Comment on “the lottery”

Great shock and deep thoughts are irrigated into my mind as soon as I finished reading the novel. I realized the tremendous power of tradition or custom. I believe that everybody shouldn’t follow the traditions without any suspicion. And it’s high time that people possess the ability of critical thinking.

Just as the old saying goes: ”Evil is terrible, but what is more terrible is the unawareness of evil”. In the novel, the villagers would sentence a person to death by lottery every year. After the result of the lottery, all the villagers would throw stones on the “lucky dog” till death. What shocks me greatly is that it has been a illogical tradition for over 70 years but all the villagers just follow it naturally. When they finished the ritual of lottery, they even went back to work immediately, like nothing had ever happened. What is more ironic is that a villager has thought of a way to improve the tradition rather than questioning its rationality. The story reminds me of the novels written by Lu Xun, a famous Chinese writer. He depicted a scene of sackless and typical “spectators” Chinese people during the period of the Republic of China. The two novels have similar ideas. Evil traditions can have terrible power that would destroy a civilization if all the people are stupid and just follow the custom without any suspicion.

Therefore, it’s vital that people should learn to think critically. It’s rather significant for today’s people. All the people should possess the ability of critical thinking nowadays. In the Age of Information, we would have access to tons of information every day. However, if we cannot judge and eliminate the bad and irrational ones, or just follow the traditions or what others do without thinking, we would be bound to be harmed greatly at length.

In general, everyone should think twice before acting. And bad tradition can do great harm to us if we don’t have the spirit of suspicion. Only in this way can we avoid this sad story from happening in our real life.

第二篇：The Lottery Preparation

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a[submittedtotheAnnalsofPureandAppliedLogic]TheLotteryPreparationJoelDavidHamkinsKobeUniversityandTheCityUniversityofNewYorkAbstract.Thelotterypreparation,anewgeneralkindofLaverpreparation,worksuniformlywithsupercompactcardinals,stronglycompactcardinals,strongcardinals,measurablecardinals,orwhathaveyou.AndliketheLaverprepa-ration,thelotterypreparationmakesthesecardinalsindestructiblebyvariouskindsoffurtherforcing.Asupercompactcardinalκ,forexample,becomesfullyindestructibleby<κ-directedclosedforcing;astrongcardinalκbecomesinde-structibleby≤κ-strategicallyclosedforcing;andastronglycompactcardinalκbecomesindestructibleby,amongothers,theforcingtoaddaCohensubsettoκ,theforcingtoshootaclubC?κavoidingthemeasurablecardinalsandtheforcingtoaddvariouslongPrikrysequences.Thelotterypreparationworksbestwhenperformedafterfastfunctionforcing,whichaddsanewcompletelygeneralkindofLaverfunctionforanylargecardinal,therebyfreeingtheLaverfunctionconceptfromthesupercompactcardinalcontext.TheLaverpreparation[Lav78],whichspectacularlymakesanysupercompactcardi-nalκindestructibleby<κ-directedclosedforcing,haslongbeenanindispensibletoolandrecognizedasanimportantmilestoneinlargecardinalsettheory.Istheresuchapreparationfortheotherlargecardinals?TheLaverpreparationdoesnotseemtoworkwithstronglycompactandothercardinals.Whilestrongcardinalsaresuccessfullytreatedin[GitShl89],thefundamentalliftingtoolscurrentlyavailablefailoutrightlywhenappliedtostronglycompactnon-supercompactcardinals.Thetechnologyhassimplynotbeenavailabletomakestronglycompactcardinalsevenpartlyindestructible.Theextentofourignoranceismadestrikinglyplainbythe

factthatthefollowingquestionhasremainedopen:

2

Question.CananystronglycompactcardinalκbemadeindestructiblebytheforcingAdd(κ,1)whichadds,byinitialsegments,aCohensubsettoκ?

InthispaperIprovideanewtechnologytoanswertheabovequestion,andtoansweritthewaythatweallhopeditwouldbeanswered:anystronglycompactcardinalκcanbemadeindestructiblebyAdd(κ,1)andmore.AndthetechniqueislimitedtoneitherstronglycompactcardinalsnortheparticularposetAdd(κ,1).Speci?cally,Ipresentherethelotterypreparation,anewkindofLaverpreparation,whichworksuniformlywithstronglycompactcardinals,supercompactcardinals,measurablecardinals,strongcardinals,orwhathaveyou,andmakesthemallin-destructiblebyavarietyofforcingnotions.

MainLotteryPreparationTheorem.Thelotterypreparationmakesavarietyoflargecardinalsindestructiblebyvariousforcingnotions.Speci?cally:

1.Thelotterypreparationofasupercompactcardinalκmakesthesupercompact-nessofκindestructiblebyany<κ-directedclosedforcing.

2.Thelotterypreparationofastronglycompactcardinalκmakesthestrongcompactnessofκindestructibleby,amongothers,theforcingAdd(κ,1)whichaddsaCohensubsettoκ,theforcingwhichshootsaclubC?κavoidingthemeasurablecardinalsandtheforcingwhichaddscertainlongPrikrysequences.

3.Thelotterypreparationofastrongcardinalκsatisfying2κ=κ+makesthestrongnessofκindestructibleby,amongothers,any≤κ-strategicallyclosedforcingandbyAdd(κ,1).

4.Withadashofthegch,level-by-levelresultsholdforpartiallysupercompactandpartiallystrongcardinals.

Theprecisedetailsareinsectionfour.Thelotterypreparation,whichisde?ned

.relativetoafunctionf..κ→κ,worksbestwhenthevaluesofj(f)(κ)canbemade

largeforthedesiredkindoflargecardinalembedding.Sincefastfunctionforcingaddsagenericfunctionfforwhichthevaluesofj(f)(κ)canbealmostarbitrarilyspeci?ed,thelotterypreparationworksespeciallywellwhenperformedafterfastfunctionforcingandde?nedrelativetothisgenericfastfunction.AninterestingrelatedresultisthefactthatfastfunctionforcingaddsanewcompletelygeneralkindofLaverfunction:

3

GeneralizedLaverFunctionTheorem.Afterfastfunctionforcingthereisa.function?..κ→(V[f])κsuchthatforanyembeddingj:V[f]→M[j(f)]withcriticalpointκ(whetherinternalorexternal)andanyzinH(λ+)M[j(f)]whereλ=j(f)(κ)thereisanotherembeddingj?:V[f]→M[j?(f)]suchthat:

1.j?(?)(κ)=z,

2.M[j?(f)]=M[j(f)],

3.j??V=j?V,and

4.IfjistheultrapowerbyastandardmeasureηconcentratingonasetinV,thenj?istheultrapowerbyameasureη?concentratingonthesamesetandmoreoverη?∩V=η∩Vand[id]η?=[id]η.

Thestandardmeasuresinclude,amongmanyothers,allnormalmeasures,allsupercompactnessmeasuresand,uptoisomorphism,allstrongcompactnessmeasures.Therestrictionthatz∈H(λ+)isnotonerousbecausethevalueofj(f)(κ)isamazinglymutable,andcanbealmostarbitrarilyspeci?ed:foranyembeddingj:V[f]→M[j(f)]andanyα<j(κ)thereisanotherembeddingj?:V[f]→M[j?(f)]suchthatj?(f)(κ)=α,j??V=j?VandM[j?(f)]?M[j(f)].Forthesereasons,thegenericLaverfunction?canbee?ectivelyusedwithalmostanykindoflargecardinalembeddingmuchasaLaverfunctionisusedwithasuper-compactnessembedding.Inthisway,fastfunctionforcingfreestheLaverfunctionconceptfromthesupercompactcardinalcontext.

Goingbackatleastto[Men74],whereseveralpreservationtheoremsareproved,settheoristshavewonderedaboutthepossibilityofmakingstronglycompactandothercardinalsindestructiblebyforcing;perhapsthelotterypreparationprovidesananswer.Thelargerquestion,though,ofpreciselyhowindestructiblethesecardi-nalscanbemadeisstillverymuchopen.ProbablythelotterypreparationprovidesmoreindestructibilitythanIwillidentifyinthispaper.Itisnaturaltohopethatanystronglycompactcardinalcanbemadefullyindestructible,perhapsbytheusualsortofreverseEastonpreparation,aniterationofclosedforcing.Thesadfact,however,withwhichIconcludethispaperisthatsuchapreparationissimplyimpossible.

ImpossibilityTheorem.Bypreparatoryforcingwhichadmitsagapbelowκ(suchasanypreparationnaivelyresemblingtheLaverpreparation),ifthemeasur-abilityofκcanbemadeindestructibleby<κ-directedclosedforcing,thenκmusthavebeensupercompactinthegroundmodel.

Thedetailsforthistheoremareinsection?ve.

§1FastFunctionForcing4

Letmequicklyexplainthestructureofthispaper.First,IintroduceWoodin’sfastfunctionforcing,showinginsectiononethatitpreservesavarietyoflargecar-dinalsandinsectiontwothatitaddsanewgeneralkindofLaverfunction.Next,Iintroducethelotterypreparation,provinginsectionthreethatitpreservesavarietyoflargecardinalsandinsectionfourthatitmakesthesecardinalsindestructiblebyvariousfurtherforcing.Lastly,insection?veIprovetheImpossibilityTheorem..Throughout,Itrytousestandardnotation,andarguefreelyinZFC.Byp..A→B,ImeanthatpisapartialfunctionfromAtoB.AndifpisaconditionintheposetP,thenbyP?pImeanthesub-poset{q∈P|q≤p}.Myfocusisalmostalwaysonκ,thelargecardinalathand,andsoinvariably,thecriticalpointofwhateverembeddingIamconcernedwithwillbedenotedbyκ.

§1FastFunctionForcing

Fastfunctionforcing,inventedbyW.HughWoodin(withaninfantformdueto.RobertSolovay),allowsonetoaddafunctionf..κ→κsuchthatthevalueofj(f)(κ)canbealmostarbitrarilyspeci?edforembeddingsj:V[f]→M[j(f)]intheextension.Thesefunctionsbehave,therefore,inacompletelygenerallargecar-dinalcontextmuchlikeLaver’sfunctiondoesinthesupercompactcardinalcontext.Indeed,inthenextsectionIwillprovethatwithafastfunctiononecanobtainacompletelygeneralkindofLaverfunctioninacompletelygenerallargecardinalsetting.AndsincetheexistenceofLaverfunctionsinthesupercompactcardinalcontexthasprovedsoindispensible—thesefunctionsappearindozensifnothun-dredsofarticles—thegeneralizedgenericLaverfunctionsheremay?ndabroadapplication.Solet’sbeginwithfastfunctionforcing.

ThefastfunctionforcingnotionFforthecardinalκconsistsofconditions.p..κ→κsuchthatdom(p)?inacchassizelessthanκandifγ∈dom(p)thenp"γ?γand|p?γ|<γ.Theconditionsareorderedbyinclusion.The(unionof.the)genericforthisforcingisthefastfunctionf..κ→κ,apartialfunctiononκ.Toemphasizetheroleofκ,IwillsometimesdenoteFbyFκ.

ByFλ,κImeantheversionoffastfunctionforcingconsistingofconditionswithdomainin[λ,κ).Itiseasytosee,bytakingtheunionofconditions,thatFλ,κis≤λ-directedclosed:theonlyapparantdi?cultyisthesupportrequirementthat|p?γ|<γforγ∈dom(p);butifγ>λisinaccessible,thenaunionofsizeλofsupportsofsizelessthanγstillhassizelessthanγ,andsothedi?cultyiseasilyaddressed.

§1FastFunctionForcing5

FastFunctionFactorLemma1.1Belowtheconditionp={?γ,α?}∈F,whereγisinaccessible,αisanordinalandλisthenextinaccessiblebeyondγandα,thefastfunctionforcingposetfactorsasF?p?=Fγ×Fλ,κ.

Proof:Ifq≤pthenq?γ∈Fγanddom(q)isdisjointfrom(γ,λ).Thus,themapq→?q?γ,q?[λ,κ)?providesthedesiredisomorphism.?Lemma

.Thus,iff..κ→κisafastfunctiononκandγ∈dom(f),thenf?γisafast

functiononγ.Moregenerally,thesameargumentshowsthatiffisafastfunctionandf"γ?γ,thenf?γ×f?[γ,κ)isgenericfortheposetFγ×Fγ,κ.Notethatifγisregular,then|Fγ|≤γ.

RemarkonGapForcing1.2Foratechnicalreasonwhichwillbemadeclearlater,IwillattimeswanttoprecedetheforcingFwithsomesmallforcing,suchasaddingaCohensubsettotheleastinaccessiblecardinal.Thiskindofforcingisgenerallybenigninthelargecardinalcontext,andIwillregardthissmallforcingasapartoffastfunctionforcingwhenevertheneedarises.TheprimaryreasontodosoisthatforcingoftheformP1?P2,where|P1|<δandP2is≤δ-strategicallyclosedinVP1issaidin[Ham98]and[Ham∞]toadmitagapatδ.TheGapForcingTheoremof[Ham∞],withaforerunnerin[Ham98],assertsthatafterforcingV[G]whichadmitsagapatδ<κ,anyembeddingj:V[G]→M[j(G)]forwhichM[j(G)]isclosedunderδ-sequencesinV[G]—andthisincludesanyultrapowerembeddingonanyset,aswellasmoststrongnessextenderembeddings—isaliftofanembeddingfromthegroundmodel.Thatis,M?Vandj?V:V→Misanembeddingwhichisde?nableinV.Additionally,ifM[G]isλ-closedinV[G]forsomeλ≥δ,thenMisλ-closedinV,andinparticular,j"λ∈M;andifjisaλ-strongnessembeddinginducedbyanaturalextender,whereλiseitherasuccessorordinalorhasco?nalitymorethanδ,thenVλ?M.Thus,theresultsof[Ham∞]showthatgapforcingcannotcreatenewmeasurablecardinals,strongcardinals,stronglycompactcardinals,supercompactcardinals,andsoon,withlevel-by-levelversionsgenerallyavailable.Inordertoappealtothistheorem,therefore,inthecontextoffastfunctionforcing,IwillintroduceaverylowgapbyprecedingFbysomeverysmallforcingandhereafterregardthissmallforcingasapartoffastfunctionforcing,thoughIwillmentionitonlywhenIwanttoapplytheGapForcingTheorem.?Remark

Woodinde?nedfastfunctionforcinganduseditwithsomethingbelowastrongcardinal(in[CumWdn]heandCummingshadanembeddingj:V→MsuchthatMκ?Mandj(κ)>κ++).Hisargument,whichIgivebelow,worksequallywell

§1FastFunctionForcing6

withmeasurableandsupercompactcardinalsand,inamodi?edform,withweaklycompactcardinals.Asigni?cantcontributionofthispaperisthatfastfunctionforcingworksalsowithstronglycompactcardinals.Forpresentationalclarity,Iwillpresentthefastfunctionliftingtechniquesinthelargecardinalorder,ratherthanthetemporalorderinwhichthetheoremswere?rstproved.

FastFunctionTheorem1.3Fastfunctionforcingpreservesallcardinalsandco-?nalitiesanddoesnotdisturbthecontinuumfunction.Consequently,fastfunctionforcingpreservesallinaccessiblecardinals.

Proof:SupposethatγisregularinVbuthasco?nalityδ<γinV[f].SinceFhassizeκ,wemayassumethatγ≤κ.Therearetwocases.First,itmayhappenthatf"δ?δ.Inthiscase,theforcingfactorsasFδ×Fδ,κ.TheinitialforcingFδ,however,istoosmalltocollapsetheco?nalityofγandthetailforcingFδ,κis≤δ-distributiveandsocannotcollapsetheco?nalityofγtoδ;thiscontradictsourassumption.Second,alternatively,itmayhappenthatf(β)>δforsomeβ<δ.Inthiscase,theforcingfactorsasFβ×Ff(β),κ,andagaintheinitialforcingistoosmalltocollapsetheco?nalityofγandthetailforcingistoodistributivetocollapseittoδ;soagainwereachacontradiction.Thus,fastfunctionforcingpreservesallcardinalsandco?nalities.

