the mitchell order below rank-to-rank

The Mitchell Order below Rank-To-RankAuthor(s): Itay NeemanSource: The Journal of Symbolic Logic, Vol. 69, No. 4 (Dec., 2004), pp. 1143-1162Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/30041780 .

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THE JOURNAL OF SYMBOLIC LOGIC

Volume 69, Number 4, Dec. 2004

THE MITCHELL ORDER BELOW RANK-TO-RANK

ITAY NEEMAN

Abstract. We show that Mitchell order on downward closed extenders below rank-to-rank type is wellfounded.

11. Introduction. Given two extenders E and F set E <1 F iff E E Ult(V, F). Intuitively < is regarded as a relation of potency on extenders. If E belongs to Ult(V, F) then F is viewed as more potent than E. The relation, known now as the Mitchell order though it need not be transitive, was introduced by Mitchell [3] in connection with his inner models for sequences of measures. One particular question about this relation has since taken on a life of its own, and this is the question of wellfoundedness.

The wellfoundedness of < on normal measures was established already as part of the work in Mitchell [3]. Steel [4] later established the wellfoundedness of < on amenable extenders,1 and conjectured that wellfoundedness should continue to hold further up, for all extenders below rank-to-rank type (see Section 2 for the definition). This is the most one could hope for, since wellfoundedness is easily violated using extenders of rank-to-rank type.

Steel obtained the wellfoundedness of < on amenable extenders by proving a much stronger result. Building on Martin-Steel [2] he showed that any non-overlapping (see Remark 5.7), length w iteration tree composed of amenable extenders has an infinite branch leading to a wellfounded direct limit. The existence of branches of this kind is crucial to current constructions in inner model theory, and it was conjectured that it too should continue to hold further up, for all extenders below rank-to-rank type.

Unfortunately the stronger conjecture fails. In Section 5 we give an example of a non-overlapping length ow iteration tree which does not have an infinite branch- indeed not even a branch of length 4--let alone an infinite branch with a wellfounded

Received September 4, 2003; accepted June 7, 2004. This material is based upon work supported by the National Science Foundation under Grant

No. DMS-0094174.

1An extender E is amenable if: (a) do(E) is a successor ordinal; and (b) strength(E) < iE (X) where x is least so that 2x > do(E) and iE denotes the ultrapower embedding by E. These two conditions are equivalents of the conditions in the definition of amenability in Steel [4, p. 934], and phrased using the notation and terminology of Section 2 of the current paper. Non-amenable extenders appear a little above superstrong extenders. For example the extenders assumed in Example 5.2 below are non-amenable; they fail to satisfy condition (a).

1 2004, Association for Symbolic Logic 0022-4812/04/6904-0008/$3.00

1143



1144 ITAY NEEMAN

direct limit. The tree uses non-amenable extenders at a level slightly above superstrong, but well below rank-to-rank type.

The original conjecture on the other hand does hold, at least with a slight restriction. In Section 4 we prove that the Mitchell order is wellfounded on downward closed extenders below rank-to-rank type. The restriction of downward closure (see Section 2 for the definition) is aesthetically annoying and we do not know whether it can be removed. But in practice it does not matter much, since the extenders which actually come up in the study of large cardinals satisfy this, and even stronger, closure conditions.

This paper is organized as follows: In Section 2 we list the relevant definitions. These include the definition of extender; the term has slightly different meanings in different papers and we want to be precise on the meaning it has here. In Section 3 we formulate and prove a lemma which is later used for the main argument on wellfoundedness, in Section 4. Finally in Section 5 we give a couple of pathological examples, including the one mentioned above of a length cw iteration tree with no branches of length 4.

12. Preliminaries. Let (*, *) denote the Gddel pairing operation on ordinals. Given sets of ordinals A and B define A x B to be {(a, f) a c A A fl E B}. Note that A x B is then a set of ordinals too. We refer to it as the product of A and B. In general define finite products of sets of ordinals as follows: For n = 0 set Hi<nAi equal to Ao; for n > 0 set Hi<nAi equal to (Hi<n-lAi) x An. Define finite sequences of ordinals similarly by setting (a) equal to a, and setting (ao, ..., an) equal to ((ao,..., an 1), an) for n > 0.

If A is a set of ordinal sequences of length n, and u: n -+ n is a permutation of n, then define aA by setting (ao..., an-i) E aA

e= (a,(0o) .... a1 (n-1)) E A. If A is a set of ordinal sequences of length n + 1, then define bp(A) to be the

set {(ao,... Ian-_1)

(3] E ao)(ao,...a,n-1,) E A}. We refer to bp(A) as the bounded projection of A.

By a fiber through a sequence of sets (Ai | i < w) we mean a sequence (ai i < co) so that (Vi < wo) (ao..., ai) E Ai.

We call an ordinal r nice if cofinally many 1 < r are closed under G6del pairing. Note that we allow the case of successor q. A successor ordinal r is nice if r - 1 is closed under Gddel pairing. Note that if r is nice then U< q 9 () is closed under finite products.

DEFINITION 2.1. An extender is a function E which satisfies the following conditions:

1. The domain of E is equal to UO< 9(1) for some nice ordinal r. 2. E sends ordinals to ordinals and sets of ordinals to sets of ordinals. E is

non-trivial in the sense that there is some ordinal in the domain of E which is not sent to itself.

3. E respects products, intersections, set differences, membership, the predicates of equality and membership, permutations, and bounded projections. More precisely this means that for all A, B E dom(E), all ordinals a E dom(E), and all permutations a of the appropriate format:



THE MITCHELL ORDER BELOW RANK-TO-RANK 1145

(a) E(A x B) = E(A) x E(B), E(A n B) = E(A) n E(B), and E(A - B) = E(A) - E(B);

(b) a E A =-* E(a) E E(A); (c) E({(a,fl) AxA I a = fl}) isequalto {(a,fl) C E(A) xE(A) I a = l},

and similarly with a cE f replacing a = Pf; (d) E(uA) = aE(A); and (e) E(bp(A)) = bp(E(A)).

4. E is wo-complete. Precisely this means that for any sequence (Ai i < wo) of sets which are each in the domain of E, if there exists a fiber through (E(Ai) I i < w) then there exists also a fiber through (Ai I i < ao).

The ordinal r of condition (1) is called the domain ordinal of E, denoted do(E). The set UA~dom(E) E(A) is called the support of E, denoted spt(E). Using condition (2) it's easy to see that the support of E is an ordinal.

By a pre-extender we mean an object E which satisfies conditions (1)-(3) in Definition 2.1, but not necessarily condition (4). The point of this distinction is that condition (4) involves second order quantification over E, whereas conditions (2) and (3) involve only bounded quantifiers over the transitive closure of E. By removing condition (4) we obtain an absolute notion:

CLAIM 2.2. Let Q be a wellfounded model of set theory and let r E Q be an ordinal. Suppose that (V1 < r) 19VG) = OQ(Q). Let E E Q and suppose that Q ="E is a pre-extender with domain ordinal q'". Then E is a pre-extender also in V. -d

The corresponding claim for extenders is false: E may be an extender in Q yet fail to be co-complete in V.

