ON ELEMENTARY EMBEDDINGS FROM AN INNER ... - CiteSeerX
The Journal of Symbolic Logic Volume 00, Number 0, XXX 0000
ON ELEMENTARY EMBEDDINGS FROM AN INNER MODEL TO THE UNIVERSE
J. VICKERS AND P. D. WELCH
Abstract. We consider the following question of Kunen: Does Con(ZFC + ∃M a transitive inner model and a non-trivial elementary embedding j : M −→ V ) imply Con(ZFC + ∃ a measurable cardinal)? We use core model theory to investigate consequences of the existence of such a j : M → V . We prove, amongst other things, the existence of such an embedding implies that the core model K is a model of “there exists a proper class of almost Ramsey cardinals”. Conversely, if On is Ramsey, then such a j, M are definable. We construe this as a negative answer to the question above. We consider further the consequences of strengthening the closure assumption on j to having various classes of fixed points.
§1. Introduction. It is quite natural to study the properties of elementary embeddings j : V −→ M for M some inner model, since many such embeddings, if they exist, have first order formulations within ZFC . The question of reversing the arrow and looking at a non-trivial j : M −→ V in general does not readily admit of such formulations. So we study in this paper what might be considered the ZFC consequences of the second order statement that there are proper classes j, M such that . . . (This was the formulation of Kunen’s question as it appears in [7], but of course, if the reader does not like this conceit, then he or she may simply translate everything into properties of elementary submodels of some Vë |= ZFC , by replacing V by Vë for some inaccessible ë. Hence we could equally well be motivated by investigating the possible elementary substructures of some Vë of size ë.) It is not hard to see that some sort of hypothesis beyond ZFC is needed to generate such situations. If there were a non-trivial j : M −→ V , then j ↾ L : L −→e L is non-trivial and so O # must exist. Indeed, by considering critical points, one can see that such an embedding could not be definable by class terms, even from parameters, or there would be an infinite regress of ordinals! Obviously more can be extracted from this, and it is natural to ask how much. As ´ the hypothesis is similar to a statement about On possessing Jonsson or Ramsey like properties, it is not surprising that it implies some kind of large cardinal property. Whatever the “core” of V is, denoting it by K, then j ↾ K M : K M −→ K V and we may ask what kind of K must there be for this to happen. Kunen’s question Received July 16, 1998; revised January 1, 2000 c 0000, Association for Symbolic Logic
0022-4812/00/0000-0000/$00.00
1
2
J. VICKERS AND P. D. WELCH
(Question 15.8 of [7]) in this vein was: can we extract an inner model with a measurable cardinal? Or more precisely, does Con(ZFC{j} ˙ + ∃j : M −→ V , j 6= id , j elementary) ⇒ Con(ZFC + ∃ a measurable cardinal )? (Here ZFC{j} ˙ means ZFC in L{∈, a language with a predicate symbol for j, whilst j : M −→ V is ˙ ˙ j} considered to be Σ1 -elementary in L{∈} ˙ alone. (The latter implies full elementarity.) This question is easily answered however. Perhaps more interestingly, in § 1.1 we consider the consequences of the existence of such elementary embeddings from an M into V , varying the basic question by adding closure requirements on j to see what K must be like. From just the existence of such a j we get (Theorem 1.4) that K must contain a proper class of “almost Ramsey” cardinal (defined below). We get full Ramseyness (Theorem 1.6) by asking that dom(j) have a class of regular fixed points. By insisting that dom(j) contain a stationary class of singular fixed points of cofinality ù, then we obtain that K must have a Mahlo class of measurables; if the fixed points have uncountable cofinality, or if M computes correctly countable cofinalities, then O sword exists (Theorem 1.10) (that is, Jensen’s core model for measures of order zero - see [3], is non-rigid). We should like to thank M. Foreman for saving us from a demonstrably inconsistent hypothesis for Theorem 1.10 (we had assumed that M was closed under ù-sequences); and for allowing us to include his counterexample as Theorem 2.4. In § 2 we make a first attempt at some converses by showing that if On is Ramsey, then we may define a j, M with j : M −→e V , j 6= id . We could formulate this as a statement about Vκ , where κ is a Ramsey cardinal, and show that definably over hVκ , ∈, I i (where I is a class of good indiscernibles for hVκ , ∈i coming from the Ramsey property) there are classes j, M satisfying our requirements. We thus have Con(ZFC + ∃ a Ramsey cardinal ) ⇒ Con(ZFC{j} ˙ + j : M −→e V, j 6= id ). Although this is not an exact equiconsistency, it could be considered close. However, overall in this direction we have left several upper bounds open. We make some comments on results obtainable by Woodin, using the stationary tower forcing. (We should like to thank the referee for prompting us to make such comparisons.) Woodin has shown [11], that if κ is a measurable cardinal, and Pκ the full stationary tower forcing, then if G is Pκ generic over V , then we do have an embedding j : Vκ −→ Vκ [G]. (Below the measurable κ there is an unbounded ´ sequence of completely Jonsson cardinals, fixed by j.) However in this situation, we cannot have that the structure hVκ [G], j, ∈i |= ZFC{j} ˙ . That is, we cannot put back the predicate Vκ . (For example, note that if V = L[ì] with ì a normal measure on κ, then Vκ |= “V = K” (with K the Dodd-Jensen core model), then V [G] is a model of the same, and hence some new “mouse” has been added, that must iterate past all those of Vκ !) This, then is slightly different from our requirements that we have hV, j, ∈i |= ZFC{j} ˙ . If we regard the relationship of V to a forcing extension as being “close”, then perhaps we need the extra strength in order to obtain such a close relationship between the “M ” and “V ”. Our results do not require such strength since our “M ” and “V ” are not at all close in this sense. In § 3.1 we reprove the Jensen Indiscernibles Lemma [2], originally done using the old-style fine-structure of the Dodd-Jensen KDJ , but now for the modern hierarchy and the K under consideration. The reasons for this are that, corresponding to ˝ (if cf(α) > ù), Jensen’s results for the Dodd-Jensen K, this shows that the α-Erdos
ON ELEMENTARY EMBEDDINGS FROM AN INNER MODEL TO THE UNIVERSE
3
almost Ramsey, Ramsey etc. properties are downwards absolute between V and any universal weasel. Secondly, this actually also gives a second proof of Theorems 1.4 and 1.6 which are proven in § 1 by using a coarse “lift-up” technique. Our results are all in the region of the consistency strength of inner models with measures of order zero, K MOZ so it is appropriate to use Jensen’s model of [3] in order to discuss them. This model will be K throughout the paper. (Although the Kunen question can be resolved with just the Dodd-Jensen K DJ , or even the Σ1 -K of [10].) We assume familiarity with the measures of order zero K, and the techniques of that paper, some of which appear in [5]. We use the strong Covering Lemma for K, [9], [4] on singular cardinals â of V being either singular, or possibly measurable in K if cf(â) = ù. Mostly however it is the theory of iteration and comparison that is useful, and fine-structural results have been hived off to § 3. In § 3.2 we sketch the outlines of Jensen’s Σ∗ fine structure theory in a general setting. It is a relatively bare-bones outline, but all the essential definitions are here. What we have omitted is a full discussion of Σ(n) 1 relations and functions, and a fuller analysis of good parameters, and of witnesses. The intention here is to give an introduction to the theory, and, in § 3.3, to that of the Σ∗ ultrapower, to provide a modicum of self-containedness for the proof of the fine-structural Upwards Extensions of Embedding Lemma of § 3.4 which is needed for Theorem 1.10. This has essentially been extracted from various parts of §3.3 of [3], but we have thought it worthwhile to give a complete version here, as firstly, the reader may not wish to do it for his or herself, (and anyway such arguments are not often in print), and secondly this is a form of “lift-up” which we wish to use elsewhere. We should like to emphasize our great debt to Jensen’s work, and in particular to [3] Some prerequisites The set-theoretic notations are standard. [6] gives an account of the basics of general elementary embeddings. Our notation for iterations of mice is that of Jensen’s: an iteration of a mouse M is of the form M I = hhMi ihði,j ihíi , αi i | i < èi with models Mi and indices híi , αi i and commuting M maps ði,j : Mi −→ Mj , with ùíi ≤ ùαi ≤ On ∩ Mi and if Ei 6= ∅ (the íi ’th filter M : M |αi −→Eíi Mi+1 where M |â denotes on the E Mi filter sequence), then ði,i+1 M
M hJâM , E M , Eùâ i. ( M ||â will denote hJâE , E M , Øi), or when there is no danger of M = id ↾ M |αi . Direct limits are taken confusion, simply |JâE |.) If Eíi = Ø then ði,i+1 as needed. A truncation occurs when ùαi < On ∩ Mi . We truncate if need be in order to allow a filter to become a full measure over the subsets of its critical point in M |αi , and thus are able to take an ultrapower. It is a fundamental fact that in any iteration only finitely many truncations can occur. The coiteration of two mice or “weasels” (= proper class mice) is performed by “iterating away points of least difference”. This results in a terminating double iteration of length some è ≤ On, with either Mè = Nè , Mè ∈ Nè , or Nè = Mè . We shall use the following well-known lemma: Basic Coiteration Lemma If (M, N ) are premice, and they are coiterated to models (Mè , Nè ) with iterations N M ii≤j≤è ii, hhNi ii≤è , hαiN ii ù then O s exists. Proof. As in the proof of the last theorem, it would suffice for a contradiction to show that K M is universal. But j(ô) = ô so ô is singular in K by the Covering lemmas of [9] and [4]. So K M |= ô is singular. The result follows by Lemma 1.8. ⊣ M Theorem 1.10. If ∃j : M −→ e V, j 6= id and ∀α(cf(α) = ù ⇒ cf (α) = ù) s then O exists. Proof. We first note that the assumption implies that there is a stationary class S of cardinals ô with cf(ô) = ù which are fixed points of j. Consider the comparison of K with K M with resulting coiteration hhNi , KiM i|i ≤ èi for some è ≤ On, with indices híi |i < èi. Let the critical points of the ultrapowers used be hκi |i < èi. Let α = crit(j). Clearly cf(α) > ù. (1) On the K side of the coiteration, the very first index used, í0 , is less than K + α =df α + , and hence involves a truncation to a set mouse N0 = K|í0 . Proof. Suppose the first index used on the K side were í0 ≥ α + . Then α + = KM α+ and by our assumption, and the Covering Lemma in M , cf(α + ) > ù. M Then P (α)K = P (α)K and if U is the derived measure from the embedding j e =df (meaning that U = {X ⊆ α | X ∈ K ∧α ∈ j(X )}), then consider ð : K −→ K + +K e Ult(K, U ). K is wellfounded as α = α , and U ù-complete. As we are below O ¶ , ð is an iteration map, ð0,ì for an iteration by full measures hEí |í ≤ ìi. The first measure, E0 , used in this iteration must have critical point α = crit(U ), and so E0 = EαK+ ; and it agrees with U on P (α) ∩ K; so U must be EαK+ . ′ Let ó : K M −→ K ′ =df Ult(K M , U ), then EαK+ = ∅, by coherence. Defining k : K ′ −→Σ0 K by k([f]U ) = j(f)(α) we have as usual that k ↾ α + + 1 = id ↾ α + + 1, ′ whilst EαK+ 6= ∅, EαK+ = ∅ - a contradiction. ⊣
9
ON ELEMENTARY EMBEDDINGS FROM AN INNER MODEL TO THE UNIVERSE
We have as usual a cub class C and N1 iterating past K M , with ðNi Nk (κi ) = κk for i < k ∈ C , whilst ðK0M ,KiM “κi ⊆ κi . However C ∩ S cannot form a stationary class of cardinals singular in K M , as then we should have K M universal - a contradiction as in the last theorem. Again the only other possibility is that it forms a class of measurables of K M . Let ù2 < κ ∈ C ∩ S with κ greater than the last truncation point on the iteration KM on the K side. Let κ ′ = κ + κ . Then KκM |κ ′ is an initial segment of Nκ , and the latter is sound above κ. Then κ ′ = κ + ù (2) ñN < κ, and κ > cf(κ ′ ) > ù. κ
Nκ
= sup ðK M ,KκM “κ +
KM
. Note
Proof. By (1) there is at least one truncation on the K-side, and from this the n first clause follows: since from the last truncation point i0 say, we have ñN ≤ κi0 io n for some n. For this n we have ñNκ < κ. As M calculates countable cofinalities correctly, by the Covering Lemma in M , cf(κ +
KM
M
) = cf(κ + ) > ù.
