Chapter 7 To infinity and beyond（第1页）

Chapter7Toinfinityandbeyond!

Innitywithininnity

Itwasthegreat16th-turyItalianpolymathGalileoGalilei（1564–1642）whowasrsttoalertustothefactthattheesisfuallydifferentfromniteones。Asalludedteofthisbook，thesizeofaissmallerthanthatofasedsetifthemembersoftherstbepairedoffwiththoseofjustaportionofthesed。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…，ahesecasestherespeeheevehehebersarealsoinnite。

&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。

Whatwehavealloenincasuallyaganydecimalexpansionistoopeowhatareknowraalhoseliebeyoarisethrougheugeometryandebrais。entshowsusthattraaland，inadditiobeinfinitelymanyofthem，foriftheyformedonlyafiheycouldbeplafrontofourlistofalgebraiumbers（thenoals），soyieldingalistingofallrealnumbers，knowisimpossible。Whatisstrikingisthatwehavediscoveredtheexisterahoutidentifyingasihem!Theirexistencelythroughpariainiioher。Thetraalsarethellthehugevoidbetweenthemorefamiliaralgebraiumbersaionofalldecimalexpansions:toborrowanastrohetraalsarethedarkmatterofthenumberworld。

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

Russell’sParadox

SupposetothetrarythatSwasitselftable，inwhichcasethesubsetsoftheumberscouldbelistedinsomeorderA1，A2，…。NowanarbitrarynumbernmayormaynotbeamemberofAussiderthesetAthatbersnsuishesetAn。NowAisasubsetoftheumbers（possiblytheemptysubset）aheaforesaidlistatsomepoiA=Ajsay。Anunaionnowarises:isjamemberofAj?Iftheahen，bytheverywayAisdened，wecludethatjisnotamemberofA，butA=Aj，sothatisself-tradictory。Thealternativeisno，jisnotamemberofAj，inwhichcase，agaiiohatjisamemberofA=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

&ivennumbers，aandb，onletcbetheiraverage。Ifcisirrational，wehaveaherequiredkind（irrational）。Ifoherhand，cisrational，putd=c+t，wheretistheirratiohepreviraph。Bywhathasgonebefore，dwillalsobeirrational，aakenlargeenough，wealwaysedissoclosetecofthetwogivennumbersaandbthatitliesbetweehisway，weseethattheirratioooformadeand，aswiththerationalnumbers，weferthatthereareinnitelymanyirrationalnumberslyiwohenumberline。