Definition 2.1 follows the fashion in fine structure where extenders are now viewed as restrictions of elementary embeddings, see for example Zeman [5, 12.1]. Earlier approaches described extenders as directed systems of measures, see for example Kanamori [1, 126]. The two fashions are easily interchangeable, and the choice of fashion is largely a matter of convenience. One point worth emphasizing is our treatment of the domain. The domain of E consists not of subsets of q = do(E) but of subsets of ordinals 1 < r. In this we differ from the treatment in Kanamori [1].

REMARK 2.3. The critical point of an extender E is the least ordinal a so that E(a) . a. An extender E is short if do(E) = crit(E) + 1. In other words an extender is short if its domain includes only subsets of its critical point and no more. (This is the smallest the domain could be.) It should be noted that many papers-including Martin-Steel [2]-use the term extender to refer to short extenders, building a restriction of shortness into their definitions. The demand of shortness in these papers limits their extenders to at most the level of superstrong (see below for the definition).

Extenders are naturally induced by elementary embeddings. Let 7n: V -- M be an elementary embedding of V into some wellfounded class model M.2 Let 2 be an ordinal closed under G6del pairing. Let r be least so that sup(n"rl)

_> 2. Suppose

that r is nice. Define the i-restriction of n

to be the map E given by:

2As a matter of convention when we say a wellfounded model of set theory we mean a transitive model equipped with the standard membership relation E.



1146 ITAY NEEMAN

(RI) dom(E) = U< 9( ); and (R2) E(x) = n(x) n A for each x E dom(E). Our choice of y here is such that the full map x H [ir(x) n ], for x E V, can be directly recovered from E. Indeed our q is the least ordinal which allows for this.

REMARK 2.4. Whenever we talk about the A-restriction of an embedding ar below, we assume that A is closed under G6del pairing and that the corresponding ordinal r is nice. We sometimes neglect to mention this assumption explicitly.

It is easy to check that the A-restriction of 7r is an extender. The items in condition (3) of Definition 2.1 for the most part follow directly from the elementarity of 7r. Condition (3e) uses also the absoluteness between M and V of formulae with only bounded quantifiers. Condition (3b) uses, among other things, our particular choice of r in defining the A-restriction of nr (if q were chosen larger the condition would fail). Condition (4) follows using the elementarity of 7r and the wellfoundedness of M. If a fiber through (E(Ai) I i < aw) exists in V then using the wellfoundedness of M such a fiber must also exist in M. Its existence can then be pulled back via 7 to yield a fiber through (Ai I i < o).

REMARK 2.5. The A-restriction makes sense also in the case of an embedding into an illfounded model M, so long as the wellfounded part of M contains 2. But ow- completeness may fail in this case, and the A-restriction need only be a pre-extender.

The description above shows how extenders are induced by elementary embeddings into wellfounded models. Extenders also give rise to such elementary embeddings, through the ultrapower construction, which we describe next.

Fix an extender E. Let 9 be the class of functions f E V so that dom(f) belongs to dom(E). Let 9 = {(f, a) If C / A a E E(dom(f))}.

For two functions f,g E set Zg - {(a, fl) | f(a) = g(fl)} and Zf = f,g

{(a, fl) I f(a) e g(fl)}. Both Z7g and Zfg are bounded subsets of / - do(E),

since dom(f) and dom(g) are bounded subsets of q and cofinally many r < r are closed under G6del pairing. It follows that Z;g and Zf, are both elements of the domain of E.

Define a relation r on by setting (f, a) - (g, b) iff (a, b) E E(Z7,g). One can check using condition (3) in Definition 2.1 that - is an equivalence relation. Let [f, a] denote the equivalence class of (f, a). Let 9* denote 1/'. Define a relation R on 9* by setting [f, a] R [g, b] iff (a, b) E E(Zg). Again using condition (3) in Definition 2.1 one can check that R is well defined.

The following property, known as Los' Theorem, can be proved from the various definitions, by induction on the complexity of p1: Let [fl, al],.., [f,, an] be elements of -*. Let

Op(vl... , Vn) be a formula. Let Z = {(al..., an) I V

p[fl(al) ..., f,(a,)]}. Then (9*, R) op[[fl,

a], ...,[fn,a,]] iff (al,..., an) belongs to E(Z).

For each set x let cx be the function with domain {0} and value cx (O) = x.

From Log' Theorem it follows that the map x - [cx, 0] is elementary, from V into (9*, R). (In particular then (9*, R) satisfies ZFC.)

Finally, using the w-completeness of E, one can check that the relation R is wellfounded: if ([fi, ai] I i < w) were an infinite descending sequence in R, then the




sequence of sets Ai = {(ao,..., ai) I fo(ao) ~ fi(al) 9 ... fi(ai)} would violate

co-completeness. The structure (9*, R) is therefore a wellfounded model of ZFC.

DEFINITION 2.6. The (internal) ultrapower of V by E, denoted Ult(V, E), is the transitive collapse of the structure (0*, R). The (internal) ultrapower embedding is the map j: V --+ Ult(V, E) defined by j(x) = [cx, 0]. (We are abusing notation here, identifying the ultrapower with the structure (9*, R) rather than its transitive collapse.)

Let 2 = spt(E). Using the various definitions one can prove the following two properties. The first relates the ultrapower embedding back to the extender E, and the second describes a certain minimality of the ultrapower:

(U1) The A-restriction of j is precisely equal to E; (U2) Every element of Ult(V, E) has the form j(f)(a) for some function f E

and some a E 2.

These properties (and the transitivity of Ult(V, E)) in fact determine the ultrapower and the embedding completely.

REMARK 2.7. The ultrapower construction makes sense also in the case of a pre- extender E. In this case too Ult(V, E) and j satisfy conditions (Ul) and (U2), but Ult(V, E) may be illfounded.

It's worthwhile seeing what happens when the two directions-from elementary embedding to extender and from extender to elementary embedding-are combined:

LEMMA 2.8. Let 7r: V --+ M be an elementary embedding of V into a wellfounded model M. Let 2 be an ordinal and let E be the A-restriction of tr. Let N = Ult(V, E) and let j: V -+ N be the ultrapower embedding.

Then there is an embedding k: N -- M with 7r = k o j (see Diagram 1) and crit(k) > 2 . -

Thus the ultrapower by the A-restriction of i captures n up to 2.

N= Ult(V, E)

k

M 7t1

V J

Diagram 1. The original map i and the ultrapower map j.

We say that 7n: V - M is a-strong just in case that 9() C M for all 1 < a. An extender E is a-strong just in case that 9(G) c Ult(V, E) for all 1 < a. Using Lemma 2.8 it is easy to obtain the following:

LEMMA 2.9. Let rn: V - M be a-strong. Let 2 be an ordinal and let E be the A-restriction of n. Suppose that 2 > (2<a)M. Then E too is a-strong. -A

The strength of ir: V -- M is defined to be the largest a so that 7r is a-strong. The strength of an extender E is defined similarly, using the ultrapower. (Notice



1148 ITAY NEEMAN

that the strength of an embedding is always a cardinal.) Lemma 2.9 shows that extenders are adequate means for capturing the strength of embeddings.

Two models Qi and Q2 are said to agree to an ordinal p just in case that for every S< p, Qi and Q2 have precisely the same subsets of 1. The level of agreement of Qi and Q2 is the largest p so that Qi and Q2 agree to p. In the case of models V and Ult(V, E) this is the same as the strength of E.

