A MODEL IN WHICH GCH HOLDS ATSUCCESSORS BUT FAILS AT LIMITSJames CummingsMathematical Sciences Research InstituteAbstract. Starting with GCH and a P3�-hypermeasurable cardinal,a model is produced in which 2� = �+ if � is a successor cardinal and2� = �++ if � is a limit cardinal. The proof uses a Reverse Eastonextension followed by a modi�ed Radin forcing.1. INTRODUCTION1. Historical remarks and statement of the main result.The continuum problem is an old one, dating back to Cantor and hisstatement of the Continuum Hypothesis in [Ca]. Put in a modern formwhich might have puzzled Cantor, the problem is to determine whichbehaviours of the continuum function � 7�! 2� are consistent with ZFC.Throughout this paper ZFC will be the base set theory, though as wesee below strong set-theoretic hypotheses will play an essential role inthe result.Before G�odel progress on the continuum problem was made by thedescriptive set theorists, who showed that certain easily de�nable setsof reals could not be counterexamples to CH. G�odel [G] took the ma-jor step forward of showing that in a certain submodel of V , the con-structible universe L, the following Generalised Continuum Hypothesisis true (here CARD denotes the class of cardinals):8� 2 CARD 2� = �+ (GCH)Then in 1963 Cohen [Co] initiated the method of forcing and showedthat the negation of CH was also consistent with ZFC, making CH the1991 Mathematics Subject Classi�cation. Primary 03E35; Secondary 03E55..Key words and phrases. Radin forcing, singular cardinals problem, large cardinals..I would like to thank Adrian Mathias and Hugh Woodin, who supervised mydoctoral thesis from which this result is taken; also the Science and EngineeringResearch Council of Great Britain for their support. Typeset by AMS-TEX1

2 JAMES CUMMINGS�rst statement of \natural" mathematics to be shown undecidable inset theory. This breakthrough was quickly followed by the work of Eas-ton [E], who showed that if GCH holds in V and F : REG �! CARD(where REG is the class of regular cardinals) is a class function suchthat 8� 2 REG F (�) > �8�; � 2 REG � < � =) F (�) � F (�)8� 2 REG cf(F (�)) > �then there exists a class forcing preserving all cardinals and co�nalitiessuch that in the extension8� 2 REG 2� = F (�)That is, the values of the continuum function at regular cardinals arehighly independent of each other, and can be \anything reasonable".It is worth noticing that all the consistency results mentioned so farare relative to ZFC, that is are results of the form \Con(ZFC) impliesCon(ZFC + �)" where � is some statement of set theory.The �rst connection between large cardinals and the continuum func-tion was made slightly prior to the invention of forcing by Scott [Sc],who used his technique of taking ultrapowers of V to show that if � ismeasurable, U is a normal measure on �, andf � < � j 2� = �+ g 2 Uthen 2� = �+. So su�ciently strong large cardinals can impose somestructure on the behaviour of the continuum function.For some time after Easton's work it was felt that Easton's resultshould generalise to any \reasonable" function F : CARD �! CARD,and that all that was needed were more sophisticated techniques forforcing over models of ZFC; but then Silver [Si1] proved some theoremsshowing that if � is a singular strong limit cardinal of uncountable co-�nality then there is a strong connection between the behaviour of thecontinuum function below � and the value of 2�. For example if GCHfails at � then it already fails on a closed unbounded set of cardinals lessthan � (the parallel with Scott's result is not coincidence; Silver's proofgoes via a generic ultrapower by an ultra�lter extending the club �lter).More surprises were in store; let SCH (Singular Cardinals Hypothesis)be the assertion that if � is singular then 2� is the least cardinal � � 2<�with cf(�) > �. This would imply that the continuum function wasdetermined by its behaviour on the class of regular cardinals. Jensen [DJ]

A MODEL IN WHICH GCH HOLDS AT SUCCESSORS BUT FAILS AT LIMITS3was inspired by seeing Silver's result to show that if 0] does not exist thenSCH is true: however by work of Silver and Prikry (see [Si2], [P]) it wasknown that if the existence of a supercompact cardinal is consistent thenso is the failure of SCH (use Reverse Easton forcing to get a measurable� such that 2� = �++, then use Prikry forcing to make � have co�nality!. � becomes a rather large singular strong limit cardinal at whichGCH fails). So large cardinals are essential for an understanding of thecontinuum function.It is natural to ask about the situation at small singular cardinals.Magidor showed in [Ma1], [Ma2] that (among other things)1) Con(ZFC +GCH+ there is � �++-supercompact) implies Con(@! isstrong limit and 2@! = @!+2)2) Con(ZFC + there exist � < � with � supercompact and � huge)implies Con(GCH holds below @! and 2@! = @!+2)From 1) SCH can fail at the smallest singular cardinal of them all;from 2) no direct analogue of the theorem from [Si1] mentioned abovecan hold.There is a large gap in strength between 0] and a supercompact car-dinal. At one end Dodd and Jensen [D] invented the core model andshowed that failure of SCH has consistency strength greater than that ofa measurable cardinal, and following this line of research Mitchell [Mi2]got the consistency of measurable cardinals of high order from the failureof SCH. At the other end Woodin [W] showed how to get Magidor's re-sult 1) from the consistency of a P2�-hypermeasurable cardinal. Shelahgot some bounds ([Sh1], Chap. XIII) for singular cardinals of co�nality!: for example if @! is strong limit then 2@! < @(2@0 )+ . Shelah alsoshowed [Sh2] that @! being strong limit was consistent with 2@! beingany successor cardinal below @!1 , using a supercompact, and Magidorextended this in unpublished work to get a similar result with GCHholding below @!.Another line of development has involved global results. Foreman andWoodin [FW] showed that starting from slightly more than a supercom-pact it is consistent that the GCH fail everywhere, and Woodin thenproduced a model where 8� 2� = �++. Woodin later showed thathypermeasurability-type hypotheses su�ce for these results.This paper, which is basically a revised version of Chapter Two of mydoctoral thesis [C], also proves a global result: if a P3�-hypermeasurablecardinal is consistent then so isZFC + GCH holds at all successors and fails at all limitsThe �nal model is of the form V�� where � is a P2�-hypermeasurablein V�, so the model contains many smallish large cardinals.

4 JAMES CUMMINGSThe proof combines Radin forcing with ideas from unpublished workof Woodin, in which he shows that Magidor's result 2) mentioned abovecan also be done from a P2�-hypermeasurable.Added in proof: Magidor and Gitik [GM] have recently determinedthe exact strength of the failure of SCH. A sample result is that failureat @! is equiconsistent with o(�) = �++. Their methods will also doMagidor's result 2) from this hypothesis (and indeed the stronger resultwith GCH up to @! and a countable jump at @! from less than a strongcardinal) but are very speci�c to cardinals of co�nality ! and do notseem to mix with Radin forcing in any straightforward way.Also Shelah has proved the surprising theorem that if 2@0 < @! then@@0! < @!4 , and shown that some new idea is needed to make a modelin which @! is strong limit and 2@! > @!1 .2. Background material.In this section we establish our terminology and notation; we also giveproofs for some facts which we need in the Reverse Easton constructionof Section Two. These facts are part of the folk literature of set theory,but we give the proofs here to make the exposition self-contained.The set-theoretic notation used is mostly that of [J], with the caveatthat the forcing terminology is that of Kunen's reference [K], in partic-ular \�-closed" means \having lower bounds for decreasing sequencesof length less than �". \�-dense" means that the intersection of fewerthan � dense open sets is dense.(set theoretic term)M means the result of computing the term in themodel M , and when we write for exampleM � cf(�) = �++we mean that cfM (�) = �++M . If P is a poset and p 2 P then P � p isused to denote f q 2 P j q �P p g with the inherited partial ordering.When we say that two posets are isomorphic we mean there is an orderpreserving map from one to the other with dense domain and range (thisis enough to make them equivalent from the point of view of forcing).Certain forcing notions will be used very heavily; if � is regular, � � �is a cardinal and � > � is inaccessible then(1) Add(�; �) is the poset of partial functions from �� � to 2 withsupports of size less than �. We regard it as adding � genericsubsets of �. It is �-closed and (2<�)+-c.c., and will make 2<�have cardinality � without adding bounded subsets to �.(2) Coll(�; �) is the poset of partial functions from � to � with sup-ports of size less than �. It will collapse � to be of cardinality�.

A MODEL IN WHICH GCH HOLDS AT SUCCESSORS BUT FAILS AT LIMITS5(3) Coll(�;< �) is the poset of partial functions f : ��� � wheref has support of size less than � and 8� 8� f(�; �) < �. It is�-closed and �-c.c., and makes � into the successor of �.Many well-known facts about iterated forcing will be used withoutcomment; we refer the reader to Baumgartner's article [B].{1. Extenders. An extender is a device for describing an elemen-tary embedding between models of set theory; there is a good accountin [MS] but our conventions di�er rather from those of Martin and Steel.The reader of [MS] should have no trouble in providing proofs for theassertions we make here.De�nition. Let a, b be �nite sets of ordinals with a � b. Supposeb = f�1; : : : ; �ng<a = f�l1 ; : : : ; �lmg<De�ne fab : [ON]n �! [ON]m byfab : f�1; : : : ; �ng< 7�! f�l1 ; : : : ; �lmg<De�nition. Let M be a transitive model of ZFC, let � be a cardinal ofM , and � > � an ordinal. Then E is a (�; �)-M-extender i�E = hEa : a 2 [�]<!iwhere for some ordinal �(1) For all a 2 [�]<! Ea is a M-�-complete M-ultra�lter on [�]jaj.(2) For all a; b 2 [�]<!, if a � b and fab : [�]jbj �! [�]jaj is de�ned asabove, then 8A A 2 Ea () f x j fab(x) 2 A g 2 Eb(3) The limit ultrapower of M by the system E (which exists by (1) and(2) and standard facts about limit ultrapowers) is well-founded.(4) If jE :M �! N is the limit ultrapower embedding then(a) crit(jE) = �.(b) � � jE(�).(c) For all a 2 [�]<! X 2 Ea () a 2 jE(X).(d) N = f jE(f)(a) j a 2 [�]<! ^ f 2M ^ f : [�]jaj �!M g.

6 JAMES CUMMINGSDe�nition. If E is an extender as above then the critical point of E is�, the length of E is � and the width of E is �.The motivation for this de�nition is that if we have transitive modelsof ZFC M , N , and an elementary embedding i : M �! N such that� � i(�), then we can de�ne a (crit(i); �)-M -extender of width � byX 2 Ea () a 2 i(X)for a 2 [�]<!.The embedding i will factor as k � jE where crit(k) � �and k : hf; aiE 7�! i(f)(a). iE is an approximation to i which improvesas � increases.We list some facts.Fact 1. If GCH holds and j : V �!M is an elementary embedding intoa transitive inner model M , with critical point �, such that �M � Mand V�+n � M (a so-called Pn�-hypermeasurable embedding) then theextender of length �+n approximating j has width � and gives rise to aPn�-hypermeasurable embedding.Fact 2. Let GCH hold, and let j : V �!M be a Pn�-hypermeasurableembedding generated by a (�; �+n)-V -extender. Factor j as k � j0 wherej0 : V �! M0 is the ultrapower of V by f X j � 2 j(X) g. Then k isgenerated by a (�++M0 ; �+n)-M0-extender of width �+nM0 .Fact 3. If E 2 V is a V -extender then for every ordinal � the �thiterate of V by E exists and is well-founded.We make the remark that GCH was critical to the truth of Facts 1 and2; later on we will construct a model in which � is P2�-hypermeasurableand 2� = �++, and there we will need a (�; �+3)-extender to describethe associated elementary embedding. This is natural enough, since inthis instance we have jV�+2j = �+3.{2. Extending embeddings. We collect some information aboutforcing and elementary embeddings, for use in the Reverse Easton con-structions of Section 2.Fact 1. Let M , N be inner models of ZFC, j :M �! N an elementaryembedding, P 2M a notion of forcing. Let G be P -generic over M , Hj(P )-generic over N , and suppose8p 2 P p 2 G =) j(p) 2 H (�)Then j can be extended to j� :M [G] �! N [H ], wherej� : iG( _� ) 7�! iH(j( _� ))