⊣
hKiM
Let the iteration | i ≤ κi with indices híi |i < κi be copied (as in Claim 1 of e 0 as hK e i | i ≤ κi with commuting Theorem 1.4) via j = ó0 to an iteration of K = K i M e maps ói : Ki −→e K and indices hói (íi )|i < κi. e κ |ä. Note that óκ (κ ′ ) = κ +Keκ = Let ä = sup óκ “κ ′ . Then óκ : KκM |κ ′ −→Σ0 K +K +V κ = κ by the Covering Lemma in V , and the fact that the copy iteration below eκ κ fixes κ and κ +K . So we see ä < κ +K . We now have to do a fine-structural lift-up argument, as this time we know that n+1 n < κ < ñN ñN for some n ≥ 0. Fix this n. κ κ f for some mouse M f with óe ⊇ óκ ↾ (3) There is a lift-up map óe : Nκ −→Σ∗ M n+1 n M ′ κ f f e Kκ |κ where M has the following properties: (i) M |ä = K |ä; (ii) ñM e < ä ≤ ñMe ; f is the Σ(n) -closure of κ ∪ {p} for some p ∈ Rn+1 . (iii) M 1
e
M
Proof. This is a somewhat standard argument (c.f., [3] §3.3.2) exploiting the fact that cf(κ ′ ) > ù to get a wellfounded fine-structural “pseudo-ultrapower”. Instead of the coarse pseudo-ultrapower of Claim 3 of Theorem 1.4 we define Nκ D = {ha, fi | dom(f) ⊂ ô < κ ′ and either f ∈ Σ(m) 1 (Nκ )(m < n) ∨ f ∈ Hñ n } Nκ
and then set e ∈ ha, fi hb, gi ⇐⇒ ha, bi ∈ óκ ({hu, vi | f(u) g(v)} I = and ˙ G(ha, bi) ⇐⇒ a ∈ óκ ({u | f(u) ∈ G}) (for G ∈ {F, E}). We then define D = hD, I, e, E˙ , F˙ i and proceed to prove a Łos Theorem for Σ0 formulae. The fact that cf(κ ′ ) > ù results in e being wellfounded, and then D is f for some premouse M f. The canonical embedding óe : Nκ −→ M f isomorphic to M f is iterable, and so a mouse, is an is defined by óe(x) = [h0, {hx, 0i}i]I . That M amplification of the argument that it is wellfounded (see [3] §3.3.3). Using that κ ′ is the Nκ -successor of κ enables one to prove that óe is a Σ(n) 1 preserving embedding, and ultimately properties (ii) and (iii) ((i) holds by construction). The full proof is in § 3.4. ⊣
10
J. VICKERS AND P. D. WELCH
f∈K e κ ; thus ñ ù = ñ n+1 = κ. (4) M e e M M
f f eκ Proof. There is a Σ(n) 1 (M ) subset of κ that codes M . As K is universal (being just a simple iterate of K), in their coiteration the first point of difference between f and K e κ is not less than ä, so we see that this code is in K e κ . As κ is obviously a M n+1 κ ù e cardinal of K we must have ñMe = ñ e = κ. ⊣ M
f) is an initial segment of K e κ . core(M f)|ä + 1 = M f|ä + 1; hence (5) core(M κ e e M K Eä = Eä .
e (n)M n+1 f). Then P ∼ Proof. Let P =df core(M = Σ1 Hull(ùñM e ∪ {pMe }). (Where pMe e M f f is the standard parameter of M .) But as Eä 6= Ø, M is an iterate of P above ä. Let e κ |ã. Then N ||ä = M f||ä = P||ä. But ã ≥ ä be least so that ñ ù ≤ κ. Let N = K e
K κ |ã
over both P and N there are Σ∗ definable subsets of κ coding wellorders of type ä. As ä = κ + (or is the height of the ordinals) in both models, neither model contains such a code. So a simple comparison argument shows that P and N coiterate without truncation to the same model. But then by soundness, in fact P = N . ⊣ f is Remark. With a little more work on parameters, one can show that in fact M eκ . itself sound and an initial segment of K KM
KM
S
Let F = Eκ′κ . Let M = |KκM ||κ ′ | = JκE′ κ . Let F be defined by: F = X ∈M óκ (X ∩ F ). Then: e κ |ä, E Keκ i ∧ F = E Keκ . (6) óκ extends to: óκ ↾ hM , F i −→Σ0 hK ä ä e κ ||ä|. And F = |K e κ ||ä| ∩ E Keκ ′ . By the initial Proof. óκ is a cofinal map into |K óκ (κ )
e
κ
segment condition F = EäK .
e
e
κ
⊣
KκM κ′
(6) shows, as ó˜ ⊇ óκ , that EäK = EäM ⇒ E KκM κ′
= EκN′κ . So the conclusion of this
consequence holds by (5). But E 6= EκN′κ by virtue of the comparison process a contradiction! ⊣ Let K ′ be a core model for sequences of measures with o(κ) < (κ)++ (for example Mitchell’s Core Model). Question. If ∃j : M −→ e V and j 6= id , α is a regular cardinal, and there exists stationary many cardinals of cofinality α fixed by j then: (a) does K M |= “∃ unboundedly many measurable cardinals κ with o(κ) = α” (where we set o(κ) = 0 if α = ù)? (b) if M computes correctly cofinalities ≤ α then is this consistent? §2. Some converses. We briefly consider here some partial converses to the above. It might be thought at first that with some extra work, one could obtain from a non-trivial elementary embedding j : M → V , an inner model with a measurable cardinal. The following indicates that this is not possible. Suppose that κ is a ´ strongly inaccessible Jonsson cardinal, and that there is A ⊆ κ with Hë = Lë [A] ´ for cardinals ë < κ, and (by the Jonsson property) that there is a non-trivial j : Lκ [A] −→ Lκ [A], then we clearly have candidates for a “j : M −→ V ” by ´ cardinal is known to taking M = Lκ [A] and V as Lκ [A]. And such a Jonsson be equiconsistent with a Ramsey cardinal. See [6]. Notice also that this argument
ON ELEMENTARY EMBEDDINGS FROM AN INNER MODEL TO THE UNIVERSE
11
shows that if κ, j : Lκ [A] −→ Lκ [A] are as above, then we can perform a “lift-up” argument (as in Claim 3 of Theorem 1.4) to extend j to j˜ : L[A] −→ L[A] (using the regularity of κ). We can also think of this argument as taking an extender ultrapower of L[A], Ult(L[A], Ej ) where Ej is the long extender derived from the embedding j; by the regularity of κ this is wellfounded. But then V cannot be L[A] since otherwise Ej ∈ V and the latter would be an ultrapower of the inner model L[A]! Hence: Lemma 2.1. Suppose V = L[A] for some set A ⊆ κ with κ strongly inaccessible. (i) If j : M −→ V , (or even if j ↾ Lκ [A] −→ Lκ [A]) then j = id . (ii) κ is not a J´onsson cardinal. ´ Instead of arguing in terms of the Jonsson property, one can frame a hypothesis of the form “Suppose On is Ramsey ....” (or again reformulation in terms of some Vκ where κ is a Ramsey cardinal etc.). Definition 2.2. (i) On is Ramsey if there is a class I ⊆ On, unbounded, of good indiscernibles for hVκ , ∈, I i, (ii) κ = On is 1-Ramsey, if the class I in (i) is Mahlo. In (ii) we mean that I has non-empty intersection with any closed and unbounded class C ⊂ On, definable (with parameters allowed) over hV, ∈, I i. Notice that if κ is a Ramsey cardinal, then we have an I ⊆ κ so that in hVκ , ∈ I i, On is Ramsey in this sense. Of course, if On is Ramsey we have, definably over hV, ∈, I i, a satisfaction relation for hV, ∈i, since α < â ∈ I ⇒ Vα ≺ Vâ . Further, using AC , we may define skolem functions, and so skolem hulls for hVα , ∈i, so that α < â ∈ I ⇒ HullVâ (ã) = HullVα (ã) for ã < α. A prototypical situation is again where κ is Ramsey in K and then all the above hold over hKκ , ∈ I i. Theorem 2.3. (i) Suppose I ⊆ On witnesses On is Ramsey. Then, definably over hV, ∈, I i, there is a transitive class M , and an elementary embedding j : hM, ∈i −→ hV, ∈i, with j 6= id . (ii) Additionally, if I witnesses On is 1-Ramsey, then j can be taken with unboundedly many regular fixed points. Proof. The proof of the proposition is straightforward. Let α0 = min I . By the remarks above we may define in hV, ∈, I i the hulls Hâ =df HullVâ ((I ∩â\{α0 })∪α0 ) for â ∈ I , and their transitive collapses with maps ðâ : Mâ −→ Hâ . By the usual properties of good indiscernibles, it is easy to verify that S crit ðâ = α0 ,Sand that α < â ∈ I ⇒ ðâ ↾ Mα = ðα . Hence we may set j = â∈I ðâ , M = â∈I Mâ . This concludes (i). For (ii), simply note that there must be, stationarily often, inaccessible ä ∈ I with j(ä) = ä. ⊣ Part (i) of the last theorem can be rephrased as a relative consistency result from the existence of a Ramsey cardinal. This shows the answer to the question of Kunen is negative. A number of questions about the possibilities of finding models which satisfy further closure properties are immediate. For example: Question 1. Can one arrange for there to be an inner model M and a non-trivial j : M −→ V , with stationary many fixed points of cofinality ≥ ù?
12
J. VICKERS AND P. D. WELCH
Question 2. The same question as for 1, but with M correctly computing the cofinality of ordinals â with cf(â) = ù? Our original Theorem 1.10 had the hypothesis that ù M ⊆ M . Matt Foreman pointed out that this was impossible. We thank him for letting us include his proof here. Theorem 2.4 (Foreman). If ∃j : M −→ e V, j 6= id then ù M 6⊆ M . Proof. Suppose for a contradiction that there exists such an M and j satisfying also the conclusion. Let α = crit(j). By the supposed closure on M it is easy to see that there are α < ë0 . . . < ën < . . . for n < ù with each ën a singular cardinal of cofinality ù and j(ën ) = ën . Then hën in κ, then ð is Σ2 -preserving. For ñ < κ N ð = id ↾ HM . ⊣ In the second stage we show that this interpretation is the correct one: i.e. Hi M does indeed equal Hùñ n . The point is now to show that the ñn , which were defined M from the Hn , are the “right” ordinals. This is stated in the next lemma. Actually for hierarchies involving measures, as here, stronger statements are provable (which are below). We have divided the preservations properties into two lemmas, as for hierarchies involving extenders further amenability assumptions are required to gain this extra strength. For measures we have in any case full amenability which we can exploit. n An embedding ð : M −→Σ(n) M is cofinal if ð is Σ(n) 0 preserving and HM = 0 n n n n . This implies that ð ↾ HM : HM −→Σ0 HM cofinally in the usual sense, ∪ð“HM and that ð : M −→Σ(n) M . 0
20
J. VICKERS AND P. D. WELCH
It only remains to show that the projecta above κ correspond to the ordinal ñn . Lemma 3.28.
m+1 m ñm = ñM if κ ≤ ùñM ;
m m . ñm ≤ ñM if κ ≤ ùñM
m Proof. By induction on m. This is trivial for m = 0. We show ñm ≤ ñM by showing that: (4) hHm , Ai is amenable for A ∈ P(ùñm )∩ Σ1(m−1) (M ). Suppose A ⊆ ùñm and is Σ(m−1),M ([ha, fi]) , say A(x) ⇐⇒ M |= ϕ(x, [ha, fi]) 1 (m−1) with ϕ ∈ Σ1 . Let w = [hb, gi] ∈ Hm . We need A ∩ w ∈ Hm . Let card(a) = n. Define h : [κ]n −→ HM m by h(u) = {t ∈ g(u) | M |= ϕ(t, f(u))}. Then h ∈ Γm . So [ha, hi] ∈ Hm , and using 3.23(iv) [ha, hi] = [hb, gi] ∩ A. m+1 m We now show ñm ≥ ñM if κ ≤ ùñM . We show:
(5) There is A ⊆ ùñm , A ∈ Σ1(m−1) (M ) with A ∈ / M. (m−1) (M ) definable in a parameter p. Suppose A is Σ1 Let A be Σ1(m−1) (M ) in ð(p) = p by the same definition. We show that A∩ùñm ∈ / M. Suppose otherwise. Let A ∩ ùñm = [ha, fi] ∈ M . Then A ∩ ùñm = x ⇐⇒ ∀z m (zm ∈ A ⇐⇒ z m ∈ x).
The right hand side here is a Π(m) 1 , in p, formula. M |= ϕ[[ha, fi], ð(p)] so {u|M |= ϕ[f(u), p]} ⊆ {u|f(u) = A} ∈ Ea , by 3.23 (iv). So ∃uf(u) = A ∈ M . A contradiction. ⊣ n+1 n n > κ. Then ð“PM ⊆ PM . Corollary 3.29. (i) Let ùñM ∗ (ii) ð : M −→E M ; in particular for f ∈ D , [f] = ð(f)(κ).
