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CHAPTER 1.INTRODUCTION .........................................1
1.1.Largecardinalsandtheconsistencystrengthhierarchy.........1
1.2.Innermodeltheory.........................................3
1.3.Miceanditerationstrategies.................................5
1.4.HODinmodelsofdeterminacy..............................6
1.5.Leastbranchhodpairs......................................8
1.6.Comparisonandthemousepairorder........................9
1.7.Hodpaircapturing.........................................12
1.8.Constructingmousepairs...................................13
1.9.Thecomparisonargument...................................15
1.10.Planofthebook............................................17
CHAPTER 2.PRELIMINARIES ........................................19
2.1.Extendersandultrapowers..................................19
2.2.Pureextenderpremice......................................21
2.3.Projectaandcores..........................................24
2.4.Elementarityofmaps.......................................33
2.5. rΣk cofinalityandnearelementarity..........................40
2.6.Iterationtreesonpremice...................................49
2.7.Iterationstrategies..........................................55
2.8.Comparisonandgenericityiterations.........................62
2.9.Coarsestructure............................................65
CHAPTER 3.BACKGROUND-INDUCEDITERATIONSTRATEGIES ........71
3.1.Fullbackgroundextenderconstructions......................72
3.2.Resurrectionmaps.........................................78
3.3.AShiftLemmaforconversionstages.........................81
3.4.Conversionsystems........................................85
3.5.Inducediterationstrategies..................................92
3.6.Internalconsistencyforiterationstrategies....................93
3.7.Measurableprojecta........................................95
3.8.Projectawithmeasurablecofinality..........................99
CHAPTER 4.MOREMICEANDITERATIONTREES .....................103
4.1.Micewithprojectum-freespaces.............................104
4.2.Othersoundnesspatterns....................................112
4.3.Elementarityforpremouseembeddings.......................115
4.4.Plustrees..................................................129
4.5.Copymaps,liftedtrees,andlevelsofelementarity.............136
4.6.Iterationstrategiesandcomparison...........................149
4.7.PFSconstructionsandtheirresurrectionmaps.................154
4.8.Conversionsystemsandinducedstrategies....................158
4.9.Backgroundsforplusextenders..............................170
4.10.SolidityinPFSconstructions................................178
4.11.TheBicephalusLemma.....................................199
CHAPTER 5.SOMEPROPERTIESOFINDUCEDSTRATEGIES .............203
5.1.Copyingcommuteswithconversion..........................203
5.2.Positionalityandstrategycoherence..........................208
5.3.Pullbackconsistency........................................211
5.4.Internalliftconsistency.....................................215
5.5.Areductionto λ -separatedtrees.............................219
CHAPTER 6.NORMALIZINGSTACKSOFITERATIONTREES ............221
6.1.Normalizingtreesoflength2................................222
6.2.Normalizing T F ........................................231
6.3.Theextendertree V ext ......................................245
6.4.Treeembeddings...........................................246
6.5.Normalizing T U ..........................................251
6.6.Thebranchesof W (T , U ) ...................................266
6.7.Quasi-normalizingstacksofplustrees........................278
6.8.Copyingcommuteswithnormalization.......................284
6.9.Normalizinglongerstacks..................................290
CHAPTER 7.STRATEGIESTHATCONDENSEANDNORMALIZEWELL ...293
7.1.Thedefinitions.............................................294
7.2.Coarse Γ-Woodinsand Γ-universality........................302
7.3.Stronguniqueiterabilityfrom UBH ..........................307
7.4.Finestrategiesthatnormalizewell...........................317
7.5.Finestrategiesthatcondensewell............................329
7.6.Pureextenderpairs.........................................336
CHAPTER 8.COMPARINGITERATIONSTRATEGIES ....................345
8.1.Iteratingintoabackgroundedpremouse......................346
8.2.Extendingtreeembeddings..................................351
8.3.Resurrectionembeddingsasbranchembeddings...............357
8.4.Iteratingintoabackgroundedstrategy........................360
CHAPTER 9.FINESTRUCTUREFORTHELEASTBRANCHHIERARCHY ..391
9.1.Leastbranchpremice.......................................392
9.2.Leastbranchhodpairs......................................398
9.3.MousepairsandtheDodd-JensenLemma....................400
9.4.Backgroundconstructions...................................405
9.5.Comparisonandthehodpairorder...........................413
9.6.Theexistenceofcores......................................418
CHAPTER 10.PHALANXITERATIONINTOACONSTRUCTION ..........443
10.1.TheBicephalusLemma.....................................443
10.2.ThePseudo-premouseLemma...............................451
10.3.ProofofLemma9.6.5......................................464
10.4.Somesuccessfulbackgroundconstructions...................483
10.5. UBH holdsinhodmice.....................................486
CHAPTER 11.HOD INTHEDERIVEDMODELOFAHODMOUSE .......499
11.1.Genericinterpretability.....................................499 11.2.Mouselimits...............................................501
11.3.HODasamouselimit......................................504
11.4.HODmicesatisfy V = K ...................................513
11.5.Furtherresults.............................................518
PREFACE
Thisbookbeganlifeasalongresearcharticletitled Normalizingiterationtrees andcomparingiterationstrategies.Ifoundthemainideasbehindthecomparison processthatmotivatesitinSpring2015,andcirculatedahandwrittenmanuscript shortlyafterward.IcirculatedapreliminaryformofthepresentbookinApril2016, andhaverevisedandexpandeditmanytimessincethen,asvarioussignificant gapsanderrorsshowedup.Thelastmajorrevisionstookplacein2020-2021.1
Beyondmakingthebookcorrect,oneofmygoalshasbeentomakeitaccessible. Iwasencouragedherebythefactthatthenewdefinitionsandresultsareactually quiteelementary.Theyrestonthetheoryof Finestructureanditerationtrees (FSIT),andcanbeseenascompletingthattheoryinacertainway.Thecomparison theoremforpureextendermicethatisattheheartofFSITisdeficient,inthathow twomicecomparedependsonwhichiterationstrategiesarechosentocompare them.Hereweremedythatdefect,bydevelopingamethodforcomparingthe strategies.Theresultisacomparisontheoremfor mousepairs paralleltotheFSIT comparisontheoremforpureextendermice.Wethenusethecomparisonprocess underlyingthattheoremtodevelopafinestructuretheoryfor strategymice parallel tothefinestructuretheoryforpureextendermiceofFSIT.
Therearepointsatwhichdescriptivesettheoryunderdeterminacyhypotheses becomesrelevant.Atthesepoints,itwouldhelptohavereadthelatersectionsof [65].However,IhaveincludedenoughmaterialthatthereaderfamiliarwithFSIT butshakyondeterminacyshouldbeabletofollowtheexposition.Ourworkhere ismotivatedbytheproblemofanalyzingordinaldefinabilityinmodelsofAxiom ofDeterminacy,buttheprerequisiteforfollowingmostofitisjustinnermodel theoryatthelevelofFSIT.
1Thisisapre-publicationcopyonly.Thefinal,publishedversionofthebookcanbepurchased throughCambridgeUniversityPressandotherstandarddistributionchannels.Thisprepublicationcopy ismadeavailableforpersonaluseonlyandmustnotbesoldorre-distributed.
INTRODUCTION
Inthisbookweshalldevelopageneralcomparisonprocessforiterationstrategies, andshowhowtheprocesscanbeusedtoanalyzeordinaldefinabilityinmodels oftheAxiomofDeterminacy.Inthisintroduction,welookatthecontextand motivationforthetechnicalresultstocome.
Webeginwithabroadoverviewof innermodeltheory,thesubjecttowhichthis bookbelongs.Eventuallywereachanoutlineoftheideasandresultsthatarenew here.Thejourneyisorganizedsothatthetechnicalbackgroundneededtofollow alongincreasesasweproceed.
1.1.Largecardinalsandtheconsistencystrengthhierarchy
Strongaxiomsofinfinity,orastheyaremoreoftencalled,largecardinalhypotheses,playacentralroleinsettheory.Thereareatleasttworeasons.
