Some remarks on the maximality ofInner ModelsP. D. WelchGraduate School of Science & Technology,Kobe University,Rokko-dai, Nada-ku,Kobe 657, Japan.welch@pascal.seg.kobe-u.ac.jpAbstract. We consider maximality properties of inner models, elemen-tary embeddings between them, and survey some connections throughthe concept of J�onsson cardinal. In particular we give proofs of:Theorem Assume there is no inner model with a strong cardinal andK is the core model. If � is a regular J�onsson cardinal, then(i) �+ = �+K ;(ii) f� < � j � regular, �+ = �+Kg is stationary in �;(iii) 8A � �, A# exists.Theorem Assume there is no inner model with a Woodin cardinal, thereis a measurable cardinal , and K is the Steel core model. If � < is aregular J�onsson cardinal, then (i) and (ii) above hold.1 IntroductionThe �rst half of this paper constitutes the lecture given at the conferencemeeting, and contains some general discussion and a survey of the area. Thesecond half gives proofs of some new results that were mentioned but not given.The reader interested only in these, (the theorems of the abstract) can turn tox2.2 Theorem 12 and x2.3. We assume at those points a working familiarity with�ne structural arguments, and in particular with [19] and [23].Themes :1. Maximality properties of inner models.2. Elementary embeddings.3. Some connections through the concept of J�onsson cardinal.De�nition 1. An Inner Model M is a transitive class of sets, a model of ZFCaxioms with On �M .We abbreviate this as IM(M). We regard V as an inner model of itself.G�odel's universe L is then a paradigm for an inner model. We write 9IM() tomean there exists an inner model with property (usually is simply \a largecardinal".)

2 Some remarks on the maximality of Inner ModelsExamples of what one could mean by \Maximality" of an inner model M �V :a) The cardinals of M are the cardinals of V .b) The re ection of various cardinal properties from V down to M .c) Certain cardinal computations are correctly done in M .E.g. if � a cardinal, is �+ = �+M?d) Other Covering properties, e,g, the Strong Covering Lemma:CL(M) : If X � �; card(X) > !, then 9Y 2M;Y � X; card(Y ) = card(X).As cardinals can be collapsed and altered so easily by forcing, we shall havealmost nothing to say about the situation in a) here, but we shall look at theother notions with particular reference to the canonical inner models known as\core models". We shall adopt the usual conventions concerning set theoreticalnotation. Closed and unbounded sets of ordinals will be called cub.De�nition 2. A class j is an elementary embedding between the inner modelsM;N if j : M �!�1 N . (By this we mean j is '-preserving for any formula' 2 �1 \ Lf _2; _=g.)De�nition 3. The critical point of j; crit(j) � �� j(�) > �:By ZFf _|g, or ZFCf _|g, we shall mean the usual axioms of ZF or ZFC, butin the language Lf _2; _=; _|g, that is with an extra predicate letter _|.Theorem 1 (Kunen 1971 [13]) ZFCfjg ` j : V �!�1 V =) j = id.Open Question 1): Is this true for ZFfjg?A.Suzuki has witten out an explicit proof that this must be true of anyZF -de�nable class term j [25].Fact 2 ZFC ` 9j : V �!�1 M ^ � = crit(j) ! � is a measurable cardinal.Question: (Kunen)[14] DoesCon(ZFCfjg + j : M �!�1 V; j 6= id) =) Con(ZFC + 9 a measurable cardi-nal)?If j : M �!�1 N IM(M); IM(N) and j 6= id, then j � LM : LM �! LN . Ofcourse by absolunteness of the construction of L, LM = LN = L. Hence V 6= L,by Kunen's result (or indeed by an earlier result of Scott proving directly thatthere were no measurable cardinals in L).But if j : L �!�1 L; j 6= id, and � = crit(j) we may de�ne a �lter measuringall constructible subsets of �: X 2 U () X 2 P(�)L^� 2 j(X). One may checkthat this is a normal �lter, and we may form an ultrapower. The ultrapower iswellfounded as we may de�ne a �0-preserving embedding k of it, back into L, by:k([f ]U ) = j(f)(�). By a result of Kunen, we may \iterate" this ultrapower of L

Some remarks on the maximality of Inner Models 3by U , and the resulting images and critical points form a closed and unboundedin On class of indiscernibles for L - known as \O# exists".But it is not necessary to start out with an elementary embedding j of all ofL in order to achieve this.Example Let j : L� �! L�; crit(j) < �, with � regular, let � > �; � be any reg-ular cardinal. Then we may directly perform a \lift-up" or \pseudo-ultrapower"�nding ~| : L� �!�1 L�with ~| � j.The construction is well known and has a long history. We give an outlineonce again here. Of course using the regularity of � one could just de�ne ameasure U as above on the critical point of j, and simply determine that all of Lhas a wellfounded ultrapower by U . (There are far stronger statements possiblethan that of the Example, but we illustrate with a procedure that will have otherapplications later.)Let � = ff 2 L� j domf = u 2 L�g. Form a domain D = fha; fi j f 2� ^ a 2 j(domf)g and de�ne relations I; e by:ha; fiIe hb; gi () ha; bi 2 j(fhu; vi j f(a)=2 g(v)g):Set D = hD; I; ei. We now use the following version of Los Theorem:For ' 2 �0 \ Lf _2; _=g : D j= '(hai; fii)() ai 2 j(fu j '[fi((u)i)]L�g)If e is wellfounded let [ ] : hD; I; eig !hM;=;2i. Then M is transitive and[x]=2 [y]() xIe y( for x; y 2 D):De�ne ~| : L� �!M by ~|(x) = [h�; fh�; xg]. One can check (i) ~| is �0 and co�nal(so �1-preserving) into M ; (ii) ~| � j.Why is e wellfounded? Suppose there are hai; fii with (�) [hai+1; fi+1i] 2[hai; fii] (i < !). Let � be su�ciently large < � so that ffig � L�. Let � < �be su�ciently large so that fdomfig � L�.Let � : L�g !X where � [ ffig � X � L� with card(X) = card(�). Then� < �. Let �(f i) = fi.By (�) hai+1; aii 2 j(fhu; vi j fi+1(u) 2 fi(v)g) ! hai+1; aii 2 j(fhu; vi j f i+1(u) 2 f i(v)g) [Elementarity of � ] ! hai+1; aii 2 fhu; vi j j(f i+1)(u) 2 j(f i)(v)g [Elementarity of j]But then : : : j(f i+1)(ai+1) 2 j(f i)(ai)!There is no need to stop at any particular � in the above: one may simplylift j to a ~| : L! L of all of L.1.1 Covering and Re ectionWe shall be considering covering properties between inner models and V . Theparadigm here is Jensen's original Covering Lemma for L.

4 Some remarks on the maximality of Inner ModelsTheorem 3 (Jensen 1974)[1] (:0#)(i) d) holds for M = L, that is CL(L). This has the following consequences:(ii) � singular �! (� singular)L(iii) L computes correctly successors of singular, weakly compact cardinals. Infact cf(( +)L) � card( ) for � !2.But note that, in general, no other successors need be correctly calculated. Otherre ection properties:{ � weakly compact �! (� weakly compact)L{ � weakly compact �! 8A(8� < � A \ � 2 L �! A 2 L). Hence if � isweakly compact and V� = L� then V�+1 = (V�+1)L.{ � singular cardinal ^ V� = L� �! V�+1 = (V�+1)L.Note that only the last of these is a genuine consequence of the (proof of the)Covering Lemma for L, whilst the �rst two are simply consequences of the weakcompactness property, or rather the tree property. There are many other exam-ples of \small" large cardinal properties that relativise to L which we could list.Back to Kunen's Question:Let �(j) denote j : M =df dom(j) �!�1 V; j 6= id.{ Note no such j de�nable by a class term ' 2 Lf _2; _=g from parameters.{ If j : M �! V , then j : L �! L; j : L[0#] �! L[0#] : : :It is thus natural to ask how far one can go with this.Theorem 4 (Vickers-Welch)[26] ZFCf _jg + �(j) =) Con(ZFC+\There is aproper class of `almost Ramsey' cardinals"):De�nition 4. � is almost Ramsey () 8� < � � �! (�)<w;� is Ramsey () � �! (�)<w.The answer to Kunen is `No'. We �rst recall the de�nition:De�nition 5. An in�nite cardinal � is J�onsson, if for any algebra A = h�; (fn)n<withere is B � A a proper subalgebra of A of cardinality �.We list some facts about J�onsson cardinals; references to these may be found in[11].Fact 5 (Rowbottom) � Ramsey =) � regular J�onsson. ( 6( Devlin);� not J�onsson =) �+ not J�onsson.(Tryba, Woodin ind.) � regular =) �+ not J�onsson.(Devlin-Rowbottom) the least J�onsson (if it exists) is weakly inaccessible or hasco�nality !.(Prikry) A singular limit of measurables is J�onsson.

