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Felix Hausdorff i n Greifswald, 1914-192 1 Universitats-und Landesbibliothek , Hss.-Abt. NL Hausdorff: Kapse l 65: Nr. 06 https://doi.org/10.1090/hmath/025

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Hausdorff o n Ordered Set s

J. M. Plotkin Editor

Translated by J. M. Plotki n

America n Mathematica l Societ y Londo n Mathematica l Societ y Editorial Boar d American Mathematica l Societ y Londo n Mathematica l Societ y Joseph W . Daube n Ale x D . D . Crai k Peter Dure n Jerem y J . Gra y Karen Parshall , Chai r Pete r Neuman n Michael I . Rose n Robi n Wilson , Chai r

Original Germa n tex t translate d b y J . M . Plotki n

The articl e Sum ofH\ sets b y Feli x Hausdorff, originall y published a s "Summe n vo n Ni - Mengen", Fundament a Mathematica e XXV I (1936 ) i s reprinted b y permissio n o f Instytu t Matematyczny Polskie j Akademi i Nauk .

2000 Subject Classification. Primar y 01A75 , 01A60 , 03-03 , 06-03 , 26-03 .

For additiona l informatio n an d update s o n thi s book , visi t www.ams.org/bookpages/hmath-25

Library o f Congres s Cataloging-in-Publicatio n Dat a Hausdorff, Felix , 1868-1942 . [Selections. English . 2005 ] Hausdorff o n ordere d set s / J . M . Plotkin , edito r ; translated b y J . M . Plotkin . p. cm . — (Histor y o f mathematics, ISS N 0899-242 8 ; v. 25 ) Includes bibliographica l references . ISBN 0-8218-3788- 5 (alk . paper ) 1. Ordered sets-History. 2 . Hausdorff, Felix , 1868-1942 . 3 . Logic, Symbolic and mathematical - History, 4 . Lattic e theorv-History . 5 . Functions-History . I . Plotkin , J . M . (Jaco b M.) , 1941 - II. Title . III . Serie s QA171.48.H38 200 5 511.3/2-dc21 200504532 8

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Felix Hausdorff i n his study a t home, Bonn, June 8-14, 192 4 Universitats-und Landesbibliothe k Bonn , Hss.-Abt. NL Hausdorff: Kapse l 65: Nr. 29 Contents

Preface Selected Hausdorf f Bibliograph y Introduction t o "Abou t a Certain Kin d o f Ordered Sets " About a Certain Kin d o f Ordered Set s [ H 1901b ] Introduction t o "Th e Concep t o f Power i n Se t Theory " The Concep t o f Power i n Se t Theor y [ H 1904a ] Introduction t o "Investigation s int o Orde r Type s I , II, III" Investigations int o Orde r Type s [ H 1906b ] I. The Power s o f s II. Th e Highe r Continu a III. Homogeneou s Type s o f the Secon d Infinit e Cardinalit y Introduction t o "Investigation s int o Orde r Type s IV , V " Investigations int o Orde r Type s [ H 1907a ] IV. Homogeneou s Type s o f o f the Continuu m V. O n Pantachie Type s Introduction t o "Abou t Dens e Orde r Types " About Dens e Orde r Type s [ H 1907b ] Introduction t o "Th e Fundamentals o f a Theory o f Ordered Sets " The Fundamentals o f a Theory o f Ordered Set s [ H 1908 ] Introduction t o "Graduatio n b y Final Behavior " Graduation b y Final Behavio r [ H 1909a ] Appendix. Sum s o f N i Set s [ H 1936b ] Bibliography This page intentionally left blank Preface

Introduction As the nineteent h centur y wa s ending, Geor g Canto r (1845-1918 ) pub - lished hi s ow n Summa Theologica fo r se t theory . Thi s two-par t work , en - titled Beitrdge zur Begriindung der transfiniten Mengenlehre, appeare d i n 1895 an d 189 7 i n volume s 4 6 an d 49 , respectively , o f Mathematisch e An - nalen. Th e secon d installmen t o f Beitrdge turne d ou t t o b e Cantor' s las t published wor k on . Thoug h Beitrdge coul d wel l serve as a primer on basi c se t theory , i t als o seeme d unfinished . Fundamenta l question s re - mained. Ar e al l cardinal s comparable ? Ca n al l set s b e well-ordered ? Ar e there just tw o possibilities fo r th e cardinalitie s o f infinite set s o f reals? Ye t in some cosmically just wa y this work marked a fitting endin g fo r a journey that bega n with the bread an d butter questio n o f uniqueness o f representa - tions o f functions b y trigonometric serie s and ended i n the realm o f pure se t theory. The yea r 189 7 als o sa w Cantor' s contribution s gai n officia l an d publi c recognition fro m hi s peers. Th e First Internationa l Congres s o f Mathemati- cians took place in Zurich in August o f 1897 , and Canto r wa s in attendance. In a major address , th e Zuric h mathematicia n Adol f Hurwit z (1859-1919) , who wa s on e o f the congress' s organizers , devote d considerabl e spac e to a n exegesis o f Cantor' s ordina l number s an d thei r applicatio n t o th e analysi s of closed point sets . Hurwit z propounde d Cantor' s wor k a s the basi s fo r a n investigation o f analytic functions i n terms o f their set s o f singularities.1 A t this same congress, Jacques Hadamard (1865-1963 ) gave a short talk entitle d Sur certaines applications possibles de la theorie des ensembles. The next international congress was held in Paris in August 1900 . There , David Hilbert (1862-1943 ) gave the famous address in which, after a provoca- tive meditatio n o n th e esthetic s o f mathematical problem s an d o n th e cri - teria fo r admissibl e solutions , h e liste d twenty-thre e problem s whos e solu - tions woul d advanc e th e entir e enterpris e o f mathematics. 2 Th e firs t thre e

1 Uber die Entwickelung der allgemeinen Theorie der analytischen Funktionen in neurer Zeit, Verhandlunge n de s ersten internationalen Mathematiker-Kongresse s i n Zuric h vom 9 . bi s 11 . August 189 7 ( : Teubner), 1898 , 91-112 . Hurwitz' s discussio n o f Cantor's wor k appear s o n pp. 94-97 . 2Because o f time constraints , Hilber t mentione d onl y ten problem s i n hi s actua l ad - dress: thre e fro m foundations , fou r fro m arithmeti c an d algebra , an d thre e fro m functio n theory. Al l twenty-three problem s ar e i n th e printe d version : Mathematische Probleme,

