Felix Hausdorff and the Hausdorff edition∗

Erhard Scholz, (Wuppertal, Germany)

January 17, 2005

Hausdorff and Mongré contemporay concepts of space and the contemporary attempts to secure time. He combined Niezschean and axiomatic foundations for Felix Hausdorff (b. November 8, Kantian views and enriched them by as premature. Working on the basis 1868, at Breslau, d. January 26, 1942, mathematical arguments in terms of of a “naive” concept of set (express- ) studied at , Cantorian set theory and stepwise gen- edly understood as a semiotic tool of Freiburg and between 1887 and eralized transformations, comparable thought), he nevertheless achieved an 1891 (dissertation) and started research to F. Klein’s approach in the Erlanger exceptionally high precision of argu- in applied mathematics related to work Programm. In Hausdorff’s perspective, mentation. Although his set theoreti- of his teacher, the astronomer H. the generalization of transformations cal studies prior to 1910 concentrated Bruns.1 After his habilitation (1895) from Euclidean via differentiable and on order structures (remember that his he taught at and a continuous to any point transforma- earliest interest in transfinite set the- local commercial school. He moved tion would lead to a general transfinite ory was triggered by the tremendous in a milieu of Leipzig intellectuals and set as symbol for some structureless amount of possible different modes of artists, strongly influenced by the early fictitious “absolute”. The progression progression of a fictitious “transcen- work of F. Nietzsche, striving for a cul- of “absolute” or “transcendent” time dent time point” in a transfinite set), tural modernization of late 19th cen- might then be perceived as any order he contributed crucial insights in foun- tury Germany. structure on the set of the “transcen- dational questions, most importantly Between 1897 and 1904 (with some dent” world content, without any per- his maximal chain principle (related to additional later contributions) Haus- ceivable relation to the order of the Zorn’s lemma, but different from it), dorff published two philosophical empirical or phenomenological time a characterization of weakly inaccessi- books, a poem collection, and a satiri- ordering. That served Mongré as an ble cardinals (in present terminology) cal theater play under the pseudonym argument that “the absolute” has to and the universality property for order Paul Mongré. The play ridiculized be considered as essentially void of structures of what he called “ eta- the honor codex of late 19th century objective meaning. He thus proudly alpha sets”. The latter became one of German Bildungsbürger adapting to proclaimed the “end of metaphysics”.4 the roots of “saturated structures” in the values of the Wilhelminean offi- During this period, Hausdorff reori- model theory of the 1960s. Moreover, cer corps. At the time it was quite ented his mathematical work towards Hausdorff hit upon the importance of successful. It had about 300 perfor- the new field of transfinite set the- the generalized in 6 mances between 1904 and 1930 at ory. He gave one of the first lecture his studies of eta-alpha sets. about 40 towns, among them Berlin, courses on the topic in summer 1901 Budapest, Prag, Strassburg, Wien, and and contributed important results to Zürich.2 Moreover, Mongré regularly it, among others the Hausdorff recur- The “Grundzüge” contributed cultural critical essays to sion for aleph exponentiation and deep the Neue Deutsche Rundschau, a lead- methods for the classification of or- In summer 1910 Hausdorff started ing intellectual journal.3 In his sec- der structures (confinality, gap types, teaching at Bonn university as “ex- ond book, The Chaos in Cosmic Selec- general ordered products, and eta- traordinarius” (associate professor) tion (Hausdorff 1898), he critically de- alpha sets).5 Hausdorff considered and broadened his perspective on set composed metaphysical remnants in theory as a general symbolical basis for

∗This article is an extended version of a contribution submitted to the Princeton Companion to Mathematics edited by T. Gowers and J. Barrow-Green. 1See (Hausdorff Werke, V (2005), forthcoming). 2Information due to a communication of U. Roth. 3Cf. (Vollhardt/Roth 2002); more in (Hausdorff Werke, vol. VIII (forthcoming)). 4Cf. (Stegmaier 2002) and (Hausdorff Werke, VII (2004), 49–61); for relations to mathematics (Hausdorff Werke, vol. VI (forthcoming)). 5(Hausdorff Werke, vol. I (forthcoming)); detailed references to Hausdorff’s publications in all volumes. 6See (Hausdorff Werke, II (2002), 600ff.), (Moore 1982, 116 etc.) and (Felgner 2002).

