LOWNESSNOTIONS,MEASUREANDDOMINATION3Theorem1.7(Kjos-Hanssen[9]).ALRBi - PDF document

LOWNESSNOTIONS,MEASUREANDDOMINATION3Theorem1.7(Kjos-Hanssen[9]).ALRBifandonlyifeveryA1classofpositivemeasurehasaB1subclassofpositivemeasure.CombiningTheorem1.7withthewell-knownresultofKurtz[10]thatevery02classhasa;02subclassofthesamemeasure,itfollowsthat;0LRAexactlycharacterizesthep.m.dominatingsets.Corollary1.8(Kjos-Hanssen[9]).Aisp.m.dominatingifandonlyif;0LRA.Asthispoint,wehavethefollowingpicture.Aisu.a.e.dominating)Aisa.e.dominating)Aisp.m.dominating,;0LRAInSection3,weclosethiscirclebyshowingthatifAisp.m.dominating,thenAisu.a.e.dominating.ThisresultisanapplicationofamoregeneraltheoremalongthelinesofTheorem1.7:everyA2classhasaB2subclassofthesamemeasureifandonlyifALRBandATB0.Asanotherapplication,weprovethatifAislowfor1-randomnessthenitislowforweak2-randomness(seealsoNies[15]).ThemaintechniqueusedinSection3givesusanewwaytoleveragetheassumptionthatALRB.ItisrstintroducedinSection2,whereweshowthatLRimpliesLK,areducibilitythatcomparesthestrengthoforaclesintermsoftheireectonprex-freeKolmogorovcomplexity.Intheremainingsections,weexaminetheimplicationoftheequivalenceofu.a.e.dominationandp.m.dominationforthereversemathematicsquestionofhowdicultitistoprovethatPOS!G-REG.InSection5,weshowthatRCA0isnotstrongenoughtoprovethisimplication,oreventhatG-"!G-REG.InSection7,weshowthatWWKL0`POS!G-REG.NoticethatsinceWKL0doesnotproveG-REG,thefactthatWWKL0|whichisweakerthanWKL0|provesthisimplicationisnottrivial.Moreover,sincemeasuretheoryisverylimitedwithoutWWKL0[18],itisreasonabletoworkoverthissystemtoprovetheequivalence.Ournotationisstandardthroughout.Weusetodenotethesubsetrelationbetweensets(orclasses),vtodenotetheinitialsegmentrelationbetween(niteorinnite)strings,andjjtodenotethelengthofanitestring.WeidentifyasetXwiththeinnitestringgivenbyitscharacteristicfunction.ForX!ands2!,X[s]denotesthestringhX(0);X(1);:::;X(s�1)i.ForY2,[Y]denotestheopenclassin2!ofallXsuchthat92Y(vX).IfZ2!,thenZc=2!nZ.Finally,ifMisanymachine(viewedasdeningapartialfunctionfrom2to2),thendom(M)denotesthesetofstringsonwhichMconverges(thatis,thedomainofthedenedfunction).2.LRimpliesLKInthissection,weexaminetherelationshipbetweenLRandLK,areducibilitybasedonaninformationtheoreticdenitionofrandomness.ThereaderwhoisnotfamiliarwithKolmogorovcomplexityisreferredtoLiandVitanyi[12]foranintroduction.IfUisauniversalprex-free(Turing)machineandisanitebinarystring,thentheprex-free(Kolmogorov)complexityofisdened(uptoanadditiveconstantdependingonthechoiceofU)byK()=minfjjjU()=g:WewillusetwobasicfactsfromthetheoryofKolmogorovcomplexity. 