ix X PREFACE problems deal t wit h foundations , an d proble m No . 1 was labelle d Cantors Problem von der Mdchtigkeit des Continuums. A s its title indicates, the firs t problem call s fo r a proof o f what i s now termed Cantor' s Wea k Continuu m Hypothesis: ther e ar e only tw o possible fo r the infinit e subset s of th e continuum , tha t o f th e se t o f natura l number s an d tha t o f th e en - tire se t o f reals . I n addition , th e firs t proble m als o call s fo r a proo f o f th e well-orderability o f the continuum , agai n a s conjectured b y Cantor. 3 Thus a s the twentieth centur y began , Cantor , whos e work i n set theor y was finall y receivin g it s due , ha d begu n a d e fact o retirement , an d Davi d Hilbert, on e o f th e tw o mathematica l great s o f th e day—th e othe r bein g Henri Poincar e (1854-1912)—ha d mad e th e positiv e resolutio n o f tw o o f Cantor's mos t importan t conjecture s hi s own number on e problem. 4 I t wa s time fo r th e appearanc e o f the secon d generatio n o f Cantorians . The firs t decad e o f the ne w centur y witnesse d a burst o f activity i n se t theory wit h man y o f the participants belongin g to a group twenty to thirt y years younger than Cantor ; amon g them wer e Feli x Bernstein (1878-1956) , Gerhard Hessenber g (1874-1925) , Phillip Jourdai n (1879-1919) , Ernst Zer - melo (1871-1953) , an d Feli x Hausdorf f (1868-1942) . Althoug h h e wa s no t of thi s younge r generation , th e vetera n mathematicia n Arthu r Schoenflie s (1853-1928) als o worked i n set theory durin g this period. H e was an arden t disseminator o f Cantoria n ideas , writin g a n encyclopedi a articl e an d book - length report s o n se t theor y an d it s applicatio n t o th e stud y o f poin t sets , i.e., subset s o f R an d R n ([Sc h 1898 ; 1900 ; 1908]). However, i n th e year s 1900-1910 , i t wa s Zermel o an d Hausdorf f wh o came to dominate the discours e o n set theor y an d wh o represented th e tw o trends that wer e to characteriz e se t theory' s future : it s abstractio n an d it s pursuit as a mathematical theory. Zermelo , in seeking to clarify the existence assumptions i n hi s proo f o f well-ordering , axiomatize d th e concep t o f se t based solel y o n th e membershi p relatio n an d a fe w simpl e operations ; h e became the fathe r o f abstract se t theory. 5 Hausdorf f carrie d o n i n Cantor' s footsteps, developin g set theory as a branch o f mathematics worthy o f study in it s ow n righ t an d capabl e o f undergirdin g bot h genera l topolog y an d

Gottinger Nachrichte n (1900) , 253-297 . Hi s entir e text , translate d int o Englis h b y Dr . Mary Winsto n Newton , appear s i n the Bulletin of the American Mathematical Society, 8 (1901-1902), 437-479 . 3 Hilbert fel t tha t th e well-orderabilit y o f the continuu m ha d a clos e connectio n wit h the continuu m proble m an d tha t perhap s i t wa s th e ke y t o it s solution . H e aske d tha t the solve r provid e a specifi c (i.e. , definable ) well-ordering : u Es erschein t mi r hochs t wiinschenswert, einen direkten Beweis dieser merkwurdigen Behauptung von Cantor zu gewinnen [hi s italics] , etw a durc h wirklich e Angab e eine r solche n Ordnun g de r Zahle n ..." ["I t appear s highl y desirabl e t o m e to produce a direct proof of Cantor's remarkable assertion, perhap s b y actuall y indicatin g on e suc h orderin g o f the [real ] numbers .. . "J 4Actually, thei r positio n o n Hilbert' s lis t i s a n artifac t o f hi s organizationa l scheme , but thei r placemen t di d hav e a psychologica l effect . 5See [K a 1996 , 9-12 ] an d [K a 2004] . Th e latte r articl e i s a penetrating examinatio n of Zermelo's plac e i n the histor y o f set theory . PREFACE XI theory ; h e wa s a founde r o f th e firs t an d a significan t contributo r to th e second . At leas t superficiall y th e education an d earl y professional career s o f both men see m quit e similar : dissertation s an d habilitation s i n applie d mathe - matics, a n interes t i n Cantor' s se t theor y a s manifeste d i n 190 1 b y th e offering o f the firs t course s devoted entirel y to se t theor y an d b y the appear - ance o f their ow n firs t publication s i n se t theory . W e paus e t o presen t brie f biographical sketche s fo r bot h (u p throug h th e mid-teen s o f the earl y nine - teen hundreds) , askin g th e reader' s indulgenc e a s w e maintai n th e concei t of parallel development . Ernst Zermel o wa s bor n i n Berli n i n 1871 . Afte r studyin g a t it s uni - versity an d the n a t Hall e an d Freiburg , h e returne d t o Berli n wher e h e received hi s doctorate i n 1894 ; his dissertatio n o n the calculu s o f variation s was directe d b y H . A . Schwar z (1843-1921) , Weierstrass' s successor . H e then becam e a n assistan t t o Ma x Planc k a t ' s Institut e fo r Theoret - ical , stayin g ther e fro m 189 4 t o 1897 . H e move d t o Gottinge n i n 1897, where h e complete d hi s Habilitation i n 189 9 with a work o n hydrody - namics. A s a Privatdozent a t Gottingen , Zermel o wa s influence d b y Hilber t to tur n hi s attention t o question s i n se t theor y an d foundations. 6 At Gottinge n i n th e winte r semeste r o f 1900-1901 , Zermel o offere d a course devote d entirel y t o se t theor y (i n th e spiri t o f Cantor' s Beitrdge). This wa s th e firs t suc h cours e offere d anywhere . I n 1901 , he publishe d hi s first se t theor y paper , Uber die Addition transfiniter Cardinalzahlen, i n th e Gottinger Nachrichten . I t wa s a modes t contributio n t o th e arithmeti c o f cardinal numbers . However , hi s second publicatio n i n se t theor y ([Z e 1904] ) ensured hi s immortality : h e prove d tha t ever y se t coul d b e well-ordere d o n the basi s o f a newl y articulate d principle , th e Axio m o f Choice. 7 Initia l reactions wer e les s than laudatory ; criticis m cam e swiftl y an d wa s no t sotto voce. Zermel o enjoye d th e privat e suppor t o f Hilbert , bu t i n publi c h e wa s largely lef t t o defen d himself . An d defen d himsel f h e did . I n [Z e 1908a] , he offere d a ne w proo f o f the Well-Orderin g Theore m togethe r wit h a bril - liantly writte n repl y t o hi s critics . I n a second , related , pape r [Z e 1908b] , Zermelo presented hi s axiomatization fo r the concep t o f set, which , with im - portant late r contributions b y Fraenkel and Skolem , i s the widely recognize d ZF (ZFC ) se t theor y o f today . Thi s firs t phas e o f Zermelo' s foundationa l

6In [P e 1990], Volker Peckhaus gives a detailed picture o f Zermelo's years at Gottingen . Though plague d b y financial an d healt h problems , Zermel o enjoye d increasin g status wit h Hilbert an d th e member s o f hi s school . Fro m 1905 , h e wa s Titularprofessor, a positio n financed b y "sof t money. " Hilbert' s manuever s t o regulariz e hi s position failed . 7 At the Third International Congres s of , hel d in in August 1904, Julius Konig presented a proof—later withdrawn—tha t th e continuum coul d no t b e well-ordered. (Se e the Introduction to "The Concept of Power in Set Theory," p . 24. ) A little ove r a mont h afte r thi s unsettlin g (fo r se t theorists ) meeting , Zermel o discovere d his proo f tha t th e continuu m an d an y othe r se t coul d b e well-ordered . Zermelo' s proo f i s contained i n a brie f lette r t o Hilber t date d Septembe r 18 , 1904 . Xll PREFACE research ended i n 191 0 when he left Gottinge n fo r a Professorship a t Zurich . In 1916 , Zermelo retired fro m thi s positio n fo r healt h reasons. 8 Felix Hausdorf f wa s born i n Breslau (no w Wroclaw) i n 186 8 and move d with hi s parent s t o Leipzi g a t a n earl y age . Afte r studyin g a t Leipzig , Freiburg, an d Berlin , h e returne d t o Leipzi g wher e h e receive d hi s doctor - ate i n 1891 ; hi s dissertatio n o n th e refractio n o f ligh t i n th e atmospher e was supervised b y (1848-1919) , a mathematical astronome r and, fro m 1882 , director o f Leipzig's observatory . I n 1893-1895 , Hausdorf f worked a t th e observator y doin g scientifi c calculations . A s a Privatdozent for astronom y an d mathematics , h e complete d hi s Habilitation a t Leipzi g in 189 5 with a wor k o n th e absorptio n o f ligh t i n the atmosphere . B y th e late 1890s , Hausdorf f bega n t o mov e awa y fro m applie d mathematic s an d towards pure mathematics wit h publications i n theory an d non - Euclidean geometry . At Leipzig , i n th e summe r semeste r o f 1901 , h e taugh t a Beitrdge- influenced cours e o n se t theory , historicall y th e secon d suc h course—afte r Zermelo. I n 1901 , h e als o publishe d hi s first pape r o n se t theory , Uber eine gewisse Art geordneter Mengen, i n the Report s o f Leipzig' s Koniglic h Sachsischen Gesellschaf t de r Wissenschaften . I n 1902 , h e becam e aussor- dentlicher Professor a t Leipzig ; shortl y thereafter , i n what ma y hav e bee n an unwis e caree r move , h e turne d dow n th e offe r o f a simila r positio n a t Gottingen.9 Hausdorf f wa s appointed Extraordinarius a t Bon n i n 1910 . H e became ordentlicher Professor a t Greifswal d i n 1913. 10 Durin g th e perio d 1900-1910, he published thirteen article s o n mathematics. Seve n o f them— some 28 1 journal pages—wer e i n se t theory , mostl y abou t ordere d sets . I n 1914, h e publishe d th e boo k Grundzuge der Mengenlehre. 11 Th e first si x chapters, approximatel y hal f o f this classi c and highly praised textbook, ar e devoted to Hausdorff's versio n o f Cantorian set theory as developed throug h