1 mathematics. In early 1912 he found Grundzüge started only after World ble metrical space is of ℵ0 a beautiful axiomatic characterization War I, and most strongly in the ris- or of the continuum. That was an of topological spaces by neighbour- ing schools of modern mathematics important step forward for a strategy hood systems and started to compose in , around the journal Fun- proposed by G. Cantor to clarify the a monograph on “basic features of set damenta Mathematicae, and the Soviet continuum hypothesis. Although this theory” (Grundzüge der Mengenlehre). Union mainly among N. Lusin’s stu- goal could not be achieved along this It was finished two years later, after dents around P. Alexandroff. Between road, it led to the development of an he had moved to Greifswald univer- the latter and Hausdorff there arose extended field of investigation on the sity in 1913 on a call to an “ordinary” a close scientific exchange and intel- border region between set theory and (full) professorship, and became his lectual friendship, interrupted only analysis, now dealt with in descriptive opus magnum (Hausdorff 1914b)7. after 1933. All in all, the Grundzüge be- set theory.13 In this book, Hausdorff showed how came one of the founding documents When Hausdorff revised his opus set theory could be used as a working of mathematical modernism in the sense magnum for a second edition in the frame for mathematics more broadly. of the 1920/30s. In a lecture course later 1920s, he rewrote the parts on It contained three parts, (I) general set in 1923 Hausdorff introduced an ax- and topologi- theory and order structures, (II) topo- iomatic basis for theory, cal spaces completely, extending the logical spaces and their basic proper- which anticipated Kolmogorov’s ax- first considerably and concentrating 10 ties, (III) theory and integra- iomatization of 1933. the second on metrical spaces. As other tion. While set theory was intro- books on general set and general topol- duced in a non-axiomatic style, al- ogy had appeared in the meantime, he though with extraordinary precision, Real analysis and descriptive set the- omitted these parts; thus the socalled topological spaces and measure the- ory “second editon” was a completely new ory were given an axiomatic presen- book on specialized topics of set theory In his own research, Hausdorff took tation. In part (II), Hausdorff pub- (Hausdorff 1927). up questions in real analysis, now in- lished his neighbourhood axioms for gen- formed by the new “basic features” of eral spaces, found two years earlier, general set theory. His introduction of introduced separation and countabil- Last years at Bonn what are now called Hausdorff measure ity axioms, studied connectivity prop- and (Hausdorff erties and other concepts. This part In 1921 Hausdorff had returned to 1919) became of long-lasting impor- of the book contained the first com- Bonn university, now as a full pro- tance in the theory of dynamical sys- prehensive treatment of the theory of fessor and colleague of E. Study and tems, geometrical measure theory and metric spaces, initiated by M. Fréchet (a little later) O. Toeplitz. After the the study of “”, which arose in 1906, and laid the basis for an im- rise to power of the Nazi regime, broad and even popular interest in the portant part of the traditon of general life and work conditions deterioriated last third of the 20th century.11 of the coming century.8 steadily and more and more drasti- Other important technical contribu- In part (III) he gave a lucid introduc- cally for Hausdorff and other people tions dealt with summation methods of tion to measure theory, building upon of Jewish origin (F. Hausdorff had dis- infinite divergent series and a gener- the work of E. Borel and H. Lebesgue. tached himself from religion during the alization of the Riesz-Fischer theorem, In a paper published shortly before the 1890s, his wife had converted to protes- which established the now well known book, and added in content as an ap- tantism). While he was still regularly relation between Lp function spaces pendix to the latter, Hausdorff gave a emeritated in early 1935, his colleague and lq series of Fourier coefficients, for negative answer to Lebesgue’s ques- O. Toeplitz was dismissed and left 1 + 1 = 1, and opened the path for tion (for n ≥ 3), whether a (finitely) ad- p q Nazi-Germany for Palestine shortly be- ditive content function invariant under later developments in harmonic anal- fore the outbreak of the second World ysis on topogical groups (Hausdorff War. Hausdorff’s attempts for emigra- congruences can be defined on all sub- 12 sets of Euclidean IRn (Hausdorff 1914a). 1923). tion came too late to be successful and Using the axiom of choice, he “con- Like in the case of his ealier stud- his contacts to local re- ies of order structures such investiga- duced essentially to one sensible and structed” a partition of the 2-sphere 14 (up to a countable residual set), in tions led Hausdorff back to founda- upright colleague, E. Bessel-Hagen. which each part is congruent to the tional questions of set theory. Already When Felix Hausdorff, his wife union of two of them. This was the in the Grundzüge he had been able to Charlotte and a sister of her were or- starting point for the later paradoxical show that certain Borel sets were ei- dered to leave their house for a local in- constructions of measure theory by Ba- ther countable or of the cardinality of ternment regime in January 1942, they nach and Tarski.9 the continuum. In 1916 Hausdorff, and opted for suicide rather than suffering An intense reception of the independently P. Alexandroff, could further persecution. At that time their show that any Borel set in a separa- (adult) daughter was living at Jena.