Some remarks on the maximality of Inner Models 5The following is a long-standing problem.Open Question 2) Can @! be J�onsson?The above is at one end of a scale of questions concerning possibly small oraccessible J�onsson cardinals. The following result(s) showed that such cardinalswould yield strong inner models.Theorem 6 (Donder-Koepke, Koepke)[4]; [12] (� regular but not weakly �-Mahlo)or (� = !� > �) then � J�onsson =) IM(Proper class of measures).The following outright ZFC theorem of Shelah (an implication of pcf theory)clinches the matter.Fact 7 (Shelah)[22] ZFC ` � regular J�onsson =) � is weakly �-Mahlo; if� singular ^ �+ J�onsson =) � is a limit of regular J�onssons. E.g. @!+1 notJ�onsson.Let � be a strongly inaccessible J�onsson, letA = hV�;2i. Let B � A; card(B) =�;B \ � 6= �. Let j : hM;2ig !B � A.Now let hV�;2; j;Mi arise as above from the de�nition of � as a strongly in-accessible J�onsson cardinal. Then hV�;2; j;Mi j= �(j). Since a Ramsey cardinalis a strongly inaccessible J�onsson we have shown:Theorem 8 (Vickers-Welch)[26]ZFC + 9 Ramsey cardinal =)Con(ZFCfjg + �(j)).�(j)'s consistency strength is thus sandwiched between that of a Ramsey andan almost Ramsey cardinal. It is interesting to ask further questions about thepossible range of j.Theorem 9 (Vickers-Welch)[26] (i) ZFCfjg+�(j)+ranj contains unboundedlymany regular �xed points =) Con(ZFC + 9 proper class of Ramseys).(ii) ZFCfjg + �(j) + ranj contains stationary many �xed ponts � of co�nality!1 ) \Oswordexists".(We explain this latter term below.)Open Question 3) What is a good upper bound for the consistency strength of(ii)?(The natural conjecture here is something like a measurable with Mitchell order!1.)Remark: It is interesting to compare (i) of the last theorem with the following.If G is a class P-generic for the proper class form of Woodin's full stationarytower forcing, over V where V contains a proper class of completely J�onssoncardinals (a notion ostensibly stronger, but actually equiconsistent with the def-inition of J�onsson above, and so with Ramsey), then in V [G] one may de�nean embedding of V �! V [G] (my thanks to A. Kanamori for emphasising thispoint). But care must be taken here, since the structure hV [G]; V;2i is not a

6 Some remarks on the maximality of Inner ModelsZFC-model. In one sense the model V [G] is \near" V here (being a forcingextension), whilst the model M of Theorem 9 (i) is \thin" inside V . Thus thelatter perhaps is an easier situation to realise, and thus requires smaller largecardinal assumptions.2 The computation of successor cardinals2.1 Beyond 0#, beyond LThe generalizations of L to other inner models, the \core models" of V , also havesome \maximality" assuming some anti-large cardinal axiom. Usually the fullestexpression of this maximality is achieved through a Covering Lemma. Such mod-els K are of the form L[E] where E is a sequence of �lters, or extenders. TheDodd-Jensen core KDJ (see [2]) was the �rst step out from the constructible uni-verse, and enjoys the full Covering Lemma, if :9IM(measurable cardinal). Formodels containing more measurable cardinals it was known by work of Mitchellthat a weaker result would have to su�ce.A Weak Covering Lemma over an inner model M :WCL(M) : singular cardinal �! + = +M :Other models have been built subsequently: Jensen's model for measuresof Mitchell order zero, KMOZ , previously Mitchell's model KMitchell for bothof which the Weak Covering Lemma held (under the appropriate assumption,that :9j : KMOZ �! KMOZ (known as :Osword) for the former model, orfor the latter, assuming there was no inner model with a measurable cardinalof maximal Mitchell order: �++; Kstrong (Jensen, assuming no # for an innermodel of a strong cardinal, or \:O{"), and Steel's models, the �rst of which weshall write as KSteel, which is built assuming both that there is no IM(Woodin),and a technical assumption to assure iterability, that there is some measurablecardinal in the universe. Such a model also enjoys Weak Covering under theseassumptions (Mitchell-Schimmerling-Steel, [18].)General Moral: The larger cardinals V might contain, the greater di�cultyin the construction of the relevant core, and the greater di�culty in proving`maximality'.De�nition 6. A cardinal � is strong if 8�9j�(crit(j�) = � ^ j� : V �!eM� ^ V� � VM�� ).Fact 10 (i) Jensen [7](No # for an IM(Strong)) WCL(Kstrong).(�) � � !2 ^ (� = �+)KMOZ �! cf(�) � card(�):(ii) (Mitchell-Schimmerling-Steel)[18]; [17];(No IM(Woodin), but 9 a measurablecardinal). WCL(KSteel), and (�) holds for KSteel.

Some remarks on the maximality of Inner Models 7Once one has larger models the door is open to prove further re ection prop-erties from V into the model, that would not have been possible for L.The following re ection properties (inconsistent with L) hold.{ Let cf(�) � !1 and suppose that � is �-Erd}os. Then (� is �-Erd}os)KDJ(Jensen)[3]{ � is J�onsson �! (� is Ramsey)KDJ (Jensen, Mitchell)[3][15]{ � is J�onsson �! (� is J�onsson or measurable)KMOZ (Jensen-Vickers)[9].{ � is �-J�onsson �! (� is �-J�onsson)KSteel (Mitchell)[16].De�nition 7. (Mitchell [16]) � is �-J�onsson (for any regular � � �) if we re-quire only in the de�nition of J�onsson that the subalgebra B have order type�.2.2 J�onsson cardinals and L[A] modelsA Ramsey cardinal is weakly compact, and this together with the fact � Ramseyimplies 8a 2 V� \a# exists" yields:� Ramsey =) 8A � �A# exists (so V 6= L[A]).Do regular J�onsson's enjoy this property? (Singular J�onssons need not: byPrikry's result in Fact 5, in L[�] where � is an !-sequence of measures, then inL[�], =df sup� is J�onsson, but 9A � (:A# exists).) The �rst observationlends plausibility to this.Theorem 11 (Mitchell [16], Welch) � regular J�onsson, then :9A � � V =L[A].Proof: Suppose for a contradiction V = L[A] for some A � �. Let j : L�[A] �!L�[A] arise from the J�onsson property, with j chosen so that crit(j) is least overall possible such A. Just as in the Introduction, perform a \lift-up" replacingL� there by L�[A] here, and using as D now the class of pairs using � = ff 2L[A] j domf = u 2 L�[A]g); thus yielding a ~| : M = L[A] �! V = L[A]. Theargument is identical. From all this we may construct a class term, de�ning amap of an inner modelM of the universe to V , which is itself impossible. Q.E.D.The next result shows that it must be true (or at least there must be innermodels with some large cardinals).Theorem 12 � regular J�onsson ^9A � �:A# =) 9 a # for an IM(StrongCardinal).We give some detail of the proof of the above. We assume some familiaritywith the notion of premouse and mouse, and adopt the formalism of [19], withtheir associated good extender hierarchies. We explain some of this nomenclaturebelow before giving a proof, which is itself a simple iteration and comparisonargument, although the reader must be referred to [19] for an explanation of allterms. For the hierarchies under consideration, that is \below O{" this materialis also summarised in [24] x1.