A MODEL IN WHICH GCH HOLDS AT SUCCESSORS BUT FAILS AT LIMITS7and j� is still elementary.Proof. Let �(x1; : : : ; xn) be a formula of set theory with free variablesamong x1; : : : ; xn. M [G] � �(iG( _�1); : : : ; iG( _�n))=) 9p 2 G p MP �( _�1; : : : ; _�n)=) 9q 2 H q Nj(P ) �(j( _�1); : : : ; j( _�n))=) N [H ] � �(iH(j( _�1)); : : : ; iH(j( _�n)))Letting � be \x1 = x2", j� is well-de�ned; replacing � by :� j� is fullyelementary. �Fact 2. Let M , N , j, P 2M be as in the statement of Fact 1. Let j begenerated by an extender with width �, and let M � \P is �+�dense".Let G be P -generic over M . ThenH = f q 2 j(P ) j 9p 2 G j(p) � q gis j(P )-generic over N , and as (*) holds j extends to an elementaryembedding j� :M [G] �! N [H ].Proof. Let D 2 N , D � j(P ) be a dense open subset. D = j(f)(a),where a 2 [ON ]<! , f 2M , and8x 2 [�]jaj f(x) is a dense open subset of PAs M � P �+�dense, D� = \x2[�]jaj f(x)is a dense open subset of P . D� 2 M , so �nd p such that p 2 D� \ G.M � 8x p 2 f(x), thus N � j(p) 2 j(f)(a) = D. So D \H 6= ;, henceH is generic. �Fact 3. Let M , N , j, P , G, H be as in the statement of Fact 1. Let(*) hold, and let j be generated by a (�; �)-M-extender. Then j� is sogenerated.Proof. Let x 2 N [H ]. x = iH( _� ) for some _� 2 N j(P ). _� = j(f)(a),where a 2 [�]<!, f 2 M , f : [�]jaj ! MP . De�ne f� 2 M [G],dom(f�) = [�]jaj, f� : y 7�! iG(f(y)). Clearly x = j�(f�)(a), soN [H ] = f j�(g)(b) j g 2M [G]; b 2 [�]<! g �{3. Building generics. We collect some facts pertinent to construct-ing generics over one model inside another.

8 JAMES CUMMINGSFact 1. Let M , N be inner models of ZFC, M � N . Let P 2M ,N � \P is ��closed"N � jfA j A is a maximal antichain in P;A 2M gj � �Then there is in N a P -generic �lter over M .Proof. In N , enumerate the set of maximal antichains of P lying in Mas hA� : � < �i. Build by induction a sequence hp� : � < �i such that� < � =) p� � p�8� 9x 2 A� p� � xThe closure enables the construction to be continued at limit stages.G = f p 2 P j 9� p� � p gis clearly a �lter on P which is generic over M . �Fact 2. Let M , N be inner models of ZFC, M � N , � 2 ON . ThenN � �M �M () N � �ON �MProof. By AC , every set in M is in bijection with some ordinal. �Fact 3. Let M , N be inner models of ZFC, M � N , N � �M � M .Let P 2 M be a forcing notion, N � \P is �+�c:c:", and let G beP -generic over N . Then N [G] � �M [G] �M [G].Proof. By Fact 2 it is enough to show that N [G] � �ON � M [G]. Butit is immediate from the chain condition of P in N that to any name inNP for a �-sequence of ordinals there corresponds an equivalent namein MP . �{4. Mutual genericity. We give some results which guarantee themutual genericity of generic objects under certain circumstances.Fact 1 (Easton's Lemma). Let P be a �-c.c notion of forcing, andQ be a �-closed one. Then P \ Q is �-dense", Q \P is �-c.c", andgenerics for P and Q are mutually generic.Proof. Let _A 2 V Q, q 2 Q be such thatq \ _A : � �! P enumerates an antichain"De�ne a descending sequence hq� : � < �i of elements of Q, anda sequence hp� : � < �i of elements of P such that q0 = q and8� < � q�+1 _A(�) = p�. Then clearly hp� : � < �i enumerates

A MODEL IN WHICH GCH HOLDS AT SUCCESSORS BUT FAILS AT LIMITS9an antichain in P lying in V ; this is a contradiction, so we conclude that Q P ��c:c:.Let G be P -generic, and H Q-generic. The mutual genericity of Gand H is immediate, for if A 2 V [H ] is a maximal antichain in P thenjAj < � so A 2 V and A \G 6= ;.Finally we show that V [G] and V [G][H ] have the same < �-sequencesof ordinals. V [G][H ] = V [H ][G], so let _� 2 V [H ]P be a term for such asequence. By a chain condition argument we can assume _� � P � ONand j _� j < �, so _� 2 V by closure. �The following fact is due to Woodin, and plays a central role in [W].Fact 2. Let � be a measurable cardinal, U some normal measure on �,j : V �!M the ultrapower by U . Let P be a �-closed partial ordering,let Q = j(P ). �M � M , and j(�) > �, so Q is �+-closed; let G be Q-generic over V , and transfer it along j to get j� : V [G] �! M [j�(G)].Then G is in fact Q-generic over M [j�(G)].Proof. Let _D 2 M j(Q) be a name for a dense subset of Q. As M is anultrapower, _D is of the form j(f)(�), where for each � f(�) 2 V Q is aname for a dense subset of P . De�ne a subset of Q,E = f q 2 Q j 9g : � �! P q = j(g)(�) ^ 8� q dg(�) 2 f(�) gThe claim is that E is a dense subset of Q. Fix r in Q, and let r berepresented as j(h)(�) for some h : � ! P . Construct by induction on� < � functions h� and elements of P p� such thath� : � �! P , and if � < � then 8 h�( ) �P h�( )j(h�)(�) cp� 2 f(�)8 � � h�( ) = p j(h0)(�) � rConstruction. At stage �, let p 2 P be some lower bound for the se-quence hh�(�) : � < �i, and let q 2 Q be a lower bound for the sequencehj(h�)(�) : � < �i. f(�) names a dense subset of P , so �nd q� � q,p� � p, such that q� bp� 2 f(�). Let p� be p�, and alter some functionrepresenting q� to get an appropriate h�. (This is possible because weonly need alter the function on a set of measure zero). De�ne h� byh�(�) = p�; then r� =def j(h�)(�) is an extension of r lying in E.h(0) h(1) : : : h(�) : : :p0 h0(1) : : : h0(�) : : :p0 p1 : : : h1(�) : : :... ... . . . ... : : :p0 p1 : : : p� : : :

10 JAMES CUMMINGSAs E lies in V and is dense, �nd q = j(h)(�) such that q 2 E \ G,where h is chosen to witness q 2 E. q 8� < � dh(�) 2 f(�), soapplying j and letting � be �, j(q) \j(h)(�) 2 j(f)(�). j(q) 2 j�(G),so q 2 ij�(G)( _D) hence G is Q-generic over M [j�(G)]. �{5. Term forcing. We describe a forcing trick which is needed forthe Reverse Easton construction of Section 2.Let P be a forcing notion, _Q 2 V P , P _Q is a forcing notion. De�nea forcing _Q=P 2 V by_� 2 _Q=P () _� 2 V P^ P _� 2 _Q_� � _Q=P _� () P _� � _Q _�_Q=P is technically a class, but can be treated as a set, as in the standardtreatment of iterated forcing (see [B] or [K]).Forcing with _Q=P adds a \universal generic object" for _Q. Moreprecisely we haveFact 1. Let H be _Q=P -generic over V , and let G be P -generic over V .Then F = f iG( _� ) j _� 2 H g is iG( _Q)-generic over V [G].Proof.Claim 1. F is a �lter.Proof of Claim 1._�; _� 2 H =) 9 _� 2 H P _� � _Q _�; _� =) iG( _� ) �iG( _Q) iG( _�); iG( _�)_� 2 H^iG( _�) � iG( _�) =) 9 _� iG( _� ) = iG( _�)^ _� � _� =) _� 2 H �Claim 2. F is generic.Proof of Claim 2. Let _D be a name for a dense subset of _Q, and de�neE = f _� 2 _Q=P j P _� 2 _D gE is dense in _Q=P ; for if _� 2 _Q=P , P 9x x 2 _D ^ x � _�so by the maximum principle there is _� such that P _� 2 _D ^ _� � _�

A MODEL IN WHICH GCH HOLDS AT SUCCESSORS BUT FAILS AT LIMITS11As H is generic and E 2 V , for some _� 2 H iG( _� ) 2 iG( _D). �The proof is done. �There are a couple of interesting cases in which, if Q is some forcingnotion de�ned in the ground model and _Q names the forcing de�ned byworking that de�nition in V P , then Q is isomorphic to _Q=P . This isuseful in Reverse Easton forcing as another way of transferring genericsbetween models.Fact 2. Let P be a forcing notion of cardinality � with the �-c.c., where� is such that �<� = �. Thena) For any � Add(�; �)V P =P is isomorphic to Add(�; �).b) If � � � is such that �<� = � then Coll(�; �)V P =P is isomorphic toColl(�; �).Proof. If � is some cardinal let P� denote the partial ordering whichhas a top element > with � incomparable extensions (which we identifywith the elements of �).For a), recall that Add(�; �) is the product of � copies of P2 with < �-support. As P is ��c:c: it is clear that Add(�; �)V P =P is just the prod-uct of � copies of P2=P with < �-support. If _� 2 P2=P , _� is determinedup to equivalence by the pair of Boolean values ([ _� = 0]; [ _� = 1]), and anypair (b1; b2) such that b1^b2 = 0 corresponds to a name: moreover the or-dering is just the restriction of the product order on R:O:(P )�R:O:(P ).This ordering has a dense subset isomorphic with P�, where the � in-comparable elements are the pairs (b;:b) for b 2 R:O:(P ).So Add(�; �)V P =P is isomorphic to the product of � copies of P� with< �-support, i.e. isomorphic to Add(�; �) as claimed. The proof for b)is exactly similar with P� playing the role of P2; �<� = � guaranteesthat P�=P is isomorphic to P�. �{6. Two facts about Cohen forcing. The following facts are alsoneeded in Section 2.Fact 1. Let � = cf(�) > ! and let P be a �-closed notion of forcingwith jP j = �. Then P is isomorphic to the Cohen forcing Add(�; 1).Proof. The proof is just like the proof (see [J] p.277) that a forcing whichcollapses � and has size � is isomorphic to Coll(!; �). �Fact 2. Let j : V �!M be the ultrapower of V by a normal measure Uon some cardinal �, where 2� = �+. Then j(Add(�; �++)) is isomorphicin V to Add(�+; �++).

12 JAMES CUMMINGSProof. As 2� = �+ and j is an ultrapower embedding, j(�++) = �++and the set S de�ned byS = f � < �++ j j(�) = � gis unbounded in �++. Let �S, the closure of the set S, be enumerated inincreasing order as h�� : � < �++i. As �S is club in �++,�++ = [�<�++[�� ; ��+1)In V the forcing Add(j(�); [�� ; ��+1))M is �+�closed and cardinal-ity �+, hence isomorphic by Fact 1 to the forcing Add(�+; [�� ; ��+1)).Choose for each � an isomorphism �� , and de�ne a function � withdomain Add(j(�); j(�++))M by� : p 7�! [�<�++ ��(p � j(�)� [�� ; ��+1))Then we claim � is a bijection onto Add(�+; �++).First we show that � maps into Add(�+; �++). Let p be a conditionin Add(j(�); �++)M . It is necessary to show that �(p) is a condition inAdd(�+; �++), which will be immediate once we check that the supportof �(p) has cardinality � �. 9� < �++ support(p) � j(�) � �� , andtherefore support(�(p)) � �+ � �� . Do an induction on � � � withthe hypothesis that �(p) � �+ � �� has support of cardinality � �. Theinduction step is trivial at successor � and those limit � with cfV (�) � �.So assume cfV (�) > �; �� 2 S, because cfV (��) = cfV (�) > � and so�� is a continuity point of j. SocfM (��) = cfM (j(��)) = j(cfV (��)) > j(�)Hence 9� < � support(p � j(�) � ��) � j(�) � �� and the inductionhypothesis goes through.To show that � is onto, let q 2 Add(�+; �++). Because �M �M , andq has support of cardinality � �, the sequenceh��1� (q � �+ � [�� ; ��+1)) : � < �++ ^ support(q) \ �+ � [�� ; ��+1) 6= ;iis inM . As � < j(�) and j(�) is regular inM , the union of this sequenceis a condition which maps to q under �. �

A MODEL IN WHICH GCH HOLDS AT SUCCESSORS BUT FAILS AT LIMITS133. The structure of the proof.As the proof is rather long and complicated we give a brief overviewof what is going on.We start with the hypotheses(1) � is P3�-hypermeasurable, as witnessed by j : V �! M with�M �M , V�+3 �M , crit(j) = �.(2) GCH.(3) j is generated by a (�; �+3)-extender.(3) is really super uous as given a model of (1), (2) we can factor throughan extender to get (3).In Section 2 we do two Reverse Easton extensions, and at the end wehave a model of the hypotheses(1) � is P2�-hypermeasurable, as witnessed by j : V �! M with�M �M , V�+2 �M , crit(j) = �.(2) For each strong inaccessible � � � 2� = �++ and for all n0 < n < ! =) 2�+n = �+n+1.(3) M � 2� = �++ ^ 8n 0 < n < ! =) 2�+n = �+n+1(4) There is F 2 V which is P -generic over N whereP = Coll(�+4; < i(�))� Coll(i(�); i(�+))Nand i : V �! N is the ultrapower of V by the ultra�lterU = fX � � j � 2 j(X) g(5) j is generated by a (�; �+3)-extenderThen in Section 3 we use these hypotheses to de�ne a modi�ed Radinforcing; this forcing will shoot a closed unbounded set of V -measurablesC through �, and in addition if �, � are successive points of C it willadd a generic for the collapsing algebra Coll(�+4; < �)� Coll(�; �+).We prove analogues of the Prikry Lemma and Mathias' criterion forPrikry genericity, and use these to show that the forcing has producedthe right behaviour for the continuum function below � and that � hasretained its hypermeasurability.2. PREPARING THE MODEL1. First stage.Recall that the initial hypotheses are(1) � is P3�-hypermeasurable, as witnessed by j : V �! M with�M �M , V�+3 �M , crit(j) = �.(2) GCH.(3) j is generated by a (�; �+3)-extender.

14 JAMES CUMMINGSFactor j as k � i where i : V �! N is the ultrapower by the ultra�lterU = fX � �j� 2 j(X)g and k : [f ]U = i(f)(�) 7�! j(f)(�). Let � be thecritical point of k. GCH implies that � = �++N , and �+ < � < �++. Aswe remarked in the introduction k is generated by a (�; �+3)-N -extenderof width �+3N .We do the Reverse Easton iteration of Add(�+; �++) for strong inac-cessible � � �. Let P� denote the forcing up to stage �, and let G� beP�-generic over V . Let Q� = (Add(�+; �++))V [G�] be the forcing we doat stage �, and let g be Q�-generic over V [G�]. Factor Add(�+; �++) as(Add(�+; �)�Add(�+; �++ n �))V [G�]and split up g as g1 � g2 correspondingly.Now consider the iterations j(P� � _Q�), i(P� � _Q�) computed by Mand N respectively. As the three models V , M , N agree to rank � theycompute the same iteration below �, so de�ning (P�)M and (P�)N inthe obvious way P� = (P�)M = (P�)NP� is �-c.c. and thereforeV [G�] � �M [G�] �M [G�]V [G�] � �N [G�] � N [G�]So �+ = �+V [G�] = �+M [G�] = �+N [G�]. Because V�+2 � M we have�++ = �++M = �++M [G�], and so if (Q�)M is the forcing M [G�] computesat � then(Q�)M = (Add(�+; �++))M [G�] = (Add(�+; �++))V [G�] = Q�On the other hand if (Q�)N is the forcing which N [G�] computes at �then because �++N [G�] = �++N = � we have(Q�)N = (Add(�+; �++))N [G�] = (Add(�+; �))V [G�]Now k(P�) = P� and 8p 2 G� k(p) = p, so we can extend k (usingFact I.2.3.1) to k : N [G�] �!M [G�]k((Q�)N ) = (Q�)M = Q�, and since crit(k) = � and g1 � g we havethat 8p 2 g1 k(p) = p 2 g. So we can extend k further to getk : N [G�][g1] �!M [G�][g]

A MODEL IN WHICH GCH HOLDS AT SUCCESSORS BUT FAILS AT LIMITS15Because the forcing notion (Q�)N is �+ closed it adds no �-sequencesof ordinals, so V [G�][g1] � �N [G�][g1] � N [G�][g1]Also, because N is an ultrapower and GCH holds in V ,V [G�][g1] � ji(�)j = �+Consider the iteration R which N [G�][g1] does between �+ 1 and i(�).Standard facts about Reverse Easton forcing imply thatN [G�][g1] � jRj = i(�), R is i(�)-c.c. and �+4-closedWe exploit these facts to build H 2 V [G�][g1] which is R-generic overN [G�][g1]. The closure of R in N [G�][g1] and the width of the extenderfor k allow us to transfer H along k, and get k(H) k(R)-generic overM [G�][g].So we can build mapsi : V [G�] �! N [G�][g1][H ]j : V [G�] �!M [G�][g][k(H)]k : N [G�][g1][H ] �!M [G�][g][k(H)]such that j = k � i.The map i : V [G�] �! N [G�][g1][H ] is de�ned in V [G�][g1]. SinceQ� is �+-closed and i is generated by an extender of width � we cantransfer g1 along i and get an internal ultrapoweri : V [G�][g1] �! N [G�][g1][H ][i(g1)] = N [i(G� � g1)]Now let Q1 =def (Add(�; �++))V [G�][g1]. As V [G�][g1] is a model of2� = �+ Fact I.2.6.2 implies that in V [G�][g1]i(Q1) ' Add(�+; �++)Let Q2 =def (Coll(�; �+))V [G�][g1]. In V [G�][g1] we also know thatji(�+)j = �+ and i(Q2) is �+-closed soi(Q2) ' Add(�+; 1)The upshot of all this is that in V [G�][g1]i(Q1 �Q2) ' Add(�+; �++ n �)

16 JAMES CUMMINGSRecall that g2 is generic over V [G�][g1] for this latter forcing (by closure,this has the same de�nition in V [G�] and V [G�][g1]) so lettingQ =def i(Q1 �Q2)we can rearrange g2 as �g a Q-generic over V [G�][g1].Transferring g2 along i, j we get mapsi : V [G�][g] �! N [i(G� � g)]j : V [G�][g] �!M [j(G� � g)]k : N [i(G� � g)] �!M [j(G� � g)]with j = k � i.Let V � = V [G�][g], N� = N [i(G� � g)], M� = M [j(G� � g)]. FromFact I.2.4.2 we have that �g is Q-generic over N�. Closure implies thatQ1 �Q2 = (Add(�; �++)� Coll(�; �+))V �and hence thatQ = (Add(i(�); i(�++))� Coll(i(�); i(�+)))N�It is also worth observing that i : V � �! N� is the ultrapower of V � bythe normal measure on � induced by j.We claim that j : V � �! M� is a P3�-hypermeasurable embedding.As GCH holds in V � it is enough to show thatV � � P(�++) �MThis follows immediately from the fact that the forcing P� � _Q� has size�++ and is �++-c.c., so that every subset of �++ in V � has a canonicalname in M .We now have a model of the hypotheses(1) � is P3�-hypermeasurable, as witnessed by j : V �! M with�M �M , V�+3 �M , crit(j) = �.(2) GCH.(3) j is generated by a (�; �+3)-extender.(4) There is in V an object �g which is i(P )-generic over N , whereP = Add(�; �++)�Coll(�; �+) and i : V �! N is the ultrapowerby fX � � j � 2 j(X) g.

A MODEL IN WHICH GCH HOLDS AT SUCCESSORS BUT FAILS AT LIMITS172. Second stage.We are given the hypotheses from the end of the �rst stage. Factor jin two steps through the modelsN = the collapse of f j(f)(�) j f : � �! V g�N = the collapse of f j(f)(a) j a 2 [�++]<! ^ f : [�]jaj �! V gN is the familiar ultrapower approximatingM , while �N corresponds tothe extender of length �++. We have mapsi : V �! Nk : N �!M�{ : N �! �N�k : �N �!Msuch that j = k � ik = �k � �{Let � = �++N , and argue as before that �+ < � < �++.Do the reverse Easton iteration where we force at inaccessible � � �with Add(�; �++). Let P� be the forcing up to �, and let G� be P�-generic over V . LetQ� = (Add(�; �++))V [G�]P = (Add(�; �++)� Coll(�; �+))VLet g be Q�-generic over V [G�], and let �g be i(P )-generic over N asguaranteed by the hypotheses.The same argument as in stage one shows that all four models computethe same forcing up to �, and it is easy to extend the maps k, �{, �k to getk : N [G�] �!M [G�]�{ : N [G�] �! �N [G�]�k : �N [G�] �!M [G�]where k = �k � �{�N [G�], M [G�] are closed under �-sequences from V [G�] and computethe true �++, so in the obvious notationQ� = (Q�) �N = (Q�)M

18 JAMES CUMMINGSAlso we have (Q�)N = (Add(�; �))V [G�]Letting g factor as g1 � g2 where g1 is (Q�)N -generic we build furtherextensions k : N [G�][g1] �!M [G�][g]�{ : N [G�][g1] �! �N [G�][g]�k : �N [G�][g] �!M [G�][g]still preserving the relation k = �k � �{The same argument as in stage one allows us to build H 2 V [G�][g1]which is R-generic overN [G�][g1], where R is the iteration between �+1and i(�) as computed by N [G�]. We transfer H along �{, k to geti : V [G�] �! N [i(G�)]k : N [i(G�)] �!M [j(G�)]�{ : N [i(G�)] �! �N [�{ � i(G�)]�k : �N [�{ � i(G�)] �!M [j(G�)]where all the maps are de�ned in V [G�][g]. Let l =def �{ � i.jP�j = � and P� is �-c.c., so by Fact I.2.6.2 the term forcing(Add(�; �++)� Coll(�; �+))V P� =P�is isomorphic in V to P . Applying i, we use the i(P )-generic �g to de�nea generic ga � g1c for the forcing(Add(i(�); i(�++))� Coll(i(�); i(�+)))N [i(G�)]over the model N [i(G�)]. Using the fact thatV [G�][g1] � �N [i(G�)] � N [i(G�)]we also build g0c generic over N [i(G�)] for the forcing(Coll(�+4; < i(�)))N [i(G�)]and observe that automatically g0c is mutually generic with ga � g1c .We transfer g0c , g1c , ga along �{ to get new generics �g0c , �g1c , �ga. We needto alter the generic �ga to get a generic ha such that8p 2 g l(p) 2 ha (y)Recall that conditions in Q� are partial functions p : �++ � � 2 withjdom(p)j < �. Therefore l(p) = l\p, and so if q 2 ha we need to alter itsvalues on the set l\�++ � �.