4BJRNKJOS-HANSSEN,JOSEPHS.MILLER,ANDREEDSOLOMONLemma2.1(Kraftinequality).IfA2isprex-free,thenP2A2�jj1.Inparticular,ifMisaprex-freeTuringmachine,thenP2dom(M)2�jj1.Theorem2.2(Kraft{ChaitinTheorem).Lethdi;iii2!beacomputablesequenceofpairssuchthatdi2!,i22andPi2!2�di1.(Therangefhdi;ii:i2!gofsuchasequenceiscalledaKraft{Chaitinset.)Thereisaprex-freemachineMandstringsioflengthdisuchthatM(i)=iforalli2!.Inparticular,theuniversalityofUimpliesthatK(i)di+O(1).AiscalledLevin-Chaitinrandomifforalln,K(A[n])n�O(1).Despitethedierenceincontext,thisnotionofrandomnesscoincideswithMartin-Lofrandom-nessdenedabove.Nies[14]denedareducibilityLKsimilartoLR,butbasedonKolmogorovcomplexity.TheideaofthisreducibilityisthatALKBifAisnomoreusefulthanBinthesensethatAcannotcompressinformationanymorethanBcan.Denition2.3(Nies[14]).ALKBif(8)KB()KA()+O(1).ItisstraightforwardtoshowthatALKBimpliesALRB;ourgoalforthissectionistoshowthattheyareequivalent.Ourproofwillrequireonebasicfactfromrealanalysis.Lemma2.4.Lethaiii2!beasequenceofrealnumberswith0ai1,foralli.ThenQi2!(1�ai)&#x]TJ/;ø 9;&#x.962; Tf;&#x 10.;ܩ ;� Td;&#x [00;0iPi2!aiconverges.Lemma2.5.Foranycomputablefunctionf:!!!thereisauniformlycom-putablecollectionofnitesetsofbinarystringsVn,n2!,suchthat[Vn]=2�f(n)andthesets[Vn],n2!,formamutuallyindependentfamilyofeventsunder.Proof.AssumethatVthasbeendenedforallts.LetkbethelengthofthelongeststringinStVtandletVs=fb0f(s):22kg.ItisclearthatVs,s2!,hastherequiredproperties.Theorem2.6.IfALRB,thenALKB.Proof.Identifyingtheelementsof!2withnaturalnumbersviaaneectivebi-jection,weletVs,s2!beasguaranteedbyLemma2.5forthefunctionf(hn;i)=n.ThisensuresthatifI!2,then�Ts2I[Vs]c
=Qhn;i2I(1�2�n),sinceeachVsisindependentfromalloftheothers.LetUAbeauniversalprex-freemachinerelativetoAanddeneI=fhjj;i:UA()=g:ThenIisA-c.e.,soP=Ts2I[Vs]cisaA1class.NotethatPhn;i2I2�nP2dom(U)2�jj1bytheKraftinequality.Also,h0;iisnotinIforany.SobyLemma2.4,(P)=Qhn;i2I(1�2�n)&#x!]TJ;&#x/F8 ; .96;& T; 11;&#x.994;&#x -3.;ؕ ;&#xTd [;0.ThereforebyTheorem1.7,thereisaB1classQPsuchthat(Q)&#x!]TJ;&#x/F8 ; .96;& T; 11;&#x.994;&#x -3.;ؕ ;&#xTd [;0.DeneJ=fhn;i:[Vhn;i]\Q=;g.NotethatJisaB-c.e.setsinceQcisgeneratedbyaB-c.e.setofstrings,Vhn;iisanitesetofstrings,and[Vhn;i]\Q=;ifandonlyif[Vhn;i]iscoveredbyanitesetofbasicintervalsfromQc.Also,bythecommentsintherstparagraphofthisproof,Qhn;i2J(1�2�n)=�Ts2J[Vs]c