8 Some remarks on the maximality of Inner ModelsDe�nition 8. (i) A premouse M of the form J EM� is below O{ i� wheneverE is a (�; �) extender on the EM -sequence and � < �, then supf j crit(EM ) =�g < �.(ii) O{ is the unique sound mouse M such that M is not below O{ , but everyproper initial seg. of M is below O{ .It is easy to see that O{ is active (i.e. its last extender predicate - in factequivalent to a �lter - is nonempty), and that if � is the critical point of thislast extender, then there is a � < � such that for co�nally many � < �, thereis a (�; �) extender on the O{ sequence. It is easy to see that a premouse Mbeing below O{ is equivalent to the condition that if crit(EM� ) = � then JM� j=\ there are no strong cardinals ". If we are restricting ourselves to premice belowO{, then the iterations arising are of the following simple kind:De�nition 9. An iteration tree T of length � is almost linear if for any � <� < � T -pred(� + 1) = � ) � = � + n for some n < ! ^ 8k � n(crit(ET� ) =crit(ET�+k)).It is easy to argue from the initial segment condition that for premice belowO{ that every iteration tree on M is almost linear (cf. x8 of [23]). In what followsthe reader will lose little by picturing the iterations to be completely linear: ina sense we have only the possibility of non-linear iterations due to the formalarrangement of our extender hierarchies. If we restrict ourselves to working belowO{, then our notion of a \universal" inner model L[E] is somewhat simpli�ed,and we have the fact that any such \universal weasel" is an iteration of the coremodel Kstrong. (Fact14 below.)Correctly computing successors of singulars characterises universal weasels.Fact 13 (cf. [19]x3) W is a universal weasel if and only if on a stationary classof �, �+ = �+W . Further, if assume there is no inner model for a strong cardinal,then W is a universal weasel if and only if �+ = �+W for arbitrarily largecardinals �.Fact 14 ([23] 8.13 for example) If :O{, then for every universal weasel W ,there is an almost linear iteration tree T on K with last model W , and such thatT does not drop in model or degree. Similarly if j : Kstrong �! M , then M isuniversal, and j is (the) iteration map taking Kstrong to M .Proof: (of Theorem 12) Suppose for a contradiction that � is a regular J�onssoncardinal and that A � � has no #. Then � is (weakly) inaccessible. Withoutloss of generality we assume A is such that Kstrong � � � (V�)L[A]. By theJ�onsson property there is A � � and j with j : L�[A] �! L�[A]. Again, usingthe regularity of �, perform a \lift-up" yielding a ~| : L[A] �! L[A]. It is easyto see that we also must have :A# (otherwise we could carry indiscernibles forL[A] via j to L[A]). By the L-Covering Lemma relativised to the models L[A]and L[A], we must have that �+ in both models has co�nality greater than orequal to �, and in fact the full covering lemma holds for subsets of ordinalsabove �. But now let K1 =df KL[A] and K2 =df KL[A]. We claim these are

Some remarks on the maximality of Inner Models 9both \universal" weasels. To see this note, by the Weak Covering Lemma for Kapplied inside L[A] (L[A]) we have that successors of singulars greater than � arecorrectly calculated (from the point of view of L[A] (respectively L[A])) by K2(resp. K1). But then unboundedly often successors are calculated correctly bythe models K1;K2, and thus by the Fact 13 they are both universal, and hencesimple (almost linear) iterates of the true core model Kstrong. Set K = Kstrong.Now run a comparison of the two models. This results in iteration trees, callthem, T on MT0 = K1 and U on MU0 = K2, with a common �nal inner model,W = MT1 = MU1 say. (By \universality" we cannot have any truncation of anymodel occurring in these trees, and the �nal models on T , U must be identical.)Note if we let � � 0 be least with MU� � � = K1 � � then a) � 2 [0;1)U (whereU is the underlying almost linear tree ordering of U); and by our assumptionon A: b) the tree T � � is trivial (that is for i � �MTi = MT0 = K1 and�Ti;j = id (i < j � �)). This is because the same is true for the comparison of(K1;K). (This is tantamount to saying that K2 � � = K � �: which it is, sincewe ensured that K � � � L�[A].)Now extend using a \lift-up" j to ~| : MU� �!W with W � K2 � � = K � �.Again one can do this just as in the Introduction: MU� is a model of the formL[E], with HL�[E]� = jL�[E]j for L[E]-cardinals �. This enables us to form adomain D using functions in MU� with domains in L�[E] = MU� � �. This givesus a wellfounded class W . Note that we get iterability of the model W virtuallyfor free: any set-sized initial segment of W is iterable by any iteration de�nablewithin W (by elementarity of ~|); but in the region of almost linear iterationsany iteration whatsoever can be \embedded" back into a \universal" iterationof W that can be de�ned internally to W . (This \universal iteration" procedure,and the construction of the \embedding back" of a general iteration is describedin full detail in [24] Def. 2.8 and Lemma 2.10.) If all initial segments of W areiterable, then clearly W is iterable. Then W is a normal iterate of K (Fact 14).So let � : K �!W be the (unique) iteration map. But then � = ~| � �U0;� by thesame Fact. But W � � = K � �, hence as crit(�) = � = crit(~|) = crit(j) this canonly happen if for some � > �; crit(EK� ) = �. By the Initial Segment Condition(part of the condition for the construction of the \good extender sequence" hi-erarchy for such L[E] models) we must have that f� j crit(EK� ) = crit(EK� )g isunbounded in any K-cardinal � � � with crit(EK� ) < �. But as we are belowO{, this means there are no measurable cardinals of K in (�; �). But this impliesthat � = 0, i.e. , K1 � � = K � �. But then ~| : K �! W is an iteration map;hence � is measurable in K; but then so is j(�). A contradiction! Q.E.D.Clearly this argument could be pushed a little further, but beyond lineariterations, one usually invokes the existence of a measurable cardinal (or atleast #'s for sets in V) to construct Kc and hence K; this somewhat obviatesthe need for any proof at all! A small corollary is immediate from the argumenthowever:

10 Some remarks on the maximality of Inner ModelsCorollary 15 (:O{) In Kstrong there is no \strong past a J�onsson". That isif � is J�onsson, then for no � < � do we have supf� < � j crit(EK� ) = �g = �.Let � be weakly compact cardinal. Suppose :9j : L �! L : then �+ = �+L.[Because �+L < �+ �! cf(�+L) = card(�+L) = � (by WCL(L))]. Weak com-pactness implies there is a �-complete L-ultra�lter on P(�)L. So Ult(L;U) iswellfounded and 9j : L �! Ult(L;U) �= L. For relatively small core modelssimilar considerations work. The following theorem however requires deeper ar-guments about the nature of justi�cation concerning which extenders are addedto the good extender sequence that is inductively de�ned to build K,Theorem 16 (Schimmerling-Steel)[21] (No IM(Woodin) +9 measurable cardi-nal ) 8� < (� weakly compact �! �+ = �+Ksteel):Much recent work has focussed on looking at 2 sequences (or weakeningsthereof).De�nition 10. Let 2 Card: 2 is the following principle:9hC� j � < +i such that(i) 8Lim(�) < + C� � � is c:u:b:(ii) 8� 2 C�� C� \ � = C�(iii) otp(C�) � The de�nition is due to Jensen [5] where it was also shown that 8� 2� holds inL; as core models were developed proofs of 2 were given for KDJ (Welch [28]),KMOZ (Wylie [29]), and for KStrong (Jensen [8]). (In fact it is not the pointof this survey to go into this matter so thoroughly, but stronger \global" 2principles are possible, and in fact these global principles were shown for themodels in the works cited for L;KDJ and KMOZ ; for Kstrong see [10],[30].)If one weakens the de�nition of 2 to allow at each � < + a non-empty �nitefamily of cub-in-� guesses C� for C 2 C� (or, weaker still, � � many guesses- thus at (ii), for any limit point � of any C 2 C� we require that C \ � 2 C�),then one obtains the weaker principles 2<! (or 2� ).Theorem 17 (Schimmerling)[20] (No IM(Woodin) +9 measurable cardinal )8� < 2<!� holds in KSteel:In fact much more than this is proven in [20]: there it is shown true in L[E]models without superstrong extenders. It now appears that full 2 would holdin the above theorem for all < , (and in fact in a wider class of L[E] modelswhich do not contain a measurable Woodin cardinal) by work of Zeman [31].{ Now a 2� or 2<!� sequence is absolute between an IM(M), and V , provided�+ = �+M . Hence{ (No IM (Woodin) +9 measurable cardinal ) ) 8� singular, or weaklycompact 2<!� holds in V .

Some remarks on the maximality of Inner Models 11Question (Vickers): What about successors of J�onssons?Remark: The point of the question (which could have been asked about KDJ(and answered) some years ago) is that a regular J�onsson cardinal can fail to beweakly compact.Theorem 18 (Vickers-Welch[27]) Let � be J�onsson.a) V� = V KMOZ� �! V�+1 = V KMOZ�+1b) Assume :Osword. Then �+ = �+KMOZ [Hence 2� in V ].2.3The last could equally well have been proven assuming :O{ and using Kstrong.We sketch the proof of:Theorem 19 (:O{) If � is a regular J�onsson cardinal, then �+ = �+Kstrong .Proof: We do not wish to repeat large segments of the proof of Theorem 13above (or of Theorem 18), so we give a sketch outline which should su�ce forthis and to see the truth of Theorem 20. Let K be Kstrong. Suppose the theoremfalse; let � be J�onsson and suppose �0 = �+K < �+ and by the Covering Lemmafor K cf(�+K) = �. let D : � �! �0 be a monotone co�nal map. Appealing tothe J�onsson property of � in V , let X � hK�0 ; EK � �0; Di where card(X) =�; X \ � 6= �. Let j : hH;E;Di �!�! hK�0 ; EK � �0; Di come from the inverseof the transitive collapse on X . Then j 6= id; j(�) = �; j\On\H is co�nal in �0.Let � = On\H . Now form the coiteration of (H;K) setting MT0 = H;MU0 = K,via almost linear trees T ;U of length �. Corollary 15 ensures that if � > � thenfor � � i < �, crit(EUi ) � �. Let � be least so that MT� � � = MU� � �.Claim 1: �U0;�\� 6� �.Remark: In the proof of Theorem 12 we had the universality of K1 to use,in order to argue that the �rst �-steps at least of the iteration involved nomovement on the K1-side. We cannot use that here; but for the moment let usassume there is no movement on the H side, and that the tree T is trivial fori < �.Proof: (of Claim 1) If this failed then MT� � � (here assumed to be H � �) is aninitial segment of MU� . In fact the latter must be a proper class: any truncationat some step i < � would result from this point onwards in an iteration of aninitial segment of the i'th model of cardinality < �, but this would contradictour assumption that no ordinal in the iteration gets sent up beyond �. But nowwe are in the same situation as the last part of the proof of Theorem 12 and geta contradiction via a lift-up argument as there. Q.E.D.(Claim 1)We therefore have (+) 9i0 < �9� < ��Ui0;�(�) � �.Claim 2

