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Main features of set theory. (Grundzüge der Mengenlehre.) (German) Zbl 1175.01034

Leipzig: Veit & Comp. 473 p., 53 Figuren (1914).
F. Hausdorff’s 1914 treatise on set theory [Grundzüge der Mengenlehre. Leipzig: Veit & Comp. (1914; JFM 45.0123.01)] is one of the great books of mathematics. A list of its eminent qualities might begin with readability, clearness, conciseness, liveliness, ingenuity, wittiness, diversity, comprehensiveness, and nonchalance. The command of language is that of a brilliant writer. Through a classical approach shines an idiosyncratic modernity. Marvellous presentations of textbook material are complemented with countless new ideas, which turned out to be seminal not only for set theory, but also for topology and measure theory.
The German term “Grundzüge” can be translated as “main features” or “outlines”. It denotes a broad treatment of a subject which might stop at a certain level of complexity, but which gives a full picture of what is considered to be characteristic. Hausdorff speaks of “Hauptsachen der Mengenlehre” (main issues of set theory) in his foreword and addresses a wide audience consisting of all “who possess some abstraction of thinking”. “Grundzüge” is definitely not to be read as “Grundlagen” (foundations), and thus the title already points at Hausdorff’s understanding of set theory, which is explained in the first chapter of the book:
“Die Mengenlehre ist das Fundament der gesamten Mathematik. Über das Fundament dieses Fundamentes ist eine vollkommene Einigung noch nicht erzielt worden. Den Versuch, den Prozeß der uferlosen Mengenbildung durch geeignete Forderungen einzuschränken, hat E. Zermelo unternommen. Da indessen diese äußerst scharfsinnigen Untersuchungen noch nicht als abgeschlossen gelten können und da eine Einführung des Anfängers in die Mengenlehre auf diesem Wege mit großen Schwierigkeiten verbunden sein dürfte, so wollen wir hier den naiven Mengenbegriff zulassen, dabei aber tatsächlich die Beschränkungen innehalten, die den Weg zu jenem Paradoxon abschneiden.”
(Set theory is the foundation of all mathematics. A complete agreement about the foundation of this foundation has not yet been reached. The attempt to delimitate the process of the boundless formation of sets by adequate postulates has been undertaken by E. Zermelo. But since this keen-witted analysis cannot be presumed to be completed, and since an introduction of the beginner into set theory along these lines should be linked with major difficulties, we want to allow the naive notion of a set here, but in doing so we in fact keep to the limitations which cut off the way to that paradox.)
So the teacher is aware of the paradoxes of naive set theory, but nevertheless teaches naive set theory. Hausdorff’s book marks the beginning of what has been done ever since: Beginners are not confronted with a – by now well-understood – axiomatic system, but they are taught naive set theory with a hint at Russell’s paradox. In 1914, when Zermelo’s first axiomatic system of 1908, including his axiom of choice, was still discussed controversially, completed and made precise, Hausdorff’s attitude is of crucial importance: While the foundations of the new foundation of mathematics had to be clarified and disseminated, an outstanding mathematician was there writing a dauntless 476 page book about set theory, substantially advancing the subject and its impact for all of mathematics. The effect was stabilizing. No one reading the book is left with the impression that set theory is something vague or inconsistent. The book is about the fascinating mathematics of infinity. After reading it, one might be eager to see how this rich theory can be given a proper foundation. Then Zermelo’s system and its extensions by Abraham Fraenkel and others naturally supply the theory presented in Hausdorff’s book with axioms. Thus Hausdorff, not interested in axiomatics himself, helped to promote axiomatic set theory.
Hausdorff’s achievement appears even greater when we look at the treatises on set theory written before 1914. Cantor presented his theory in two lengthy journal articles in 1895 and 1897, and these remained the main sources of knowledge for a long time. Besides there was, among some others, Arthur Schoenflies’s “Entwicklung von der Lehre von den Punktmannigfaltigkeiten” [Deutsche Math. Ver. 8, No. 2, 1–250; F. d. M. 31, 70, (1900; JFM 31.0070.08); Leipzig: B. G. Teubner. (1908; JFM 39.0095.16); reworked “Entwicklung der Mengenlehre und ihrer Anwendungen.” Leipzig und Berlin: Teubner. (1913; JFM 44.0087.18)], Gerhard Hessenberg’s “Grundbegriffe der Mengenlehre” [Göttingen: Vandenhoeck & Ruprecht. VIII u. 220 S (1906; JFM 37.0067.03)] and William and Grace Chisholm Young’s “The theory of sets of points” [Cambridge: University Press. XII u. 316 S. \(8^{\circ}\) (1906; JFM 37.0070.01)]. Commendable as they are, they now look of only historic value when compared to Hausdorff’s book.
