E:\CPBR\RUSSJOUR\TYPE3201\russell 32,1060 red.wpd June 25, 2012 (9:21 pm) Trans hisintroductiontoG.Cantor, and 1991) York: Dover,1955). he made his living from a few scholarships and from freelance writing as an as writing freelance andfrom scholarships afew livingfrom his he made as aWranglerandsocouldno graduate reich’s ataxia”.Itpreventedhimfrom ta History ofSetTheoryandLogics(1906–1918) Columbia U.P.,1977);hereafter“ Based onHisCorrespondencewithPhilipJourdain The BertrandRussellResearchCentre,McMasterU. russell: forms thecoreofmybook Russelluntilhisdeath in1919;theexchange He correspondedatlengthwith of logic, and aspects of the history and history of logic,andaspectsthe set theory and itshistory,th and alsoon Russell was of majorconsequenceforhis Russell, the logico attended aspecialCollegecourseinmathematical P 2 1 su Jourdain youth From his Jourdain’s principalhistoricalwr I.Grattan-Guinness, Cambridge in 1899 with a scholarship in mathematics. Two years laterhe in years 1899 Cambridge Two with a scholarship in mathematics. hilip EdwardBertrandJourdain(18 W nite Numbers the JournalofBertrand RussellStud W Centre forPhilosophyofNa rst of its kind ever taught in a British university. This contact with contact kind evertaughtinaBritishuniversity.This its of rst JOURDAIN, RUSSELLAND THE AXIOMOFCHOICE: , trans. anded. Jourdain(LaSalle, A NEWDOCUMENT Dear Russell Hendon, London I. Grattan-Guinness Middlesex U. Business School U.Business Middlesex ontheirrelationship. London ivor2  RJ T ered fromacreepingparalysiscalled“Fried- y itings are gathered together in together in gathered are itings i. z ”. — ocuments w @ z Dear Jourdain: aCommentaryonRussell career wc Contributions totheFoundingofTheory mdx.ac.uk philosophy ofmechanicsandscience. tural and Social Science, Science, tural andSocial e n.s.32(summer 2012):69–74 ies , ed. I. Grattan-Guinness(Bologna: I. , ed. t competeforafellowshipatTrinity.So e historiesofmathematical analysis and 2 king the Part 2 Tripos; thus he did not hedid thus king thePart2Tripos; intellectual career,forhefocusedupon a 79–1919) wentuptoTrinityCollege nw 2 ae, uk ae, 4 (London:Duckworth;NewYork: bt 2 Ill.: Open Court,1915;repr.New Ill.: , uk issn 0036-01631; T Selected Essays onthe l.s.e. ered byBertrand online1913-8032 y ’ clueb s Logic, 1 , E:\CPBR\RUSSJOUR\TYPE3201\russell 32,1060 red.wpd June 25, 2012 (9:21 pm) (Amsterdam: North-Holland,1985). Annalen mathematical context involved (namely, de (namely, mathematical contextinvolved called axiom shortlybeforeZermelo,and choices withinhis facedtheadditionaldi sell F. (1821) àGödel(1940)”,2vols.(U.ofToulousedouble andM.Guillemot,“L’axiome du Cassinet early inOctober.Thoselastmonths itsfounder of death inFebruary1919 soonaftertheeditor general particular Russell.Heseemstohave served asits The Monist lishing Companyemployedhim,especially independent scholar.Inparticular, th i.grattan-guinness 70 ics, asErnstZermelonameditafew focus uponthedenumerable“axiomof to Jourdain’s principalresearchinterest valent totheaxiom,andeventually many came tolight. hopefully morecongenialassumptionsthatwerelogicallyequi- several sought others triedtoreprovetheoremsthatus all) set-theorists andnot logicians.Some forming asetroused considerable doubt regard be always of chosenmemberscan each setinindependentactions;th from in late 1910s,sensinghis forthcoming demi thata deductions or bydrawingupon failed eitherbyrequiringassu provable fromtheotheraxioms,andpublish mathematicalcontexts.But and theoretic the unavoidabilityof shoulder-shrugging, in thecontextsofcorrespondence. in Moore, y W A. Medvede 5 4 3 As theyearspassedcommunity nite ensembleofnon-empty and pairwise disjoint setsand choose amember The history of this/these axioms has been has axioms ofthis/these Thehistory E. Zermelo, “Beweis, dass jede Menge wohlgeordnet werden kann”, dass jede Mengewohlgeordnetwerdenkann”, Zermelo, “Beweis, E. See especially H. Rubin and J. and SeeespeciallyH.Rubin Zermelo’s AxiomofChoice… 59 (1904): 514–16. 