1TobepreciseifjjnthenthereareuniversalconstantscandcsuchthatthinkingofpsometimesasastringK0pKKpc2jj2jpjc2n2logpcnpcprovidedp2logpnccwhichistrueforppncthatwecan ID: 127432
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Numberingsandrandomness5denotethefair-coinCantor-Lebesguemeasureon2!.BytheproofofSchnorr'sTheoremwehave(fx:(8n)K(xn)n�cg)1�2�(c+1):Consequently,ifc0,thenA[0;c]�0andA[0;c]6=?.Theorem6.Letc0.Thereisnoe ectiveenumerationofR[0;c].Proof.Supposethatf ege2!issuchanenumeration,withauniformlycom-putableapproximation e;ssuchthat e=lims!1 e;sand e;s e;s+1.NotethatA[0;c]=fx:(8n)K(xn)n�cgisa01class.Lets=maxf e;s:esg.Then=lims!1sisleft-c.e.,andsincetheleft-c.e.membersofA[0;c]aredenseinA[0;c],istherightmostpathofA[0;c].Howevertherightmostpathofa01classisalsoright-c.e.,denedintheobviousway.ThusisaMartin-Lofrandomrealthatiscomputable,acontradiction.Theorem7.Foreachcthereisane ectivenumberingofR[c+1;1).Proofsketch.Letfmege2!beane ectiveenumerationofallleft-c.e.randomreals,withtheadditionalpropertythatforeachethereareinnitelymanye0suchthatforalls,me;s=me0;s.Wewilldeneane ectivenumberingf ege2!ofR[c+1;1).WesaythatastringsatisesrandomnessconstantcatstagetifKt()jj�c;otherwise,wesaythatfailsrandomnessconstantcatstaget.Weproceedinstagest2!,monitoringeachme;tforetatstaget.Ifforsomet0,n,e,weobservethatme;t0[n]failsrandomnessconstantc,thenwewanttoassignaplaceformeinourenumerationofR[c+1;1).Soweletdbeminimalsothat dhasnotyetbeenmentionedintheconstruction,andlet d;s=me;sforallstagesst0untilfurthernotice.Ifme;t1[n]atsomestaget1t0satiesrandomnessconstantc,thenweregrethavingassignedmeaplaceinourenumerationf ege2!.Tocompensateforthisregret,wechoosealargenumberp=pc;nandforallstagesst1let d;s=me;s[n]_0p_ s.Thelargenessofpguaranteesthatme;s[n]_0pdoesnotsatisfyrandomnessconstantc.1Ifmeactuallydoesfailrandomnessconstantc,butatalargerlengthn0�n,thenbecausethereareinnitelymanye0withme0=mewewilleventuallyassignsome d0tosomesuchme0atastaget2thatissolargethatme0;t2[n0]=me0[n0].Thus,eachrealinR[c+1;1)willeventuallybeassignedapermanent d0. 1Tobeprecise,ifjj=nthenthereareuniversalconstants^cand~csuchthat,thinkingofpsometimesasastring,K(_0p)K()+K(p)+^c2jj+2jpj+~c=2n+2logp+~cn+p�cprovidedp�2logpn+~c+c;whichistrueforp=pn;cthatwecannde ectively. Numberingsandrandomness7Proof.(1)implies(2):LetfTege2!begiven,anddeneS=f(X;e):8nXn2Teg:(2)implies(1):LetSbegiven,letabeaTuringfunctionalsuchthat(X;e)2S,Xa(e)",andletTe=f22:a;jj(e)"g.InlightofProposition1,wemayuseeithernotion.NotethatifCbelongstoafamilyasinProposition1thenCisa01class.3.1ExistenceofnumberingsTheorem8.LetP2!,letCPbethecollectionofall01classescontainedinP,andletNPbethecollectionofallnonempty01classescontainedinP.AssumePhasthefollowingproperties:(i)Pisco-dense:nocone[],22,iscontainedinP;(ii)Pisclosedundershifts:ifx2Pthen_x2P;(iii)NP6=;.Thenthereisnoe ectivenumberingofeitherCPorNP.Proof.IfthereisanumberingofNPthenthereisoneofCP,becauseif;2CP(asisalwaysthecase)wemaysimplyaddanindexof;tothenumbering.Thusitsucestoshowthatthereisnoe