Let M be a model of ZFC and let E be an extender in the sense of M. Let P be another model of ZFC. Suppose that M and P agree to do(E). Let jP be the class of functions f in P so that dom(f) E dom(E), and modify the ultrapower construction described earlier to use

-'V instead of F. The sets Zg and Z E

which come up during the construction are now elements of P (this is because the functions f and g belong to P). Both are bounded subsets of do(E), and the assumed agreement between P and M is enough to imply that they belong to M. It follows that they belong to the domain of E, so that the reference to E (Zfg) and E (Z g) made during the ultrapower construction continues to make sense in the current, more general, settings. The construction thus gives rise to an (external) ultrapower of P by E, denoted Ult(P, E), and an (external) ultrapower embedding j: P -* Ult(P, E). The word "external" here indicates that E is applied to a model to which it does not belong. Note that the proof of wellfoundedness does not adapt from the internal to the external settings. The w-completeness of E in M is not enough to secure the wellfoundedness of Ult(P, E).

The ability to take an extender in one model, and use it to form the ultrapower of another model, is key to the concept of iteration trees, defined in Martin-Steel [2]. We shall make such uses in Section 5. In Section 4 we shall only need ultrapowers in the more special situations of M C P, and in such cases the ultrapowers are really internal to P. But already there we shall need the following fact:

FACT 2.10. Let P and M be models of ZFC. Let E be an extender in the sense of M. Let a be the level of agreement between P and M. Suppose that a > do(E) so that Ult(P, E) makes sense. Let j: P -- Ult(P, E) be the ultrapower embedding of P by E, and let j*: M -+ Ult(M, E) be the ultrapower embedding of M by E. Then:

1. j[a =j*[a; and

2. Ult(P, E) and Ult(M, E) agree to sup{j(]) + 1 I r < a}.

The fact is proved by observing that for each 1 < a: (a) only functions f with range(f) C 1 are relevant to the definition of j(c); (b) only functions f with range(f) c (1) are relevant to the definition of Ult(P, E) n 9 (j(])); and (c) the agreement between P and M is sufficient to imply that FP and FM have precisely the same such functions.

Notice that sup(j" do(E)) > spt(E) (indeed, do(E) is the least ordinal with this property). If do(E) is a successor ordinal, then from this and condition (2) in Fact 2.10 it follows that Ult(P, E) and Ult(M, E) agree to spt(E) + 1. But if do(E) is a limit ordinal then the two ultrapowers need only agree to spt(E). This restricted level of agreement between Ult(P, E) and Ult(M, E) appears in cases where the level of agreement between M and P is precisely equal to do(E). Situations of this kind can be manufactured using large cardinals slightly beyond the level of superstrong




(defined in the next paragraph). It is precisely these situations that hold back the proof of the wellfoundedness of the Mitchell order in Steel [4], as the argument there uses agreement to spt(E) + 1.

Let E be an extender and let j be the ultrapower embedding by E. The critical sequence of E is defined as follows: Ko = crit(E); and n,+1 = j(K,) for each n < co. E is said to be superstrong if strength(E) > .1. E is said to be of rank-to-rank type if strength(E) > supn<. En. (It is easy to check in this case that spt(E) must also be greater than supn< ,, . This in turn implies that do(E) > supn<w in too.) If strength(E) <

supn<o , then E is said to be below rank-to-rank type. CLAIM 2.11. Let E be an extender and let j be the ultrapower embedding by E.

Suppose E is below rank-to-rank type. Then for every a c (crit(E), strength(E)] there exists some inaccessible cardinal C so that C < a and j (C) > a.

PROOF. Let (rn, n < co) be the critical sequence of E. Since E is below rank-to- rank type we have strength(E) < sup,,<. , so a <

supn<o Kn~. Let n be least so that Kn+ 1 a. If Kn+1 > a take ( = Kn. If ,n+1 = a then a is a limit of inaccessible cardinals and letting ( be the first inaccessible cardinal above ri works. -4

CLAIM 2.12. Let M be a transitive model of set theory and let a be the level of agreement of M and V. Let E be an extender in M and let fl be the strength of E in M. Suppose do(E) < a, so that E is a pre-extender in V (by Claim 2.2). Suppose further that E is co-complete in V.

Suppose that E is below rank-to-rank type. Then the level of agreement between V and Ult(V, E) is equal to min{a, fl}.

PROOF. If E were of rank-to-rank type in the sense of M then its domain ordinal would have to be at least the supremum of its critical sequence. The agreement between M and V would then imply that E is of rank-to-rank type in the sense of V, contradicting the assumption of the claim. So M '"E is not of rank-to-rank type".

Let j* be the ultrapower embedding of M by E, and using Claim 2.11 fix C < min{a, fl} so that j*(C) > min{a, /#}. By Fact 2.10, Ult(V, E) and Ult(M, E) agree to j*(C) + 1, and hence agree past min{a, /f}. From this, the fact that V and M agree precisely to a, and the fact that M and Ult(M, E) agree precisely to P, it follows that V and Ult(V, E) agree precisely to min{a, fi}. -

Let us finally turn to the matter of downward closure.

DEFINITION 2.13. A pre-extender E is downward closed just in case that for every y < do(E), there exists some A E dom(E) and some a E spt(E), so that {( (a, 1) c E(A)} is precisely equal to E"y.

Our formulation in Definition 2.13 should make it clear that downward closure involves only bounded quantifiers over the transitive closure of E, and is therefore absolute between transitive models of set theory.

For extenders, downward closure is easily seen to be equivalent to the condition that for every cardinal y < do(E), E"y belongs to Ult(V, E). Put another way, downward closure states that E cannot begin to measure subsets of y unless E"y is put into the ultrapower. This is a weak form of an initial segment condition, a condition stating that certain strict restrictions of E must belong to the ultrapower by E.



1150 ITAY NEEMAN

CLAIM 2.14. Let E be an extender and let j: V -* Ult(V, E) be the ultrapower embedding by E. Let P be a transitive set with card(P) strictly less than the domain ordinal of E. Suppose that E is downward closed. Then P and j [P belong to Ult(V, E).

PROOE Let y = card(P). Let h: y - P be a bijection. Then j(h) maps j"y onto j"P. Since both j(h) and j"y belong to Ult(V, E) (the latter by downward closure), so does j"P. P and j [P can be recovered from j"P, since P is the transitive collapse of j"P, and j r P is the anti-collapse embedding. -d

REMARK 2.15. It follows from Claim 2.14 that the strength of a downward closed extender E is at least its domain ordinal. To see this, let 1 < do(E) and let X be a subset of 1. Use Claim 2.14 with P = U {X} to get X E Ult(V, E).

13. A note on co-completeness. Let R be a wellfounded model of set theory and let F be an extender in the sense of R. Let q = do(F). Suppose that R agrees with V to r. F is then a pre-extender in V by Claim 2.2. We present here a condition on R sufficient to guarantee that F is co-complete (hence an extender) in V.

LEMMA 3.1. (Under the settings above.) Suppose that (ql+)R is equal to (q+)v. Then F is co-complete in V.