Proof. (i) This was shown in the last part of the last proof. For (ii) all that we have left to show is (iv) in Def 3.19, that M is the closure of ranð under good n+1 Σ(n) 1 -functions, for ùñM < κ. We’ve extended ð to such functions so it is enough to show M = {ð(f)(a) | f ∈ D , a ∈ [í]n , dom(f) = [κ]n }. By (3) above [cα ] = α for α < κ. As {u | M |= f(u) = f(id (u))} ∈ Ea by the Łos theorem for f ∈ Γ, we have D |= f = cf (id ) and so M |= [ha, fi] = ð(f)(a). ⊣ If we are using measures to take ultrapowers, we can build filter hierarchies with the measures fully amenable with respect to the structures concerned. This enables us to improve the last inequality of Lemma 3.28, and indeed, continue the analysis down below the critical point. However we shall continue to assume that we have an extender ultrapower and impose some additional requirements on E in order to do this. We now use weak amenability to enable us to improve the last lemma and corollary at the point where the projectum crosses the critical point. n+1 n+1 n+1 n > κ ≥ ùñM , and that P (ùñM )∩M = ùñM ∩M . Lemma 3.30. Suppose ùñM Then: n+1 n+1 n n ≤ ùñM = ùñM ; (iii) ùñM (i) ð : M −→Σ(n) M cofinally; (ii) ùñM . 0
Proof. Assume n > 0. n+1 Claim Let p ∈ PM . Then there is a set A ∈ Σ(n−1)M ({ð(p)}) such that A ⊆ ùñn 1 with A ∈ / M.
ON ELEMENTARY EMBEDDINGS FROM AN INNER MODEL TO THE UNIVERSE (n−1) (M ) and M Proof. Let A = Ap(n−1) n−1,p↾n−1 . Then A ∈ Σ1
n,p↾n
M
21
M = hHùñ n , Ai is
amenable. Let B ⊆ Hn be defined by the same Σ1(n−1) formula ϕ in parameter ð(p). M Let H = Hùñ n . ð ↾ H : H −→Σ0 Hn cofinally. SM (1) B = x∈H ð(x ∩ A). Proof. Pick x n ∈ H . Let y n = x n ∩ A ∈ H . Then
M |= ∀z n ∈ x n (z n ∈ y n ←→ ϕ(z n , p)) ⇐⇒ M |= ∀z n ∈ ð(x n )(z n ∈ ð(y n ) ←→ ϕ(z n , ð(p)) (As ð is Σ0 preserving in the pseudo-interpretation.) This then simply says ð(x n ∩ A) = ð(x n ) ∩ B. As ð ↾ H is cofinal (1) follows. ⊣ (2) B ∈ /M n+1 Proof. Assume B ∈ M . Then hHn , Bi ∈ M (by Corollary 3.27.) As p ∈ PM , n,p↾n
n+1 )∩P (ùñM ) with D ∈ / M . As ð ↾ hH , Ai −→Σ0 ,cof hHn , Bi, there is D ∈ Σ1 (M n+1 n+1 and ð ↾ ùñM = id , this D is Σ1 (hHn , Bi) ⊆ M . But by assumption P (ùñM ) is ⊣ the same in both M and M . Contradiction! A′ Now just code B onto ùñn to get a suitable A. As ñn is p.r.closed, and Hn = Jñn , let f be Σ1 (M ) with f : ùñn −→ Hn be onto; let A = {î | f(î) ∈ B}. Then A is as required. ⊣ n From the Claim we see that ùñM ≤ ùñn . The reverse inequality we already established in the last lemma. This gives us (ii) here; and we can now assert (i). The D of (2) is Σ1(n)M (p), and the same definition over M in ð(p) yields a D ′ with n+1 n+1 n+1 =D∈ / M . Hence ùñM ≤ ùñM which is (iii). ⊣ D ′ ∩ ùñM
Remark. It is possible to replace the assumption on weak amenability with just n that RM 6= Ø and get the same conclusions. Definition 3.31. E is Σ1 -amenable with respect to M , if Ea is Σ1 (M ) for any a ∈ [í] κ ≥ ùñM Lemma 3.32. Suppose ùñM , and that E is Σ1 -amenable. Then
(n) P (κ) ∩ Σ(n) 1 (M ) ⊆ Σ1 M .
n ′ n Proof. Let B ∈ P (κ) ∩ Σ(n) 1 (M ). Suppose B(z) ←→ ∃x B (x , z, ð(f)(a)) (n) (n) ′ with B ∈ Σ0 (M ). Using Σ0 cofinality, we can write n n B(z) ←→ ∃u ∈ HM ∃x ∈ ð(u)B ′ (x, z, ð(f)(a)) ←→ ∃u ∈ HM D(ð(u), z, ð(f)(a)) (n) (where D ∈ Σ(n) 0 (M )) Let D ∈ Σ0 (M ) have the same definition. n B(z) ←→ ∃u ∈ HM {v | D(u, z, f(v))} ∈ Ea ←→
∃u n ∃x n (x n = {v | D(u n , z, f(v))} ∧ x n ∈ Ea )
22
J. VICKERS AND P. D. WELCH
(0) for any z < κ. The matrix here is Σ(n) 0 ∩ Σ1 .
⊣ n
Remark. There is a non-uniformity in the above due to the final “x ∈ Ea ”, but that is the only one. We now go down below κ. m Lemma 3.33. Let E be Σ1 -amenable, and assume ùñM ≤ κ. Then m m (i) HM = HM ; (ii) ð : M −→Σ(m) M ; 1
(m) m m (iii) Σ(m) 1 (M ) ∩ P (HM ) = Σ1 (M ) ∩ P (HM ); m m (iv) ð“PM ⊆ PM . m Proof. By induction on m. We first show (i). Note that HM by definition is m A′ ′ and ùñM is a cardinal of M = hJα ′ , B i. Similarly for M = hJαA , Bi. We
A′ Jùñ m , M
′
m m also know that JκA = JκA . So it suffices for (i) to show that ùñM = ùñM . This will m also show (ii), since ð ↾ HM = id . Suppose that m = k + 1.
m m m ≥ ñM : since there is B ⊆ ùñM , B ∈ Σ(k) ñM 1 (M )\M . By the inductive hypoth-
m esis, as ð : M −→Σ(k) M and ð ↾ HM = id , B is thus also Σ(k) 1 (M ). But also 1
m B∈ / M , since P (ùñM ) is the same in both models. m m m m < ùñM with B ∈ Σ(k) ñM ≥ ñM : suppose not. Then there is B ⊆ ùñM 1 (M )\M .
k But then B ∈ Σ(k) 1 (M ) by the previous lemma, if ùñ > κ, and by the inductive hypothesis (iii) for k otherwise. But B cannot be in M , since P (κ) ∩ M ⊆ M . So m m . This proves (i) and as remarked, (ii). ùñM 6< ùñM m m For (iii): let H = HM = HM and suppose B ⊆ H . (k) B ∈ Σ(m) 1 (M ) ←→ ∃D ∈ Σ1 (M ) B ∈ Σ1 (hH, Di) (m) ←→ ∃D ∈ Σ(k) 1 (M ) B ∈ Σ1 (hH, Di) ←→ B ∈ Σ1 (M ).
where the second equivalence uses the inductive hypothesis (iii) or the last lemma. m , p = ð(p). We need to show: This is (iii) for m. We consider (iv). Let p ∈ PM
k,p↾k Claim AM ∈ / M for 1 ≤ k ≤ m. We prove this by induction on m − k. If m = k, then as ð ↾ H = id , and k,p↾k k,p↾k ð : M −→Σ(m) M , so AM = AM , and the latter is not in M , and hence not 1
in M either, because P (H ) ∩ M = P (H ) ∩ M . Suppose it holds for k + 1 ≤ m. k+1,p↾k+1 k,p↾k k+1 k Then AM = B ∩ HM where B ∈ Σ1 (hHM , AM i) in parameter p(k). But k,p↾k k,p↾k k AM ∈ M ⇒ hHM , AM i ∈ M which contradicts the inductive hypothesis for k. ⊣ Collecting together the pieces of the above one obtains (with a small amount m swept under the carpet in (ii) for ùñM < κ, as the argument requires some more (n) analysis of the Σ1 relations than we have catered for here): Corollary 3.34. Let E be Σ1 -amenable. Then (i) ð : M −→Σ∗ M (n) (ii) P (κ) ∩ Σ(n) 1 (M ) = P (κ) ∩ Σ1 (M ) for all n < ù. 3.4. An Upwards Extension of Embeddings Lemma. M
M
Lemma 3.35. Suppose M = hJâE , E M , F M i is a mouse, and Q = hJíE , E M ↾ íi and there is a cofinal embedding ó : Q−→Σ0 Q. We suppose there is κ < í with
23
ON ELEMENTARY EMBEDDINGS FROM AN INNER MODEL TO THE UNIVERSE
Q |= “κ is the largest cardinal”, and either â = í or M |= í = κ + . Suppose that M cf(í) > ù. Then there is ó˜ ⊇ ó and M = hJâE , E M , F M i with M a mouse and n+1 ≥ κ, where κ = ó(κ). Further, if M is sound ó˜ : M −→Σ(n) M , for any n with ùñM 1
m+1 m+1 m m , then ùñM ≤ κ < ùñM , M is ≤ κ < ùñM above κ, and for some m < ù, ùñM
the closure of κ ∪ {p} under good Σ(m) ˜ : M −→Σ(m) M . 1 -functions for a p ∈ M , and ó 1
We first remark that in many situations one may use the lemma with κ a cardinal (or at least the cardinal of a model) which we are not assuming here. One can often improve the conclusion to require that M is sound above κ (and that ó(p ˜ M ) = pM \κ where pM is the standard parameter of M ) - but this requires more work on parameters, which we have eschewn. The proof of this lemma involves forming a “pseudo-ultrapower” M of M that has also a fine-structure preserving extension of ó mapping M into M . The construction is very similar to that for forming an ultrapower by an extender (more precisely it is that for the “long” extender or “derived” extender from the map ó). So naturally it bares many similarities to what we have just done in the previous subsection. Much of what we shall do is redefinition of the previous notions, and many of the changes in the proofs require only a change of notation. We omit some detail therefore. We shall first redefine the class of functions Γ, the stratified functions Γn , and form the term model D , and the “strata” Hn . There is little that can be said about the Hn for n > m + 1 and indeed we shall not define these. The “lift-up” structure M is wellfounded (and iterable) by the crucial use of the uncountable cofinality of í. Definition 3.36. Γ is the set of functions f with dom (f) ⊆ u for some u ∈ Q n+1 n and either f is a good Σ(n) 1 (M ) function for some n with ùñM ≥ í, or f ∈ HM n where ùñM ≥ í. ˙ and D as indicated § 1.1. We define e, I, E, Definition 3.37. D = {ha, fi|f ∈ Γ ∧ a ∈ ó(f)}; set: ∈ ha, bi Ie hb, gi iff ha, bi ∈ ó({hu, vi|f(u) = g(v)}) ˙ ˙ F˙ i. G(ha, fi) iff a ∈ ó({u|f(u) ∈ G}) for G = E, F ; set: D = hD, I, e, E, One can then again prove a Łos theorem for Σ0 formulae: (1) Let ha1 , f1 i, · · · , han , fn i ∈ D and let ϕ be a Σ0 formula. Then − → → −−−→ → → u )]}) D |= ϕ[hai , fi i] ⇐⇒ − ai ∈ ó({− u |M |= ϕ[ fi (− Then I satisfies the axioms for equality, and D satisfies extensionality. If e is wellfounded then there is an isomorphism [ ] : D ←→M g where M is transitive and I [x] = [y] iff x y. We can define ó ˜ : M −→ M by ó(x) ˜ = [h0, {h0, xi}i]. M will Σ 0 ∈ e be of the form hJâE , E, F i, and we shall show that it is a mouse and ó˜ satisfies the properties of the Lemma. Since wellfoundedness of e can be considered a special case of the proposed iterability of M , we omit the proof of the wellfoundedness of e, and simply assume it for the moment. We first note: (2) ó˜ ↾ Q = ó Proof. Note that for f, g ∈ Q [ha, fi] ∈ [hb, gi] ⇔ ó(f)(a) ∈ ó(g)(b). Hence for such f we may define a map [ha, fi] 7→ ó(f)(a). This map is onto and
24
J. VICKERS AND P. D. WELCH
so by the above is the identity. Hence [ha, fi] = ó(f)(a) = ó(f)(a) ˜ and so ó(x) ˜ = [h0, {h0, xi}i] = ó(x). ⊣ Define n n+1 ≥í Γn = {f ∈ Γ | ran(f) ⊆ HM } if ùñM n+1 n n < í ≤ ùñM } if ùñM . = {f ∈ Γ | ran(f) ∈ HM
Hn = {[ha, fi] | f ∈ Γn ∧ ha, fi ∈ D} if Γn is defined. ùñn = On ∩ Hn . Note that Hn is transitive as before. n ≥ í. (3) ó˜ : M −→Σ(n) M cofinally for any n with ùñM 0
Proof. We shall prove (3) in three stages. The first two are just the manoeuvres of Lemma 3.28. The third stage will use the extra assumption on the cofinality of í to check that (3) holds for the m of the crossing projectum of Lemma 3.35. In the k ≥ í on the right hand side first step we interpret the variables of type k with ùñM as varying over Hk . We prove (3) using this pseudo-interpretation. Again we then shall show in the second step that this interpretation is “correct” in that Hk in fact k is HM for any k ≤ m ⊣ The proofs of (4),(5) & (6) are, with minor alterations, those of the last subsection. −−→ n (4) Let hai , fi i ∈ D and let ϕ ∈ Σ(n) 0 where ùñM ≥ í. Then −−→ − →− − → − → M |= ϕ[[ha , f i]] ⇐⇒ a ∈ ó({ u | M |= ϕ[ f ( → u )]}). i