First,largecardinalhypothesescanbeusedtodecideinanaturalwaymany questionswhichcannotbedecidedonthebasisof ZFC (thecommonlyaccepted systemofaxiomsforsettheory,andhenceallofmathematics).Manysuch questionscomefrom descriptivesettheory,thetheoryofsimplydefinablesetsof realnumbers.Forexample,thehypothesisthatthereareinfinitelymanyWoodin cardinalsyieldsasystematicanddetailedtheoryofthe projective setsofreals,those thataredefinableinthelanguageofsecondorderarithmeticfromrealparameters. ZFC byitselfyieldssuchatheoryatonlythesimplestlevelsofsecondorder definability.
Second,largecardinalhypothesesprovideawayoforganizingandsurveying allpossiblenaturalextensionsof ZFC.Thisisduetothefollowingremarkable phenomenon:foranynaturalextension T of ZFC whichsettheoristshavestudied, thereseemstobeanextension S of ZFC axiomatizedbylargecardinalhypotheses suchthattheconsistencyof T isprovably(in ZFC)equivalenttothatof S.The consistencystrengthsofthelargecardinalhypothesesarelinearlyordered,and usuallyeasytocompare.Thusallnaturalextensionsof ZFC seemtofallinto
ahierarchylinearlyorderedbyconsistencystrength,andcalibratedbythelarge cardinalhypotheses.2
Thesetwoaspectsoflargecardinalhypothesesareconnected,inthattheconsistencystrengthorderonnaturaltheoriescorrespondstotheinclusionorderon thesetoftheir“sufficientlyabsolute”consequences.Forexample,if S and T arenaturaltheoriesextending ZFC,and S hasconsistencystrengthlessthanor equaltothatof T ,thenthearithmeticconsequencesof S areincludedinthose of T .Ifinaddition, S and T haveconsistencystrengthatleastthatof“thereare infinitelymanyWoodincardinals”,thentheconsequencesof S inthelanguageof secondorderarithmeticareincludedinthoseof T .Thispatternpersistsatstill higherconsistencystrengths,withstillmorelogicallycomplicatedconsequences aboutrealsandsetsofrealsbeingbroughtintoauniformorder.Thisbeautifuland suggestivephenomenonhasapracticaldimensionaswell:onewaytodevelopthe absoluteconsequencesofastrongtheory T istocomputeaconsistencystrength lowerbound S for T intermsoflargecardinalhypotheses,andthenworkinthe theory S.Foroneofmanyexamples,theProperForcingAxiom(PFA)yieldsa canonicalinnermodelwithinfinitelymanyWoodincardinalsthatiscorrectfor statementsinthelanguageofsecondorderarithmetic,andtherefore PFA implies allconsequencesoftheexistenceofinfinitelymanyWoodincardinalsthatcanbe statedinthelanguageofsecondorderarithmetic.
Onecanthinkoftheconsistencystrengthofatheoryasthedegreetowhich itiscommittedtotheexistenceofthehigherinfinite.Largecardinalhypotheses maketheircommitmentsexplicitly:theysimplysayoutrightthattheinfinitiesin questionexist.Itisthereforeusuallyeasytocomparetheirconsistencystrengths. Othernaturaltheoriesoftenhavetheircommitmentstotheexistenceoftheinfinite wellhidden.Nevertheless,settheoristshavedevelopedmethodswherebythese commitmentscanbebroughttothesurface,andcompared.Thesemethodshave revealedtheremarkablephenomenondescribedinthelastparagraph,thatnatural theoriesappeartobewellorderedbythedegreestowhichtheyarecommittedto theinfinite,andthatthisdegreeofcommitmentcorrespondsexactlytothepower ofthetheorytodecidequestionsaboutconcreteobjects,likenaturalnumbers,real numbers,orsetsofrealnumbers.
Weshouldemphasizethattheparagraphsabovedescribeageneralpatternof existingtheorems.Therearemanyexamplesofnaturaltheorieswhoseconsistency strengthshavenotyetbeencomputed,andperhapsthey,orsomenaturaltheory yettobefound,willprovidecounterexamplestothepatterndescribedabove. Thepervasivenessofthepatternwhereweknowhowtocompareconsistency strengthsisevidencethatthiswillnothappen.3 Thetwomethodswherebyset
2Let con(T ) besomenaturalformalizationoftheassertionthat T isconsistent.Theconsistency strengthorderisgivenby: S ≤con T iff ZFC provescon(T ) → con(S)
3Thepatternextendstoweaksubtheoriesof ZFC aswell.Thisbookisconcernedonlywith theorieshavingverystrongcommitmentstoinfinity,andsoweshallignoresubtheoriesof ZFC,butthe linearityoftheconsistencystrengthsbelowthatof ZFC isevidenceoflinearityhigherup.
theoristscompareconsistencystrengths,forcingandinnermodeltheory,seemto leadinevitablytothepattern.Inparticular,thewellorderofnaturalconsistency strengthsseemstocorrespondtotheinclusionorderoncanonicalminimalinner modelsforlargecardinalhypotheses.Forcingandinnermodeltheoryseem sufficientlygeneraltocompareallnaturalconsistencystrengths,butatthemoment, thisisjustinformedspeculation.Soonereasonableapproachtounderstanding thegeneralpatternofconsistencystrengthsistodevelopourcomparisonmethods further.Inparticular,innermodeltheoryisingreatneedoffurtherdevelopment, astherearequiteimportantconsistencystrengthsthatitdoesnotyetreach.
1.2.Innermodeltheory
Theinnermodelprogramattemptstoassociatetoeachlargecardinalhypothesis H acanonicalminimaluniverseofsets MH (an innermodel)inwhich H istrue. Thestronger H is,thelarger MH willbe;thatis, G ≤con H ifandonlyif MG ⊆ MH Someofourdeepestunderstandingoflargecardinalhypothesescomesfromthe innermodelprogram.
Theinnermodelswehavesofarconstructedhaveaninternalstructurewhich admitsasystematic,detailedanalysis,a finestructuretheory ofthesortpioneered byRonaldJensenaround1970([16]).Thusbeingabletoconstruct MH givesus averygoodideaastowhatauniversesatisfying H mightlooklike.Innermodel theorytherebyprovidesevidenceoftheconsistencyofthelargecardinalhypotheses towhichitapplies.(Theauthorbelievesthatthiswillsomedayincludeallthe largecardinalhypothesescurrentlystudied.)Sinceforcingseemstoreduceany consistencyquestiontotheconsistencyquestionforsomelargecardinalhypothesis, itisimportanttohaveevidencethatthelargecardinalhypothesesthemselvesare consistent!Noevidenceismoreconvincingthananinnermodeltheoryforthe hypothesisinquestion.
Thesmallestofthecanonicalinnermodelsistheuniverse L ofconstructiblesets, isolatedbyKurtG¨odel([14])inhis1937proofthat CH isconsistentwith ZFC.It wasnotuntilthemid1960’sthatJ.SilverandK.Kunen([57],[23])developedthe theoryofacanonicalinnermodelgoingproperlybeyond L,byconstructing MH for H = “thereisameasurablecardinal”.4 Sincethen,progressivelylarger MH forprogressivelystronger H havebeenconstructedandstudiedindetail.(Seefor example[7],[27],and[28].)Atthemoment,wehaveagoodtheoryofcanonical innermodelssatisfying“thereisaWoodincardinal”,andevenslightlystronger hypotheses.(See[26],[30],and[61],forexample.)Oneofthemostimportantopen problemsinsettheoryistoextendthistheorysignificantlyfurther,withperhaps
4ZFC isofcoursetooweak,consistency-wise,toprovethatthereissuchamodel.Silverand Kunenworkedinthetheory ZFC+ “thereisameasurablecardinal”.Inthemid1970s,Doddand Jensendevelopedgeneralmethodsforconstructingthecanonicalinnermodelwithameasurableunder awideassortmentofhypotheses.See[7].
themostwell-knowntargetbeingmodelssatisfying“thereisasupercompact cardinal”.