Asimilarargumentshowsthatfastfunctionforcingpreservesthevaluesof2δcalculatedinthegroundmodel.Againwesplitintothetwocases.Iff"δ?δ,thenwemayfactortheforcingasFδ×Fδ,κ.Theinitialforcinghassizeδandthetailforcingis≤δ-distributive;soneithercana?ectthevalueof2δ.Alternatively,iff(β)≥δforsomeβ<δ,thenwemayfactortheforcingasFβ×Ff(β),κandmakethesameargument.Sothevalueof2δispreserved,andthetheoremisproved.?TheoremFastFunctionTheorem1.4Fastfunctionforcingpreserveseveryweaklycom-pactcardinal.Indeed,ifκisweaklycompactinV,thenafteraddingafastfunction.f..κ→κthereareweaklycompactembeddingsj:N[f]→M[j(f)]suchthatj(f)(κ)isanydesiredordinaluptoκ+.

.Proof:SupposeκisweaklycompactinVandf..κ→κisafastfunction.Since

theprevioustheoremestablishesthatκremainsinaccessibleinV[f],inordertoprovethatκisweaklycompactthereitsu?cesonlytoverifythatκhasthetree

˙fforsomenameT˙.InVproperty.SosupposeTisaκ-treeinV[f].Thus,T=T

letNbeatransitiveelementarysubstructureofH(κ+)ofsizeκwhichcontainsF

˙andisclosedunder<κ-sequences.SinceκisweaklycompactinVthereisanandT

embeddingj:N→Mwithcp(j)=κ.ImayassumethatM={j(h)(κ)|h∈N}

§1FastFunctionForcing7

(byreplacingMwiththeMostowskicollapseofthissetifnecessary).ItfollowsthatM<κ?MinVsinceif?a=?aα|α<β?isasequenceinVofelementsofMforsomeβ<κ,thenbuildthefunctionH(δ)=?hα(δ)|α<β?,whereaα=j(hα)(κ)andhα∈N.BytheclosureassumptiononNitfollowsthatH∈Nandconsequentlyj(H)∈M,so?a=?j(hα)(κ)|α<β?∈M,asdesired.Returningtothemainargument,now,foranyα<j(κ)theconditionp={?κ,α?}isinj(F).ThetailforcingFλ,j(κ),whereλisthenextinaccessibleofMbeyondκandα,is<κ-closedinM.SincethereareonlyκmanydensesetsforthisforcinginM,wemaylinethemupintoaκ-sequenceanddiagonalizetomeettheminordertoproduceinVanM-genericftail?Fλ,j(κ).Thecombinationj(f)=f∪p∪ftailisM-genericforj(F)andconsequentlytheembeddingliftstoj:N[f]→M[j(f)].

˙fitfollowsthatT∈N[f].Byconstruction,j(f)(κ)=p(κ)=α.NowsinceT=T

SinceTisaκ-treeitfollowsthatj(T)isaj(κ)-tree.Anyelementontheκthlevelofj(T)providesaκ-branchthroughT.SoκhasthetreepropertyinV[f],asdesired.Andsincetheorginaljcanbechosensothatj(κ)isaslargebelowκ+asdesired,andαcanbeaslargebelowj(κ)asdesired,thevalueofj(f)(κ)canbeanyordinaluptoκ+.

Byemployingfactorargumentsasintheprevioustheorem,itiseasytoseemoregenerallythatallweaklycompactcardinalsarepreserved.?Theorem

FastFunctionTheorem1.5(Woodin)If2κ=κ+,thenfastfunctionforcingwithκpreservesthemeasurabilityofκ.Indeed,everyultrapowerj:V→MbyameasureonκinVliftstoanultrapowerj:V[f]→M[j(f)]suchthatj(f)(κ)isanydesiredordinaluptoj(κ).Consequently,theliftedembeddingcanbetheultrapowerbyanormalmeasure,evenwhentheoriginalembeddingwasnot.

.Proof:Supposethatκismeasurableand2κ=κ+inV,thatf..κ→κisafast

functionandthatj:V→Mistheultrapowerembeddingbyameasure?onκandα<j(κ).Belowtheconditionp={?κ,α?},factortheforcingasF×Ftail,whereFtail=Fλ,κforthenextinaccessibleλinMbeyondκandα.TheforcingFtailis≤κ-closedinM.Since2κ=κ+,asimplecountingargumentshowsthat|j(κ+)|V=κ+.Consequently,sinceMκ?M,wecanlineupallthedensesetsofMfortheforcingFtailanddiagonalizeagainstthemtoproduceinVanM-genericftail.Thus,inV[f]wemaylifttheembeddingtoj:V[f]→M[j(f)]wherej(f)=f∪p∪ftail.Soκremainsmeasurablethere,asdesired.Byconstructionj(f)(κ)=p(κ)=α.

Somereadersmaybesurprisedbythe?nalconclusionofthetheorem.Supposeα=[id]?.Bythestandardseedtechniques(e.g.see[Ham97]),itfollowsthat

§1FastFunctionForcing8

M={j(h)(α)|h∈V}.Ifasabovewearrangetheliftj:V[f]→M[j(f)]insuchawaythatj(f)(κ)=α,thenitiseasytoseethatM[j(f)]={j(h)(κ)|h∈V[f]},i.e.theseedκgeneratesthewholeembedding.Fromthis,itfollowsthattheliftedembeddingjistheultrapowerbythenormalmeasureν={X|κ∈j(X)}inV[f],asdesired.Ielaboratedonthisphenomenonin[Ham94].?Theorem

Thefactorargumentsemployedin1.3easilyextendtoshowthatfastfunctionforcingforκpreservesallmeasurablecardinalsatwhichthegchholds.Thegeneralphenomenonthatthevalueofj(f)(κ)canbeanyordinaluptoj(κ)isfurtherexplainedintheFastFunctionFlexibilityTheorembelow.

Recallthatacardinalκisstrongwhenforeveryλitisλ-strong,sothatthereisanembeddingj:V→MwithcriticalpointκsuchthatVλ?M.Ifthereissuchanembedding,thenbyfactoringthroughbythecanonicalextender,thereisonesuchthatM={j(h)(s)|h∈V&s∈Vλ};onesimplyreplacesjwithπ?j,whereπistheMostowskicollapseofthisset.Furthermore,ifλiseitherasuccessorordinalorhasco?nalityatleastκ,thenforsuchanembeddingMisclosedunderκ-sequencesinV.

FastFunctionTheorem1.6(Woodin)If2κ=κ+,thenfastfunctionforcingpreservesthestrongnessofκ.

Proof:Theresultiscompletelylocal,sinceIwillshowthatifκisλ-stronginVthenthisispreservedtothefastfunctionextensionV[f].Supposej:V→Mwitnessestheλ-strongnessofκ,sothatVλ?M.Letδ=|Vλ|.Usingthecanonicalextender,ImayassumethatM={j(h)(s)|h∈V&s∈δ<ω}.Letp={?κ,δ?}betheconditionwhichjumpsuptoδatκ.Thus,bytheFactorLemma1.1,belowptheforcingj(F)factorsasF×Ftail,whereFtailis≤δ-closedinM.Nowusethepair?κ,δ?asaseedtoformtheseedhullX={j(h)(κ,δ)|h∈V}?Mandobtainthefactorembedding

V

k-M

wherek:M0→MistheinverseofthecollapseofX.SinceκandδareinX,itfollowsthatk(δ0)=δforsomeδ0<j0(κ),thatk(p0)=pforp0={?κ,δ0?}∈j0(P)

§1FastFunctionForcing9

andthatcp(k)>κ.Theembeddingj0:V→M0,beinggeneratedbytheseed?κ,δ0?,issimplyanultrapowerbyameasureonκ.Inparticular,since2κ=κ+,thediagonalizationargumentof1.5providesaliftj0:V[f]→M0[j0(f)]belowthe

M0M0conditionp0.Itmustbethatj0(f)=f∪p0∪ftail,whereftailisM0-genericforthe

M0≤δ0-closedforcingFtail.

IclaimthatkliftstoM0[j0(f)].First,sincecp(k)>κ,certainlyweknowthatkliftstok:M0[f]→M[f].Inordertoliftktherestofthewayitsu?cestoshow

M0?FtailisM-generic.So,supposeD∈MisopenanddenseinFtail.thatk"ftail?)(s)forsomeSinceM={k(h)(s)|h∈M0&s∈δ<ω},itfollowsthatD=j(D

?=?Dσ|σ∈δ<ω?inM0ands∈δ<ω,whereeveryDσisanopendensesubsetD0M0M0ofFtail.SinceFtailis≤δ0-closedinM0,itfollowsthat

M0M0D)?D.Thus,sinceftailisM0-generic,k"ftail

meetsD,asdesired.Consequently,kliftsfullytok:M0[j0(f)]→M[k(j0(f))],wherek(j0(f))isthe?ltergeneratedbyk"j0(f).Thecompositionk?j0providesaliftofjtoj:V[f]→M[j(f)].SinceVλ?Mandf∈M[j(f)],itfollowsthat(V[f])λ?M[j(f)],andsoκisstillλ-stronginV[f],asdesired.?Theorem

Thenexttheoremprovidesthe?rstnontrivialexampleofthepreservationofanarbitrarystronglycompactcardinalofwhichIamaware.IwillmakeakeyuseofanoldtechniqueofMenas[Men74],usedalsoin[Apt98],inordertoknowthatthecardinalremainsstronglycompactafterforcing(MenasandApterbothneedastronglycompactlimitofsupercompactcardinals).ArthurApterhaspointedoutthatMenas’stechnique,anachronisticallypresentedinthe‘dark’agesbeforetheLaverpreparation[Lav78],probablyhadmuchunrealizedpotential.Ihopethattheresultsinthispapertendtocon?rmhisview.

FastFunctionTheorem1.7Fastfunctionforcingpreservesthestrongcompact-nessofκ.Indeed,everystrongcompactnessmeasurefromVextendstoastrongcompactnessmeasureintheextension.

.Proof:Supposethatf..κ→κisaV-genericfastfunction,thatλ≥κ,andthat?0<κisa?nemeasureonPκλinV.Letθ≥2λ,andletj:V→Mbeanyθ-strongly

compactembedding,theultrapowerbya?nemeasureηonPκθinV.Bythecoverpropertyforstronglycompactembeddings,thereisasetY∈Msuchthatj"?0?Yand|Y|M<j(κ).ImayassumethatY?j(?0),andconsequently∩Y∈j(?0).Anyelements0∈∩Yisaseedfor?0inthesensethatX∈?0?s0∈j(X)foranyX?PκλinV.Fixsuchans0.Lets=[id]ηandδ=|s|M,andpickanyγ≥δ.Thus,sinceηisa?nemeasureonPκθ,wehavej"θ?s∈j(Pκθ)andθ≤δ<j(κ).

§1FastFunctionForcing10

Now,inj(F),letpbethecondition{?κ,γ?}.BytheFastFunctionFactorLemma,theforcingj(F)factorsbelowthisconditionasF×Ftail,whereFtailis≤γ-closedinM.ForcetoaddaV[f]-genericftail?Ftail,andletj(f)=f∪p∪ftail.Bythefactorization,thisisM-genericforj(F),andconsequentlytheembeddingliftsinV[f][ftail]toj:V[f]→M[j(f)].Byconstruction,j(f)(κ)=p(κ)=γ.SincetheforcingFtailwas≤γ-closedinM,itis≤γ-distributiveinM[f];inparticular,itaddsnonewsubsetstoδoverM[f].Let??0bethemeasuregerminatedbytheseeds0viatheliftedembedding,sothatX∈??0?s0∈j(X)forX∈V[f].Itiseasytoseethat??0measureseverysetinV[f],thatitextends?0,and,sincej"λ?s0,thatis?ne.Itremainsonlyformetoshowthat??0∈V[f].ForthisIwilluseMenas’skeyideain

˙α|α<θ?forthesubsetsofPκλin[Men74].EnumerateinVthenicenamesu=?X

<κV[f](asimplecountingargumentshowsthatthereare2λmanyofthem).Thus,

˙β|β∈s?,j(u)∈Mandconsequentlyalsoj(u)?s∈M.Enumeratej(u)?s=?Y

˙β)j(f)}.˙j(α)=j(X˙α)forα<θ.Lett={β∈s|s0∈(YandobservethatY

Thus,t?sandt∈M[j(f)].SinceshassizeδandFtailis≤δ-distributiveinM[f],itfollowsthatt∈M[f],andthereforet∈V[f].Nowsimplyobservethat˙α)f∈???s0∈j((X˙α)f)=j(X˙α)j(f)=(Y˙j(α))j(f)?j(α)∈t.So??isde?nable(X00

inV[f]fromtandj?θ.Thus,??0∈V[f]asdesired.?Theorem

FastFunctionTheorem1.8Fastfunctionforcingpreservesthesupercompact-nessofκ;andeverysupercompactnessmeasurefromthegroundmodelextendstoasupercompactnessmeasureintheextension.

Proof:Toseethatthesupercompactnessofκispreserved,onecansimplytakeηtobeaθ-supercompactnessembeddinginthepreviousargumentand?0theλ-supercompactnessmeasuregerminatedviajbytheseeds0=j"λ.Theresultingmeasure??0iseasilyseentobenormaland?ne.SoκremainssupercompactinV[f].

Sonowletmeshowabitmore;namely,thateverysupercompactnessmeasurefromVextendstoasupercompactnessmeasureinV[f].SupposeinVthat?0isaλ-supercompactnessmeasureonPκλandη0isaθ-strongcompactnessmeasure<κonPκθforsomeθ≥2λ.Itisnotdi?culttoargue(seetheargumentprecedingTheorem4.2)thatη=?0×η0isisomorphictoaθ-strongcompactnessmeasurewhoseembeddingj:V→Misclosedunderλ-sequences.Furthermore,s0=j"λisaseedfor?0viaj.Thepreviousargumentshowshowtoliftthisembeddingsothatthemeasure??0germinatedbys0viaj:V[f]→M[j(f)]liesinV[f].Again,itisnotdi?culttoarguethat??0isnormaland?ne,asdesired.?Theorem

§1FastFunctionForcing11

Thepreviousargumentactuallyestablishesthefollowingtheorem:

LocalVersion1.9Ifκis2λ-stronglycompactthenfastfunctionforcingpre-servestheλ-strongcompactnessofκ.Thesameholdsforsupercompactness.In-<κdeed,ifκis2λ-stronglycompactandλ-supercompact,thenfastfunctionforcingpreservestheλ-supercompactnessofκ.

Bypayingaslightgchpenalty,wecanemploythediagonalizationargumenttoobtainacompletelylocalversion:

CompletelyLocalVersion1.10(Woodin)Ifκisλ-supercompactand2λ=λ+,thenthisispreservedbyfastfunctionforcing.Indeed,everyλ-supercompact-nessembeddinginthegroundmodelliftstotheforcingextension.

