Denabilityintherecursivelyenumerabledegrees

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1 Denabilityintherecursivelyenumerabledegrees CornellUniversity,IthacaNY14853USA UniversityofChicago,ChicagoIL60637 TheodoreA.Slamanz RichardA.Shorex AndreNiesy 1.Introduction Naturalsetsthatcanbeenumeratedbyacomputablefunction(therecursively UniversityofChicago,ChicagoIL60637 sameturingdegreeask? enumerableorr.e.sets)alwaysseemtobeeitheractuallycomputable(recursive) [1944]andsincethencalledPost'sProblemisthenjustwhethertherearer.e. setswhichareneithercomputablenorcomplete,i.e.,neitherrecursivenorofthe Problem,thecompleter.e.setK.Theobviousquestion,rstposedinPost orofthesamecomplexity(withrespecttoturingcomputability)asthehalting 0,thedegree(equivalenceclass)ofthecomputablesets,andgreatestelement1 structureisapartialorder(indeed,anuppersemilatticeorusl)withleastelement ofequicomputablewiththepartialorderinducedbyturingcomputability.this inmadison,march1996inalecturegivenbythesecondauthor. LetRbether.e.degrees,i.e.,ther.e.setsmodulotheequivalencerelation throughmsi,cornelluniversity,daal-03-c UniversityofSiena. xpartiallysupportedbynsfgrantsdms ,dms ,dms andaro ypartiallysupportedbynsfgrantdms MostofthematerialinthispaperwaspresentedtotheAssociationatitsannualmeeting zpartiallysupportedbynsfgrantdms andacnrvisitingprofessorshipatthe

2 or00,thedegreeofk.post'sproblemthenasksifthereareanyotherelements ofr. ablepartialorderingorevenuppersemilatticecanbeembeddedintor. [1956]wasfollowedbyvariousalgebraicorordertheoreticresultsthatwereinterpretedassayingthatthestructureRwasinsomewaywellbehaved: The(positive)solutionofPost'sproblembyFriedberg[1957]andMuchnik Theorem1.1.(Embeddingtheorem;Muchnik[1958],Sacks[1963])Everycount- e.degreesa<bthereisanr.e.degreecsuchthata<c<b. degreeatherearer.e.degreesb;c<asuchthatb_c=a. Theorem1.2.(SacksSplittingTheorem[1963b])Foreverynonrecursiver.e. Theorem1.3.(SacksDensityTheorem[1964])Foreverypairofnonrecursiver. andeveryembeddingf:p!r,thereisanextensiongofftoanembeddingof moreformally,acountablysaturated)uslwithleastandgreatestelements: Conjecture1.4.(Shoeneld[1965])ForeverypairP,!Qofniteuslswith0;1 was\nice"asthesweepingconjecturethatther.e.degrees,r,area\dense"(or QintoR. TheseresultsledShoeneldin1963toformulatetheviewthatthestructure neous(everystructurepreservingmapfromonenitesubsettoanothercanbe algebraswhichsatisfythecorrespondingpropertyfortheappropriatefamilyof thefamiliarpropertiesofstructureslikedenselinearorderingsoratomlessboolean andso(ifaxiomatizable)havedecidabletheories.theyarecountablyhomoge- categorical(i.e.,thereisauniquesuchcountablestructureuptoisomorphism) structures(linearorderingsandbooleanalgebras).suchstructuresarecountably Iftrue,thisconjecturewouldhaveimpliedthatther.e.degreeshadmanyof degrees. constitutedanessentiallycompletecharacterizationofthestructureofther.e. ofthestructure.apositivesolutiontoshoeneld'sconjecturewouldthushave c>0whichisbelowbothaandb.thustheconstructionofaminimalpairsof extendedtoanautomorphism)andsotherearecontinuummanyautomorphisms r.e.degrees,i.e.,nonzeror.e.aandbsucha^b=0,refutedtheconjecture. Theconjectureclearlyimplies,forexample,thatforanya;b>0thereisa 2