Zermelo's mov e to Zuric h terminated wha t Peckhau s call s his first pro-Hilbert phas e [Pe 2002]. Afte r hi s retirement, h e moved bac k to i n 192 1 and too k up residenc e in the Black Forest nea r Freiburg. I n 1926 , he was appointed t o an honorary Professorshi p at Freiburg , whic h wa s a n unsalarie d position . H e returne d t o foundationa l researc h i n the perio d 1927-1935 ; Peckhau s label s thi s Zermelo' s co?7ira-Hilber t phase , becaus e h e totally oppose d Hilbert' s finitistic proo f theory . Zermel o los t hi s positio n a t Freibur g i n 1935 fo r failin g t o giv e th e Naz i salut e properly ; h e wa s reinstate d i n 1946 . H e die d i n Freiburg o n Ma y 21 , 1953. 9See [D i 1967 , 52 ] on this episode . 10Hausdorff returne d t o Bon n a s Ordinarius i n 1921 . H e wa s forcibl y retire d b y th e Nazis i n 193 5 because h e wa s a Jew . H e die d i n Bon n o n Januar y 26 , 1942 . Hausdorff , his wife , an d hi s wife' s siste r committe d suicid e just befor e the y wer e t o b e interne d a t the forme r cloiste r Zur Ewigen Anbetung i n Endenich. Endenic h wa s the first sto p befor e deportation t o th e Eas t an d almos t certai n death . A photocop y an d tex t transcriptio n of Hausdorff' s stoi c an d poignan t farewel l letter , date d Januar y 25 , 1942 , appears i n [B r 1996, 263-267] ; an Englis h translation o f this lette r appear s i n [E i 1992 , 101-102] . 1 *Hausdorff's dedicatio n i n Grundzuge reads : "De m Schopfe r de r Mengenlehr e Herr n i n dankbarer verehrun g gewidmet " [ [ "Dedicated t o the creator o f set theor y Herr Geor g Canto r i n gratefu l admiration"]] . PREFACE xin a decade of his own research; the last four chapters form an important found - ing text fo r genera l . This end s ou r brie f attemp t a t comparativ e biography . A n intriguin g question remain s unanswered : fo r Zermelo , i t wa s Hilbert' s influenc e tha t brought hi m t o se t theor y an d foundations , bu t wha t le d Hausdorf f t o se t theory? Bot h Cantor an d Hausdorff attende d th e 189 7 Zurich congress, an d the mathematicians a t Hall e and Leipzi g met frequently . Thes e gatherings , known a s Kranzchen, alternate d betwee n th e tw o locations , an d som e a t Halle wer e even hel d a t Cantor' s hom e ([Ko w 1950 , 106]) . Bu t socia l inter - action alon e doe s no t accoun t fo r intellectua l affinity . Fo r som e importan t clues, w e need t o conside r th e work s o f the write r Pau l Mongre . Mongre's firs t book , whic h showe d th e influenc e o f both Nietzsch e an d Schopenhauer, wa s the aphoristic Sant'llario-Gedanken aus der Landschaft Zarathustras] i t appeared i n 1897 . A second book followe d i n 1898 . Entitle d Das Chaos in kosmischer Auslese-Ein erkenntniskritischer Versuch, it wa s critical o f metaphysics an d disparage d th e abilit y o f the empirica l worl d a s recognized by our consciousness to reveal anything about the transcendenta l world.12 I n 1900 , Mongre published Ekstasen, a book o f poetry, an d i n 190 4 his on e ac t pla y Der Arzt seiner Ehre, a comedy , wa s performe d i n Berli n and Hamburg . I n th e perio d 1897-1910 , Pau l Mongre , th e frien d o f avan t garde artists an d writer s an d th e quasi-discipl e o f Nietzsche, als o publishe d essays and review s in various cultural periodicals . Th e variet y an d quantit y of Mongre' s outpu t alon e migh t piqu e ou r interest , bu t fo r u s hi s mos t remarkable attribut e i s his true identity—h e wa s Feli x Hausdorff. 13 It seem s tha t i t wa s Hausdorff' s philosophical-literar y alte r ego , Pau l Mongre, who inspired hi s interest i n set theory . Walte r Purkert , i n the firs t section o f hi s ver y informativ e introductio n t o th e secon d volum e o f Haus- dorff's Collecte d Work s [P u 2002 , 2-7] , present s stron g interna l evidenc e that Hausdorf f becam e acquainte d wit h Cantoria n se t theor y somewher e i n the perio d 1897-1898 . I n particular , Purker t establishe s tha t betwee n th e writing o f Sant'llario an d Das Chaos Hausdorff-Mongre learne d o f Cantor's

12We refe r t o thes e book s a s Sant'llario an d Das Chaos fo r short . Th e Englis h translations o f their title s are: Saint llario-Thoughts from Zarathustra's Countryside an d Chaos in Cosmic Selection-A Critical Epistemological Essay, respectively . 13 A complete Hausdorff-Mongr e bibliograph y i s available fro m http://www.aic.uni-wuppertal.de/fb7/hausdorff/schriver.htm. All ou r reference s t o Hausdorff' s publication s follo w th e designation s assigne d b y thi s bibliography. A reprin t o f Das Chaos appeare d i n 197 6 with a n altere d title : Zwischen Chaos und Kosmos oder Vom Ende der Metaphysik, wit h a n introductio n b y Ma x Bens e (Baden - Baden: Agis Verlag) . Bot h Sant'llario an d Das Chaos ar e reprinte d i n Felix Haus- dorff, Gesammelte Werke, Band VII, Philosophisches Werk, W . Stegmaie r (ed. ) (Berlin - Heidelberg-New Yor k : Springer-Verlag), 2004 . Thi s volume also includes line-by-lin e com - mentaries fo r SanVllario an d Das Chaos. Werne r Stegmaier' s introductio n describe s Hausdorff's philosophica l antecedent s an d th e mai n idea s i n hi s philosophica l works . I n addition, thre e shor t article s b y Hausdorf f o n Nietzche' s theor y o f return J Wiederkunft des Gleichen\ an d hi s Will to Power \Der Wille zur Macht\ ar e reprinted . XIV PREFACE