12 Some remarks on the maximality of Inner Models(i) � = �,(ii) There is a cub C � � so that i < j 2 C =) �Uij(�i) = �j = j where�i = crit(EUi ).Proof: Due to Claim 1, if � < � we should have to have an extender EUiwith critical point < � but of length > �; but there are no \strong past aJ�onsson" such extenders. Hence (i) holds. (ii) is a standard regressive set argu-ment. Q.E.D.(Claim 2)Now consider the �'th model in U : MU� . For i 2 C we have that �Ui;�(�i) = �,and moreover sup �Ui;�\(�+i )MUi = (�+)MU� . (As always for a transitive model Nwith � the largest N -cardinal, we adopt the convention that (�+)N is On \N .)Call the latter �. Then cf(�) < � whilst cf(�) = �. But � > � (otherwise the nextstep of the coiteration would require a truncation on the T -side, contradictingthe universality of K. (Note as no extenders overlap � this would really require atruncation rather than simply using an extender EMT�� with critical point below�.) But in that case there is N a proper initial segment of MU� and some n < !with �n+1N = � < � � �nN . We now invoke the procedure of constructing a \�nestructure" preserving lift-up, (a considerable re�nement of the previous lift-upprocedure for providing wellfounded and iterable models which we shall not gointo here). This method provides an extension k of the embedding j and aniterable model P of the form L [F ] with �n+1P = � < �0 � �nP , where k is a �ne-structure preserving embedding k : N �! P with k � j. The remark to makeis that P is su�ciently sound and is coiterable with K in order to argue that infact there is a code for P in K. The two crucial points for the proof to go throughhere are: (i) j is a co�nal map of � into �0; (ii) H j= \� is the largest cardinal"and cf(�) > !. These ensure that the P constructed has On \ P � �0, and isiterable. But de�nably over P there is a map of � onto �0! But this contradictsthe assumption that �0 = (�+)K . (The reader may �nd the further detail of thisargument in [27] Thm. 1.)Now consider what happens to this argument if T is nontrivial below �.Claim 3 �T0;�\� � �.Proof: Suppose otherwise; if �U0;�\� � �, the embedding j would yield a con-tradiction to the Dodd-Jensen Lemma (cf. [19] 5.3 - or in other words it wouldimply that K � � <� H � � where <� is the mouse ordering). Hence (+) holds.But a standard regressive set argument would then show that on a cub D � �we should have agreement between the extenders used on the two sides, contra-dicting the de�nition of comparison. Q.E.D.(3)We thus have again MT� � � a proper initial segment of MU� . We also havethat (1) ~� =df (�+)MT� (= On \MT� ) = �T0;�(�) = sup�T0;�\�and hence has co�nality �. We copy the iteration T � � via j to an iteration jTon K � �0 (again cf. [27] or [19] 5.2) with resulting maps ji : MTi �! M jTi

Some remarks on the maximality of Inner Models 13so that we have a �nal map | =df j�, | : MT� �! eK where eK = M jT� .We use this as the base for our lift-ups. In our proof of Claim 1 we embed-ded the proper class MU� �! W via a lift-up map ~| extending j � (H � �).Now we should lift-up | � (MU� � �) to a ~| : MU� �! W with W � eK � �.With this variation Claim 1 can be concluded in the same way. For the �nalargument (of the paragraph following Claim 2) we should have some initialsegment N of MU� whose n + 1'st projectum drops to � but whose height is� ~�. We then lift-up (�ne-structurally) | : MT� �! eK to a �ne-structure pre-serving map k : N �! P . Note that | is co�nal into On \ eK, and both eK,MT� j= \� is the largest cardinal" and cf(~�) = � > !. The crucial points men-tioned above thus still hold true. Q.E.D.(Theorem 19)In fact it simply re ects :Theorem 20 (:O{) If � regular J�onsson, then f� < � j � regular, �+ = �+Kgis stationary in �.Proof: Let D � � be cub. Let � > � be su�ciently large, and by the J�onssonproperty, let j : hV ;2;K;Di �! hV� ;2;K � �;Di (where K = Kstrong) be anelementary embedding with card(ran j \ �) = �, j � � 6= id. The point is thatthe Claims of the last proof hold here (they were really only based on the univer-sality of K). Thus, if we consider the coiteration of MT0 = K and MU0 = K � �,some point will be sent up past � on the U-side of the coiteration, and there willbe a cub in � set C with the properties of Claim 2. But note that for i < j 2 C,�Ui;j((�i)+)MUi ) = (�j)+)MUj and in fact �Ui;j is continuous between these two or-dinals. Hence all such successors have the same �xed co�nality � for some � < �.By the Mahloness of � (cf. [22]), there is a regular 2 D \ C, with � < andwith the additional property that �T0; \ � (and hence �T0; ( ) = ). Thenwe shall have that �T0; takes ( +)K co�nally to ( +)MT . Observe now thatwe simply cannot have ( +)V > ( +)K : by the Weak Covering Lemma appliedinside V , we should have V j= cf(( +)K � . But by the previous commentcf(( +)K) = cf(( +)MT ) = cf(( +)MU ) = �. Therefore we must have thatV j= + = ( +)K . But then j( )+ = (j( )+)K . And j( ) 2 D. Q.E.D.Extensions of these arguments show:Theorem 21 Assume there is no IM(Woodin) ^9( a measurable cardinal).Let K = KSteel, and suppose that � is J�onsson . Thena) �+ = �+K ;b) f� < � j � regular ; �+ = �+Kg is stationary below �.Proof: We work towards a proof of a) �rst. We shall need some notation forkind of phalanx that we shall use. We adopt that of De�nition 9.6 of [23]. Thisruns as follows:

14 Some remarks on the maximality of Inner ModelsDe�nition 11. A phalanx � of premice of length � + 1 is a pair(h(M ; k ) j � �i; h� j < �i)such that for all � �(1) M is a premouse, k � !, and if k 6= 0 then M is a k -soundpremouse;(2) if < � < �, and �� 6= 0 then � < ��;(3) if < � � �, �k� (M�) > � ;M agrees withM� below � .Actually the de�nition cited allows for a more exotic choice of creaturesthan just premice, but we shall not need them. Nor do we need to consider theprotomice of [18]. [23] also uses the notation deg�( ), and �( ; �) for k ; � .There is a further coordinate � , that we have suppressed which is used forhandling Type III premice. We allow for models with �� = 0. The point of this isonly to cater for some \dummy" models that have arisen from deriving a phalanxfrom a comparison tree with padding steps. They play no dynamic role in theconstructions and we could simply cut them out, but this would only involveanother step at some later point. We make the further simpli�cations:(i) if any k = 0, it is omitted (for example if M is a weasel);(ii) if the phalanx is of length 2 and the �rst model is a weasel, we shall thusjust write ((M0; (M1; k1)); �0) and even omit k1 if it is understood. This will beespecially true when k1 is just n(M1; �0) =df maxfn j �nM1 > �0g (if the latterexists, otherwise set it to !). We note that for the structures arising in our proofsthe M for < � will all be weasels, with only the last \starting model" M�possibly being a less than sized premouse (although this is a fact we shouldneed to demonstrate).(iii) If � is a phalanx of length � and ((Q;n); ) a suitable pair to enlarge thephalanx one point, we write �b((Q;n); ) for this extension of length � + 1.De�nition 12. An iteration tree of length � on the phalanx � of length � + 1is an !-maximal padded tree on �:for � � �; degT (�) = k�, and for � < � � �; degT (�) is the unique k � ! suchthat MT� = Ultk((MT� )�; ET� ) if � = � + 1 , and k and (MT� )� are chosen ac-cording to the rules of [19] Def. 6.1.2 for forming !-maximal trees, and ET� 6= ;;it is set to degT (�) otherwise. If � is a limit, then degT (�) is the eventual valueof degT ( ) for su�ciently large with T�.Given an iteration tree on a model we can derive a phalanx from it as follows:De�nition 13. Let T be an !-maximal padded iteration tree of length � + 1.Then �(T ) is the unique phalanx of length � + 1 so that:(i) M�(T )� = MT� and k�(T )� = degT (�) for all � � �;(ii) ��(T )� = �T� .