3 recursive. greesaandbsuchthata^b=0,i.e.,anydegreerecursiveinbothaandbis Theorem1.5.(Lachlan[1966],Yates[1966]):Therearenonrecursiver.e.de- tother.e.degrees". senseofhomogeneityforthenotionofr.e.inthesensethat\foreach(notnecessarilyr.e.)degreed,theorderingofdegreesr.e.indanddisorderisomorphic toconjecturein[1966]thatthetheoryofrisdecidableandthatthereisastrong turedin[1963]thattherewereminimalpairsandthatrisnotalattice,continued byshoeneld'sconjecturecontinuedtoholdsway.evensacks,whohadconjec- Manyothercounterexamplesfollowed.Nonetheless,theparadigmsuggested Shelah[1982];Shore[1982])andintheinterveningyearstherecontinuedtobea growinglistofexamplesofvarioustypesofdegreesandexamplesofcomplexity inthestructure: nonzerobranchingdegrees(nontrivialinma)andnonbranchingdegrees Bothoftheseconjectureseventuallyturnedouttobefalse(Harringtonand cappabledegrees(halvesofminimalpairs)andnoncappabledegrees(yates (Lachlan[1966]); cuppabledegrees,i.e.,thosewhichjoin(cup)to00,(bythesackssplitting alldistributivenitelattices(lachlan,lerman,thomason;seesoare[1987, Soare[1980]); beddableinrbutnotallnitelatticesaresoembeddable(lachlanand degreeswhichsplitovereverysmallerdegree(anylowdegreea,i.e.,a0=00, theorem)andnoncuppabledegrees(lachlan[1966a]); p.157])andthetwobasicnondistributivelattices(lachlan[1972])areem- degreesoverwhich00splits(anylowdegreebyrobinson[1971])anddegrees degreeswhichboundparticularlattices(lachlan[1972])anddegreesthat [1990])anddegreeswhichdonot(Lachlan[1975]); overwhichitdoesnot(harrington;jockuschandshore[1983]); byrobinson[1971],anylow2degreea,i.e.a00=000,byshoreandslaman donot(weinstein[1988],downey[1990]); 3

4 notuntilsometwentyyearaftertherefutationofshoeneld'sconjecturethata dramaticallydierentviewofthestructureofther.e.degrees(aswellasofthe ::: complexitywouldcompletelycharacterizethestructure. tocharacterizingther.e.degrees,itsuggeststhatasucientlystrongproofof complexityofthestructurebut,ratherthanseeingthiscomplexityasanobstacle degreesasawhole)becametheprevailingparadigm.thisviewstartsfromthe Theseresults,andthestructureitself,wereoftenviewedaschaoticanditwas degreestolookthesameandfortheretobemanyautomorphisms,onecould looktoprovethatthetheoryisascomplicatedaspossible,thereareasmany dierenttypesofdegreesaspossible(eventhatnotwoarealikebutrathereach isdenable)andthatthestructurehasnoautomorphisms. Insteadofexpectingthestructuretobedecidableandhomogeneous,forall crystallizedthenewparadigmofcomplexityasaroutetocharacterization: thestudyofr.therstuseddenablerepresentationsofpartialorderingsand thesecondembeddingsofnitelygeneratedpartiallattices.itistheultimate Shelah[1982])andhomogeneity(Shore[1982])introducedcodingtechniquesinto expressionofsuchcodingproceduresthatisembodiedintheconjecturethat TherstrefutationsofSacks'sconjecturesaboutdecidability(Harringtonand Woodin;seeSlaman[1991]):Thereisadenablecodingofastandardmodelof Conjecture1.6.(BiinterpretabilityConjectureforR,Harrington;Slamanand arithmetic,n0,inrforwhichtherelationassociatingeachr.e.degreedtothe (codesinthemodelof)setsofitsdegreeisalsodenable. fromrinton0.however,inthedegreesasawholeandeveninconsidering givenistheappropriateoneingeneralsettings.thispointwillbediscussed inn0forasetofthatdegreeoreventothedenabilityofanyone-onemap isequivalenttothedenabilityofamaptakingeachr.e.degreetoanindex relativizationsofr,simpleindicesforsetsofthedegreesbeingconsideredarenot usuallyavailableandothercodingsforsetsmustbeused.thustheformulation (InthecontextofjustthestructureR,thedenabilityoftherelationdescribed completeinformation,forexample,aboutdenabilityinr(everydegreeinr furtherinx2.) wouldbedenableaswouldeveryrelationonrwhichisdenableinarithmetic) providesastrongcharacterizationofthestructureofr.iftrueitwouldgive Morethansimplysayingthatther.e.degreesarecomplicated,thisconjecture 4