ectivenumberingofCP.Supposetothecontrarythate7![Te]enumeratesthefamilyof01classesinCP.By(iii),wemayassume[T0]6=;.By(i),T0hasinnitelymanydeadends.LetthedeadendsofT0belistedinacomputableway(forinstance,bylength-lexicographicorder),asn,n2!.By(i)again,wemaylet nbetheleastextensionofnwhichextendsadeadendofTn.DeneacomputabletreeTbyputtingT0above n.Thatis,let[T]\[ n]=[ nT0]and[T]=[T0][Sn[ nT0].By(ii),theresultingclass[T]belongstoCP.Since[T0]6=?,[T]\[ n]6=;=[Tn]\[ n],so[T]isnotcontainedinorequaltoany[Tn].Allassumptions(i),(ii),(iii)ofTheorem8arenecessary:considerP=2!,P=fxg,wherexisasinglecomputablereal,andP=;,respectively.Corollary1.Thefollowingfamiliesof01classeshavenoe ectivenumbering:1.01classescontainingonlyMartin-Lofrandomreals;2.special01classes(thosecontainingonlynon-computablereals);3.01classescontainingonlyrealsxsuchthattheMuchnikdegree[7]offxgisaboveaxednonzeroMuchnikdegree;4.01classescontainingonlynite(oronlyco-nite)subsetsof!.Proposition2.(1)Thefamilyofall01classescontainingonlyrealsthatareMartin-Lofrandomwithrespecttoaxedrandomnessconstantise ectivelyenumerable. Numberingsandrandomness9Theorem10.Anye ectivelyenumerablefamilyof01classesFwithFf[O]:O2AghasaFriedbergnumbering.Proof.ForasetZ2,wesaythatZislterclosedifZisclosedunderextensions(2Z)_ 2Z)andsuchthatwheneverboth_0and_1areinZthen2Z.ThelterclosureofYistheintersectionofalllterclosedsetscontainingYandisdenotedbyY".SinceFise ectivelyenumerable,wemaylete7!Yebeanumberingofalllterclosedsetsofstringswith[Ye]2F.SinceYe6=Ye0implies[Ye]6=[Ye0],itsucestoinjectivelyenumeratethesesetsYe.LetL1=O":O2A andL2=fYe:Ye62L1g:ItisclearthatL1isinjectivelyenumerable.Bytheassumptionofthethe-orem,each[O"]2F.ItisalsoclearthateachnitesubsetofanyY2L2iscontainedininnitelymanyO"2L1.WeclaimthatL2hasane ectivelyenumerationfYege2!,tobeconstructedbelow.FixeandletYe=fngn2!inorderofenumeration.S2isanacceptablefamilyifitsoptimalcoveringOhasniteevencardinality.InparticularO62A.WesaythatstagenisgoodifnhasgreaterlengththananymemberofOn=OSnforSn=f0;:::;n�1ganddoesnotextendanymemberofOn.Construction.WewillconstructYeasYe=Sn2!Ye;nforuniformlycomputablesetsYe;n.WesetYe;�1=?.Supposen0.Ifstagenisnotgood,wekeepYe;n=Ye;n�1.Ifstagenisgood,therearetwocases.Casea.Snisanacceptablefamily.ThenletYe;nbethelterclosureofOn.Caseb.Otherwise.ThenletYe;nbethelterclosureofOn[fng.Weseparatelyenumerateallsetsgeneratedfromanyacceptablefamilywhoseoptimalcoveringhasniteevencardinality.(*)EndofConstruction.Verication.NotethatinbothCaseaandCaseb,Ye;nisthelterclosureofanacceptablefamily,sowedonotenumerateanymemberofL1.By(*),itthereforesucestoshowthatweenumerateallsetsgeneratedfromaninnitefamily,i.e.non-clopensets,andthateachYeissomeYe0.IfYeisnotclopenthenthereareinnitelymanygoodstages.ThenintheendYe=Ye,becauseniscoveredeitherrightaway(caseb)oratthenextgoodstage(caseb).Corollary3.Thefamilyofall01classesofmeasurelessthanone,orequiva-lently01classesofpositivemeasure,hasaFriedbergnumbering.AcknowledgmentsThesecondauthorwaspartiallysupportedbyNSFgrantDMS-0652669. 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