PROOF. Suppose that F is not co-complete in V. Fix a sequence (Ai I i < wo) E V so that each Ai is in the domain of F, there is a fiber through (F(Ai) I i < wo), and there is no fiber through (Ai i < wc). We shall use the wellfoundedness of Ult(R, F) in conjunction with the assumption that (r+)R = (r+)v to argue for a contradiction.

Let 60 = sup Ao and for each i > 1 let 6i = max{6il, sup Ai }. Since Ai belongs to the domain of F, 6i is smaller than r. If supi< 6i is smaller than r then the entire sequence (Ai I i < co) can be coded by a bounded subset of q, and therefore belongs to R. But having (Ai i < co) in R immediately contradicts the w-completeness of F in R. We may therefore assume that supi<, 6i is equal to q.

Let T be the tree of attempts to construct a fiber through (Ai I i < wc). More precisely, nodes of length I in T are sequences (ao, ..., al-1) so that for each n < 1, (ao, ..., an) e A,. Since there is no fiber through (Ai I i < co), T is a wellfounded tree. Let v be its rank.

T is a tree involving ordinals smaller that supi< ,, 6i = . It follows that the rank of T is at most (qr+)v. Using the assumption in Lemma 3.1 that (q+)R - (r+)V we get v < (q+)R. We may therefore fix in R a wellordering w of 17 so that the order type of w is equal to the rank of T. Working in V let

po: T -+ w be an order

preserving embedding. For each 1 < co let T, be the part of T consisting of nodes of length at most 1. We

have then T = Ul<o T1. T, can be defined using just the sets Ao,..., Ai-1. Each of

these sets is in the domain of F, hence in R. So Tl

ER for each I < ow. For each I < w let cp = {(&, op(c)) i e T1 and

p(dc) < 6}. We have then

o = U1<~ pt.

l is coded by a subset of

1i. Since R and V agree to rq > 6i we have

Ol E R for each l. In sum we have the entire wellordering w in R, we have

Io and

Tl in R for each 1,

and we know that U,<o cp is an order preserving embedding of U1<m Tl into w.




Let R* = Ult(R, F) and let j: R - R* be the ultrapower embedding. Note that R* is wellfounded, since R ="F is w-complete". Let T* = Ul<. j(Ti) and let S

* = Ut<,

j(pl). These definitions make sense since T, and

1p belong to R.

CLAIM 3.2. T* is the tree of attempts to create afiber through (F (Ai) I i < wc). PROOF. Nodes of length 1 in T, are sequences of the form (ao, ... al-1) so that

for each n < 1, (ao,..., a, ) E An. For a fixed I this statement can be phrased inside R. By the elementarity of j then, nodes of length 1 in j(Ti) are sequences of the form (ao..., al-1) so that for each n < 1, (ao,..., an) E j(An) = F(An). This is true for each I < co. The claim follows.

COROLLARY 3.3. T* is illfounded. PROOF. Remember that we are working with a sequence (Ai I i < co) so that there

is a fiber through (F(Ai) I i < o). -1 CLAIM 3.4. The domain of(p* is equal to T*.

PROOF. Fix 1 < co. We prove that j(Tl)

is contained in the domain of 1o*. For each k < wm let Ck =

-f' E Tl Ip(a) < 6k}. Then T, = Uk<w Ck since

the 6k-s are cofinal in qr. Each Ck is a subset of Tl

and hence can be coded by a subset of 3i. Thus the entire sequence (Ck I k < Co) can be coded by a subset of o. x 61. The sequence therefore belongs to R. We may thus apply the elementarity of j to the fact that T, = Uk<, Ck and conclude that

j(Tl) = Uk< J (Ck). Now

d E T n Ck ==- -

e dom(pomax {l,k}). Applying j to this and using our previous conclusion that j(Ti) = Uk<, J(Ck) it follows that j(Ti) C dom(Uk< i (Ok)). -i

CLAIM 3.5. (p* is order preserving from T* into j(w). PROOF. Given the previous claim it is enough to show that each j('pl) is order

preserving from its domain into j(w). But this is immediate using the elementarity of j, since the statement "'p is order preserving from its domain into w " is true and can be phrased in R. -

Remember that w is a wellordering inside R. Using the elementarity of j we have R* ""j(w) is a wellordering". Since R* itself is a wellfounded model, it follows that j(w) is wellfounded in V. But Corollary 3.3 and Claim 3.5 together show that j(w) is illfounded, giving a contradiction. -1 (Lemma 3.1)

14. The Mitchell order. We take this section to prove: THEOREM 4.1. The Mitchell order is wellfounded on downward closed extenders

below rank-to-rank type. PROOF. Let (En I n < o-) be a sequence of downward closed extenders, all below

rank-to-rank type. Suppose for contradiction that the sequence is descending in the Mitchell order. In other words suppose that En+1 E Ult(V, E,) for each n < c. We work to derive a contradiction.

For each n < co let r, = do(En). Let a = limsupn<,,,n. In other words let a = infm<, supn>m in. We refer to a as the level ordinal of the sequence (En in < c).

Replacing (En | n < w) we may assume that its level ordinal is minimal among counterexamples to Theorem 4.1. In other words we may assume that:

1. There is no counterexample to Theorem 4.1 with level ordinal smaller than a.



1152 ITAY NEEMAN

Our strategy is to try and absorb a tail-end of (En I n < co) into one of the models Mn for n > 0. Then inside Mn there will a counterexample to Theorem 4.1 with level ordinal a. This would contradict the shift of condition (1) to Mn since we'll have a < jo,n (a).

We divide the argument into three main parts. First we make some observations on the sequence (E, I n < co) and bring it to more convenient form by dropping initial segments of the sequence if needed. Then we define a chain of elementary substructures with specific properties in V which can be reflected into M1. Finally we use a chain with these properties in M1 to construct there a counterexample to Theorem 4.1 with level ordinal a.

The most important property of (En, n < ow) that we wish to capture and then reflect into M1 is the co-completeness of the extenders En. But w-completeness is too complex a property for reflection arguments. We will get around this problem through a use of Lemma 3.1 later on. We set the foundation for this future use in condition (5) below.

Let us begin the argument. Recall that a = lim supno1

rq. Removing a finite initial segment from the sequence (En I n < co) if necessary, we may assume that:

2. r , a for all n.

Let Mo = V. For each n E o w let Mn+1 = Ult(V, En) and let jo,n+l: V -- Mn+1 be the ultrapower embedding by En.

Let fin be the strength of En, that is the level of agreement between V and Mn+1. By Steel [4, p. 933] (also by Claim 2.12) strength(En+1) 1 strength(En), so that the sequence (fin I n < co) stabilizes. Let # be its eventual value. By dropping an initial segment of the sequence (En I n < co) we may assume that ft = fp already for n = 0.

CLAIM 4.2. a < P. PROOF. Otherwise there are arbitrarily large n < cw so that r, > P. Fix some

such n with n > 1. En is an extender in the sense of V, with domain ordinal r,. Since # < r, it follows that all subsets of /f belong to the domain of En. Since En belongs to Mn it follows that all subsets of P (in V) belong to Mn. But then the level of agreement between V and M, is then at least /f + 1 > fl, contradiction. -

The last claim tells us that a < strength(En) for each n < co. Since a > do(En) we have further a > crit(En). We may therefore use Claim 2.11, and obtain for each n < co an inaccessible cardinal ~, so that:

3. (n < a and jo,n+1(Wn) > a.