5 andautomorphismsforr(noneotherthantheidentitywouldexist).(clearly ifwecandenablyrelateeachdegreetothesetsofthatdegree,therecanbe mappingfromrtothestandardmodelofarithmeticandtranslatethedenitions inarithmetic.) 2.ResultsandRelativizations noautomorphismofr.asforthedenabilityclaims,justusethedenable (Corollary2.4)arethensimilartothosedescribedfromthefullconjecturebut preciseinthetheorembelow,ourresultsarewithintwojumpsoftheconjecture. Thecorollariesthatwecanderiveaboutrigidity(Corollary2.3)anddenability inthedirectionofprovingthebiinterpretabilityconjecture.inasensemade Cooper[1996]hasannouncedtheexistenceofanautomorphismofRandhence only\uptotwojumps": thefailureofthebiinterpretabilityconjecture(aswediscussfurtherinx4).on theotherhand,theresultswearereportingonhereshowhowfarwehavecome Theorem2.1.InRthereisadenablecopyN0ofthestructure(N;+)and adenablerelationassociatingeachdegreeawithcodesforsetsofdegreea00. Indeed,thereisadenablemapf:R!N0suchthat,foreverya,f(a)is(the codefor)theleastindexofanr.e.setwforwhichw002a00. Denition2.2.Ann-aryrelationP(x1;:::;xn)onRisinvariantunderthe doublejumpif,wheneverrj=p(x1;:::;xn)andx00 truethatrj=p(y1;:::;yn).pisinvariantinrifwheneverrj=p(x1;:::;xn) and'isanautomorphismofr,rj=p('(x1);:::;'(xn)).pisdenablein arithmeticifthesetofn-tuplesofindicesofr.e.setswhosedegreessatisfypis Thefollowingnotionshelpmaketheideaof\uptotwojumps"precise. denablein(n;+;). Thefollowingcorollariesaboutdenability(exceptforthelastone)allfollow 1Ty00 1;:::;x00nTy00 n,itisalso (Therstone,althoughalsoformallyaconsequenceoftheTheorem,isactually inarithmetic(onindices)toonesinn0andthenusingthedenablefunction aningredientinitsproof.) immediatelyfromthetheorembysimplytranslatingtheappropriatedenitions fgivenbythetheoremtoassociatetheindiceswiththecorrespondingdegrees. 5

6 Corollary2.3.AnyrelationonRwhichisinvariantunderthedoublejumpis invariantinr. Corollary2.4.AnyrelationonRwhichisdenableinarithmeticandinvariant underthedoublejumpisdenableinr. Corollary2.5.Foreachk2therelationxskydenedbyx(k)Ty(k)is denableinr. Corollary2.6.Foreachcr.e.inandabove000,thesetofr.e.degreesawith doublejumpcisdenableinr. Corollary2.7.ThejumpclassesLn=faja(n+1)=0(n+1)g(thelown+1degrees) andhn=faja(n)=0(n+1)g,(thehighndegrees)aredenableinrforn2. Corollary2.8.ThejumpclassH1=faja0=000g(thehighdegrees)isdenable inr. Proof(ofCorollary2.8):ItfollowsfromtheRobinsonJumpInterpolationTheorem[1971]that,forxr.e.,x0=000ifandonlyifforeverycr.e.inandabove000 thereisab<xwithb00=c.aseverysuchcisa00forsomer.e.abythesacks JumpTheorem[1963a],H1=fxj(8a)(9b<x)(as2b)gwhilethisclassisclearly relativization.ifzisanarbitrarydegree,wedenotetherelativizationofther.e. denablebycorollary2.5.2 aboutdegreesrelativize.indeedallthestructuralresultsaboutrmentionedin degrees,thestructureofdegreesr.e.inandabovez,byrz.nowalmostallresults x1aretrueineverystructurerz.ontheotherhand,wehavelearnedfromthe Shore[1979],[1982a]andthedegreesbelow00inShore[1981]thatitisprecisely variousrefutationsofsuchhomogeneityprinciplesforthedegreesasawholein BeforedescribingtheproofofTheorem2.1,wewanttodiscusstheissueof torzformostdegreesz. relativizationsthatarepossibletoshowthatrisnotevenelementarilyequivalent relativizemost,butnotall,ofourresultstoeveryrz.indeed,wecanusethe usedinshore[1982]toshowthat,ingeneral,risnotisomorphictorz.wecan homogeneity.inther.e.degrees,codingsandembeddingsofpartiallatticeswere thetypesofresultsthatwehaveestablishedthatleadtocounterexamplesto 6