Proof:Supposej:V→Misaλ-supercompactembeddinginV,andthatfisaV-genericfastfunction.Letpbethecondition{?κ,λ?},sothatbelowptheforcingj(F)factorsasF?FtailandFtailis≤λ-closedinM.Sinceasimplecountingargumentshows|j(2κ)|=λ+,thereareatmostλ+manyopendensesubsetsofFtailinM,countedinV.Thus,usingtheclosureofMandtheclosureoftheforcing,ImaylinethemupanddiagonalizeagainstthemtoconstructinVanM-generic?lterftail?Ftail.Consequently,inV[f]theembeddingliftstoj:V[f]→M[j(f)]wherej(f)=f∪p∪ftail,anditisnotdi?culttoverifythatthisembeddingisaλ-supercompactembeddinginV[f].?Theorem

Becausetheprevioustheoremsshowthatfastfunctionforcingpreserveslargecardinals,oneexpectsmanyembeddingsj:V[f]→M[j(f)]inthefastfunctionex-tension.Whatismore—andthisisthefundamentalfactwhichmakesfastfunctionforcinguseful—thenexttheoremshowsthattheseembeddingsaresoeasilymodi-?edthatthevalueofj(f)(κ)canbealmostarbitrarilyspeci?ed.Letmede?nethatameasureηinV[f],oranyforcingextension,isstandardwhenthecriticalpointκoftheinducedembeddingj:V[f]→M[j(f)]isde?nableinM[j(f)]froms=[id]?andparametersinran(j?V).Thus,anynormalmeasureonκisstandard,asisanysupercompactnessmeasure(sinceκistheleastelementnotinj"λ).Also,Lemma

2.7belowshowsthatinthetypeofforcingextensionsofthispaper,everyθ-strongcompactnessmeasureisisomorphictoastandardstrongcompactnessmeasure.<κ<κ

FastFunctionForcing12

.FastFunctionFlexibilityTheorem1.11Supposethatf..κ→κisafast

functionaddedgenericallyoverVandthatj:V[f]→M[j(f)]isanembedding(eitherinternalorexternaltoV[f])withcriticalpointκ.Thenforanyα<j(κ)thereisanotherembeddingj?:V[f]→M[j(f)]suchthat:

1.j?(f)(κ)=α,

2.j??V=j?V,

3.M[j?(f)]?M[j(f)],and

4.Ifαisnottoomuchlargerthanκ(seebelow),thenM[j?(f)]=M[j(f)].Inthiscase,ifjistheultrapowerbyastandardmeasureηconcentratingonasetinV,thenj?istheultrapowerbyastandardmeasureη?concentratingonthesameset,andmoreoverη∩V=η?∩Vand[id]η=[id]η?.

Proof:Fixjandα.Letκ?bethenextinaccessibleabovebothκandα,andletγbethenextelementofdom(j(f))aboveκ?.Thus,γisnotalimitofinaccessiblecardinals.BytheFastFunctionFactorLemma,ftail=j(f)?[γ,j(κ))isM-genericforFγ,j(κ).Nowconsidertheembeddingj?V:V→M(whichperhapsmaynotbede?nableinV),andtheconditionp={?κ,α?,?κ?,β?}whereβ<γislargerthaneveryinaccessiblebelowγ.§1

Belowthiscondition,theforcingj(F)factorsasF×Fγ,j(κ).SincewehaveM-genericsfortheseposets,wecanletj?(f)=f∪p∪ftailandlifttheembeddingtoj?:V[f]→M[j?(f)].Byconstructionwehavej?(f)(κ)=p(κ)=αandj??V=j?V.Also,j?(f)iseasilyconstructedfromj(f),soM[j?(f)]?M[j(f)].Finally,inthecasethatαdoesnotexceedthenextinaccessibleclusterpointofdom(j(f))beyondκ,thenitfollowsthatthe‘missing’partofj(f),namelyj(f)?[κ,γ),issimplyaconditioninj(P),andthereliesinM.InthiscaseM[j(f)]=M[j?(f)].

Finally,supposeinthiscasethatjistheultrapowerbyastandardmeasureηconcentratingonasetD∈V.Lets=[id]η.ThisisaseedforηinthesensethatX∈η?s∈j(X).SinceD∈ηitfollowsthats∈j(D)andconsequentlys∈M.Foratechnicalreason,Iwillchooseβinthepreviousargumenttobeanindexoftheconditionj(f)?[κ,γ)∈Mwithrespecttoj(?a)where?aisanenumerationofVκin

§1FastFunctionForcing13

Vsuchthatforeveryξ<κeveryelementofVξappearsunboundedlyoftenamongthe?rst?ξmanyelementsof?a.Letη?bethemeasuregerminatedbytheseedsviaj?;i.e.X∈η??s∈j?(X).Inordertoarguethatj?istheultrapowerbyη?,itsu?cestoshowthateveryelementofM[j?(f)]hastheformj?(h)(s)forsomeh∈V[f](see[Ham98]foranelementaryintroductiontotheseseedtechniques).LetXbetheseedhullofs,thatis,thesetoftheelementsinM[j?(f)]havingthisform.ItiseasytoverifytheTarski-Vaughtcriterion,andsoX?M[j?(f)].Furthermore,sincethemeasureηwasstandard,itfollowsthatκ∈X(andthisistheonlyreasonforthatassumption).Consequently,β∈Xandsobythetechnicalchoiceofβthemissingpartofj(f)alsoliesinX.Thus,fromj?(f)∈Xwecanreconstructj(f),andsoj(f)∈X.Now,supposex∈M[j?(f)]=M[j(f)].Sincejistheultrapowerbyηweknowthatx=j(h)(s)forsomefunctionh∈V[f].Thisfunctionhasa

˙∈V.Sox=j(h˙f)(s)=j(h˙)j(f)(s).Sinceallthesetsinthislastexpressionnameh

areinX,itmustbethatx∈Xalso;soj?istheultrapowerbyη?.Therestofthetheoremfollowsbecause[id]η?=s=[id]ηandj??V=j?V.?Theorem

Inthecontextofastronglycompactcardinalκ,Menaswasveryconcernedin.[Men74]withthesituationinwhichthereisafunctionf..κ→κwithwhatIwillcalltheMenasproperty,namely,thatforeveryλthereshouldbea?nemeasure?onPκλwithultrapowerembeddingj:V→Msuchthatj(f)(κ)≥|[id]?|M.Thesefunctions?guredcruciallyinhispreservationarguments.Menasprovedthateverystronglycompactlimitofstronglycompactcardinalshassuchafunction,butconjecturedthatthiswouldnotbethecaseforeverystronglycompactcardinal.Iwillprovehere,however,thatonecanhavesuchafunctionforanystronglycompactcardinal.

Theorem1.12EveryfastfunctiononastronglycompactcardinalhastheMenasproperty.

Proof:ThisalmostfollowsdirectlyfromtheFlexibilityTheorem,exceptforthedi?cultythatforlargeαtheembeddingj?producedintheFlexibilityTheoremmaynotitselfbeaλ-strongcompactnessembedding;soanadditionalfactorargumentisneeded.Supposethatj:V[f]→M[j(f)]isaλ-stronglycompactembeddingbysomemeasure?inV[f].Letγ=|s|wheres=[id]?.Sinces∈M[j(f)]andshassizeγitfollowsthats∈M[j(f)?γ],andsoithasanames˙∈Mofsizeγ.Sincej"λ?sby?neness,wemayusethenames˙tobuildasets?∈Mofsizeγsuchthatj"λ?s?.Furthermore,wemayassumeκistheleastelementnotins?,bysimplyremovingitifnecessary.Nowletj?:V[f]→M[j?(f)]beanembeddingasintheFlexibilityTheoremsuchthatj?(f)(κ)=αforsomeα>γ.Let??be

§1FastFunctionForcing14

themeasuregerminatedbytheseeds?viaj?,sothatX∈???s?∈j?(X).Sincej?"λ=j"λ?s?∈j?(Pκλ),thismeasureisa?nemeasureonPκλ,andsinceitwasobtainedbyaseedviaj?,weobtainthefollowingfactordiagram:

V[f]

k-M[j?(f)]

wherej0istheultrapowerby??andkistheinversecollapseoftheseedhullX={j?(h)(?s)|h∈V[f]}?M[j?(f)].Sinceκistheleastelementnotins?,itfollowsthatκ∈Xandhencealsoα=j?(f)(κ)∈X.Lets0andα0bethecollapsesofs?andα,respectively,sothatk(s0)=s?andk(α0)=α.Itfollowsthat[id]??=s0andj0(f)(κ)=α0>|s0|,sofhastheMenaspropertywithrespectto??,asdesired.?Theorem

TheMenaspropertyhasanaturalanalogueforsupercompactandstrongcar-.dinals.Speci?cally,Ide?neforasupercompactcardinalκthatf..κ→κhasthesupercompactMenaspropertywhenforeveryλthereisaλ-supercompactnessembeddingjforwhichj(f)(κ)>λ.Thus,forexample,everyLaverfunctionhastheMenasproperty.Forastrongcardinalκ,Ide?nethatfhasthestrongMenaspropertywhenforeveryλthereisaλ-strongembeddingjforwhichj(f)(κ)>?λ.Suchfunctionsarerelatedtothehigh-jumpingfunctionsof[Ham98].

Theorem1.13Everyfastfunctiononasupercompactcardinalhasthesupercom-pactMenasproperty.

Proof:Thistheoremistruelevel-by-levelforpartiallysupercompactcardinals.Supposethatj:V[f]→M[j(f)]isaλ-supercompactnessembeddinginV[f],theultrapowerbyanormal?nemeasureηonPκλ.ByRemark1.2,weknowthatM?Vandinfactj:V→Misde?nableinVandMisλ-closedthere(thoughitneednotbetheultrapowerbyanormalmeasureonPκλthere).Iclaimthatj(f)?[κ,λ)isinM.Ifnot,thenpartofitmustbegenericoverMforsomenontrivial≤κ-closedforcingofsizeatmostλ,namely,Fκ,γ,whereγisthe?rstinaccessibleclusterpointofdom(j(f))beyondκ.SincethisposetisthesameinMasinV,withthesamedensesets,the?ltergeneratedbyj(f)?[κ,γ)inFκ,γmustbeV-generic.Butthisisimpossible,sinceitwasaddedbytheκ-c.c.forcing

§1FastFunctionForcing15

F.Soj(f)?[κ,λ)∈M.Consequently,bytheFlexibilityTheorem,wemaymodifytheembeddingtoj?:V[f]→M[j?(f)]sothatj?(f)(κ)=αforsomeα>λandM[j?(f)]=M[j(f)].Furthermore,wemayassumethatj?istheultrapowerbyameasureη?with[id]η=[id]η?.Sinceηisnormaland?ne,itfollowsthat[id]η=j"λ,so[id]η?=j"λ=j?"λ,andsoη?isalsoanormal?nemeasureonPκλ.Finally,sincej?(f)(κ)=α>λ,themeasureη?exhibitsthatfhastheMenaspropertyforaλ-supercompactnessembedding.?Theorem

Thepreviousproofinfactshowsthatthefunctionh:κ→κ,whereh(γ)isthenextinaccessibleclusterpointofdom(f)beyondγ,isahigh-jumpingfunctionintheterminologoyof[Ham98].Itfollows,byTheorem3.4of[Ham98],thatfastfunctionforcingmustdestroythealmosthugenessofκ.

Theorem1.14EveryfastfunctiononastrongcardinalhasthestrongMenasproperty.

Proof:Thistheoremisalmosttruelevel-by-level.Speci?cally,Iwillshowthatifκis(λ+1)-stronginV[f]thenfhastheMenaspropertywithrespectto(λ+1)-strongembeddingsinV[f].Itfollows,usingtheusualfactorargumentandtheinducedλ-strongextender,thatfhastheMenaspropertywithrespecttoaλ-strongembeddingalso.So,supposej:V[f]→M[j(f)]isa(λ+1)-strongembedding,sothatVλ+1[f]?M[j(f)].ByfactoringthroughbythenaturalextenderImayassumethatM[j(f)]isclosedunderκ-sequencesinV[f],andconsequently,byRemark1.2,thatM?VandfurthermoreMλ+1=Vλ+1.Iwillargueasintheprevioustheoremthatj(f)?[κ,λ)∈M.Ifthisfails,thentheremustbesomeγ≤λwhichisaninaccessibleclusterpointofdom(j(f)),andj(f)?[κ,γ)isM-genericforforFMκ,γ.SinceFκ,γisthesamewhethercomputedinVorM,andVandMhavethesamedensesetsforit,itfollowsthatj(f)?[κ,γ)isactuallyV-genericforFκ,γ.Butthisisimpossiblesincej(f)∈V[f],aκ+-c.c.forcingextensionofV,andFκ,γis≤κ-closed.Soj(f)?[κ,γ)mustjustbeaconditioninj(F)andhenceanelementofM.Now,wecontinueasintheprevioustheorem.BytheFlexibilitytheorem,thereisanotherembeddingj?:V[f]→M[j?(f)]suchthatj?(f)(κ)>?λ+1andM[j?(f)]=M[j(f)].Thereisnotroublemakingj?(f)(κ)largerthan?λ+1sincetheproofoftheFlexibilityTheoremshowsthatitcaneasilybepushedupbeyondthenextinaccessibleaboveλ.Thus,j?:V[f]→M[j?(f)]has(V[f])λ+1?M[j?(f)]andj?(f)(κ)>?λ+1,asdesired.?Theorem

Letmeconcludethissectionwithaquickapplicationoffastfunctionforcing.KunenandParis[KunPar71]werethe?rsttoshowthatameasurablecardinalκcan

§1FastFunctionForcing16

havemanynormalmeasuresinaforcingextension.Thefollowingargumentshowsthatfastfunctionforcingworksnicelytoseethesamefactforavarietyoflargecardinals.

ManyMeasuresTheorem1.15Fastfunctionforcingwithκaddsmanymea-sures.Speci?cally,

1.Every(su?cientlynice)weakcompactness?lteronκinVextendstoκmanyweakcompactness?ltersinV[f].

2.If2κ=κ+,theneverymeasureonκinVextendsto22manymeasuresinV[f],themaximumconceivablenumber.Indeed,everymeasureinVextends

κto22manymeasures,eachisomorphicinV[f]toadistinctnormalmeasure.

3.Ifκis2λ-stronglycompactinVthenthereareλ+manynon-isomorphicλ-strongcompactnessmeasuresinV[f].Thus,ifalso2κ=κ+,thenthereareκ22·λ+many.

4.If=λtheneveryλ-supercompactnessmeasureinVextendsto

manyλ-supercompactnessmeasuresinV[f],themaximumconceivablenumber.Proof:Thoughitisabitmoreworktogettheoptimalbounds,thistheoremfollowsinspiritfromtheFlexibilityTheorem;essentially,thefactthatj(f)(κ)canhavemanydi?erentvaluesmeansthattheremustbemanydi?erentmeasures.Thus,1holdsforthenice?ltersImanagedtoliftintheprevioustheorems,becauseforeachweakcompactnessembeddingthereareκmanypossiblevaluesforj(f)(κ).

Letmeprove2.Thesimpleideaoflookingatthepossiblevaluesofj(f)(κ)

κeasilygives2κmanymeasures;inordertoget22manymeasures,Iwillconsider

thepossiblevaluesofj(f).SupposeκismeasurableinVand?isanymeasureonκwithembeddingj?:V→M.FastFunctionTheorem1.5showsthattherearemanyliftsofj?toj:V[f]→M[j(f)].Howmanyarethere?Well,theproofproceededbydiagonalizingagainstthedensesetsofM,andtherearediversewaystocarryoutthisdiagonalization.Speci?cally,belowanyconditioninj(F)thereisanantichainofsizej(κ),whichhassizeκ+inV.Thuswecanbuildatreeofheightκ+ofdescendingconditonsinFtailsuchthateverynodesplitsintoanantichainofsizeκ+onthenextlevel.Furthermore,wecanarrangethateverynodeontheαthlevelofthistreeisintheαthdensesetofM,sothatanyκ+-branchthroughthistree+κwillproduceanM-genericforFtail.Sincethereareκ+κ=22manyκ+-branches

κthroughthistree,thereare22manywaystoperformthediagonalization,and

eachoftheresultinggenericsproducesadi?erentj(f),andconsequentlyadi?erentmeasureinV[f].SowehavemanymeasuresinV[f].Now,letmearguethatwe<κλ2+λ<κ22κ<κ

§1FastFunctionForcing17

canarrangeforalloftheseembeddingstobeultrapowersbynormalmeasuresinV[f].Ifwebuildthetreebelowtheconditionwhichensuresj(f)(κ)=α,whereα=[id]?isthecanonicalseedfor?,thenwithrespecttotheliftedembeddingj:V[f]→M[j(f)],theseedhullofκgeneratestheoldseedαandconsequentlyallofM[j(f)](seetheOldSeedLemmaof[Ham97]).Thus,sincetheentireembeddingisintheseedhullofκ,theembeddingisanembeddingbythenormalmeasureηinducedbyκ.Ifνisthemeasuregerminatedbytheseedαwithrespecttoj,then?extendstoνsinceweliftedtheembedding.Butsinceκgeneratesαandviceversa,themeasuresνandηareisomorphic,so2holds.