59(1904): , commissioning papers (and also somebooks)fromcolleagues,in , commissioningpapers(and T , Rannyaya istoriyaaksiomivibora W nitary predicate calculus;hehadspottedtheneedfor predicate nitary U culty ofexpressingthein mptions thatdid mptions not hold insu y 2. E Rubin, (NewYork:Springer,1982). w obsession years afterintroducing it in 1904. form thetime-frameofthispaper. e American house The Open Court Pub- e AmericanhouseTheOpenCourt in the foundations of mathematicscame of in thefoundations ssumed someformofthe axiom.Bythe editor, PaulCarus,buthediedhimself ed the axiombyproofsthat avoided it; s andoppositionamongmany(though en theaxiomassertsthatcollection ed asagenuineset.Butthismanner of of them rejected the axiom altogether; ofthemrejectedtheaxiom choix dans les mathématiques de Cauchy choix danslesmathématiquesde Equivalents oftheAxiomChoice se, he produced a proofthathepub- se, heproduced it the“multiplicativeaxiom” after the the axiominquiteawiderangeofset- choice” in set theory and mathemat- and theory choice”inset W charted inmuchdetail.SeeespeciallyJ. Jourdain alwaysbelievedthatitwas accepted, withvariousdegreesof from 1912 on their principal journal from 1912ontheirprincipal ning in ed a variety of “proofs”, allofwhich (Moscow: Nauka, 1982); andG. 1982); (Moscow: Nauka, docteur d’étatdessciences W nite productsofnumbers). W 4 nitude ofindependent Logicistssuch asRus- RJ U discusses theaxioms discusses cient generality, Mathematische 3 Takean , 2nded. , 1983); y H. 5 E:\CPBR\RUSSJOUR\TYPE3201\russell 32,1060 red.wpd June 25, 2012 (9:21 pm) ematica dozens oftheRusselllettersmentionedea a translationofZermelo’s1904title. gress very widelyinvariou lished 111–30. Jourdain”, the Archivesinlate1980sfrom attention tomy beendrawn ithas hands; prepared speci October. However,adocumentexistsin beyond communication, and no visitby who eventuallyagreedto a visit late in Hampshire tohearthenewproof.Hewas In hislastdaysJourdainbeggedcolle ed manyofhisreplies.Aswell as the Cantor letters, stuck intwothicknotebooksuponthepa May 1922Childsentthelettersinma ematician J. A. executors, thephilosopher Jourdain’s So Jourdain. with correspondence own his from Cantor to writeabiography,andknewofJourdain’scontactswith planned Le papers. WhenCantordiedin1918, from thelettersofitschieffounderGeorgCantorinsomehisown quoted Jourdain’s spondence, isconserved do methefavourofconsideringit”.Mittag-Le had overcomethelastlogicaldi con died hespoke other informationfromCantor.Chil some contained quitealotofhistorical which expressed his reluctanceinanoteatitshead. V 8 7 6 In earlier yearsJourdainhadcorresponde Child alsoincludedthema P. I.Grattan-Guinness,“TheCorrespon ForJourdainbibliographysee , andeventheFrenchAcademyofScie er, who had published Cantor in his journal who hadpublishedCantorinhis er, y E. 43(1922):239–61.Child’slettertoMittag-Le y B. Jourdain, “A Proof ThatAny “A B. Jourdain, Nachlass Jahresbericht derDeutschen Mathematiker-Vereinigung y M. Child (whoalsotookmajorpositionson Child M. W cally for Russell’s attention, and indeedattention, musthaveRussell’s comeintohiscally for is lost. W dently to me about the matter, and seemed to thinkthathe seemed and dently tomeaboutthematter, at theInstitutMittag-Le s levelsofelaborationin Jourdain, RussellandtheAxiomofChoice nuscript of Jourdain’s lastpaperontheproof, of nuscript 3. U RJ w culties oftheproof , pp.197–8,orMoore(n.5),356–7. document the SwedishmathematicianGöstaMittag- 6 September 1919;butbythenJourdainwas Aggregate Can Be Well-Ordered”, BeWell-Ordered”, Can Aggregate Russell Estate, a legal entity that had been Russell