7(Purkert 2002) 8See (Epple/Herrlich e.a. 2002) and further commentaries in (Hausdorff Werke, II (2002), 745–772). 9(Chatterji 2002) 10(Hausdorff Werke, V (2005)), cf. (Hochkirchen 1999). 11Commentaries in (Hausdorff Werke, IV (2001), 44– 54, ) and in (Brieskorn 1996). 12Cf. S. Chatterji’s commentaries in (Hausdorff Werke, IV (2001), 163– 171, 182–190). 13(Koepke/Kanovei 2002) 14(Neuenschwander 1996) She could escape from deportation and tellectuals of the turn to the 20th cen- Hausdorff, Felix. 1923. “Eine Ausdehnung des managed to hide in the Harz region un- tury; but much of the correspondence Parsevalschen Satzes über Fourierreihen.” Mathematische Zeitschrift 16:163–169. In til the end of the war and the downfall seems to be lost. Most of what is (Hausdorff Werke, IV (2002), 173–162). of the Nazi regime. known, is due to patient and labori- Hausdorff, Felix. 1927. Mengenlehre, zweite ous collecting activities of E. Brieskorn, umgearbeitete Auflage. Berlin: de Gruyter. the driving force behind the editorial Revised 3rd edition 31935. English transla- The Nachlass and the Hausdorff edi- project. tion by J.R. Auman e.a., New York: Chelsea tion 1957, 1962. In (Hausdorff Werke, III (forth- Readers of this note who know about coming)). letters from or to Felix Hausdorff are Hausdorff’s voluminous Nachlass was Hausdorff, Felix. 1969. Nachgelassene Schriften. 2 handed over to a local friend of his. heartily invited to get into contact with Bände, ed. G. Bergmann. Stuttgart: Teub- It survived the end of the war with the editorial office and to communicate ner. only minor damages (Hausdorff n.d.). their findings to its scientific coordina- Hausdorff, Felix. Werke. Gesammelte Werke ein- It now lies at the University library tor: schließlich der unter dem Pseudonym Paul Prof. Dr. Walter Purkert Mongré erschienenen philosophischen und lit- at Bonn and has been made ac- erarischen Schriften. Ed. E. Brieskorn, F. cessible for research by a detailed Mathematisches Inst., Hausdorff Ed. Hirzebruch, W. Purkert, R. Remmert, E. catalogue [ http://www.aic.uni- Universität Bonn Scholz. Berlin etc.: Springer: 2001ff. [9 vol- wuppertal.de/fb7/hausdorff/findbuch. Beringstr. 1, D-53115 Bonn, Germany umes planned; II (2002), IV (2001), V (2005 forthcoming), VII (2004). asp]. e-mail: [email protected] Hausdorff, Felix. n.d.. “Nachlass.” Hand- An interdisciplinary group of 21 per- schriftenabteilung Universitätsbibliothek sons from four countries, under the Bonn. Findbuch [ http://www.aic.uni- direction of E. Brieskorn and W. Purk- wuppertal.de/fb7/hausdorff/findbuch.asp]. ert (Bonn), is preparing a collected References Hochkirchen, Thomas. 1999. Die Axiomatisierung edition of the scientifc, literary, and der Wahrscheinlichkeitsrechnung und ihre Kon- Brieskorn, Egbert (Hrsg.). 1996. Felix Haus- texte: Von Hilberts sechstem Problem zu Kol- philosophical works of Felix Hausdorff dorff zum Gedächtnis. Aspekte seines Werkes. mogoroffs Grundbegriffen. Göttingen: Van- in nine volumes (Hausdorff Werke). Braunschweig: Vieweg. denhoek und Ruprecht. This edition has been generously sup- Chatterji, Srishti. 