DegreesQuestionsAgainExamplesDMaximalSetsClassi2cationQuestionsYetAgainMainOpenQuestionAgainLookattheindexsetofallbAintheorbitofAwiththehopesof2ndingsomeanswersTheindexsetofsuchbAisin611QuestionCompl ID: 872707
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1 Questions Degrees QuestionsAgain Example
Questions Degrees QuestionsAgain Examples D-MaximalSets Classication QuestionsYetAgainMainOpenQuestion,AgainLookattheindexsetofallbAintheorbitofAwiththehopesofnd
2 ingsomeanswers.TheindexsetofsuchbAisin&#
ingsomeanswers.TheindexsetofsuchbAisin11. Question(Completeness)Whichc.e.setsareautomorphictocompletesets? Theorbitsintheprevioustheoremallcontaincompletesets. Quest
3 ionIstheabovequestionanarithmeticalquest
ionIstheabovequestionanarithmeticalquestion? Questions Degrees QuestionsAgain Examples D-MaximalSets Classication QuestionsYetAgainPromptSets DenitionAispromptly
4 simpleithereiscomputablefunctionpsu
simpleithereiscomputablefunctionpsuchthatforalle,ifWeisinnitethenthereisaxandswithx2„We;ats\Ap„s……. DenitionAc.e.setAispromptithere
5 iscomputablefunctionpsuchthatforalle,ifW
iscomputablefunctionpsuchthatforalle,ifWeisinnitethenthereisaxandswithx2We;atsandAsux”Ap„s…ux. Theorem(Cholak,Downey,Stob)Allpromptsimplesetsareautomo
6 rphictoacompleteset. Questions Degrees Q
rphictoacompleteset. Questions Degrees QuestionsAgain Examples D-MaximalSets Classication QuestionsYetAgainDegreesofn-TardysetsSplitsof2-tardysare3-tardybut TheoremT
7 hereisa3-tardythatisnotcomputableinany2-
hereisa3-tardythatisnotcomputableinany2-tardy. QuestionIsthereann‚1-tardysetthatisnotcomputedbyanyn-tardyset?Isthereaverytardysetswhichisnotcomputedbyanyn-tardy? Qu
8 estions Degrees QuestionsAgain Examples
estions Degrees QuestionsAgain Examples D-MaximalSets Classication QuestionsYetAgainMainOpenQuestions,Localized Question(D-MaximalCompleteness)DoestheorbitofeveryD-m
9 aximalsetcontainacompleteset? QuestionIs
aximalsetcontainacompleteset? QuestionIstheabovequestionanarithmeticalquestion? QuestionWhyD-maximal? Questions Degrees QuestionsAgain Examples D-MaximalSets Classic
10 ation QuestionsYetAgainFriedbergSplitsof
ation QuestionsYetAgainFriedbergSplitsofMaximalSets DenitionS1tS2ƒMisaFriedbergsplitofM,forallW,ifW&MisinnitethenW&Siisalso. Lemma(Downey,Stob)Anontrivialsp
11 litofamaximalset(ahemimaximalset)isaFrie
litofamaximalset(ahemimaximalset)isaFriedbergSplit. Theorem(Downey,Stob)TheFriedbergsplitsofthemaximalsetsformadenableorbit.Thisorbitincludesacompleteset. Proof.LetM
12 1tM2ƒM.IfW[Mƒ!thenW&Misin&
1tM2ƒM.IfW[Mƒ!thenW&Misinnite. „W…ƒ� 1„RW\M1…\ 2„RW\M2…t1„RW\W\M1…t1„RW\W\M2&#
13 133;. Questions Degrees QuestionsAgain E
133;. Questions Degrees QuestionsAgain Examples D-MaximalSets Classication QuestionsYetAgainHerrmannSets DenitionAsetHisHerrmanniforallc.e.setsDdisjoin
14 tfromHthereisacomputablesetRsuchthatRisd
tfromHthereisacomputablesetRsuchthatRisdisjointfromH,DR,andR�Disinnite.forallc.e.setsBthereisac.e.setDdisjointfromHsuchthateitherBH\BorB[D[Hƒ!