(Q)&#x!]TJ;&#x/F8 ; .96;& T; 11;&#x.994;&#x -3.;ؕ ;&#xTd [;0.ThereforebyLemma2.4,Phn;i2J2�nconverges.Furthermore,weclaimthatIJ.Ifhn;i2I,then[Vhn;i]\P=;.SinceQP,[Vhn;i]\Q=;andhencehn;i2J. LOWNESSNOTIONS,MEASUREANDDOMINATION5SincePhn;i2J2�nconverges,xc2!suchthatthissumisboundedby2c.ThenbJ=fhn+c;i:hn;i2JgisaKraft{ChaitinsetrelativetoB.ThereforebytheKraft{ChaitinTheorem,hn;i2J=)hn+c;i2bJ=)KB()n+c+O(1)n+O(1):SinceIJ,wehavehKA();i2Jforeach22!.ThusKB()KA()+O(1).Inotherwords,ALKB.Corollary2.7.ALRBifandonlyifALKB.Proof.Asnotedpreviously,ALKBimpliesALRB.Theorem2.6suppliestheotherimplication.WeoeroneapplicationofTheorem2.6basedonaspecialcaseofLRandLK.Aislowfor1-randomnessifALR;,thatis,ifeveryrandom(inthemeasuretheoreticsense)remainsrandomrelativetoA.Similarly,AiscalledlowforKifALK;,thatis,everystringcontainsasmuchinformationrelativetoAasitdoeswithnooracle.Corollary2.8(Nies[14]1).Aislowfor1-randomnessifandonlyifAislowforK.Proof.ThiscorollaryfollowsfromCorollary2.7bysettingB=;.3.PreservingMeasureInthissection,weshowthatp.m.dominationimpliesu.a.e.domination,therebyshowingtheequivalenceofthethreedominationnotionsintroducedinSection1.Lemma3.1.IfATB0andALRB,theneveryA1classhasaB2subclassofthesamemeasure.Proof.TheproofwillbesimilartothatofTheorem2.6.Identifyingnowtheele-mentsof22withnaturalnumbersviaaneectivebijection,weletfVsgs2!beasguaranteedbyLemma2.5forthefunctionf(h;i)=jj.Asbefore,ifI22,then�Ts2I[Vs]c
=Qh;i2I(1�2�jj).LetXbeaA1class.Assume,withoutlossofgenerality,thatX6=;.LetSA2beaprex-freeA-c.e.setofstringssuchthatX=2!r[SA];notethatSAdoesnotcontaintheemptystring.LetI=fh;i:2SAwithuseg.ConsidertheA1classP=Ts2I[Vs]c.NotethatPh;i2I2�jj=P2SA2�jj1bytheKraftinequality.SobyLemma2.4,(P)=Qh;i2I(1�2�jj)&#x!]TJ;&#x/F8 ; .96;& T; 14;&#x.373;&#x -3.;ؖ ;&#xTd [;0.ThereforebyTheorem1.7,thereisaB1classQPsuchthat(Q)&#x!]TJ;&#x/F8 ; .96;& T; 14;&#x.373;&#x -3.;ؖ ;&#xTd [;0.DeneJ=fh;i:[Vh;i]\Q=;g.AsintheproofofTheorem2.6,JisaB-c.e.set,IJ,andQh;i2J(1�2�jj)=�Ts2J[Vs]c
(Q)&#x!]TJ;&#x/F8 ; .96;& T; 14;&#x.373;&#x -3.;ؖ ;&#xTd [;0.ThereforebyLemma2.4,Ph;i2J2�jjconverges.ByassumptionATB0,soletfAsgs2!beaB-computablesequenceapproxi-matingA.DeneTs=fh;i2J:(9ts)2SAttwithuseg 1Yetanotherproof|onebasedonworkofHirschfeldt,NiesandStephan[8]|canbefoundinNies[15]. 8BJRNKJOS-HANSSEN,JOSEPHS.MILLER,ANDREEDSOLOMONCombiningCorollary3.4andTheorem3.5togetherwiththefactthatlowforweak2-randomtestsimplieslowforweakfor2-randomnessyieldsthefollowingcorollary.Corollary3.6.ForanysetA,thefollowingconditionsareequivalent:(1)Aislowfor1-randomness,(2)Aislowforweak2-randomtests,and(3)Aislowforweak2-randomness.Corollary3.6canalsobeprovedusingthegoldenrunmachineryofNies[14].Thiswasdiscoveredindependently,andearlier,byNiesandaproofalongtheselinesisgiveninNies[15].4.MeasureDefinitionsinReverseMathematicsIntheremainderofthispaper,weconsiderthereversemathematicsquestionofhowdicultitistoprovePOS!G-REG.Webeginwithdenitionsofcodesforopen,closed,GandFsubsetsof2NinRCA0.