This property of ~, is sometimes easier to use when formulated internally to Mn. Let jn: Mn -+ Ult(Mn, En) be the ultrapower embedding of Mn by En. Since Mn and V agree to a, ji and jo,n+l are the same up to a, and so certainly the same on (n. We get then:

4. ~n < a and j (~n) > a.

CLAIM 4.3. For each n < co, 9(a) n Mn+1 C Mn.

PROOF. This is a standard argument using Fact 2.10. Since V and Mn agree past 5n, Ult(V, En)-which is equal to Mn+1-and Ult(Mn, En) agree past jn(~n), and therefore past a. Since the ultrapower Ult(Mn, En) is internal to Mn it is




,contained in M,. Putting these two facts together it follows that '9(a) n M+I

,9(a) n Ult(Mn, En) C Mn. -

Let p, be the cardinal successor of a in Mn. The last claim implies that Pn+1 1 Pn for each n < w, so that the sequence (p, I n < wc) stabilizes. Let p be its eventual value. By discarding an initial segment of (En I n < ow) if needed we may assume that already pl reaches the stable value. In other words:

5. (a+)Mn = p for every n > 1.

We intend to use this later to set the grounds for an application of Lemma 3.1. Recall that each of the domain ordinals rn is at most a. If these ordinals are

all strictly smaller than a then certainly they are strictly smaller than fl, and in that case an argument of the kind used in Steel [4] converts the iteration tree given by (M,,E, n < ow) into an infinite E-descending chain of models, yielding a contradiction.3 We may therefore assume that there is some n so that r, = a. By discarding an initial segment of (En, n < o) if needed we may in fact assume that:

6. /o = a. This concludes the first part of our argument, involving properties of the sequence

(En I n < co) recorded for future use. We now pass to the second part, constructing in V a certain chain of elementary substructures with properties which can be reflected into M1.

The minimality in condition (1) implies that a has cofinality < w. For otherwise using a closure argument one could find & < a so that the sequence ((En 0 ) n < w) is a counterexample to Theorem 4.1, where En | denotes the restriction of E to bounded subsets of &. But this counterexample has level ordinal & < a, contradicting condition (1).

Fix then a non-decreasing sequence of ordinals 6i, i < w, cofinal in a. Let 0 be some regular cardinal larger than the rank of (E, n < a)). Fix an

increasing chain of substructures Ho - H1 -/ H2 ... -I Vo so that:

(a) (E, I n < co) and (6i 1 i < wo) belong to Ho; (b) Ho is countable; (c) card(Hi) < a for i > 1; (d) 6i u {6i} c Hi for i > 1 ; and (e) All bounded subsets of Ci belong to Hi and Ci U {Ci} c Hi for i

_ 1.

Such a chain can be obtained easily. Let us only note that Ci < a is inaccessible, so there are exactly Ci bounded subsets of (i. This is needed to reconcile condition (e) with condition (c).

Let N be the transitive collapse of Ho and let 7c: N -+ Ho be the anticollapse embedding. Let E,, in, ,n, etc. be the collapsed images of En, 7n, i,, etc.

For each n < o let N0+1 = Ult(N, En) and let jo,n+1: N --+ N+1 be the ultrapower embedding. We have then the picture of Diagram 2. Our plan is to reflect a similar situation into M1.

For each i < o let Pi be the transitive collapse of Hi. For i K k < a) let 7i,k : Pi -* Pk be the map induced by the identity embedding from Hi to Hk.

3We shall not go into that argument here, but only say that the assumption rl1 < fl for each n is crucial there. Through Fact 2.10 it implies that Ult(V, En) and Ult(Mn, En) agree past spt(En), so that Skolem hulls of size spt(En) in Mn+ = Ult(V, En) belong to Ult(Mn, En), and therefore to Mn.



1154 ITAY NEEMAN

N3 10.3

7r

M3

10.3 E2

E2_ 3 N2

10.2

7r

M2 j0.2

El

El

3

3

7r

Nl1 -1 L

Eo

M1 1o 0.

Eo

7r

W

N

Diagram 2. The picture in V.

Notice that Pi and 7ti,k are all objects of hereditary size less than a. They therefore belong to M1.

Let P, be the transitive collapse of Ui<w Hi. For each k < w let nk,,: Pk -4 P" be the map induced by the identity embedding from Hk to Ui< Hi. P. is of size a and need not belong to M1. But we will reflect its properties to find a sufficiently similar model inside M1.

Working inside M1 let T be the tree of attempts to construct transitive models Qi for i < w; elementary embeddings i,k: Qi '-- Qk for i < k < w; maps V/i: ON n Qi -- 0; and an elementary embedding V: N91 -- Mi1 l so that:

(i) Qo = N; (ii) card(Qi) < a for all i > 1; (iii) The embeddings 'i,k

commute in the natural way; (iv) All bounded subsets of ao,i (ti) belong to Qi and crit(ai,k) > 0ao,i(i) for all

i < k < o with i 2 1; (v) crit(oi,k) >

TO,i(6i) for all i < k < w with i > 1;

(vi) V/i is order preserving for all i; (vii) V/k o Ui,k = Vi for all i < k < w;

(viii) For every i > 1, and every X C i in N1, (1X) = o0,i(f) (ix) c1(&a) = a, 1o(fi) = p, and pc(E1) = El. To be more precise: Let {ei)i<(} enumerate N1 with eo = (1, j , E1). Nodes of

length I in T consist of models Qi for i < 1; embeddings ai,k for i < k < 1; maps /i for i < 1; and a map [ {e I i < l), all in M1, satisfying the demands of conditions (i)-(viii) for i, k < 1 and I E {eo,..., e-l }, and satisfying condition (ix).

CLAIM 4.4. In V there exists an infinite branch through T.

PROOF. We show that the objects Qi = Pi; Ci,k = 7ti,k ; i = (jo,01 o ti,) ON; and W = nr [ N1 form an infinite branch through T.

It's easy to check that these objects satisfy conditions (i)-(vii) and (ix). Let us just comment that condition (iv) follows from condition (e), and condition (v) follows from condition (d).

For condition (viii) notice that n0,i and 71 are the same on subsets of ji-this follows from condition (d).

We have so far verified conditions (i)-(ix). But this is not quite enough. Remem- ber that T is defined inside M1, and only elements of M1 are allowed as nodes. To complete the proof of the claim we must check that each of the objects Pi, 71i,k, and




Vi = (jo, o 0ri,oo) [ON belongs to M1. (With regards to 1 there is nothing to check; its finite restrictions trivially belong to M1.)

Pi and 7ti,k are objects of size less than a and belong to M1 because M1 and V agree to a. V/i can be recovered from jo,1 (i,oo) and jo,i [Pi; in fact it is equal to [jo,1 (7i,oo)]o (jo, [ Pi n ON). jo,1 (7ri,o) belongs to M1 trivially, and jo,l [Pi belongs to M1 by Claim 2.14 which we can use because Eo0 is downward closed. For the application of downward closure in the last part note that the cardinality of Pi is strictly less than a, and that ro = a by condition (6). -1

REMARK 4.5. The application of Claim 2.14 in the last part of the proof above is our only use of downward closure. We used downward closure to make sure that the maps (jo,1i on ,oo) [ ON belong to MI. These maps are needed in the tree T to witness the wellfoundedness of the direct limit of (Qi, oi,k I i < k < aw). Downward closure is thus ultimately used in reflecting the wellfoundedness of P to the wellfoundedness of the structure Q which we construct below.

