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Chapter 7 To infinity and beyond（第1页）

Chapter7Toinfinityandbeyond!

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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