15 . Theorem(Cholak,Downey,andHerrmann)TheH
. Theorem(Cholak,Downey,andHerrmann)TheHerrmannsetsformadenable03orbitcontainingacompletesetdierentfromthemaximalandhemimaximalsets. Thereisa03listofdis
16 jointcomputablesetsRisuchthatforallBdisj
jointcomputablesetsRisuchthatforallBdisjointfromHthereisannwithBFinRi. Questions Degrees QuestionsAgain Examples D-MaximalSets Classication QuestionsYetAga
17 inComplexityRestrictions Theorem(Cholak,
inComplexityRestrictions Theorem(Cholak,Harrington)IfAisD-hhsimpleandAandbAareinthesameorbitthenED„A…›03ED„bA…. Doesnotprovideananswertothef
18 ollowing: Question(D-maximalCompleteness
ollowing: Question(D-maximalCompleteness)WhichD-maximalsetsareautomorphictocompletesets? QuestionIstheabovequestionanarithmeticalquestion? QuestionCanweclassifytheD-maxi
19 malsets? Questions Degrees QuestionsAgai
malsets? Questions Degrees QuestionsAgain Examples D-MaximalSets Classication QuestionsYetAgainTheClassicationD-maximalsets DenitionForaD-maximalset,alistofc
20 .e.setsfXigi2!generatesD„A…i&#
.e.setsfXigi2!generatesD„A…iforallDifDisdisjointfromAthenthereisansuchthatDSinXi.(Thislistneednotbecomputable.) Lemmaf;ggeneratesD„
21 ;A…iAismaximal.Foranycomp
;A…iAismaximal.ForanycomputablesetR,fRggeneratesD„A…iAismaximalon R.Foranynoncomputablec.e.setW,fWggeneratesD„A…iAishem
22 imaximal.Inallothercases,thelistofg
imaximal.Inallothercases,thelistofgeneratorsisinnite. Questions Degrees QuestionsAgain Examples D-MaximalSets Classication QuestionsYetAgainNestedNoncomputa
23 bleGenerators DenitionHisr-maximali&
bleGenerators DenitionHisr-maximalinocomputablesetsplits Hintotwoinnitesets.HisatomlessiHhasnomaximalsuperset. TheoremThereareD-maximalsplitsofato
24 mlessr-maximalsets.TheclassofD-maxi
mlessr-maximalsets.TheclassofD-maximalsplitsofatomlessr-maximalsetsbreaksintoinnitelymanyorbits.(Usesasimilarresultaboutatomlessr-maximalsetsbyCholakandNies.)&#
25 15;AisaD-maximalsplitofatomlessr-maximal
15;AisaD-maximalsplitofatomlessr-maximalsetithereisalistofnestednoncomputablegeneratorsfXigi2!forD„A…,i.e.forallithenXiXi‚1. QuestionDotheseorbi
26 tscontaincompletesets? Questions Degrees
tscontaincompletesets? Questions Degrees QuestionsAgain Examples D-MaximalSets Classication QuestionsYetAgainAnti-FriedbergSplits DenitionA0isananti-Friedbergspl
27 itofAithereisanA1suchthatA0tA1ƒ
itofAithereisanA1suchthatA0tA1ƒAand,forallW,eitherW�A0isc.e.orW[Aƒ!. LemmaAllD-maximalsplitsofmaximal,hhsimple,Herrmannandatomlessr-maximalsetsarea
28 nti-Friedberg.Allanti-Friedbergsplitsare
nti-Friedberg.Allanti-FriedbergsplitsareD-maximal. QuestionDotheanti-Friedbergsplitsofaset(orbit)formanorbit? Questions Degrees QuestionsAgain Examples D-MaximalSets Cla
29 ssication QuestionsYetAgainMainOpenQ
ssication QuestionsYetAgainMainOpenQuestions,Localized Question(D-MaximalCompleteness)DoestheorbitofeveryD-maximalsetcontainancompleteset? QuestionIstheabovequestion