COROLLARY 4.6. In M1 there is an infinite branch through T. PROOF. Immediate from the last claim and the wellfoundedness of M1: If T

were wellfounded in M1 it would also be wellfounded in V, contradicting the last claim.

-- From now on we work inside M1. Using the last corollary fix an infinite branch

through T, given by Qi, 7i,k, q/i, and 1 say. Let Q be the direct limit of the system (Qi, Qi,k I i < k < w). Let a: N = Qo -- Q be the direct limit map.

CLAIM 4.7. Q is wellfounded. PROOF. The maps Vi can be combined, using the commutativity of condition

(vii), to form an order preserving map from the ordinals of Q into 0. CLAIM 4.8. For each n > 1, all bounded subsets of a(r,) in M1 belong to Q. PROOF. This follows immediately from condition (iv). CLAIM 4.9. a and p agree on bounded subsets of &. PROOF. This follows immediately form conditions (viii) and (v), and the fact that

the ordinals Si are cofinal in &. CLAIM 4.10. po(6) = a, ( P() = p, and (E11) = El. PROOF. Immediate from condition (ix). REMARK 4.11. The properties of Q, n7, and (o given by Claims 4.7 through 4.10

ultimately trace back to properties of P and nr. The tree of attempts T was designed to capture these specific properties, and used to reflect them to a structure inside M1.

We begin now to construct inside M1 a descending chain in the Mitchell order, consisting of models and extenders (N,, Fn I 1 < n < o). We work to obtain the picture of Diagram 3.

To start we have a, an elementary embedding of N into Q,4 and 4 , an elementary embedding of N11 into M1 6I 0. The relationship of a and

(o given by Claim 4.9

is sufficient for copying the ultrapower of N by E1 to an ultrapower of M1 by

4Note that the maps are presented only schematically in Diagram 3, without their exact targets. a is not elementary into a rank initial segment of M1, but only elementary into Q which belongs to M1. The same is true for the embeddings rn,. r,, is elementary not into Nn, but into il,n (Q) which belongs to N,.



1156 ITAY NEEMAN

Jo.4 NT4

T4

N4 i1.4

F3 3

E3 3 10s3

N3

73

N3 i1,3 F2 3

E2_ 3

10.2

N2

T2

N2 F1 =El

11,2

El 3 NI1

(Po

M1

a

NA

7r

V

Diagram 3. The picture in M1.

p (EA) = El, allowing us to obtain T2 as displayed in Diagram 3, elementary into il,2(Q) which belongs to N2. Having obtained T2 we set F2 = r2(E2)

and obtain T3

through copying the ultrapower of N by E-2 to an ultrapower of M1 by F2. Having obtained T3 we set F3 = z3(E3), etc.

The claims which follow verify that a construction of this kind makes sense, and most importantly that the objects used, namely the F,-s, are extenders in M1. The delicate part of course is w-completeness. The elementarity of r, is sufficient only to tell us that F,7 is o-complete in i1,, (Q) (the target of z[,). We will need Lemma 3.1 and the properties collected above to reason further that F, is Co-complete in M1.

Let F1 - EI, let N2 = Ult(MI, E1), and let il,2: M1 -+ N2 be the ultrapower embedding. Recall that N2 is equal to Ult(N, E1), and so every element of N2 has the form Jo,2(f)(d) for some f E T with dom(f) E dom(E1) and some o E spt(E1). Define 2: N2 --+ N2 by:

T2(o,2(f)()) - il,2(a(f))(p( ))

(To move jo,2(f)(a): first move f using a followed by io,2, then apply the resulting function to the shift of a by p.) This is the usual definition used in the copying of ultrapowers. The argument showing that it makes sense is standard using the agreement given by Claim 4.9 and the fact that do(Ei) is at most 5. The standard argument also show that a, il,2, jo,2, and T2 commute in the manner of Diagram 3. Since a is elementary into Q, z2 is elementary into il,2(Q).

CLAIM 4.12. N2 and M1 agree to a.

PROOF. By Claim 2.12 the strength of E1 in M1 is at least its strength in V, which is greater than or equal to a by Claim 4.2. -1

Let ji: N1 -- Ult(A1, E1) be the ultrapower embedding of N1 by E1. This is the collapse of the internal ultrapower map jl introduced just before condition (4) above.

CLAIM 4.13. T2 and p agree on subsets of jl (lI). PROOF. This follows by a standard copying argument form the fact that (-

which is smaller than a by condition (4)-is below the level of agreement between p and a.

COROLLARY 4.14. T2 agrees with a on bounded subsets of - COROLLARY 4.15. z2(5)

= a andz2(iP) = p.




REMARK 4.16. In passing from Claim 4.13 to Corollaries 4.14 and 4.15 we are using the fact that jI (1,) is greater than i. This is the collapse of condition (4) above, and traces back to the fact that E1 is below rank-to-rank type. We shall make another use of this fact in Claim 4.17, and make similar uses of the fact that the extenders E, for n > 1 are below rank-to-rank type later on.

CLAIM 4.17. il,2(Q) and MI agree to a.

PROOF. Since M1 and N2 agree to a it's enough to check that il,2(Q) and N2 agree to a. Using Claim 4.8 and the elementarity of il,2 we see that il,2(Q) and N2 agree to il,2(a (l)). Thus it's certainly enough to check that il,2(a (l)) is greater than a. Now il,2 (a(l)) is equal to 2 (o70,2 (ci)) by commutativity, and jo,2 (1) is greater than

6 by condition (3). So il,2(a(91)) > T2(a) = a. - We continue now to define Fn, il,,+l, and rn+l for n > 2, working by induction

on n. At the start of stage n we assume the inductive conditions:

(I1) il,n(Q) and M1 agree to a; (12) rn is elementary from N9 into il, (Q) and commutes with a, il,,, and jo,n in

the manner of Diagram 3; (13) In agrees with a on bounded subsets of 5; (14) rz,(5) = a and Tn(p() = p; At the end of stage n we will prove these conditions for n + 1.

Notice that for the base case n = 2 these conditions hold: condition (I1) is Claim 4.17, condition (12) is standard in copying arguments, condition (13) is Corollary 4.14, and condition (14) is Corollary 4.15.

Let us begin stage n. Set Fn = i, (E]). Using the elementarity given by condition (12) we see that il, (Q) 1"F, is an extender with domain ordinal zC (zr)". ijn is at most & by condition (2), and in (5) = a by condition (14). So the domain ordinal of F , is at most a.

Since il,n(Q) and M1 agree to a we know that Fn is a pre-extender in M1. We intend to appeal to Lemma 3.1 to see that Fn is w-complete in M1. But first we need:

CLAIM 4.18. (do(Fn,))i ),(Q) = (do(Fn,))M' PROOF. If do(Fn) < a this follows from the fact that il,, (Q) and M1 agree to a.

So suppose that do(Fn) = a. We have to show that (a+)i',(Q) = (aC+)M. Now (a+)M' is equal to p by condition (5). So we must show that (a+)i',(Q) - p.

