the lottery读后感

篇一：the lottery>读后感

昨天查资料时看到有篇文章这样说 Jackson写的这篇>故事：美国大多数人应该都知道这篇文章，即使不知道是谁写的，也应该知道“ The Lottery（摸彩）”这个故事。

我感觉，这篇《 The Lottery（摸彩）》的性质，应该和《皇帝的新装》差不多吧。

看完那篇《 The Lottery》之后，心里嗟吁不已。那个小镇有个上百年来一直沿袭着的传统，每年六月里的一天，总会把小镇上的人们聚集在一起，摸彩。

随着故事散漫地进展，我也散漫地读着。天气如何地好，女孩子们如何地聚在一起聊些无聊的话，男孩子们如何地搞闹追逐着玩石块。大人们如何有一句没一句地拉家常，等着摸彩。镇长如何地捧了大盒子过来，如何准备工作都做好了。然后怎样一个人一个人地被叫上去摸彩。

故事就这么有一搭没一搭地进展着。等所有人都摸了彩以后，镇长才让大家一起打开摸到的纸片。 我呢，仍是不在意地读着。

直到读到结尾，抽到彩的那人原来是要被全村人用石头打死。于是从刚刚散漫的故事进展中我忽地一惊，吓了一跳。心里嗟吁不已。

摸彩是这个镇上人上百年来的习俗，每年都要摸彩摸出一个人来，然后其余的人用石块将他打死。人们在摸彩前后及扔石块时竟没有一点哀痛，只是忙忙碌碌地想快快打完了收工，赶着回家继续各自没干完的活儿。因为对于这样一个“传统”，镇上的每个人都已是根深蒂固地习惯了，在他们的概念里，摸彩是理所当然的，摸到彩的人要被众人打死也是理所当然的。

而作为局外人，读完之后只觉的哭笑不得，好不可思议！因为这个传统本身就是如此地没道理，如此地荒谬，也如此地恶心。

我开始时不喜欢这个故事，可后来越想便越感受到它寓意的丰蕴醇厚。

事实自然是如此，坚持着一个传统的人自然是觉的自己所坚持的传统是理所当然的。我们也是如此。而问题是：你如何知道在这些你认为是理所当然的传统中，哪些是真理，哪些仅仅是由传统和文化影响所成的定式思维呢？哪些是该坚持的，哪些是不该坚持的呢？不光是“传统”，其实是推到我们所信之事的每一个层面。

我不是说该怀疑所有，我是说，总有些是该被怀疑推止的，也总有些是该经的起怀疑的洗礼后更加坚稳的。

篇二：the lottery读后感

he lottery译为《摸彩》，说的俗一点就是抽彩票，但其性质不同，结果也不同。这篇文章相当复杂，不 是它太难懂而是里面的事情实在是与中国的文化没有一点交接，用中国话说就是封建迷信，还是相当封建的。

我大概读了3遍以后，差不多看懂说什么了，又去查查资料才了解实质，写作背景就是反乌托邦，这个反乌托邦更是相当复杂，有兴趣自己去查吧，然后继续这个作者是个美国人叫Shirley Jackson。据说这个人相当强悍，很会讲故事，被人称为天才小说家，而且她讲的故事都很幽默，而且不经意之间就流露出来，要们心神领会，怪不得读着很有难度，之后就是她的小说结尾往往很令人意外，这个确实是。自这篇文章出版以后她的粉丝就是越来越多了，还被誉为“作家的作家”，因为有很多作家都崇拜她，果然很无敌。文章开头就埋伏笔设悬念，从一帮小小孩子堆石子开始引出抽奖的整个过程，其中细节描写相当入微。

例如，故事里的人物姓名，就具有丰富的象征。负责摸彩活动的萨莫思（Summers），英文意思是“夏天”，其复数形式暗指年头或曰时光的流逝；他的助手格瑞午思

（Graves），则意为“坟墓”，这既暗示了每次摸彩的结果都是某一个人的死亡，也暗示了摸彩活动本身最终应该的去处——这应当也是作者本人的意图吧。德拉克柔

（Delacroix）这个名字的原意为“十字架的”；可是，作者在故事的开头处就专门说明，村民们总是把这个名字念错，并且完全忘记了正确的读法。这其中的深意自然是关涉基督教的——村民们早就无法正确理解基督教的真正教义了。此外，那个摇摇欲坠的三条腿的破凳子，被看作是暗指失去权威的三位一体的神权。

那个破旧的黑色盒子， 则既象征着死亡，也象征传统的陈旧以及村民对传统的混沌与盲从；此外，黑盒子是用多年以前的“老盒子的残余木板”拼成的，这个细节也暗示了摸彩活动所代表的传统已经变质并远远落后于时代。真是相当的复杂。

在所有的象征和寓意里，含意最丰富的还是故事的情节——作为仪式的杀人。《摸彩》在故事开始时， 描写了一幅伊甸园般的美好>景色，村民们也相处和平。但是在故事的结尾，人们读到的却是一场与美好环境格格不入的杀戮。尤其令人发指的是，这是一次和平时期亲人和邻里间的残杀。