Statement4holdssimilarly.Supposethat2λ=λ+andj:V→Misaλ-supercompactnessembedding.Thediagonalizationtechniqueof1.10showsthatwemaylifttheembeddingtoj:V[f]→M[j(f)].Again,wecanbuilda≤λ-closedtreeofheightλ+andλ+branchingateachnodesuchthatanybranchthroughthistreeprovidesadi?erentgenericj(f)withwhichtolifttheembedding.BytheOldSeedLemmaof[Ham97],theseedj"λstillgeneratesthewholeembedding,andconsequentlyeachoftheseliftsprovidesadi?erentλ-supercompactnessmeasureliftingandextendingtheoriginalmeasure.Thus,thereare(λ=manymeasuresextendingtheoriginalmeasure,asdesired.Byworkingbelowaconditionwhichforcesj(f)(κ)=λ+1,wecanarrangethatallthesemeasureswitnesstheMenaspropertyoff.

Finally,letmeprove3.By1.12,thereare?nemeasures?onPκλinV[f]witnessingtheMenaspropertyoff,sothatthecorrespondingembeddingj:V[f]→M[j(f)]hasj(f)(κ)>|s|wheres=[id]?.Furthermore,wemayassumethatκistheleastelementnotins,bysimplyremovingitifnecessaryandworkingwiththeinducedisomorphicmeasure;sowemayassumethatthemeasure?isstandard.Thus,bystatement4intheFlexibilityTheorem,wemayforanyα<λ+?ndanembeddingj?:V[f]→M[j?(f)],theultrapowerbyameasure??with

[id]??=s,suchthatj?(f)(κ)=α,j??V=j?VandM[j?(f)]=M[j(f)].Sincej?"λ=j"λ?s∈j?(Pκλ),itfollowsthat??isa?nemeasureonPκλ.Andsincedi?erentchoicesofαprovidedi?erentembeddingsj?,thesemeasuresareallpairwisenon-isomorphic,andsowehaveλ+manymeasures.Finally,sincetheargumentbeforeTheorem4.2showsthattheproductofanormalmeasurewithastrongcompactnessmeasureisisomorphictoastrongcompactnessmeasure,by2if

κ2κ=κ+thereareatleast22manystrongcompactnessmeasuresonPκλinV[f],

andsothetheoremisproved.?Theorem++)λλ<κ22<κ

§2GeneralizedLaverFunctions18

Previously,itwasnotknownevenhowtoforcetwonon-isomorphicλ-strongcompactnessmeasuresforastronglycompactcardinal.Nevertheless,inthecaseofstrongcompactness,thetheoremisnotthestrongestconceivableresult,sincethefollowingquestionremainsopen.

Question1.16Supposeκisstronglycompact.Isthereaforcingextensioninwhichforeveryλtherearethemaximumconceivablenumberofnon-isomorphic

λ<κ2?nemeasuresonPκλ,namely2many?

§2GeneralizedLaverFunctions

TheexistenceofLaverfunctionsinthesupercompactcardinalcontexthasprovedindispensible;thesefunctionsappearindozensifnothundredsofarticles.Becauseofthis,wewouldreallyliketohaveLaverfunctionsforotherkindsoflargecardi-nals.Iampleased,therefore,toproveherethatfastfunctionforcingaddsanewcompletelygeneralkindofLaverfunctiontoanylargecardinal,therebyfreeingthenotionofLaverfunctionfromthesupercompactcardinalcontext.

VκisageneralizedLaverfunctionunderSpeci?cally,Ide?nein

.Mwithcriticalpointκthefunctionf..κ→κwhenforanyembeddingj:

andanyz∈H(λ+)V→

M)suchthatj?(?)(κ)=zandj??ord=j?ord.Thisde?nitionis

perfectlysensiblewhetherκismeasurable,strong,stronglycompact,supercompact,orhuge,andsoon.Ofcourse,Imakethisde?nitiononlyinthenontrivialcasethatj(f)(κ)>0ispossible;naturallythefunction?wouldbethemostusefulwhenthefunctionf,likeafastfunction,hasthepropertythatλ=j(f)(κ)canbeverylarge.Often,thefunctionfwillinfactbeafastfunctionoratleasthavetheMenasproperty.Inthiscase,everygeneralizedLaverfunctiononasupercompactcardinalisaLaverfunctioninLaver’soriginalsense.Buttheconverseneednothold;indeed,theremaybenogeneralizedLaverfunctionsatall:

Observation2.1IfV=hodandκisatleastmeasurable,thenthereisnogeneralizedLaverfunctionforκ.Inparticular,thereisnogeneralizedLaverfunctioninL[?]orinthecoremodels.

Proof:SupposeV=hodandκismeasurable.Letj:V→Mbeanyembeddingwithcriticalpointκ.Ifj?:V→Misanembeddingsuchthatj?ord=j??ordthensinceeverysetishereditarilyordinalde?nable,itfollowsthatj=j?.Consequently,thereisnofreedomtochoosej?(?)(κ);itmustbeequaltoj(?)(κ).SothereisnogeneralizedLaverfunction.?Observation

§2GeneralizedLaverFunctions19

GeneralizedLaverFunctionTheorem2.2Fastfunctionforcingaddsageneral-izedLaverfunction.Speci?cally,afterfastfunctionforcingV[f],thereisafunction.?..κ→(V[f])κwiththepropertythatforanyembeddingj:V[f]→M[j(f)]withcriticalpointκ(whetherinternalorexternal)andforanyz∈H(λ+)M[j(f)],whereλ=j(f)(κ),thereisanotherembeddingj?:V[f]→M[j(f)]suchthat:

1.j?(?)(κ)=z,

2.M[j?(f)]=M[j(f)],

3.j??V=j?V,and

4.IfjistheultrapowerbyastandardmeasureηconcentratingonasetinV,thenj?istheultrapowerbyastandardmeasureη?concentratingonthesamesetandmoreoverη?∩V=η∩Vand[id]η?=[id]η.

Proof:Theideaisquitesimple,giventheFlexibilityTheoremforfastfunctionforcing.InVenumerateVκas?a=?aα|α<κ?withthepropertythatforeveryξ<κeveryelementofVξappearsunboundedlyoftenamongthe?rst?ξmanyelementsoftheenumeration.InV[f]let?(γ)=(af(γ))f?γ,providedthatthismakessense,i.e.,thatγ∈dom(f)andaf(γ)isanFγ-name.Supposej:V[f]→M[j(f)]isgivenandz∈H(λ+)M[j(f)]whereλ=j(f)(κ).Bytheclosureofthetailforcing,z∈M[f]andsoz=z˙fforsomenamez˙∈M.Thenamez˙mustbej(?a)(α)forsomeindexα,andbytheassumptionon?asuchanαcanbefoundbelowthenextinaccessiblebeyondλandκ.Therefore,bytheFlexibilityTheorem,thereisanotherembeddingj?:V[f]→M[j?(f)]=M[j(f)]satisfyingtheconclusionsoftheFlexibilityTheorem,withj?(f)(κ)=α.Inparticular,statements2,3and4hold.Itfollows,bythede?nitionof?,thatj?(?)(κ)=z˙f=z,asdesiredforstatement1.Sothetheoremisproved.?Theorem

NoticethattherestrictionthatzisinH(λ+)isnotonerous,becausebytheFlex-ibilityTheoremthevalueofλ=j(f)(κ)ishighlymutableandcanbemadetobeanydesiredordinaluptoj(κ).Certainly,anyzinH(δ)M[j(f)]canbeaccomodatedwithoutmodi?cationforanyδuptothenextinaccessibleclusterpointofdom(j(f))beyondκ.Andtheargumentsof1.13and1.14showthatforλ-supercompactor(λ+1)-strongembeddings,thisisalwaysatleastλ.Moregenerally,though,anyele-mentofMj(κ)[f]isapossiblevalueofj?(?)(κ),becausegivenanyj:V[f]→M[j(f)]onecan?rstapplytheFlexibilityTheoremtogetj?:V[f]→M[j?(f)]?M[j(f)]suchthatj?(f)(κ)islarge,andthenapplytheGeneralizedLaverFunctionTheoremtomakej?(?)(κ)whateverelementofMj(κ)[f]wasdesired.

Fortheremainderofthissection,letmesimplyspellouttheparticularcon-

§2GeneralizedLaverFunctions20

sequencesoftheprevioustheoremforvariouslargecardinals.Henceforthinthissection,therefore,let?bethegeneralizedLaverfunctionoftheprevioustheoremcomputedinthefastfunctionextensionV[f]relativetothe?xedenumeration?aofVκ.

Theorem2.3SupposeinVthatκismeasurableand2κ=κ+.Thenforanyultrapowerembeddingj:V→Mbyameasureonκandanyz∈H(κ+)V[f]thereisaliftj:V[f]→M[j(f)]suchthatj(?)(κ)=z.Furthermore,theliftcanbearrangedtobeanormalultrapower.

Proof:Thisiswhatfallsoutofthepreviousarguments.Beginningwithanyultrapowerj:V→Mbythemeasure?andanyz∈H(κ+)V[f]itfollowsthatz∈M[f]andsowecanlifttheembeddingtoj:V[f]→M[j(f)]insuchawaythatj(f)(κ)picksouttheordinalindexofanameforz,sothatj(?)(κ)=z.Byusinganame,say,whichalsocodestheordinalα=[id]?,itfollowsthatforsomefunctiong∈V[f]wehaveα=j(g)(κ).SinceeveryelementofMhastheformj(g′)(α)whereg′isafunctioninV,itfollowsfromthisthateveryelementofM[j(f)]hastheformj(h)(κ)forsomefunctionh∈V[f],andconsequentlytheliftedembeddingisanormalultrapower.?Theorem

Essentiallythesameargumentworksforweaklycompactcardinals:

Theorem2.4SupposeinVthatκisweaklycompact.Thenforanysetz∈H(κ+)inV[f]thereisaweaklycompactembeddingj:N[f]→M[j(f)]suchthatj(?)(κ)=z.Inparticular,?κholds.

Proof:Sincez∈V[f]thereisanamez˙∈Vofhereditarysizeκ.Pickatransitive

˙a∈NandNisclosedunder<κ-N?H(κ+)ofsizeκinVsuchthatz,˙F,?,?

sequences.SinceκisweaklycompactinVthereisanembeddingj:N→Mwithcriticalpointκ.Asin1.4,wecanassumethatMisalsoclosedunder<κ-sequences.TheusualargumentshowsP(κ)N?Mandsoz˙∈M.Bythediagonalizationargument,becausethereareonlyκmanydensesubsetsofFinN,wecanlifttheembeddingtoj:N[f]→M[j(f)],andfurthermore,wecandosoinsuchawaythatj(f)(κ)=αwhereαistheindexofz˙withrespecttoj(?a).Consequently,j(?)(κ)=z˙f=z,asdesired.Itiseasynowtodeducethatfisapowerfulkindof?sequence.?Theorem

Letmenowgraduallymoveupwardsthroughthelargecardinalhierarchy.

Theorem2.5Ifj:V[f]→M[j(f)]isaλ-strongembedding(withthenaturalextender)andz∈(V[f])λthenthereisanotherλ-strongembeddingj?:V[f]→M[j?(f)]suchthatj?(?)(κ)=z,j??V=j?VandM[j?(f)]=M[j(f)].

§2GeneralizedLaverFunctions21

Proof:Supposethatj:V[f]→M[j(f)]isaλ-strongembeddinggeneratedbythenaturalextender,sothatM[j(f)]={j(h)(s)|h∈V[f]&s∈(V[f])λ},andthatz∈(V[f])λ.Ifλisasuccessorordinalthentheargumentof1.12showsthatj(f)?[κ,λ)∈M,andsoz=z˙fforsomez˙∈M.Alternatively,ifλisalimitordinal,thenz∈(V[f])βforsomemuchsmallerβ,andconsequentlytheargumentof1.14appliedtotheinducedfactorembeddingshowsj(f)?[κ,β)∈M.Thus,z=z˙fforsomez˙∈Mλ.Ineithercase,thenamez˙has,belowthenextinaccessible,someindexαwithrespecttoj(?a),andsobytheFlexibilityTheorem,thereisanotherembeddingj?:V[f]→M[j?(f)]suchthatj?(f)(κ)=α,j??V=j?VandM[j?(f)]=M[j(f)].Bythechoiceofαitfollowsthatj?(?)(κ)=z,andsothetheoremisproved.?Theorem

Asbefore,thetheoremcanbemodi?edtoallowforzwhichappearhigherinthehierarchyifwearewillingtogiveuptheequalityofM[j?(f)]andM[j(f)].Speci?cally,ifz∈Mj(κ)[f],thentherewillbeanembeddingj?:V[f]→M[j(f)]suchthatj?(?)(κ)=z,j??V=j?VandM[j?(f)?M[j(f)].Theembeddingj?willstillbeaλ-strongembeddingbecause(M[j(f)])λ=(M[f])λbytheargumentshowingj(f)?[κ,β)∈M.Thenextinaccessibleclusterpointofdom(j(f))beyondκmustbeatleastλ.

Next,Itreatthecaseofstronglycompactcardinals.

Theorem2.6Iftheembeddingjof2.2isaθ-strongcompactnessembedding,whereθ<κ=θ,thentheembeddingj?mayalsobechosentobeaθ-strongcom-pactnessembedding.

Sinceeveryθ-stronglycompactmeasureisinfactisomorphictoaθ<κ-stronglycompactmeasure,weseebysimplyreplacingθwithθ<κthattheassumptionthatθ<κ=θishardlyarestrictionatall.Andbecauseameasure?isaθ-strongcompactnessmeasureexactlywhens=[id]?isacoverofj?"θwithasubsetofj?(θ)ofsizelessthanj?(κ),thetheoremfollowsbystatement4of2.2andthefollowinglemma.De?nethataforcingextensionV[G]ismildifeverysetofhereditarysizelessthanκinV[G]isaddedbyaposetofsizelessthanκinV.Certainlyfastfunctionforcingismild,becauseallthetailforcingsFλ,κare≤λ-closed.

Lemma2.7Everyθ-strongcompactnessmeasureinamildforcingextensionV[G],whereθ<κ=θ,isisomorphictoastandardθ-strongcompactnessmeasurewhichconcentrateson(Pκθ)V.

Proof:Supposethatj:V[G]→M[j(G)]istheultrapowerbya?nemeasureηonPκθinV[G].Sincemeasuresareisomorphicexactlywhentheyinducethe

§2GeneralizedLaverFunctions22

sameembedding(see[Ham97]),itsu?cestoshowthatjistheultrapowerbyastandard?nemeasureconcentratingon(Pκθ)V.Lets=[id]η.Thus,j"θ?sand

?forforcingof?]forsomegenericG??P|s|M[j(G)]<j(κ).Thus,bymildness,s∈M[G

somesizeγsuchthat|s|≤γ<j(κ).Thus,shasanames˙∈Mofsizeγ.Usingthis

|snameitispossibletoconstructasets?∈Mofsizeγsuchthatj"θ?s?andκ∈?.