Innermodeltheoryisacrucialtoolincalibratingconsistencystrengths:inorder toprovethat H ≤con T ,where H isalargecardinalhypothesis,onegenerally constructsacanonicalinnermodelof H insideanarbitrarymodelof T .Because wedonothaveafullinnermodeltheoryveryfarpastWoodincardinals,welack themeanstoprovemanywell-knownconjecturesoftheform H ≤con T ,where H issignificantlystrongerthan“thereisaWoodincardinal”.Broadlyspeaking,there aregreatdefectsinourunderstandingoftheconsistencystrengthhierarchybeyond Woodincardinals.
Innermodeltheoryisalsoacrucialtoolindevelopingtheconsequencesforreal numbersoflargecardinalhypotheses.Indeed,thebasicsofinnermodeltheory forWoodincardinalswerediscoveredin1985-86byD.A.Martinandtheauthor, atroughlythesametimetheydiscoveredtheirproofofProjectiveDeterminacy, or PD.(Martin,Moschovakis,andothershadshowninthe1960’sand70’sthat PD decidesinanaturalwayalltheclassicalquestionsaboutprojectivesetsleft undecidedby ZFC alone.)Thissimultaneousdiscoverywasnotanaccident,as thefundamentalnewtoolinbothcontextswasthesame: iterationtrees,andthe iterationstrategies whichproducethem.Sincethen,progressininnermodeltheory hasgivenusadeeperunderstandingofpuredescriptivesettheory,andthemeans tosolvesomeoldproblemsinthatfield.
Thefundamentalopenproblemofinnermodeltheoryistoextendthetheoryto modelssatisfyingstrongerlargecardinalhypotheses.“Thereisasupercompact cardinal”isanoldandstillquitechallengingtarget.Oneverywellknowntest questionhereiswhether (ZFC+“thereisasupercompactcardinal”) ≤con ZFC + PFA.Theanswerisalmostcertainlyyes,andtheproofalmostcertainlyinvolves aninnermodeltheorythatisfiringonallcylinders.5 Thatkindofinnermodel theorywehavenowonlyatthelevelofmanyWoodincardinals,butsignificant partsofthetheorydoexistalreadyatmuchhigherlevels.6
5Aparallel,andstillolder,questioniswhether(ZFC +“thereisasupercompactcardinal”) ≤con ZFC+ “thereisastronglycompactcardinal”.
6J.Baumgartnershowedintheearly1980sthat ZFC + PFA ≤con ZFC+“thereisasupercompact cardinal”.SupercompactsarefarbeyondWoodincardinals,inthesensethattherearemanyinteresting consistencystrengthsstrictlybetweenthetwo,andinthesensethatconstructingcanonicalinner modelsforsupercompactspresentssignificantnewdifficulties.Manysettheoreticprincipleshave beenshownconsistentrelativetotheexistenceof(sometimesmany)supercompactcardinals,so inner-model-theoreticevidenceoftheirconsistencywouldbevaluable.
1.3.Miceanditerationstrategies
Thecanonicalinnermodelsweseekareoftencalled mice.Therearetwo principalvarieties,thepureextendermiceandthestrategymice.7
Apureextenderpremouseisamodeloftheform Lα [E] where E isacoherent sequenceofextenders.Hereanextenderisasystemofultrafilterscodingan elementaryembedding,andcoherencemeansroughlythattheextendersappearin orderofstrength,withoutleavinggaps.ThesenotionswereintroducedbyMitchell inthe1970s8,andtheyhavebeenafoundationforworkininnermodeltheory sincethen.
Inthisbook,weshallassumethatourpremicehavenolongextendersontheir coherentsequences.9 Suchpremicecanmodelsuperstrong,andevensubcompact, cardinals.Theycannotmodel κ +-supercompactness.Longextendersleadtoan additionalsetofdifficulties.
An iterationstrategy isawinningstrategyforplayer II intheiterationgame.For anypremouse M,theiterationgameon M isatwoplayergameoflength ω1 + 1 10 Inthisgame,theplayersconstructatreeofmodelssuchthateachsuccessivenode onthetreeisobtainedbyanultrapowerofamodelthatalreadyexistsinthetree. I istheplayerthatdescribeshowtoconstructthisultrapower.Hechoosesan extender E fromthesequenceofthelastmodel N constructedsofar,thenchooses anothermodel P inthetreeandtakestheultrapowerof P by E.Iftheultrapoweris ill-foundedthenplayer I wins;otherwisetheresultingultrapoweristhenextnode onthetree.Player II movesatlimitstages λ bychoosingabranchofthetreethat hasbeenvisitedcofinallyoftenbelow λ ,andissuchthatthedirectlimitofthe embeddingsalongthebranchiswell-founded.Ifhefailstodoso,heloses.IfII managestostayinthecategoryofwellfoundedmodelsthroughall ω1 + 1 moves, thenhewins.Awinningstrategyfor II inthisgameiscalledan iterationstrategy for M,and M issaidtobe iterable justincasethereisaniterationstrategyforit. Iterablepureextenderpremicearecalled pureextendermice
Pureextendermicearecanonicalobjects;forexample,anyrealnumberbelongingtosuchamouseisordinaldefinable.Letussaythatapremouse M is pointwise definable ifeveryelementof M isdefinableover M.Foranyaxiomatizabletheory T ,theminimalmousesatisfying T ispointwisedefinable.Thecanonicityofpure extendermiceisduetotheiriterability,which,viathefundamental Comparison Lemma,impliesthatthepointwisedefinablepureextendermicearewellordered byinclusion.Thisisthe mouseorder onpointwisedefinablepureextendermice.
7Strategymicearesometimescalled hodmice,becauseoftheirroleinanalyzingthehereditarily ordinaldefinablesetsinmodelsoftheAxiomofDeterminacy.
8See[27]and[28].
9Anextenderisshortifallitscomponentultrafiltersconcentrateonthecriticalpoint.Otherwise,it islong.
10Iterationgamesofotherlengthsarealsoimportant,butthislengthiscrucial,soweshallfocuson it.
Theconsistencystrengthof T isdeterminedbytheminimalmouse M havinga genericextensionsatisfying T ,andthustheconsistencystrengthorderonnatural T ismirroredinthemouseorder.However,inthecaseofthemouseorder,we have proved thatwehaveawellorder;whatwecannotyetdoistienatural T at highconsistencystrengthstoit.Asweclimbthemouseorder,themicebecome correct(reflectwhatistrueinthefulluniverseofsets)athigherandhigherlevels oflogicalcomplexity.
Iterationstrategiesforpointwisedefinablepureextendermicearealsocanonicalobjects;forexample,apointwisedefinablemousehasexactlyoneiteration strategy.11 Theexistenceofiterationstrategiesisattheheartofthefundamental problemofinnermodeltheory,andforapointwisedefinable M,toprovethe existenceofaniterationstrategyistodefineit.Inpractice,itseemsnecessary togiveadefinitioninthesimplestpossiblelogicalform.Aswegohigherinthe mouseorder,thelogicalcomplexityofiterationstrategiesmustincrease,inaway thatkeepspacewiththecorrectnessofthemicetheyidentify.
Ourmostpowerful,all-purposemethodforconstructingiterationstrategiesis the coremodelinductionmethod.Becauseiterationstrategiesmustactontrees oflength ω1,theyarenotcodedbysetsofreals.Nevertheless,thefragmentof theiterationstrategyforacountablemousethatactsoncountableiterationtrees is codedbyasetofreals.Ifthissethappenstobeabsolutelydefinable(thatis, UniversallyBaire)thenthestrategycanbeextendedtoactonuncountableiteration treesinauniqueway.Thereisnootherwayknowntoconstructiterationstrategies actingonuncountabletrees.Thus,havinganabsolutelydefinableiterationstrategy forcountabletreesistantamounttohavingafulliterationstrategy.Thekey ideainthecoremodelinductionistousetheconceptsofdescriptivesettheory, underdeterminacyhypotheses,toidentifyanextrelevantlevelofcorrectnessand definabilityforsetsofreals,atargetlevelatwhichthenextiterationstrategyshould bedefinable.