i
i
n (5) ó˜ : M −→Σ(n) M cofinally, for ùñM ≥ í. 0
Proof. We mean here in the pseudo-interpretation. By cofinally, we mean ó˜ ↾ m m HM is cofinal into HM , in case m is defined. But this is straightforward. ⊣ As ó˜ is Σ0 and cofinal into M , acceptability of M translates to acceptability of n n n M , and so Hn = JñMn for ùñM for ùñM ≥ í. It remains to show that ñn = ñM ≥ í. n+1 n n n (6) ñn = ñM if í ≤ ùñM ; ñn ≤ ñM if í ≤ ùñM . Proof. The reader is referred to (4) and (5) of Lemma 3.28. The notational changes are simply that in (4) the function h now has domain dom(g). The map between the two structures is now ó˜ (not ð); and in (5) we need to conclude with: a ∈ ó({u|M |= ϕ[f(u), p]}) = ó({u|f(u) = A}) rather than claim that the set is in Ea . ⊣ We now use the extra condition on Q, í. m+1 m (7) If ùñM then < í ≤ ùñM m+1 m (i) ùñM < í ≤ ùñM ; m (ii) ñm = ñM . m+1 (iii) M is the closure of κ ∪ {r} under good Σ(m) 1 (M )-functions, for some r ∈ RM . Proof. We first show ó(í) ˜ = í = sup ó“í ˜ (setting ó(On ˜ ∩ M ) = On ∩ M ). Let î = ó(f)(a) ˜ < ó(í) ˜ with [ha, fi] ∈ D. Without loss of generality we can assume ran(f) ⊆ í. By the definition of Γ, and using í is a cardinal of M , ran(f) cannot be cofinal in í. Hence there is ç with ran(f) ⊆ ç < í and so ó(f)(a) ˜ < ó(ç) < í. m If ó(κ) = κ, then one may simply check that κ +M = í. By (6) í ≤ ùñM . m+1 m+1 m ˜ = p and p ∈ PM . Suppose ùñM < í ≤ ùñM . Let óp)
25
ON ELEMENTARY EMBEDDINGS FROM AN INNER MODEL TO THE UNIVERSE
m For clause (ii) we only have left to show ñM ≤ ñm . We do this now. Let m,p m A be defined by the cofinal map ó˜ ↾ HM : hM m,p , AM i −→Σ0 hHm , Ai. That S m,p m,p ˜ ∩ AM ). Then hHm , Ai ≺Σ0 M . By the Downwards is, let A = x∈H m ó(x
f, pe such that M fm,pe = hHm , Ai Extensions of Embeddings Lemma, there are M m f −→ (n) M with and pe ∈ RM e . We have the further properties that there is ð : M Σ M
0
m e = p. Since p ∈ RM ð ↾ Hm = id and ð(p) we can factor through, and see that f such that óˆ ↾ H m = ó˜ ↾ H M and ó(p) e Then there is óˆ : M −→ (n) M = p. ˆ Σ1
M
M
ð(ó(f)(a)) ˆ = ó(f)(a) ˜ for ha, fi ∈ D . Hence ðóˆ = ó˜ and ð is onto; hence ð = id , m f óˆ = ó, ˜ and M = M . Then A cannot be in M whilst it is Σ(m) 1 (M ). Hence ñM ≤ ñm . m+1 m Let p ∈ PM and let p = ó(p). ˜ Any x ∈ HM has the form ó(f)(a) ˜ where f ∈ Γm (M ) and a ∈ ó(dom(f)) If f ∈ hM m,p (i, hî, p(m)i) some î < κ, then ó(f)(a) ˜ = hM m,p (j, hhî, ai, p(m)i) for some j < ù, î = ó(î); hence M m,p ⊆ hM m,p (í ∪ {p(m)}). But as at the beginning of this proof, hM m,p (κ ∪ {p(m)}) m,p is cofinal in í. As í = (κ + )M , (or equals On ∩ M m,p ), this suffices to show m,p hM m,p (κ∪{p(m)}) ⊇ M . This shows that both that M is the closure of κ∪{p(m)} m+1 under good Σ(m) ≤ κ, which are clauses (i) and (iii). ⊣ 1 −functions, and that ùñM Claim M is wellfounded and iterable above κ. Proof. Suppose not; let I = hMα |α < èi be an iteration of M0 = M above κ that cannot be continued. Then there is a set {xi }i∈ù ⊆ M of supports of functions coding an illfounded ∈è -chain through the ordinals of a model Mè . We treat wellfoundness of M as a special case of iterability. In this case set xi+1 ∈ xi as a descending e-chain through M = M0 . Let H be some sufficiently large ZFC − -model that is transitive and contains I, {xi }. Let I ∪ {xi } ⊆ Y ≺ H where Y is countable. Let k : H ←→Y g be the inverse of the transitive collapse. Let k(hN, κ ′ , x i , I i) = hM, κ, xi , I i. Then in H , {x i } is a set of supports for functions coding illfoundedness of the last model M è of I = hM α |α < èi. Let {yi }i∈ù enumerate N . n , or fi a good Σ(n) Then each k(yi ) is of the form ó(f ˜ i )(ai ) for some fi ∈ HM 1 (M ) function with domfi ∈ Q. We define by cases a model P and a map ð : P −→ M . n Case 1. n ≥ 1 and í < ñM . Let ì < í be chosen so that n,p
(i) P = df hM |ì, An,p |ìi is amenable, ñP1 ≤ κ. (ii) supi domfi < ì n,p n,p i (iii) id : P −→Σ1 hM , AM n (i) & (ii) are clearly possible since cf(í) > ù. As í < ñM , í is a cardinal of n,p
n,p
n,p
M , so hM |í, An,p |íi ≺Σ1 hM , An,p i. But there are arbitrarily large ì < í n,p with such a P ≺Σ1 hM | í, An,p |íi. Hence P exists. Now any fi is either in M n,p (k) n H = M or has a good Σ (M ) definition involving parameters î~i < ùñ k . By M
1
M
the soundness of M above κ, either way such an fi has a good Σ(n) 1 (M ) definition ~ involving parameters îi below κ.
26
J. VICKERS AND P. D. WELCH
By the Extensions of Embeddings Lemma there is a premouse P, iterable above κ, and q ∈ RPn , and a ð : P −→Σ(n) M , with P = P n,q and ð(q) = p, and 0
(n) n,p n n An,q |ì. As An,q P = A P ∈ HM , P ∈ HM . Let f i have the same good Σ1 definition over P from î~i , q as fi did over M in î~i , p. Note that ð is in fact Σ(n) preserving. 1
Let ó(p) = p. ˜ But as domfi = domf i and ð is Σ(n) 1 -elementary ð(f i (u)) = fi (u) for any u ∈ domf i . n Case 2. n ≥ 1 and í = ñM . ì may be chosen as in Case 1, however in (iii) we may only have that P is Σ0 n,p n,p embedded into hM , AM i by id. However this suffices to still extend id to a ð : P −→Σ(n) M , but which is nevertheless Σ1(n−1) preserving. (This is just part of 0
the Extensions of Embeddings Lemmas.) However we may now at no additional cost impose on ì, using cf(í) > ù, and the Case hypothesis: n (iv) Any fi that has a good Σ1(n−1) (M ) definition in parameters î~ < ñM , then such a î~ is below ì. Again we have ð : P −→Σ(n) M , with P a premouse iterable above κ, P ∈ HíM . 0
As ð is Σ1(n−1) preserving, we can let f i be the functions defined over P with the same definitions as that of fi over M , with supi domfi < On ∩ P. In both cases ˜ where P˜ = ó(P), by l (yi ) = ó(f we define l : N −→ P, ˜ i )(ai ). l will embed, in particular, the supports for a descending sequence of ordinals of an iterate of P˜ above κ. This will contradict the latter’s iterability above κ. (9) l is Σ(n) 0 preserving. Proof. P˜ |= ϕ(l (yi )) ⇐⇒ P˜ |= ϕ(ó(f ˜ i )(ai )) ⇐⇒ ai ∈ {u | P˜ |= ϕ(ó(f i )(u))} ⇐⇒ ai ∈ ó({u | P |= ϕ(f˜i (u))}) (as ó ↾ P is Σù ) ⇐⇒ ai ∈ ó({u | M |= ϕ(f i (u))}) (n) (as ð is Σ(n) 0 or Σ1 depending on the case)
˜ i )(u))} (as ó˜ is Σ(n) ⇐⇒ ai ∈ {u | M |= ϕ(ó(f 0 ) ⇐⇒ M |= ϕ(ó(f ˜ i )(ai )) ⇐⇒ N |= ϕ(yi ) (as k is Σù ).
⊣
Note that P˜ is truly iterable above κ, since “P is iterable above κ” holds in some admissible set A with P ∈ A ∈ M , and hence by elementarity holds in A˜ = ó(A ) ˜ But ù1 ⊆ A˜ so this is absolute. But this is now a contradiction since of P. {l (x i )}i∈ù contains supports coding an illfounded iteration of P˜ above κ. Case 3. n = 0. A standard argument shows that cf(í) > ù =⇒ cf(On ∩ M ) > ù. As ó˜ is Σ0 -cofinal the same is true of M . There is thus ô˜ = ó(ô) with (i) ∀y k(y) ∈ Jô˜M (ii) ∃ fn ∈ JôM ∃an ∈ Q
ó(f ˜ n )(an ) = k(yn ).
ON ELEMENTARY EMBEDDINGS FROM AN INNER MODEL TO THE UNIVERSE
27
We can assume that ô˜ is chosen sufficiently large so that each fn can be taken ˜ . in hM |ô (κ ∪ pM ). Let X ≺Σ1 JôM be such that í ∪ {pM } ⊆ X , let ð : P ←→X P ~ Let ð(q) = pM , and let f n have the same Σ1 ({q, în }) definition as fn had in ~ Note that sup domfi ≤ í ∩ X . Also note that P ∈ J M (since ΣM |ô ({p , î}). M
i
í
A1P ⊂ JκP , as ñP1 ≤ κ, and then a simple comparison argument shows A1P ∈ M ). Again define l : N −→ P˜ =df ó(P) by l (yn ) = ó(f ˜ n )(an ). (10) l is Σ0 -preserving. Proof. The equivalences are exactly those of (9), noting in this case that ð : ⊣ P −→Σ1 M |ô, and so ð : P −→Σ0 M . But P˜ is iterable above κ as before, so l embeds a supposedly non-iterable premouse above κ ′ into one that is iterable above l (κ ′ ) = κ! This is a contradiction. ⊣ REFERENCES
[1] A. J. Dodd, The Core Model, Lecture Notes in Maths, vol. 61, London Mathematical Society, 1982. [2] H-D. Donder, R. B. Jensen, and B. J Koppelberg, Some applictions of the Core Model, Set Theory and Model Theory, Springer Lecture Notes in Maths, vol. 872, 1981, pp. 55–97. [3] R. B. Jensen, The Core Model for Measures of Order Zero, 1989, circulated manuscript. [4] R. B. Jensen and J. Vickers, A covering lemma part 2, manuscript. [5] , Absoluteness of J´onsson Cardinals, 1997, preprint. [6] A. Kanamori, The Higher Infinite, Perspectives in Mathematical Logic, Springer Verlag, Berlin, 1994. [7] A. Miller, Some interesting problems, Set Theory of the Reals, (Proceedings of the Israeli Mathematical Conference), vol. 6, American Mathematical Society, 1993, pp. 645–654, http://www.math.wisc.edu/∼ miller. [8] J. R. Steel, The Core Model Iterability Problem, Springer Lecture Notes in Maths, vol. 8, Springer, 1988. [9] J. Vickers, A covering lemma part 1, 1990, manuscript. [10] P. D. Welch, Countable unions of simple sets in the Core Model, this Journal, vol. 61 (1996), no. 1, pp. 293–312. [11] W. H Woodin, Σ21 -absoluteness and supercompact cardinals, May 1985, handwritten notes. UNIVERSITY OF BRISTOL DEPARTMENT OF MATHEMATICS BRISTOL BS8 1TW, ENGLAND KOBE UNIVERSITY GRADUATE SCHOOL OF SCIENCE & TECHNOLOGY ROKKO-DAI, NADA-KU, KOBE 657, JAPAN
E-mail: [email protected]
ON ELEMENTARY EMBEDDINGS FROM AN INNER MODEL TO THE UNIVERSE
J. VICKERS AND P. D. WELCH
Abstract. We consider the following question of Kunen: Does Con(ZFC + ∃M a transitive inner model and a non-trivial elementary embedding j : M −→ V ) imply Con(ZFC + ∃ a measurable cardinal)? We use core model theory to investigate consequences of the existence of such a j : M → V . We prove, amongst other things, the existence of such an embedding implies that the core model K is a model of “there exists a proper class of almost Ramsey cardinals”. Conversely, if On is Ramsey, then such a j, M are definable. We construe this as a negative answer to the question above. We consider further the consequences of strengthening the closure assumption on j to having various classes of fixed points.