7 isprescribedbytheorem3.7.thuswemustadjustourdenitionof\denable precisemethodusedtointerpretpairsofdegreesascodesforsetsinn0ornz0 thenotionofacodeforasetcannolongerbeviewedassimplyanindex.the ofcorollary2.3relativizeandsothendoestherstversionoftheorem2.1and thedoublejumparethesameforrzasforr.however,aswementionedbefore, almostallthecorollariesmentioned.thenotionsofinvariantandinvariantunder Allthetechnicallemmasdiscussedinx3leadinguptoandincludingtheproof inarithmetic"accordingly.wenowallowfreesetvariablesinourformulas andtheusualbinaryrelationsymbol2formembership(i.e.,themembershipof degreesisthensaidtobedenableinarithmeticifthereissuchaformula adegreecodinganaturalnumberinthesecodedsets).ann-aryrelationpon thatp=fhdeg(x1);:::;deg(xn)ijnj= withthepreviousdenitionwhenallthesetsxiarer.e.) Theorem2.9.Foreverydegreez,thereisadenablecopyNz0ofthestructure zwithcodesforsetsofdegreea00. (N;+)inRzandadenablerelationassociatingeachdegreear.e.inandabove (X1;:::;Xn)g.(Ofcourse,thisagrees such Corollary2.10.Foreverydegreez,anyrelationonRzwhichisinvariantunder thedoublejumpisinvariantinrz. Corollary2.11.Foreverydegreez,anyrelationonRzwhichisdenablein inr. Corollary2.12.Foreverydegreez,andforeachk2therelationxsky arithmetic(asredenedabove)andinvariantunderthedoublejumpisdenable denedbyx(k)ty(k)isdenableinrz. Corollary2.13.Foreverydegreez,thejumpclassesLzn=faja(n+1)=z(n+1)g andhzn=faja(n)=0(n+1)garedenableinrzforn2. Corollary2.14.Foreverydegreez,thejumpclassHz1=faja0=000gisden- tofailureasanyfunctiondenableinrz(andsoarithmetic)takingdegreesdto (unique)representativesofdwouldcontradictarithmeticdeterminacy.thesame istrueevenifwetrytoassociatedegrees(r.e.inandabovez)withintegers(in toindicesorevenanyformofuniquecodesforsetsofgivendegreesisdoomed 2.6donotrelativize.Indeed,anyattemptattalkingaboutmapsfromdegrees thestandardmodelofarithmeticdenedinrz)uptoanyjump: Ontheotherhand,theproofsofthelastpartofTheorem2.1andofCorollary 7

