ToAppearVol0No02008000045GeneralizingCantorsAgumentInthissectionweshall rstseehowthebasicideaofCantorsargumentintheconstructionofthesetDcanbegeneralizedinanalogywithYablosnonselfreferentia ID: 324842
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ToAppear,Vol.0,No.0(2008)00{002itspowerset.Thetraditionaldiagonalizationproofconstructssuchasubsetusingthenegationoperator.WeintroduceYablo'snon-self-referentialLiar'sparadox,andpresentadi erentproofofCantor'stheoreminspiredbyYablo'sparadox.Thisproofconstructsanotherleft-oversubsetwhichdoesnotrequireinvokingthenega-tionoperationforitsdenition.Wethendiscussvariousaspectsoftheproof,andindicatesimilaritieswithtwootherparadoxes.WeoutlineotherproofsprovidedbyCantor;andnallyshowyetanotherproofwhichmayinsomesensebeconsidereddualtothenegation-freeproof.2Cantor'sDiagonalizationProofWerecallCantor'sdiagonalizationproofofhiseponymoustheorem.Theorem2.1Cantor'sTheorem:Foranyset,thereisnofunctionmap-pingitsmembersontoallitssubsets.Proof[2,3]:ForanysetX,letP(X)denotethepowersetofX,i.e.P(X)=fTjTXg.SupposethatthecardinalityofXisequaltothecardinalityofP(X).Thismeansthataone-to-onecorrespondencecanbeestablishedbetweenXandP(X).Foranyx2XandTx2P(X),let(x;Tx)denotepairsofelementsestablishedbytheone-to-onecorrespondence.NowconsiderthesetD=fyjy62Tyg.ClearlyD2P(X),andDdi ersfromeverysetTywithrespecttotheelementy.Thusanyone-to-onecorrespondenceomitsthesetD.Q.E.D.NoticethattheconstructionofthesetD,whichisleft-overbyanyone-to-onemapping,involvestheuseofnegationinstatingy62Ty.3ParadoxwithoutCircularityYablo'sparadox[13,14,15]isanon-self-referentialLiar'sparadox.BeforetheformulationofYablo'sparadox,allknownparadoxesinlogicseemedtorequirecircularityinanunavoidableway.Eachofthemusedeitherdirectself-reference,orindirectloop-likeself-reference.So,itappearedasthoughself-referencewasanecessaryconditionfortheconstructionofparadoxicalsentences.Yablo'sparadoxdemonstratedthatthiswasnotthecase.Weprovideabriefoutlineofitinthissection. ToAppear,Vol.0,No.0(2008)00{0045GeneralizingCantor'sAgumentInthissectionweshallrstseehowthebasicideaofCantor'sargumentintheconstructionofthesetDcanbegeneralizedinanalogywithYablo'snon-self-referentialliar'sparadox.SuchageneralizationwouldgiverisetothesetNinthenegation-freeproof.Extendthedenitionofasimpleelementtothenotionofak-simpleelementasfollows.Denet2Xtobeak-simpleelementwhenfork�0,thereisnosequences1;:::;sksuchthats12M(t);:::;sk2M(sk�1)andt2M(sk).LetNkbethesetofallk-simpleelementsofX.Thenaneasyargumentshowsthattherecanbenon2XsuchthatNk=M(n).Cantor'sargumenthasN0,wheretis0-simplewhent62M(t).Inthenegation-freeproof,thesetsNkareintermediaries,beforeleadinguptothesetN.WehavereplacedthecirclesmentionedinNkby`omega',i.e.inanalogywithYablo'sparadox,wehaveopenedthesecircles,toconstructN.DoesthesetNusenegation?ItisperhapsnotimmediatelyobviousthatthesetNdoesnotusenegation.Itisdenedas:N=ft2XjtisasimpleelementgBydenition,t2Xisasimpleelementwhenallpossibletracesbeginningwithtterminate.Isthestatement\allpossibletracesbeginningwithttermi-nate"negation-free?Tosettlethisquestionwewouldhavetorewriteitasarst-orderformula.Anobviousrewriting,whichcomesoutofthedenitionofk-simpleabove,wouldcorrespondto:\thereisnonon-terminatingse-quences1;s2;:::suchthats12M(t);:::;sk2M(sk�1);:::"Negationseemstooccurexplicitlyintheaboveformula.Butthereisalsoanothernegationimplicitinthenotionofanon-terminatingsequenceabove.ThatthesetNisnegation-freecanbeshownasfollows.Wecanalsorewritethestatement\allpossibletracesbeginningwiths0terminate"asarst-orderformulainanotherway.Fork0,letanelements0inXbecalledsimplewhenforeachpossiblesequence(beginningwiths0):s0;s1;:::;sk(8isi+12M(si))thereexistsajsuchthatM(sj)= .Thischaracterizationisnegation-free. ToAppear,Vol.0,No.0(2008)00{0066.2Mirimano 