Sinceθ<κ=θ,themeasureηisisomorphictoaκ-completemeasureonθ.Theremustthereforebeanordinalδ<j(θ)suchthatM[j(G)]={j(h)(δ)|h∈V[G]}.Wemayassume,bysimplyaddingsuchapointifnecessary,thatthelargestelementofs?hastheform?β,δ?,usingasuitablede?nablepairingfunction,forsomeordinalβ<j(θ).Letη?bethemeasuregerminatedbys?viaj.Sinces?isasubsetofj(θ)ofsizeγ<j(κ)andj"θ?s?,itfollowsthatη?isa?nemeasureonPκθinV[G].Furthermore,sinces?∈M,itconcentratesonthePκθofthegroundmodelV.Iclaimthatηisisomorphictoη?.Toprovethis,itsu?cesbytheseedtheoryof[Ham97]toshowthattheseedhullofs?,namelyX={j(h)(?s)|h∈V[G]}?M[j(G)],isallofM[j(G)].Bytheassumptiononthelargestelementofs?,weknowδ∈Xandsincealsoran(j)?X,itfollowsthatM[j(G)]?X,asdesired.Themeasureη?isstandardbecauseκistheleastelementnotins?=[id]η?.SoIhaveprovedthateveryθ-strongcompactnessmeasureηinV[G]isisomorphictoastandardθ-strongcompactnessmeasureη?inV[G]concentratingonthePκθofthegroundmodel.?Lemma

SoTheorem2.6isproved.Asusual,bygivinguptheequalityofM[j?(f)]andM[j(f)]itispossibletoaccomodatelargerzthanstatedinthetheorem,asIwill

Mhastheθ-strongcompactnessprovenext.De?nethatanembeddingj:

Mcoverpropertywhenthereisasets∈<j(κ).Thus,

scanbeusedtogerminateviaja?nemeasureonPkθ.Intheeventthatjisanultrapowerbyameasureonsomeset,itfollowsbyaneasyargumentthatevery

Mofsize|s|subsetof

§3TheLotteryPreparation23

s∈Msuchthatj"θ?sand|s|M<j(κ).Thissetalsoworksasacover,therefore,inM[j?(f)].?Theorem

Theorem2.9Iftheembeddingjof2.2isaθ-supercompactnessembedding,thentheembeddingj?mayalsobechosentobeaθ-supercompactnessembedding.Proof:Thisisimmediatebyproperty4of2.2andRemark1.2,sincetheremarkshowsthatj"θ∈M,andthisis[id]η.Thatis,supercompactnessmeasuresinV[f]arealwaysstandard,andtheyalwaysconcentrateonthePκθofthegroundmodelV.?Theorem

Thenexttheoremimprovesonthis;evenwhenλ=j(f)(κ)issmallitispossibleforθ-supercompactnessembeddingstohavej?(?)(κ)=zforanyz∈H(θ+).

Theorem2.10Ifj:V[f]→M[j(f)]isaθ-supercompactnessembeddinginV[f]andz∈H(θ+)V[f]thenthereisanotherθ-supercompactnessembeddingj?:V[f]→M[j?(f)]suchthatj?(?)(κ)=z,j??V=j?VandM[j?(f)]=M[j(f)].

Proof:Thisfollowsbythesameideaasin1.13;thepointisthatj(f)?[κ,θ)mustbeaconditioninM,andsothevalueofj(f)(κ)canbefreelychangedsoastopickouttheindexofthenameofanyelementinH(θ+)M[j(f)].?Theorem

Theprevioustheoremistruelevel-by-levelinthesensethatitistrueevenwhenκisonlypartiallysupercompact;forexample,κmaybeonlymeasurable.What’smore,itisasbeforepossibleforthefunction?tocapturemorezthanjustthoseinH(θ+).Speci?cally,ifj:V[f]→M[j(f)]isaθ-supercompactembeddingandz∈Mj(κ)[f]thenthereisanembeddingj?:V[f]→M[j?(f)]suchthatj?(?)(κ)=z,j??V=j?VandM[j?(f)]?M[j(f)].Inthiscase,however,theembeddingj?maynotbeaθ-supercompactnessembedding,thoughbyRemark1.2itwillhavej?"θ=j"θ∈M.

§3TheLotteryPreparation

Iaimheretopresentthelotterypreparation,anewgeneralkindofLaverprepa-ration,whichworksuniformlywithavarietyoflargecardinals—suchasweaklycompactcardinals,measurablecardinals,strongcardinals,stronglycompactcardi-nalsandsupercompactcardinals—andmakesthemindestructiblebyvariousfurtherforcing,dependingonthestrengthofthecardinal.

Letmebeginbyde?ningmyterms.ThebasicbuildingblockiswhatIcallalotterysum.Speci?cally,thelotterysumofacollectionAofforcingnotionsis

§3TheLotteryPreparation24

theforcingnotion?A={?Q,p?|Q∈A&p∈Q}∪{1l},orderedwith1laboveeverythingand?Q,p?≤?Q′,q?whenQ=Q′andp≤Qq.BecausecompatibleconditionsmusthavethesameQ,theforcinge?ectivelyholdsalotteryamongalltheposetsinA,alotteryinwhichthegeneric?lterselectsa‘winning’posetQandthenforceswithit.

Notethatthelotterysumoftheemptysetisthetrivialposet{1l}.Iwillde?ne.thelotterypreparationofκrelativetoa?xedfunctionf..κ→κ.Thoughthede?nitionworks?newithanyfunction,theforcingworksbestwhenusedwithafunctionhavingtheMenasproperty,suchasafastfunction.

Thelotterypreparationofκwillbeaκ-iterationwhichatmanystagesγ<κwillperformthelotterysumofthecollectionofposetswhichareallowedatstageγ.Thus,atstageγ,thegeneric?lterwille?ectivelyselectaparticularsuchposetasthewinnerofthelotteryandthenforcewithit.Generically,awidevarietyofposetswillbechoseninthelotteriesbelowκ,therebyre?ectingthepossibilitiesatstageκonthej-side.TheessentialideaisthatratherthanconsultingaLaverfunctionaboutwhichparticularforcingistobedoneatstageγ,thelotterypreparationinsteadusesthelotterysumofallposetswhichwemightliketoseeatstageγ,andletsthegeneric?lterdecidegenericallyamongstthem.

O?cially,letmesaythataposetQisallowedatstageγwhenforeveryδ<γtheposetQis<δ-strategicallyclosed(thatis,thesecondplayerhasastrategyenablinghertoplayadescendingδ-sequencefromtheposet,wheretheplayersalternatelyplayelementsdescendingthroughtheposet,andthesecondplayerplaysatlimitstages).Thisrequirement,whilebroadlyinclusive,isenoughtoensurethatthetailforcingisdistributive.

Letmenowgivethede?nition.Thelotterypreparationofκrelativetothe.functionf..κ→κisthereverseEastonsupport*κ-iterationwhichhasnontrivialforcingatstageγonlywhenγ∈dom(f)andf"γ?γ.Atsuchstages,theforcing

§3TheLotteryPreparation25

QγisthelotterysuminVPγofallposetsinH(f(γ)+)whichareallowedatstageγ.Otherwise,theforcingatstageγistrivial.

WhileIhaveprovedintheprevioussectionthatfastfunctionforcingaddsageneralizedLaverfunction,pleaseobservethatIamnotusingthisgeneralizedLaverfunctiontode?nethelotterypreparation.CertainlyonecouldusethegeneralizedLaverfunctionstode?neakindofgeneralizedLaverpreparation,andsuchaprepa-rationwouldhavemanyofthesamefeatures(byessentiallythesamearguments)thatIidentifyhereforthelotterypreparation.Butitseemsconceptuallysimplertome,andmoretothepoint,touselotterysumsinordertoallowthegeneric?ltertodecidewhichforcingistobedoneateachstage.Doingsoavoidstheneedtocarefullycon?guretheembeddingsothatj(?)(κ)isasrequired;withalottery,onesimplyworksbelowtheconditionwhichoptsforthedesiredforcingatstageκ.Indeed,thisabilitytoselectarbitrarilythewinnerofthestageκlotteryiswhatallowsustogetbywithoutanyLaverfunction.InasensethelotterypreparationshowsthatwhatwastrulyimportantaboutLaver’sfunctionwasnotthatj(?)(κ)couldbearrangedtobeanydesiredset—sincealotterysumcangenericallypickoutanydesiredsetatstageκ—butratherthattheLaverfunctioncouldbearrangedsothatthenextelementofthedomainofj(?)beyondκwasaslargeasyoulike.Thisiswhatsupportsthecrucialtailforcingarguments;aslongasonehasawayofreachinguphigh(e.g.bytheMenasproperty),onecanusealotterysumtoallowthegeneric?ltertoselectanydesiredsetorposet,andthetailforcingwillbesu?cientlyclosed.Thus,thelotterypreparationde?nedrelativetoafastfunctionworkse?ectivelywithawidevarietyoflargecardinals.

LotteryFactorLemma3.1Foranyγ<κwhichisclosedunderf,thelotterypreparationPκfactorsasPγ?Pγ,κwherePγisthelotterypreparationde?nedusingf?γandPγ,κisthelotterypreparationde?nedinVPγusingf?[γ,κ).

Proof:Thisfollowsbytheusualiteratedforcingfactorarguments.ThepointisthatthePγ+α-namesappearinginthestageγ+αlotterycanbeiterativelytransformed,byrecursiononα,intoPγ?Pγ,γ+α-names.Thereisnoproblemwiththesupportsbecausewetookaninverselimitatallbuttheinaccessiblestages.?LemmaLemma3.2Ifinthelotterypreparationthereisnonontrivialforcinguntilbeyondstageγ,thenthepreparationis≤γ-strategicallyclosed.

Proof:Sincetheforcingateachstageα>γisα-allowed,itis≤γ-strategicallyclosed,witha(nameofa)strategyσα.Givenapartialplay,adescendingsequenceofconditions?pβ|β<γ′?forsomeγ′<γ,wherepβ=?p˙βα|γ<α<κ?,oneapplies

§3TheLotteryPreparation26

thestrategiesσαcoordinate-wisetoobtainσ(?pβ|β<γ′?)=?q˙α|γ<α<κ?,whereq˙αisthenamefortheconditionobtainedbyapplyingthestrategyσαto

′?p˙βα|β<γ?.Recursively,sinceeachofthestrategiesσαcansuccessfullynegotiateallthelimitsuptoγ,sodoesthisstrategyσ,andsothelemmaisproved.?Lemma

TheconsequenceofthislemmaisthatwhenonefactorsthelotterypreparationasPγ?Pγ,κ,thenPγ,κis<γ-strategicallyclosedinVPγ.Thetwolemmastogethershowthatifwearetryingtoliftanembeddingj:V→MandQissomeforcingwhichwhichisallowedinthestageκlotteryofj(P),thenbysimplyworkingbelowaconditionpwhichoptsforQinthestageκlotterywemayfactortheforcingasj(P)?p?=P?Q?Ptail,wherePtailhastrivialstagesuntilbeyondj(f)(κ).Throughthissimplelotterytechnique,weobtainthecrucialfactorizationthatoneordinarilyneedsaLaverfunctiontoobtain,andwehavedonesoinacompletelygenerallargecardinalcontext,withnosupercompactnessassumptions.Thisistheideawhichwillsupporttheindestructibilityresultsofthenextsection.

Letmenowprovethatthelotterypreparationpreservesavarietyoflargecar-dinals,beginningatthebottomandmovingupwards.Inthetheoremsbelow,ifnoassumptionisexplicitymadeconcerningf,thenthetheoremholdsforthelotterypreparationde?nedusinganyfunctionf.Toavoidthetrivialityofsmallforcing,letmeassumethatthedomainoffisunboundedinκ.

LotteryPreparationTheorem3.3Thelotterypreparationofaninaccessiblecardinalκpreservestheinaccessibilityofκ.

Proof:SupposethatκbecomessingularinV[G].Letγ∈dom(f)beaclosurepointofflargerthanδ=cof(κ)V[f]andfactortheforcingatstageγasPγ?Pγ,κ.TheforcingPγhassizelessthanκandsocannothavecollapsedtheco?nalityofκ.TherestoftheforcingPγ,κis≤δ-strategicallyclosedinVPγ,andsoalsocannothavecollapsedtheco?nalityofκtoδ,acontradiction.Thus,κmustberegularinV[G].Similarly,theforcingPγ,sinceithassizelessthenκ,cannotforce2δ≥κforanyδ<κ,andthetailforcingPγ,κaddsnofurthersubsetstoδifδ<γ.SoκremainsinaccessibleinV[G].?Theorem

LotteryPreparationTheorem3.4Thelotterypreparationofaweaklycompactcardinalκpreservestheweakcompactnessofκ.

Proof:Sincetheprevioustheoremshowsthatκremainsinaccessible,itsu?cestoshowthatκhasthetreepropertyinV[G].So,supposeTisaκ-treeinV[G].

˙∈VsothatT˙G=T.InVpickatransitiveN?H(κ+)ChooseanameT

˙,P∈NandN<κ?N.SinceκisweaklycompactinVofsizeκsothatκ,T

§3TheLotteryPreparation27

thereisanembeddingj:N→Mwithcriticalpointκ.ImayassumethatM={j(h)(κ)|h∈N},byreplacingMwiththecollapseofthissetifnecessary.Itfollows,asin1.4,thatM<κ?M.InM,theforcingj(P)factorsasP?Ptail,wherePtailisthelotterypreparationusingj(f)?[κ,j(κ));thisis<κ-strategicallyclosedinM[G].Furthermore,itisnotdi?culttoestablishthat(M[G])<κ?M[G].Thus,sinceM[G]hassizeκwemaybydiagonalizationconstructinVanM[G]-genericGtail?Ptail,andtherebylifttheembeddingtoj:N[G]→M[j(G)]where

˙G=TweknowT∈N[G].Thus,j(T)isaj(κ)-treej(G)=G?Gtail.SinceT

inM[j(G)].Anyelementontheκthlevelofj(T)providesaκ-branchthroughTinM[j(G)],andhenceinV[G].SoκhasthetreepropertyinV[G]andthereforeremainsweaklycompact.?Theorem

LotteryPreparationTheorem3.5Thelotterypreparationofameasurablecardinalκsatisfying2κ=κ+preservesthemeasurabilityofκ.

Proof:Thisargumentissimilarto1.5.Supposeκismeasurableand2κ=κ+inV,andthatGisV-genericforthelotterypreparationofκde?nedrelativetosomefunctionf.Iwillshowthateveryultrapowerembeddingj:V→MbyameasureonκinVliftstoanembeddinginV[G].Givensuchanembedding,factortheforcingj(P)asP?Ptail,wherePtailisthelotterypreparationde?nedinM[G]usingj(f)?[κ,j(κ)).Belowaconditionwhichoptsfortrivialforcinginthestageκlottery,theforcingPtailis≤κ-strategicallyclosedinM[G].Furthermore,standardargumentsestablishthatM[G]κ?M[G]inV[G]andacountingargumentestablishesthat|j(κ+)|V=κ+.Thus,bydiagonalization,wecaninV[G]constructanM[G]-genericGtail?Ptail;onesimplylinesupallthedensesetsintoaκ+-sequenceandmeetsthemone-by-one,followingthestrategyinordertogetthroughthelimitstages.Thus,inV[G]theembeddingliftstoj:V[G]→M[j(G)]wherej(G)=G?Gtail,asdesired.?Theorem

LotteryPreparationTheorem3.6Thelotterypreparationofastrongcardinalκsatisfying2κ=κ+,de?nedrelativetoafunctionwiththestrongMenasproperty,preservesthestrongnessofκ.