8 Theorem2.15.Therearedegreeszsuchthatthereisnok2!andnomapf fromrztonz0;theisomorphiccopyofndenableinrz,whichisdenablein RzsuchthatakTbkimpliesthatf(a)=f(b). thesetofdegreesinrzwithdoublejumpcisnotdenableinrz. Theorem2.16.Therearedegreeszandcwithcr.e.inandabovez00,suchthat showsthattheanalogofcorollary2.6alsofails: Rz.Theproofagaininvolvesdeterminacyconsiderations.Asimilarargument Thus,ingeneral,noanalogofthesecondpartofTheorem2.1ispossiblefor Theorem2.18.Ifz006000thenRz6R. Theorem2.17.Ifz006w00thenRz6=Rw. elementarilyequivalenttor. formostzandwthestructuresrzandrwarenotisomorphicandarenot Wecan,infact,usetherelativizedresultsabovethatdoholdtoshowthat innw0(n0)withinrw(r)orviceversa. mentaryequivalence)involvecodingsetsinnz0withinrzthatcannotbecoded Asusual,theproperties(sentences)demonstratingnonisomorphism(nonele- jumpind(00)canthenbeusedtogiveanewproofofslamanandwoodin's resultthateverydegreeabove000isxedundereveryautomorphismofd. double)andsoderivesimilarresultsford(00).theinvarianceofthedouble denabilityresultsestablishedthereford(00)byonejump(fromtripleto becombinedwiththemethodsofshore[1988]toimprovetheinvarianceand WealsonotethatthecodingstructuresusedfortheaboveresultsonRcan 3.LemmasandProofs Wewillnowoutlinetheproofoftheseresultsandstatethetechnicallemmas dardmodelofarithmetic,itimmediatelygivesaninterpretationoftruearith- metic,th(n;+;),inr.thus,thetheoryofrisatleastascomplicatedas oftherststepsalongtheroadindicatedbythebiinterpretabilityconjectureand twotheorieshavethepreciselysamedegree.)itisnotsurprisingthenthatsome neededalongtheway.sincetheorem2.1includesthedenabilityofastan- Th(N;+;).(Indeed,asthestructureRisobviouslydenableinarithmetic,the 8

9 itsundecidability(harringtonandshelah[1982];slamanandwoodin;ambos- actuallyleadingtoourresultwerethecodingofarithmeticintorusedtoprove SpiesandShore[1993]).ItwasevenknownthatthetheoriesofRandNwere sivetranslationss(t)takingsentences( biinterpretable: Theorem3.1.(HarringtonandSlaman;SlamanandWoodin)Therearerecur- provideatranslationofthetheoryofpartialorderingsintor.asthetheoryof codingsofpartialorderingsinrdevelopedtoproveitsundecidability.theyeach sentencess; andrj= Eachproofofthistheorem(includingournewone)beginswithoneofthe $Nj= Tofpartialorderings(arithmetic)suchthatNj=$Rj=S T. )ofarithmetic(partialorderings)to partialorderingsisrichenoughtocodeallofpredicatelogic,wecanviewthe trueinthosemodelswhichareisomorphiccopiesofn,thestandardmodelsof codingsasprovidinguswithmodelsofsomeniteaxiomatizationofarithmetic. Therealproblem,now,istodenablydeterminethe(translationsof)sentences standardmodeloratleastaclassofmodelsallofwhicharestandard.onewould arithmetic.themostnaturalapproachtothisproblemwouldtobetodenea do.ourapproachbeginswithslamanandwoodin'scodingofpartialorderings: thensimplysaythatasentenceofarithmeticistrue(inn)itheappropriate Theorem3.2.(SlamanandWoodin):GivenanyrecursivepartialorderingP= theorem.thustheyinterpretedthetheoryofnbutnotthestructureitselfaswe translationistruein(anyof)thedenablestandardmodel(s).theproofsof h!;itherearer.e.degreesp;q;r;landgi(fori2!)suchthat thistheorembyharringtonandslamanandlaterbyslamanandwoodindidnot managetodenestandardmodelsandtookmuchmoreindirectapproachestothe 2.fori;j2!,ijifandonlyifgilTgj; 1.thegiaretheminimaldegreesxrsuchthatqx_p; andwoodin'sworkthatweneedlater.)9 3.rpqislow,i.e.(rpq)0=00 4.Ifa>0isanygivenr.e.degree,wecanalsomaker<a. (Parts3and4arerelativelystraightforwardtechnicalimprovementsofSlaman