2002. “Mesure and integra- Koepke, Peter; Kanovei, Vladimir. 2002. ported by the Deutsche Forschungs tion theory.” In (Hausdorff Werke, II (2002), “Deskriptive Mengenlehre in Haus- Gemeinschaft and the Akademie der 788–800). dorffs Grundzügen der Mengenlehre.” In Wissenschaften Nordrhein-Westfalen. It Epple, Moritz; Herrlich, Horst; Hušek, Mirek; (Hausdorff Werke, II (2002), 773–787). is now run as a long-term project Preuß, Gerhard; Purkert Walter; Scholz, Moore, Gregory. 1982. Zermelo’s Axiom of Choice. Erhard. 2002. Zum Begriff des topologi- Its Origins, Development and Influence. Berlin of the latter. Four volumes have schen Raumes. In (Hausdorff Werke, II etc.: Springer. already been published or are in (2002), 675–744). [http://www.aic.uni- Neuenschwander, Erwin. 1996. “Felix Haus- preparation for immediate publica- wuppertal.de/fb7/hausdorff/Topologi- scher−Raum.pdf]. dorffs letzte Lebensjahre nach Dokumenten tion; the remaining five are sched- aus dem Bessel-Hagen Nachlaß.” In uled to follow during the next few Felgner, Ulrich. 2002. “Die Hausdorffsche The- (Brieskorn 1996). orie der -Mengen und ihre Wirkungs- years [http://www.aic.uni-wupper- geschichte.” In (Hausdorff Werke, II (2002), Purkert, Walter. 2002. “Grundzüge der tal.de/fb7/hausdorff/baende.htm]. 645–674). [http://www.aic.uni-wupper- Mengenlehre – Historische Ein- tal.de/fb7/hausdorff/Eta−Alpha.pdf]. führung.” In In (Hausdorff Werke, II (2002),1–90). [http://www.aic.uni- Hausdorff, Felix (Mongré, Paul). 1898. Das Chaos Call for support in retrieving Haus- wuppertal.de/fb7/haus- in kosmischer Auslese. Leipzig: Naumann. In dorff/HistEinfuehrung.pdf]. dorff correspondence (Hausdorff Werke, VII (2004), 587–807). Stegmaier, Werner. 2002. “Ein Mathematiker Hausdorff, Felix. 1914a. “Bemerkung über den in der Landschaft Zarathustras. Felix Volume IX of the edition will contain Inhalt von Punktmengen.” Mathematische Hausdorf als Philosoph.” Nietzsche Studien the available scientific, literary, and Annalen 75:428–433. In (Hausdorff Werke, 31:195–240. [Similarly editor’s introduction personal correspondence of F. Haus- IV (2001), 3–10). in (Hausdorff Werke, VII, 1–83)] dorff. Very informative parts of the Hausdorff, Felix. 1914b. Grundzüge der Mengen- Vollhardt, Friedrich; Roth. 2002. Die Signifikanz lehre. Leipzig: Veit. In (Hausdorff Werke, II des Außenseiters. Der Mathematiker Felix exchange between P. Alexandroff and (2002), 91–576). F. Hausdorff have been preserved, and Hausdorff und die Weltanschauungsliteratur Hausdorff, Felix. 1919. “Dimension und äußeres um 1900. In Literatur und Wissen(schaften) also letters of Mongré/Hausdorff to Maß.” Mathematische Annalen 79:157–179. In 1890–1935, ed. C. Mailllard; M. Titzmann. the Nietzsche archive and to literary in- (Hausdorff Werke, IV (2001), 21–43). Stuttgart/Weimar: Metzler pp. 213–234.