1878 result tha t al l the R n hav e the sam e cardinality . H e als o cite s severa l places i n Das Chaos wher e connection s betwee n philosophica l argumenta - tion an d se t theoreti c concept s an d theorem s ar e made. 14 Philosophical interests, as expressed i n his writings as Paul Mongre, may have brought Hausdorf f t o Cantor , bu t severa l commentators hav e seen th e solution o f Cantor' s continuu m proble m a s the drivin g forc e behin d Haus - dorff's wor k o n ordere d set s i n th e year s 1901-1909. 15 I t i s no t har d t o imagine that an y wh o was worth his or her salt and who had more tha n a passin g acquaintanc e wit h Cantor' s wor k woul d b e tantalize d by the challeng e se t b y Hilbert an d woul d tak e u p the continuu m problem . With respect to this challenge, starting i n 1904 , Hausdorff, the skilled math- ematician, extende d Cantor' s concep t o f cardina l exponentiatio n t o creat e the tool o f ordered powers o f ordered sets; he discovered the fundamental re - lation o f cofinality, which led him to isolate an important clas s of invariants, the regular cardinal s and their inverses . H e then use d ordered products an d his invariants t o develo p a representation theory , ab initio, fo r ordere d set s in general and the important subfamil y o f densely ordered sets in particular. His representatio n theor y ofte n bumpe d int o th e continuu m problem , bu t he seeme d t o us e suc h occurrence s t o direc t hi s attention t o mor e realisti c goals or to measure the difficult y o f a specifi c problem . For th e intereste d scholar , ther e i s no better plac e to begi n t o see k an - swers t o meta-questions , suc h a s th e rol e playe d b y th e continuu m prob - lem i n Hausdorff' s research—motivationa l goa l o r cautionar y symbo l o f intractability—than i n the articles themselves. Thes e same articles are also a treasure trove fo r those wanting to find the sources o f concepts and method s that hav e hitherto bee n see n a s originating i n Grundzuge der Mengenlehre. These range fro m th e perhaps mundane identificatio n o f sets with function s via characteristic functions to the famous back-and-forth constructio n o f iso- morphisms, fro m the extremely fruitfu l concep t o f 77-se t to what w e now call Hausdorff's maxima l principle . O f course , w e also fin d thing s tha t di d no t make i t int o Grundzuge: worthwhil e ideas , such a s the concep t o f type ring and importan t results , suc h a s th e calculatio n o f a shar p lowe r boun d fo r the cardinalit y o f irreducible, continuou s c^-sets an d th e proo f that Haus - dorff gap s exis t i n certain partiall y ordere d sets . The n ther e ar e the firsts: for example , th e firs t eliminatio n o f C H fro m a proo f ([ H 1907a]) , the firs t statement o f GC H ([ H 1908]) , an d th e firs t us e o f a maxima l principl e i n algebra ([ H 1909a]) . Finally , through thes e articles we are afforde d glimpse s of Hausdorff' s opinion s o n foundation s a s h e trie s t o d o battle agains t th e ineffability o f the uncountable .

14Das Chaos i s discusse d i n connectio n wit h Hausdorff' s mathematica l wor k i n [E i 1992, 85-91; 1994 , 3-10]; Erhard Schol z echoes this title in his article Logischen Ordnungen im Chaos ([Sch o 1996] ) an d use s i t thematicall y i n commentin g o n Hausdorff' s wor k u p to 1919 . 15In particular , se e [M o 1989 , 109 ; 1996 , 127 , 130] , [B r 1996 , 5] , [Ko e 1996 , 86] , and [Scho 1996 , 109 , 114]. PREFACE xv

Our mai n ai m i n translatin g Hausdorff' s paper s o n ordere d set s i s t o make the m availabl e t o a large r audienc e fo r whateve r purpose . A t th e same time , w e hop e t o expos e mor e peopl e t o th e earl y wor k o f a creativ e and productive mathematician o f the firs t rank . Th e period covere d i s truly an amazin g on e i n the histor y o f se t theory : " a hundred flower s bloomed, " and man y wilte d i n the intens e light . However , unde r th e aegi s o f Hilbert' s school, Zermel o create d th e ne w branc h o f abstrac t se t theory , whil e sep - arately th e inner-directe d Hausdorf f explode d upo n th e scen e an d usurpe d the positio n o f the era's numbe r on e Cantorian. 16 The Translation s an d Introductor y Essay s The article s translated i n this volume are: Uber eine gewisse Art geord- neter Mengen \About a Certain Kind of Ordered Sets\ [ H 1901b] ; Der Potenzbegriff in der Mengenlehre \The Concept of Power in Set Theory^ [H 1904a] ; Untersuchungen uber Ordnungstypen I, II, III ^Investigations into Order Types I, II, J77] ] [ H 1906b]; Untersuchungen uber Ordnungstypen IV, V ^Investigations into Order Types IV, V 1^ [ H 1907a]; Uber dichte Ord- nungstypen [[O n Dense Order Types\ [ H 1907b] ; Grundziige einer Theorie der geordneten Mengen [[T/z e Fundamentals of a Theory of Ordered Sets\ [H 1908]; Die Graduierung nach dem Endverlauf ^Graduation by Final Be- havior1^ [ H 1909a]; and in the Appendix, Summen von K i Mengen \Sums of Ki Sets\ [ H 1936b] . Furthe r bibliographica l dat a fo r thes e article s appear s in our Selected Hausdorff Bibliography. The provenanc e o f these translation s require s som e explanation . I wa s led to Hausdorff's Grundziige der Mengenlehre an d then to his earlier paper s on ordere d set s i n th e earl y 1990 s a s I tried t o discove r th e origin s o f th e back-and-forth construction . Thi s resulted i n [P I 1993] and in my realization of the wealth o f material i n [ H 1906b; 1907a] . I was determined t o produc e translations o f both . I n th e summe r o f 1995 , I finishe d a firs t versio n o f [H 1906b] . Th e projec t la y dorman t unti l 1997 , whe n a t th e AM S win - ter meeting s i n Sa n Diego , I spok e abou t Hausdorff' s wor k o n orde r types . Afterwards, Mario n Scheeper s introduce d himself , an d I learne d o f hi s in - terest i n these sam e papers . Mario n ha d don e som e partial translation s o f Hausdorff whil e pursuing hi s study o f gap theorems (se e [S c 1993]). Shortly afte r ou r meetin g i n Sa n Diego , w e bega n a n extensiv e e-mai l correspondence i n which w e decided to work together o n translations o f the remaining paper s o n ordere d sets . A s amateu r translator s w e wer e bot h feeling our way. W e agreed that th e translations were to be fairly literal an d that anachronisti c terminolog y wa s to b e avoided . Eac h o f u s had primar y responsibility fo r certai n papers : I too k [ H 1901b ; 1907b ; 1908 ; 1909 a §1 ] and Mario n too k [ H 1904a ; 1907a ; 1909 a §§2-5] . Translation s wer e passe d