Proof:Whatismore,theresultiscompletelylocal;following1.6,Iwillshowthatifκisλ-stronginVthenthisispreservedtothelotterypreparationV[G].Supposej:V→Mwitnessestheλ-strongnessofκandtheMenaspropertyoff,sothatVλ?Mandδ=|Vλ|<j(f)(κ).Byusingtheinducedcanonicalextenderifnecessary,ImayassumethatM={j(h)(s)|h∈V&s∈δ<ω}.Sinceδ<j(f)(κ),belowtheconditionp∈j(P)whichoptsfortrivialforcinginthestage

§3TheLotteryPreparation28

κlotterytheforcingfactorsasP?Ptail,wherePtailis≤δ-strategicallyclosed.LetX={j(h)(κ,δ)|h∈V}?Mbetheseedhullof?κ,δ?,andfactortheembeddingas

V

k-M

wherek:M0→MistheinverseofthecollapseofX.SinceκisinX,itfollowsthatp∈X,andsok(p0)=pforsomep0∈j0(P).Similarly,sinceδisinX,weknowthatk(δ0)=δforsomeδ0<j0(κ).Also,sinceκisinXweknowthatcp(k)>κ.Theembeddingj0:V→M0,beinggeneratedbytheseed?κ,δ0?,issimplyanultrapowerbyameasureonκ,andthereforeliftsbythediagonalizationargumentof1.5toanembeddingj0:V[G]→M0[j0(G)]belowtheconditionp0.

M0M0?P0isM0[G]-genericforItmustbethatj0(G)factorsasG?GM,whereGtailtailtail

≤δ0-strategicallyclosedforcing.

ItremainstolifttheembeddingktoM0[j0(G)].Sincecp(k)>κ,certainlyk

0?Ptailisliftstok:M0[G]→M[G].Fortherestitsu?cestoshowthatk"GMtail?)(s)M[G]-generic.ButeveryopendensesetD∈M[G]forPtailhastheformk(D

?=?Dσ|σ∈δ<ω?inM0[G]ands∈δ<ω,whereeachDσisanopenforsomeD0M0M0densesubsetofPtail.SincePtailis≤δ0-distributiveinM0[G],theintersection

D)?D.Thus,since

M00GMtailisM0[G]-generic,k"GtailmeetsD,asdesired.Consequently,kliftsfullyto

k:M0[j0(G)]→M[k(j0(G))],wherek(j0(G))isthe?ltergeneratedbyk"j0(G).Thecompositionk?j0providesaliftofjtoj:V[G]→M[j(G)].SinceVλ?MandG∈M[j(G)],itfollowsthat(V[G])λ?M[j(G)],andsoκisstillλ-stronginV[G].?Theorem

LotteryPreparationTheorem3.7Thelotterypreparationofastronglycom-pactcardinalκ,de?nedrelativetoafunctionwiththeMenasproperty,preservesthestrongcompactnessofκ.

Proof:Whatismore,following1.7Iwillshowthateverystrongcompactnessmeasureinthegroundmodelextendstoameasureintheforcingextension.SupposethatfhastheMenaspropertyinV(e.g.perhapsfwasaddedbyfastfunctionforcingoverasmallermodel),thatG?PisV-genericforthelotterypreparation

§3TheLotteryPreparation29

<κrelativetof,thatλ≥κandthat?0isa?nemeasureonPκλinV.Letθ≥2λ,

andpickj:V→Maθ-stronglycompactembedding,theultrapowerbya?nemeasureηonPκθinVwitnessingtheMenaspropertyonf.Asin1.7,?xaseeds0for?0,sothatX∈?0?s0∈j(X)forX?PκλinV.Inparticular,j"λ?sby?neness.Lets=[id]η,andδ=|s|M.Thus,j"θ?s∈j(Pκθ)andθ≤δ<j(κ).Now,inj(P),letpbetheconditionwhichoptsinthestageκlotteryforatrivialposet.BytheMenaspropertyweknowj(f)(κ)>δ,andsothenextnontrivialstageofforcingliesbeyondδ.Inparticular,belowtheconditionptheforcingfactorsasP?Ptail,wherePtailis≤δ-strategicallyclosedinM[G].ForcebelowptoaddGtail?PtailgenericallyoverV[G],andinV[G][Gtail]lifttheembeddingtoj:V[G]→M[j(G)],wherej(G)=G?Gtail.Let??0bethemeasuregerminatedbytheseeds0viatheliftedembedding,sothatX∈??0?s0∈j(X)forX∈V[G].Itiseasytoseethat??0measureseverysetinV[G],thatitextends?0,and,sincej"λ?s0,thatis?ne.Itremainsonlyformetoshowthat??0∈V[G].<κAsin1.7thereareagainonly2λmanynicenamesinVforsubsetsofPκλin

˙α|α<θ?.Thus,j(u)∈M,andV[G],andwemayenumeratethemu=?X

˙β|β∈s?∈M,withY˙j(α)=j(X˙α)forα<θ.Letconsequentlyalsoj(u)?s=?Y

˙β)j(G)}.Thus,t?sandt∈M[j(G)].SinceshassizeδandPtailt={β∈s|s0∈(Y

is≤δ-strategicallyclosedinM[G],itfollowsthatt∈M[G],andthereforet∈V[G].

˙α)G)=j(X˙α)j(G)=(Y˙j(α))j(G)?j(α)∈t,itfollows˙α)G∈???s0∈j((XSince(X0

?that??0isde?nableinV[G]fromtandj?θ,andso?0∈V[G]asdesired.?Theorem

Theorem3.8Thelotterypreparationofasupercompactcardinalκ,de?nedrel-ativetoafunctionwiththeMenasproperty,preservesthesupercompactnessofκ.

Proof:Ifinthepreviousargumentonetakesjtobeaθ-supercompactembeddingands0=j"λ,thenitiseasytoseethattheresultingmeasure??0isnormaland?neonPκλ,andsoκisλ-supercompactinV[G],asdesired.?Theorem

Theprevioustwotheoremsareglobalinthesensethattheyassumeκisfullystronglycompactorfullysupercompactinthegroundmodelandconcludethatκremainsfullystronglycompactorsupercompactafterthelotterypreparation.Butitiseasytoextractfromtheproofsthefollowingmorelocalfacts,whereweassumethelotterypreparationismaderelativetoafunctionwiththeappropriateamountoftheMenasproperty:

§3

<κTheLotteryPreparation30LocalVersion3.9Ifκis2λ-stronglycompactinV,thenafterthelottery

<κpreparationκremainsλ-stronglycompact.Ifκis2λ-supercompactinV,then,

<κafterthelotterypreparationκremainsλ-supercompact.Indeed,ifκis2λ-strongly

compactandλ-supercompactinV,thenafterthelotterypreparationκremainsλ-supercompact.

Acompletelylocalresult,inwhichtheverysamelargecardinalassumptionmadeinVispreservedtoV[G],ispossibleifoneiswillingtopayaslightgchpenalty:

CompletelyLocalVersion3.10Thelotterypreparationofaλ-supercompact<κcardinalκwith2λ=λ+,de?nedrelativetoafunctionwiththeMenasproperty,preservestheλ-supercompactnessofκ.

Proof:Thisargumentfollowsthediagonalizationtechniqueusedin1.10.Supposethatj:V→Misaλ-supercompactembeddinginVsuchthatj(f)(κ)>λandthatV[G]isthelotterypreparationofκ.Byoptingfortrivialforcinginthestageκlottery,wemayfactortheforcingj(P)asP?PtailwherePtail=Pλ,j(κ)is≤λ-strategicallyclosedinM[G].StandardargumentsestablishthatM[G]isclosedunderλ-sequencesinV[G],andasimplecountingargumentshowsthatthereare

<κatmost2λ=λ+manysubsetsofPtailinM[G],countedinV[G].Thus,ImaylineuptheopendensesubsetsofPtailinM[G]intoaλ+-sequence,andconstructbydiagonalizationadescendingλ+-sequenceofconditions,accordingtothestrategy,whichmeetseveryopendensesetonthelist.EveryinitialsegmentofthesequenceisinM[G],andsothestrategyensuresthattheconstructioncanproceedthroughanylimitstage.Thus,inV[G]IconstructanM[G]-genericGtail?Ptailandtherebylifttheembeddingtoj:V[G]→M[j(G)].Soκremainsλ-supercompactinV[G],asdesired.?Theorem

ThemeasureswhichexistafterthelotterypreparationV[G]enjoyaspecialrelationshipwiththemeasuresfromthegroundmodel.Namely,Ihaveshownthatundersuitablehypothesiseverysupercompactnessorstrongcompactnessmeasureinthegroundmodelextendstoameasureintheforcingextension;amazingly,theconversealsoholds.

Forthefollowingtheorem,assumethatthe?rstelementofthedomainofthefunctionfusedtode?netheiterationisverysmall,say,belowtheleastweaklycompactlimitofweaklycompactcardinals,andthatf(β)≥β.

§3TheLotteryPreparation31

Theorem3.11Belowacondition,thelotterypreparationcreatesnonewmeasur-able,strong,Woodin,stronglycompact,orsupercompactcardinals.Inaddition,itdoesnotincreasethedegreeofstrongcompactnessorsupercompactnessofanycardinal.Andexceptpossiblyforcertainlimitordinalsofsmallco?nality,itdoesnotincreasethedegreeofstrongnessofanycardinal.Thereasonforeachofthesefactsisthateverymeasureintheforcingextensionwhichconcentratesonasetinthegroundmodelextendsameasurefromthegroundmodel.

Proof:SupposethatGisgenericforthelotterypreparationbelowtheconditionwhichopts,inthevery?rstlotteryatstageβ,toaddaCohensubsettoβ.Ifγisthenextelementofthedomainbeyondf(β),thentheforcingfactorsasAdd(β,1)?Pγ,κ,wherePγ,κistheremainderofthepreparation.Thus,thisisforcingofsizeβfollowedbyforcingwhichis≤β+-strategicallyclosed.Suchforcingissaidin[Ham98]toadmitagapatγ=β+.AsIexplainedinRemark1.2,TheGapForcingCorollaryof[Ham∞]assertsthataftersuchforcingeveryembeddingj:V[G]→M[j(G)]withthepropertythatM[j(G)]isclosedunderγ-sequences—andthisincludesanyultrapowerembeddingwithcriticalpointκbyameasureonanyset,becausesuchembeddingsarealwaysclosedunderκ-sequences—liftsanembeddingfromthegroundmodel.Thatis,M?Vandj?V:V→Misde?nableinV.Ifjwastheultrapowerbysomemeasure?concentratingonasetD∈V,thens=[id]?∈j(D)∈Misaseedfor?inthesensethatX∈??s∈j(X).Consequently,?∩Visde?nableinVfromj?V.If?ismeasureonκ,orastrongorsupercompactnessmeasureinV[G],thenitisnotdi?culttoseethat?∩VisthecorrespondingkindofmeasureinV.Iftheoriginalembeddingjwasanaturalλ-strongnessextenderembeddingforλeitherasuccessorordinaloralimitordinalofco?nalityaboveγ,then[Ham∞]showsthattherestrictedembeddingj?Vwitnessestheλ-strongnessofκinV.ItfollowsthatWoodincardinalsalsocannotbecreated.Sothetheoremisproved.?Theorem

Thus,thelotterypreparationisagentleone;measuresinthelotterypreparationextensionarecloselyrelatedtomeasuresinthegroundmodel.AndRemark1.2showsthatthesameistrueoffastfunctionforcing,providedthatitisprefacedbysomesmallforcing.

§4IndestructibilityAftertheLotteryPreparation32

§4IndestructibilityAftertheLotteryPreparation

NowIcometothemaincontributionofthispaper,namely,thatthelotteryprepara-tionmakeslargecardinalsindestructible.Laver’s[Lav78]originalpreparation—myinspiration,ofcourse—showedspectacularlythatanysupercompactcardinalcanbemadehighlyindestructible.GitikandShelah[GitShl89]extendedtheanalysistostrongcardinals.Here,thelotterypreparationuni?esandgeneralizestheseresultsbyprovidingauniformpreparationwhichworkswithanylargecardinal,whetheritissupercompact,stronglycompact,strong,partiallysupercompact,par-tiallystronglycompact,ormerelymeasurable,andsoon.Ineachofthesecases,thelotterypreparationmakesthecardinalindestructiblebyavarietyofforcingnotions,dependingonthestrengthofthecardinalinthegroundmodel.

Ofcourse,thelotterypreparationwillmakeasupercompactcardinalκfullyindestructibleby<κ-directedclosedforcing,andastrongcardinalκ(suchthat2κ=κ+)indestructibleby≤κ-distributiveforcing,andmore.Thesearguments,givenbelow,essentiallyfollowthecorrespondingresultsin[Lav78]and[GitShl89].Thelevel-by-levelresult,lackingintheLaverpreparationbecauseLaverfunctionsarenotavailablelevel-by-level,isneverthelesspossiblewiththelotterypreparation,whichrequiresnoLaverfunction.Thus,withabitofthegch,evenpartiallysupercompactcardinalscanbemadeindestructible.

Themostsigni?cantcontributionofthispaper,however,concernsthestronglycompactcardinals.BecausetheliftingargumentsinvolvedinLaver’stheoremsim-plyfailwhenusedwithastronglycompactembedding,ithasbeenanopenquestionforsometimetodeterminethedegreetowhichastronglycompactcardinalcanbepreservedbyforcing.Indeed,Menas[Men74]seemsveryconcernedwiththisquestion.Apterconcludeshispaper[Apt97]withquestionsaskinghowmuchinde-structibilityispossiblewithastronglycompactnon-supercompactcardinal.Untilnow,therehavebeennonontrivialinstancesofanarbitrarystronglycompactcar-dinalbeingpreservedbyforcing.Progresshadbeenmadeinthespecialcaseofastronglycompactlimitofsupercompactcardinals(forwhichaMenasfunctionalwaysexists):Apter[Apt96],usingMenas’stechnique,showedhowtomakesuchcardinalsκindestructiblebyAdd(κ,1);Menas[Men74]himselfalsoseemsalsoveryclosetoprovingthis.Apter[Apt97]showshowtomakeanystronglycompactlimitofsupercompactcardinalsindestructiblebyany<κ-directedclosedforcingwhichdoesnotaddasubsettoκ.Recently,ApterandGitik[AptGit97]proved,impressively,thatanysupercompactcardinalcanbemadeintostronglycompactcardinalwhichis

§4IndestructibilityAftertheLotteryPreparation33

simultaneouslytheleastmeasurablecardinalandfullyindestructibleby<κ-directedclosedforcing.Here,Iaspiretoeliminatesuchsupercompactnessassumptionsandworkwithanarbitrarystronglycompactcardinal.Howmuchindestructibilityispossible?

Fortheremainderofthissection,assumethatV[G]isthelotterypreparationofκrelativetoafunctionfwiththesuitableMenaspropertyinV.Forexample,perhapsfwaspreviouslyaddedbyfastfunctionforcing.Occasionally,forafewofthetheorems,Iwillmaketheadditionalassumptionthatinfactfwasaddedinthisway.

IndestructibilityTheorem4.1Afterthelotterypreparation,astronglycompactcardinalκbecomesindestructiblebyCohenforcingAdd(κ,1).