(Weswitchfrom!toNasitisstandardtouseNtodenotetherstorderpartofanygivenmodelofsecondorderarithmetic.)Acodeforanopensetin2NisasetO2N.WecanassumewithoutlossofgeneralitythatOisprexfree.WewriteX2[O](andsaythatXisinthesetcodedbyO)ifthereisastring2OsuchthattvX.Itisoftenusefultothinkofanopensetastheunionofasequenceofclopensets.Fort2N,weletOt=f2Ojjjtgandnotethat[O]=St[Ot].Equivalently,wecanspecifyanopensetbya01formula(allowingparameters)9s'(x),where'(x)containsonlyboundedquantiers.Inthiscontext,wesaythatXisinthecodedopensetif9s'(X[s]).Lateritwillbeconvenienttothinkofthecollectionofstringssatisfying(orenumeratedby)suchaformulaeventhoughthiscollectionneednotbeasetinRCA0.Weusetheterm01classofstrings(orsimply01class,relyingoncontexttodierentiatebetweenthisnotionofclassandtheoneusedinthecontextofsetsofreals)todenotethecollectionofstringssatisfyingaparticular01formula.Thisterminologyallowsustousesetnotationforsuchcollections,althoughanysuchstatementisunderstoodasstandingfortheappropriatetranslationofthedeningformulas.IfOisthe01classofstringscorrespondingtotheformula9s'(x),thenOt=fjjjt^9st'()g.Asabove,eachOtisclopenand[O]=St[Ot].Inthiscontext,wecannotassumethatthe01classofstringsOisprexfree.However,abusingnotation,wecanassume(byremovingstringsfromOtinauniformmanner)thatthenitesetsOtareprexfree.InsystemsweakerthanACA0,wecannotassumethatboundedincreasingse-quencesofrationalsconverge.Therefore,ratherthanassumingthatopensetshavedenitemeasures,weworkwithcomparativestatementssuchas(O)qforq2Q.TodenethesenotionsinRCA0,letObea(prex-free)codeforanopenset.Fort2N,dene(Ot)=P2Ot2�jj,andforq2Q,dene(O)q,8t((Ot)q)(O)&#x-386;q,9t((Ot)&#x-386;q)(O)q,8r2Q(rq!(O)&#x-277;r) LOWNESSNOTIONS,MEASUREANDDOMINATION11Corollary5.5.RCA00POS!G-REGandRCA00G-"!G-REG.Proof.ThiscorollaryfollowsimmediatelyfromCorollaries5.2and5.4.6.LogarithmPropertiesWehavenowestablishedthatalthoughpositivemeasuredominationisequivalenttouniformalmosteverywheredomination,RCA0isnotstrongenoughtoprovePOS!G-REG.Inthelasttwosections,weshowthatWWKL0isstrongenoughtoprovethisimplication.Inthissection,wesketchthedevelopmentofthenaturallogarithminRCA0andgiveananalogueofLemma2.4.Wewishtodenethenaturallogarithmusingtheusualintegralformln(x)=Zx11 udu:Becausethefunctionf(u)=1=udoesnothaveamodulusofuniformcontinuity,wedonotautomaticallygetacodeforln(x)asacontinuousfunctioninRCA0.(SeeSimpson[17],DenitionIV.2.1,LemmaIV.2.6,andTheoremIV.2.7fortherelevantbackgroundonintegralsinsubsystemsofsecondorderarithmetic.)Letq2Q+.FollowingthestandardprocedureforestimatingRq11 udubyrectan-gles,wesubdividetheinterval[1;q](or[q;1]ifq1)intonequalpieces.Becausef(u)=1=uisadecreasingfunction,weobtainupperandlowerestimatesoftheintegralusingtheleftandrightendpointsofeachintervaltodenetheheightoftheapproximatingrectangle.AshortcalculationshowsthatUpperSum�LowerSum=jq�1j n1�1 q;whichgoesto0asn!1.InRCA0,wedenethefollowingcodeforln(x).