'sParadoxMirimano 'sParadox,alsoknownastheParadoxoftheClassofAllGroundedClasses,wasformulatedbyDmitriMirimano [6,7,8,16],insettheory.Denition6.3(GroundedClass):AclassXissaidtobeagroundedclasswhenthereisnoinniteprogressionofclassesX1,X2,:::(notneces-sarilyalldistinct)suchthat:::2X22X12X.Denition6.4(ClassofallGroundedClasses):LetYbetheclassofallgroundedclasses.Mirimano 'sParadoxisbroughtoutbythequestion:IsY,theclassofallgroundedclasses,itselfgrounded?LetusassumethatYitselfisagroundedclass.HenceY2Yandsowehave:::Y2Y2Y2YcontrarytogroundednessofY.ThereforeYisnotagroundedclass.IfontheotherhandYisnotgrounded,thenthereisaninniteprogressionofclassesX1,X2,:::suchthat,:::2X22X12Y.SinceX12Y,X1isagroundedclass.Butthen:::2X22X1,whichmeansX1inturnisnotgrounded,whichisimpossiblesinceX12Y.7Cantor'sOtherProofsInthissection,webrie ysketchCantor'stwootherproofsfortheuncount-abilityofthecontinuum[1,2,3,10].Theorem7.1(UncountabilityoftheContinuum)Therecannotbeanyone-to-onecorrespondencebetweenthenaturalnumbersandtherealnumbers.Cantor'sProofbyDiagonalization[2,3]:Considertherealnumbersbe-tweenzeroandone,representedbyinnitedecimalexpansions.Anyattempttoconstructaone-to-onecorrespondencebetweenthethenaturalnumbersandtherealswillfailforthefollowingreason.Foranyone-to-onecorrespon-dencewecanconstructarealnumberthatisaninniteexpansionwhichisdi erentfromeveryotherrealnumberintherangeofthemapping.Thiscanbedonebymakingthenumberconstructeddi erfromtherstnumberofthemappingintherstdecimalplace;di erfromthesecondnumberofthemappingintheseconddecimalplace;andbycontinuinginthiswayto ToAppear,Vol.0,No.0(2008)00{008Lemma8.1(Kurepa'sLemma)IfP=(P;P)isaposet,andthetreeP=(P;P)isdenedasthesetofascendingsequencesofelementsofPorderedbyend-extension,thenthereisnoorderpreserving,one-to-onemappingf:P!P.Proof[4,12]:ConsultfromKurepa[4].Foraproofofamoregen-eralversionoftheabovetheorem,consultfromTodorecevic{Vaananen[12].Q.E.D.Theorem8.2(UncountabilityoftheContinuum)Therecannotbeanyone-to-onecorrespondencebetweenthenaturalnumbersandtherealnumbers.Proof:LetX=fa0;a1;:::gbeanyset,andP(X)denotethepowersetofX.Assumethatitispossibletodeneaone-to-onemappingM:P(X)!X.UseMtoconstructchainsinP(X)andXsuchthat:a0=M( )a1=M(fa0g)a2=M(fa0;a1g):::a!=M(fa0;a1;:::g)a!+1=M(fa0;a1:::;a!g):::ByKurepa'slemma,Xhastobeaproperclass.ThiscontradictstheassumptionthatXisaset,henceaone-to-onemap-pingMcannotexist.Q.E.D.9ConclusionFundamentaltheoremsarefascinatingphenomenaontheirownright.Equallyfascinatingarepathswhichreconstructtheirproofsusingaminimalorevenanon-standardrepertoireofbasicconstructsandreasoningmechanisms.Wehavetraversedsuchapathinthispaper,andpresenteddi erentproofsofatheoremwhichmarkedthebeginningsoftransnitesettheory.Wehaveindicatedhowtwooftheproofsbearaspecialrelationshiptooneanother,andthatitmaybepossibletotounifytheminacommonframework.Weareexploringsuchaframework,anditsfurtherimplicationssuchasapossiblemechanismwhichmightautomaticallygeneratethemfromeachother. 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