Some remarks on the maximality of Inner Models 15(Recall that in a padded iteration tree arising from the comparison processfor example, there are ordinals �T� de�ned as the natural length of ET� if thelatter is non-empty, but can possible be 0 otherwise; as mentioned above this ishow models with ��(T )� = 0 arise in the derived phalanx).By WCL(K) we may assume that � is regular and, using the notation ofTheorem 19, that �0 has co�nality �. Again let D : � ! �0 be monotone andco�nal. Let W be the \very soundness" witness for JK� obtained by iteratingeach order zero measure � > �0 (cf. [23] Ch. 8). Let X � hV� ; EW ; D;2i where� is some su�ciently large cardinal, card(X) = �, but X \ � 6= �. X exists bythe J�onsson property at �. Let j : hH;E;D;2ig !hX;EW \X;D\X;2i be theinverse of the transitive collapse. Let j(�) = �0, and thus j is continuous at �by virtue of D. Then j : W !�! W , where W = JE� , � = On \ H . FollowingTheorem 19 we should like to compare (W;W ) at least up to agreement below�, via (padded) trees T ;U , and prove a version of Claim 1, where some � < �is sent up to � by �U0;�.We �rst suppose for simplicity that T is trivial, and there is no movementon the W side, and that the statement of Claim 1 of Theorem 19 fails. Thenagain there can be no truncation on the branch [0; �]U (as otherwise we'd have astructure of size< � iterating up to one of size �).ThusMU� is a proper class, i.e. aweasel. We could perform a \lift-up" of j to ~| : MU� ! W 0 with W 0 � W � �.Wellfoundedness of W 0 is again not an issue, but su�cient co-iterability of W 0with W is: in particular we should want the phalanx ((W;W 0); �) to be coiterablewith W .We adopt Mitchell's solution to this iterability problem (in a slightly moregeneral situation) from [16], which essentially is to look ahead as the iterationtree U grows, and to check if we were to perform a similar lift-up of ~| : MUi !W 0ithat ((W;W 0i ); �(i)) has no \badly behaved" iteration trees on it. (This means,no trees with an ill-founded last model, or, if of limit length, that the only co�nalbranches through the tree are ill-founded.) If there are such bad trees, then thereare witnesses to this in a small substructure Y �MUi of size < �. We set MUi+1to be the collapse of this Y , and we continue the comparison by dropping tothis set-sized premouse. The tree U built in this way using this kind of \specialdrop" Mitchell calls a quasi-iteration tree.The point of this man�uvre is that �nally one is guaranteed either a fullyiterable phalanx ((W;MU� ); �), or a small structure of size < � iterating up tothe cardinal �. In the former case [16] can still show some ordinal is sent up to�, and in the latter the quasi-iteration tree guarantees this, and in both casesthe critical points (on a tail) form a cub in � set of indiscernibles for J E� (whichis what [16] is looking for). We shall need somewhat more than this.A comparison of the arguments that follow with [16] will reveal the extent ofour indebtedness to Mitchell's proof where this iterability problem is overcome.Indeed, the reader may view much of the current proof as an account of hisargument, but within a changed overall framework of a proof such as that ofTheorem 19. We should like to thank him for letting us use these ideas in thisarticle.

16 Some remarks on the maximality of Inner ModelsOne query that may arise is that of the continuability properties of the quasi-iteration tree U if there is such a special drop. We need a guarantee of uniquewellfounded branches at limit stages. [16] constructs an auxiliary regular treeU and an embedding of U into U : the unique branches of U can be used toguarantee a good branch in U . There is thus a \derived iteration strategy" forthe tree U (cf. the kind of argument on p.75 of [19]). Our ultimate aim is, as inTheorem 19's proof, to have a set-mouse N , an initial segment of MU� , to sendvia a lift-up k : N ! P , where P is some premouse which may be comparedwith W . To ensure the su�cient iterability of ((W;P ); �) now, we add a secondargument to Mitchell's recipe of special drops, and thus look for initial segmentsof our current structure MUi which could lift to badly behaved phalanges.We de�ne the comparison trees T ;U arising from coiteratingW;W as follows.T will be an entirely standard tree, arising just as in the standard comparisonprocess, but U will be augmented by the non-standard special drop outlinedabove. To ensure the continuability of the process we embed the tree U into aniteration tree U via maps ��. The existence of the \uniqueness strategy" for Uwhich picks out for us the unique co�nal wellfounded branch, can be used togive us a strategy for forming branches in U . We use the comparison notation of[19] Sect. 5. We de�ne by induction T � �+1;U � �+1;U � �+1 and a degU(�)embedding �� : MU� !MU� as follows:� = 0. MT0 = W ; MU0 = W = MU0 ; �0 = id; �U0 = 0 = �T0 .� = � a limit. U � � is then an !-maximal iteration tree, and the \uniquenessstrategy" picks for us the unique co�nal branch b � [0; �)U with a wellfoundeddirect limit model. Let < � be such that no drops of any kind have occurred inbn � [0; �)U (there will only be �nitely many of these along any such putativebranch), and so that no � + 1 2 [ ; �) is from a special pair (see below in CaseIIa) for this latter notion). Set MU� = MUb - the direct limit model. Then sethMU� ; h�U�;�ii =df Lim�! hhMU� i; h�U�;� 0i <�<U� 02bi. Set degU(�) = degU (�) to bethe eventual value of degU(�) as � ! �; � 2 b. �� is de�ned to be the limit ofthe embeddings �� for � 2 b, de�ned to ensure commutativity.� = � + 1. We are given U � � + 1;U � � + 1; T � � + 1 and so have modelsMU� ;MU� ;MT� . For most � we shall extend T and U using the normal comparisonprocess. We shall ensure that the comparison process will come to a clean haltwith models MU� ;MT� at some inaccessible cardinal stage � � � with the twomodels agreeing up to � (in fact � will be �). (If it would \normally" haltbefore, we simply put in some \padding steps" to make it nominally halt at thenext inaccessible stage - as can be surmised, this is an inessential part of theproof. This is somewhat unnecessarily to ensure we only look for special dropsat notationally convenient stages � 2 � below.)De�nition 14. � = f� < � j � is inaccessible ^ j\� � �g.

Some remarks on the maximality of Inner Models 17Case I � =2 � or � < sup <� �U .If MT� is an initial segment of MU� or vice versa, then as indicated in thelast parenthetical remark above, we pad out one more step everywhere settingMW�+1 = MW� , �W�;� = id, �W� = 0, for W 2 fT ;U ;Ug, and �� = �� . Otherwisethere is some least ordinal � witnessing some di�erence in the hierarchies EMT� ,EMU� . Suppose EU� 6= ; in this de�nition. Then we set EU� = EMU���(�). If MU�� is (aninitial segment of) MU then MU�� is (an initial segment of) MU . �� is again de-�ned to commute through the ultrapowers: ��([ha; fi]MU EU� = [h��(a); � (f)i]MU EU�(stating only the \n = 0" case). Then �� is a weak degU -embedding. (Note thatactually �� is fully elementary until there is a \normal" drop � 2 DU .) If EU� = ;then set �� = �� . �U� and �T� are de�ned as for a normal comparison process step.Case II Otherwise.Case IIa) MU� is a weasel and fM =df Ult(MU� ; Ej � �) yields a badly behavedtree on the phalanx ((W;fM); �) for some � 2 �; � � �, where MU� ;MT� agreeup to �.Note that if V is such a bad tree for some least � � �, then there are initialsegments W 0;M 0 of W;fM respectively, which are properly small in the sense of[23] Def.6.12, on which we can construe V as a bad tree (see the arguments par-ticularly of 6.14 op.cit.). As our phalanx is properly small and we are assumingthere is no IM(Woodin), V is simple.Let X � Y � V+1 be chosen with X countable, j;V ; T � �+1;U � �+1; � 2X . Let � : Rg !X with �(V ;W ;M) = V ;W 0;M 0 etc. The absoluteness argu-ment of 6.14 cited shows that in V V is truly a bad tree on ((W;M); ) where�( ; ) = �; �. We assume Y chosen so that � + 1 � Y , card(Y ) � �. Let� : Sg !Y , and : R ! S be given by = ��1 � �. If (M) = eQ, theneQ = Ult(Q;Ej � �), where Q = ��1(MU� ). The bad tree V can be copied via to yield a bad tree on (( (W ); (M )); ( )) = (( (W ); eQ); �), and using theembedding � � (W ) : (W )!W 0 we see:(1) There is a bad tree on ((W; eQ); �).We shall then take MU�+1 as Q, �U� = 0. This is essentially Mitchell's tacticto ensure a su�ciently iterable phalanx in the case his �nal modelMU� is a weasel.Remark: 1 Suppose in the above � = � were such that � was Mahlo in �,and the case hypotheses held, then there is a � 2 �, � < �, � 2 [0; �)U , alreadywith the property that there is a bad tree on the phalanx ((W; fM); �), wherefM = Ult(MU� ; Ej � �).