16The creativ e outpourin g fro m Hausdorff-Mongr e i n th e spa n 1897-191 0 i s trul y remarkable, a s i s its intellectua l scope . Th e Hausdorff-Mongr e bibliograph y list s twenty - two items fo r Hausdorf f an d twenty-one item s fo r Mongre during this era; fo r Mongre , thi s includes thre e book s an d fo r Hausdorff , thre e memoir-lengt h articles . XVI PREFACE back an d fort h electronically , an d th e tas k o f smoothin g roug h spot s an d maintaining uniformit y acros s variou s iteration s wa s m y responsibility . B y September 1 , 1998, the first version s o f all these papers had been completed . In the sprin g o f 1999 , Marion decide d to ad d a translation o f [ H 1936b ] be- cause of its connection with [ H 1909a] and its relevance to the gap literature; a firs t versio n wa s finished b y June. Th e joint phas e o f our translating wor k ended i n the fal l o f 1999 . Sinc e then, I have gon e back to the original s an d reworked th e translations . O f course , havin g th e version s fro m ou r join t efforts mad e th e resultin g iteration s muc h les s burdensome, an d i t allowe d me to concentrate on passages that ha d earlier caused u s difficulty. Multipl e passes o f proofreadin g helpe d t o increas e th e accurac y o f the translations . This reworkin g took plac e fro m th e winter o f 200 0 through th e fal l o f 2002. In Januar y o f 2003 , I began writin g introductor y essays fo r th e papers . Each translation , excep t tha t o f [ H 1936b ] i n the Appendix , i s preceded b y an essay, prosaically entitled Introduction to — . Thes e essays, each equipped with it s ow n en d notes , ar e "programs " tha t tak e th e reade r throug h th e highlights o f eac h article , mak e connection s amon g articles , an d offe r com - mentary o n relationships to the wor k o f others. (Al l translations fro m Ger - man source s appearin g i n the introduction s wer e don e b y me. ) Th e intro - ductions ca n b e rea d before , during , o r afte r th e relevan t article—o r the y can be ignored. I f before, I hope that the y inspire a possibly wavering reader to take u p the actua l work . I wis h t o offe r furthe r comment s o n th e particular s o f the translation s and introductions . Th e number s appearin g i n th e margin s o f the page s o f the translation s refe r t o th e correspondin g page s i n the originals . Al l pag e references t o HausdorfPs article s that appea r i n the introductions ar e to the relevant page or pages in the original article, which can be easily found i n the translation by using the numbers in the margins. Usuall y the first occurrenc e of a translated technical term in a given article is accompanied by the original German ter m i n double brackets , fo r example : initia l segmen t [[Abschnitt]] . This is not done for certain ubiquitous terms. Th e "set " words always receive the same translation: Menge — set, Inbegriff — aggregate, and Gesamtheit = totality] th e term Machtigkeit, whic h Cantor appropriated fro m the geometer Jacob Steiner , i s alway s translated a s cardinality. Doubl e bracket s ar e als o used t o enclos e clarifyin g information . I n th e introductions , I hav e freel y used standard se t theor y abbreviations : C H = continuu m hypothesis ; GC H = generalize d continuu m hypothesis ; A C = axio m o f choice ; Z F (ZFC ) = Zermelo-Fraenkel se t theor y (Zermelo-Fraenke l se t theor y plu s th e Axio m of Choice) . Through th e miracle o f WF^K.2s1 I have attempted t o maintain the loo k of the originals, even making the fon t size s in proper name s a little larger o r adding spac e betwee n thei r letter s a s wa s the custo m o f the times . I hav e duplicated som e o f Hausdorff 's one o f a kind notation s an d hav e continue d his practic e o f elidin g th e lette r "i " o r th e lette r "j " (but no t both ) whe n itemizing a lis t b y letters . Th e organizatio n int o paragraph s i s exactly a s PREFACE xvn in the originals ; however , sentence s hav e bee n punctuate d t o reflec t Englis h usage. Mino r an d obviou s misprints , whic h wer e amazingl y ver y fe w i n number, ar e correcte d withou t comment . I hav e kep t th e origina l indexin g style fo r footnote s (numeral s o r asterisks ) and , wher e possible , th e sam e values fo r indices ; an y deviation s fro m thi s ar e noted b y placin g the origina l designation i n doubl e bracket s withi n th e curren t footnote . Note s fro m th e translator ar e rare , bu t whe n the y occur , the y ar e indexe d b y smal l roma n letters. Acknowledgements In preparin g m y essays , I hav e benefite d fro m scholarl y work s emanat - ing fro m Germany , i n particula r th e book s edite d b y Euge n Eichor n an d Ernst-Jochen Thiel e ([Ei T 1994] ) an d b y Egber t Brieskor n ([B r 1996]). 17 These volume s ar e stil l both importan t an d useful . Bu t currentl y th e great - est boo n t o Hausdorf f scholarshi p i s (o r wil l be ) th e Hausdorf f Edition , a project undertake n b y a consortium o f scholars to publish the collected work s of Hausdorff-Mongre . Thi s nine-volum e editio n i s als o t o includ e excerpt s from hi s Nachlafi, alon g wit h editoria l essay s an d commentary . Thre e vol - umes hav e alread y appeared . O f particula r importanc e t o m e durin g th e preparation o f thi s boo k wa s th e publicatio n o f th e Hausdorf f Edition' s volume containin g Grundziige der Mengenlehre. I t wa s th e firs t volum e published (designate d officiall y a s volum e II) , an d I profite d greatl y fro m reading Walte r Purkert' s introductor y essay ; hi s essa y i s availabl e on-lin e ([Pu 2002]) . I t wa s als o ver y helpfu l t o hav e th e catalo g (Findbuch ) o f Hausdorff's Nachlafi prepare d b y Walte r Purkert ; i t to o i s available on-lin e ([Pul995]).18 Sources i n Englis h tha t mentio n Hausdorff , othe r tha n i n passing , ar e rather sparse . A n exceptio n i s [M o 1982] , in whic h th e work s i n th e perio d 1904-1909 tha t w e hav e translate d ar e covered . I foun d Gregor y Moore' s book ver y helpful , no t onl y fo r hi s commentary o n specifi c articles , but als o for th e ric h portrai t tha t h e paint s o f a n importan t era . Moor e ha s als o written a shor t articl e o n Hausdorf f highlightin g th e year s 1901-190 8 ([M o 1996]). In a mor e persona l vein , I wan t t o than k Mario n Scheeper s whos e par - ticipation cam e a t a crucia l time . Hi s enthusias m fo r thi s projec t provide d the wil l t o pursu e wha t seeme d lik e a dauntin g task . I als o wan t t o than k

17The first appraisal s o f Hausdorff' s lif e an d wor k appeare d i n German y twenty-fiv e years afte r hi s deat h wit h th e traditional , bu t belated , essay s o f remembranc e i n th e Jahresbericht o f the DMV [D i 1967] , [L o 1967] . N o doub t th e necessit y t o acknowledg e the circumstance s tha t force d Hausdorf f fro m hi s professorshi p an d eventuall y le d hi m to choos e suicid e ha d bee n a difficult psychologica l barrie r t o societa l recognitio n o f thi s great Germa n mathematician . 18The miraculous surviva l o f the Secon d Worl d Wa r b y Hausdorff's scientifi c Nachlafi is recounted i n [B o 1967] and [Be r 1967] ; also see Bergmann's articl e in [B r 1996 , 271-281]. The forwar d t o [P u 1995 ] provides a complet e histor y o f the Nachlafi. XV111 PREFACE my colleague s a t Michiga n Stat e University : Susa n Schuur , Christe l Rot - thaus an d Marti n Fuch s i n the Departmen t o f Mathematics; an d Elizabet h Mitt man i n the Germa n Department . Professo r Schuu r rea d m y essays an d made helpfu l comment s an d suggestions . Professor s Rotthau s an d Fuch s provided emergenc y hel p whe n I wa s overcom e b y a Germa n languag e co - nundrum, an d Professo r Mit t man allowe d m e to atten d he r classe s an d b e a student o f German onc e more. As with an y undertaking o f long duration, ther e were peaks and valleys , and I gratefull y acknowledg e th e deb t I ow e t o friend s wh o offere d thei r encouragement an d provide d a sympathetic ea r when I needed it . Foremos t among these i s my wif e an d colleagu e Susa n Schuur . It goe s withou t sayin g tha t an y o f thi s volume' s shortcoming s ar e m y responsibility an d n o one else's.

