Hausdorff o n Ordered Set s a 3 o x: OH Felix Hausdorff i n Greifswald, 1914-192 1 Universitats-und Landesbibliothek Bonn, Hss.-Abt. NL Hausdorff: Kapse l 65: Nr. 06 https://doi.org/10.1090/hmath/025 zwvrvwc^ • SOURCE S 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 Mathematics 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 Copying an d reprinting . Individua l reader s o f thi s publication , an d nonprofi t librarie s acting fo r them , ar e permitte d t o mak e fai r us e o f the material , suc h a s to cop y a chapter fo r us e in teachin g o r research . 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The Londo n Mathematica l Societ y i s incorporated unde r Roya l Charte r and i s registered wit h th e Charit y Commissioners . Visit th e AM S hom e pag e a t http://www.ams.org / 7 6 5 4 3 2 1 1 0 09 0 8 0 7 0 6 0 5 To the memory of Felix Hausdorff ^>« i ^r '% *Jk 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 Order Type 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 Cardinality 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 set theory. 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 (Leipzig : 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 cardinalities 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 .