§1. Introduction. It is quite natural to study the properties of elementary embeddings j : V −→ M for M some inner model, since many such embeddings, if they exist, have first order formulations within ZFC . The question of reversing the arrow and looking at a non-trivial j : M −→ V in general does not readily admit of such formulations. So we study in this paper what might be considered the ZFC consequences of the second order statement that there are proper classes j, M such that . . . (This was the formulation of Kunen’s question as it appears in [7], but of course, if the reader does not like this conceit, then he or she may simply translate everything into properties of elementary submodels of some Vë |= ZFC , by replacing V by Vë for some inaccessible ë. Hence we could equally well be motivated by investigating the possible elementary substructures of some Vë of size ë.) It is not hard to see that some sort of hypothesis beyond ZFC is needed to generate such situations. If there were a non-trivial j : M −→ V , then j ↾ L : L −→e L is non-trivial and so O # must exist. Indeed, by considering critical points, one can see that such an embedding could not be definable by class terms, even from parameters, or there would be an infinite regress of ordinals! Obviously more can be extracted from this, and it is natural to ask how much. As ´ the hypothesis is similar to a statement about On possessing Jonsson or Ramsey like properties, it is not surprising that it implies some kind of large cardinal property. Whatever the “core” of V is, denoting it by K, then j ↾ K M : K M −→ K V and we may ask what kind of K must there be for this to happen. Kunen’s question Received July 16, 1998; revised January 1, 2000 c 0000, Association for Symbolic Logic
0022-4812/00/0000-0000/$00.00
1
2
J. VICKERS AND P. D. WELCH
(Question 15.8 of [7]) in this vein was: can we extract an inner model with a measurable cardinal? Or more precisely, does Con(ZFC{j} ˙ + ∃j : M −→ V , j 6= id , j elementary) ⇒ Con(ZFC + ∃ a measurable cardinal )? (Here ZFC{j} ˙ means ZFC in L{∈, a language with a predicate symbol for j, whilst j : M −→ V is ˙ ˙ j} considered to be Σ1 -elementary in L{∈} ˙ alone. (The latter implies full elementarity.) This question is easily answered however. Perhaps more interestingly, in § 1.1 we consider the consequences of the existence of such elementary embeddings from an M into V , varying the basic question by adding closure requirements on j to see what K must be like. From just the existence of such a j we get (Theorem 1.4) that K must contain a proper class of “almost Ramsey” cardinal (defined below). We get full Ramseyness (Theorem 1.6) by asking that dom(j) have a class of regular fixed points. By insisting that dom(j) contain a stationary class of singular fixed points of cofinality ù, then we obtain that K must have a Mahlo class of measurables; if the fixed points have uncountable cofinality, or if M computes correctly countable cofinalities, then O sword exists (Theorem 1.10) (that is, Jensen’s core model for measures of order zero - see [3], is non-rigid). We should like to thank M. Foreman for saving us from a demonstrably inconsistent hypothesis for Theorem 1.10 (we had assumed that M was closed under ù-sequences); and for allowing us to include his counterexample as Theorem 2.4. In § 2 we make a first attempt at some converses by showing that if On is Ramsey, then we may define a j, M with j : M −→e V , j 6= id . We could formulate this as a statement about Vκ , where κ is a Ramsey cardinal, and show that definably over hVκ , ∈, I i (where I is a class of good indiscernibles for hVκ , ∈i coming from the Ramsey property) there are classes j, M satisfying our requirements. We thus have Con(ZFC + ∃ a Ramsey cardinal ) ⇒ Con(ZFC{j} ˙ + j : M −→e V, j 6= id ). Although this is not an exact equiconsistency, it could be considered close. However, overall in this direction we have left several upper bounds open. We make some comments on results obtainable by Woodin, using the stationary tower forcing. (We should like to thank the referee for prompting us to make such comparisons.) Woodin has shown [11], that if κ is a measurable cardinal, and Pκ the full stationary tower forcing, then if G is Pκ generic over V , then we do have an embedding j : Vκ −→ Vκ [G]. (Below the measurable κ there is an unbounded ´ sequence of completely Jonsson cardinals, fixed by j.) However in this situation, we cannot have that the structure hVκ [G], j, ∈i |= ZFC{j} ˙ . That is, we cannot put back the predicate Vκ . (For example, note that if V = L[ì] with ì a normal measure on κ, then Vκ |= “V = K” (with K the Dodd-Jensen core model), then V [G] is a model of the same, and hence some new “mouse” has been added, that must iterate past all those of Vκ !) This, then is slightly different from our requirements that we have hV, j, ∈i |= ZFC{j} ˙ . If we regard the relationship of V to a forcing extension as being “close”, then perhaps we need the extra strength in order to obtain such a close relationship between the “M ” and “V ”. Our results do not require such strength since our “M ” and “V ” are not at all close in this sense. In § 3.1 we reprove the Jensen Indiscernibles Lemma [2], originally done using the old-style fine-structure of the Dodd-Jensen KDJ , but now for the modern hierarchy and the K under consideration. The reasons for this are that, corresponding to ˝ (if cf(α) > ù), Jensen’s results for the Dodd-Jensen K, this shows that the α-Erdos
ON ELEMENTARY EMBEDDINGS FROM AN INNER MODEL TO THE UNIVERSE
3
almost Ramsey, Ramsey etc. properties are downwards absolute between V and any universal weasel. Secondly, this actually also gives a second proof of Theorems 1.4 and 1.6 which are proven in § 1 by using a coarse “lift-up” technique. Our results are all in the region of the consistency strength of inner models with measures of order zero, K MOZ so it is appropriate to use Jensen’s model of [3] in order to discuss them. This model will be K throughout the paper. (Although the Kunen question can be resolved with just the Dodd-Jensen K DJ , or even the Σ1 -K of [10].) We assume familiarity with the measures of order zero K, and the techniques of that paper, some of which appear in [5]. We use the strong Covering Lemma for K, [9], [4] on singular cardinals â of V being either singular, or possibly measurable in K if cf(â) = ù. Mostly however it is the theory of iteration and comparison that is useful, and fine-structural results have been hived off to § 3. In § 3.2 we sketch the outlines of Jensen’s Σ∗ fine structure theory in a general setting. It is a relatively bare-bones outline, but all the essential definitions are here. What we have omitted is a full discussion of Σ(n) 1 relations and functions, and a fuller analysis of good parameters, and of witnesses. The intention here is to give an introduction to the theory, and, in § 3.3, to that of the Σ∗ ultrapower, to provide a modicum of self-containedness for the proof of the fine-structural Upwards Extensions of Embedding Lemma of § 3.4 which is needed for Theorem 1.10. This has essentially been extracted from various parts of §3.3 of [3], but we have thought it worthwhile to give a complete version here, as firstly, the reader may not wish to do it for his or herself, (and anyway such arguments are not often in print), and secondly this is a form of “lift-up” which we wish to use elsewhere. We should like to emphasize our great debt to Jensen’s work, and in particular to [3] Some prerequisites The set-theoretic notations are standard. [6] gives an account of the basics of general elementary embeddings. Our notation for iterations of mice is that of Jensen’s: an iteration of a mouse M is of the form M I = hhMi ihði,j ihíi , αi i | i < èi with models Mi and indices híi , αi i and commuting M maps ði,j : Mi −→ Mj , with ùíi ≤ ùαi ≤ On ∩ Mi and if Ei 6= ∅ (the íi ’th filter M : M |αi −→Eíi Mi+1 where M |â denotes on the E Mi filter sequence), then ði,i+1 M
M hJâM , E M , Eùâ i. ( M ||â will denote hJâE , E M , Øi), or when there is no danger of M = id ↾ M |αi . Direct limits are taken confusion, simply |JâE |.) If Eíi = Ø then ði,i+1 as needed. A truncation occurs when ùαi < On ∩ Mi . We truncate if need be in order to allow a filter to become a full measure over the subsets of its critical point in M |αi , and thus are able to take an ultrapower. It is a fundamental fact that in any iteration only finitely many truncations can occur. The coiteration of two mice or “weasels” (= proper class mice) is performed by “iterating away points of least difference”. This results in a terminating double iteration of length some è ≤ On, with either Mè = Nè , Mè ∈ Nè , or Nè = Mè . We shall use the following well-known lemma: Basic Coiteration Lemma If (M, N ) are premice, and they are coiterated to models (Mè , Nè ) with iterations N M ii≤j≤è ii, hhNi ii≤è , hαiN ii ù then O s exists. Proof. As in the proof of the last theorem, it would suffice for a contradiction to show that K M is universal. But j(ô) = ô so ô is singular in K by the Covering lemmas of [9] and [4]. So K M |= ô is singular. The result follows by Lemma 1.8. ⊣ M Theorem 1.10. If ∃j : M −→ e V, j 6= id and ∀α(cf(α) = ù ⇒ cf (α) = ù) s then O exists. Proof. We first note that the assumption implies that there is a stationary class S of cardinals ô with cf(ô) = ù which are fixed points of j. Consider the comparison of K with K M with resulting coiteration hhNi , KiM i|i ≤ èi for some è ≤ On, with indices híi |i < èi. Let the critical points of the ultrapowers used be hκi |i < èi. Let α = crit(j). Clearly cf(α) > ù. (1) On the K side of the coiteration, the very first index used, í0 , is less than K + α =df α + , and hence involves a truncation to a set mouse N0 = K|í0 . Proof. Suppose the first index used on the K side were í0 ≥ α + . Then α + = KM α+ and by our assumption, and the Covering Lemma in M , cf(α + ) > ù. M Then P (α)K = P (α)K and if U is the derived measure from the embedding j e =df (meaning that U = {X ⊆ α | X ∈ K ∧α ∈ j(X )}), then consider ð : K −→ K + +K e Ult(K, U ). K is wellfounded as α = α , and U ù-complete. As we are below O ¶ , ð is an iteration map, ð0,ì for an iteration by full measures hEí |í ≤ ìi. The first measure, E0 , used in this iteration must have critical point α = crit(U ), and so E0 = EαK+ ; and it agrees with U on P (α) ∩ K; so U must be EαK+ . ′ Let ó : K M −→ K ′ =df Ult(K M , U ), then EαK+ = ∅, by coherence. Defining k : K ′ −→Σ0 K by k([f]U ) = j(f)(α) we have as usual that k ↾ α + + 1 = id ↾ α + + 1, ′ whilst EαK+ 6= ∅, EαK+ = ∅ - a contradiction. ⊣
9
ON ELEMENTARY EMBEDDINGS FROM AN INNER MODEL TO THE UNIVERSE
We have as usual a cub class C and N1 iterating past K M , with ðNi Nk (κi ) = κk for i < k ∈ C , whilst ðK0M ,KiM “κi ⊆ κi . However C ∩ S cannot form a stationary class of cardinals singular in K M , as then we should have K M universal - a contradiction as in the last theorem. Again the only other possibility is that it forms a class of measurables of K M . Let ù2 < κ ∈ C ∩ S with κ greater than the last truncation point on the iteration KM on the K side. Let κ ′ = κ + κ . Then KκM |κ ′ is an initial segment of Nκ , and the latter is sound above κ. Then κ ′ = κ + ù (2) ñN < κ, and κ > cf(κ ′ ) > ù. κ
Nκ
= sup ðK M ,KκM “κ +
KM
. Note
Proof. By (1) there is at least one truncation on the K-side, and from this the n first clause follows: since from the last truncation point i0 say, we have ñN ≤ κi0 io n for some n. For this n we have ñNκ < κ. As M calculates countable cofinalities correctly, by the Covering Lemma in M , cf(κ +
KM
M
) = cf(κ + ) > ù.