The collapse of condition (5) to N tells us that (1+)N" = t. Applying z,, which is elementary from N, into il,n(Q), it follows that (rn()+)in,(Q) = -r,( ). Using condition (14) we get

(aO+)i'(Q) - p.

COROLLARY 4.19. MI ="Fn is an extender". PROOF. This is a direct application of Lemma 3.1 inside M1, with io,n(Q) for R

and F, for F. Note that the use of Lemma 3.1 is dependent on Claim 4.18 which tells us that the cardinal successor of do(F,) is absolute between il,, (Q) and M1. Claim 4.18 in turn is dependent on condition (5) which we arranged earlier. -

Now that we know that F, is an extender in M1 we can set N,,+1 = Ult(M1, F,,) and let il,,,+1: M1 -+

N,,+11 be the ultrapower embedding. This amounts to copying the

ultrapower of N by E, to an ultrapower of M1 by z, (E,) = F,. Standard copying



1158 ITAY NEEMAN

arguments (as in the case of El above) allow us to define zn+1 along the lines of Diagram 3, commuting with jO,n+l, a, and il,n+l, and elementary into

io,n+l(Q). This immediately gives condition (12) for n + 1. As in the case of Claim 4.13 the copying produces an agreement between n+l1 and rn:

CLAIM 4.20. zn+1 and zn agree on subsets offn (n). - Conditions (13) and (14) for n + 1 follow from this last claim, using the same

conditions for n and using the fact, given by condition (4), that jn (in) > 5.

CLAIM 4.21. Nn+1 and M1 agree to a. PROOF. The strength of En in Mn is at least its strength in V by Claim 2.12. The

strength of En in V is at least a by Claim 4.2. Thus certainly Mn "the strength of En is at least a". Collapsing this to N we see that N, ="the strength of En is at least 5". Applying ,n it follows that the strength of Fn in il,,n(Q) is at least In(&) = a. Thus Ult(il,n(Q),Fn) and il,n(Q) agree to a. Now il,n(Q) and M1 agree to a by condition (I1). It follows that M1, Nn+1 = Ult(MI, F,), il,n(Q), and Ult(il,n(Q), Fn) all agree to a. -d

Condition (I1) for n + 1 follows from this last claim using an argument similar to that in Claim 4.17. This completes stage n of the construction and puts us in the position to start stage n + 1.

Having completed all w stages of the construction we obtain, inside MI, a sequence (Fn I 1 < n < w) of extenders descending in the Mitchell order. Using absoluteness, the elementarity of the anticollapse map 7r, the elementarity of z,, and the fact that z, (&) = a, it's easy to verify that: Fn is below rank-to-rank type if En is below rank-to-rank type; Fn is downward closed if En is downward closed; and do(Fn) < a if do(En) < a. Since the extenders En are all below rank-to-rank type and downward closed, and since lim supn<, do(En) = a, we get:

* M1 "(Fn I I n < w) is a sequence of downward closed extenders below rank-to-rank type, descending in the Mitchell order, with level ordinal at most a".

Condition (3) tells us that Co < a and jo,1(0o) > a. So certainly a < jo,1(a). Folding this into the statement above we get:

* M1 "(Fn I 1 < n < wo) is a sequence of downward closed extenders below rank-to-rank type, descending in the Mitchell order, with level ordinal strictly smaller than jo,1 (a)".

But this contradicts the shift to M1 (via jo,1) of the minimality given by condition (1) at the start of the section. -d (Theorem 4.1)

15. Pathologies. An infinite descending chain in the Mitchell order is the same as a length co iteration tree with no branches of length greater than 2. The result of the last section can therefore be rephrased as stating that every length wo iteration tree made of downward closed extenders below rank-to-rank type must have a branch of length 3. This is far short of the infinite branch one gets in the context of amenable extenders, through the work of Steel [4]. But in general it's impossible to get more. We give here an example of an iteration tree, using non-amenable extenders obtained from large cardinals slightly beyond superstrong, with no branches of length 4.




We also give an example of an iteration tree of length 3, with its final model illfounded. This again contrasts sharply with the situation in the context of amenable extenders, where Steel [4] shows that illfoundedness does not occur at successor models.

REMARK 5.1. For simplicity we assume the GCH in this section. Examples similar to the ones we give can be constructed without the GCH, essentially by replacing uses of the cardinal successor operation with uses of the operation 1 - 2f.

EXAMPLE 5.2. Suppose there are K, z, and E so that: E is an extender with crit(E) = K' and E(r.) = z, the domain ordinal of E is K+('), and E is r+(")-strong. Then there is an iteration tree of length 3 with an illfounded last model.

Extenders of the kind assumed in Example 5.2 require large cardinals beyond superstrong, but well below rank-to-rank embeddings.

CONSTRUCTION FOR EXAMPLE 5.2. Fix r, z, and E satisfying the assumptions in Example 5.2, with z smallest possible. We have then:

(*) There do not exist 6 and F so that F is an extender with crit(F) = K., F (r) = 6, do(F) = .+ ("1), and strength(F) = +("), with 6 < z.

Set Mo = V. Set M1 = Ult(V, E) and let jo,1: V -+ M1 be the ultrapower embedding. Let E* = jo,1 (E). E* is an extender in the sense of M1. The domain ordinal of E* is r+("). Now V and M1 agree to z+(O) because of the strength of E. It follows using Claim 2.2 that E* is a pre-extender in V. We can therefore set M2 = Ult(V, E*) and let j0,2: V -+ M2 be the ultrapower embedding. Diagram 4 illustrates the iteration tree we produced. E* is only known to be a pre-extender in V, so Ult(V, E*) need not be wellfounded. In fact we will show that it is illfounded.

M2 E* 10,2

M1 E o0.1 V

Diagram 4. Example 5.2.

For each n < w let En be the restriction of E to subsets of .t+("). Then E,+1 D En for each n and E = Un<. En.

Working in V let T be the tree of attempts to construct B E (K, z), Fn for n < w, and F* for n < c, so that:

(i) Each F, is a6+(n)-strong extender with crit(F,) = K, F,(r.) = 6, and do(F,) = S +(") + 1;

(ii) Each F,*

is an extender with crit(F,*) = 6, F,1*(6)

= r, and do(F*) = 6+(") + 1; (iii) Fn+1 F, and

Fn*+l F,* for each n < wo; and

(iv) F,* o F, = E, for each n < w. A node of length I > 0 in T consists of 6, Fn for n < 1, and F* for n < 1.

CLAIM 5.3. T is wellfounded. PROOF. Suppose for contradiction that 6, (Fn n < o), and (F* I n < co) form

an infinite branch through T. Let F =

U,,< Fn and let F* = U,,n< F"*.

Using condition (iii) it's easy to see that F and F* are pre-extenders. From condition (iv) it follows that F* o F is equal



1160 ITAY NEEMAN

to E. From this and the w-completeness of E it follows that F is co-complete. F is thus an extender. Using condition (i) we see that crit(F) = n, F(K) = 6, the domain ordinal of F is r,+("), and F is 5+(")-strong. But this contradicts the

minimality of z expressed in condition (*) above. -d

REMARK 5.4. In general the increasing union of extenders need not be co-complete. The w-completeness of F in the proof of Claim 5.3 hinges on the fact that F embeds into E (precisely, F* o F = E). This allows reducing the co-completeness of F to the wc-completeness of E.