Proof:SupposethatV[G]isthelotterypreparationofκrelativetothefunctionfwiththestronglycompactMenasproperty,andthatg?κisV[G]-genericforAdd(κ,1)V[G].Iwillshowthatevery?nemeasure?0onPκλinVextendstoa

<κmeasureinV[G][g].Letθ≥2λ,andsupposej:V→MwitnessestheMenas

propertyoff,sothatitistheultrapowerbya?nemeasureηonPκθinVandj(f)(κ)>δ=|s|wheres=[id]η.BelowtheconditionpwhichoptsinthestageκlotteryforAdd(κ,1)M[G]=Add(κ,1)V[G],theforcingj(P)factorsasP?Add(κ,1)?Ptail,wheretheforcingPtailis≤δ-strategicallyclosedinM[G][g].ForcetoaddGtail?PtailoverV[G][g],andlifttheembeddingtoj:V[G]→M[j(G)],wherej(G)=G?g?Gtail.Nowconsidertheforcingj(Add(κ,1)V[G])=Add(j(κ),1)M[j(G)].Thesetg?κisaconditioninthisposet,soIcanforcebelowittoaddagenericg??j(κ).Sinceg?pullsbacktogviaj,thatistosay,sincegisamastercondition,theembeddingliftsinV[G][g][Gtail][?g]toj:V[G][g]→M[j(G)][j(g)]wherej(g)=g?.Selectaseeds0for?0asinTheorem1.7,andlet??0betheV[G][g]-measuregerminatedbys0viatheliftedembedding,sothatX∈??0?s0∈j(X).Iwantto

˙showthat??0isinV[G][g].Enumerateu=?Xα|α<θ?thenamesforsubsets

˙β)j(G)?j(g),ofPκλinV[G][g].Again,lettbethesetofallβ∈ssuchthats0∈(Y

˙β|β∈s?.Sincet?sandthetailforcingis≤δ-strategicallywherej(u)?s=?Y

closed,itfollowsthatt∈M[G][g],andconsequentlyt∈V[G][g].Usingtandj?θasinTheorem3.7,Iconcludethat??0isinV[G][g],asdesired.?Theorem

Letmeintroducenowanotherkindofforcingforwhichstronglycompactcar-dinalswillbecomeindestructible.ForanysetSwhichisinanormalmeasureonκ,theclubforcingQSwilladdaclubC?κsuchthatC∩inacc?S;conditionsareclosedboundedsetsc?κsuchthatc∩inacc?S,orderedbyend-extension.

§4IndestructibilityAftertheLotteryPreparation34

Foreveryβ<κ,thesetofsuchcwhichmentionanelementaboveβisa≤β-closedopendenseset,sinceonecansimplytaketheunionofaβ-chainofsuchcondi-tionsandaddthesupremumtoobtainastrongercondition;thesupremumcannotbeinaccessiblesinceitisaboveβbutwasreachedbytheβ-sequence.ThusQS?S,ispreservesallcardinalsandco?nalities.Avariant,thecoherentclubforcingQmeantdirectlytofollowalotterypreparationorotheriteration,andimposestheadditionalrequirementthatwheneverδisaninaccessibleclusterpointofC,thentheprecedingiterationaddedtheclubCδ=C∩δbyforcingwithQS∩δatstageδ.Thisforcingaddsacoherentsystemofclubswhichre?ectattheirinaccessibleclusterpoints.

InthenexttheoremIwillneedthesimplefact(provedalsoin[Men74])thatif?isanormal?nemeasureonPκλandηisa?nemeasureonPκθforsomeθ≥λofco?nalityatleastκ,thentheproductmeasure?×ηisisomorphictoa?nemeasureonPκθ,andtheresultingθ-stronglycompactembeddingj:V→Misclosedunderλ-sequences;inparticular,j"λ∈M.Toseewhythisistrue,considerthecommutativediagramcorrespondingtotheproductmeasure?×η:

V

k-M

wherekistheultrapowerofM?byj?(η).EveryelementofM?hastheformj?(h)(j?"λ)forsomeh∈V,andeveryelementofMhastheformk(F)(s),whereF∈M?ands=[id]j?(η).Thus,sincek(j?"λ)=j"λ,everyelementofMhastheformj(h)((j"λ),s).Lettbetheelementofj(Pκθ)whichisobtainedinMbysimplyplacingacopyofj"λatthetopofs,separatedbyabriefgap.Fromtonecanrecoverbothsandj"λ,soeveryelementofMhastheformj(h)(t)forsomefunctionh∈V.Intheseedterminologyof[Ham97],theseedtgeneratesallofM.Itfollowsthatjistheultrapowerbythecorrespondingmeasureη?,de?nedbyX∈η??t∈j(X),andconsequentlythatη?isisomorphicto?×η.Sincej"θ?s?t,itfollowsthatη?isa?nemeasureonPκθ.Thus,jisaθ-stronglycompactembedding,asdesired.Sincej"λ∈M,itfollowsthatMisclosedunderλ-sequences,sinceanyλ-sequence?j(fα)(t)|α<λ?isequaltoj(F)(t,j"λ)where

§4IndestructibilityAftertheLotteryPreparation35

F(σ,τ)=?fα(σ)|α∈τ?,andisthereforeinM.SoIhaveprovedthefactthatIneed.Theargumentalsoworkstoshowthatif?ismerelya?nemeasureonPκλ,thenstill?×ηisisomorphictoa?nemeasureonPκθ.

Forthepurposesofthenexttheorem,letmesaythatasubsetSofthestronglycompactcardinalκisspecialwhenforarbitrarilylargeθitisintheinducednormalmeasureofaθ-strongcompactnessembeddingwitnessingtheMenaspropertyoff.If2κ=κ+andthefunctionfwasaddedbyfastfunctionforcing,thenthespecialsetsincludeanysetinanormalmeasureonκintheoriginalgroundmodel.Tosee

V[f]isthewhythisisso,supposethatSisinanormalmeasureonκin

fastfunctionextension.By1.5weknowthatSisinanormalmeasureinV,andinVwecantakeaproductofthismeasurewithanyθ-strongcompactnessmeasuretogetaθ-strongcompactnessmeasurewithκ∈j(S).Then,bytheFlexibilityTheorem

1.11,wecanmodifyj(f)andfactortheembeddinginthemannerofTheorem1.12toensurethatj(f)(κ)islargeenoughtowitnesstheMenasproperty;sinceS∈

C=C∪{κ}isavailableasaconditionin

?S,observesimilarlyj(QS).Inthecoherentclubcontext,inwhichCisgenericforQ

thatC∩κ=C,andthisisthegenericwhichwasusedatstageκinj(G).Ineithercontext,letj(C)beV[G][C][Gtail]-genericbelow

§4IndestructibilityAftertheLotteryPreparation36

Forany?nemeasure?0onPκλinV,wecan?ndbythecoverpropertyforjanelements0suchthatX∈?0?s0∈j(X)forX?PκλinV.Usethissameseedtogerminateameasurewithrespecttotheliftedembeddingjaccordingtotherule

?X∈??0?s0∈j(X)forX?PκλinV[G][C].Itisclearthat?0extends?0,and

theargumentofTheorem3.7,usingtheenumerationuofthenamesforsubsetsofPκλ,showsthat??0isinV[G][C],asdesired.?Theorem

IwouldlikenowtoamusethereaderbyapplyingtheideaoftheprevioustheoremwiththesetSofnon-measurablecardinals.

Corollary4.3Assumethatthegchholds.Then,whilepreservingthestrongcompactnessofanystronglycompactcardinalκ,onecanaddaclubC?κwhichcontainsnomeasurablecardinals.Furthermore,thiscanbedonewhilepreservingallcardinalsandco?nalities,andwhileneithercreatingnordestroyinganymeasurablecardinals.

.Proof:We?rstaddafastfunctionf..κ→κoverV.Thispreservesallcardinalsand

co?nalitiesandneithercreatesnordestroysanymeasurablecardinals.Next,wewillforceoverthemodi?edlotterypreparationP,inwhichforcingisallowedinthestageγlotteryonlywhen,inadditiontotheearlierrequirementthatitisinH(f(γ)+)andforeveryβ<γitis<β-strategicallyclosed,butalsothatitpreservesallcardinalsandco?nalitiesanddoesnotdestroyanymeasurablecardinals.SupposethatG?PisV[f]-genericforthismodi?edpreparation.Sincewecanarrangethepreparationtoadmitaverylowgap,byRemark1.2theforcingdoesnotcreateanymeasurablecardinals.Bytheremarksprecedingtheprevioustheorem,thesetSofnon-measurablecardinalsinVisspecialinV[f],andconsequently,sincethemodi?cationstothelotterypreparationheredonotcreateanydi?cultiesintheliftingargumentoftheprevioustheorem,itfollowsthatκremainsstronglycompactininV[f][G][C],whereC?κistheclubaddedbyforcingwithQSoverV[f][G].SinceC∩inacc?S,itfollowsthatCcontainsnocardinalswhicharemeasurableinV.Sincenomeasurablecardinalsarecreated,CcontainsnocardinalswhicharemeasurableinV[f][G]orinV[f][G][C].Becausetheforcingateverystagepreservescardinalsandco?nalities,thestandardreverseEastoniterationargumentsestablishthattheentireiterationalsopreservescardinalsandco?nalities.Finally,Iwillshowthatallmeasurablecardinalsarepreserved.CertainlythemeasurablecardinalsinVaboveκarepreserved.Also,Ihavearguedthatthestrongcompactnessofκitselfispreserved.Sosupposethatγ<κisameasurablecardinalinV,andhencealsoV[f].SincetheforcingafterstageγinPisstrategicallycloseduptothenextinaccessiblecardinal,itcannota?ectthemeasurabilityofγ.Also,forcingatstage

§4IndestructibilityAftertheLotteryPreparation37

γisonlyallowedwhenitpreservesthemeasurabilityofγ.Thus,itsu?cestoshowthatγismeasurableinV[f][Gγ].Therearetwocases.First,itmayhappenthatf"γ?γ.Inthiscase,theforcinguptostageγisexactlythemodi?edlotterypreparationofγ,whichbytheargumentof3.5preservesthemeasurabilityofγ.Second,alternatively,itmayhappenthatforsomeβ<γwehavef(β)≥γ.Sothereisnoforcingbetweenstageβandγ.Inthiscase,theforcinguptostageβissmallrelativetoγ,andthereforepreservesthemeasurabilityofγ,andtheforcingatstageβwasonlyallowedprovidedthatitalsopreservedthemeasurabilityofγ,soγismeasurableinV[f][Gγ],asdesired.?Corollary

Inthepreviousargument,ifoneusescoherentclubforcingoneobtainsalsoawholesequenceofclubsCγ?γforlotsofγ≤κ,alldisjointfromthemeasurablecardinals,withthecoherencyproperty,sothatwheneverβisaninaccessibleclusterpointofCγ,thenCβexistsandCγ∩β=Cβ.

Letmeintroduceanotherforcingnotionforwhichstronglycompactcardinalsbecomeindestructible.ThelongPrikyforcingposetQF,whereFisaκ-complete?lteronκ,consistsofconditions?s,A?,wheres∈[κ]<κandA∈F,orderedinthePrikrymanner,sothat?s,A?≤?t,B?whensend-extendst,A?B,ands?t?B.Thisforcingaddsasinglesetg?κsuchthateverysetinFcontainsatailofg.Itis<κ-directedclosedandhastheκ+-chaincondition;soallcardinalsandco?nalities

Mwhenarepreserved.De?nethatasetzisaccessibletoanembeddingj:

z∈

§4IndestructibilityAftertheLotteryPreparation38

M[j?(f)]?M[j(f)]suchthatj?(f)(κ)>δ.SinceRwasallowedatstageκ,itisinH(j(f)(κ)+)M[j(f)][G],andconsequently,bytheclosureofftail,itisinM[f][G].

g]andtheposetR?QFsatis?estherequiredclosureAndsincealsoQF∈M[f][G][?

conditions,itisallowedtoappearinthestageκlotteryofj?(P).LetpbetheconditionwhichoptsforR?QFatstageκ,sothatbelowptheforcingj?(P)factorsasP?(R?QF)?Ptail,wherePtailis≤δ-strategicallyclosedinM[j?(f)][G][?g][g].

?PtailoverV[f][G][g]andlifttheembeddingtoj?:V[f][G]→ForcetoaddG?tail?M[j?(f)][j?(G)]wherej?(G)=G?(?g?g)?G?tail.Nowconsiderthej(QF)forcing.

EnumerateF=?Xα|α<2κ?.Sincej"θ?s,itfollowsthatj?(F)?s=?Yβ|β∈s?j(2κ)?providesacoverofj?"Fofsizeatmostδ.SincethesetsYβareallinj?(F),andj?(F)isaj?(κ)-complete?lterinM[j?(f)][j?(G)],IcanintersectthemalltoobtainasetY=∩{Yβ|β∈s?j(2κ)}∈j?(F).Sincej?(Xα)=Yj?(α),itfollowsthatY?j?(X)foranyX∈F.Thus,?g,Y?isaconditioninj?(QF)withthepropertythatany?t,A?∈ghas?g,Y?≤?t,j?(A)?=j?(?t,A?);thatis,itisamaster

?condition.Forcebelowittoaddthegenericj?(g),andinV[f][G][g][G?tail][j(g)]lift

theembeddingtoj?:V[f][G][g]→M[j(f)][j?(G)][j?(g)].If?0isany?nemeasureonPκλinV,thenwemay?ndaseeds0∈Mfor?0asin3.7,andlet??0bethemeasuregerminatedbys0viaj?.Certainly??0extends?0andtheargumentof3.7involvingtheenumerationuofthenicenamesinVforsubsetsofPκλinV[f][G][g]showsthat??0liesinV[f][G][g],asdesired.?Theorem

Inthecasethatastronglycompactcardinalhassomenontrivialdegreeofsupercompactness,thispartialsupercompactnesscanbeusedtoobtainmoreinde-structibilityforthefullstrongcompactness.

IndestructibilityTheorem4.5Ifκisstronglycompactandλ-supercompactinthegroundmodel,thenafterthelotterypreparationrelativetoafastfunction,bothofthesepropertiesareindestructiblebyany<κ-directedclosedforcingofsizelessthanorequaltoλ.

Proof:FirstletmeshowthepreliminaryclaiminV[f]thatforanyθthereisaλ-closedθ-stronglycompactembeddingwitnessingtheMenaspropertyoff.BeginwiththeargumentprecedingTheorem4.2,whichproducesaθ-stronglycompactembeddingj:V[f]→M[j(f)],theultrapowerbya?nemeasureηonPκθ,whichisclosedunderλ-sequences.Inparticular,j"λ∈M[j(f)].ByRemark1.2,itmustbethatj"λ∈M.ByLemma2.7,wemayassumethats=[id]η∈M,andmoreoverthatj"λisdirectlycodedintothetopelementsofs.Now,bytheFlexibilityTheorem1.11,thereisanotherembeddingj?:V[f]→M[j?(f)]?M[j(f)]with

§4IndestructibilityAftertheLotteryPreparation39

j?(f)(κ)>|s|.LetX={j?(h)(s)|h∈V[f]}?M[j?(f)]betheseedhullofswithrespecttoj?,andj0:V[f]→M0[j0(f)]theinducedfactorembedding,withj0=π?jwhereπistheMostowskicollapseofX.Itfollowsthats0=π(s)generatesallofM0[j0(f)],andsoj0istheultrapowerbytheθ-stronglycompactmeasureη0germinatedbysviaj(orbys0viaj0).Sincej"λ∈X,itfollowsthatj0"λ∈M0[j0(f)].Furthermore,j0(f)(κ)>|s0|.Inparticular,j0isaλ-closedθ-stronglycompactembeddingwhichwitnessestheMenaspropertyoff,sothepreliminaryclaimisproved.