Absolutedefinabilityleadstodeterminacy.Thusatreasonablyclosedlimitsteps inacoremodelinduction,onehasamodel M of AD + V = L(P(R)) thatcontains therestrictionstocountabletreesoftheiterationstrategiesalreadyconstructed. UnderstandingthestructureofHODM isimportantforgoingfurther.
1.4. HOD inmodelsofdeterminacy
HOD istheclassofallhereditarilyordinaldefinablesets.Itisamodelof ZFC12 , butbeyondthat, ZFC doesnotdecideitsbasictheory,andthesameistrueof ZFC augmentedbyanyoftheknownlargecardinalhypotheses.Theproblemisthat thedefinitionsonehasallowedarenotsufficientlyabsolute.Incontrast,thetheory
11ThisfollowsfromTheorem4.11of[65],andthefactthatanyiterationstrategyforapointwise definable M hastheWeakDodd-Jensenpropertywithrespecttoallenumerationsof M
12See[31].
of HOD indeterminacymodelsiswell-determined,notsubjecttothevagariesof forcing.13
Thestudyof HOD inmodelsof AD hasalonghistory.Thereadershould see[67]forasurveyofthishistory. HOD wasstudiedbypurelydescriptiveset theoreticmethodsinthelate70sand80s,andpartialresultsonbasicquestions suchaswhether HOD |= GCH wereobtainedthen.Itwasknownthenthatinner modeltheory,ifonlyonecoulddevelopitinsufficientgenerality,wouldberelevant tocharacterizingtherealsin HOD.Itwasknownthat HODM iscloseto M in variousways;forexample,if M |= AD+ + V = L(P(R))14,then M canberealized asasymmetricforcingextensionof HODM ,sothatthefirstordertheoryof M is partofthefirstordertheoryofitsHOD. 15
Justhowrelevantinnermodeltheoryistothestudyof HOD inmodelsof AD becameclearin1994,whentheauthorshowedthatifthereare ω Woodincardinals withameasurableabovethemall,then HODL(R) upto θ L(R) isapureextender mouse.16(See[60].)Shortlyafterward,thisresultwasimprovedbyHughWoodin, whoreduceditshypothesisto ADL(R),andidentifiedthefull HODL(R) asamodel oftheform L[M, Σ],where M isapureextenderpremouse,and Σ isapartial iterationstrategyfor M. HODL(R) isthusanewtypeofmouse,sometimescalled a strategymouse,sometimescalleda hodmouse.See[77]foranaccountofthis work.
Sincethemid-1990s,therehasbeenagreatdealofworkdevotedtoextending theseresultstomodelsofdeterminacybeyond L(R).Woodinanalyzed HOD in modelsof AD+ belowtheminimalmodelof ADR finestructurally,andSargsyan extendedtheanalysisfurther,firsttodeterminacymodelsbelow ADR + “θ is regular”(see[37]and[38]),andmorerecently,tomodelsofstillstrongerformsof determinacy. 17 Partofthemotivationforthisworkisthatitseemstobeessential inthecoremodelinduction:ingeneral,thenextiterationstrategyseemstobea strategyforahodmouse,notforapureextendermouse.Thisideacomesfrom workofWoodinandKetchersidaround2000.(See[21]and[47].)
13Wemeanheredeterminacymodelsoftheform M = L(Γ, R),where Γ isaproperinitialsegment oftheuniversallyBairesets.IftherearearbitrarilylargeWoodincardinals,thenforanysentence ϕ , whether ϕ istrueinallsuch HODM isabsoluteundersetforcing.ThisfollowseasilyfromWoodin’s theoremonthegenericabsolutenessof (Σ2 1)uB statements.See[64,Theorem5.1].
14AD+ isatechnicalstrengtheningof AD.Itisnotknownwhether AD ⇒ AD+ ,butinevery modelof AD constructedsofar, AD+ alsoholds.Inparticular,themodelsof AD thatarerelevantin thecoremodelinductionsatisfy AD+
15ThisisatheoremofWoodinfromtheearly1980s.Cf.[67].
16Inadeterminacycontext, θ denotestheleastordinalthatisnotthesurjectiveimageofthereals.
17See[39].Partofthisworkwasdoneincollaborationwiththeauthor;see[68],[74],and[69].The determinacyprinciplesdealtwithhereareallweakerthanaWoodinlimitofWoodincardinals.
1.5.Leastbranchhodpairs
Thestrategymiceusedintheworkjustdescribedhavetheform M = L[E, Σ], where E isacoherentsequenceofextenders,and Σ isaniterationstrategyfor M
Thestrategyinformationisfedintothemodel M slowly,inawaythatisdictated inpartbythedeterminacymodelwhose HOD isbeinganalyzed.Onesaysthatthe hierarchyof M is rigidlylayered,or extenderbiased.Theobject (M, Σ) iscalleda rigidlylayered(extenderbiased) hodpair.
Perhapsthemainmotivationfortheextenderbiasedhierarchyisthatitmakesit possibletoproveacomparisontheorem.Thereisnoinnermodeltheorywithout suchatheorem.Comparingstrategymicenecessarilyinvolvescomparingiteration strategies,andcomparingiterationstrategiesissignificantlymoredifficultthan comparingextendersequences.Rigidlayeringletsoneavoidthedifficulties inherentinthegeneralstrategycomparisonproblem,whileprovingcomparisonfor aclassofstrategymiceadequatetoanalyze HOD intheminimalmodelof ADR + “θ isregular”,andsomewhatbeyond.Thekeyisthatinthisregion, HOD doesnot havecardinalsthatarestrongpastaWoodincardinal.
Unfortunately,rigidlayeringdoesnotseemtohelpincomparingstrategymice thathavecardinalsthatarestrongpastaWoodin.Moreover,ithasseriouscosts. Thedefinitionof“hodpremouse”becomesverycomplicated,andindeeditisnot clearhowtoextendthedefinitionofrigidlylayeredhodpairsmuchpastthatgiven in[39].Thedefinitionof“rigidlylayeredhodpremouse”isnotuniform,inthatthe extentofextenderbiasdependsonthedeterminacymodelwhose HOD isbeing analyzed.Finestructure,andinparticularcondensation,becomemoreawkward. Forexample,itisnottrueingeneralthatthepointwisedefinablehullofalevelof M isalevelof M.(Theproblemisthatthehullwillnotgenerallybesufficiently extenderbiased.)
Themorenaivenotionofhodpremousewouldabandonextenderbias,and simplyaddtheleastmissingpieceofstrategyinformationatessentiallyevery stage.ThiswasoriginallysuggestedbyWoodin.18 Thefocusofthisbookisa generalcomparisontheoremforiterationstrategiesthatmakesitpossibletouse thisapproach,atleastintherealmofshortextenders.Theresultingpremiceare called leastbranchpremice (lpm’s),andthepairs (M, Σ) arecalled leastbranch hodpairs (lbrhodpairs).Combiningresultsofthisbookand[73],onehas THEOREM 1.5.1([73]). Assume AD++ “thereisan (ω1, ω1) iterationstrategy forapureextenderpremousewithalongextenderonitssequence”;then (1) forany Γ ⊆ P(R) suchthat L(Γ, R) |= ADR+ “thereisno (ω1, ω1) iteration strategyforapureextenderpremousewithalongextenderonitssequence”, HODL(Γ,R) isaleastbranchpremouse,and
18Therearesomefine-structuralproblemswiththeprecisemethodforinsertingstrategyinformation originallysuggestedbyWoodin.Themethodforstrategyinsertionthatiscorrectindetailisdueto SchlutzenbergandTrang.Cf.[56].
(2) thereisa Γ ⊆ P(R) suchthat L(Γ, R) |= ADR+ “thereisno (ω1, ω1) iterationstrategyforapureextenderpremousewithalongextenderonits sequence”,and HODL(Γ,R) |= “thereisasubcompactcardinal”.