A MODEL IN WHICH GCH HOLDS AT SUCCESSORS BUT FAILS AT LIMITS19Claim 1. If q 2 l(Q�) then jdom(q) \ (l\�++ � �)j � � in the modelV [G�][g].Proof. Let q = l(f)(a), wheref : [�]jaj �! Q� ^ f 2 V [G�] ^ a 2 [�++]<!De�ne Xx = f � < �++ j 9 < � (�; ) 2 dom(f(x)) gand let X = Sx2[�]jajXx. Clearly jX j � � andf � < �++ j 9 < � (i(�); ) 2 dom(q) g � X �Since V [G�][g] � � �N [l(G�)] � �N [l(G�)]we can alter each condition without getting out of the model �N [l(G�)],and the set of altered conditions ha is in V [G�][g]. It remains to beshown that ha is still l(Q�)-generic. Let D 2 �N [l(G�)] be a dense opensubset of l(Q�).Claim 2. De�ne an equivalence relation on l(Q�) byp v q () dom(p) = dom(q) ^ jf z j p(z) 6= q(z) gj � �Let E = f q 2 l(Q�) j 8p 2 l(Q�) p v q =) p 2 D gThen E is dense in l(Q�).Proof. Work in �N [l(G�)].We �rst prove the following fact.Subclaim 1. If r 2 l(Q�) then there exists s � r such that8p p v r =) p [ (s n r) 2 DProof. Enumerate f X � dom(r) j jX j � � g as hX� : � < �i where� < l(�). De�ne by induction a decreasing sequence of conditionshr� : � < �i. r0 = r, and r�+1 is de�ned as follows; let q be givenby q(z) = ( r�(z)1 n r�(z) z 2dom(r�) nX�z 2X�

20 JAMES CUMMINGSlet �q � q be in D, and let r�+1 = r� [ (�q n q). The forcing l(Q�) isl(�)-closed so de�ne s as a lower bound for hr� : � < �i. s clearlyworks. �To �nish the proof of the claim start with r 2 l(Q�) and apply thesubclaim �+ times to produce a decreasing sequence hs� : � < �+isuch that if p v s� then p [ (s�+1 n s�) 2 D. Let q be a lower boundfor hs� : � < �+i. We claim this works. If dom(p) = dom(q) andX = fz jp(z) 6= q(z)g has cardinality � � then for some � X � dom(s�).So p 2 D. �This su�ces to show that ha is generic, and a small variation on theargument shows that it is mutually generic with �g0c � �g1c .We now transfer ha along �k to get Ha j(Q�)-generic over M . (y)implies that 8p 2 g j(p) 2 Ha, so we can build mapsj : V [G�][g] �!M [j(G� � g)]�k : �N [l(G� � g)] �!M [j(G� � g)]l : V [G�][g] �! �N [l(G� � g)]where j = �k � lDe�ne V � = V [G�][g],M� =M [j(G��g)], N� = �N [l(G��g)]. We arguethat l : V � �! N� is just the ultrapower approximating j : V � �!M�.To do this factor l through ly : V �! Ny the ultrapower approximatingl, say l = ky � ly. PV �(�) � Ny and N y � 2� = �++ so �++Ny = �++and crit(ky) > �++. Since �++ � rge(ky) and N� is generated by a(�; �++)-extender Ny = N� and we are done.It remains to be argued that �g0c � �g1c is generic for an appropriatecollapse ordering, but this is immediate by the closure of l(Q�).So we have produced a model of the hypotheses(1) � is P2�-hypermeasurable, as witnessed by j : V �! M with�M �M , V�+2 �M , crit(j) = �.(2) For each strong inaccessible � � � 2� = �++ and for all n0 < n < ! =) 2�+n = �+n+1.(3) M � 2� = �++ ^ 8n 0 < n < ! =) 2�+n = �+n+1(4) There is F 2 V which is P -generic over N whereP = Coll(�+4; < i(�))� Coll(i(�); i(�+))Nand i : V �! N is the ultrapower of V by the ultra�lterU = fX � � j � 2 j(X) g(5) j is generated by a (�; �+3)-extender

A MODEL IN WHICH GCH HOLDS AT SUCCESSORS BUT FAILS AT LIMITS21We make some incidental remarks(1) F 2M , as it can be coded by an element of V�+2.(2) �++N = �++ and �+3M = �+3(3) crit(�k) = �+3N < �+3(4) F can be transferred along �k because it is generic for su�cientlyclosed forcing.3. THE RADIN FORCING1. Measure sequences.We use the results of the last section to de�ne measure sequences ,which will be the building blocks for the modi�ed Radin forcing.De�nition. If � and � are inaccessibles,P (�; �) =def Coll(�+4; < �)� Coll(�; �+)Retaining the notation used in the last paragraph of the precedingsection, let i12 : N �! i(N) be i(i), and let i02 : V �! i(N) be i(i) � i.Since N � i(�)i(N) � i(N)P = i(P )(�; i(�)) = i02(P )(�; i(�))and so we can sensibly de�neP �� = f f : [�]2 �! V� j dom(f) 2 U � U ^ 8�; � f(�; �) 2 P (�; �) gF �� = f f 2 P �� j i02(f)(�; i(�)) 2 F gNotice that U can be read o� from F �� asU = fX � � j 9f 2 F �� dom(f) = X �X gWe use j and F �� to de�ne our prototypical measure sequence u:u(0) =def crit(j) = �u(1) =def F ��u(�) =def fX 2 V�+1 j u � � 2 j(X) g (� > 1)where the construction continues for as long as u � � 2M . If the readeris familiar with Radin forcing she will notice that u has in second placethe object F ��, where putting�U =def fX � V� j f � j h�i 2 X g 2 U gwould give a standard Radin sequence as in [R].The idea behind the de�nitions that follow is that we use the strongre ection properties of the cardinal � to show that V� contains manyobjects which resemble u. In fact each measure on the sequence u shouldconcentrate on the set of u-like sequences.

22 JAMES CUMMINGSDe�nition. (k; g) is a constructing pair i�(1) k : V �! k(V ) is an elementary embedding into a transitiveinner model, and if � is the critical point then �k(V ) � k(V ).(2) g is k0(P )(�; k0(�))-generic over k0(V ), where k0 is the ultra-power approximating k (as usual we factor k through k0, sayk = l � k0).(3) g 2 k(V ).(4) g can be transferred along l to give a k(P )(�; k(�))-generic overk(V ).De�nition. If (k; g) is a constructing pair as above then g�� is de�nedfrom g in the same way as F �� was de�ned from F , that isg�� =def f f j k02(f)(�; k0(�)) 2 g gwhere k02 = k0(k0) � k0.De�nition. If (k; g) is a constructing pair as above, then a sequence wis constructed by (k; g) i�(1) w 2 k(V ).(2) w(0) = � = crit(k).(3) w(1) = g��.(4) w(�) = fX 2 V�+1 j w � � 2 k(X) g (1 < � < lh(w)).(5) k(V ) � jlh(w)j � w(0)++.(j; F ) is the archetypal constructing pair, and for every � < �+3 itconstructs w � �.De�nition. If w is constructed by some (k; g) then de�ne�w = w(0)If lh(w) � 2 de�ne alsog��w =def w(1)�w =def fX � �w j 9f 2 g��w dom(f) = [X ]2 g��w =def fX � V�w j f � j h�i 2 X g 2 �w ggw =def f [f ]�w��w j f 2 g��w gFw =def ��w \\f w(�) j 1 < � < lh(w) gwhere [f ]�w��w is the set represented by the function f in the ultrapowerV �w��w=�w � �w.Of course, given a particular (k; g) which constructs w then gw = gand so forth, but the point is that these de�nitions are independent ofthe choice of a constructing pair for w.

A MODEL IN WHICH GCH HOLDS AT SUCCESSORS BUT FAILS AT LIMITS23De�nition. De�ne inductively classes of sequencesU0 =def f w j 9(k; g) (k; g) constructs w gUn+1 =def f w 2 Un j Un \ V�w 2 Fw gU1 =def \n2!UnLemma 1. For every � < �+3 u � � exists and is a member of U1.Proof. It is routine to check that if � < �+3 then u � � can be coded byan element of V�+2, and hence u � � 2M and u(�) exists.Before proving the second half we unravel the de�nition of U1.Claim 1. 8� < �+3 u � � 2 U1if and only if 8� < �+3 8n < ! Mn � u � � 2 U0where Mn is the nth iterate of V by j.Proof. This is quite easy by induction.8� < �+3 u � � 2 Un+1() 8� < �+3 Un \ V� 2 u(�)() 8� < �+3 u � � 2 j(Un)() 8� < �+3 M � u � � 2 Un() 8� < �+3 Mn+1 � u � � 2 U0 �Fix � < �+3. We prove that M � u � � 2 U0, and then use theagreement between M and Mn for n > 1 to argue that this is enough.Fix � such that, if j� : V �!M� is the elementary embedding had byrestricting the extender E which describes j to E� = E � [�]<!, then(1) �++ < � < �+3 and � > �.(2) �M� �M�.(3) u � � and F are in M�.By the choice of � �++M� = �++ and � < �+++M� . Also E� 2 M , and ifj�� : M �! j��(M) is the embedding which M computes from E� thenj�� = j� �M and the models M�, j��(M) agree up to rank �.Using this information it is routine to check thatM � \ (j�� ; F ) constructs u � �"

24 JAMES CUMMINGSTo �nish the proof observe that if j1n :M �!Mn is the map arisingfrom the iteration of j thencrit(j1n) = j(�) > �+3 > rank(u � �)so Mn � u � � 2 U0 �Because the measures on a measure sequence are countably complete,if w 2 U1 and lh(w) � 2 then U1 \ V� 2 Fw.2. The forcing poset.Given w 2 U1 with lh(w) � 2 we de�ne a poset Qw. The forcingwhich we eventually use to establish the theorem will be Qu�� for a well-chosen value of �, but it is advantageous to do the proofs uniformly forall Qw.De�nition. p is a condition in Qw i�p = hXi : i < niwhere n > 1 and1) X0 = h�0i for some �0 < �.2) For 0 < i < n� 1, Xi isEITHER a pair (Pi; �i) where�i < �.Pi 2 P (�i�1; �i).OR a quadruple (wi; Ai; Hi; hi), wherewi 2 U1 and lh(wi) > 1.�i =def �wi .Ai 2 Fwi .Hi 2 g��wi .hi is a function, dom(hi) = f �v j v 2 Ai g n (�i�1 + 1).8� 2 dom(hi) hi(�) 2 P (�i�1; �).dom(Hi) = [dom(hi)]2.3) �0 < �1 < � � � < �n�1 < �.4) Xn is a quadruple (wn; An; Hn; hn) with the same properties as above,such that wn = w.Before giving the formal de�nition of the ordering on the conditionswe give some informal motivation. The intended generic object is a pairof sequences (~v; ~F ) where(1) lh(~v) = lh(~F ).

A MODEL IN WHICH GCH HOLDS AT SUCCESSORS BUT FAILS AT LIMITS25(2) 8� < lh(~v) v(�) 2 U1. (Let �� = �v(�))(3) � 7�! �� is monotone increasing, and C = f �� j � < lh(~v) g isclosed unbounded in �w.(4) For � < lh(~v) F (�) is a P (��; ��+1)-generic �lter.The typical condition p = hXi : i < ni of the last de�nition is intendedput the following constraints on (~v; ~F ):(1) v0 = �0.(2) If 0 < i < n� 1 and Xi is a pair (Pi; �i) then for some � < lh(~v)�i�1 = ��, �i = ��+1, and Pi 2 F (�).(3) If 0 < i < n and Xi is a quadruple (wi; Ai; Hi; hi) then �i is alimit point �� in C. Let �i�1 = ��. Then hi(��+1) 2 F (�), andif � < � < � we have v(�) 2 Ai and H(��; ��+1) 2 F (�).De�nition. Let X be a tuple.(1) uX = h�i if X = h�i or X = hP; �i.(2) uX = w if X = hw;A;H; hi.(3) �X = �uXDe�nition. Let p = hXi : i < miq = hYj : j < nip extends q i�1) m � n2) f�X0 ; : : : �Xm�1g � f�Y0 ; : : : �Yn�1g3) �X0 = �Y0 , X0 = Y04) For 0 < i < m, if Xi is a pair hP; �i thenEITHER there is j < n such that Yj = hQ;�i, and P � Q (thiswill make sense because other clauses of the de�nition will ensurethat in this case �Xi�1 = �Yj�1 , so P and Q are in the same partialordering)OR there is no j with � = �Yj and for the least j such that � < �YjYj is a quadruple hu;A;H; hi� 2 AIF �Xi�1 = �Yj�1 P � h(�)IF �Xi�1 > �Yj�1 P � H(�Xi�1 ; �)5) For 0 < i < m, if Xi is a quadruple hu;A;H; hi thenEITHER there is j < n Yj = hu;B; I; ii, in which caseA � B8(�; �) 2 dom(h) H(�; �) � I(�; �) andIF �Xi�1 = �Yj�1 8� 2 dom(h) h(�) � i(�)IF �Xi�1 > �Yj�1 8� 2 dom(h) h(�) � I(�Xi�1 ; �)

26 JAMES CUMMINGSOR there is no j with �Xi = �Yj , and for the least j such that�Xi < �YjYj is a quadruple hv; C; J; jiu 2 CA � C8(�; �) 2 dom(H) H(�; �) � J(�; �)IF �Xi�1 = �Yj�1 8� 2 dom(h) h(�) � j(�)IF �Xi�1 > �Yj�1 8� 2 dom(h) h(�) � J(�X�1; �)Remark. Qw has �++w -c.c.Proof. Let hp� : � < �++w i be an antichain in Qw, and let us writep� = x� _ hw;A�; H�; h�iSince x� 2 V�w we may assume that x� is some �xed sequence x. Incom-patibility can only arise from disagreement between the h�, and p� ? p�implies h� and h� represent incompatible conditions in the ultrapowerby �w. Since they represent conditions in a forcing notion of size �++wwe have a contradiction and the claim follows. �3. The generic object.De�nition. A sequence u 2 U1 occurs in a condition p = hXi : i < mii� there is i < m� 1 such that one of the following holds:(1) i = 0 and u = X0(2) Xi = hP; �i and u = h�i(3) Xi = hu;A;H; hiDe�nition. P occurs in p = hXi : i < mi i� there is i with 0 < i < msuch that Xi = hP; �i for some �.Notice that w does not occur in the conditions of Qw.De�nition. Let G be a Qw-generic �lter. De�ne a pair of sequences(hv(�) : � < �i; hF (�) : � < �i)(1) hv(�) : � < �i enumerates f v 2 U1 j 9p 2 G v occurs in p g sothat � 7�! �� = �v(�) is increasing.(2) F (�) = f P 2 P (��; ��+1) j 9p 2 G P occurs in p gIt is routine to check that the generic �lter G and the pair of sequences(~v; ~F ) are each de�nable from the other.We brie y develop the basic properties of the forcing; the proofs arevery similar to those for Radin forcing(see [R]).