?≥λandanyθ≥22λ,Continuingwiththemainargumentnow,?xanyλ

andsupposethatj:V[f]→M[j(f)]isaλ-closedθ-stronglycompactembeddingwitnessingtheMenaspropertyoff.Wemayassumej"λ∈Mandj(f)(κ)>δ=|s|wheres=[id]η∈M.Supposeg?QisV[f][G]-genericforthe<κ-directedclosedforcingQofsizeatmostλ.BythetechniquesusedpreviouslyIcanlifttheembeddingtoj:V[f][G]→M[j(f)][j(G)]inV[f][G][g][Gtail]suchthatthegenericj(G)=G?g?GtailoptsfortheforcingQatstageκ,andthenextstageofforcingisbeyondδ.IknowthatQ∈M[j(f)][G]sinceM[j(f)]isclosedunderλ-sequencesandQhassizeatmostλ.Itisallowedtoappearinthestageκlotterybecausej(f)(κ)>δ≥θ.Considernowtheforcingj(Q).Fromgandj"λwecanconstructj"ginM[j(f)][j(G)].Andsincethissethassizeλ<j(κ)andisdirected,thereisaconditionp∈j(Q)beloweveryelementofj"g.Forcebelowptoaddj(g),andlifttheembeddingtoj:V[f][G][g]→M[j(f)][j(G)][j(g)]inV[f][G][g][Gtail][j(g)].Let?bethesetofallX?PκλinV[f][G][g]suchthatj"λ∈j(X).Itiseasytoseethatthisisnormaland?ne.Furthermore,theargumentsoftheprevioustheoremsshowthat?∈V[f][G][g].Consequently,κisλ-

?extendstoa?nemeasuresupercompactthere.Finally,any?nemeasure?0onPκλ

??0inV[f][G][g]bytheargumentsgivenpreviously,soκisstronglycompactthereaswell,asdesired.Indeed,Ihaveshownthateveryλ-supercompactnessmeasureinVextendstoameasureinV[f][G][g],andeverystrongcompactnessmeasureinVextendstoameasureinV[f][G][g].?Theorem

Noticethatwhilethelotterypreparationusesstrategicallyclosedforcingateverystage,Ionlyclaimpreservationby<κ-directedclosedforcinginthepre-vioustheorem.Thiscannotbegeneralizedtoincludeall<κ-strategicallyclosedforcing,becausetheforcingwhichaddsanon-re?ectingstationarysubsettoκis<κ-strategicallyclosed,butalwaysdestroyseventheweakcompactnessofκ.Thereasonforusingstrategicallyclosedforcinginthelotterypreparationistoallow?<κ

§4IndestructibilityAftertheLotteryPreparation40

fortheforcingsuchasQS,whichisnotgenerally<κ-closed,whilesimultaneouslyretainingthedistributivityofthetailforcingPtail.

Inanycase,itfollowsfromthepreviousargumentthatforasupercompactcardinal,thelotterypreparationaccomplisheseverythingthattheoriginalLaverpreparationwasmeanttoaccomplish:

Corollary4.6Afterthelotterypreparation,asupercompactcardinalκbecomesindestructiblebyany<κ-directedclosedforcing.

Proof:Ifoneusesafastfunction,thiscorollaryfollowsimmediatelyfromthepre-vioustheorem.Letmeillustrate,nevertheless,howonecandirectlyfollowLaver’s

[Lav78]originalargumentinthelotterycontext,whileassumingonlytheMenaspropertyonf.Supposeg?QisV[G]-genericforsome<κ-directedclosedforc-ingQ.Fixanyλandletj:V→Mbeaθ-supercompactembeddingforsome<κθ≥2λ,|Q|whichwitnessestheMenaspropertyoff,sothatj(f)(κ)>θ.Be-lowaconditionwhichoptsforQinthestageκlottery,theforcingj(P)factorsasP?Q?Ptail,wherePtailis≤θ-strategicallyclosedinM[G][g].ForcetoaddagenericGtail?Ptail,andinV[G][g][Gtail]lifttheembeddingtoj:V[G]→M[j(G)]wherej(G)=G?g?Gtail.Now,usingj"θ,itfollowsthatj"g∈M[j(G)],andsobythedirectedclosureofj(Q)thereisamasterconditionq∈j(Q)beloweveryelementofj"g.Forcetoaddj(g)belowqandlifttheembeddinginV[G][g][Gtail][j(g)]toj:V[G][g]→M[j(G)][j(g)].Let?bethenormal?nemeasureonPκλgermi-natedviajbyj"λ.Since?measureseverysetinV[G][g],itsu?cestoshowthat?∈V[G][g].Certainly?isinV[G][g][Gtail][j(g)].SincetheforcingGtail?j(g)was≤θ-directedclosedinV[G][g],itcouldnothaveadded?.So?isinV[G][g],asdesired.?Corollary

ImprovedVersion4.7Afterthelotterypreparationrelativetoafastfunctionandanyfurther<κ-directedclosedforcing,everysupercompactnessmeasureonκfromthegroundmodelextendstoameasureintheforcingextension,andeverysupercompactnessmeasureintheforcingextensionextendsameasurefromthegroundmodel.Furthermore,ifthegchholds,theneverysu?cientlylargesuper-compactnessembeddingfromthegroundmodelliftstotheextension.

ThisisanimprovementovertheLaverpreparation,throughwhichonecanliftanembeddingj:V→Monlywhenj(?)(κ)isappropriate.

Proof:The?rsthalfofthe?rstsentencefollowsimmediatelyfromtheproofof

4.5.ThesecondhalffollowsfromRemark1.2.ThesecondsentencefollowsbythediagonalizationtechniqueofTheorems3.10and1.10.Speci?cally,ifV[f][G]

§4IndestructibilityAftertheLotteryPreparation41

isthelotterypreparationrelativetothefastfunction,g?Qis<κ-directedclosed

<κforcingofsizeatmostλand2λ=λ+,thenanyλ-supercompactnessembedding

j:V→Mliftstotheextensionj:V[f][G][g]→M[j(f)][j(G)][j(g)].?Theorem

ThepreviousargumentadmitsacompletelylocalanalogueinawaythatLaver’soriginalpreparationdoesnot.Ingeneral,onecannotperformtheLaverpreparationofaλ-supercompactcardinalκunlessonehasλ-supercompactnessLaverfunction;<κbutLaver’sproofthatsuchafunctionexistsrequiresthatκis2λ-supercompactinthegroundmodel.Thus,ithasbeenopenwhetheranypartiallysupercompactcardinalcanbemadeindestructible,evenassumingthegch.Thisquestionisansweredbythefollowingtheorem.

Level-by-levelPreparation4.8Ifκisλ-supercompactinVand2λ=λ+,thenafterthelotterypreparationtheλ-supercompactnessofκisfullyindestructibleby<κ-directedclosedforcingofsizeatmostλ.

Proof:ThisisessentiallywhatIactuallyarguedintheprevioustheorem.Tosup-portthediagonalizationargument,oneonlyneedstheMenaspropertyonf.?Theorem

Ihaveshownbytheprevioustheoremsthatthelotterypreparationmakesanystronglycompactcardinalκpartiallyindestructible;butperhapsthereismuchmoreindestructibilitythanIhaveidenti?ed,soitisnaturaltoask:

Question4.9Forwhichothernaturalforcingnotionsdoesastronglycompactcardinalκbecomeindestructibleafterthelotterypreparation?

Letmeconsidernowthelotterypreparationofastrongcardinalκ.Recallthatκisstrongwhenforeveryλthecardinalκisλ-strong,sothatthereisanembeddingj:V→MwithcriticalpointκsuchthatVλ?M.GitikandShelah[GitShl89],usingWoodin’s[CumWdn]techniqueforpreservingastrongcardinal,showedhowtomakeanystrongcardinalindestructibleby≤κ-directedclosedforcing(indeed,theyimprovethistoweaklyκ-closedposetswiththePrikryproperty).IwouldliketoshowthatsuchindestructibilityisalsoachievedbytheLotterypreparation.Theorem4.10Afterthelotterypreparationofastrongcardinalκsuchthat2κ=κ+,thestrongnessofκbecomesindestructibleby≤κ-strategicallyclosedforcing.

Proof:ThisissimilartotheproofofTheorem3.6,exceptthatIwilloptfortheappropriateforcingatstageκinj(P).SupposethatV[G]isthelotterypreparationde?nedrelativetoafunctionfwiththeMenaspropertyforstrongcardinalsinV.Theresultislocalinthatonlytheλ-strongnessofκinVisneededtoknowthatthe<κ

§4IndestructibilityAftertheLotteryPreparation42

λ-strongnessofκisindestructibleoverV[G]byanyforcingnotionofrankbelowλ.SupposeH?Q∈(V[G])λisgenericoverV[G]for≤κ-strategicallyclosedforcingQ.Fixaλ-strongembeddingj:V→Mfromthegroundmodelsuchthatj(f)(κ)>λ.AsinTheorem3.6,ImayassumethatM={j(h)(s)|h∈V&s∈δ<ω},whereδ=?λ<j(κ).Letpbetheconditioninj(P)whichoptstoforcewithQinthestageκlottery.Sincethenextinaccessiblebeyondbeyondλmustbebeyondδ,theforcingj(P)factorsbelowpasP?Q?Ptail,wherePtailis≤δ-strategicallyclosedin

˙=j(h)(β)M[G][H].Theremustbesomeordinalβ<δandfunctionhsuchthatQ

˙forQ.LetforsomenameQ

X={(z˙)G?H|z˙=j(g)(κ,β,δ)forsomefunctiong∈V}.

Itisnotdi?culttoverifytheTarski-Vaughtcriterion,sothatX?M[G][H].Also,Xisclosedunderκ-sequencesinV[G][H].Notethatκ,β,δ,p,QandPtailareallinX.Furthermore,sincePtailisj(κ)-c.c.inM[G][H]andthereareonly2κ=κ+manyfunctionsg:κ→VκinV,thereareatmostκ+manyopendensesubsetsofPtailinX.SincePtailis≤κ-closed,onecanperformthediagonalizationargumenttoconstructinV[G][H]a?lterGtail?PtailwhichisX-generic.LetmearguenowthatGtailisalsoM[G][H]-generic.IfDisanopendensesubsetofPtailinM[G][H],

˙G?HforsomenameD˙∈M.Consequently,D˙=j(g)(κ,κ1,...,κn)thenD=D

forsomefunctiong∈Vandκ<κ1<···<κn<δ.Let

Dremainsopenanddense.Furthermore,

D,andsince

§5ImpossibilityTheorem43

Theorem4.11Afterthelotterypreparationofastrongcardinalκrelativetoa

?Sfastfunction,thestrongnessofκisindestructiblebyAdd(κ,1)andbyQSandQ

wheneverκ∈j(S)forarbitrarilylargeλ-strongembeddingsj.

§5ImpossibilityTheorem

OnemighthopetogeneralizetheprevioustheoremsbyprovingthatthelotterypreparationorsomeotheralternativetotheLaverpreparationcanmakeanystronglycompactcardinalfullyindestructible.Butthishopewillnotbeful?lled;thesadfactwhichIwillnowproveisthatnopreparationwhichnaivelyresemblestheLaverpreparationcanmakeastronglycompactnon-supercompactcardinalfullyindestructible.

ImpossibilityTheorem5.1Thelotterypreparationwillalwaysfailtomakeastronglycompactcardinalfullyindestructibleunlessitwasoriginallysupercompact.Infact,anyforcingwhichresemblestheLaverpreparation—aniterationofstrate-gicallyclosedforcinginwhichthenextnontrivialstageofforcingliesbeyondthesizeofthepreviousone—willfailtomakeastronglycompactnon-supercompactcardinalfullyindestructible.Indeed,afteraddingasingleCohenreal,thereisno≤ω1-strategicallyclosedpreparatoryforcingwhichmakesastronglycompactnon-supercompactcardinalκfullyindestructible.

Thistheoremreliesonmyrecentworkin[Ham∞]and[Ham98],inwhich,asImentionedinRemark1.2,Ide?nedthatanotionofforcingadmitsagapbelowκwhenitfactorsasP1?P2where,forsomeδ<κ,|P1|<δand?P2is≤δ-strategicallyclosed.AnykindofLaverpreparation,obtainedbyiteratingtheclosedforcingprovidedbysomekindofLaverfunction,admitsnumerousgapsbelowκ.Thelotterypreparationadmitsagapbetweenanytwolotterystages.TheImpossibilityTheorem5.1,therefore,isanimmediateconsequenceofthefollowingtheorem.Theorem5.2Forcingwhichadmitsagapbelowastronglycompactcardinalκcannotmakeitindestructibleunlessitwasoriginallysupercompact.

Thistheoremisanimmediateconsequenceofthefollowingtheorem,wherecoll(κ,θ)istheusualforcingnotionwhichcollapsesθtoκ.

Theorem5.3IfV[G]admitsagapbelowκandκismeasurableinV[G]coll(κ,θ),thenκwasθ-supercompactinV.

Proof:SupposethatV[G]admitsagapbelowκ,thatκremainsmeasurableafterthecollapseofθtoκ.Imustshowthatκisθ-supercompactinV.Fixan

§5ImpossibilityTheorem44

embeddingj:V[G][g]→M[j(G)][j(g)]withcriticalpointκ,witnessingthatκremainsmeasurableinV[G][g].NoticethattheforcingG?gadmitsagapbelowκ.Sincemoreoverjisanultrapowerembedding,M[j(G)][j(g)]isclosedunderκ-sequencesinV[G][g].Sinceθhasbeencollapsed,itisthereforealsoclosedunderθ-sequences.ItfollowsdirectlynowfromtheGapForcingTheoremof[Ham∞],explainedinRemark1.2,thatj?Visde?nableinVandthatMisclosedunderθ-sequencesinV.Thus,κisθ-supercompactinV,asdesired.?Theorem

Corollary5.4Thefollowingareequivalent:

1.κissupercompact.

2.κismeasurableinaforcingextensionwhichadmitsagapbelowκandinthisextensionthemeasurabilityofκisindestructiblebycoll(κ,θ)foranyθ.

Proof:Certainly1implies2becausetheLaverpreparation(ortheLotteryprepa-ration)ofasupercompactcardinalκmakesκindestructibleandadmitsagapbelowκ.Conversely,2implies1bytheprevioustheorem.?Corollary

Infact,ifthegchholds,thentheresultiscompletelylocal:κisθ-supercompactifandonlyifthereisaforcingpreparationwhichadmitsagapbelowκwhichmakesthemeasurabilityofκindestructiblebycoll(κ,θ).Fortheforwarddirection,onecanuseTheorem4.8.

Whileonemightsupposefromtheseresultsthateveryindestructiblestronglycompactcardinalissupercompact,thiscannotberightbecausethetheoremofApterandGitik[AptGit97],whichImentionedearlier,saysthatitispossibletohaveafullyindestructiblestronglycompactcardinalwhichisalsotheleastmea-surablecardinal.Suchacardinalcouldneverbesupercompact.Beginningwithasupercompactcardinal,ApterandGitik’spreparationinvolvesiteratedPrikryforcingandconsequentlydoesnotadmitagapbelowκ.

Thetheoremabovedoesshow,however,thatonecannothopetomakestronglycompactnon-supercompactcardinalsindestructiblewithforcingthatnaivelyre-semblestheLaverpreparation,sinceallsuchforcingswouldadmitagapbelowκ.Inparticular,onecannotprovethatanystronglycompactcardinalcanbemadeindestructibleby≤ω1-closedpreparatoryforcing,oreven≤ω1-strategicallyclosedpreparatoryforcing,sinceifsuchforcingwereprefacedbyaddingaCohenreal,thenthecombinedforcingwouldadmitagap.Thus,whenitcomestomakinganystronglycompactcardinalfullyindestructible,weevidentlyneedacompletelynewtechnique.Atthemoment,thefollowingquestionsareopen:

Bibliography45

Question5.5Supposeκisstronglycompact.Isthereapreparatoryforcingto

makethestrongcompactnessofκindestructiblebyforcingoftheformAdd(κ,δ)?

orjustbyAdd(κ,κ+)?oroftheformcoll(κ,δ)?orjustcoll(κ,κ+)?

KobeUniversity,Kobe,Japan,and

TheCityUniversityofNewYork

hamkins@postbox.csi.cuny.edu

http://www.library.csi.cuny.edu/dept/users/hamkins

Bibliography

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