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1UNIT1MATHEMATICSPARTIESTREADINGREADING1HTTP/WWWSCIENTIFICAMERICANCOM/ARTICLECFMIDWHATISRUSSELLSPARADOXSECTIONAPREREADINGTASKWARMUPQUESTIONSWORKINPAIRSANDDISCUSSTHEFOLLOWINGQUESTIONS1WHOISBERTRANDRUSSELLBERTRANDARTHURWILLIAMRUSSELLB1872D1970WASABRITISHPHILOSOPHER,LOGICIAN,ESSAYISTANDSOCIALCRITICBESTKNOWNFORHISWORKINMATHEMATICALLOGICANDANALYTICPHILOSOPHYHISMOSTINFLUENTIALCONTRIBUTIONSINCLUDEHISDEFENSEOFLOGICISMTHEVIEWTHATMATHEMATICSISINSOMEIMPORTANTSENSEREDUCIBLETOLOGIC,HISREFININGOFTHEPREDICATECALCULUSINTRODUCEDBYGOTTLOBFREGEWHICHSTILLFORMSTHEBASISOFMOSTCONTEMPORARYLOGIC,HISDEFENSEOFNEUTRALMONISMTHEVIEWTHATTHEWORLDCONSISTSOFJUSTONETYPEOFSUBSTANCETHATISNEITHEREXCLUSIVELYMENTALNOREXCLUSIVELYPHYSICAL,ANDHISTHEORIESOFDEFINITEDESCRIPTIONSANDLOGICALATOMISMRUSSELLISGENERALLYRECOGNIZEDASONEOFTHEFOUNDERSOFMODERNANALYTICPHILOSOPHY,ANDISREGULARLYCREDITEDWITHBEINGONEOFTHEMOSTIMPORTANTLOGICIANSOFTHETWENTIETHCENTURY2WHATISRUSSELLSPARADOXRUSSELLDISCOVEREDTHEPARADOXTHATBEARSHISNAMEIN1901,WHILEWORKINGONHISPRINCIPLESOFMATHEMATICS1903THEPARADOXARISESINCONNECTIONWITHTHESETOFALLSETSTHATARENOTMEMBERSOFTHEMSELVESSUCHASET,IFITEXISTS,WILLBEAMEMBEROFITSELFIFANDONLYIFITISNOTAMEMBEROFITSELFTHEPARADOXISSIGNIFICANTSINCE,USINGCLASSICALLOGIC,ALLSENTENCESAREENTAILEDBYACONTRADICTIONRUSSELLSDISCOVERYTHUSPROMPTEDALARGEAMOUNTOFWORKINLOGIC,SETTHEORY,ANDTHEPHILOSOPHYANDFOUNDATIONSOFMATHEMATICS3WHATEFFECTDIDRUSSELLSPARADOXHAVEONGOTTLOBFREGGSSYSTEMATFIRSTFREGEOBSERVEDTHATTHECONSEQUENCESOFRUSSELLSPARADOXARENOTIMMEDIATELYCLEARFOREXAMPLE,“ISITALWAYSPERMISSIBLETOSPEAKOFTHEEXTENSIONOFACONCEPT,OFACLASSANDIFNOT,HOWDOWERECOGNIZETHEEXCEPTIONALCASESCANWEALWAYSINFERFROMTHEEXTENSIONOFONECONCEPTSCOINCIDINGWITHTHATOFASECOND,THATEVERYOBJECTWHICHFALLSUNDERTHEFIRSTCONCEPTALSOFALLSUNDERTHESECONDBECAUSEOFTHESEKINDSOFWORRIES,FREGEEVENTUALLYFELTFORCEDTOABANDONMANYOFHISVIEWS4WHATISRUSSELLSRESPONSETOTHEPARADOXRUSSELLSOWNRESPONSETOTHEPARADOXCAMEWITHTHEDEVELOPMENTOFHISTHEORYOFTYPESIN1903ITWASCLEARTORUSSELLTHATSOMERESTRICTIONSNEEDEDTOBEPLACEDUPONTHEORIGINALCOMPREHENSIONORABSTRACTIONAXIOMOFNAIVESETTHEORY,THEAXIOMTHATFORMALIZESTHEINTUITIONTHATANYCOHERENTCONDITIONMAYBEUSEDTODETERMINEASETORCLASSRUSSELLSBASIC2IDEAWASTHATREFERENCETOSETSSUCHASTHESETOFALLSETSTHATARENOTMEMBERSOFTHEMSELVESCOULDBEAVOIDEDBYARRANGINGALLSENTENCESINTOAHIERARCHY,BEGINNINGWITHSENTENCESABOUTINDIVIDUALSATTHELOWESTLEVEL,SENTENCESABOUTSETSOFINDIVIDUALSATTHENEXTLOWESTLEVEL,SENTENCESABOUTSETSOFSETSOFINDIVIDUALSATTHENEXTLOWESTLEVEL,ANDSOONUSINGAVICIOUSCIRCLEPRINCIPLESIMILARTOTHATADOPTEDBYTHEMATHEMATICIANHENRIPOINCAR,ANDHISOWNSOCALLED“NOCLASS“THEORYOFCLASSES,RUSSELLWASABLETOEXPLAINWHYTHEUNRESTRICTEDCOMPREHENSIONAXIOMFAILSPROPOSITIONALFUNCTIONS,SUCHASTHEFUNCTION“XISASET,“MAYNOTBEAPPLIEDTOTHEMSELVESSINCESELFAPPLICATIONWOULDINVOLVEAVICIOUSCIRCLEONRUSSELLSVIEW,ALLOBJECTSFORWHICHAGIVENCONDITIONORPREDICATEHOLDSMUSTBEATTHESAMELEVELOROFTHESAME“TYPE“5HAVEYOUEVERHEARDOFZERMELOFRAENKELSETTHEORYCANYOUGIVEANACCOUNTOFITCONTRADICTIONSLIKERUSSELLSPARADOXAROSEFROMWHATWASLATERCALLEDTHEUNRESTRICTEDCOMPREHENSIONPRINCIPLETHEASSUMPTIONTHAT,FORANYPROPERTYP,THEREISASETTHATCONTAINSALLANDONLYTHOSESETSTHATHAVEPINZERMELOSSYSTEM,THECOMPREHENSIONPRINCIPLEISELIMINATEDINFAVOUROFSEVERALMUCHMORERESTRICTIVEAXIOMSAAXIOMOFEXTENSIONALITYIFTWOSETSHAVETHESAMEMEMBERS,THENTHEYAREIDENTICALBAXIOMOFELEMENTARYSETSTHEREEXISTSASETWITHNOMEMBERSTHENULL,OREMPTY,SETFORANYTWOOBJECTSAANDB,THEREEXISTSASETUNITSETHAVINGASITSONLYMEMBERA,ASWELLASASETHAVINGASITSONLYMEMBERSAANDBCAXIOMOFSEPARATIONFORANYWELLFORMEDPROPERTYPANDANYSETS,THEREISASET,S1,CONTAININGALLANDONLYTHEMEMBERSOFSTHATHAVETHISPROPERTYTHATIS,ALREADYEXISTINGSETSCANBEPARTITIONEDORSEPARATEDINTOPARTSBYWELLFORMEDPROPERTIESDPOWERSETAXIOMIFSISASET,THENTHEREEXISTSASET,S1,THATCONTAINSALLANDONLYTHESUBSETSOFSEUNIONAXIOMIFSISASETOFSETS,THENTHEREISASETCONTAININGALLANDONLYTHEMEMBERSOFTHESETSCONTAINEDINSFAXIOMOFCHOICEIFSISANONEMPTYSETCONTAININGSETSNOTWOOFWHICHHAVECOMMONMEMBERS,THENTHEREEXISTSASETTHATCONTAINSEXACTLYONEMEMBERFROMEACHMEMBEROFSGAXIOMOFINFINITYTHEREEXISTSATLEASTONESETTHATCONTAINSANINFINITENUMBEROFMEMBERSWITHTHEEXCEPTIONOFB,ALLTHESEAXIOMSALLOWNEWSETSTOBECONSTRUCTEDFROMALREADYCONSTRUCTEDSETSBYCAREFULLYCONSTRAINEDOPERATIONSTHEMETHODEMBODIESWHATHASCOMETOBEKNOWNASTHE“ITERATIVE”CONCEPTIONOFASETHTTP/PLATOSTANFORDEDU/ENTRIES/RUSSELL/SECTIONCPOSTREADINGTASKREADINGCOMPREHENSION1DIRECTIONSWORKONYOUROWNANDFILLINTHEBLANKSWITHTHEMAINIDEAPART1PARA1BRIEFINTRODUCTIONTORUSSELLSPARADOX3PART2PARAS25THEEFFECTOFRUSSELLSPARADOXONGOTTLOBFREGESSYSTEMPARA2RUSSELLSPARADOXDEALTAHEAVYBLOWTOFREGESATTEMPTSTODEVELOPAFOUNDATIONFORALLOFMATHEMATICSUSINGSYMBOLICLOGICPARA3ANILLUSTRATIONOFRUSSELLSPARADOXINTERMSOFSETSPARA4CONTRADICTIONFOUNDINTHESETPARA5FREGENOTICEDTHEDEVASTATINGEFFECTOFRUSSELLSPARADOXONHISSYSTEMANDINABILITYTOSOLVEITPART3PARAS68SOLUTIONSOFFEREDBYMATHEMATICIANSTORUSSELSPARADOXPARA6RUSSELLSOWNRESPONSETOTHEPARADOXWITHHIS“THEORYOFTYPES“PARA7ZERMELOSSOLUTIONTORUSSELLSPARADOXPARA8WHATBECAMEOFTHEEFFORTTODEVELOPALOGICALFOUNDATIONFORALLOFMATHEMA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TLYREPRESENTEDWITHTHEMATHEMATICALFORMULAZZ2C2ITAPPEAREDTHATTHINGSWERESETTLINGINTOAPLEASANTANDFRUITFULROUTINE,WITHHISSCHOOLLESSONSSUPPLEMENTEDBYLONGTALKSWITHHISUNCLEABOUTCLASSICALANALYSIS,THEITERATIVEWORKOFPIERREFATOUANDTHEEQUALLYFASCINATINGJULIASETSGENERATEDBYGASTONJULIA3JUSTASTHEYOUTHFULMANDELBROTHADPASSEDHISCOLLEGEENTRANCEEXAMSBYTRANSLATINGALGEBRAICPROBLEMSINTOGEOMETRICALPROBLEMS,ANDSOLVINGTHEMBYINTUITIVELYDEDUCINGTHE“PERFECTED“SHAPE,HEHEREREALIZEDTHEREWASSOMETHINGDEEPER,SOMETHINGMATHEMATICAL,BEHINDTHESESTRANGEPATTERNS4BUTTHEBEAUTYINMANDELBROTSMODELSWASNOTTHATTHEYGENERATEDADECEITFULRANDOMNESS,BUTTHATTHEYCOULDGENERATEGRAPHEDDATAWHOSEVISUALPATTERNACCURATELYMIMICKEDTHE9VISUALPATTERNSCREATEDBYREALPHENOMENA。5BUTDESPITEBEINGTHEINSPIRATIONFORSUCHMETAPHYSICS,MANDELBROT,WHENASKEDIFFRACTALSDONTPOINTTOASINGLERULEUNDERLYINGREALITY,HASSIMPLYSTATED,“THEREISNOSINGLERULETHATGOVERNSTHEUSEOFGEOMETRYIDONTTHINKONEEXISTS“3TRANSLATETHESENTENCESINTOCHINESE1HEISBESTKNOWNFORCOININGTHETERMFRACTALTODESCRIBEPHENOMENASUCHASCOASTLINES,SNOWFLAKES,MOUNTAINSANDTREESWHOSEPATTERNSREPEATTHEMSELVESATSMALLERANDSMALLERSCALES他主要是因为用分形这个概念来描述（海岸线，雪花，山脉和树木）等不规则形状等现象而闻名于世，这些不规则形状在越来越小的规模上不断重复同一模式。2ACLOSERLOOKREVEALSTHATTHEBORDERSOFTHESETDONOTFORMCRISPLINESBUTSEEMTOSHIMMERLIKEFLAMES如果再仔细观察，就可以发现集的边界并没有呈波纹线，而是像火焰一样闪光。3KRANTZINTRODUCEDANEWELEMENTINTOTHEDEBATE,HOWEVER,BYSTATINGTHATTHEMANDELBROTSET“WASNOTINVENTEDBYMANDELBROTBUTOCCURSEXPLICITLYINTHELITERATUREACOUPLEOFYEARSBEFORETHETERMMANDELBROTSETWASCOINED“但是，克朗兹在这场辩论中引入了一个新东西，他说曼德布洛特集不是曼德布洛特发明的，而是早在“曼德布洛特集”这个术语出现几年以前就已经明确地在数学文献中出现了。4MANDELBROTALSOSUGGESTEDTHATEVENIFBROOKSANDMATELSKISPUBLICATIONHADPRECEDEDHIS,THEYSTILLCOULDNOTBECONSIDEREDDISCOVERERSOFTHESET,BECAUSETHEYDIDNOTAPPRECIATEITSSIGNIFICANCE曼德布洛特同时也暗示即使布鲁克斯和马特尔斯基的论文先于他发表，但因为他们没有领会到其价值，仍然不能将他们看作是曼德布洛特集的发现者。5INRESPONSETOHUBBARDANDDOUADYSCHARGETHATHEISSTINGYINGRANTINGCREDIT,MANDELBROTSAYSHEHASALSOBEENACCUSEDOFOVERCITATION对胡巴德和杜阿迪指责他对论文中材料来源的说明上做得非常少，曼德布洛特回应说也有人也指责过他过分引用别人的成果。4TRANSLATETHESENTENCESINTOENGLISH1他的生活和工作过程正如使他成名的几何学一样，既不是线性的，也不具备简单的形状。LIKETHEGEOMETRYTHATMADEHIMFAMOUS,NEITHERHISLIFENORTHECOURSEOFHISWORKWASLINEARORSIMPLISTICINSHAPEANDFORM2曼德布洛特说在随后的的两年里他在多个领域中摸索，却没有明显的相联系的线索。MANDELBROTSAYSHESPENTTHENEXTTWOYEARSGROPING,EXPLORINGFIRSTONEFIELDANDTHENANOTHER,WITHOUTANYCLEARSENSEOFTHECONNECTINGTHREAD3更奇怪的是，他发现噪音周期与清晰传送周期的比率是恒定的，与用于绘制这个现象10的时间大小无关。STRANGERSTILL,HEFOUNDTHATTHERATIOOFPERIODSOFNOISETOPERIODSOFCLEANTRANSMISSIONREMAINEDCONSTANT,REGARDLESSOFTHESCALEOFTIMEUSEDTOPLOTTHEPHENOMENON4通过查询一直到1900年的记录，他开始发现了一个令人惊异的模式，这个模式使他明白了他10年前有关线路噪声的研究工作。USINGRECORDSDATINGBACKTO1900,HEBEGANTOPE