A MODEL IN WHICH GCH HOLDS AT SUCCESSORS BUT FAILS AT LIMITS27Lemma 1. Let G be Qw-generic and let (~v; ~F ) and ~� be de�ned asabove. Then C = f �� j � < lh(~v) g is a closed unbounded subset of �w.Proof.1. C is closed.Proof. Let p 2 Qw be a condition, � < �w an ordinal such that p � =2 C.No sequence v with �v = � occurs in p, so consider X the �rst entry ofp with �X > �. If X is a pair then p forces � is not a limit point ofC. If X is a quadruple hu;A;H; hi then de�ne p0 � p by shrinking A toA0 = A n V�, so that p0 forces � is not a limit point of C.2. C is unbounded.Proof. Let � < �w, p 2 Qw, say p = x _ hw;A;H; hi. Since A 2 ��wthere is h�i 2 A with � > �, so p can be extended top0 = x _ hh(�); �i _ hw;A;H;H(�)iand p0 forces that � is not a bound for C.The lemma is proved. �Wemake the remark that if lh(w) � 3 then we could also have done the\unbounded" clause by showing that many quadruples could be addedto the condition p. (See the next Lemma)Lemma 2. With the same assumptions as lemma 1(1) If � < � is such that lh(v(�)) > 1 then the pair (~v � �; ~F � �) isgeneric for the forcing Qv(�).(2) 8A 2 V�w+1 (A 2 Fw () 9� 8� > � v(�) 2 A)(3) 8H 2 g��w 9� 8� > � H(��; ��+1) 2 F (�)Proof. (1) follows immediately from the observation that if p is a con-dition in which u� appears then Qw � p factors into two parts, of whichthe �rst is Qu� � p0 for p0 2 Qu� an initial segment of p.For (2), �rst assume that B 2 Fw. Let p = x _ hw;A;H; hi be somecondition; then the conditionp0 = x _ hw;A \ B;H; hiis an extension of p forcing that a �nal segment of ~v lies in B. For theother direction, assume towards a contradiction that B contains a �nalsegment of ~v but B =2 Fw. The \unbounded" clause of the last lemmashows that V� nB =2 ��w, so we must have V� nB 2 w(�) for � > 1.

28 JAMES CUMMINGSSubclaim 1. If � > 1 and B 2 w(�) then v(�) 2 B for an unboundedset of � < �.Proof. We show that if � < �w and q = x _ hw;A;H; hi is a conditionthen the set of v for which �v > � and there exist B, I , i such that(v;B; I; i) can be added to q is a measure one set for w(�). This clearlysu�ces.Fix (k; g) a constructing pair for w. By de�nition, it is enough toshow that in k(V ) there are are B, I , i such that hw � �;B; I; ii can beadded to k(q) = x _ hk(w); k(A); k(H); k(h)iWe claim that A, H , h will work (they exist in k(V ) because they areall elements of V�w+1).(1) A 2 w(�), so w � � 2 k(A).(2) A = i(A) \ V�w � i(A).(3) H = i(H) � �w � �w.(4) h = i(h) � �w.The subclaim is proved. �This gives a contradiction, so (2) is proved.Finally, let I 2 g��w . From (2) (��; ��+1) 2 dom(I) for large enough �,and an argument parallel to that for a) shows that any condition re�nesto one which forces I(��; ��+1) 2 F� on a �nal segment. The lemma isproved. �4. The Prikry property.We prove that questions about the generic extension by some Qw canbe decided in a simple way; this will enable us to prove that the forcingdoes not do too much damage to the universe.De�nition. If p and q are conditions in Qw then q is a direct extensionof p i� q �Qw p and lh(q) = lh(p). We write q �d p in this case.Theorem 1. Let b 2 RO(Qw) be a Boolean value, and let p 2 Qw besome condition. Then there is q �d p such that q k b.Proof. The strategy of the proof is to write p as z _ hw;A;H; hi, showthat b is decided by some extension z0 _ hw;A0; H 0; h0i � p in whichno sequences have been added past the top of z, and then make anappeal to induction and the uniform de�nition of the forcing. So �xp = p0 = z0 _ hw;A0; H0; h0i. Also �x (k; g) a constructing pair for w.We factor k through k0 the ultrapower by �w, to getk = l � k0

A MODEL IN WHICH GCH HOLDS AT SUCCESSORS BUT FAILS AT LIMITS29De�nition.(1) P �1 = ff 2 k0(V ) jdom(f) 2 k0(�w)^8� f(�) 2 k0(P )(�w; �)g.(2) g� = f f 2 P �1 j [f ]k0(�w) 2 g g.(3) P � = l(P �1 ).(4) P �� = f F 2 V j 8(�; �) 2 dom(F ) F (�; �) 2 P (�; �) g.Equip P �1 with the pre-orderingf � g () f � j f(�) � g(�) g 2 k0(�w)and equip P � similarly. By the de�nition of constructing pair g� isP �1 -generic over k0(V ) and l\g� generates a P �-generic over k(V ).De�nition. x is a lower part i� x is a proper initial segment of somecondition in Qw.De�nition. If x is a lower part then �x =def �X for X the last tuplein x.De�nition. Let F , G be elements of P ��.(1) F � G i� 8(�; �) 2 dom(F ) F (�; �) � G(�; �).1(2) F �� G i� f (�; �) j F (�; �) � G(�; �) g 2 �w � �w.In the �rst case we say that F re�nes G, in the second that F re�nes Galmost everywhere.Similarly, let f , g be functions with domain in �w such that for some� < � 8� f(�); g(�) 2 P (�; �).(1) f � g i� 8� 2 dom(f) f(�) � g(�).(2) f �� g i� f � j f(�) � g(�) g 2 �w.Step One. We �nd A1 � A0, H1 � H0, h1 � h0, such thatp1 = z0 _ hw;A1; H1; h1ihas the following property; if z _ X _ hw;B; I; ii is an extension of p1deciding b and having �X > �z0 then there exist �A � A1, �H � H1 suchthat z _ X _ hw; �A; �H; �H(�X)i decides b.Construction. Consider conditions in k(Qw) of the formz _ X _ hk(w); A;H; hiwhere z is a lower part, X is a tuple such that uX = w � � for some0 < � < lh(w). Notice that h 2 P �.1So implicitly we are demanding dom(F ) � dom(G).

30 JAMES CUMMINGSSay that h 2 P � is good if for all lower parts z 2 V�w and all X suchthat 9� < lh(w) uX = w � �9A 9H 9h0 � h z _ X _ hk(w); A;H; h0i k k(b) =)=) 9A0 9H 0 z _ X _ hk(w); A0; H 0; hi k k(b)Claim 1. The set of good h is a dense subset of P � lying in k(V ).Proof of Claim. Work in k(V ). There are �w possibilities for z. Fromthe de�nition of constructing pair w 2 k(V ) andk(V ) � jlh(w)j � �++wAlso for each � there are only 2�w = �++w X such that uX = w � �.So enumerate the possible pairs (z;X) as h(z ; X ) : < �++w i. Fixf 2 P �, and de�ne a descending sequence in P � by setting f0 = f andde�ning f +1 to be some g � f such that9A 9H z _ X _ hk(w); A;H; gi k k(b)if such exists, and f +1 = f otherwise. Let f 0 2 P � be a lower boundfor the sequence hp : < �++w i. Now let z, X , A, H , h0 � f 0 be suchthat z _ X _ hk(w); A;H; h0i k k(b)(z;X) is enumerated as (z ; X ) for some . f +1 was de�ned, so forsome B, I z _ X _ hk(w); B; I; f +1i k k(b)f 0 � f +1, so B, I witness that f 0 is good. �Since g� is generic we can �nd F 2 g�� such that F � H0 andk(F )(�w) is good. Now the set R of those u such that8z 8X [uX = u^9A 9H 9h0 � F (�u) z _ X _ hw;A;H; h0i k b =)=) 9A0 9H 0 z _ X _ hw;A0; H 0; F (�u)i k bis by construction a member of the �lter Fw.De�ne A1 = A0 \ Rh1 = h0 � f �v j v 2 A1 g n (�z0 + 1)H1 = F � [dom(h1)]2We prove that p1 has the right behaviour. If z _ X _ hw;B; I; ii is anextension of p1 deciding b and having �X > �z0 then uX is chosen fromR and i � F (�X ) so we can �nd A0 and H 0 as in the de�nition of R. Itis easy to alter A0, H 0 to �A, �H with the right properties. �

A MODEL IN WHICH GCH HOLDS AT SUCCESSORS BUT FAILS AT LIMITS31Step Two. We �nd A2 � A1, H2 � H1, h2 � h1, such thatp2 = z0 _ hw;A2; H2; h2ihas the following property; if z _ X _ hw;B; I; ii is an extension ofp2 deciding b and having �X > �z0 then z _ X _ hw;A2; H2; H2(�X )idecides b.Construction. For each � < �w there are fewer than �w lower parts zwith �z = �. Fix � for the moment, and for each z such that �z = �and 9B � A1 9G � H1 z _ hw;B;G;H1(�z)i k bchoose an appropriate pair (Bz ; Gz). Using genericity �nd F� 2 g�� suchthat F� � Gz for all z with �z = �. Find C� 2 with [C�]2 � dom(F�).E� =def A1 \ \z;�z=�Bz \ f v j h�vi 2 C� gBy the usual argument using the constructing pair E� 2 Fw. By gener-icity again �nd F 2 g��, F � H1, such that8� < �w F �� F�Find D� 2 �w such that8( ; �) 2 [D�]2 F ( ; �) � F�( ; �)Let E0� = E� \ f v j h�vi 2 D� gA2 = f v 2 A1 j 8� < �w v 2 E0� gh2 = h1 � f �v j v 2 A2 gH2 = F � [dom(h2)]2p2 = z0 _ hw;A2; H2; h2iWe claim that p2 has the property we demanded. To show this letz _ X _ hw;B; I; ii be an extension of p2 deciding b and having�X > �z0 . Let x = z _ X . Now by the work we did in Step Oneand the de�nitions abovez _ X _ hw;Bx; Gx; H2(�X )i k bTherefore it will be enough to show thatz _ X _ hw;A2; H2; H2(�X)i � z _ X _ hw;Bx; Gx; H2(�X)i

32 JAMES CUMMINGSwhich amounts to showing that anything which can be added betweenX and the top on the LHS can also be added on the RHS.If v 2 A2, and �v > �X = �x, then v 2 E0�x and therefore v 2 Bx.If v; w 2 A2, and �x < �v < �w, then v; w 2 E0�x and so by de�nition�v; �w 2 C�x \D�x . ThereforeH2(�v; �w) = F (�v; �w)� F�x(�v ; �w) because (�v ; �w) 2 [D�x ]2� Gx(�v ; �w) because (�v ; �w) 2 [C�x ]2So the constraint imposed by A2 and H2 is more stringent than thatimposed by Bx and Gx, and the claim is proved. �Step Three. We �nd A3 � A2, H3 � H2, h3 � h2 such that ifp3 = z0 _ hw;A3; H3; h3iand z0 _ X _ hP; �i _ hw;A0; H 0; h0i � p3is a condition deciding b, �z0 < �X < �, thenz0 _ X _ hH3(�X ; �); �i_ hw;A3; H3; H3(�)i k bConstruction. We de�ne embeddings k12 = k(k), k02 = k(k) � k. Con-sider conditions in k02(Qw) of the formz _ X _ hP; k(�w)i_ hk02(w); k02(A2); k02(H2); k02(H2)(k(�w))iwhere z is a lower part and uX = w � � for some � with 0 < � < lh(w).P 2 P (�w; k(�w))k02(V ), and by the de�nition of constructing pair l\ggenerates a generic over k(V ) for this forcing.An argument exactly parallel to that of Step One enables us to de�neA3, H3, h3. �Step Four. We �nd A4 � A3, H4 � H3, h4 � h3 such that ifp4 = z0 _ hw;A4; H4; h4iand z0 _ X _ hw;A0; H 0; h0i � p4