18 Some remarks on the maximality of Inner ModelsTo see this we just have to observe in the above proof that we could usethe Mahloness of � below �, to �nd an inaccessible � < � with � 2 � so thatwe could have taken a Y with X � Y � V+1, with card(Y ) = �; � \ Y = �.Note that such a Y , if � : Sg !Y as above, with Q = ��1(MU� ), then thereis a map � : Q ! MU� given by � = (�U�;�)�1 � �. That this makes sense isbecause � is on the branch [0; �]U (as this is cub below � by elementarity) andhMU� ; h�U�;�ii =df Lim�! hhMU� i�<U� ; h�U�;�0i�<U�0<U�i. Thus if x 2 MU� \ S ) x =�U�;�(x) for some � <U �; x 2 MU� . So �(x) = �U�;�(x) 2 MU� . But then the trulybad tree V on ((W; eQ); �) (in the notation of the case) can be copied via the in-duced map ~� derived from � between the lift-up ultrapowers eQ = Ult(Q;Ej � �)and fM =df Ult(MU� ; Ej � �), to give a bad tree on ((W;fM); �).We shall follow [16] and call f�; � + 1g a \special pair" and say that � + 1is a \special drop". To complete the de�nition of the case we put DU+2 =DU+1 [ f� + 1g. We intend now to continue the comparison using MU�+1 ratherthan MU� . We put �U� = 0 and degU(� + 1) = !, set � <U � + 1; � <U � + 1.Set ��+1 : MU�+1 ! MU�+1 with MU�+1 = MU� . Note ��+1 is fully elementary,and that MU�+1 continues to agree with MT� up to sup <�f�T ; �U g � � (and thelatter is only o�cially < � if for some < � MU is an initial segment of MT - or vice versa - which we shall later remark cannot happen). We thus pad onemore step in T to keep the indices correct: MT�+1 = MT� ; �T� = 0; �T�;�+1 = id.Case IIb) There are no such badly behaved \lift-up" models.If one of MU� , MT� is an initial segment of the other, or if � = �, we halt thecomparison. Otherwise we look for the least � witnessing a di�erence in theirhierarchies and proceed just as in Case I.This completes the de�nition of T ;U ;U . We note that since � is Mahlo in �there are plenty of opportunities to look for badly behaved trees in Case II.Let � � � be the length of the comparison process.Claim 0 a) If MU� is a weasel, and fW =df Ult(MU� ; Ej � �), then ((W;fW ); �) isco-iterable with W .b) If MU� is a set premouse, Q is a proper initial segment of MU� , andeQ = Ultn(Q;Ej � �) where n = n(Q; �) < ! exists, then ((W; ( eQ;n)); �) isco-iterable with W .Proof: In a) note that clearly DU \ [0; �]U = ; (as MU� must be a weasel). Then� must be � (otherwise we have not �nished our coiteration!). If the Claim wasfalse, then as � is Mahlo in �, the remark above shows there is an earlier stage� < � at which we should be compelled to take a special drop to a set-sizedpremouse!

Some remarks on the maximality of Inner Models 19For b) note again that if the conclusion failed we must have � = �. If weconsider the branch b = [0; �]U , then DU \ b is �nite (even if it contains a specialdrop). So there must be a < � so that DU \ b is contained in . We shall needthe following:Sublemma Suppose MU� has size < � but there is Q, a proper initial segment ofMU� with a badly behaved tree on ((W; ( eQ;n)); �) where eQ = Ultn(Q;Ej � �) andn = n(Q; �) =df minfm j �Qm+1 � �g exists. Suppose further � 2 [0; �]U with� the U-predecessor of + 1 with � <U + 1 <U �. Then (MU +1)� is a properinitial segment of Q. Hence + 1 2 DU .Proof: Let G = EU be the �rst extender used on [�; �]U . Then we claim thatlh(G) � On \ Q. For, if it were greater we should know that Q was an initialsegment of MT and hence so of MT� . But then Ultn(Q;Ej � �) can be embeddedinto fM =df Ult(MT� � � + 1; Ej � �); and then we should conclude ((W; fM ); �)had a badly behaved tree on it. But this is absurd since, (under our assumptionon the triviality of T ) fM is an initial segment of W !This means that + 1 2 DU : by speci�cation �!Q � � and hence � <(crit(G)+)MU +1 with the latter a cardinal of MU +1, but not of MU� .Q.E.D.(Sublemma)So suppose V witnesses the falsity of part b) of the Claim and is a badlybehaved tree on � = ((W; eQ); �) (dropping the n here). Suppose lh(V) = � andLim(�), (we leave the successor case to the reader) and that there are no co�nalwellfounded branches b.Let X � Y � V+1 be chosen with X countable, j;V ; Q; T � �+1;U � �+1 2X , and Y \� = � 2 � with card(Y ) = � > . Let � : Rg !X with �(V ;W ;M) =V ;W;M etc. Let � : Sg !Y , and : R ! S be given by = ��1 � �. By thearguments of Section 2 of [23] V is a countable, simple, ill-behaved tree on thecountable phalanx � = ((W;Q0); �) where �(�) = �;Q0 = Ult(Q;E| � �), whereQ is some initial segment of MU� as, by elementarity of �, �(Q) = Q. Noten(Q; �) = n(Q; �). Consider the pair of elementary maps�0 : W ! (W ) = ��1(W ); �1 : Q0 ! (Q0) = ��1( eQ)given by � W; � Q0 respectively. As V is bad on �, if we use the elementarypreserving maps � ��0; �1, V can be copied to a bad tree on ((W; (Q0); �) (with� = ��1(�) = (�). Further (Q0) = Ultn(Q0; (E| � �)) = Ultn(Q0; Ej � �)where Q0 is an initial segment of �(MU� = ��1(MU� ). As � 2 [ ; �]U we know thathMU� ; h�U�;�ii = Lim�! hhMU� i�<U�; h�U�;�0 i�<U�0<U�i. So MU� = ��1(MU� ). Thus Q0is an initial segment of MU� with a badly behaved tree on �0 = (W;Ultn(Q;Ej ��); �).As � 2 � and is on the main branch to � we have by the sublemma that + 1 2 DU , where � <U + 1 <U � and � is the immediate U-predecessor of + 1. This contradicts our de�nition of . Q.E.D.(Claim 0)

20 Some remarks on the maximality of Inner ModelsClaim 1 �U0;�\� 6� �.Proof: Suppose this failed. Then there can be no truncation at all on the branchb = [0; �]U . This again is obvious by the Dodd-Jensen Lemma (cf. [19] 5.3) ifall such drops are standard ones, as MT0 = W has cardinality �. But supposef ; + 1g is the special drop on b, where < � � �. Without loss of generalitywe can assume that f + 1g = DU \ b. Then MU +1 is a structure of size < �.If the Claim fails then we must have that � < �; that is, MU� is a proper initialsegment of MT� (here assumed to be just W ). Also if eQ = Ultn(MU +1; Ej � ),by speci�cation we must have that ((W; eQ); ) has a badly behaved tree on it.But as �U +1;� is de�ned on all of MU +1, and as we are able to embed eQ intoW 0 = Ultn(MT� ; Ej � �), we should conclude there is a badly behaved tree on((W;W 0); �). But as above this is absurd as W 0 = W in our assumed situationof T 's triviality! Hence MU� is a weasel.By Claim 0a) we may form the coiteration of W with ((W;fW ); �) wherefW = Ult(MU� ; Ej � �), via trees V on W , and W on the phalanx ((W;fW ); �),with corresponding maps �Vi;j ; �Wi;j with �nal models MV1 = P;MW1 = Q. Let kbe the ultrapower map above k : MU� ! fW . (By our temporary assumption onthe triviality of T , here k � j � �.) As k � �U0;� : W ! fW is an elementary mapof the universal weasel W into fW , irrespective of whether the �nal model Q ofW is above W , or fW , by Dodd-Jensen we must have that P = Q, and there hasbeen no dropping of any kind on either side. But then:(2) The main branch b of W is b = [1;1]W ; that is, Q is above fW not W .Proof: If Q were above W , we should have �W0;1 : W ! Q, �V0;1 : W ! Q,and �W0;1\Def (W;A0) = Def (Q;A0) = �V0;1\Def (W;A0) (in the notation of[23] Section 5). As JK� � Def (W;A0) this would imply =df crit(�V0;1) =crit(�W0;1) < �, and if E;F are the �rst extenders used on the branches [0;1]W ,[0;1]V respectively, then = crit(E) = crit(F ) and if � = minf�(E); �(F )gthen E � � = F � � (cf. the argument of 5.1 of [23]). But this can never happenin such a comparison process. So Q is above fW . Q.E.D.(2)Set = crit(�V0;1). Set l = �W0;1 � k � �U0;�. Then crit(l) � crit(k) = � =dfcrit(j). Then l\Def (W;A0) = Def (Q;A0) = �V0;1\Def (W;A0). By the de�-nition of W , JK� � Def (W;A0); we thus conclude l � � = �V0;1 � �. Butcrit(l) � crit(k) = � < �. So = crit(�V0;1) = crit(l) � � < �. We completeClaim 1 by showing that �U0;�( ) � �. As crit(�W1;1) � � (as now P = Q is abovefW ), and k\� � �, it su�ces to show that �V0;1( ) � �. But �V0;1( ) < � isabsurd, since if E were the �rst extender used on the main branch, we shouldhave = crit(E) < �(E) < �V0;1( ) < �, whilst fW and W agree up to �!Q.E.D.(Claim 1)