Estate,alegalentitythathad background includingaquotationand agues to come to his home in Fleet, dence between Georg Cantor and Philip and dence betweenGeorgCantor nner in which Jourdain nner inwhich had kept them, y the Russell ArchivesthatJourdainhad E. Heath and the historian andmath- Heathandthehistorian E. very recently. Apparently itarrivedin Russell occurred before his deathon1 occurredbeforehis Russell ges of which hehadwrittenorsketch- rlier, andseveralothersofinterest. d recalledthat“Justaweekbeforehe he asked for the original letters from letters original he askedforthe especially anxiousto talk toRussell, nces, each papercarryingasitstitle d with severalset-theorists, and had V 8 V er, Djursholm, Sweden.Therestof er, Djursholm, Acta Mathematica er publisheditinthe V Mind, Nature, Science Pro- 7 z ”; soheasked“wouldyou er, andsomerelatedcorre- thenotebookscontained The Monist 73 (1971): part1,pp. (1971): 73 inthe1880s, Acta Math- Acta z ). On12 , but 71 E:\CPBR\RUSSJOUR\TYPE3201\russell 32,1060 red.wpd June 25, 2012 (9:21 pm) is, witha Annalen symbols. anonymous amanuensisseemstohave very had become which hand, notinhis is which echoes someunfortunatesarcas which byareferenceto“you(andWhiteheadandHardy)”, end the indicatedat is is listed without an author in the 1992 is listedwithoutanauthorinthe ofthedo history The 1978. in Edith died ments that Russellhadit waswound upafterhis stipulated widow inhis will; to light in his last home in Wales andelsewhere, set up after Russell’s death in 1970tohandle materialsthat cameposthumously and to ful i.grattan-guinness 72 valent totheaxiom. was scepticaloftheaxiom,soonpointed proof hadbeen found before Zermelo’scontroversialproposal;EmileBorel,who and to each sequence out in“wellorder”,likethe laid be could set any his “well-orderingprinciple”,thatallthemembersof nounced of settheory,especiallybasicproperti original motivationheldaspecialattrac While theaxiomofchoicewa a chain (Cantor jargon) of name) (his all“chains” heconstructed document this make to set ofordinalswithoutneeding of anyset members and April1919butwhichappearedtohavebeenlost. referred inletterstoRussellMarch Jourdain be oneofthepaperstowhich in his it place While undated,itscontentssurely titled, thedocumentelaboratesJourdain’s isomorphic totheordinalsfrom1 isomorphic 28 (1919):382–4;comparehimin 12 11 10 9 the lastproofoftheprincipleJourdainattemptedtoshowthatall his In isunsigned,Jourda document While the B&S, p.354.The B&S, P. F.

RJ y y E. E. , pp.146,149. 60 (1905): 194–5. (1905): 60 k y y 1 B. Jourdain,“AProof That Any J. of type 1, each ordered couple of membersgave a chain W y rst member, and a uniquesuccessor bothtoeachnon- and member, rst E. Borel,“Quelquesremarques W nite orin n , that is, well-orderedis, that sequencesof ofthe , members M 12 W ve-leaf documentisin couldbearrangedinone- W nite subset that does not contain the contain nite subsetthatdoesnot s needed inawidevarietyofcontexts,Cantor’s needed s RJ 4. , pp.149–51. w argument catalogueofthesecond Russell Archives. n ms that endhissecondpiecein Aggregate Can Be Well-Ordered”, Well-Ordered”, Be Can Aggregate . Each individual member of Eachindividualmember . tion. In ordertoguaranteethegenerality anychoices.Intheversion presented in sur la théorie des functions”, sur lathéoriedes es suchascomparability,hehadan- cument after1919isnotknown,butit out thattheprincipleitselfwasequi- been usedtowritingmathematical O last proof; itseemstobecomplete. last ra z of positive ordinals 1, 2, 3, …; that …; 3, 2, 1, ordinals positive of in isclearlytheauthor.Whileun- hard to read in his last years; his 2 910,box11.69. one correspondence withsome W nal periodoflife;itmaywell 10 His intended audience Hisintended M W of“ordinaltype” W l certainrequire- nal member.No W Mathematische nal member k 2 M M of type 2, Mind Mind formed thatare , n.s. . 