Canonical structure for the universe of set theory

CANONICALSTRUCTUREINTHEUNIVERSEOFSETTHEORY:PARTONEJAMESCUMMINGS,MATTHEWFOREMAN,ANDMENACHEMMAGIDORAbstract.Westartbystudyingtherelationshipbetweentwoinvariantsiso-latedbyShelah,thesetsofgoodandapproachablepoints.Aspartofourstudyoftheseinvariants,weproveaformof“singularcardinalcompactness”forJensen’ssquareprinciple.Wethenstudytherelationshipbetweeninter-nallyapproachableandtightstructures,whichparallelstoacertainextenttherelationshipbetweengoodandapproachablepoints.Inparticularwecharac-terizethetightstructuresintermsofPCFtheoryanduseourcharacterisationtoprovesomecoveringresultsfortightstructures,alongwithsomeresultsontightnessandstationaryreflection.Finallyweprovesomeabsolutenesstheo-remsinPCFtheory,deduceacoveringtheorem,andapplythattheoremtothestudyofprecipitousideals.1.IntroductionItisadistinguishingfeatureofmodernsettheorythatmanyofthemostinter-estingquestionsarenotdecidedbyZFC,thetheoryinwhichweprofesstowork;toputitanotherway,ZFCadmitsalargevarietyofmodels.Anaturalresponsetothisistoidentifyinvariantswhichmaytakedifferentvaluesindifferentmodels,andwhichcodifyalargeamountofinformationaboutamodel.Ofparticularinterestareinvariantswhicharecanonical,inthesensethattheAxiomofChoiceisneededtoshowthattheyexist,butonceshowntoexisttheyareindependentofthechoicesmade.Forexampletheuncountableregularcardinalsarecanonicalinthissense.Shelahdiscoveredalargeclassofcanonicalinvariants,thestudyofwhichhelabeledPCFtheory.Theseinvariantsincludetwowhicharecentralinthispaper;Shelah[24,26](undersomemildcardinalarithmeticassumptionsonthesingularcardinalμ)definedtwostationarysubsetsofμ+,thesetsofgoodandapproachablepoints.Thedefinitionsofthesesetsappeartodependoncertainarbitrarychoices,but(modulotheclubfilter)areinfactindependentofthesechoices.Othercanonicalstructureswestudyinthispaperincludethestationarysetsoftightandinternallyapproachablestructures,andthecollectionofgoodpointsonascale.Itisknownthateveryapproachablepointisgoodandthatweakformsofsquare,forexampleJensen’sweaksquareprinciple∗μ,implythateverypointisapproach-able.ForemanandMagidor[16]showedthattheirprinciple“Veryweaksquare”,1991MathematicsSubjectClassification.Primary03E35,03E55;Secondary03E05.Keywordsandphrases.PCFtheory,goodordinal,approachableordinal,theidealI[λ],in-ternallyapproachablestructure,tightstructure,squaresequence,coveringproperties,precipitousideal,mutualstationarity,stationaryreflection.ThefirstauthorwaspartiallysupportedbyNSFGrantsDMS-9703945andDMS-0070549.ThesecondauthorwaspartiallysupportedbyDMS-9803126andDMS-0101155.ThesecondandthirdauthorswerepartiallysupportedbytheUS-IsraelBinationalScienceFoundation.12JAMESCUMMINGS,MATTHEWFOREMAN,ANDMENACHEMMAGIDORwhichcapturessomeoftheapproachabilityimpliedby∗μ,impliesmanyoftheinterestingconsequencesof∗μ.Cummings[5]showedthattheassertionthateverypointisgoodseverelyconstrainsallμ+-preservingextensionsofV.Ourfirstmotivationfortheworkinthispaperistheproblemoftherelationshipbetweenthesetsofgoodandapproachablepoints.Thisproblemistrivialwhenweaksquaresexist,butnon-trivialingeneral.Forexampleitisconsistentrelativetolargecardinalsthatnoteverypointofcofinalityℵ1inℵω+1isgood.Wehavespeculatedthatperhapsthesetsofgoodandapproachablepointscoincide,andinSection3weprovethatundersomestrongstructuralhypothesesthisisthecase.Theconceptofanapproachableordinaliscloselylinkedtothatofaninternallyapproachable(IA)structure.Tobemoreprecise,thesetofapproachableordinalsofcofinalityηcanbecharacterized[16]asthesetofordinalswhichhavetheformsup(N∩μ+)forsomeinternallyapproachableNoflengthandcardinalityη.ForemanandMagidor[17]isolatedtheconceptoftightstructureintheirworkonmutualstationarityandthenon-saturationofthenon-stationaryidealonPκλ,andtightnessturnsouttobecloselyrelatedtotheissuesofgoodnessandapproacha-bility.Inparticularinternallyapproachablestructuresaretight,andifNistightthensup(N∩μ+)isgood.OursecondmotivationfortheworkinthispaperistheanalogyTightstructuresIAstructures=GoodordinalsApproachableordinalsHereisanoutlineofthepaper.Section2containssomebackgroundmaterial.•Insection3weproveatechnicalresultaboutsquare-likesequencesusingthemachineryofPCFtheory.Weusethistoshowthatundersomestruc-turalhypothesesallgoodpointsinℵω+1ofcofinalitygreaterthanℵ1areapproachable,andalsotoshowakindof“singularcardinalcompactness”forsquaresequences.ForexampleweshowthatifCHholdsandℵnholdsforallnω,thenthereisasequencehCγ:γ∈ℵω+1∩cof(ℵ2)iwhereCγisaclubsubsetofγwithordertypeℵ2andtheCγcohereatcommonlimitpointsofuncountablecofinality.•Insection4westudytheimportantpropertyofuniformityforastructure,andshowthatsufficientlyuniformstructurescanbereconstructedfromtheircharacteristicfunctions.•Insection5wecharacterizetightstructuresintermsofPCFtheory.WealsoshowthatthepropertiesofuniformityandtightnesscansometimesbepropagatedfromasetofregularcardinalsKtothesetpcf(K).•Insection6weexploretherelationshipbetweentightness,coveringprop-ertiesandinternalapproachability.Weprovetheoremsshowingthatundersomecircumstancestightnessandinternalapproachabilityareequivalent.Wealsorecordaremarksontheconnectio