⊣
hKiM
Let the iteration | i ≤ κi with indices híi |i < κi be copied (as in Claim 1 of e 0 as hK e i | i ≤ κi with commuting Theorem 1.4) via j = ó0 to an iteration of K = K i M e maps ói : Ki −→e K and indices hói (íi )|i < κi. e κ |ä. Note that óκ (κ ′ ) = κ +Keκ = Let ä = sup óκ “κ ′ . Then óκ : KκM |κ ′ −→Σ0 K +K +V κ = κ by the Covering Lemma in V , and the fact that the copy iteration below eκ κ fixes κ and κ +K . So we see ä < κ +K . We now have to do a fine-structural lift-up argument, as this time we know that n+1 n < κ < ñN ñN for some n ≥ 0. Fix this n. κ κ f for some mouse M f with óe ⊇ óκ ↾ (3) There is a lift-up map óe : Nκ −→Σ∗ M n+1 n M ′ κ f f e Kκ |κ where M has the following properties: (i) M |ä = K |ä; (ii) ñM e < ä ≤ ñMe ; f is the Σ(n) -closure of κ ∪ {p} for some p ∈ Rn+1 . (iii) M 1
e
M
Proof. This is a somewhat standard argument (c.f., [3] §3.3.2) exploiting the fact that cf(κ ′ ) > ù to get a wellfounded fine-structural “pseudo-ultrapower”. Instead of the coarse pseudo-ultrapower of Claim 3 of Theorem 1.4 we define Nκ D = {ha, fi | dom(f) ⊂ ô < κ ′ and either f ∈ Σ(m) 1 (Nκ )(m < n) ∨ f ∈ Hñ n } Nκ
and then set e ∈ ha, fi hb, gi ⇐⇒ ha, bi ∈ óκ ({hu, vi | f(u) g(v)} I = and ˙ G(ha, bi) ⇐⇒ a ∈ óκ ({u | f(u) ∈ G}) (for G ∈ {F, E}). We then define D = hD, I, e, E˙ , F˙ i and proceed to prove a Łos Theorem for Σ0 formulae. The fact that cf(κ ′ ) > ù results in e being wellfounded, and then D is f for some premouse M f. The canonical embedding óe : Nκ −→ M f isomorphic to M f is iterable, and so a mouse, is an is defined by óe(x) = [h0, {hx, 0i}i]I . That M amplification of the argument that it is wellfounded (see [3] §3.3.3). Using that κ ′ is the Nκ -successor of κ enables one to prove that óe is a Σ(n) 1 preserving embedding, and ultimately properties (ii) and (iii) ((i) holds by construction). The full proof is in § 3.4. ⊣
10
J. VICKERS AND P. D. WELCH
f∈K e κ ; thus ñ ù = ñ n+1 = κ. (4) M e e M M
f f eκ Proof. There is a Σ(n) 1 (M ) subset of κ that codes M . As K is universal (being just a simple iterate of K), in their coiteration the first point of difference between f and K e κ is not less than ä, so we see that this code is in K e κ . As κ is obviously a M n+1 κ ù e cardinal of K we must have ñMe = ñ e = κ. ⊣ M
f) is an initial segment of K e κ . core(M f)|ä + 1 = M f|ä + 1; hence (5) core(M κ e e M K Eä = Eä .
e (n)M n+1 f). Then P ∼ Proof. Let P =df core(M = Σ1 Hull(ùñM e ∪ {pMe }). (Where pMe e M f f is the standard parameter of M .) But as Eä 6= Ø, M is an iterate of P above ä. Let e κ |ã. Then N ||ä = M f||ä = P||ä. But ã ≥ ä be least so that ñ ù ≤ κ. Let N = K e
K κ |ã
over both P and N there are Σ∗ definable subsets of κ coding wellorders of type ä. As ä = κ + (or is the height of the ordinals) in both models, neither model contains such a code. So a simple comparison argument shows that P and N coiterate without truncation to the same model. But then by soundness, in fact P = N . ⊣ f is Remark. With a little more work on parameters, one can show that in fact M eκ . itself sound and an initial segment of K KM
KM
S
Let F = Eκ′κ . Let M = |KκM ||κ ′ | = JκE′ κ . Let F be defined by: F = X ∈M óκ (X ∩ F ). Then: e κ |ä, E Keκ i ∧ F = E Keκ . (6) óκ extends to: óκ ↾ hM , F i −→Σ0 hK ä ä e κ ||ä|. And F = |K e κ ||ä| ∩ E Keκ ′ . By the initial Proof. óκ is a cofinal map into |K óκ (κ )
e
κ
segment condition F = EäK .
e
e
κ
⊣
KκM κ′
(6) shows, as ó˜ ⊇ óκ , that EäK = EäM ⇒ E KκM κ′
= EκN′κ . So the conclusion of this
consequence holds by (5). But E 6= EκN′κ by virtue of the comparison process a contradiction! ⊣ Let K ′ be a core model for sequences of measures with o(κ) < (κ)++ (for example Mitchell’s Core Model). Question. If ∃j : M −→ e V and j 6= id , α is a regular cardinal, and there exists stationary many cardinals of cofinality α fixed by j then: (a) does K M |= “∃ unboundedly many measurable cardinals κ with o(κ) = α” (where we set o(κ) = 0 if α = ù)? (b) if M computes correctly cofinalities ≤ α then is this consistent? §2. Some converses. We briefly consider here some partial converses to the above. It might be thought at first that with some extra work, one could obtain from a non-trivial elementary embedding j : M → V , an inner model with a measurable cardinal. The following indicates that this is not possible. Suppose that κ is a ´ strongly inaccessible Jonsson cardinal, and that there is A ⊆ κ with Hë = Lë [A] ´ for cardinals ë < κ, and (by the Jonsson property) that there is a non-trivial j : Lκ [A] −→ Lκ [A], then we clearly have candidates for a “j : M −→ V ” by ´ cardinal is known to taking M = Lκ [A] and V as Lκ [A]. And such a Jonsson be equiconsistent with a Ramsey cardinal. See [6]. Notice also that this argument
ON ELEMENTARY EMBEDDINGS FROM AN INNER MODEL TO THE UNIVERSE
11
shows that if κ, j : Lκ [A] −→ Lκ [A] are as above, then we can perform a “lift-up” argument (as in Claim 3 of Theorem 1.4) to extend j to j˜ : L[A] −→ L[A] (using the regularity of κ). We can also think of this argument as taking an extender ultrapower of L[A], Ult(L[A], Ej ) where Ej is the long extender derived from the embedding j; by the regularity of κ this is wellfounded. But then V cannot be L[A] since otherwise Ej ∈ V and the latter would be an ultrapower of the inner model L[A]! Hence: Lemma 2.1. Suppose V = L[A] for some set A ⊆ κ with κ strongly inaccessible. (i) If j : M −→ V , (or even if j ↾ Lκ [A] −→ Lκ [A]) then j = id . (ii) κ is not a J´onsson cardinal. ´ Instead of arguing in terms of the Jonsson property, one can frame a hypothesis of the form “Suppose On is Ramsey ....” (or again reformulation in terms of some Vκ where κ is a Ramsey cardinal etc.). Definition 2.2. (i) On is Ramsey if there is a class I ⊆ On, unbounded, of good indiscernibles for hVκ , ∈, I i, (ii) κ = On is 1-Ramsey, if the class I in (i) is Mahlo. In (ii) we mean that I has non-empty intersection with any closed and unbounded class C ⊂ On, definable (with parameters allowed) over hV, ∈, I i. Notice that if κ is a Ramsey cardinal, then we have an I ⊆ κ so that in hVκ , ∈ I i, On is Ramsey in this sense. Of course, if On is Ramsey we have, definably over hV, ∈, I i, a satisfaction relation for hV, ∈i, since α < â ∈ I ⇒ Vα ≺ Vâ . Further, using AC , we may define skolem functions, and so skolem hulls for hVα , ∈i, so that α < â ∈ I ⇒ HullVâ (ã) = HullVα (ã) for ã < α. A prototypical situation is again where κ is Ramsey in K and then all the above hold over hKκ , ∈ I i. Theorem 2.3. (i) Suppose I ⊆ On witnesses On is Ramsey. Then, definably over hV, ∈, I i, there is a transitive class M , and an elementary embedding j : hM, ∈i −→ hV, ∈i, with j 6= id . (ii) Additionally, if I witnesses On is 1-Ramsey, then j can be taken with unboundedly many regular fixed points. Proof. The proof of the proposition is straightforward. Let α0 = min I . By the remarks above we may define in hV, ∈, I i the hulls Hâ =df HullVâ ((I ∩â\{α0 })∪α0 ) for â ∈ I , and their transitive collapses with maps ðâ : Mâ −→ Hâ . By the usual properties of good indiscernibles, it is easy to verify that S crit ðâ = α0 ,Sand that α < â ∈ I ⇒ ðâ ↾ Mα = ðα . Hence we may set j = â∈I ðâ , M = â∈I Mâ . This concludes (i). For (ii), simply note that there must be, stationarily often, inaccessible ä ∈ I with j(ä) = ä. ⊣ Part (i) of the last theorem can be rephrased as a relative consistency result from the existence of a Ramsey cardinal. This shows the answer to the question of Kunen is negative. A number of questions about the possibilities of finding models which satisfy further closure properties are immediate. For example: Question 1. Can one arrange for there to be an inner model M and a non-trivial j : M −→ V , with stationary many fixed points of cofinality ≥ ù?
12
J. VICKERS AND P. D. WELCH
Question 2. The same question as for 1, but with M correctly computing the cofinality of ordinals â with cf(â) = ù? Our original Theorem 1.10 had the hypothesis that ù M ⊆ M . Matt Foreman pointed out that this was impossible. We thank him for letting us include his proof here. Theorem 2.4 (Foreman). If ∃j : M −→ e V, j 6= id then ù M 6⊆ M . Proof. Suppose for a contradiction that there exists such an M and j satisfying also the conclusion. Let α = crit(j). By the supposed closure on M it is easy to see that there are α < ë0 . . . < ën < . . . for n < ù with each ën a singular cardinal of cofinality ù and j(ën ) = ën . Then hën in κ, then ð is Σ2 -preserving. For ñ < κ N ð = id ↾ HM . ⊣ In the second stage we show that this interpretation is the correct one: i.e. Hi M does indeed equal Hùñ n . The point is now to show that the ñn , which were defined M from the Hn , are the “right” ordinals. This is stated in the next lemma. Actually for hierarchies involving measures, as here, stronger statements are provable (which are below). We have divided the preservations properties into two lemmas, as for hierarchies involving extenders further amenability assumptions are required to gain this extra strength. For measures we have in any case full amenability which we can exploit. n An embedding ð : M −→Σ(n) M is cofinal if ð is Σ(n) 0 preserving and HM = 0 n n n n . This implies that ð ↾ HM : HM −→Σ0 HM cofinally in the usual sense, ∪ð“HM and that ð : M −→Σ(n) M . 0
20
J. VICKERS AND P. D. WELCH
It only remains to show that the projecta above κ correspond to the ordinal ñn . Lemma 3.28.
m+1 m ñm = ñM if κ ≤ ùñM ;
m m . ñm ≤ ñM if κ ≤ ùñM
m Proof. By induction on m. This is trivial for m = 0. We show ñm ≤ ñM by showing that: (4) hHm , Ai is amenable for A ∈ P(ùñm )∩ Σ1(m−1) (M ). Suppose A ⊆ ùñm and is Σ(m−1),M ([ha, fi]) , say A(x) ⇐⇒ M |= ϕ(x, [ha, fi]) 1 (m−1) with ϕ ∈ Σ1 . Let w = [hb, gi] ∈ Hm . We need A ∩ w ∈ Hm . Let card(a) = n. Define h : [κ]n −→ HM m by h(u) = {t ∈ g(u) | M |= ϕ(t, f(u))}. Then h ∈ Γm . So [ha, hi] ∈ Hm , and using 3.23(iv) [ha, hi] = [hb, gi] ∩ A. m+1 m We now show ñm ≥ ñM if κ ≤ ùñM . We show:
(5) There is A ⊆ ùñm , A ∈ Σ1(m−1) (M ) with A ∈ / M. (m−1) (M ) definable in a parameter p. Suppose A is Σ1 Let A be Σ1(m−1) (M ) in ð(p) = p by the same definition. We show that A∩ùñm ∈ / M. Suppose otherwise. Let A ∩ ùñm = [ha, fi] ∈ M . Then A ∩ ùñm = x ⇐⇒ ∀z m (zm ∈ A ⇐⇒ z m ∈ x).
The right hand side here is a Π(m) 1 , in p, formula. M |= ϕ[[ha, fi], ð(p)] so {u|M |= ϕ[f(u), p]} ⊆ {u|f(u) = A} ∈ Ea , by 3.23 (iv). So ∃uf(u) = A ∈ M . A contradiction. ⊣ n+1 n n > κ. Then ð“PM ⊆ PM . Corollary 3.29. (i) Let ùñM ∗ (ii) ð : M −→E M ; in particular for f ∈ D , [f] = ð(f)(κ).