The book emerged from research and teaching in equal measure. It is very likely that Hausdorff met Cantor regularly in Leipzig and Halle before the turn of the century. In 1901, Hausdorff gave a course on set theory to three students in Leipzig. His first set theoretic publication was a note on cardinal arithmetic in 1904. In 1905, he wrote a review of Russell’s influential “The principles of Mathematics”. Between 1906 and 1909 he wrote a series of highly original papers extending Cantor’s systematic analysis of well-orderings to the more general theory of linear orderings. The Mathematische Annalen paper of 1908 has an unusual length of 70 pages and its introduction hints at a book about the subject. Between 1909 and 1914 Hausdorff published mainly non-mathematical writings under the pseudonym Paul Mongré, which he had used since 1897. It is Paul Mongré who is behind the remarkable eloquence of the Grundzüge. Concerning teaching, Hausdorff lectured on set theory in 1910 and 1912 at Bonn. In 1912 he began to write the book, which appeared in April 1914 at von Veit in Leipzig. The reception was slow, also because of the First World War. But then the book was very well received by the mathematicians of the next generation, many from Poland and Russia, among them Pavel Alexandrov, Stefan Banach, Kazimierz Kuratowski, Wacław Sierpinski, Hugo Steinhaus, Alfred Tarski, Andrei Tikhonov, Stanisław Ulam, and Paul Urysohn. Hausdorff once wrote to Alexandrov that “my star indeed rises in the east”. An in-depth review of the book was written by Henry Blumberg for the Bulletin of the American Mathematical Society in 1920. Blumberg is full of admiration and praise: “It would be difficult to name a volume in any field of mathematics that surpasses the Grundzüge in clearness and precision.” This is no longer true: Using mathematical logic, the preciseness of the Grundzüge can easily be surpassed. But unless foundational matters are at stake, Hausdorff’s level of preciseness is perfectly balanced and still an – if not the – ideal. And also in other respects there is basically nothing to complain about. Blumberg only notes that “little is left to the reader’s imagination” and that there could have been “a more emphatic message”, but he adds that “such remonstrance would be like quarreling with Beethoven for having written symphonies instead of operas”.
The first of the ten chapters of the book introduces the approach of naive set theory and then defines all basic set theoretical operations with sets and systems of sets. Notably Hausdorff studies, in modern terminology, \(\sigma\)-rings, lattices of sets, and the ring generated by a lattice. In the second chapter functions are defined in the now standard way as certain sets of ordered pairs. Hausdorff notes parenthetically that an ordered pair \((a,b)\) could be defined as \(\{ \{ a, 1 \}, \{ b , 2 \} \}\). (Today, C. Kuratowski’s more intrinsic definition \(\{ \{ a \}, \{ a, b \} \}\) [Fundamenta Math. 4, 151–163 (1923; JFM 49.0409.02)] is preferred, but Hausdorff’s definition is a good example of the many small gems appearing in the book.) The rest of the chapter is devoted to operations with functions, including general Cartesian products.
Chapters three through six deal with main themes of Cantor’s set theory: cardinals and powers (chapter three), ordered sets and order types (chapter four), well-ordered sets and ordinal numbers (chapter five), and relations between ordered and well-ordered sets (chapter six). Hausdorff’s fondness of ordered sets becomes apparent, and indeed in his foreword he admits that this material is dealt with relatively broadly. His attitude to foundations is particularly important for these chapters: He assigns unspecified symbols to sets such that two sets \(M\) and \(N\) get the same symbol if and only if they are equipollent. The symbols are then called cardinals and the symbol of \(M\) is the cardinality of the set \(M\). Order-types of linear orderings are introduced in the same way. From a modern point of view, Hausdorff does not define cardinals and order-types (which is a nontrivial task working in an axiomatic system). But he stresses the important properties of his symbols and achieves a rich mathematical theory, which can, a posteriori, be equipped with a formal definition. Hausdorff would not regard this last step as important, in contrast to John von Neumann, who gave the first formal definition of an ordinal number in 1923. The same applies to transfinite recursion, which Hausdorff takes for granted, while von Neumann proves transfinite recursion.