J.M. Plotki n East Lansing , 200 4 Bibliography

[BeP 1987 ] Beckert, Herber t an d Purkert , Walte r (eds.) . Leipziger mathe- matische Antrittsvorlesungen, Teubne r Archiv e zu r Mathemati k (Leipzig :Teubner), v . 8, 1987 . [Ber 1967 ] Bergmann, Giinter . Vorldufiger Bericht uber den wissenschaft- lichen Nachlafi von Felix Hausdorff, Jahresberich t de r Deutsche n Mathematiker-Vereinigung 6 9 (1967) , 62-75. [Bern 1942 ] Bernays, Paul . A System of Axiomatic Set Theory, Part III, Journal o f Symboli c Logi c 7 (1942) , 65-89. [Berns 1901 ] Bernstein, Felix . Untersuchungen aus der Mengenlehre (Ph.D. dissertation: Gottingen; printe d a t Halle) . [Berns 1905a ] Bernstein, Felix . Zur Mengenlehre, Jahresberich t de r Deutsche n Mathematiker-Vereinigung 1 4 (1905) , 198-199 . [Berns 1905b ] Bernstein, Felix . Uber die Reihe der transfiniten Ordnungszahlen, Mathematische Annale n 6 0 (1905) , 187-193 . [Berns 1905c ] Bernstein, Felix . Zur Mengenlehre, Mathematisch e Annale n 60 (1905) , 463-464 . [Berns 1905d ] Bernstein, Felix . Untersuchungen aus der Mengenlehre, Mathematische Annale n 6 1 (1905) , 117-155 ; reprint o f [Berns 1901 ] with alterations . [Bo 1967 ] Bonnet, Hans . Felix Hausdorff zum Gedachtnis. Geleitwort, Jahresbericht de r Deutsche n Mathematiker-Vereinigun g 6 9 (1967), 75-76 . [Br 1996 ] Brieskorn, Egber t (ed.) . Felix Hausdorff zum Gedachtnis I: Aspekte seines Werkes (Braunschweig/Wiesbaden : Vieweg), 1996 . [Ca 1978 ] Campbell, Pau l J . The Origin of "Zorn's Lemma," Histori a Mathematica 5 (1978) , 77-89 . [Can 1879 ] Cantor, Georg . Uber unendliche, lineare Punktmannichfaltig- keiten, Mathematisch e Annale n 1 5 (1879) , 1-7 . [Can 1884 ] Cantor, Georg . Uber unendliche, lineare Punktmannichfaltig- keiten, Mathematisch e Annale n 2 3 (1884) , 453-488. [Can 1891 ] Cantor, Georg . Uber eine elementare Frage der Mannigfaltigkeitslehre, Jahresbericht de r Deutschen Mathematiker-Vereinigun g 1 (1891), 75-78. [Can 1895 ] Cantor, Georg . Beitrdge zur Begriindung der transfiniten Mengenlehre I, Mathematische Annale n 4 6 (1895) , 481-512 ; translate d i n [Ca n 1915] , 85-136. [Can 1897 ] Cantor, Georg . Beitrdge zur Begriindung der transfiniten Mengenlehre II, Mathematische Annale n 4 9 (1897) , 207-246 ; translate d i n [Ca n 1915] , 137-201.

317 318 BIBLIOGRAPHY

Cantor, Georg . Contributions to the Founding of the Theory of Trans- finite Numbers (Chicago : Open Court) , translate d an d edite d b y Phili p Jourdain; reprinte d (Ne w York: Dover), 1952 . Dauben, Josep h W . Georg Cantor: His Mathematics and of the Infinite (Cambridge : Harvard Universit y Press) , 1979 ; reprinte d (Princeton : Princeton Universit y Press) , 1990 . Dierkesmann, Magda . Felix Hausdorff. Fin Lebensbild, Jahresberich t de r Deutschen Mathematiker-Vereinigun g 6 9 (1967) , 51-54 . Du Bois-Reymond , Paul . Sur la grandeur relative des infinis des fonctions, Annal i d i matematic a pur a e d applicat a 4 (series 2 ) (1870) , 338-353. Du Bois-Reymond , Paul . Fine neue Theorie der Convergenz und Divergenz von Reihen mit positiven Gliedern, Journa l fii r di e rein e un d angewandte Mathemati k 7 6 (1873) , 61-91 . Du Bois-Reymond , Paul . Erlduterung zu den Anfangsgriinden der Vari- ationsrechnung, Mathematisch e Annale n 1 5 (1879) , 282-315, 564-576 . Du Bois-Reymond , Paul . Die allgemeine Funktionentheorie I (Tubingen :Laupp), 1882 . Eichorn, Eugen . Felix Hausdorff-Paul Mongre: Some Aspects of His Life and the Meaning of His Death, i n Recent Developments of General Topology and its Applications; International Conference in Memory of Felix Hausdorff (1868-1942), W . Gahle r e t al . (eds. ) (Berli n Akademie) , 85-117, 1992 ; translated b y Mitc h Cohen . Eichorn, Eugen . In memoriam Felix Hausdorff (1868-1942). Fin biographischer Versuch, i n Vorlesungen zum Gedenken an Felix Haus- dorff, E. Eichor n an d E . Thiel e (eds. ) (Berlin : Heldermann), 1-49 , 1994 . Eichorn, Euge n an d Thiele , Ernst-Joche n (eds.) . Vorlesungen zum Gedenken an Felix Hausdorff (Berlin : Heldermann), 1994 . Erdos, Pau l an d Hajnal , Andras . On a Classification of Denumerable Order Types and an Application to the Partition Calculus, Fundament a Mathematicae 5 1 (1962) , 117-129 .

Feigner, Ulrich . Die Hausdorffsche Theorie der r) a-Mengen und ihre Wirkungsgeschichte i n Felix Hausdorff, Gesammelte Werke, Band II, Grundzilge der Mengenlehre, E . Brieskor n e t al . (eds. ) (Berlin - Heidelberg-New York : Springer-Verlag), 634-674 , 2002 ; available i n pdf format : http://www.aic.uni-wuppertal.de/fb7/hausdorf f /Eta-Alpha.pdf. Ferreiros, Jose . Labyrinth of Thought: A History of Set Theory and its Role in Modern Mathematics (Basel : Birkhauser), 1999 . Fisher, Gordon . The Infinite and Infinitesimal Quantities of du Bois- Reymond and Their Reception, Archiv e fo r Histor y o f Exact Science s 24, no . 2 (1981) , 101-163 . Fraenkel, A . A. Abstract Set Theory (Amsterdam : NorthHolland), 1968 . Frewer, M . Felix Bernstein, Jahresberich t de r Deutsche n Mathematiker - Vereinigung 8 3 (1981) , 84-95. Garciadiego, Alejandr o R . Betrand Russell and the Origins of the Set-theoretic 'Paradoxes' (Basil : Birkhauser), 1992 . Gottwald, Siegfrie d an d Kreiser , Lothar . Paul Mahlo —Leben und Werk, NTM Schriftenreih e fii r Geschicht e der Naturwissenschaften Techni k un d Medizin 2 1 (1984 ) no . 2 , 1-22 . BIBLIOGRAPHY 319

