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Chapter 7 To infinity and beyond（第1页）

Chapter7Toinfinityandbeyond!

Infinitywithininfinity

Itwasthegreat16th-turyItalianpolymathGalileoGalilei（1564–1642）whowasfirsttoalertustothefactthatthenatureofiionsisfuallydifferentfromfiniteones。Asalludedtoeofthisbook，thesizeofafiissmallerthanthatofasedsetifthemembersofthefirstbepairedoffwiththoseofjustaportionofthesed。However，isbytrastbemadetodinthiswaytosubsetsofthemselves（wherebythetermsubsetImeahiself）。Weneedgohesequeuralumbers1，2，3，4，···ioseethis。Itiseasytodesynumberofsubsetsofthisthatthemselvesforma，andsoareio-onedehefullset（seeFigure8）:theoddnumbers，1，3，5，7，···，thesquarenumbers，1，4，9，16，···and，lessobviously，theprimenumbers，2，3，5，7，···，ahesecasestherespeeheevehehebersarealsoinfinite。

&Hotel

Thisratherextraordinaryhotel，whichisalwaysassociatedwithDavidHilbert（1862–1943），theleadihematiofhisday，servesttolifethestraheischieffeatureisthatithasinfinitelymanyrooms，numbered1，2，3，···，andboaststhatthereisalwaysroomatHilbert’sHotel。

8。Theevensandthesquarespairedwiththenaturalnumbers

&，however，itisinfactfull，whichistosayeadeveryroomisoccupiedbyaguestandmuayofthedeskoreerfrontsupdemandingaroom。AnuglyseisavoidedwhenthemaerveakestheclerkasidetoexplainhowtodealwiththesituatioofRoom1tomovetoRoom2sayshe，thatofRoom2tomoveintoRoom3，andsoon。Thatistosay，weissueaglobalrequestthattheerinRoomnshouldshiftintoRoomn+1，andthiswillleaveRoom1emptyfentleman!

Andsoyousee，thereisalwaysroomattheHilbertHotel。Buthowmu?

&evening，theclerkistedwithasimilarbutmsituation。Thistimeaspaceshipwith1729passengersarrives，alldemandingaroominthealreadyfullyoccupiedhotel。Theclerkhas，however，learnedhislessonfromthepreviousnightaoexteocopewiththisadditionalgroup。HetellsthepersoninRoom1togotoRoom1730，thatofRoom2toshifttoRoom1731，andsoon，issuingtheglobalrequestthattheerinRoomnshouldmoveintoRoomnumbern+1729。ThisleavesRhto1729freeforthenewarrivals，andhtlyproudofhimselffwiththisnewversionoflastnight’sproblemallbyhimself。

Thefinalnight，however，theclerkagaihesamesituation–afullhotel，butthistime，tohishorror，notjustafewextraersshowupbutaninfinitespacecoafinitelymanypassengers，oheumbers1，2，3。···。Theoverwhelmedclerktellsthecoachdriverthatthehotelisfullandthereisnoceivablewayofdealingwiththemall。Hemightbeabletosqueezeiwomore，anyfinitesurelynotinfinitelymaisplainlyimpossible!

Amighthaveeagaiimelyiionofthemanagerwho，beingwellversedinGalileo’slessonsos，informsthecoachdriverthatthereisall。ThereisalwaysroomatHilbert’sHotelforanyoneandeveryoakeshispanigdeskclerkasideforanotherlesson。Allwehis，hesays。Wetelltheo1toshiftintoRoom2，thatinRoom2toshifttoRoom4，thatinRoom3togotoRoom6，andsooheglobalinstruisthattheonshouldmoveintoRoom2n。Thiswillleavealltheoddnumberedroomsemptyforthepasseheinfinitespaceatall!

Themaohaveitallurol。However，evenhewouldbecaughtoutifaspaceshiptursomehowhadtheteologytohaveonepasseiinuumoftherealline。Onepersonforeverydeumberwouldtotallyoverruel，andweshallseewhyiion。

parisons

Allthismaybesurprisiimeyouthinkaboutit，butitisnotdifficulttoacceptthatthebehaviourofismaydifferisfromfihispropertyofhavingthesamesizeasossubsetsisthereforeapoiury，htor（1845–1918）wentmuchfurtheranotallisberegardedashavingequallymahisrevelatioediisnothardtoappreceyourattentionisdrawntoit。

torasksustothinkaboutthefollowing。SupposewehaveanyiLofnumbersa1，a2，···thoughtofasbeinggiveninde。ThenitispossibletowritedownanotherdoesnotappearahelistL:wesimplytakeatobedifferentfroma1iplaceafterthedet，differentfroma2inthesealplace，differentfroma3ihirddecimalpladsoon–inthisway，wemaybuildsureitisoahelist。ThisobservationlooksinnocuousbutithastheimmediatecethatitisabsolutelyimpossibleforthelistLtoallnumbers，becausethenumberawillbemissingfromL。Itfollowsthatthesetofallrealisalldecimalexpansions，otbewritteninalist，orinotherutio-onedehenaturalumbers，theihislineisknownastument，astheliesoutsidethesetLisstructedbyimaginingalistofthedecimaldisplaysofLasinFigure9anddefinihediagonalofthearray。