Some remarks on the maximality of Inner Models 21Just as in Theorem 19 we have (+) 9i0; < � �Ui0;�( ) � �.Claim 2 (i) & (ii) of Claim 2 of Theorem 19 hold.Proof: (i). Suppose � < �. Just as in Theorem 19, for this to happen an exten-der EUi would have to have been used with length larger than �; but U only usesextenders of length smaller than �. Again (ii) is standard. Q.E.D.(Claim 2)We then have the same phenomenon occurring as in Theorem 19: by thedi�ering co�nalities argument there is some least initial segment N of the �'thmodel of U , MU� , over which we can de�nably collapse �. By Claim 0 b) - with thisN in place of Q there, n(N; �), the least n with �n+1N = � < � � �nN exists, so weshall be able to perform the comparison of ((W;P ); �) with W , via n-maximaltrees R;S, where P = Ultn(N;Ej � �), P the �ne structural lift-up ultrapowerof N as before. j is co�nal from � into �0 = j(�) = �+K by assumption. Then + 1'st mastercode An+1P is de�nable over the �nal model MR1, if the latter isabove the model P rather than above W . If this is so, �R1;1 : P !MR1, and themap is without any drops and is �n+1-preserving. There is thus a code for thestructure P in W ; but then �0 is de�nably collapsed in W , whilst �0 = �+W ! Toshow that MR1 is above P is a standard argument; as otherwise we must haveMR1 = MS1, by Dodd-Jensen and properties of W . Then �R0;1 = �S0;1 and weshould argue that we have compatibility of the �rst extenders used on either side- as usual a contradiction. This su�ces.We now consider the complications involved in removing the assumptionthat T was trivial. Essentially no new ideas are needed. Just as in the conclu-sion of Theorem 19, if MU� were a weasel we should want to compare a lift-upof MU� ! W where now instead of comparing the phalanx ((W;fW ); �) with W(where fW was Ult(MU� ; Ej � �)) we have to take into consideration the ultra-powers that have been taken in T , as MU� �MT� . We should thus want to copyT to jT with intermediate copy maps j� : MT� ! M jT� for � < �. We shallcall j� ~| and now shall lift-up MU� using E~| � �. We set fW = Ult(MU� ; E~|) andcompare �(jT )bhfW;�i with �(jT ). This will also require some adjustments tothe comparison process, since we shall now be looking ahead to see if there arebadly behaved trees on �(jT � � + 1)bhUlt(MU� ; Ej� � �); �i for those � 2 �(that are additionally now closed under j�). We add these features to the com-parison de�nition: Case I also holds if j�\� 6� �.Case IIa)MU� is a weasel, and fM = Ult(MU� ; Ej� � �) yields a badly behaved treeon �(jT � � + 1)bhfM;�i where MU� ;MT� agree up to � � �; j�\� � �; � 2 �.The arguments given, but now replacing the phalanx of length 1, W , with�(jT � � + 1) shows there is a countable � 2 R; �(�) = �(jT � � + 1), with Va bad tree on �bhM; i and so on. We then have:

22 Some remarks on the maximality of Inner Models(3) There is a bad tree on �(jT � � + 1)bh eQ; �i where eQ = Ult(Q;Ej� � �).The remark in this case also holds.Remark: 2 If � = � in the above, and is such that Case IIa) hypotheseshold, and that T =df f� < � j � 2 � ^ j�\� � �g is Mahlo in �, then there isa � < �; � 2 T , � 2 [0; �)U with the property that there is a bad tree on thephalanx �(jT � � + 1)bhfM; �i where fM = Ult(MU� ; Ej� � �).The revamped Claim 0 reads:Claim 3a) If fW =df Ult(MU� ; E � ~| � �) is a weasel, then �(jT )bhfW;�i is coit-erable with �(jT ).b) If Q is a proper initial segment ofMU� , n = n(Q; �) < !, and eQ = Ultn(Q;E~| ��) then �(jT )bh( eQ;n); �i is coiterable with �(jT ).The proof of Claim 3a) goes through using remark 2. For part b), the Sub-lemma now reads:Sublemma Suppose MU� has size < � but there is Q, a proper initial segmentof MU� with a badly behaved tree on �(jT � � + 1)bh( eQ;n)); �i where eQ =Ultn(Q;Ej� � �) and n = n(Q; �) =df minfm j �Qm+1 � �g exists. Supposefurther � 2 [0; �]U with � the U-predecessor of +1 with � <U +1 <U �. Then(MU +1)� is a proper initial segment of Q. Hence + 1 2 DU .In the proof of this, using j� � � = j� � � the conclusion there now saysthat �(jT ) � �bhfM;�i = �(jT ) has a badly behaved tree on it (where fM =Ult(MT� � � + 1; Ej � �)). But this can be construed as simply the derivedphalanx from a tree on the iterable W , and so there can be no badly behavedtrees on this! The conclusion that + 1 2 DU follows as before.We take appropriate substructures X � Y � V+1 as before with now Va badly behaved tree on �(jT )bh eQ; �i, and both the latter in X . If �(�) =�(jT ), instead of a single �0, we have for our copy construction �0i =df �W i; �0i : W i ! (W i) for i < lh(�). Thus:�0i : �� ��1(�(jT )) ; �1 : Q0 ! ��1( eQ):Q.E.D.(Claim 3)Claim 4 �U0;�\� 6� �; i.e.Claim 1 above still holds.Proof: of Claim 4. We argued that if the Claim failed there could be no drop-ping on b = [0; �]U : even if the latter had special drops, as MU� would be aproper initial segment of MT� , we could embed eQ into Ult(MT� ; Ej � �). In ourcurrent situation we should say that we could embed eQ = Ultn(MU +1; Ej � )into W 0 =df Ult(MT� ; E~| � �) and conclude there is a badly behaved tree on