10 structurebeingcoded.)thekeyweusetodenablyselectasetofsuchmodels modelof(anitelyaxiomatizedversionof)arithmetic.(wereferthereaderto Hodges[1993,5.3]forprecisedenitionsofwhatitmeanstodenablycode(oras hesays,interpret)onestructureortheoryinanother.roughlyspeaking,itmeans togiveasequenceofformulaswhichdenerstthedomainofthecodedstructure andthenthevariousrelationsandfunctionsonitthatprovidethe\copy"ofthe Asexplainedabove,wearethinkingofthepartialorderingPascodinga thatareallstandardistheabilitytouniformlydenecomparisonmapsbetween aninitialsegmentofeverymodel.)thecrucialtechnicallemmaneededtodene andpermitting: Theorem3.3.GivenanyrecursivepartialorderingP=h!;iandlowr.e. suchmapsisonethatcombinesslamanandwoodincodingwithconeavoiding modelsarethemodelsmsuchthateachinitialsegmentofmcanbemappedinto degreesq0;:::;qm;r0;r1therearer.e.degreesp;q;r;landgi(fori2!)asin (nite)initialsegmentsofcertainsuchmodels.(theideahereisthatthestandard iinthemodelcodedbyp,thengf(i)tqiandqi6tqj)gf(i)6tqjfor Theorem3.2suchthatifgf(i)isthedegreecorrespondingtothenaturalnumber betweentherstnelementsofm1andthoseofm2. withthestructureinherentinm,thesemapsdenethedesiredisomorphism i;j<mwhilegf(k)6tr0;r1fork>m. modelsarelow,weusethistheoremtointerpolateathirdmodelmsothatwe betweentherstnnumbersofm2andthesecondnnumbersofm.together candeneisomorphismsbetweentherstnnumbersofm1andthoseofmand Wenowgiveasucientconditionforamodeltobestandardandindicate GiventwocodedlowmodelsM1;M2,i.e.,allthedegreesinthedomainofthe initialsegmentofeverymodelwhoseelementsarebelowcbytheschemedescribed elementsallbelowsomecisgoodwithrespecttocifmcanbeembeddedintoan class: Denition3.4.AmodelMofarithmetic(codedbyparametersp;q;r;l)with Thuswecandeneaclassofmodelswhichareallstandardandsuchthatthere aredenableisomorphismsbetweenthenaturalnumbersofanytwomodelsinthe howtogetadenableschemeformapsbetweeninitialsegmentsofsuchmodels. above. isstandard(asitcanbemappedintosomestandardmodel).moreover,givenany Now,everymodelallofwhoseelementsarelowisgoodandeverygoodmodel 10

11 twogoodmodelswecandeneanisomorphismbetweenthembyinterpolatingtwo otherlowmodels.thuswecandeneanequivalencerelationonthe(codesfor) ofarithmetic. Theorem3.5.ThereisacodingschemeinterpretingarithmeticinRsuchthat naturalnumbersinthesemodelsandinterpretationsofthelanguageofarithmetic ontheseequivalenceclassesthatmakethestructuresodenedastandardmodel allthemodelssodenedarestandard.moreover,thereisadenableequivalence relationontheparameterscodingthesemodelsandthedegreescodingthenatural numbersinthesemodelssuchthatthecodingschemedenesastandardmodel N0ofarithmeticontheequivalenceclasses. parametersandthentranslatethischaracterizationofisomorphismtypeintoour modelofarithmetic. typeofr(a)(theorderingofr.e.degreesbelowa)relativetocertainother conjecture.wenextwanttocomeascloseaswecantoassociatingeachdegreea withsomekindofcode(orevenastandardr.e.index)forsetsofthatdegree.the ideaistorstcharacterize,totheextentpossible,adegreeabytheisomorphism WenowhavethedenablecopyN0ofNinRrequiredbythebiinterpretability uralnumberscanbeenumeratedrecursivelyin000.theparticularmethodof thesuccessorfunctionsothatitis3inthesensethatthe(codesfor)thenat- generatingsuchstructuresistakenfromshore[1981]. Theorem3.6.Givenanya>0andanynoncappableu,therearedegreesb,e0, e1,f0,f1,p,q,r,landuniformlyr.e.degreesgi(fori2!)withp;q<uand TherstingredientisacodingschemeforacopyofNwhichecientlycodes alltheotherdegreesbelowbothaandusuchthat theminimaldegreesx,b<x<rsuchthatqx_ptogetherwith standardmodelofarithmeticasdescribedabovewiththegiasthe thepartialorderingonthemdenedbyxy,xlydenea (Ambos-Spiesetal.[1984]).) thecharacterizationofthenoncuppabledegreesasthepromptlysimpledegrees (Inadditiontotheconstructionobviouslyneedtoprovethistheorem,weuse foreachi2!,(g2i_e1)^f1=g2i+1and(g2i+1_e0)^f0=g2i+2. elementsi; 11