(SeeSimpson[17],DenitionII.6.1,fortheformaldenitionofacodeforacontinuousfunctioninasubsystemofsecondorderarithmetic.)LetlnNQ+Q+Q+Q+begivenby(n;a;r;b;s)2lnifandonlyif0a�r,theuppersumfortheestimateofln(a+r)usingnintervalsisb+s,andthelowersumfortheestimateofln(a�r)usingnintervalsis&#x-278;b�s.Sincethedierencebetweentheupperandlowersumsconvergesto0,lnisacodeforacontinuousfunctionandthefunctionln(x)denedbytheseconditionscoincideswithRx11=udu.Theproofthat1=xisthederivativeofln(x)canbecarriedoutinastraightforwardmannerwithinRCA0.Lemma6.1(RCA0).Thefollowingresultshold.(1)TheMeanValueTheorem.(2)IffisadierentiablefunctiononanopenintervalinR,thenf0=0onthisintervalifandonlyiffisconstant.Iff00onthisinterval,thenfisnondecreasing,andiff00onthisinterval,thenfisnonincreasing.(3)Foralla;b2R+,ln(ab)=ln(a)+ln(b).(4)Forallk2Nandallsequencesofpositiverationalnumbersa0;:::;ak,ln(Qki=0ai)=Pki=0ln(ai).Proof.Part(1)isprovedbyHardinandVellemanin[7].Parts(2)and(3)followbytheirclassicalproofsusingtheMeanValueTheorem.Part(4)followsby01inductiononksincetheequalitypredicatebetweenrealsis01. LOWNESSNOTIONS,MEASUREANDDOMINATION13Proof.If0ai1 2,thenbyLemma6.3,0�ln(1�ai)2ai.ThuskXi=0ln(1�ai)kXi=0(�2ai)=(�2)kXi=0ai�4soasinProposition6.6,Qki=0(1�ai)e�41=81(usingthefactthate3).7.WorkinginWWKL0Throughoutthissection,weworkinWWKL0toprovePOS!G-REG.OurproofwillroughlybeaformalizationoftheargumentsinLemma3.1andCorollary3.3withoneimportantdierence.IntheproofsleadingtoCorollary3.3,weusedthefactthatevery02classcontainsa;02classofthesamemeasure.Thisfactallowedustoswitchfromworkingwitha02classtoworkingwithclosedclasseswithoracles.BecauseWWKL0cannotprovetheexistenceof;0,weneedtoworkdirectlywiththegivenGsetandapproximateitsmeasurewithinWWKL0.ThroughoutthissectionweworkinWWKL0(infact,exceptforLemma7.8,weworkinRCA0),assumePOSandproveG-REG.LetX=hXiji2NibeacodeforaGsetofpositivemeasure.EachXiisanonemptyprex-freesubsetof2NandXi;sdenotesthesetofallstrings2Xisuchthatjjs.Wewillbenotationallysloppyaboutthedistinctionbetweencodingsets,suchasXandXi,andthesubsetsof2Ntheycode,relyingonthecontexttoindicatewhichistheintendedmeaning.Ifthecontextisnotclear,wewillusesquarebrackets[X]todenotethecodedsubsetof2N.Foreachpairi;n2N,wedeneafunctionmi;n(t)byprimitiverecursion(uni-formlyiniandn)toapproximate(Xi).Setmi;n(0)=0andmi;n(t+1)=mi;n(t)if(Xi;t+1�Xi;mi;n(t))2�n�i�1;t+1otherwise.Lemma7.1.Thefollowingpropertiesholdforeachi;n2N.(1)8t;u(tu!mi;n(t)mi;n(u)).(2)8t;u(mi;n(t)mi;n(u)!