Some remarks on the maximality of Inner Models 23�(jT )bhW 0; �i. This is again an absurdity as such a tree can be construed asone on the �rst member of �(jT ) which is the iterable W itself! Continuing byClaim 3a) we may form the coiteration of �(jT )bhfW;�i with �(jT ) via treesW and V respectively. With the notation above k � ~| � � (rather than justj � �), the argument that the �nal models P = MW1 and Q = MV1 are equal isas before. We now have:(4) The main branch b of W is b = [� + 1;1]W ; that is, root(P ) = � + 1 and Pis above fW .Proof: (Note here, lh(�(jT )) = � + 1, and fW is then the � + 1'st model ofthe starting phalanx �(jT )bhfW;�i.) Suppose rootb(P ) = �; rootc(P ) = �0 with�; �0 � �. Then if E;F are the �rst extenders used on [�; �]W and [�0; �]V , bothlh(E); lh(F ) > �, that is, E;F do not come from any extender used in buildingjT .Subclaim � = �0.Proof: If not, without loss of generality, say � < �0, and let � be the greateststage with � �jT �0; �. Then � <jT �0. A similar argument over the de�nabilityhulls shows: (�W�;1��jT0;�)\Def (W;A0) = Def (P;A0) = (�V�0;1��jT0;�0)\Def (W;A0).Hence �W�;1 � �jT�;� � �jT0;� = �V�0;1 � �jT�;�0 � �jT0;� . If G is the �rst extender used onthe path [�; �0]jT then crit(G) = crit(�W�;1 � �jT�;�). If � = � then crit(G) =crit(�W�;1), which contradicts E 2 W as the �rst extender used with crit(E) =crit(�W�;1), since then G is compatible with E. But if � < � and H is the �rstextender used on [�; �]jT , then again crit(G) = crit(H), G � �h 6= H � �H andthen they must be incompatible (cf. [19] p.49 at a)), and the overall maps intoW must di�er on the two sides. Q.E.D.(Subclaim)But now �V�;1 = �W�;1, and so E;F are compatible. But this does not happenin comparisons! Hence rootb(P ) = � + 1. Q.E.D.(4)Set h = �V�;1 � �jT0;� (where � = root(P ) � �). Set = crit(h). As beforeset l = �W�+1;1 � k � �U0;�. Now k � � � ~| � �, but we conclude as before thatl � � = h � � and crit(l) � crit(j) < �.Subclaim h( ) � �.Proof: Suppose not. Notice that this implies that � = root(P ) = �. For, if� < � then the �rst extender used in [�;1]V , would have length � �, whilst itwould have critical point < �, and the Subclaim would be proven. We thus haveh � � = �jT0;� � � = l � � = k � �U0;� � �.As crit(h) = crit(l) = ; crit(h) = crit(E) where E = EjT� where � + 1 2[0; �]T ^ T -pred(� + 1) = 0, and thus E is the �rst extender used on the copiediteration on [0; �]jT . Let F = ET� . Let G = EU� be the corresponding �rst ex-

24 Some remarks on the maximality of Inner Modelstender used on the iteration U with U-pred(� + 1) = 0 ^ � + 1 2 [0; �]U . Thencrit(G) = also. Let � = inff�(F ); �(G)g. Then both(5) crit(�T�+1;�); crit(�U�+1;�) > �(6) j� � � = j�+1 � � = ~| � � = k � �.Also = crit(E) < � (as E = j� (F ) whilst � =2 ran(j� ) for any � < �). Hencefor X � 2 dom(j), j(X) = k(X) = X . Since G � �; F � � are incompatible onP( ) \MT0 \MU0 , we can �nd a 2 [�]<!; X � so that(7) a 2 �T0;�+1(X)() a 62 �U0;�+1(X):But by (6), the left hand side here holds if and only if:j� (a) = j�+1(a) 2 j� (�T0;�+1(X) \ �) = j�+1(�T0;�+1(X) \ �)= �jT0;�+1(j(X) \ j�+1(�)) (By the copy map construction )= h(X) \ j�+1(�)) = l(X \ �) (By (5))= j� (�U0;�+1(X) \ j� (�)): (By (5), (6))But then by (7) we have deduced a contradiction:j� (a) 2 j� (�U0;�+1(X) \ �)() a =2 �U0;�+1(X) Q.E.D.(Subclaim)But this now completes Claim 4: if h( ) � �, as h( ) = l( ), this can onlyarise if �U0;�( ) � �. Q.E.D.(Claim 4)Since U only uses extenders of length < � we have that Claim 2 above stillholds, and we can �nish the argument using the di�ering co�nalities of the \�+'s"in the di�erent models as outlined above.For the second part of the theorem argue just as in Theorem 20, as again weknow some point is sent up to �. Q.E.D.(Theorem 21)We remark that by the argument above of the second part, which is reallyabout the failure of K-condensation, there will be weaker cardinals than J�onsson(such as the \�-J�onsson " of Def. 8 above) below which many successors of reg-ular cardinals will also be correctly computed in K.

Some remarks on the maximality of Inner Models 25We conclude with some further open problems.Open Questions4) Does ZFC ` regular J�onsson �! 8A � A# exists? [True if a) Ramsey,or, b) No # for IM(Strong) see Theorem 12 above. False if singular.]5) Does ZFC ` strongly inaccessible J�onsson =) weakly compact? [Con-jecture : No.]6) Is there a forcing notion that kills o� the Ramsey property whilst preserv-ing the J�onsson property and strong inaccessibility? [ccc forcing preservesJ�onssonness so the emphasis here is on strong inaccessibility.]7) Let be regular J�onsson, K = KSteel. Does V = V K �! V +1 = V K +1?[True for Kstrong.]8) Does regular J�onsson +:2 =) IM(Woodin)? [NB < = =) 2 .]9) Suppose no IM(Woodin). Let � be J�onsson. Show that there is a regularcardinal < � such that 2 holds. [The results here do not quite give this:it is another question about removing the measurable (or larger cardinal)needed for the K construction.]10) Is it consistent, relative to large cardinals, that the �rst regular for which2 holds be greater than (or equal) to the �rst J�onsson ?References1. K. Devlin & R.B. Jensen Marginalia to a theorem of Silver in Proc. of Kiel LogicConference 1974, Lecture Notes in Mathematics, 499, Springer Verlag, Berlin-NewYork, 1975, 115-142.2. A.J. Dodd, The Core Model, Lond. Math. Soc. Lecture Notes in Maths, 61, CUP,1982.3. H-D. Donder, R.B. Jensen, & B.J.Koppelberg, Some applications of the CoreModel, in Set Theory and Model Theory, Springer Lecture Notes in Maths., 872,1981, 55-97.4. H-D. Donder & P.Koepke, On the Consistency strength of `accessible' J�onssoncardinals and of the weak Chang conjecture, Annals of Pure & Applied Logic, 25(1983), 233-261.5. R.B. Jensen, The Fine Structure of the Constructible Hierarchy, in Annals of Math-ematical Logic, 4, (1972), 229-308.6. , The Core Model for Measures of Order Zero, circulated manuscript, 1989.7. , Non-overlapping Extenders, circulated manuscript, 199?.8. , Some remarks on 2 below O{, circulated manuscript.9. R.B. Jensen & J. Vickers, Absoluteness of J�onsson Cardinals, accepted for the J.Symbolic Logic.10. R.B. Jensen & M. Zeman, Smooth categories and global 2, submitted to Annals ofPure & Applied Logic.11. A. Kanamori, The Higher In�nite, Perspectives in Mathematical Logic, 1994,Springer Verlag, Berlin.12. P. Koepke, Some applications of short core models, in Annals of Pure & AppliedLogic 37, (1988), 179-204.

26 Some remarks on the maximality of Inner Models13. K. Kunen, Elementary Embeddings and In�nitary Combinatorics, in J. of SymbolicLogic 36, (1971), 407-413.14. A. Miller, Some Interesting Problems, in Set Theory of the Reals, Ed. H. Judah,Israeli Mathematical Conference Proceedings, 6 , 1993, 645-654, Amer. Math. Soc.or http://www.math.wisc.edu/ miller15. W.J. Mitchell, Ramsey cardinals and Constructibility, in J. of Symbolic Logic, 44,1979, 260-266.16. , J�onsson Cardinals, Erd}os Cardinals, and the Core Model submitted to theJ. for Symbolic Logic.17. W.J. Mitchell & E. Schimmerling,Weak Covering without countable closure, Math.Res. Lett. 2 (1995), 595-609.18. W.J. Mitchell, E. Schimmerling, & J.R. Steel, The Covering Lemma up to a Woodincardinal , Annals of Oure & Applied Logic, 84, (1997), 219-255.19. W.J. Mitchell & J.R. Steel, Fine Structure and Iteration Trees, Springer LectureNotes in Logic, 3 Springer Verlag, 1994.20. E. Schimmerling, 2<!� holds in L[E], submitted to the J. of Symbolic Logic.21. E. Schimmerling, & J.R.Steel, On the Maximality of the Core Model. to appear inthe Trans. of Amer. Math. Soc.22. S. Shelah, Cardinal Arithmetic, Oxford Logic Guides 29, (1994), OUP.23. J.R. Steel, The Core Model Iterability Problem, Springer Lecture Notes in Logic,8 Springer Verlag, 1996.24. J.R. Steel & P.D. Welch, �13-Absoluteness and the Second Uniform Indiscernible,in Israeli J. of Math. 104 (1998), 157-190.25. A. Suzuki, Submitted to the J. of Symbolic Logic.26. J. Vickers & P.D. Welch, Elementary Embeddings of a submodel into the Universe,submitted to the J.Symbolic Logic.27. , On Successors of J�onsson cardinals, to appear in the Archive for Mathe-matical Logic.28. P.D. Welch, Combinatorial principles in the Core Model, D.Phil. thesis, Oxford1979.29. D. Wylie, Ph.D. thesis, MIT. ,1990.30. M. Zeman, Global 2 below O{, in preparation.31. , Towards 2� in Higher Core Models, manuscript, 1998.