《圣经》里描写的人类第一次杀人，就发生在兄弟之间。那是人类始祖亚当和夏娃的长子该隐对兄弟亚伯的残杀。值得注意的是，上帝接下来的警告是，如果有人因此想杀该隐，则“必遭报七倍”，可见杀人罪之严重。然而，如此严厉的警告依然没有使人类停止杀戮。整个人类历史记满了人与人的自相残杀。死去的人就像替罪羊，或是为了“玉米快熟”一类的眼前利益，或是为了其他更丰厚的经济目的。

为了让读者不至于忽略这个重要的寓意，杰克逊还利用了石头杀人的典故。这个典故也出自《圣经》一篇著名寓言。耶稣的敌人要求耶稣依照摩西之律，用石头砸死淫妇。可是，当耶稣说“你们当中谁没有罪，谁先拿石头砸她！”时，人们“便从年老的开始，一个接一个溜走了”。不过，在《摸彩》中，人们读到的情节却恰恰相反：最年长的沃内没有丝毫的反省意识，反而是他在带头招呼人们去拿石头砸人。当然，所有的村民都和他一样，盲从并自以为是。因此，在集体参与下，在和平时期，全村人联手杀害了一

个自己人。这个问题相当严重啊，就好比是那个FLG，不仅危害自己，简直就是迫害全人类。从上看出相信科学有多么重要啊！！！

这篇文章看到这就差不多了，但我还在网上看到了更深一层的了解，还是从作者的背景入手介绍杰克逊从小生长在富裕的中产阶级家庭。她的母亲笃信美丽是女孩>幸福之源的传统观念，一心要把她培养成和自己一样的社交圈中人，美丽并讨人喜欢。可是，杰克逊虽相貌端正， 却对当可爱的洋娃娃毫无兴趣。为了反叛母亲的传统观念，她把自己吃得胖胖的，然后嫁给了一个犹太知识分子的大学教授。可是，在美国东部佛蒙特的偏僻小大学里，反犹、反知识的传统同样令杰克逊感到窒息般的压抑。在她心里，她，乃至她的家人，似乎都是那个城市的异类，而她所做的，就是保持自我，拒绝依顺社会习俗。她和环境及邻里的紧张对峙，不仅化作她的作品情节，同时也成为她的精神生活的写照。 还有人从社会层面来分析在《摸彩》中的深层意义。二十世纪四十年代，世界动荡，战争频繁。帝国主义，殖民主义，法西斯主义和极权主义泛滥成灾。西方文坛因之出现了一批“反乌托邦”作家和作品，探索人类世界种种悲剧性的未来。不论是从科学技术的角度去预见人类社会的暗淡前景，还是从政治体制入手去否定国家机器的违反人性，所有这些作品都是从社会制度或统治形式的角度，对现实提出质疑和批判。《摸彩》延续并深化了这个主题。它所质疑的不是众人皆知的帝国主义、殖民主义或法西斯主义的罪恶，而是被寄予希望的“多数人统治”——民主制度。《摸彩》一针见血，直触事物本质：即使在集体参与的民主形式下，在所谓的人人有份和机会平等的社会活动中，民主方法也仍然可能是摧残个体乃至杀人的集体暴行，甚至还充当掩盖暴行的堂皇借口，一如《摸彩》中集体杀死哈太太的决定。《摸彩》就这样毫不留情地瓦解了人们对民主政体的幻想，冷酷地证实了“民主是除了已经先后尝试过的其他形式之外，最糟糕的统治形式”。也难怪对《摸彩》愤怒的不仅来自右派，还有信奉民主理想的左派。

但我觉得最重点还是《摸彩》还毋庸置疑地表明，多数人统治的民主方法不可信赖，是因为它无法制止或消除人性深处对权力和金钱的贪欲，对他人受苦受难的漠然置之，以及对自己逃脱灾难的幸灾乐祸。因此，《摸彩》虽然也批判了不合理的传统，各种权力形式，以至于男权主义，但是它的靶标中心是连民主也无法制止的人性黑暗。不少读者无法接受的一个细节是，就连传统、权力和男权主义的多重受害者哈太太，也是一个自私的人——为了减少自己抽中彩票的机会，她竟违背惯例，要求嫁出去了的大女儿也参加第二轮摸彩。这个细节曾经激怒众多读者，包括杰克逊的母亲。

《摸彩》一针见血，直触事物本质：即使在集体参与的民主形式下，在所谓的人人有份和机会平等的社会活动中，民主方法也仍然可能是摧残个体乃至杀人的集体暴行。读完以后也颇有些体会首先还是发现迷信对人的迫害，接着了解社会的黑暗，最后是对作者表示相当佩服，在社会的重压下仍能够执着的向前，为自己的小说而挖掘社会最黑暗最真实一面，却是令我刮目相看。

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

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