Proof. (i) This was shown in the last part of the last proof. For (ii) all that we have left to show is (iv) in Def 3.19, that M is the closure of ranð under good n+1 Σ(n) 1 -functions, for ùñM < κ. We’ve extended ð to such functions so it is enough to show M = {ð(f)(a) | f ∈ D , a ∈ [í]n , dom(f) = [κ]n }. By (3) above [cα ] = α for α < κ. As {u | M |= f(u) = f(id (u))} ∈ Ea by the Łos theorem for f ∈ Γ, we have D |= f = cf (id ) and so M |= [ha, fi] = ð(f)(a). ⊣ If we are using measures to take ultrapowers, we can build filter hierarchies with the measures fully amenable with respect to the structures concerned. This enables us to improve the last inequality of Lemma 3.28, and indeed, continue the analysis down below the critical point. However we shall continue to assume that we have an extender ultrapower and impose some additional requirements on E in order to do this. We now use weak amenability to enable us to improve the last lemma and corollary at the point where the projectum crosses the critical point. n+1 n+1 n+1 n > κ ≥ ùñM , and that P (ùñM )∩M = ùñM ∩M . Lemma 3.30. Suppose ùñM Then: n+1 n+1 n n ≤ ùñM = ùñM ; (iii) ùñM (i) ð : M −→Σ(n) M cofinally; (ii) ùñM . 0
Proof. Assume n > 0. n+1 Claim Let p ∈ PM . Then there is a set A ∈ Σ(n−1)M ({ð(p)}) such that A ⊆ ùñn 1 with A ∈ / M.
ON ELEMENTARY EMBEDDINGS FROM AN INNER MODEL TO THE UNIVERSE (n−1) (M ) and M Proof. Let A = Ap(n−1) n−1,p↾n−1 . Then A ∈ Σ1
n,p↾n
M
21
M = hHùñ n , Ai is
amenable. Let B ⊆ Hn be defined by the same Σ1(n−1) formula ϕ in parameter ð(p). M Let H = Hùñ n . ð ↾ H : H −→Σ0 Hn cofinally. SM (1) B = x∈H ð(x ∩ A). Proof. Pick x n ∈ H . Let y n = x n ∩ A ∈ H . Then
M |= ∀z n ∈ x n (z n ∈ y n ←→ ϕ(z n , p)) ⇐⇒ M |= ∀z n ∈ ð(x n )(z n ∈ ð(y n ) ←→ ϕ(z n , ð(p)) (As ð is Σ0 preserving in the pseudo-interpretation.) This then simply says ð(x n ∩ A) = ð(x n ) ∩ B. As ð ↾ H is cofinal (1) follows. ⊣ (2) B ∈ /M n+1 Proof. Assume B ∈ M . Then hHn , Bi ∈ M (by Corollary 3.27.) As p ∈ PM , n,p↾n
n+1 )∩P (ùñM ) with D ∈ / M . As ð ↾ hH , Ai −→Σ0 ,cof hHn , Bi, there is D ∈ Σ1 (M n+1 n+1 and ð ↾ ùñM = id , this D is Σ1 (hHn , Bi) ⊆ M . But by assumption P (ùñM ) is ⊣ the same in both M and M . Contradiction! A′ Now just code B onto ùñn to get a suitable A. As ñn is p.r.closed, and Hn = Jñn , let f be Σ1 (M ) with f : ùñn −→ Hn be onto; let A = {î | f(î) ∈ B}. Then A is as required. ⊣ n From the Claim we see that ùñM ≤ ùñn . The reverse inequality we already established in the last lemma. This gives us (ii) here; and we can now assert (i). The D of (2) is Σ1(n)M (p), and the same definition over M in ð(p) yields a D ′ with n+1 n+1 n+1 =D∈ / M . Hence ùñM ≤ ùñM which is (iii). ⊣ D ′ ∩ ùñM
Remark. It is possible to replace the assumption on weak amenability with just n that RM 6= Ø and get the same conclusions. Definition 3.31. E is Σ1 -amenable with respect to M , if Ea is Σ1 (M ) for any a ∈ [í] κ ≥ ùñM Lemma 3.32. Suppose ùñM , and that E is Σ1 -amenable. Then
(n) P (κ) ∩ Σ(n) 1 (M ) ⊆ Σ1 M .
n ′ n Proof. Let B ∈ P (κ) ∩ Σ(n) 1 (M ). Suppose B(z) ←→ ∃x B (x , z, ð(f)(a)) (n) (n) ′ with B ∈ Σ0 (M ). Using Σ0 cofinality, we can write n n B(z) ←→ ∃u ∈ HM ∃x ∈ ð(u)B ′ (x, z, ð(f)(a)) ←→ ∃u ∈ HM D(ð(u), z, ð(f)(a)) (n) (where D ∈ Σ(n) 0 (M )) Let D ∈ Σ0 (M ) have the same definition. n B(z) ←→ ∃u ∈ HM {v | D(u, z, f(v))} ∈ Ea ←→
∃u n ∃x n (x n = {v | D(u n , z, f(v))} ∧ x n ∈ Ea )
22
J. VICKERS AND P. D. WELCH
(0) for any z < κ. The matrix here is Σ(n) 0 ∩ Σ1 .
⊣ n
Remark. There is a non-uniformity in the above due to the final “x ∈ Ea ”, but that is the only one. We now go down below κ. m Lemma 3.33. Let E be Σ1 -amenable, and assume ùñM ≤ κ. Then m m (i) HM = HM ; (ii) ð : M −→Σ(m) M ; 1
(m) m m (iii) Σ(m) 1 (M ) ∩ P (HM ) = Σ1 (M ) ∩ P (HM ); m m (iv) ð“PM ⊆ PM . m Proof. By induction on m. We first show (i). Note that HM by definition is m A′ ′ and ùñM is a cardinal of M = hJα ′ , B i. Similarly for M = hJαA , Bi. We
A′ Jùñ m , M
′
m m also know that JκA = JκA . So it suffices for (i) to show that ùñM = ùñM . This will m also show (ii), since ð ↾ HM = id . Suppose that m = k + 1.
m m m ≥ ñM : since there is B ⊆ ùñM , B ∈ Σ(k) ñM 1 (M )\M . By the inductive hypoth-
m esis, as ð : M −→Σ(k) M and ð ↾ HM = id , B is thus also Σ(k) 1 (M ). But also 1
m B∈ / M , since P (ùñM ) is the same in both models. m m m m < ùñM with B ∈ Σ(k) ñM ≥ ñM : suppose not. Then there is B ⊆ ùñM 1 (M )\M .
k But then B ∈ Σ(k) 1 (M ) by the previous lemma, if ùñ > κ, and by the inductive hypothesis (iii) for k otherwise. But B cannot be in M , since P (κ) ∩ M ⊆ M . So m m . This proves (i) and as remarked, (ii). ùñM 6< ùñM m m For (iii): let H = HM = HM and suppose B ⊆ H . (k) B ∈ Σ(m) 1 (M ) ←→ ∃D ∈ Σ1 (M ) B ∈ Σ1 (hH, Di) (m) ←→ ∃D ∈ Σ(k) 1 (M ) B ∈ Σ1 (hH, Di) ←→ B ∈ Σ1 (M ).
where the second equivalence uses the inductive hypothesis (iii) or the last lemma. m , p = ð(p). We need to show: This is (iii) for m. We consider (iv). Let p ∈ PM
k,p↾k Claim AM ∈ / M for 1 ≤ k ≤ m. We prove this by induction on m − k. If m = k, then as ð ↾ H = id , and k,p↾k k,p↾k ð : M −→Σ(m) M , so AM = AM , and the latter is not in M , and hence not 1
in M either, because P (H ) ∩ M = P (H ) ∩ M . Suppose it holds for k + 1 ≤ m. k+1,p↾k+1 k,p↾k k+1 k Then AM = B ∩ HM where B ∈ Σ1 (hHM , AM i) in parameter p(k). But k,p↾k k,p↾k k AM ∈ M ⇒ hHM , AM i ∈ M which contradicts the inductive hypothesis for k. ⊣ Collecting together the pieces of the above one obtains (with a small amount m swept under the carpet in (ii) for ùñM < κ, as the argument requires some more (n) analysis of the Σ1 relations than we have catered for here): Corollary 3.34. Let E be Σ1 -amenable. Then (i) ð : M −→Σ∗ M (n) (ii) P (κ) ∩ Σ(n) 1 (M ) = P (κ) ∩ Σ1 (M ) for all n < ù. 3.4. An Upwards Extension of Embeddings Lemma. M
M
Lemma 3.35. Suppose M = hJâE , E M , F M i is a mouse, and Q = hJíE , E M ↾ íi and there is a cofinal embedding ó : Q−→Σ0 Q. We suppose there is κ < í with
23
ON ELEMENTARY EMBEDDINGS FROM AN INNER MODEL TO THE UNIVERSE
Q |= “κ is the largest cardinal”, and either â = í or M |= í = κ + . Suppose that M cf(í) > ù. Then there is ó˜ ⊇ ó and M = hJâE , E M , F M i with M a mouse and n+1 ≥ κ, where κ = ó(κ). Further, if M is sound ó˜ : M −→Σ(n) M , for any n with ùñM 1
m+1 m+1 m m , then ùñM ≤ κ < ùñM , M is ≤ κ < ùñM above κ, and for some m < ù, ùñM
the closure of κ ∪ {p} under good Σ(m) ˜ : M −→Σ(m) M . 1 -functions for a p ∈ M , and ó 1
We first remark that in many situations one may use the lemma with κ a cardinal (or at least the cardinal of a model) which we are not assuming here. One can often improve the conclusion to require that M is sound above κ (and that ó(p ˜ M ) = pM \κ where pM is the standard parameter of M ) - but this requires more work on parameters, which we have eschewn. The proof of this lemma involves forming a “pseudo-ultrapower” M of M that has also a fine-structure preserving extension of ó mapping M into M . The construction is very similar to that for forming an ultrapower by an extender (more precisely it is that for the “long” extender or “derived” extender from the map ó). So naturally it bares many similarities to what we have just done in the previous subsection. Much of what we shall do is redefinition of the previous notions, and many of the changes in the proofs require only a change of notation. We omit some detail therefore. We shall first redefine the class of functions Γ, the stratified functions Γn , and form the term model D , and the “strata” Hn . There is little that can be said about the Hn for n > m + 1 and indeed we shall not define these. The “lift-up” structure M is wellfounded (and iterable) by the crucial use of the uncountable cofinality of í. Definition 3.36. Γ is the set of functions f with dom (f) ⊆ u for some u ∈ Q n+1 n and either f is a good Σ(n) 1 (M ) function for some n with ùñM ≥ í, or f ∈ HM n where ùñM ≥ í. ˙ and D as indicated § 1.1. We define e, I, E, Definition 3.37. D = {ha, fi|f ∈ Γ ∧ a ∈ ó(f)}; set: ∈ ha, bi Ie hb, gi iff ha, bi ∈ ó({hu, vi|f(u) = g(v)}) ˙ ˙ F˙ i. G(ha, fi) iff a ∈ ó({u|f(u) ∈ G}) for G = E, F ; set: D = hD, I, e, E, One can then again prove a Łos theorem for Σ0 formulae: (1) Let ha1 , f1 i, · · · , han , fn i ∈ D and let ϕ be a Σ0 formula. Then − → → −−−→ → → u )]}) D |= ϕ[hai , fi i] ⇐⇒ − ai ∈ ó({− u |M |= ϕ[ fi (− Then I satisfies the axioms for equality, and D satisfies extensionality. If e is wellfounded then there is an isomorphism [ ] : D ←→M g where M is transitive and I [x] = [y] iff x y. We can define ó ˜ : M −→ M by ó(x) ˜ = [h0, {h0, xi}i]. M will Σ 0 ∈ e be of the form hJâE , E, F i, and we shall show that it is a mouse and ó˜ satisfies the properties of the Lemma. Since wellfoundedness of e can be considered a special case of the proposed iterability of M , we omit the proof of the wellfoundedness of e, and simply assume it for the moment. We first note: (2) ó˜ ↾ Q = ó Proof. Note that for f, g ∈ Q [ha, fi] ∈ [hb, gi] ⇔ ó(f)(a) ∈ ó(g)(b). Hence for such f we may define a map [ha, fi] 7→ ó(f)(a). This map is onto and
24
J. VICKERS AND P. D. WELCH
so by the above is the identity. Hence [ha, fi] = ó(f)(a) = ó(f)(a) ˜ and so ó(x) ˜ = [h0, {h0, xi}i] = ó(x). ⊣ Define n n+1 ≥í Γn = {f ∈ Γ | ran(f) ⊆ HM } if ùñM n+1 n n < í ≤ ùñM } if ùñM . = {f ∈ Γ | ran(f) ∈ HM
Hn = {[ha, fi] | f ∈ Γn ∧ ha, fi ∈ D} if Γn is defined. ùñn = On ∩ Hn . Note that Hn is transitive as before. n ≥ í. (3) ó˜ : M −→Σ(n) M cofinally for any n with ùñM 0