CLAIM 5.5. j0,2(T) is illfounded PROOF. Recall that E, denotes the restriction of E to subsets of K+("). En is then

an extender, with domain ordinal n+(n) + 1. Notice that E, is precisely equal to the

z+(")-restriction of the ultrapower embedding by E. Using the strength of E and Lemma 2.9 it follows that En is r+(n)-strong.

E, is an object of hereditary size r+(n), and can therefore be coded as a subset of z+('). Since M1 and V agree to z+(') we have E, E M1 for each n. M2 and M1 agree to

jo.1(r+(W)), so we also have E, E M2.

Let E,

be the restriction of E* to subsets of r+("). E* is an object of hereditary size jo, (T)+(n) in Mi. Since M1 and M2 agree past this level we have E E M2 for each n.

It is now easy to check that the objects z, (E, I n < wo), and (E, In < w) form an infinite branch through j0,2(T). -1

Claims 5.3 and 5.5 together imply that M2 is illfounded. We have then an iteration tree of length 3 with its last model illfounded. -i (Example 5.2)

EXAMPLE 5.6. Suppose there are K, r, and E so that: E is an extender with

crit(E) = . and E(r.) = r, the domain ordinal of E is K +(w2), and E is z+(wC2)- strong. Then there is an iteration tree of length co with no branches of length 4.

REMARK 5.7. An iteration tree 7-

is called non-overlapping if spt(En) _

crit(E,) whenever (n + 1) T (m + 1). (In other words the ordinals in the support of each extender used on each branch of the tree are not moved by later extenders used on the branch.) It is easy to check that the tree we construct below for Example 5.6 satisfies this condition. So in fact we get a non-overlapping iteration tree of length wc with no branches of length 4.

CONSTRUCTION FOR EXAMPLE 5.6. Let F be the restriction of E to bounded subsets of 1,+(o). F is then an extender of the kind assumed in Example 5.2. The iteration tree for Example 5.6 will be constructed through applications of F and its images. The fact that F can be extended to E will be used only for showing that all the models along the tree are wellfounded.

Set no = 0 and nk+1 = nk + k + 1 for k > 0. Let T be the following tree structure

on w: the T-predecessor of nk for k > 0 is 0; and the T-predecessor of nk + h for k > 0 and 0 < h < k is nk-h.

CLAIM 5.8. T has no branches of length 4.

PROOF. T has the trivial branch (0); branches of length 2 in the form (0, nk) for k > 0; and branches of length 3 in the form (0, nk, nk, + h') for k > 0, k' > k, and h' = k' - k. T has no branches other than these.




We now work to construct an iteration tree with its tree order precisely T. Define T = (Mn, Fn, Im,n Im T n < co) inductively as follows:

(a) Mo = V and jo,o = id; (b) F, = jo,n (F) for each n < co; (c) Mn+l = Ult(Mm, F,) where m is the T-predecessor of n + 1, and jm,n+l is the

ultrapower embedding.

The structure of - is illustrated in Diagram 5. The diagram presents the extenders schematically, drawing for each Fn a line from its domain ordinal to its strength in Mn. (The strength and domain ordinal are given precisely in Claim 5.9.) The dotted frames on the bottom line indicate those models which are equal to M6k for some k. (The diagram runs from Mn0 = Mo to Mn4 = M10.) The arrows show part of the tree structure T. Specifically they indicate the predecessors of the models M6 through M9, namely the models M,, and the models Mn3+h for 0O < h < 3. We work now to check that this structure makes sense.

P4

f13 = -14

f2 = a3

fl = a2

#0 = a1

CO

Fo

F,

F2

F3

F4

Fs5

F6

F7

F8

F9

Flo

Mo0 Ml M2 M3 M4 M5 M6 M7 M8 M9 MIoi

Diagram 5. The structure of 53.

Set ao - nK(") and let ak -= jo,~ (ao) for k > 0. Similarly set flo - r+(o) and let

fk = jO,nk (fo) for k > 0.

CLAIM 5.9. For each k and each h with 0 < h < k: Fnk+h is an extender in Mnk+h with domain ordinal ak-h and strength flk in Mnk+h. The ultrapower embedding by Fnk+h sends ak-h to ik .



1162 ITAY NEEMAN

PROOF. This is a simple matter of using the properties of F and chasing through the various embeddings to see where they send ao and flo. We leave the exact details to the reader.

The claim has the following immediate corollaries: COROLLARY 5.10. flk = ak+l for each k <t. COROLLARY 5.11. The models Mnk ,

Mnk+1,...., Mnk+1 all agree to fik .

COROLLARY 5.12. For k > 0 and0 < h < k, Mnk+h and Mnk-h-_ agree to flk-h-

ak-h. Claim 5.9 and Corollary 5.12 together tell us that Mnk+h and

Mnk-h-_ agree to the domain ordinal of

Fnk+h. It follows from this that Ult(Mnk-h-_, Fnk+h) makes

sense. This is important for the construction of T since we are setting Mnk+h+l to be precisely this ultrapower. (The claim and corollaries should thus be verified in steps during the construction.)

We know now that 3- as defined above makes sense, and has no branches of length 4. To complete the example we must check that each of the models in 9 is wellfounded. This is not given automatically; Example 5.2 shows that wellfoundedness may fail.

For each i < w let Fi be the restriction of E to subsets of n+(wO+i). We have then

Fi+l D Fi for each i and F0 D F. For each 1 < wo let ' =-

(Mn, Fn,Jm,n m T n < 1) be the iteration tree of

length l defined following the construction of T [1 except that instead of taking F,i to be j ,n(F) as in condition (b) above, take

Fn = j ,(F'-1) /' should be

thought of as a background certificate for - [ 1, constructed using extenders which extend the ones used in T [1 and provide some additional strength.

The domain ordinals of the extenders in -i are all successor ordinals. (This distinguishes P' from 37, where the domain ordinals were always limit ordinals.) The arguments of Martin-Steel [2] apply in this context and show that the models in Ti are wellfounded.

It's easy to see, using the fact that Fi D F for each i, that the models of 5 -[1 embed into the models of 3:. Since the latter models are wellfounded, so are the former models. This can be seen for each 1 < co, so the models in ~ are wellfounded.

-A (Example 5.6)

REFERENCES

[1] AKIHIRO KANAMORI, The higher infinite, 2nd ed., Perspectives in Mathematical Logic, Springer, 1997.

[2] DONALD A. MARTIN and JOHN STEEL, Iteration trees, Journal of the Americal Mathematical

Society, vol. 7 (1994), pp. 1-73.

[3] WILLIAM MITCHELL, Sets constructiblefrom sequences of ultrafilters, this JOURNAL, vol. 39 (1974), pp. 57-66.

[4] JOHN STEEL, The well-foundedness of the Mitchell order, this JOURNAL, vol. 58 (1993), pp. 931-940.

[5] MARTIN ZEMAN, Inner models and large cardinals, Logic and Its Applications, no. 5, de Gruyter, 2001.

DEPARTMENT OF MATHEMATICS, UNIVERSITY OF CALIFORNIA AT LOS ANGELES

LOS ANGELES, CA 90095-1555, USA E-mail: [email protected]



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