A MODEL IN WHICH GCH HOLDS AT SUCCESSORS BUT FAILS AT LIMITS33is a condition deciding �, lh(uX) > 1, and �X > �z0 then for some �and for w(�)-many u9B 9h z0 _ hu;B;H4 � [�u]2; hi_ hw;A4; H4; H4(�u)i k �Construction. For each lower part x and � < lh(w) letXx;1 = f h�i j h�i 2 A3 gXx;�0 = f v j 9B 9I 9i x _ hv;B; I; ii_ hw;A3; H3; H3(�v)i k � gXx;�1 = (Xx;�0 )cXx;� = Xx;�i for the unique i such that Xx;�i 2 w(�) � > 1Now for each x let Xx = [0<�<lh(w)Xx;�and de�ne X = f v j 8x �x < �v =) v 2 Xx gThe point of this de�nition is that X 2 Fw, and if v 2 X , lh(v) > 1,and 9B 9I 9i x _ hv;B; I; ii_ hw;A3; H3; H3(�v)i k � (�)then for some � < lh(w) the relation (*) holds for w(�)-many v.For each x, if there exists such a v, �nd one such that the value of �associated with it is minimal. By the de�nition of w(�) there are Bx,Ix, ix in k(V )x _ hw � �;Bx; Ix; ixi_ hk(w); k(A3); k(H3); k(H3)(�w)i k k(b)Let I 2 g�� be such that 8x I �� Ixand �nd C 2 �w such that8(�; �) 2 [C]2 8x �x < � < � =) I(�; �) � Ix(�; �)De�ne A4 � A3 \fu j�u 2 C g, H4 � H3; I , and h4 � h3 appropriately.By construction p4 has the right property. �

34 JAMES CUMMINGSStep Five. We �nd A5 � A4, H5 � H4, h5 � h4 such that ifp5 =def z0 _ hw;A5; H5; h5iandz0 _ hv;A;H5 � [�v]2; H5(�z0) � �vi_ hw;A5; H5; H5(�v)i � p5lh(v) > 1, �v > �z0 , is a condition which forces b then for some �9A z0 _ hv;A;H5 � [�v]2; H5(�z0) � �vi_ hw;A5; H5; H5(�v)i k bfor w(�)-many v (and similarly for :b)Construction. De�ne Cx;1 = f h�i j h�i 2 A4 gThen for 1 < � < lh(w) letBx0 = f v j 9A x _ hv;A;H4 � [�v]2; H4(�x) � �vi__ hw;A4; H4; H4(�v)i b gBx1 = f v j 9A x _ hv;A;H4 � [�v ]2; H4(�x) � �vi__ hw;A4; H4; H4(�v)i :b gBx2 = f v j 9A x _ hv;A;H4 � [�v]2; H4(�x) � �vi__ hw;A4; H4; H4(�v)i , b gCx;� = Bxi for the unique i such that Bxi 2 w(�)Finally de�ne Cx = [0<�<lh(w)Cx;�C = f v j 8x �x < �v =) v 2 Cx gLet A5 = A4 \ C and re�ne H4, h4 appropriately. �

A MODEL IN WHICH GCH HOLDS AT SUCCESSORS BUT FAILS AT LIMITS35Step Six. We �nd A6 � A5, H6 � H5, h6 � h5 such that ifp6 = z0 _ hw;A6; H6; h6iand z0 _ hH6(�z0 ; �); �i _ hw;A6; H6; H6(�)i � p6forces b, and � > �z0 , thenz0 _ hH6(�z0 ; ); i_ hw;A6; H6; H6( )i bfor all with h i 2 A6. Notice that this implies thatz0 _ hw;A6; H6; H6(�z0)i bas any extension of this condition is comparable with a condition forcingb. (Of course the same claim is also true with :b in the place of b.)Construction.Bx0 = f � j x _ hH5(�x; �); �i_ hw;A5; H5; H5(�)i b gBx1 = f � j x _ hH5(�x; �); �i_ hw;A5; H5; H5(�)i :b gBx2 = f � j x _ hH5(�x; �); �i_ hw;A5; H5; H5(�)i , b gBx = Bxi for the unique i with Bxi 2 �wB = f v j 8x �x < �v =) �v 2 Bx gLet A6 = A5\B and de�ne H6, h6 by an appropriate shrinking of thedomains of H5, h5. This �nishes the construction phase of the proof. �Step Seven. We prove that p6 is an extension which decides b \up tolower parts". That is, b can be decided by an extension in which noobject X with �X > �z0 is inserted.Proof. Find an extension r of p6 deciding b such that the lower part xhas a minimal number of entries Y with �Y > �z0 , and such that amongsuch conditions �x has the minimal value.Claim 1. �x = �z0 (i.e the minimal number of entries Y with �Y > �z0is in fact zero ).Proof of claim. Suppose that �x > �z0 , and let x = z0 _ X where X isa single entry. By the work done in Steps One and Twoz0 _ X _ hw;A6; H6; H6(�X)i k bSuppose that in fact it forces b (the argument for :b will be identical ).

36 JAMES CUMMINGSSubclaim 1. X is a pair, rather than a quadruple.Proof of subclaim. Let X be a quadruple hv;B; I; ii. The constructionof Step Five guaranteesf v j 9B 9i z0 _ hv;B;H6 � [�v]2; ii_ hw;A6; H6; H6(�v)i b g 2 w(�)for some � > 1. For each such v choose appropriate B, i, and let hP; �ibe some pair such that h�i 2 B and P � i(�). An easy normalityargument (v 7! � < �v, so �x � on a large set; now use completeness to�x P ) gives that the choice of hP; �i can be stabilised on a set in w(�).Fix some hP; �i such that for w(�) many v there exists Bz0 _ hP; �i_ hv;B;H6 � [�v]2; H6(�) � �vi_ hw;A6; H6; H6(�v)i bFix f a choice function giving for each such v an appropriate B. Thesituation is now essentially the same as that in the last stage of the proofthat ordinary Radin forcing has the Prikry property (see [R] ). De�nesetsA01 = f v j f t 2 dom(f) j v 2 f(t) g 2 w(�) gA11 = f v j f t 2 dom(f) j f(t) \ V�v 2 Fv g 2 w(�) gA1 = A01 \A11A2 = f t 2 dom(f) j 8v 2 A1 �v < �t =) v 2 f(t) gA3 = f v j 8C 2 Fv 9� < lh(v) f t 2 A2 \ C j f(t) \ C 2 Fv g 2 v(�) gUsing the constructing pair argument it is routine to check that(1) A1 2 Fw��.(2) A2 2 w(�).(3) A3 2 Tf w(�) j � < � < lh(w) g.So A1[A2[A3 2 Fw. De�ne A0 = A6\(A1[A2[A3). The followingsubclaim gives us what we need.Subclaim 2. Any extension ofz0 _ hP; �i _ hw;A0; H6; H6(�)iis comparable withz0 _ hP; �i _ hv; f(v); H6 � [�v ]2; H6(�) � �vi_ hw;A6; H6; H6(�v)ifor some v 2 dom(f).Proof of subclaim. Let the extension bep = �z _ X1 _ X2 � � �_ Xn _ hw;A�; H�; h�i

A MODEL IN WHICH GCH HOLDS AT SUCCESSORS BUT FAILS AT LIMITS37where ��z = �. 8i 9j uXi 2 Aj . Let v be the �rst sequence with v 62 A1.Then v 2 A2, v 2 A3, or v = w.Case 1) v 2 A2. Say v = uXj . By constructionv 2 dom(f)8i < j f(v) \ V�Xi 2 FuXiSo de�ning (for i � j)X 0i = ht; A \ f(v); H; hi if Xi = ht; A;H; hihQ; �i if Xi = hQ; �ithe condition�z _ X 01 _ � � �_ X 0j _ Xj+1 � � �_ Xn _ hw;A�; H�; h�iis a common extension of p andz0 _ hP; �i _ hv; f(v); H6 � [�v ]2; H6(�) � �vi_ hw;A6; H6; H6(�u)iCase 2) v = uXj 2 A3. Say Xj = hv;A;H; hi. By construction �ndv0 with �v0 > �Xj�1 , v0 2 A \ A2, f(v0) \ A 2 Fv0 . As before8i < j f(v0) \ V�Xi 2 FuXi , so de�ning X 0i as before for i < j�z _ X 01 _ � � �_ X 0j�1 _ hv0; f(v0); H6 � [�v0 ]2; H6(�Xj�1 ) � �v0i__ Xj _ � � �_ Xn _ hw;A�; H�; h�iis the sought-after common extension.Case 3) Like Case 2). Exploit the fact thatf v 2 A2 j f(v) \B� 2 Fw g 2 w(�)to produce an appropriate v0 with �v0 > �Xn .The subclaim is proved. �So the conditionr0 = z0 _ hP; �i _ hw;A0; H6; H6(�)ihas the property that any extension of it is compatible with some condi-tion which forces b. Thus by elementary facts about forcing the condition

38 JAMES CUMMINGSitself forces b. However this contradicts the assumption that r was cho-sen to minimise the last ordinal mentioned in the lower part, amongstconditions extending p5, deciding b, and having the same number of en-tries Y with �Y > �z0 as does x. The subclaim is therefore proved. �If �x > �z0 then by the subclaim x = z0 _ hP; �i, where �z0 = �z0 .For if not thenz0 _ hH(�z0 ; �); �i_ hw;A6; H6; H6(�)i bby the work done in Step Three, thenz0 _ hw;A6; H6; H6(�z0)i bby Step Six. This contradicts the choice of x to contain as few entriespast �z0 as possible.Now de�neC = f � j 9z� 9P � h6(�) z� _ hP; �i_ hw;A6; H6; H6(�)i k b gSubclaim 3. C 2 �w.Proof. Suppose not, and let A� = A6 n f w j �w 2 C g, de�ne H�, h� byrestriction of H6, h6, and argue as above that extension of the conditionz0 _ hw;A�; H�; h�i in the same \minimal" way to decide b demandsadding at worst one pair Y with �Y > �z0 ; this gives an immediatecontradiction. �Pick for each � 2 C an appropriate z� . Find C 0 2 �w on which z� isconstant, with value z� say, and such that every element chosen decidesb the same way. Shrinking A6 toA0 = A6 \ f w j �w 2 C 0 gde�ning h0 so that for each � in Cz� _ hh0(�); �i _ hw;A6; H6; H6(�)i k band shrinking H6 to H 0 appropriately the proof is done. For the condi-tion z� _ hw;A6; H6; h0idecides b. �

A MODEL IN WHICH GCH HOLDS AT SUCCESSORS BUT FAILS AT LIMITS39Step Eight. We conclude the proof by showing that b can be decided bya direct extension of p0.Proof. We just use induction on the length of conditions and the follow-ing obvious fact about the forcing Qw;Fact 1. If p 2 Qw is a condition of the formx _ hv;B; I; ii_ hP0; �0i : : : hPn; �ni_ hw;A;H; hithen Qw � p factors as P1 � P2 � P3, where(1) P1 = Qv � x _ hv;B; I; ii(2) P2 = P (�v0 ; �0) � P0 � � � � � P (�n�1; �n) � Pn(3) P3 = Qw � h�ni_ hw;A;H; hiThe theorem is proved. �Using this theorem we can go some way towards showing that thecardinals of the extension by Qw are exactly those which we have nottaken steps to collapse.Theorem 2. Let ~v; ~F be Qw-generic and as usual de�ne �� = �v(�).Then for every � the V -cardinals ��, �++� , �+3� , �+4� , are cardinals inthe generic extension.Proof. We divide the proof into two cases.Case 1) � is a limit. One application of the theorem shows that if Xis a bounded subset of �� in the extension then X is genericfor forcing of cardinality less than ��; hence �� survives as acardinal. Similarly if X is a bounded subset of �+4� then X is inthe extension by Qv� , which is �++� -c.c. and therefore preserves�+2� , �+3� , �+4� .Case 2) � is a successor. Say � = �+n for � limit. As before it is easy toargue that �� is preserved. Now we have that bounded subsetsof �+4� are in the extension byQv(�) � P (��; ��+1)� : : : P (��+n�1; ��)which is �++� -c.c. forcing.The theorem is proved. �Of course we have explicitly collapsed �+� for � successor, so the ques-tion left open here is the fate of �+� when � is limit. The answer to thismust await the �ne analysis of Qw-genericity which we undertake in thenext section.