12 ingthersteightdegreesandg0asparameters.forexample,g1=(g0_ isrecursiveonindiceswecanmakethisgeneratingprocedurerecursivein000by e1)^f1andsog1istheonlydegreexsuchthat1(x)holdswhere1(x)says 9x(1(x)&yx_e0&yf0&qy_p).Similarly,wecandeneeachgiby suchaformula.astheorderingofturingreducibilitytoanysetbisb3andjoin xg0_e1&xf1&qx_p.next,g2istheonlydegreeysuchthat Giventheseproperties,eachgicanbedenedbyanexistentialformulaus- degreesbelowaisa3(andjoinisrecursiveonindices)thiswouldmaketheset bydegreesbelowa.itsproofusesmethodsfromnies[1992].astheorderingon choosingutobelow. Theorem3.7.Ifhgiji2!iisauniformlyr.e.antichaininR,giislow, codeda3aswell(andnothingbetterispossible). A3setcanbecodedonsuchasetofdegreesgiinapositivewayusingand_ Thenextingredientinthedesiredcodingisaprocedurethatshowsthatevery a=deg(a)anda6tgiforeachi2!,then,foreacha3sets,therearec;da suchthats=fijctgi_dg. fora)setofdegreea00inourstandardmodelandsoanisuchthatw00 convertthischaracterizationofa00toaformuladeningfromthedegreeaa(code wecantranslatethecodingsintocodingsinourdenablestandardmodelandso amenabletothecomparisonsdescribedabovebetweenourmodelsofarithmetic, typeofainrdeterminesa00.thisprovescorollary2.3.asthecodingschemeis way.asthisclassofsetsdeterminesa00,wehaveshownthattheisomorphism Together,theseresultsshowthatpreciselytheA3setscanbecodedinthis pleandthentriplejumpclassesandhopesofcharacterizingmuchmore.however, provestheorem2.1andsoalsocorollaries2.4{ ProblemsandConjectures Atvariousearlierpointsinourworkwehadschemesfordeningrstthequadru- i2a00.this denableinr.clearly,theexistenceofsuchanautomorphismimpliesthatour denabilityresultisthebestpossible.givensuchresults,itiseasytolistthe nextquestionsalongtheselines.hereareafewpossibilities: thatmovesalowdegreetoanonlowdegreesothattheclassoflowdegreesisnot Cooper[1966])thathehadconstructedanautomorphismofRandindeedone evenbeforewegotasfardownasthedoublejumpclasses,cooperannounced(see 12

