Die hteransch-essaylstlschen Schriften des Mathematlkers Felix Hausdorff (1868-1942) Vorl~ufige Bemerkungen in systematlscher Abslcht In: Huber, M., Lauer, G. (Hrsg.): Nach der SozJalgeschlchte--Konzepte fur eine Literaturwissenschaft zwJschen Histonscher Anthropologie, Kulturgeschichte und Medlentheone Max Niemeier Verlag, TQbingen, S.551-573. Wagon, S. 1993. The Banach-Tarski Paradox Cambridge Univ. Press, Cambridge. Young, W. H. 1903. Zur Lehre der nlcht abgeschlossenen Punktmengen. Berichte 0ber die Verhandlungen der K6nigl. SAchs. Ges. der Wiss. zu Leipzig, Math.-Phys. Klasse 55, 287-293.