Grattan-Guinness, Ivor . An Unpublished Paper by Georg Cantor: Principien einer Theorie der Ordnungstypen. Erste Mitteilung, Act a Mathematica 12 4 (1970) , 65-107. Hadamard, Jacques . Sur les caracteres de convergence des series a termes positifs et sur les fonctions indefiniment croissantes, Act a Mathematica 1 8 (1894) , 319-336 . Hartogs, Friedrich . Uber das Problem der Wohlordnung, Mathematisch e Annalen 7 6 (1915) , 436-443. Hechler, Stephe n H . On the Existence of Certain Subsets of ^u, in Axiomatic Set Theory, Proceedings of Symposia in Pure Mathematics, Vol. XIII, Part 2 (Providence : American Mathematica l Society) , 1974 , 155-173. Hessenberg, Gerhard . Grundbegriffe der Mengenlehre (Gottingen : Vandenhoek & Ruprecht); als o i n Abhandlungen der Fries'schen Schule N. S. 1 Hef t 4 , 1906 . Hessenberg, Gerhard . Potenzen transfiniter Ordnungszahlen, Jahresbericht de r Deutsche n Mathematiker-Vereinigun g 1 6 (1907) , 130-137. Huntington, Edwar d V . The Continuum as a Type of Order: An Exposition of the Modern Theory., Annal s o f Mathematics (2 ) 6 (1905) , 151-184; reprinted i n [H u 1917] . Huntington, Edwar d V . The Continuum and Other Types of Serial Order (Cambridg e : Harvard Universit y Press) ; reprinted (Ne w Yor k : Dover), 1955 . Jensen, Ronal d B . Independence of the Axiom of Dependent Choices from the Countable Axiom of Choice (abstract) , Journa l o f Symboli c Logic 31 (1966) , 294 . Jourdain, Phili p E . B. On a Proof That Every Aggregate Can Be Well-Ordered, Mathematisch e Annale n 6 0 (1905) , 465-470 . Kanamori, Akihiro . The Mathematical Development of Set Theory from Cantor to Cohen, Bulleti n o f Symboli c Logi c 2 (1996) , 1-71 . Kanamori, Akihiro . The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings (Berlin : Springer Verlag) , 2003 ; 2n d edition . Kanamori, Akihiro . Zermelo and Set Theory, Bulleti n o f Symbolic Logi c 10 (2004) , 487-553. Knopp, Konrad . Theorie und Anwendung der unendlichen Reihen (Berlin : Springer Verlag) , 1947 . Konig, Julius . Zum Kontinuum-Problem, i n Verhandlungen des dritten internationalen Mathematiker-Kongresses in Heidelberg von 8. bis 13. August 1904 (Leipzig : Teubner), 1905 , 144-147 . Konig, Julius . Zum Kontinuumproblem, Mathematisch e Annale n 6 0 (1905), 177-180 , 462 . Koepke, Peter . Metamathematische Aspekte der Hausdorffschen Mengenlehre, i n Felix Hausdorff zum Geddchtnis I: Aspekte seines Werkes, E . Brieskor n (ed. ) (Braunschweig/Wiesbaden : Vieweg), 1996 , 71-106. Korselt, Alwin . Uber einen Beweis des Aquivalenzatzes, Mathematische Annale n 7 0 (1911) , 294-296 . 320 BIBLIOGRAPH Y

[Kow 1950 ] Kowalewski , Gerhard. Bestand und Wandel. Mein Lebensweg zugleich ein Beitrag zur neueren Geschichte der Mathematik (Munich: Oldenbourg), 1950 . [La 2002 ] Laugwitz , Detlef . Debates About Infinity in Mathematics Around 1890: the Cantor-Veronese Controversy, Its Origins and Its Outcome, NT M (N.S) Schriftenreih e fu r Geschicht e der Naturwissenschaften Techni k un d Medizin 1 0 (2002) , no. 2, 102-126 . [Lo 1967 ] Lorentz , G . G . Das mathematische Werk von Felix Hausdorff, Jahresbericht de r Deutsche n Mathematiker-Vereinigun g 6 9 (1967) , 54-62. [Ma 1909 ] Mahlo , Paul. Uber homogene Teilmengen des Kontinuums, Bericht e iiber di e Verhandlungen de r Koniglic h Sachsische n Gesellschaf t de r Wissenschaften z u Leipzig , Mathematisch-Physisch e Klass e 6 1 (1909) , 121-124. [Ma 1911 ] Mahlo , Paul. Uber lineare transfinite Mengen, Bericht e iibe r di e Verhandlungen de r Koniglic h Sachsische n Gesellschaf t de r Wissen - schaften z u Leipzig, Mathematisch-Physische Klass e 63 (1911) , 187-225 . [Ma 1912 ] Mahlo , Paul. Zur Theorie und Anwendung der go-Zahlen, Bericht e iiber di e Verhandlungen de r Koniglic h Sachsische n Gesellschaf t de r Wis - senschaften z u Leipzig , Mathematisch-Physisch e Klass e 64 (1912) , 108-112. [Ma 1913 ] Mahlo , Paul. Zur Theorie und Anwendung der go-Zahlen II , Bericht e iiber di e Verhandlungen de r Koniglic h Sachsische n Gesellschaf t de r Wis - senschaften z u Leipzig , Mathematisch-Physisch e Klass e 65 (1913) , 268-282. [Mo 1982 ] Moore , Gregor y H . ZermeWs Axiom of Choice: Its Origins, Development, and Influence (Ne w Yor k : Springer), 1982 . [Mo 1989 ] Moore , Gregor y H . Towards a History of Cantor's Continuum Problem, in The History of Modern Mathematics, Volume I, Davi d E . Row e an d John McClear y (eds. ) (Boston : Academic Press) , 1989 , 78-121 . [Mo 1996 ] Moore , Gregor y H . Felix Hausdorff and the Emergence of Order: 1900-1908, Proceeding s o f the Canadia n Societ y fo r th e Histor y and Philosoph y o f Mathematics 9 (1996) , 121-135 . [Pe 1990 ] Peckhaus , Volker . Teh habe mich wohl gehiltet, alle Patronen auf einmal zu verschiefien'. in Gottingen, Histor y an d Philosophy o f Logi c 1 1 (1990) , 19-58 . [Pe 2002 ] Peckhaus , Volker . Pro and Contra Hilbert: Zermelo's Set Theories, lecture give n to PIL M Internationa l Symposium , Octobe r 2002 ; available i n pd f format : http://www-fakkw.upb.de/institute/philosophie/Personal/Peckhaus/. [PI 1993 ] Plotkin , J . M. Who Put the "Back" in Back-and-Forth, i n Logical Methods: In Honor of Anil Nerode's Sixtieth Birthday, Joh n N . Crossle y et al . (eds. ) (Boston : Birkhauser), 1993 , 705-712. [Pr 1890 ] Pringsheim , Alfred . Allgemeine Theorie der Divergenz und Convergenz von Reihen mit positiven Gliedern, Mathematisch e Annale n 3 5 (1890) , 297-394. [Pu 1995 ] Purkert , Walter . Nachlafi Felix Hausdorff: Fmdbuch, 1995 ; available i n pd f o r htm l format : http://hss.ulb.un i -bonn.de:90/ulb_bonn/veroeffentlichungen/hausdorff_felix. BIBLIOGRAPHY 321