(tu^(Xi;mi;n(u)�Xi;mi;n(t))2�i�n�1)).(3)9t8ut(mi;n(u)=mi;n(t)).Proof.Properties(1)and(2)followdirectlyfromthedenitions.ToproveProperty(3),weproceedbycontradiction.IfProperty(3)failsforaparticulariandn,thenbyProperty(1),forallt,thereisau&#x-278;tsuchthatmi;n(u)&#x-278;mi;n(t).Wedeneafunctionfsuchthatf(0)=0andf(j+1)=theleastu&#x-278;f(j)suchthatmi;n(u)&#x-278;mi;n(f(j)).ByProperty(2),wehavethat(Xi;mi;n(f(j)))j
2�i�n�1,whichforj&#x-278;2i+n+1givesthedesiredcontradiction.Weletm1i;n=limsmi;n(s).(Soinasensem1i;nisthelaststagethatissignicantforthepair(i;n).)AsweareworkinginWWKL0,wecannotformafunctiontakingeachpairhi;nitom1i;n,soweunderstandeachstatementm1i;n=ktobeanabbreviationforthe
02formulagivenbytheequivalentformulations9t8ut(mi;n(u)=k)and8t9ut(mi;n(u)=k).Wesaythath;ni2NNNiscorrectatsifjjs,ns,and(i)=mi;n(s)forallijj.(Thecollectionoftriplesh;n;sisuchthath;niiscorrectatsisaset.)Wesaythath;niiscorrectif(i)=m1i;nforallijjandweletC1ndenotethe
02classofallstringssuchthath;niiscorrect.(Tohelpmaintain 14BJRNKJOS-HANSSEN,JOSEPHS.MILLER,ANDREEDSOLOMONthedistinctionbetweensetsofstringsandclassesofstrings,weuseboldfacelettersforclasses.Anystatementinvolvingaclassistoberegardedasshorthandforthestatementgivenbysubstitutinginthedeningformulafortheclass.)Noticethatinadditiontobeinga
02class,C1nisalsod.c.e.(adierenceoftwocomputablyenumerablesets)inthesensethatifh;nibecomescorrectats,theneitherh;niremainscorrectatallfuturestages(and2C1n)orh;niceasestobecorrectatsomet�sandisnevercorrectatanystaget.WeneedtodenetheappropriateversionofthesetIfromLemma3.1forourargument.Consideranarbitraryn,astages,andavalueks.Thestring=hm0;n(s);m1;n(s);:::;mk�1;n(s)iistheuniquestringoflengthksuchthath;niiscorrectats.Itgivesrisetothefollowingsequenceofclopensets(X0;(0))c(X0;(0)\X1;(1))c


0@\jjjXj;(j)1Ac:Thedierence(TjjjXj;(j))c�(Tjjj�1Xj;(j))cisaclopensetgeneratedbyanitesetofminimallengthstrings(sothesestringsformanantichain).WedenethesetINN2NNNbyh;;n;si2Iifandonlyifh;niiscorrectatsandisaminimumlengthstringusedtocover(TjjjXj;(j))c�(Tjjj�1Xj;(j))c.WewillbeinterestedinthefollowingprojectionsandrestrictionsofI.I;n;s=fjh;;n;si2IgIs=fh;;nijh;;n;si2IgI9n;s=fj9(h;;n;si2I)gI1=fh;;nij9t8st(h;;n;si2IgI1;n=fj9s(h;;n;si2IgI;n;s,IsandI9n;sareallnitesets,whileI1isa