Ofcourse,onewouldliketoremovethemouseexistencehypothesisof1.5.1, andproveitsconclusionunder AD+ alone.Findingawaytodothisisone manifestationofthelongstandingiterabilityproblemwehavediscussedabove. Althoughwedonotyetknowhowtodothis,thetheoremdoesmakeithighlylikely thatinmodelsof ADR thathavenotreachedaniterationstrategyforapureextender premousewithalongextender, HOD isaleastbranchpremouse.Italsomakesit verylikelythattherearesuch HOD’swithsubcompactcardinals.Subcompactness isoneofthestrongestlargecardinalpropertiesthatcanberepresentedwithshort extenders.19
AlthoughweshallnotproveTheorem1.5.1here,weshallproveanapproximationtoitthatmakesthesamepoints.ThatapproximationisTheorem11.3.13 below.
Leastbranchpremicehaveafinestructuremuchclosertothatofpureextender modelsthanthatofrigidlylayeredhodpremice.Inthisbookwedevelopthe basics,includingthesolidityanduniversalityofstandardparameters,andaformof condensation.In[76],theauthorandN.Tranghaveprovedasharpercondensation theorem,whosepureextenderversionwasusedheavilyintheSchimmerlingZemanwork([44])on inpureextendermice.Itseemslikelythattherestofthe Schimmerling-Zemanworkextendsaswell.
Thusleastbranchhodpairsgiveusagoodtheoryof HOD intheshortextender realm,providedthereareenoughsuchpairs.20 Below,weformulateaconjecture thatwecall HodPairCapturing,or HPC,thatmakesprecisethestatementthat thereareenoughleastbranchhodpairs. HPC isthemainopenprobleminthe theorytowhichthisbookcontributes.
1.6.Comparisonandthemousepairorder
Letusfirstsaymoreaboutthenatureofleastbranchhodpairs (M, Σ).There arefourrequirementson Σ inthedefinition:stronghullcondensation,quasinormalizingwell,internalliftconsistency,andpushforwardconsistency.Weshall describetheserequirementsinformally,omittingsomeofthefinepoints,andgive thefulldefinitionslater.
Recallthataniterationtreeonapremouse M is normal ifftheextenders EW α usedin W havelengthsincreasingwith α ,andeach EW α isappliedtothelongest
19Untilnow,therewasnoverystrongevidencethatthe HOD ofadeterminacymodelcouldsatisfy thattherearecardinalsthatarestrongpastaWoodincardinal.
20Atleastinthecasethatthebackgrounddeterminacymodelsatisfies ADR + V = L(P(R)).Some formofextenderbiasmaybeappropriateinothercases.
possibleinitialsegmentoftheearliestpossiblemodelin W.Fortechnicalreasons weneedtoconsideraslightweakeningofthelength-increasingrequirement;we calltheresultingtrees quasi-normal.Ouriterationstrategieswillactonfinite stacks ofquasi-normaltrees,thatis,sequences s = T0,..., Tn suchthatforall k ≤ n 1, Tk+1 isaquasi-normaltreeonsomeinitialsegmentofthelastmodelin Tk .Wewrite M∞(s) forthelastmodelof Tn,ifthereisone.
DEFINITION 1.6.1. Let Σ beaniterationstrategyforapremouse P. (1) (Tailstrategy)If s isastackby Σ and Q ✂ M∞(s),then Σs,Q isthestrategyfor Q givenby: Σs,Q(t)= Σ(s Q, t ). 21 (2) (Pullbackstrategy)If π : N → P iselementary,then Σπ isthestrategyfor N givenby: Σπ (s)= Σ(π s),where π s istheliftof s by π toastackon P
In(2),elementaritymustbeunderstoodfinestructurally;ourconventionisthat everypremouse P hasadegreeofsoundnessattachedtoit,andelementaritymeans elementarityatthatquantifierlevel.
Perhapsthemostimportantregularitypropertyofiterationstrategiesis strong hullcondensation.Todefineitweneedthenotionofa treeembedding Φ : T→U , where T and U arenormaltreesonthesame M.Theideaofcourseisthat Φ should preserveacertainamountoftheiterationtreestructure,butsomecareisneededin spellingoutexactlyhowmuch. Φ isdeterminedbyamap u :lh(T ) → lh(U ) and maps πα : MT α →MU u(α ) havingvariousproperties.See §6.4.
DEFINITION 1.6.2. Let Σ beaniterationstrategyforapremouse M;then Σ has stronghullcondensation iffwhenever s isastackofnormaltreesby Σ and N ✂ M∞(s),and U isanormaltreeon N by Σs,N ,and Φ : T→U isatreeembedding, withassociatedmaps πα : MT α →MU u(α ),then
(a) T isby Σs,N ,and (b) forall α < lh(T ), Σs N,T α +1 =(Σs N,U u(α )+1 )πα
Stronghullcondensationisastrongerversionofthehullcondensationproperty isolatedbySargsyanin[37].
Thesecondimportantpropertyisquasi-normalizingwell.Givenan M-stack T , U withlastmodel N suchthat T and U arenormal,shufflingtheextenders of U into T inaminimalwayproducesanormaltree W = W (T , U ).If U has alastmodel R,wegetnearlyelementarymap π : N → R.Wecall W (T , U ) the embeddingnormalization of T , U .Theideaissimple,buttherearemany technicaldetails.22 Itprovesusefultoconsideraslightlylessminimalshuffling V (T , U ) thatwecallthe quasi-normalization of T , U .Evenif T and U are normal, V (T , U ) maynotbelength-increasing,butitisnearlyso.Thereader shouldseeChapter6forfulldefinitions.
21Forpremice Q and R, Q ✂ R iffthehierarchyof Q isaninitialsegmentofthatof R
22MuchofthegeneraltheoryofnormalizationwasdevelopedindependentlybySchlutzenberg.See [54].Seealso[19]and[58].
DEFINITION 1.6.3. Let Σ beaniterationstrategyforapremouse M.Wesay that Σ quasi-normalizeswell iffwhenever s isan M-stackby Σ,and T , U isa 2-stackby Σs suchthat T and U arenormaltreeshavinglastmodels,then
(a) V (T , U ) isby Σs,and (b) letting V = V (T , U ) and π : MU ∞ →MV ∞ bethemapgeneratedbyquasinormalization,wehavethat Σs T ,U =(Σs V )π .
Thefinalbasicregularitypropertyofiterationstrategiesforpureextender premiceis internalliftconsistency.Supposethat s isastackby Σ and P ✂ Q ✂ M∞(s).Stacks t on P canbeliftedtostacks t + on Q inanaturalway.Wesaythat Σ isinternallyliftconsistentiffforallsuch s, P, and Q, Σs,P(t)= Σs,Q(t +).See §5.4.
Forpairs (M, Σ) suchthat M isastrategymouse,werequirealsothatthe internalstrategypredicateof M beconsistentwith Σ.Moregenerally,letting ˙ Σ bethepredicatesymbolusedtorecordstrategyinformation,wesaythat (M, Σ) is pushforwardconsistent iffwhenever s isastackby Σ and N ✂ M∞(s),then ΣN ⊆ Σs,N If M isapureextenderpremouse,and Σ isastrategyfor M thathasstronghull condensation,quasi-normalizeswell,andisinternallyliftconsistent,thenwecall (M, Σ) a pureextenderpair.If M isaleastbranchpremouse,and Σ isastrategy for M thathasstronghullcondensation,quasi-normalizeswell,isinternallylift consistentandpushforwardconsistent,thenwecall (M, Σ) a leastbranchhodpair. Apairofoneofthetwotypesisa mousepair.
If (M, Σ) isamousepair,and s isastackby Σ withlastmodel N,thenwecall (N, Σs) an iterateof (M, Σ).Ifthebranch M-to-N of s doesnotdrop,wecallita non-droppingiterate.Inthatcase,wehaveaniterationmap is : M → N.Letus write
(M, Σ) ✂ (R, Λ) iff M ✂ R and Σ = ΛM .
Wehavenohopeofshowinganythingaboutmousepairs (M, Σ) unlesswe assumeabsolutedefinabiltyfortheiterationstrategy.Hereweassume Σ hasscope HC,i.e.that M iscountableand Σ isdefinedonfinitestacksofcountabletrees, andweassumethatweareinamodelof AD+ . 23 Thefollowingisthemainnew resultofthebook.