Proof. We shall prove (3) in three stages. The first two are just the manoeuvres of Lemma 3.28. The third stage will use the extra assumption on the cofinality of í to check that (3) holds for the m of the crossing projectum of Lemma 3.35. In the k ≥ í on the right hand side first step we interpret the variables of type k with ùñM as varying over Hk . We prove (3) using this pseudo-interpretation. Again we then shall show in the second step that this interpretation is “correct” in that Hk in fact k is HM for any k ≤ m ⊣ The proofs of (4),(5) & (6) are, with minor alterations, those of the last subsection. −−→ n (4) Let hai , fi i ∈ D and let ϕ ∈ Σ(n) 0 where ùñM ≥ í. Then −−→ − →− − → − → M |= ϕ[[ha , f i]] ⇐⇒ a ∈ ó({ u | M |= ϕ[ f ( → u )]}). i
i
i
n (5) ó˜ : M −→Σ(n) M cofinally, for ùñM ≥ í. 0
Proof. We mean here in the pseudo-interpretation. By cofinally, we mean ó˜ ↾ m m HM is cofinal into HM , in case m is defined. But this is straightforward. ⊣ As ó˜ is Σ0 and cofinal into M , acceptability of M translates to acceptability of n n n M , and so Hn = JñMn for ùñM for ùñM ≥ í. It remains to show that ñn = ñM ≥ í. n+1 n n n (6) ñn = ñM if í ≤ ùñM ; ñn ≤ ñM if í ≤ ùñM . Proof. The reader is referred to (4) and (5) of Lemma 3.28. The notational changes are simply that in (4) the function h now has domain dom(g). The map between the two structures is now ó˜ (not ð); and in (5) we need to conclude with: a ∈ ó({u|M |= ϕ[f(u), p]}) = ó({u|f(u) = A}) rather than claim that the set is in Ea . ⊣ We now use the extra condition on Q, í. m+1 m (7) If ùñM then < í ≤ ùñM m+1 m (i) ùñM < í ≤ ùñM ; m (ii) ñm = ñM . m+1 (iii) M is the closure of κ ∪ {r} under good Σ(m) 1 (M )-functions, for some r ∈ RM . Proof. We first show ó(í) ˜ = í = sup ó“í ˜ (setting ó(On ˜ ∩ M ) = On ∩ M ). Let î = ó(f)(a) ˜ < ó(í) ˜ with [ha, fi] ∈ D. Without loss of generality we can assume ran(f) ⊆ í. By the definition of Γ, and using í is a cardinal of M , ran(f) cannot be cofinal in í. Hence there is ç with ran(f) ⊆ ç < í and so ó(f)(a) ˜ < ó(ç) < í. m If ó(κ) = κ, then one may simply check that κ +M = í. By (6) í ≤ ùñM . m+1 m+1 m ˜ = p and p ∈ PM . Suppose ùñM < í ≤ ùñM . Let óp)
25
ON ELEMENTARY EMBEDDINGS FROM AN INNER MODEL TO THE UNIVERSE
m For clause (ii) we only have left to show ñM ≤ ñm . We do this now. Let m,p m A be defined by the cofinal map ó˜ ↾ HM : hM m,p , AM i −→Σ0 hHm , Ai. That S m,p m,p ˜ ∩ AM ). Then hHm , Ai ≺Σ0 M . By the Downwards is, let A = x∈H m ó(x
f, pe such that M fm,pe = hHm , Ai Extensions of Embeddings Lemma, there are M m f −→ (n) M with and pe ∈ RM e . We have the further properties that there is ð : M Σ M
0
m e = p. Since p ∈ RM ð ↾ Hm = id and ð(p) we can factor through, and see that f such that óˆ ↾ H m = ó˜ ↾ H M and ó(p) e Then there is óˆ : M −→ (n) M = p. ˆ Σ1
M
M
ð(ó(f)(a)) ˆ = ó(f)(a) ˜ for ha, fi ∈ D . Hence ðóˆ = ó˜ and ð is onto; hence ð = id , m f óˆ = ó, ˜ and M = M . Then A cannot be in M whilst it is Σ(m) 1 (M ). Hence ñM ≤ ñm . m+1 m Let p ∈ PM and let p = ó(p). ˜ Any x ∈ HM has the form ó(f)(a) ˜ where f ∈ Γm (M ) and a ∈ ó(dom(f)) If f ∈ hM m,p (i, hî, p(m)i) some î < κ, then ó(f)(a) ˜ = hM m,p (j, hhî, ai, p(m)i) for some j < ù, î = ó(î); hence M m,p ⊆ hM m,p (í ∪ {p(m)}). But as at the beginning of this proof, hM m,p (κ ∪ {p(m)}) m,p is cofinal in í. As í = (κ + )M , (or equals On ∩ M m,p ), this suffices to show m,p hM m,p (κ∪{p(m)}) ⊇ M . This shows that both that M is the closure of κ∪{p(m)} m+1 under good Σ(m) ≤ κ, which are clauses (i) and (iii). ⊣ 1 −functions, and that ùñM Claim M is wellfounded and iterable above κ. Proof. Suppose not; let I = hMα |α < èi be an iteration of M0 = M above κ that cannot be continued. Then there is a set {xi }i∈ù ⊆ M of supports of functions coding an illfounded ∈è -chain through the ordinals of a model Mè . We treat wellfoundness of M as a special case of iterability. In this case set xi+1 ∈ xi as a descending e-chain through M = M0 . Let H be some sufficiently large ZFC − -model that is transitive and contains I, {xi }. Let I ∪ {xi } ⊆ Y ≺ H where Y is countable. Let k : H ←→Y g be the inverse of the transitive collapse. Let k(hN, κ ′ , x i , I i) = hM, κ, xi , I i. Then in H , {x i } is a set of supports for functions coding illfoundedness of the last model M è of I = hM α |α < èi. Let {yi }i∈ù enumerate N . n , or fi a good Σ(n) Then each k(yi ) is of the form ó(f ˜ i )(ai ) for some fi ∈ HM 1 (M ) function with domfi ∈ Q. We define by cases a model P and a map ð : P −→ M . n Case 1. n ≥ 1 and í < ñM . Let ì < í be chosen so that n,p
(i) P = df hM |ì, An,p |ìi is amenable, ñP1 ≤ κ. (ii) supi domfi < ì n,p n,p i (iii) id : P −→Σ1 hM , AM n (i) & (ii) are clearly possible since cf(í) > ù. As í < ñM , í is a cardinal of n,p
n,p
n,p
M , so hM |í, An,p |íi ≺Σ1 hM , An,p i. But there are arbitrarily large ì < í n,p with such a P ≺Σ1 hM | í, An,p |íi. Hence P exists. Now any fi is either in M n,p (k) n H = M or has a good Σ (M ) definition involving parameters î~i < ùñ k . By M
1
M
the soundness of M above κ, either way such an fi has a good Σ(n) 1 (M ) definition ~ involving parameters îi below κ.
26
J. VICKERS AND P. D. WELCH
By the Extensions of Embeddings Lemma there is a premouse P, iterable above κ, and q ∈ RPn , and a ð : P −→Σ(n) M , with P = P n,q and ð(q) = p, and 0
(n) n,p n n An,q |ì. As An,q P = A P ∈ HM , P ∈ HM . Let f i have the same good Σ1 definition over P from î~i , q as fi did over M in î~i , p. Note that ð is in fact Σ(n) preserving. 1
Let ó(p) = p. ˜ But as domfi = domf i and ð is Σ(n) 1 -elementary ð(f i (u)) = fi (u) for any u ∈ domf i . n Case 2. n ≥ 1 and í = ñM . ì may be chosen as in Case 1, however in (iii) we may only have that P is Σ0 n,p n,p embedded into hM , AM i by id. However this suffices to still extend id to a ð : P −→Σ(n) M , but which is nevertheless Σ1(n−1) preserving. (This is just part of 0
the Extensions of Embeddings Lemmas.) However we may now at no additional cost impose on ì, using cf(í) > ù, and the Case hypothesis: n (iv) Any fi that has a good Σ1(n−1) (M ) definition in parameters î~ < ñM , then such a î~ is below ì. Again we have ð : P −→Σ(n) M , with P a premouse iterable above κ, P ∈ HíM . 0
As ð is Σ1(n−1) preserving, we can let f i be the functions defined over P with the same definitions as that of fi over M , with supi domfi < On ∩ P. In both cases ˜ where P˜ = ó(P), by l (yi ) = ó(f we define l : N −→ P, ˜ i )(ai ). l will embed, in particular, the supports for a descending sequence of ordinals of an iterate of P˜ above κ. This will contradict the latter’s iterability above κ. (9) l is Σ(n) 0 preserving. Proof. P˜ |= ϕ(l (yi )) ⇐⇒ P˜ |= ϕ(ó(f ˜ i )(ai )) ⇐⇒ ai ∈ {u | P˜ |= ϕ(ó(f i )(u))} ⇐⇒ ai ∈ ó({u | P |= ϕ(f˜i (u))}) (as ó ↾ P is Σù ) ⇐⇒ ai ∈ ó({u | M |= ϕ(f i (u))}) (n) (as ð is Σ(n) 0 or Σ1 depending on the case)
˜ i )(u))} (as ó˜ is Σ(n) ⇐⇒ ai ∈ {u | M |= ϕ(ó(f 0 ) ⇐⇒ M |= ϕ(ó(f ˜ i )(ai )) ⇐⇒ N |= ϕ(yi ) (as k is Σù ).
⊣
Note that P˜ is truly iterable above κ, since “P is iterable above κ” holds in some admissible set A with P ∈ A ∈ M , and hence by elementarity holds in A˜ = ó(A ) ˜ But ù1 ⊆ A˜ so this is absolute. But this is now a contradiction since of P. {l (x i )}i∈ù contains supports coding an illfounded iteration of P˜ above κ. Case 3. n = 0. A standard argument shows that cf(í) > ù =⇒ cf(On ∩ M ) > ù. As ó˜ is Σ0 -cofinal the same is true of M . There is thus ô˜ = ó(ô) with (i) ∀y k(y) ∈ Jô˜M (ii) ∃ fn ∈ JôM ∃an ∈ Q
ó(f ˜ n )(an ) = k(yn ).
ON ELEMENTARY EMBEDDINGS FROM AN INNER MODEL TO THE UNIVERSE
27
We can assume that ô˜ is chosen sufficiently large so that each fn can be taken ˜ . in hM |ô (κ ∪ pM ). Let X ≺Σ1 JôM be such that í ∪ {pM } ⊆ X , let ð : P ←→X P ~ Let ð(q) = pM , and let f n have the same Σ1 ({q, în }) definition as fn had in ~ Note that sup domfi ≤ í ∩ X . Also note that P ∈ J M (since ΣM |ô ({p , î}). M
i
í
A1P ⊂ JκP , as ñP1 ≤ κ, and then a simple comparison argument shows A1P ∈ M ). Again define l : N −→ P˜ =df ó(P) by l (yn ) = ó(f ˜ n )(an ). (10) l is Σ0 -preserving. Proof. The equivalences are exactly those of (9), noting in this case that ð : ⊣ P −→Σ1 M |ô, and so ð : P −→Σ0 M . But P˜ is iterable above κ as before, so l embeds a supposedly non-iterable premouse above κ ′ into one that is iterable above l (κ ′ ) = κ! This is a contradiction. ⊣ REFERENCES
[1] A. J. Dodd, The Core Model, Lecture Notes in Maths, vol. 61, London Mathematical Society, 1982. [2] H-D. Donder, R. B. Jensen, and B. J Koppelberg, Some applictions of the Core Model, Set Theory and Model Theory, Springer Lecture Notes in Maths, vol. 872, 1981, pp. 55–97. [3] R. B. Jensen, The Core Model for Measures of Order Zero, 1989, circulated manuscript. [4] R. B. Jensen and J. Vickers, A covering lemma part 2, manuscript. [5] , Absoluteness of J´onsson Cardinals, 1997, preprint. [6] A. Kanamori, The Higher Infinite, Perspectives in Mathematical Logic, Springer Verlag, Berlin, 1994. [7] A. Miller, Some interesting problems, Set Theory of the Reals, (Proceedings of the Israeli Mathematical Conference), vol. 6, American Mathematical Society, 1993, pp. 645–654, http://www.math.wisc.edu/∼ miller. [8] J. R. Steel, The Core Model Iterability Problem, Springer Lecture Notes in Maths, vol. 8, Springer, 1988. [9] J. Vickers, A covering lemma part 1, 1990, manuscript. [10] P. D. Welch, Countable unions of simple sets in the Core Model, this Journal, vol. 61 (1996), no. 1, pp. 293–312. [11] W. H Woodin, Σ21 -absoluteness and supercompact cardinals, May 1985, handwritten notes. UNIVERSITY OF BRISTOL DEPARTMENT OF MATHEMATICS BRISTOL BS8 1TW, ENGLAND KOBE UNIVERSITY GRADUATE SCHOOL OF SCIENCE & TECHNOLOGY ROKKO-DAI, NADA-KU, KOBE 657, JAPAN
E-mail: [email protected]