02classofstrings(viatheequivalentcondition8t9st(h;;n;si2I))andI1;nisa01classofstrings.(ToseethatI9n;sisaniteset,noticethatI9n;sistheunionofthenitesetsI;n;soverthenitelymanysuchthath;niiscorrectats.)Thefollowingpropertiesareeasilyveriedfromthedenitions.Inthecurrentargument,Property(7)playstheroleoftheKraftinequalityinLemma3.1.Lemma7.2.Thefollowingpropertiesholdforall,,nands.(1)Ifh;niisnotcorrectats,thenI;n;s=;.(2)Ifh;niiscorrectats,thenI;n;sI9n;s.(3)h;;ni2I1ifandonlyifh;niiscorrectand2I1;n.Furthermore,ifh;niiscorrectandiscorrectats,thenI1;n=I;n;s.(4)Foreachnandk,thereisauniquestringsuchthatjj=kandh;niiscorrect(thatis,2C1n).Foreachik,hi;niiscorrect, \iXi!c \iXi;(i)!c= \iXi;m1i;n!c=[i[I1i;n]and [i[I1i;n]� \iXi!c!Xi2�n�i�1: LOWNESSNOTIONS,MEASUREANDDOMINATION17Lemma7.6.(P)�0.Proof.Weneedtoshowthatthereisan"2Q+suchthat8j0@0@\ijPi1A"1A:Weproceedbycontradiction.Supposethatforevery"�0,thereisajsuchthat(TijPi)".Fixanarbitrary"andthecorrespondingj.FixusuchthatPi;u=Piforallij.Asabove,weassumei=hi;i;ni;tii.Foreachij,Pi;u=[Vi;i;ni]cimplieshi;i;nii2Iu\I1,andPi;u6=[Vi;i;ni]cimpliesPi;u=2N.Furthermore,becauseeachPi;uisanitesetofstrings,wecantellwhichofthesecasesapplies.FormthenitesetK=fhi;i;niijij^Pi;u=[Vi;i;ni]cgIu:Calculatingmeasures,wehaveYhi;i;nii2K(1�2�jij�ni)=0@\ijPi;u1A=0@\ijPi1A":Furthermore,wehaveXhi;i;nii2K2�jij�niXh;;ni2Iu2�jj�n2:(TherstinequalityfollowsbecauseKIuandthesecondinequalityfollowsfromProperty(7)ofLemma7.2.)Forasmallenoughvalueof",thefactthatQhi;i;nii2K(1�2�jij�ni)"andPhi;i;nii2K2�jij�ni2contradictsPropo-sition6.7.Lemma7.7.Forall,andn,[V;;n]\P=;ifandonlyifh;niiscorrectand2I1;n.Proof.Supposethath;niiscorrectand2I1;n.ByProperty(3)ofLemma7.2,h;;ni2I1.ByLemma7.5,[V;;n]cisoneoftheintersectedsetsformingPandtherefore[V;;n]\P=;.Nowassumethatitisnotthecasethath;niiscorrectand2I1;n.AgainbyProperty(3)ofLemma7.2,wehaveh;;ni=2I1.So[V;;n]cdoesnotoccurintheintersectionformingP.Lets=h;;ni.RecallhowthesetsVtwereformedinLemma2.5.LetkbethelengthofthelongeststringinStVt.ConsiderthesequenceX=1k0f(s)1N.ItfollowsfromtheconstructionofthesetsVt,t2N,thatX2[Vs]butX2[Vt]cforeveryt6=s.Therefore,X2[V;;n]\P,so[V;;n]\P6=;.ByLemma7.7,wecanwriteZasA2Z,8n9;([V;;n]\P=;^A2[]):ByPOS,wecanxaclosedsetQPsuchthat(Q)&#xs]TJ;&#x/F11;&#x 9.9;ئ ;&#xTf 1;.15; 2.;উ ;&#xTd [;0.FollowingtheproofofLemma3.1,itwouldmakesensetodeneJtobetheclasscontainingalltriplesh;;nisuchthat[V;;n]\Q=;.TheproblemisthatwithoutWKL0,thiswould 18BJRNKJOS-HANSSEN,JOSEPHS.MILLER,ANDREEDSOLOMONnotnecessarilybea01condition.SincewewanttoworkinWWKL0,weneedaslightlydierentdenitionofJ.Takek2Nsuchthat(Q)�2�k.LetJ=fh;;nij([V;;n]\Q)2�h;;ni�k�2g:InSection4wesawthatifOisanopensetandq2Q,then(O)&#x]TJ/;ø 9;&#x.962; Tf;&#x 10.;Ԗ ;� Td;&#x [00;qisa01statement.Thus,Jisa01class.Lemma7.8.If[V;;n]\Q=;,thenh;;ni2J.Proof.ThisfollowsfromWWKL0anditisouronlyuseoftheprinciple.Ifh;;ni=2J,then([V;;n]\Q)&#x]TJ/;ø 9;&#x.962; Tf;&#x 10.;Ԗ ;� Td;&#x [00;0.ButthenWWKL0impliesthat[V;;n]\Q6=;.Lemma7.9.ThesumPh;;ni2J2�jj�nisboundedabove.Proof.BecauseJisa01class,thissumcanbeexpressedasPaiwherethese-quenceai2QisdeterminedbytheenumerationofJ.Thatis,ai=2�jj�nifthei-thelementenumeratedintoJish;;ni.