THEOREM 1.6.4. (ComparisonLemma)Assume AD+,andlet (P, Σ) and (Q, Ψ) bestronglystable24 mousepairswithscope HC ofthesamekind;thenthereare iterates (R, Λ) of (P, Σ) and (S, Ω) of (Q, Ψ),obtainedbynormaltrees T and U , suchthateither (1) (R, Λ) ✂ (S, Ω) andP-to-Rdoesnotdrop,or
23Onecouldrequirethattheybedefinedoncountablestacks.
24Strongstabilityisamildfinestructuralrequirement.Onecanavoiditbyslightlycomplicating thenotionof iterate.See4.4.5and4.6.12.
(2) (S, Ω) ✂ (R, Λ) andQ-to-Sdoesnotdrop.
Evenforpureextenderpairs,thistheoremisnew,becauseoftheagreement betweentailstrategiesitrequires.Infact,itisnoeasiertoprovethetheoremfor pureextenderpairsthanitistoproveitforleastbranchhodpairs.Theproofin bothcasesisthesame,anditmakesuseofthepropertiesoftheiterationstrategies wehaveisolatedinthedefinitionofmousepair.
WorkinginthecategoryofmousepairsenablesustostateageneralDodd-Jensen lemma.Letussay π : (P, Σ) → (Q, Ψ) iselementaryiff π iselementaryfrom P to Q,and Σ = Ψπ .Weshallshowthatanelementarysubmodelofamousepairisa mousepair,andthattheiterationmapsassociatedtonon-droppingiterationsofa mousepairareelementary.25
THEOREM 1.6.5(Dodd-Jensenlemma). Let (P, Σ) beamousepair,and (Q, Ψ) beaniterateof (P, Σ) viathestack s.Suppose π : (P, Σ) → (Q, Ψ) iselementary; thensdoesnotdrop,andforallordinals η ∈ P,is(η ) ≤ π (η ).
TheproofisjusttheusualDodd-Jensenproof;thepointisjustthatthelanguage ofmousepairsenablesustoformulatethetheoreminitspropergenerality.There isnoneedtorestricttomicewithuniqueiterationstrategies,asisusuallydone. Similarly,wecandefinethemouseorderinitspropergenerality,withoutrestrictingtomicewithuniqueiterationstrategies.If (P, Σ) and (Q, Ψ) arepairsof thesametype,then (P, Σ) ≤∗ (Q, Ψ) iff (P, Σ) canbeelementarilyembeddedinto aniterateof (Q, Ψ).TheComparisonandDodd-Jensentheoremsimplythat ≤∗ is aprewellorderoneachtype.
1.7.Hodpaircapturing
Leastbranchhodpairscanbeusedtoanalyze HOD inmodelsof AD+,provided thatthereareenoughsuchpairs.
DEFINITION 1.7.1(AD+). (a) HodPairCapturing (HPC) istheassertion: foreverySuslin-co-Suslinset A,thereisaleastbranchhodpair (P, Σ) such that A isdefinablefromparametersover (HC, ∈, Σ). (b) L[E] capturing (LEC) istheassertion:foreverySuslin-co-Suslinset A,there isapureextenderpair (P, Σ) suchthat A isdefinablefromparametersover (HC, ∈, Σ)
Anequivalent(under AD+)formulationwouldbethatthesetsofrealscoding strategiesofthetypeinquestion,undersomenaturalmapoftherealsonto HC, areWadgecofinalintheSuslin-co-Suslinsetsofreals.TherestrictiontoSuslinco-Suslinsets A isnecessary,for AD+ impliesthatif (P, Σ) isapairofoneofthe
25Neitherisobvious.Thatiterationmapsareelementaryisapropertyoftheiterationstrategy knownaspullbackconsistency.Itfollowsfromstronghullcondensation.
twotypes,thenthecodesetof Σ isSuslinandco-Suslin.Thisisthemainresultof [73],whereitisalsoshownthattheSuslinrepresentationconstructedisofoptimal logicalcomplexity.
Remark 1.7.2. HPC isacousinofSargsyan’s GenerationofFullPointclasses. See[37]and[38], §6.1.
Assuming AD+ , LEC isequivalenttothewellknownMouseCapturing:for reals x and y, x isordinaldefinablefrom y iff x isinapureextendermouseover y.Thisequivalenceisshownin[63].(SeeespeciallyTheorem16.6.)Weshow inTheorem10.4.3belowthatunder AD+ , LEC implies HPC.Wedonotknow whether HPC implies LEC.
Granted ADR and HPC,wehaveenoughhodpairstoanalyzeHOD.
THEOREM 1.7.3([73]). Assume ADR and HPC;then Vθ ∩ HOD istheuniverse ofaleastbranchpremouse.
Sometechniquesdevelopedin[59]and[73]areneededtoprovethetheorem,so weshallnotproveithere.
Thenaturalconjectureisthat LEC and HPC holdinallmodelsof AD+ thathave notreachedaniterationstrategyforapremousewithalongextender.Becauseour capturingmicehaveonlyshortextendersontheirsequences, LEC and HPC cannot holdinlargermodelsof AD+
DEFINITION 1.7.4. NLE (“Nolongextenders”)istheassertion:thereisno countable, ω1 + 1-iterablepureextenderpremouse M suchthatthereisalong extenderonthe M-sequence.
CONJECTURE 1.7.5. Assume AD+ and NLE;then LEC
CONJECTURE 1.7.6. Assume AD+ and NLE;then HPC.
Asweremarkedabove,1.7.5implies1.7.6.Conjecture1.7.5isequivalentto aslightstrengtheningoftheusualMouseSetConjecture MSC.(Thehypothesis of MSC isthatthereisnoiterationstrategyforapureextenderpremousewitha superstrong,whichisslightlystrongerthan NLE.) MSC hasbeenacentraltarget forinnermodeltheoristsforalongtime.
1.8.Constructingmousepairs
Thebasicsourceformousepairsisabackgroundconstruction.Inthesimplest case,suchaconstruction C buildspairs (Mν ,k , Ων ,k ) inductively,puttingextenders onthe Mν ,k -sequencethatarerestrictionsofniceextendersin V .Theiteration strategy Ων ,k isinducedbyaniterationstrategyfor V ,andifweareconstructing strategypremice,therelevantinformationabout Ων ,k isinsertedinto Mν ,k atthe
appropriatepoints. Mν ,k+1 isthecoreof Mν ,k .Theconstructionbreaksdownif thestandardparameterof Mν ,k behavespoorly,sothatthereisnocore.
Thereisofcoursemoretosayhere,andweshalldosolaterinthebook.For now,letusnotethatthebackgrounduniverseforsuchaconstructionshouldbea modelof ZFC thathaslotsofextenders,andyetknowshowtoiterateitself.Inthe AD+ context,thefollowingtheoremofWoodinapplies.26
THEOREM 1.8.1(Woodin). Assume AD+,andlet Γ beagoodpointclasssuch thatallsetsin Γ areSuslinandco-Suslin;thenforanyreal x thereisacoarse Γ-Woodinpair (N, Σ) suchthatx ∈ N.
Here,roughlyspeaking, N isacountabletransitivemodelof ZFC withaWoodin cardinalandatermforauniversal Γ set,and Σ isaniterationstrategyfor N that movesthistermcorrectly,andissuchthat Σ ∩ N isdefinableover N.SeeDefinition 7.2.3.
ThefollowingisessentiallyTheorem10.4.1tofollow.Ittooisoneofthemain newresultsofthebook.
THEOREM 1.8.2. Assume AD+,andlet (N, Σ) beacoarse Γ-Woodinpair.Let C bealeastbranchconstructionin N;then C doesnotbreakdown.Moreover,each ofitslevels (MC ν ,k , ΩC ν ,k ) isaleastbranchhodpairin N,andextendscanonically toaleastbranchhodpairinV.
Backgroundconstructionsofthesortdescribedinthistheoremhaveanimportant roletoplayinourcomparisonprocess.Assume AD+,andlet (M, Ω) and (N, Σ) bemousepairsofthesametype.Wecompare (M, Ω) with (N, Σ) byputting M and N intoacommon Γ-Woodinuniverse N∗,where Σ and Ω arein Γ ∩ ˇ Γ.We theniterate (M, Σ) and (N, Ω) intolevelsofafullbackgroundconstruction(ofthe appropriatetype)of N∗.Herearesomedefinitionsencapsulatingthemethod.