40 JAMES CUMMINGSWe also make the remark that the same kind of analysis will show thatwe have not added too many subsets to any cardinal of the extension;in fact, once we have shown that �+� is preserved for � limit it will beeast to show that in the extension GCH holds at all successors and failsat all limits.5. The geometric condition.We prove a simple criterion for Qw genericity, analogous to Mathias'criterion for Prikry genericity [M]. Recall that if (~u; ~F ) is generic withlh(~u) = lh(~F ) = � thena) If � < � is such that lh(u�) > 1 then the pair (~u � �; ~F � �) is genericfor the forcing Qu� .b) 8A 2 V�w+1 (A 2 Fw () 9� 8� > � u� 2 A).c) 8H 2 w(1) 9� 8� > � H(��; ��+1) 2 F�.Theorem 1. The pair (hu� : � < �i; hF� : � < �i) is generic if andonly if it satis�es the conditions a), b), c).Proof.Lemma 1. Let D be an open set, let A, H, � be such that D is densebelow h�i_ hw;A;H;H(�)iBy the methods of the Prikry Property proof, Steps One and Two it maybe assumed that ifx _ hw;A0; H 0; h0i � h�i_ hw;A;H;H(�)iis a member of D and �x > � thenx _ hw;A;H;H(�x)i 2 DThen there exists j < ! such that ( introducing a quanti�er 8��, where8��x �(x) is to be interpreted as \�(x) holds on a set in the ultra�lter�" ) 9n0 8�Un0 (�01 ; : : : �0n0) 9(P 01 ; : : : P 0n0) 9�0 > 1 8�w(�0)u0 9A0 9H0: : :9nj 8�Unj (�j1; : : : �jnj ) 9(P j1 ; : : : P jnj ) 9�j > 1 8�w(�j)uj 9Aj 9Hjh�i_ hP 01 ; �01i_ � � �_ hP 0n0 ; �0n0i_ hu0; A0; H0; H0(�0n0)i_: : :_ hP j1 ; �j1i_ � � �_ hP jnj ; �jnj i_ huj ; Aj ; Hj ; Hj(�jnj )i__ hw;A;H;H(�uj )iextends h�i_ hw;A;H;H(�)i and is in D

A MODEL IN WHICH GCH HOLDS AT SUCCESSORS BUT FAILS AT LIMITS41Proof of lemma. Suppose not. For the sake of brevity abbreviate theformula\h�i_ hP 01 ; �01i_ � � �_ hP 0n0 ; �0n0i_ hu0; A0; H0; H0(�0n0)i_: : :_ hP j1 ; �j1i_ � � �_ hP jnj ; �jnj i_ huj ; Aj ; Hj ; Hj(�jnj )i__ hw;A;H;H(�uj )iextends h�i_ hw;A;H;H(�)i and is in D"as �j(~�; ~P ; ~w; ~A; ~H), and the quanti�er sequence8ni 8�Uni (�i1; : : : �ini) 8(P i1; : : : P ini) 8�i > 1 8�w(�i)ui 8Ai 8Hias Qi. So for all j Q0 Q1 : : : Qj :�j(~�; ~P ; ~w; ~A; ~H)Fix j and i � j for the moment, and de�ne for eachs = h�01 ; : : : �0n0 ; P 01 ; : : : P 0n0 ; u0; A0; h0; : : : P i1; : : : P iniisuch that8�i > 1 8�w(�i)ui 8Ai 8Hi Qi+1 : : : Qj :�j(~�; ~P ; ~w; ~A; ~H)Y si;j = f ui 2 A0 j lh(ui) = 1 _ 8Ai 8Hi Qi+1 : : :Qj :�j(~�; ~P ; ~w; ~A; ~H) gYi;j = f u j 8s �ini < �u =) u 2 Y si;j gNow let i, j vary and de�neY =def \j<! \i<! Yi;jFor each t = h�01 ; : : : ; Ai; Hiisuch that8ni+1 8Uni+1 h�i+11 ; : : : ; �i+1ni+1i : : :8Hi+1 Qi+2 : : : Qj:�(~�; ~P ; ~w; ~A; ~H)de�ne for each ni+1 Zt;ni+1 to bef ~�i+1 2 [�]ni+1 j 8~P i+1 8Hi+1 Qi+2 : : : Qj:�(~�; ~P ; ~w; ~A; ~H) g

42 JAMES CUMMINGSand �nd Xt 2 U such that 8ni+1 [Xt]ni+1 � Zt;ni+1 . Find Y 0 � Y ,Y 0 2 Fw such thath�i 2 Y 0 ^ rank(t) < � =) � 2 Xth�i_ hw; Y;H;H(�)i has some extension in D, sayy0 _ hu0; A0; H0; h0i_: : :_ yn _ hun; An; Hn; hni__ yn+1 _ hw; Y;H;H(�yn+1)i 2 DAs D is open it is possible to extend this condition toy0 _ hu0; A0; H0; h0i_: : :_ yn _ hun; An; Hn; hni__ yn+1 _ hun+1; An+1; Hn+1; hn+1i__ hw; Y;H;H(�un+1)i 2 Dcontradicting the de�nition of Y 0. �Fix j with the property guaranteed by the lemma above. Using thesame idea as in Step Three of the proof of the Prikry property re�ne A,H , to A�, H� such that9n0 8�Un0 (�01 ; : : : �0n0) 9�0 > 1 8�w(�0)u0 9A0 2 Fu0 9H0 2 hu0: : :9nj 8�Unj (�j1; : : : �jnj ) 9�j > 1 8�w(�j)uj 9Aj 2 Fuj 9Hj 2 hujh�i_ hH�(�; �01); �01i_ � � �_ hH�(�0n0�1; �0n0); �0n0i__ hu0; A0; H0; H0(�0n0)i_ : : :� � �_ hH�(�uj�1 ; �j1); �j1i_ � � �_ hH�(�jnj�1; �jnj ); �jnj i__ huj ; Aj ; Hj ; Hj(�jnj )i_ hw;A�; H�; H�(�uj )iextends h�i_ hw;A�; H�; H�(�)i and is in DAbbreviate this formula asR0R1 : : : Rj (~�; ~w; ~A; ~H)At the same time arrange that the �ab drawn from A� all come fromappropriate measure one sets, as in the proof of the previous lemma.

A MODEL IN WHICH GCH HOLDS AT SUCCESSORS BUT FAILS AT LIMITS43Lemma 2. With the same hypotheses and notation as above there existA��, H�� re�ning A�, H�, such that if (hv : < �i; hG : < �i)satis�es the conditions a), b), c), v0 = h�i, and in additionforall 8><>: v 2 A��(�v ; �v +1) 2 dom(H��)H��(�v ; �v +1) 2 G then D meets the �lter on Qw generated by (hv : < �i; hG : < �i).Proof. For each t = h�01 ; : : : �0n0 ; u0; A0; H0; : : : �i1; : : : �iniisuch that9�i > 1 8�w(�i)ui 9Ai 2 Fui 9Hi 2 hwi Ri+1 : : : Rj (~�; ~w; ~A; ~H)�x an appropriate �t and functions f t, gt de�ned on a set Xt 2 w(�t)such that8ui 2 Xt Ai = f t(ui) ^Hi = gt(ui) =) Ri+1 : : : Rj (~�; ~w; ~A; ~H)Now perform the following construction (which is reminiscent of theproof of a subclaim in the proof of the Prikry property, and indeedwould prove a slightly stronger result in which \H-parts" are dependenton the sequence chosen in a more arbitrary way)tA01 = f v j f u j v 2 f(u) g 2 w(�t) g= j(f)(w � �t)tA02 = f v j f u j f(u) \ V�v 2 Fv g 2 w(�t) g= f v jA01 \ V�v 2 Fv gtA03 = f v j f u j g(u) � [�v]2 2 hv g 2 w(�t) g= f v j j(g)(w � �t) � [�v ]2 2 hv gHt = j(gt)(w � �t)tA0 = [0<i<3 tA0itA1 = f u j 8v 2 tA0 �v < �u =)=) f u j v 2 f(u) ^ f(u) \ V�v 2 Fv ^ g(u) � [�v]2 2 hv g 2 w(�t) gtA2 = f x j 8C 2 Fx 9� < lh(x) C \ tA1 2 x(�) g

44 JAMES CUMMINGSAs in the Prikry property proof it is easy to showtA0 2 �w \ \1< <�tw( )tA1 2 w(�t)tA3 2 \�t< <lh(w)w( )tA =def [i<2 tAi 2 FwDe�ne A��, H�� by taking the diagonal intersection of tA, Ht over suit-able t. To show that this construction works, select an n0 such that8~�n0 2 [f� jh�i 2 A��g]n0 9�0 8�w(�0)u0 9A0H0 R1 : : : Rj (~�; ~w; ~A; ~H)De�ne �0i by vi = h�0i i for 1 � i � n0. Set t0 = h�01 ; : : : �0i i. Find �0the least � with v� 2 t0A1. The existence of such a � is guaranteed byconditions a), b), c), for if ~v avoids t0A on a �nal segment then by b)(t0A)c 2 Fw which is absurd. Set u0 = v�.Claim 1. The conditionh�i_ hH��(�; �01); �01i_ � � �_ hH��(�0n0�1; �0n0); �0n0i__ hu0; f t0(u0); gt0(u0); gt0(u0)(�0n0)i_ hw;A��; H��; H��(�u0)iis in the �lter generated by (~v; ~G).Proof of Claim. It will su�ce to show that for � < �v� 2 f t0(u0)gt0(u0)(�v� ; �v�+1) 2 G�By construction v� 62 t0A1. If v� 2 t0A0 then v� 2 f t0(u0) by thede�nition of t0A1 . Suppose that v� 2 t0A2. Then for some �f t0(v�) \ t0A2 2 v�(�)but the sequence hv� : � < �i avoids this set contradicting the localgenericity condition a).So v� 2 f t0(u0). To get the second clause of the claim notice that byconstruction H��(�v� ; �v�+1) � gt0(u0)(�v� ; �v�+1)

A MODEL IN WHICH GCH HOLDS AT SUCCESSORS BUT FAILS AT LIMITS45but H��(�v� ; �v�+1) 2 G� so gt0(u0)(�v� ; �v�+1) 2 G� . The claim fol-lows. �Now select n1 and so on : : : after j stages this process will produce acondition in the �lter which also lies in D. �Now �x E a dense subset of Qw. Let x be a lower part of the formz _ hv;B; I; ii, i.e a lower part which is a member of some Qv.If q = h�vi_ y _ hw;C; J; ji then de�nex� q =def x _ y _ hw;C; J; jiLet Ex =def f q 2 Qw j x� q 2 E gFix �, and use the Prikry property to �nd A� ; H� such thath�i_ hw;A� ; H� ; H�(�)i k 9q x� q 2 Efor each x with �x = �. Say that x is good or bad depending on whetherq is forced to exist or not. If x is good it is easy to see that Ex is densebelow h�xi _ hw;A�x ; H�x ; H�x(�x)i. Apply lemma 6.3 to produceappropriate A��x , H��x for each good x, and de�ne Ay, Hy by taking thediagonal intersection over all good x. Because of a), b), c) there is some� < � such that lh(u�) > 1 and for all � > �u� 2 Ay(�u� ; �u�+1) 2 dom(Hy)Hy(�u� ; �u�+1) 2 F�Lemma 3. Good x are dense in Qu� .Proof. Fix x 2 Qu� , and extendx _ hw;A�u� ; H�u� ; H�u� (�u� )ito x0 _ s _ hw;A�u� ; H�u� ; H�u� (�s)i 2 Dwhere x0 2 Qu� . x0 must be good. �So �nd a good x in the �lter on Qu� generated by (~u � �; ~F � �). Byconstruction there is q 2 Ex which also lies in the �lter generated by(hu : � � < �i; hF : � � < �i). So x� q 2 E, and this also lies inthe �lter generated by (hu : < �i; hF : < �i). So the latter �ltermeets every dense set and is therefore generic. �

46 JAMES CUMMINGS6. The forcing preserves successors.We are now able to answer the question which was left unsolved bythe Prikry lemma, namely whether the successors of limit points on thegeneric sequence are collapsed. By the Prikry lemma it is enough todecide whether Qw collapses �+w . The proof given here was suggestedby Hugh Woodin, and replaces a much clumsier one which used thecanonical generic object described in the next section.Theorem 1. If G is Qw-generic then �+w is a cardinal in V [G].Proof. Suppose not, and let _� be a term, p 2 Qw a condition such thatp _� : �w � �+wFind � some large regular cardinal such that p, _� ,Qw are in H�. FindX � H� such that V�w � N and p, _� , Qw are in X , and such thatjX j = �w. Let � : X �! N be the Mostowski collapse of X to atransitive set N . Force over H� with Qw � p to get a generic (~v; ~F ).Claim 1. The pair (~v; ~F ) is �(Qw)-generic over the model N .Proof. This is immediate using the geometric criterion for genericity andthe observation that we have�(Fw) = Fw \N�(w(1)) = w(1) \N �Let G be the generic �lter on Qw corresponding to (~v; ~F ), and let�G be the �lter on �(Qw) similarly de�ned. Observing that ��1 mapselements of �G to elements of G, we build an elementary embeddingk : N [ �G] �! H�[G]But �(p) 2 �G so the interpretation of _� by �G must be f : �w � (�w)+N .But (�w)+N is the critical point of k so k(f) = f , while by elementarityk(f) : �w � �+w . This gives a contradiction. �7. A canonical generic sequence.Radin showed in [R] that generics for Radin forcing can be generatedby iterated elementary embeddings; the same is true for Qw, with asimilar proof (Radin's iterative scheme for choosing the ordinals to whichsequences are restricted has the same e�ect as the one given here).Fix w 2 U1, with constructing pair (k; F ), where lh(w) < k(�w).Iterate k to get a sequence of modelshM� : � 2 ONi

A MODEL IN WHICH GCH HOLDS AT SUCCESSORS BUT FAILS AT LIMITS47(whereM0 = V ) and embeddings k�� for � < � in the usual way. De�neF� = k0�(F ), �� = k0�(�w) for each � 2 ON . It is easily seen that F�is P (��; ��+1)M� -generic overM� for � > �; also if � < �0 < � then F�and F�0 are mutually generic over M�. De�ne by induction a sequencehu�i u� = h��iu� = k0�(w) � �� � 0 or a successor� limitwhere in the limit case �� is the least � < lh(k0�(w)) such that forsome X 2 k0�(w)(�) the sequence hu� : � < �i avoids X , if such �exists, and �� = lh(k0�(w)) otherwise. The construction ends at � inthe latter case.Lemma 1. For all � such that u� is de�ned and lh(u�) > 1 the pair(hu� : � < �i; hF� : � < �i) is generic over M� for the forcing Qu� .Proof. By induction on �. The non-trivial part is to show that thesequence hu� : � < �i generates the �lter Fu� . One direction is easy;suppose thatX 2M�, X contains a �nal segment of ~u � �, butX 62 Fu� .So for some < �� Xc 2 k0�(w)( ). But by the minimal choice of ��f � < � j u� 2 Xc g is unbounded in � and this is absurd.For the other direction let X 2 Fu� . Fix Y 2 k0�(w)(��) such that8� < � u� 62 Y , and de�neZ = f v j 9� Y \ V�v 2 v(�) gA familiar style of argument using k��+1 (which generates the sequencek0�(w)) shows T = X [ Y [ Z 2 Fk0�(w)� is limit, so by the de�nition of direct limit for all large � < �T = k��(T�) where T� 2 Fk0�(w). But k��+1 generates k0�(w), thereforefor such � 8 < lh(k0�(w)) k0�(w) � 2 T�+1k�+1� �xes the sequence k0�(w) � , becauselh(k0�(w)) < k0�+1(�0) = crit(k0�+1)So 8 < lh(k0�(w)) k0�(w) � 2 T

48 JAMES CUMMINGSAs the construction did not halt at � u� is a proper initial segment ofk0�(w), so u� 2 TBut now it must be the case that u� 2 X ; for certainly u� 62 Y , whileif u� 2 Z then (as by induction hu : < �i is Qu� -generic over M�)there is < � u 2 Y . �Theorem 1. The construction of the u� terminates at some stage.Proof. Suppose not. Fix � a large enough regular cardinal and selectfor each limit � � � X� 2 k0�(w)(��) such that ~u � � avoids X�. Afamiliar argument from the theory of inner models (see e.g. lemma 4 ofMitchell's paper [Mi1]) shows that for some stationary set S � �8(�; �) 2 [S]2 k0�(��) = �� ^ k0�(X�) = X�Fix such a pair (�; �). X� 2 k0�(w)(��), so arguing as aboveu� = k0�(w) � �� 2 k��(X�) = X�contradicting the choice of X�. �8. Repeat points and the preservation of large cardinals.De�nition. Let w 2 U1. � < lh(w) is a repeat point for w i�8X X 2 w(�) =) 9� < � X 2 w(�)Lemma 1. If lh(w) � (2�w)+ then w has a repeat point.Proof. For each � < lh(w) which is not a repeat, �x (by the de�nition ofrepeat point) X� 2 w(�)n S�<� w(�). There are only 2�w possible X� . �Lemma 2. If � is a repeat point for w and A 2 Fw�� then A 2 w(�).Proof. Suppose not: then Ac 2 w(�), so as � is a repeat point there is� < � with Ac 2 w(�). But this is absurd. �Lemma 3. Let w 2 U1, with (i : V �! M; g) a constructing pair andlh(w) < i(�w). Let � < lh(w) be a repeat for w, and v = w � �. Letp = x _ hv;A;H; hi 2 Qv. Thenp _ hi(v); i(A); i(H); i(H)(�v)i �i(Qv) i(p)Proof. It is required to show that hv;A;H; hi can be added in to i(p) = x _ hi(v); i(A); i(H); i(h)i.A 2 Fv, so by lemma 2 A 2 w(�) and therefore v = w � � 2 i(A). Also

A MODEL IN WHICH GCH HOLDS AT SUCCESSORS BUT FAILS AT LIMITS49A = i(A) \ V�v , H = i(H) � [�v]2, h = i(h) � �v. The lemma isproved. �Now we use the canonical generic sequence constructed in the lastsection to show that the forcingQw can preserve large cardinal propertiesof �w. It is possible to do this proof in the style of [R], using the PrikryLemma to de�ne extenders in the generic extension, but the proof wegive here is rather shorter and more perspicuous.Theorem 1. With the same assumptions as in lemma 3 let G be genericfor Qw��, and do the construction from the last section to produce an �and an H such that H is i0�(Qw��)-generic overM�, where i0� : V �!M�is the �th iterate of i. Write G = (~x; ~E) where ~x is a sequence of mea-sure sequences, and ~E is an appropriate sequence of generic collapses;similarly let H = (~u; ~F ), and de�ne a new pair G+H = (~v; ~I) where~v = ~x _ hw � �i_ ~u � (dom(~u) n f�wg)~I = ~E _ ~FThen G+H is i0�(Qw��)-generic over M�, and8p p 2 G () i0�(p) 2 G+HProof. The genericity is immediate by the geometric condition, so �xp = x _ hw � �;A;H; hi with p 2 G. By the geometric condition itsu�ces to show that p _ hi0�(v); i0�(A); i0�(H); i0�(H)(�w)i 2 G+H ,which amounts to showing that the sequences (excepting the �rst) from~u and the generics from ~F are controlled by the top part of this condition.By construction each sequence occurring on ~u is a proper subsequence ofsome i0 (w � �) for some , say i0 (w) � �. As A is in all the measures onw � �, i0 (A) 2 i0 (w)(�) by elementarity, hence i0 (w) � � 2 i0 +1(A);then i0 (w) � � 2 i0�(A) because i +1� has large critical point. Thatthe entries of ~F are controlled by the top part is immediate from theobservation that i02(H)(�w; i(�w)) 2 g. �Now it is possible to extend the map i0� : V �! M� to a mapi� : V [G] �! M�[G + H ], so that certainly the measurability of �w ispreserved. In fact more is true, for observe that if (�w)+ < � < i(�w)and P� � M then P� � M�, because crit(i1�) = i(�w) > �. An easychain condition argument shows that if this is the case then(P�)V [G] �M�[G+H ]so that we can preserve any strength that the embedding i may happento have.

50 JAMES CUMMINGS9. The �nal model.Main Theorem. If it is consistent that GCH holds and there is a P3�-hypermeasurable cardinal, then it is consistent that GCH hold at everysuccessor and fail at every limit.Proof. The proof of the Main Theorem is now straightforward. Weperform the Reverse Easton construction of section 2 to get the initialhypotheses of section 3. Then using the constructing pair (j; F ) we builda sequence u with one repeat point �. We force with Qu��, and thentruncate the model V [G] (in which � is P2�-hypermeasurable) at �. Theresult is a set model of ZFC in which GCH holds at every successor andfails at every limit, containing many moderately large cardinals.References[[B]]J. Baumgartner, Iterated forcing, Surveys in set theory, LMS Lecture Notes 87,Cambridge University Press, Cambridge, 1983, pp. 1{55.[[Ca]]G. Cantor, Ein Beitrag zur Mannigfaltigskeitslehre, Journal f�ur dir reine undangewandte Mathematik 84 (1878), 242{258.[[C]]J. Cummings, Consistency results on cardinal exponentiation, Cambridge Uni-versity, 1988.[[D]]A. Dodd, The core model, LMS Lecture Notes 61, Cambridge University Press,Cambridge, 1982.[[DJ]]K. Devlin and R. Jensen,Marginalia to a theorem of Silver, Logic conference Kiel1974, Lecture Notes in Mathematics 499, Springer, Berlin, 1974, pp. 115{142.[[E]]W. Easton, Powers of regular cardinals, Annals of Mathematical Logic 1 (1964),139{178.[[FW]]M. Foreman and W. H. Woodin (to appear).[[GM]]M. Gitik and M. Magidor (to appear).[[J]]T. Jech, Set theory, Pure and Applied Mathematics 79, Academic Press, NewYork, 1978.[[K]]K. Kunen, Set theory, North-Holland, Amsterdam, 1980.[[M]]A. Mathias, Sequences generic in the sense of Prikry, Journal of the AustralianMathematical Society 15 (1973), 409{414.[[M1]]M. Magidor, On the singular cardinals problem I, Israel Journal of Mathematics28 (1977), 1{31.[[M2]]M. Magidor, On the singular cardinals problem II, Annals of Mathematics 106(1977), 517{549.[[Mi1]]W. Mitchell, Sets constructed from sequences of measures: revisited, Journal ofSymbolic Logic 48 (1983), 600{609.[[Mi2]]W. Mitchell, The core model for sequences of measures I, Mathematical Pro-ceedings of the Cambridge Philosophical Society 95 (1984), 229{260.[[P]]K. Prikry, Changing measurable into accessible cardinals, Dissertationes Mathe-maticae 68 (1970), 5{52.[[R]]L. Radin, Adding closed co�nal sequences to large cardinals, Annals of Mathe-matical Logic 22 (1982), 243{261.[[Sc]]D. Scott, Measurable cardinals and constructible sets, Bulletin de l' Acad�emiePolonaise des Sciences (S�erie des sciences math.) 9 (1961), 521{524.

A MODEL IN WHICH GCH HOLDS AT SUCCESSORS BUT FAILS AT LIMITS51[[Sh1]]S. Shelah, Proper forcing, Lecture Notes in Mathematics 940, Springer, Berlin,1982.[[Sh2]]S. Shelah, The singular cardinals problem: independence results, Surveys in settheory, LMS Lecture Notes 87, Cambridge University Press, Cambridge, 1983,pp. 116{134.[[Si1]]J. Silver, On the singular cardinals problem, Proceedings of the InternationalCongress of Mathematicians, Vancouver, 1974, pp. 115{142.[[Si2]]J. Silver, Unpublished notes on reverse Easton forcing (1971).[[W]]W. H. Woodin (to appear).MSRI1000 Centennial DriveBerkeley CA 94720USA