Purkert, Walter . Grundzuge der Mengenlehre-Historische Einfiihrung in Felix Hausdorff, Gesammelte Werke, Band II, Grundzuge der Mengenlehre, E . Brieskor n e t al . (eds. ) (Berlin-Heidelberg-Ne w York : Springer-Verlag), 2002 , 1-89 ; availabl e i n pd f format : http://www.aic.uni-wuppertal.de/fb7/hausdorff/HistEinfuehrung.pdf. Purkert, Walte r an d Ilgauds , Hans-Joachim . Georg Cantor 1845-1918 (Basel :Birkhauser), 1987 . Sageev, Gershon . An Independence Result Concerning the Axiom of Choice, Annal s o f Mathematical Logi c 8 (1975) , 1-184 . Scheepers, Marion . Gaps in u u, i n Set Theory of the Reals, Hai m Juda h (ed.), Israe l Math . Conf . Proa, 6 , Bar-Ilan University , 1993 , 439-561. Schoenflies, Arthur . Mengenlehre, i n Encyklopddie der Mathematischen Wissenschaften rait Einschluss ihrer Anwendungen, v . I, A 5 (Leipzig : Teubner), 1898 , 184-207 . Schoenflies, Arthur . Die Entwickelung der Lehre von den Punkt- mannigfaltigkeiten, Jahresberich t de r Deutsche n Mathematiker - Vereinigung 8(ii) , (1900) , 1-251 . Schoenflies, Arthur . Uber wohlgeordnete Mengen, Mathematisch e Annalen 6 0 (1905) , 181-186 . Schoenflies, Arthur . Die Entwickelung der Lehre von den Punkt- mannigfaltigkeiten, Zweiten Teil, Jahresbericht de r Deutsche n Mathematiker-Vereinigung, Erganzungsban d 2 (1908) , 1-331 . Schoenflies, Arthur . Entwickelung der Mengenlehre und ihrer Anwend- ungen, Erster Hdlfte: Allgemeine Theorie der unendlichen Mengen und Theorie der Punktmengen (Leipzig : Teubner), 1913 . Schoenflies, Arthur . Zur Erinnerung an Georg Cantor, Jahresberich t der Deutsche n Mathematiker-Vereinigun g 3 1 (1922) , 90-106 . Scholz, Erhard . Logischen Ordnungen im Chaos: Hausdorffs fruhe Beitrdge zur Mengenlehre, i n Felix Hausdorff zum Gedachtnis I: Aspekte seines Werkes, Egber t Brieskor n (ed. ) (Braunschweig / Wiesbaden :Vieweg), 1996 , 107-134 . Sierpihski, Wraclaw . Sur une propriete des ensembles ordonnes, Fundamenta Mathematica e 3 6 (1949) , 56-67 . Sierpihski, Wraclaw . Cardinal and Ordinal Numbers, 2n d ed . revised (Warsaw : Polish Scientifi c Publishers) , 1965 . Sierpihski, Wracla w an d Tarski , Alfred . Sur une propriete caracter- istique des nombres inaccessibles, Fundament a Mathematica e 1 5 (1930) , 293-300. Skolem, Thoralf . Logisch-kombinatorischeUntersuchungen uber die Erfilllbarkeit und Beweisbarkeit mathematischen Sdtze nebst einem Theoreme uber dichte Mengen, Videnskapsselskapet s skrifter , I . Matematisk-naturvidenskabelig klasse , no . 4 (1920) , 1-36 ; reprinte d i n [Sk 1970] , 103-136 . Skolem, Thoralf . Selected Works In Logic by Th. Skolem, J.E . Fensta d (ed.) (Osl o : Universitetsforlaget), 1970 . Tarski, Alfred . Sur quelques theorernes qui equivalent a Vaxiome du choix, Fundament a Mathematica e 5 (1924) , 147-154 . Tarski, Alfred . Quelques theoremes sur les alephs, Fundament a Mathe - maticae 7 (1925) , 1-14 . 322 BIBLIOGRAPH Y

[Ta 1948 ] Tarski , Alfred . Axiomatic and Algebraic Aspects of Two Theorems on Sums of Cardinals, Fundament a Mathematica e 3 5 (1948) , 79-104 . [Ta 1951 ] Tarski , Alfred . A Decision Method for Elementary Algebra and Geome- try (Berkele y an d Lo s Angeles: University o f Californi a Press) , 1951. [vdW 1963 ] va n de r Waerden , B . L. Modern Algebra v. I (Ne w York:Ungar) , 1963 ; translated fro m 2n d revise d Germa n editio n b y Fre d Blum . [We 1895 ] Weber , Heinrich . Lehrbuch der Algebra v. I (Braunschweig : Vieweg), 1895; 2n d ed . reprinted wit h correction s (New York: Chelsea), 1955 , AMS Chelse a 2002 . [Ze 1904 ] Zermelo , Ernst. Beweis dafl jede Menge wohlgeordnet werden kann, Mathematische Annale n 59 , 514-516 ; translated b y Stefa n Bauer - Mengelberg i n From Frege to Godel, Jea n va n Heijenoor t (ed. ) (Cambridge : Harvard Universit y Press) , 1967 , 139-141 . [Ze 1908a ] Zermelo , Ernst . Neuer Beweis fur die Moglichkeit einer Wohlordnung, Mathematische Annale n 6 5 (1908) , 107-128 ; translated b y Stefa n Bauer-Mengelberg i n From Frege to Godel, Jean va n Heijenoor t (ed. ) (Cambridge: Harvard Universit y Press) , 1967 , 183-198 . [Ze 1908b ] Zermelo , Ernst . Untersuchungen iiber die Grundlagen der Mengenlehre I, Mathematisch e Annale n 6 5 (1908) , 261-281 ; translated b y Stefa n Bauer-Mengelberg i n From Frege to Godel, Jean va n Heijenoor t (ed. ) (Cambridge : Harvard Universit y Press) , 1967 , 199-215. [Zo 1935 ] Zorn , Max . A Remark on a Method in Transfinite Algebra, Bulletin o f the America n Mathematica l Societ y 41 (1935) , 667-670 . Titles i n Thi s Serie s

25 J . M . Plotkin , Editor , Hausdorf f o n ordere d sets , 200 5 24 Han s Niel s Jahnke , Editor , A histor y o f analysis, 200 3 23 Kare n Hunge r Parshal l an d Adrai n C . Rice , Editors , Mathematic s unbound : Th e evolution o f a n internationa l mathematica l researc h community , 1800-1945 , 200 2 22 Bruc e C . Bernd t an d Rober t A . Rankin , Editors , Ramanujan : Essay s an d surveys , 2001 21 Arman d Borel , Essay s i n the histor y o f Li e groups an d algebrai c groups , 200 1 20 Kolmogoro v i n perspective , 200 0 19 Herman n Grassmann , Extensio n theory , 200 0 18 Jo e Albree , Davi d C . Arney , an d V . Frederic k Rickey , A station favorabl e t o th e pursuits o f science : Primary material s i n th e histor y o f mathematics a t th e Unite d State s Military Academy , 200 0 17 Jacque s Hadamar d (Jerem y J . Gra y an d Ab e Shenitzer , Editors) , Non-Euclidea n geometry i n the theor y o f automorphic functions , 199 9 16 P . G . L . Dirichle t (wit h Supplement s b y R . Dedekind) , Lecture s o n numbe r theory , 1999 15 Charle s W . Curtis , Pioneer s o f representation theory : Frobenius , Burnside , Schur , an d Brauer, 199 9 14 Vladimi r Maz'y a an d Tatyan a Shaposhnikova , Jacque s Hadamard , a universa l mathematician, 199 8 13 Lar s Garding , Mathematic s an d mathematicians : Mathematic s i n Swede n befor e 1950 , 1998 12 Walte r Rudin , Th e wa y I remember it , 199 7 11 Jun e Barrow-Green , Poincar e an d th e thre e bod y problem , 199 7 10 Joh n Stillwell , Source s o f hyperbolic geometry , 199 6 9 Bruc e C . Bernd t an d Rober t A . Rankin , Ramanujan : Letter s an d commentary , 199 5 8 Kare n Hunge r Parshal l an d Davi d E . Rowe , Th e emergenc e o f the America n mathematical researc h community , 1876-1900 : J . J . Sylvester , Feli x Klein , an d E . H . Moore , 1994 7 Hen k J . M . Bos , Lecture s i n the histor y o f mathematics, 199 3 6 Smilk a Zdravkovsk a an d Pete r L . Duren , Editors , Golde n year s o f Mosco w mathematics, 199 3 5 Georg e W . Mackey , Th e scop e an d histor y o f commutative an d noncommutativ e harmonic analysis , 199 2 4 Charle s W . McArthur , Operation s analysi s i n th e U.S . Army Eight h Ai r Forc e i n Worl d War II , 199 0 3 Pete r L . Dure n e t al. , Editors , A century o f mathematics i n America , par t III , 198 9 2 Pete r L . Dure n e t al. , Editors , A century o f mathematics i n America , par t II , 198 9 1 Pete r L . Dure n e t al. , Editors , A century o f mathematics i n America , par t I , 198 8