DEFINITION 1.8.3. Let (M, Σ) and (N, Ω) bemousepairsofthesametype; then
(a) (M, Σ) iteratespast (N, Ω) iffthereisa λ -separatediterationtree T by Σ on M whoselastpairis (N, Ω)
(b) (M, Σ) iteratesto (N, Ω) iffthereisa λ -separated T asin(a)suchthatthe branch M-to-N of T doesnotdrop.
(c) (M, Σ) iteratesstrictlypast (N, Ω) iffititeratespast (N, Ω),butnotto (N, Ω). λ -separationisasmallstrengtheningofnormalitythatisdefinedin4.4.8.One reasonthatitisimportantisthatif T is λ -separatedand U isanormaltreeon MT ∞ ,then W (T , U )= V (T , U ).Thatis,embeddingnormalizationcoincideswith quasi-normalizationinthiscase.
DEFINITION 1.8.4(AD+). Let (P, Σ) beamousepair;then (*)(P, Σ) isthe followingassertion:Let (N, Ψ) beanycoarse Γ-Woodinpairsuchthat P ∈ HCN∗ ,
26See[22],and[66,Lemma3.13].
and Σ ∈ Γ ∩ ˇ Γ.Let C beabackgroundconstructiondonein N∗ oftheappropriate type,andlet (R, Φ) bealevelof C.Supposethat (P, Σ) iteratesstrictlypastall levelsof C thatarestrictlyearlierthan (R, Φ);then (P, Σ) iteratespast (R, Φ).
If (M, Ω) isamousepair,and N isaninitialsegmentof M,thenwewrite ΩN fortheiterationstrategyfortreeson N thatisinducedby Ω.Wecanunpackthe conclusionof1.8.4asfollows:supposethecomparisonof P with R hasproduced anormaltree T on P withlastmodel Q,with T by Σ,and S isaninitialsegment ofboth Q and R;then ΣT ,S = ΦS.Thustheleastdisagreementbetween Q and R isanextenderdisagreement.Moreover,if E on Q and F on R aretheextenders involvedinit,then F = ∅.
Weshallshow(cf.Theorems8.4.3and9.5.6below)
THEOREM 1.8.5. Assume AD+;then (*)(P, Σ) holds,forallstronglystable mousepairs (P, Σ).
Thistheoremletsuscomparetwo(ormore)mousepairsofthesametype indirectly,bycomparingthemtothelevelsofanappropriateconstruction,done ina Γ-Woodinmodel,wherebothstrategiesarein Γ ∩ ˇ Γ.Onecanshowusingthe Woodinnessthat C reachesnon-droppingiteratesofbothpairs27.Thisgivesusa stage (M, Ω) of C suchthatoneofthepairsiteratestoit,whiletheotheriterates pastit.
1.9.Thecomparisonargument
Inwhatfollows,weshallgivefairlycompleteproofsofthetheoremsabove. Thebookislong,partlybecausewewantedtomakeitaccessible,andpartly becauseweshallbeforcedtorevisethebasicdefinitionsof[30]and[81]invarious ways,sothereisalimittowhatwecansimplyquote.Inaddition,theneed tocomparestrategiesaddsalayerofcomplexitytotheproofsofthemainfine structuraltheoremsaboutstrategymice.Nevertheless,themainnewideasbehind thestrategy-comparisonprocessitselfarereasonablysimple.Wedescribethem now.
Thefirststepistofocusonproving (*)(P, Σ).Thatis,ratherthandirectly comparingtwostrategies,weiteratethembothintoacommonbackgroundconstructionanditsstrategy.Inthecomparison-of-micecontext,thismethodgoes backtoKunen([23]),andwasfurtherdevelopedbyMitchell,Baldwin([5]),and theauthor.28 WoodinandSargsyanhadusedthemethodforstrategycomparison inthehodmousecontext.Allthesecomparisonscouldbereplacedbydirect comparisonsofthetwomiceorstrategiesinvolved,butinthegeneralcaseof comparisonofstrategies,thereareseriousadvantagestotheindirectapproach.
27See8.1.4.
28Inunpublished1985notestitled“Largecardinalsand ∆1 3 wellorders”.
Thereisnoneedtodecidewhattodoifoneencountersastrategydisagreement, becauseoneisprovingthatthatneverhappens.Thecomparisonprocessisjustthe usualoneofcomparingleastextenderdisagreements.Insteadofthedualproblems ofdesigningaprocessandprovingitterminates,onehasagivenprocess,and knowswhyitshouldterminate:nostrategydisagreementsshowup.Theproblem isjusttoshowthis.Theseadvantagesledtheauthortofocus,since2009,ontrying toprove (*)(P, Σ).
ThemainnewideathatmakesthispossibleismotivatedbySargsyan’sproofin [37]thatif Σ hasbranchcondensation,then (*)(P, Σ) holds.29Branchcondensation istoostrongtoholdonce P hasextendersoverlappingWoodincardinals;we cannotconcludethat Σ(T )= b fromhavingmerelyrealized MT b intoa Σ-iterate of P.Weneedsomekindofrealizationoftheentirephalanx Φ(T b) inorderto concludethat Σ(T )= b.Thisleadstoaweakeningofbranchcondensationthat onemightcall“phalanxcondensation”,inwhichoneasksforafamilyofbranchcondensation-likerealizationshavingsomenaturalagreementwithoneanother. Phalanxcondensationisstillstrongenoughtoimply (*)(P, Σ),andmightwell betrueingeneralforbackground-inducedstrategies.Unfortunately,Sargsyan’s constructionofstrategieswithbranchcondensationdoesnotseemtoyieldphalanx condensationinthemoregeneralcase.Foronething,itinvolvescomparison arguments,andinthegeneralcase,thislookslikeaviciouscircle.Itwasduring oneoftheauthor’smanyattemptstobreakintothiscirclethatherealizedthat certainpropertiesrelatedtophalanxcondensation,namelynormalizingwelland stronghullcondensation,couldbeobtaineddirectlyforbackground-induced strategies,andthatthesepropertiessufficefor (*)(P, Σ)
Letusexplainthislastpartbriefly.Supposethatweareinthecontextof Theorem1.8.5.Wehaveapremouse P withiterationstrategy Σ thatnormalizes wellandhasstronghullcondensation.Wehave N apremouseoccuringinthe fullybackgroundedconstructionof N∗,where P ∈ HCN∗ and N∗ captures Σ.We compare P with N byiteratingawaytheleastextenderdisagreement.Ithas beenknownsince1985thatonly P willmove.Wemustprovethatnostrategy disagreementshowsup.
Supposewehaveproduceda λ -separatediterationtree T on P withlastmodel Q,that Q|α = N|α ,andthat U isanormaltreeon R = Q|α = N|α oflimitlength playedbyboth ΣT ,R (thetailof Σ)and ΩR,where Ω isthe N∗-inducedstrategyfor N.Wewishtoshowthat ΣT ,R(U )= ΩR(U ).Because Σ isinternallyliftconsistent, wecanreducetothecasethat Q = R.
Let b = ΩR(U ).Wemustsee b = ΣT ,R(U ),thatis,that b = Σ( T , U ).Since T is λ -separated,embeddingnormalizationcoincideswithquasi-normalization.
29Roughly,aniterationstrategy Σ for M has branchcondensation iffwhenever T isaniterationtree oflimitlengthby Σ, b isacofinalbranchof T withassociatediterationmap ib : M → MT b , π : M → N isaniterationmapby Σ,andthereisa k : MT b → N suchthat π = k ◦ ib ,then Σ(T )= b.See[37]for moredetail.
Letusconsider
Wc = W (T , U c)= V (T , U c)
forarbitrarycofinalbranches c of U .Weshallsee:
(1) ΣT ,R(U )= c iff Wc isby Σ.The ⇒ directionfollowsatoncefromthefact that Σ quasi-normalizeswell,andthe ⇐ directionisprovedin §6.6.