Thereissomesubtletyhere，fhtsuggestthatweeasilygetaroundthisdifficultybysimplyplagthemissihefrontofL。ThiseingMgtheannoyingnumbera。However，theunderlyigonealytor’sstruagaintointroduceafreshisheM。Weoftioaugmelistasbeforeaimes，buttor’spointremainsvalid:althoughgliststhatadditiowerepreviouslyoverlooked，thereeverbeonespecificlistthatseveryrealnumber。

9。beradiffersfromeathekthdecimalplace

&ionofallrealhereferiheofallpositiveihoughbothareisotbepairedofftogetherthewaytheevennumbersbepairedwiththelistofallumbers。Indeed，iflytumenttoaputativelistofallheio1，themissingnumberawillalsolieinthisraherefore，welikewisecludethatthiswillalsodefyeveryattemptatlistingitinfull。Imentionthisasweshallmakeuseofthatfactshortly。

tor’sresultisrehembythefactthatmasofnumbersbeputintoa，ingtheGreeks’euumbers。Alittleiyisionceacoupleoftricksarelearisnothardtoshowthatmasofnumbersaretable，whichisthetermweusetomeabelistedinthesamefashioiherwiseaistable。

Whatwehavealloenincasuallyaganydecimalexpansionistoopeowhatareknowraalhoseliebeyoarisethrougheugeometryandebrais。entshowsusthattraaland，inadditiobeinfinitelymanyofthem，foriftheyformedonlyafiheycouldbeplafrontofourlistofalgebraiumbers（thenoals），soyieldingalistingofallrealnumbers，knowisimpossible。Whatisstrikingisthatwehavediscoveredtheexisterahoutidentifyingasihem!Theirexistencelythroughpariaiioher。Thetraalsarethefillthehugevoidbetweenthemorefamiliaralgebraiumbersaionofalldecimalexpansions:toborrowanastrohetraalsarethedarkmatterofthenumberworld。

Inpassingfromtherationalstothereals，wearemovingfromoherofhigheralityasmathematisputit。Twosetshavethesamealityiftheirmembersbepairedoff，otheother。Whatbeshownusiumentisthatahasasmalleralitythaformedbytakingallofitssubsets。Thisisobviousforfiions:indeed，ilaiifwehaveasetofhereare2edinthisway。ButheisthesetSofallsubsetsoftheiuralnumbers，{1，2，3，···}?Thisquestionisnotialsointhemannerinwhichwearriveattheanswer，whichisthatSisiable。

Russell’sParadox

SupposetothetrarythatSwasitselftable，inwhichcasethesubsetsoftheumberscouldbelistedinsomeorderA1，A2，···。NowanarbitrarynumbernmayormaynotbeamemberofAussiderthesetAthatbersnsuishesetAn。NowAisasubsetoftheumbers（possiblytheemptysubset）aheaforesaidlistatsomepoiA=Ajsay。Anunaionnowarises:isjamemberofAj？Iftheahen，bytheverywayAisdefined，wecludethatjisnotamemberofA，butA=Aj，sothatisself-tradictory。Thealternativeisno，jisnotamemberofAj，inwhichcase，agaiiojisamemberofA=Aj，andoncemorewehavetraditradiisunavoidable，inalassumptiosoftheumberscouldbelistedinatablefashionmustbefalse。Ihisargumentworkstoshowthatthesetofallsubsetsofanytablebutiisuntable。

Thisparticularself-referentialstyleargumentwasirandRussell（1872–1970）inaslightlydiffereledtowhatisknownasRussell’sParadox。Russellappliedittothe‘setofallsetsthatarehemselves’，askingtheembarrassiiothatsetisamemberofitself。Again，‘yes’implies‘no’and‘no’implies‘yes’，fRusselltocludethatthissetotexist。

Inthe1890s，selfdisimplitradiingfromtheideaofthe‘setofallsets’。Indeed，Russellaowledgedthattheargumentofhisparadoxiredbytheworkoftor。Theupshotofallthis，however，isthatlyimagihematicalsetstroduymasoever，butsomerestriustbeplaaybespecified。SettheoristsandlogishavebeelingwiththecesofthiseversiisfactoryresolutionofthesedifficultiesisprovidedbythenowstandardZFCSetTheory（theZermelo-FraeheorywiththeAxiomofChoice）。

Thenumberlihemicroscope

10。Ratioeaionsonthenumberline