Chapters seven through ten turn to “applications” of set theory and they have been of enormous impact. The first three of the four chapters give a detailed and comprehensive introduction to “point sets in general spaces” (chapter 7), “point sets in special spaces” (chapter 8), and “mappings or functions” (chapter 9), spanning almost 200 pages. The presented concepts include neighbourhood, topological space, boundary of a set, compact set, relative topology, connectedness, density, separability, first and second countability, metric space, complete space, Euclidean space, continuous function, dimension, convergence of a sequence of functions. One might ask what was known before, what had to be systemized, and what is completely new. But condensed to one sentence, the three chapters are the birth of modern set-theoretic topology. Moreover, they also contain important advances in descriptive set theory: Hausdorff continues the study of definable sets of reals that had begun with Cantor’s analysis of closed sets. Transfinite hierarchies figure prominently here, and in particular the book introduces the transfinite Borel hierarchy of sets which stratifies the \(\sigma\)-algebra generated by all open sets. Hausdorff would prove in 1916 that every Borel set is countable or of the cardinality of the continuum. In an appendix of the Grundzüge a partial result is established using methods apt to prove the later general theorem.
The final chapter of the book is devoted to measure theory and integration. Hausdorff rediscovers G. Vitali’s now famous example of a non-measurable set [Bologna: Gamberini e Parmeggiani. 5 S. \(8^\circ\) (1905; JFM 36.0586.03)]. He presents the Peano-Jordan content and Lebesgue’s measure and integration theory for the Euclidean spaces. In the appendix to chapter 10 we find Hausdorff’s first presentation of his paradoxical decomposition of the sphere. It was later generalized by Banach and Tarski to what is now known as the Banach-Tarski-Paradox, and it was the root of von Neumann’s theory of amenable groups. Moreover, Hausdorff proves in the appendix that every content on a lattice of sets can be uniquely extended to the generated ring. This theorem, overlooked and reproved by several mathematicians, forms the basis of a measure theory different from Carathéodory’s, as it has been advanced by Heinz König.
In 1923, the Grundzüge were out of print and Hausdorff was asked for a new edition by the Walter de Gruyter press in Berlin, who had bought von Veit after the war. The book should appear in the series “Göschens Lehrbücherei” and the extent was, according to the guidelines of the series, limited to 320 pages. Thus Hausdorff had to rewrite the book. It appeared in 1927 with the title “Mengenlehre. Zweite, neubearbeitete Auflage” [Set Theory. Second, reworked edition, Walter de Gruyter & Co., Göschens Lehrbücherei Gruppe I Band 7 (1927; JFM 53.0169.01)]. Hausdorff streamlined the discussion of basic set theoretical notions and omitted most of order theory. He also sacrificed Lebesgue’s measure and integration theory “because there is no lack of other presentations”. The severe truncation that will “perhaps be more regretted” was the concentration on metric instead of topological spaces. Concerning foundations, Hausdorff’s interests did not change: “I could not, now as then, convince myself to a discussion about paradoxes and foundational criticism.” At least the references contain E. Zermelo’s 1908 paper [Math. Ann. 65, 261–281 (1908; JFM 39.0097.03)] and A. Fraenkel’s second edition of his “Einleitung in die Mengenlehre” [Berlin: J. Springer. (1923; JFM 49.0136.02)]. Mockeries like “we have to leave it to philosophy to fathom the ‘true being’ of cardinal numbers” are still funny, but in view of John von Neumann’s work they also begin to look out of date, misguiding readers about the necessity and possibility of a proper mathematical definition.
But the second edition also contains a lot of new material, as it comprises a clear, thorough, and up-to-date account of descriptive set theory. Borel sets are studied in great detail, as are the more general Suslin or analytic sets. Hausdorff presents his own contributions, together with recent work by Pavel Alexandrov, Henri Lebesgue, Nikolai Lusin, Wacław Sierpiński, Mikhail Suslin, and others. Hausdorff’s approach is general and establishes the results not only for the reals numbers, but for Borel and Suslin sets in Polish spaces, which are the basic structure of descriptive set theory today.
Hausdorff published a slightly extended third edition of “Mengenlehre” in 1935 [Zbl 0012.20302]. A Russian translation, edited by Alexandrov and Kolmogorov, appeared in 1937. It tries to merge the advantages of the different editions by presenting what might be called a modernized compilation of the books. An English translation of the third edition appeared in 1957 [Set theory. Translated from the German by John R. Aumann et al. New York: Chelsea Publishing Company. (1957; Zbl 0081.04601)], and was reprinted many times. Recently, the “Grundzüge” of 1914 as well as the “Mengenlehre” of 1927 and its additions of 1935 were photomechanically reprinted, annotated and commented in volumes II and III of Hausdorff’s collected works [eds. Egbert Brieskorn et al., Berlin: Springer (2002; Zbl 1010.01031); (2008; Zbl 1149.01022)]. These volumes are an invitation to read and compare Hausdorff’s books in their original shape, with the help of essays providing historical and mathematical background.

MSC:

01A75 Collected or selected works; reprintings or translations of classics
03-03 History of mathematical logic and foundations
03-02 Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations
03-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mathematical logic and foundations
01A60 History of mathematics in the 20th century
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