11 It 9 E:\CPBR\RUSSJOUR\TYPE3201\russell 32,1060 red.wpd June 25, 2012 (9:21 pm) such as Cantor’ssmallestone, such out anyneedtomakechoices. types” less than classes of d[irect]c[ontinuation]seach previously formed”byarulethat“isan a chain “ inductive ordinal” mines uniquelyachainof contained three members: “Each ofthecl ing ment atitsright.Hepermittedmultipleuseofmembers;ine from oneofitsdirectcontinuationsthatiswrittenatleftandaseg- a chain where commas mark out the members ofachain, and each semicolonseparates cluded, trespassers arepros all proposition that thatwedonotconcludether (so arbitrary selections;wedonot avoid to order Form the class ing for then proof of1904. own “the directmethod” hemeantsomeversionimitatingZermelo’s by Maybe const can that itisprovedwe choice-free, asdesired:“Wehavetoconstruct method isnottheonlyone. of myprotests—Ido,whichdirectmethod to solvetheprobleminway a of each from term ing aruletopickoutone k member of eight members { all ( n –1) Jourdain then claimed that these properties hold evenif properties Jourdain thenclaimedthatthese Jourdain explainedthisconstruction 13 k z inductive su all M Compare in ( n k –1) asamultiset. 3 chains and direct continuations at each stage deliveredtheorder-type ateachstage anddirectcontinuations chains , …,uptotakingallmembersof z oftype K on ( n –1) k k thatisawell-ordered“segment”ofit,andalsotaketheadditional n ( n ; then RJ –2) 13 K n n , p.149. , …down to a bcdefgh n U . Further,thesechainscanbe in a determined way to inadeterminedwayto of n xes” ( k z (forexample,when c k n , asa“directcontinuation”of n d z s foranyordinal n , z h arelaidout,itsclassofchainstype , M e Jourdain, RussellandtheAxiomofChoice , y }, then one ofitsclassesdirectcontinuationsis you (and Whitehead and Hardy) you (andWhiteheadandHardy) k of type 3.” This situation obtains also for “any for also oftype3.”Thissituationobtains b v ruct theoreticallychains that exhaust 2 zz on ; , that succeeded , that c , d k 1 , . Forexample(mine),if h , seems hopelesstomealso.Butthisfruitless e is a prosecuted trespasser from the e isaprosecutedtrespasserfrom n e n whichcontainschainsofall inductive in somewhat pedantic detailwhen in somewhat ecuted).…” “Ithink,then,”hecon- asses ofd[irect]c[ontinuation]sdeter- zz , andcorrelateeach , set of(quitegeneral)classes.Ididnottry ; =526).Similarly,whenthechainsof each intension M c , assume (whenthechain“exhausts” d z classofd[irect]c[ontinuation]s well-ordered byinclusion,with- , many h O zz ; , sothat his method ofform- z k c whichdeterminesasetof z the , ( z n [direct continuations]in [direct d –1) z ; . Repeatthecorrelation many c zz ), n z wasalimitordinal will z T is k M ect he wastreat- n withthechain z consistsofthe all possibleones M think—inspite n “ without merely adds W M nd- 73 M y v ). z E:\CPBR\RUSSJOUR\TYPE3201\russell 32,1060 red.wpd June 25, 2012 (9:21 pm) boarders; in particular, thesmallestlimitordinal in boarders; of not sittingattheendofsomesequenceordinalsawaitingarrival launched by the previous one, and by thepreviousone, launched W for thisnewproofalso.Cantorhadspeci elusive However, Jourdain’scon i.grattan-guinness 74 equivalent toit—butnotavoidit! some algebrasthatwerefoundlaterto were structurallysimilartosomeofthe maximalprinciplesinsettheory and none iseverywhere.Hisconstructionsof by choosingfromamongthechainsav proof ofthewell-orderingprinciplere a type to distinguishfailed his systematicallyex direct continuationswoulddeliverit of form ofquestion-beggingwould sure to some prove that the same fate awaits a setoflesserordinalitythan theory: assumethat somein number byin method principle thatemulatedPierreFermat’sproof anin him with Jourdain hadrehearsed nite ordinal In his letter of 1922 toMittag-Le of In hisletter n