(Recallthatwethinkofa01classsuchasJenumeratedinstageswithJsequaltothenitesetoftuplesh;;niswhichareinJwithanexistentialwitnesss.)WedeneanopensetRasfollows.Atthestageswhenh;;nigoesintoJ,wehave([V;;n]\Qs)2�h;;ni�k�2.Enumeratetheclopenset[V;;n]\QsintoR.Notethat(R)Ph;;ni2J2�h;;ni�k�22�k�1.Alsonotethatifh;;ni2J,then[V;;n]R[Qc.Therefore,Q�RTh;;ni2Js[V;;n]c.Foranys2N,wehaveYh;;ni2Js(1�2�jj�n)=0@\h;;ni2Js[V;;n]c1A(Q�R)(Q)�(R)&#x]TJ/;ø 9;&#x.962; Tf;&#x 10.;Ԗ ;� Td;&#x [00;2�k�2�k�1=2�k�1&#x]TJ/;ø 9;&#x.962; Tf;&#x 10.;Ԗ ;� Td;&#x [00;0:andthereforetheproductQh;;ni2J(1�2�jj�n)isboundedawayfrom0.Hence,byProposition6.6,Ph;;ni2J2�jj�nisboundedabove.ToapproximatethedeningconditionforZgivenimmediatelyafterLemma7.7,welookatthe01predicateh;;ni2J^9ts(h;;ni2It):DeneTn;s=fh;;nijh;;ni2J^9ts(h;;ni2It)g;Un;s=fj9(h;;ni2Tn;s)g:NotethatTn;sandUn;sare01classesandforanyxedn,wehaveTn;0Tn;1Tn;2


;andUn;0Un;1Un;2


:Wenallydeneourdesired02classYY=\n2N\s2N[Un;s]:Lemma7.10.ZY. LOWNESSNOTIONS,MEASUREANDDOMINATION19Proof.LetA2Zandxanyn.WeshowthatA2Ts2N[Un;s].SinceA2Z,therearestringsandsuchthat[V;;n]\P=;andA2[].SinceQP,wehave[V;;n]\Q=;,soh;;ni2J.ByLemma7.7,wehavethath;niiscorrectand2I1;n.Therefore,foralls,thereistssuchthath;;ni2It.(Infact,thisistrueforalmostallts.)Itfollowsthatforalls,h;;ni2J^9ts(h;;ni2It);andhencethath;;ni2Tn;sand2Un;sforalls.SinceA2[],wehavethatA2Ts2N[Un;s]asrequired.Lemma7.11.(Y�Z)=0.Proof.Fork2NweletZk=[2C1k[I1;k];Yk=\s2N[Uk;s]:TheproofofLemma7.10showsthatZkYk.SinceZ=TkZkandY=TkYk,itsucestoshowthat(Yk�Zk)=0.Toprovethismeasurestatement,weneedtoprovethatforevery"2Q+,thereisacsuchthat(Uk;c�Zk)".Fixk2Nand"2Q+.ByLemma7.9,xmsuchthatXh;;ni2Jh;;nim2�jj�n"
2�k:(Inthissum,,andnvary.)Fixingn=kinthissummationandmultiplyingby2k,wehave(nowlettingonlyandvary)Xh;;ki2Jh;;kim2�jj":Foreachtupleh;;ki2Tk;0suchthath;;ki=2I1,theremustbeancsuchthatforalluc,h;;ki=2Iu,andhenceh;;ki=2Tk;c.Wewouldliketoobtainasinglewitnesscwhichworksforallsuchh;;kim.Considertheboundedquantierstatement'(;;k;u)whichsaysthatuisawitnessforh;;ni2J,that9tu(h;;ki2It),andthath;;ki=2Iu.Fixanyh;;kisuchthat9u'(;;k;u),xthewitnessuforthisstatementandxtuthatwitnessesthesecondconjunctof'.Becauseh;;ni2Jandh;;ki2It,wehavethath;;ki2Tk;0.Becauseh;;ki=2Iuandtu,wehavethat8vu(h;;ki=2Iv)andhenceh;;ki=2Tk;u.Furthermore,bythepreviousparagraph,ifh;;ki2Tk;0andh;;ki=2I1,then9u'(;;k;u).Thestrong01boundingscheme(whichholdsinRCA0,seeSimpson[17]ExerciseII.3.14)impliesthat9c8h;;kim(9u'(;;k;u)!9uc'(;;k;u)):Fixsuchac.Foranyh;;kim,ifh;;ki2Tk;0andh;;ki=2I1,thenh;;ki=2Tk;c.