13 Perhaps(indeed,presumably)thereareonlycountablymanyautomorphismsofR. Conjecture4.1.(BiinterpretabilityforRwithparameters):Therelationassociatingeachr.e.degreedtothe(codesinNof)setsofitsdegreeisdenablein Rfromparameters. appealingconjectureistoweakenthebiinterpretabilityconjecturebyallowing PerhapsnoindividualdegreeisdenableinR(andonecouldconstruct automorphismstoprovethis). easytoformulate,itisnotatallclearyetwhatnewvisionwemightadopt.one sosuggeststhatitistimeforanewparadigm.whileindividualproblemsare evenoneindividualdegreeeachcontradictsthebiinterpretabilityconjectureand Perhapseachautomorphismisdenableinsomeniceway. Ofcourse,theexistenceofautomorphismsofRandthenondenabilityof imageoftheparametersdeningtherequiredrelationormap.italsoimpliesthat atmostcountablymanyautomorphismsofraseachwouldbedeterminedbythe thattakeseachr.e.degreeatothe(least)indexofanr.e.setofthatdegree. morphismsanddenabilityinr.forexample,itobviouslyimpliesthatthereare Eventhisweakenedformoftheconjecturehasimportantimplicationsforautoabilityfromparametersofanyone-onemapfromRintoNorofthespecicmap Again,intheunrelativizedsetting,thisconjectureisequivalenttotheden- shouldalsopointoutthatslamanandwoodin(seeslaman[1991])haveshown relationsonrasthosethataredenableinarithmeticandinvariantinr.we eachtypeisprincipleinthestructureofrextendedbyconstantsymbolsnaming theseparametersandsothatristheprimemodelofitstheory(withoutthe thatnisbiinterpretablewithparametersinthestructureofalldegreesbelow00 biinterpretability). parameters).(seehodges[1993,p.336].)finally,itcharacterizesthedenable aswellasinthedegreesasawhole(withanappropriatesecondorderversionof arithmeticandinvarianceinr.cooper'sclaimthatl1isnotinvariantimplies thattherstdoesnotimplythesecond.theseconddoesnotimplytherstby ourresults.corollary2.3easilyimpliesthattherearecontinuummanyinvariant Remark:ItisobviousthatdenabilityinRimpliesbothdenabilityin 13

14 denableinarithmeticassolovayhasshownthattheyareboth!+1complete theclassesl!=[lnandh!=[hnareinvariantbycorollary2.3butarenot (seesoare[1987,p.265]).thisanswerstwoquestionsraisedincooper[1996]. 5.Bibliography subsetsofrandsonotallofthemaredenable.morespecically,wenotethat 281, e.degrees,ann.pureandappliedlogic,63,3-37. algebraicdecompositionoftherecursivelyenumerabledegreesandthecoincidence ofseveraldegreeclasseswiththepromptlysimpledegrees,trans.am.math.soc. Ambos-Spies,K.,Jockusch,C.G.Jr.,Shore,R.A.andSoare,R.I.[1984],An Ambos-Spies,K.andShore,R.A.[1993],Undecidabilityand1-typesinther. greesofunsolvability,proc.nat.ac.sci.43, mationcontent,universityofleeds,departmentofpuremathematicspreprint cursivelyenumerabledegrees,ann.pureandappliedlogic49, Series,No.4. Friedberg,R.M.[1957],Tworecursivelyenumerablesetsofincomparablede- Cooper,S.B.[1996],BeyondGodel'stheorem:thefailuretocaptureinfor- enumerabledegrees(researchannouncement),bull.am.math.soc.,n.s.6,79-80.hodges,w.[1993],modeltheory,cambridgeuniversitypress,cambridge, Harrington,L.andShelah,S.[1982],Theundecidabilityoftherecursively Downey,R.G.[1990],Latticenonembeddingsandinitialsegmentsofthere- England. e.case,trans.am.math.soc.275, Jockusch,C.G.Jr.andShore,R.A.[1983].Pseudo-jumpoperatorsI:ther. Hodgesed.,LNMS255,Springer-Verlag,Berlin, recursivelyenumerabledegrees,j.symb.logic31, enumerabledegrees,inconferenceinmathematicallogic,london,1970,w. grees,proc.londonmath.soc.16, Lachlan,A.H.[1972],Embeddingnondistributivelatticesintherecursively Lachlan,A.H.[1966a],Theimpossibilityofndingrelativecomplementsfor Lachlan,A.H.[1966],Lowerboundsforpairsofrecursivelyenumerablede- overalllesserones,ann.math.logic9, Lachlan,A.H.[1975],Arecursivelyenumerabledegreewhichwillnotsplit

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