(2) Letting i∗ b : N∗ → N∗ b comefromlifting iU b to N∗ viatheiteration-strategy constructionof[30],thereisatreeembeddingof Wb into i∗ b(T ).Thisisthe keystepintheproof.ItiscarriedoutinChapter8.
(3) i∗ b(Σ) ⊆ Σ because Σ wascapturedby N∗,so i∗ b(T ) isby Σ.
(4) Thus Wb isby Σ,because Σ hasstronghullcondensation.
(5) Soby(1), ΣT ,R(U )= b.
Hereisadiagramofthesituation:
IGURE 1.9.1.Proofof (*)(P, Σ). Wb isapsuedo-hullof i∗ b(T ).
1.10.Planofthebook
Chapters2and3collectandorganizesomestandarddefinitionsandresultsfrom innermodeltheory.Thebookisaimedatpeoplewhohavealreadyencountered
thismaterial,via[30],[65],or[81]forexample,butthesechapterswillserveasa bridgetotherestofthebook.
In §3.6weexplainwhythisstandardtheoryisnotcompletelyadequatetothe problemofcomparingiterationstrategies.Roughlyspeaking,theproblemisthat theinducediterationstrategiesforthelevelsofabackgroundconstructionarenot connectedsufficientlywelltotheiterationstrategyforthebackgrounduniverse. §3.7and §3.8analyzeoneofthetwosourcesofthisshortfall,andChapter4 removesbothofthem.Thisinvolvesrevisingthenotionsof premouse and iteration tree slightly,andre-provingthestandardfinestructuralresultsinthenewsetting.
Chapter5showsthatthenewdefinitionsleadtobackground-inducediteration strategiesthatarebetterbehavedinseveralways.Chapters6and7pushfurtherin thisdirection,leadingultimatelytoTheorem7.6.2,whichsaysthatpureextender backgroundconstructions,doneinanappropriatelyiterablebackgrounduniverse, producepureextenderpairs.
InChapter8weprovethemaincomparisontheoremforpureextenderpairs, Theorem8.4.3.Weshalladapttheproofof8.4.3toleastbranchhodpairsand tophalanxcomparisonsinChapters9and10,butthemainstepsallshowupin thissimplersituation,sowehavebegunwithit.Whenweusetheproofagainin Chapters9and10,weshallcondenselongstretchesbypointingtotheproofof 8.4.3.
Chapters9and10usethestrategy-comparisonprocesstodevelopthetheory ofleastbranchhodpairs.Chapter11usesthistheorytoanalyze HOD incertain modelsof ADR,andconcludeswithadiscussionoffurtherresultsthathavebeen provedbythemethodswedevelophere.
PRELIMINARIES
Innermodeltheorydealswithcanonicalobjects,butinnermodeltheoristshave presentedtheminvariousways.Theconventionsweusehereare,forthemost part,fairlycommon.Forbasicfinestructuralnotionssuchasprojecta,cores, standardparameters,fineultrapowers,anddegreesofelementarity,weshallstay closetoMitchell-Steel[30]andthepaper[49]bySchindlerandZeman.We shalluseJensenindexingforthesequencesofextendersfromwhichpremiceare constructed;seeforexampleZeman’sbook[81].InChapter4weshallmodifythe notionofpremouseslightly,byenlargingthestandardparametersandassociated cores.Untilwegettothatpoint,ournotionofpremouseisjustthestandardone determinedbytheconventionsof[30],[49],and[81].30
Mostofourterminologytodowithiterationtreesanditerationstrategiestraces backtoMartin-Steel[26]andMitchell-Steel[30],andisbynowprettystandard. Wedoneedtoconsidercarefullyiterationstrategiesdefinedonawiderclassof iterationtreesthaniscommon,andsothereissomelessfamiliarterminology definedinsections2.6and2.7.
2.1.Extendersandultrapowers
Ournotationforextendersisstandard.
DEFINITION 2.1.1. Let M betransitiveandrudimentarilyclosed;then E = Ea | a ∈ [θ ]<ω isa (κ , θ )-extenderover M withspaces µa | a ∈ [θ ]<ω ifand onlyif
(1) Each Ea isan (M, κ )-completeultrafilterover P([µa]|a|) ∩ M,with µa being theleast µ suchthat [µ ]|a| ∈ Ea.
(2) (Compatibility)For a ⊆ b and X ∈ M, X ∈ Ea ⇐⇒ X ab ∈ Eb
(3) (Uniformity) µ{κ } = κ
(4) (Normality)If f ∈ M and f (u) < max(u) for Ea a.e. u,thenthereisa β < max(a) suchthatfor Ea∪{β } a.e. u, f a,a∪{β }(u)= u{β },a∪{β } .
30Thenotionofpremousein[81],anditsrelatedfinestructure,originateinJensen’smanuscripts [17]and[18].
Theunexplainednotationherecanbefoundin[49, §8].Weshalloftenidentify E withthebinaryrelation (a, X ) ∈ E iff X ∈ Ea.Onecanalsoidentifyitwith theothersection-functionofthisbinaryrelation,whichisessentiallythefunction X → iM E (X ) ∩ θ .Wecall θ the length of E,andwrite θ = lh(E).The spaceof E is
sp(E)= sup{µa | a ∈ [lh(E)]<ω }.
The domainof E isthefamilyofsetsitmeasures,thatis, dom(E)= {Y |∃(a, X ) ∈
E(Y = X ∨ Y =[µa]|a| X )}.If M isapremouseofsomekind,wealsowrite M|η = dom(E),where η isleastsuchthat ∀(a, X ) ∈ E(X ∈ M|η ).Byacceptability, η = sup({µ +,M a | a ∈ [θ ]<ω }).Weshallfurtherabusenotationbywriting η = dom(E) when M isdeterminedbycontext.
Thecriticalpointofa (κ , θ )-extenderis κ ,andweuseeither crit(E) or κE to denoteit.Givenanextender E over M,weformthe Σ0 ultrapower
Ult0(M, E)= {[a, f ]M E | a ∈ [lh(E)]<ω and f ∈ M}, asin[49,8.4].Our M willalwaysberudimentarilyclosedandsatisfytheAxiom ofChoice,sowehaveLos’theoremfor Σ0 formulae,andthecanonicalembedding
iM E : M → Ult0(M, E) iscofinaland Σ0 elementary,andhence Σ1 elementary.By(1)and(3), κE = crit(iM E ).Bynormality, a =[a, id]M E ,so lh(E) isincludedinthe(alwaystransitivized)wellfoundedpartofUlt0(M, E).Moregenerally,
[a, f ]M E = iM E ( f )(a)
If X ⊆ lh(E),then E X = {(a,Y ) ∈ E | a ⊆ X } E X hasthepropertiesofan extender,exceptpossiblynormality,sowecanform Ult0(M, E X ),andthereisa naturalfactorembedding τ :Ult0(M, E X ) → Ult0(M, E) givenby τ ([a, f ]M E X )=[a, f ]M E
Inthecasethat X = ν > κE isanordinal, E ν isanextender,and τ ν isthe identity.Wesay ν isa generatorof E iff ν isthecriticalpointof τ ,thatis, ν =[a, f ]M E whenever f ∈ M and a ⊆ ν .Let
ν (E)= sup({ν + 1 | ν isageneratorof E }).
So ν (E) ≤ lh(E),and E isequivalentto E ν (E),inthatthetwoproducethesame ultrapower.
Wewrite
λ (E)= λE = iM E (κE ).
Notethatalthough E maybeanextenderovermorethanone M, sp(E), κE , lh(E), dom(E), ν (E),and λ (E) dependonlyon E itself.If N isanothertransitive, rudimentarilyclosedset,and P(µa) ∩ N = P(µa) ∩ M forall a ∈ [lh(E)]<ω ,then E isalsoanextenderover N;moreover iM E agreeswith iN E on dom(E).However, iM E and iN E maydisagreebeyondthat.Wesay E is short iff ν (E) ≤ λ (E).Itiseasyto
