The Core Model Iterability Problem

Lecture Notes in Logic

8

John R. Steel

The Core Model Iterability Problem

Rf

Springer

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Addresses: Professor S. R. Buss Dept. of Mathematics University of California, San Diego La Jolla, CA 92093-0114, USA e-mail: [email protected] Professor J.-Y. Girard Laboratoire de Mathematiques Discretes Equipe Logique et Informatique 163, Avenue de Luminy - case 930 13288 Marseille - Cedex 9, France e-mail: [email protected] Professor A. Lachlan Dept. of Mathematics Simon Fraser University Burnaby, British Columbia V5A 1S6 Canada e-mail: [email protected] Professor T. Slaman Dept. of Mathematics University of Chicago Chicago, IL 60637, USA e-mail: [email protected] Professor A. Urquhart Dept. of Philosophy University of Toronto Ontario M5S 1A1, Canada e-mail: [email protected] Professor H. Woodin Dept. of Mathematics University of California at Berkeley Berkeley, CA 94720, USA e-mail: [email protected] Springer-Verlag, Mathematics Editorial Tiergartenstr. 17 D-69121 Heidelberg Federal Republic of Germany Tel.: *49 (6221) 487-250

Lecture Notes in Logic Editors: S. Buss, San Diego J.-Y. Girard, Marseille A. Lachlan, Burnaby T. Slaman, Chicago A. Urquhart, Toronto H. Woodin, Berkeley

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John R. Steel

The Core Model Iterability Problem

Springer

Author John R. Steel Department of Mathematics University of California at Berkeley Berkeley, CA 94720-3840, USA e-mail: [email protected]

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Die Deutsche Bibliothek - CIP-Einheitsaufnahme Steel, John R.: The core model iterability problem / John R. Steel. - Berlin Heidelberg New York Barcelona Budapest Hong Kong London Milan Paris Santa Clara Singapore Tokyo : Springer, 1996 (Lecture notes in logic ISBN 3-540-61938-0 NE: GT

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Mathematics Subject Classification (1991): 03E45, 03E55, 03E10, 04A10, 04A15 ISBN 3-540-61938-0 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1996 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready by the authors SPIN: 10555497 46/3142-543210 - Printed on acid-free paper

Table of Contents

§0. Introduction

1

§ 1. The construction of Kc

5

§2. Iterability

10

§3. Thick classes and universal weasels

25

§4. The hull and definability properties

29

§5. The construction of true K

35

§6. An inductive definition of A"

43

§7. Some applications

53

A. Saturated ideals

53

B. Generic absoluteness

56

C. Unique branches

59

D. 173 correctness and the size of 1/2

63

§8. Embeddings of K

73

§9. A general iterability theorem

89

References

109

Index of definitions

Ill

The Core Model Iterability Problem

J. R. Steel These notes develop a method for constructing core models, that is, canonical inner models of the form L[E], where E is a coherent sequence of extenders. They extend the earlier work in this area of Dodd and Jensen ([DJ1], [DJ2], [DJ3]) and Mitchell ([Ml], [M?]). The Dodd-Jensen theory produces models having measurable cardinals, and Mitchell's extension of it produces models having measurable cardinals K of order κ++. Here we shall extend this theory so that it can produce core models having Woodin cardinals. The extent of our debt to Dodd, Jensen, and Mitchell will become apparent; nevertheless, we shall not assume that the reader is familiar with their work. We shall, however, assume that he is familiar with the fine structure theory for core models having Woodin cardinals which is developed in [FSIT]. Our work here goes beyond [FSIT] in that it involves a construction of L[E] models which makes no use of extenders over V. Many of the applications of core model theory require such a construction. The authors of [FSIT] use extenders over V in order to show that the inner model L[E] they construct is sufficiently iterable: roughly, they demand that the extenders put onto E be the restrictions of background extenders over V, then use this fact to embed iteration trees on L[E] into iteration trees on V, and then quote the results of [IT] concerning iteration trees on V'. Here we shall describe a weakened background condition on the extenders put onto E which does not require full extenders over V, and yet suffices to carry out something like the old proof of the iterability of L[E]. The result is a solution to what is called the "core model iterability problem" in [FSIT]. The notes are organized as follows. As in Mitchell's work on the core model for sequences of measures ([Ml], [ M?]), we construct the model K in which we are ultimately most interested in two steps. In §1 we construct a model, which we call Kc, whose extenders have "background certificates". These background certificates guarantee the iterability of K°, its levels, and various associated structures. (In Mitchell's work, the background condition is countable completeness, but here we seem to need more.) In order to show that Kc and the K we derive from it are large enough to be useful, we seem to need something like the existence of a measurable cardinal. We fix a normal measure μ0 on a measurable cardinal Ω throughout this paper; we shall have OR Π K° = OR Π K = Ω. We use the measurability of Ω to show in §1 that either Kc \= there is a Woodin cardinal, or (α+)κ° = a+ for μo- a.e. a < Ω. Since in applications we are seeking an inner model with a Woodin cardinal, we assume through most of the paper that there is no such model, and thus we have (α+)χc = a+ for μ0- a.e. a < Ω. As in Mitchell's work, this "weak covering property" of Kc is crucial. We also use the measurability of Ω in a different way in §4. Further, we use the tree property of Ω to show that iteration trees of length Ω are well

2

The Core Model Iterability Problem

behaved. The fact that we do not develop the basic theory of K within ZFC may be a defect in our work. c In §2 we sketch the main new ideas in the proof that K is iterable. In §9 we give a full proof of a general iterability theorem which covers iteration c trees and psuedo-iteration trees on A" , its levels, and the associated bicephali and psuedo-premice. As in Mitchell's work, the "true core model" K is a Skolem hull of K°. In §3 and §4 we develop some concepts, derived from Mitchell's work, which are useful in the construction of this hull. In §5 we do the construction: given a stationary S C Ω with certain properties, we construct a model K(S) ^ Kc. We show (a+)κ(sϊ = α+ for μo- a.e. a. We also show that K(S) is invariant under small forcing; that is, K(S) — K(S)V^G* whenever G is V generic over some F £ VΩ Finally, we show that K(S) is independent of 5, and define K to be the common value of K(S) for all 5. We have then that (a~*~)κ — a+ for μo - a.e. α, and that K is invariant under set forcing. In §6 we give an optimally simple inductive definition of K: it turns out that K Π EC is ΣΊ(L ωι (M)). (Woodin has shown that no simpler definition is possible in general. Mitchell showed in [M?] that if no initial segment of K satisfies (3/c)(o(/c) = κ++), then KΓ\HC is Σ\ in the codes.) In §7 we use the machinery we have developed to obtain the consistency strength lower bound of one Woodin cardinal for various propositions. In §8 we return to the pure theory, and obtain some information concerning embeddings of K. We show, for example, that if there is no inner model with a Woodin cardinal, then there is no nontrivial elementary j : K —+ K. In contrast to the situation for "smaller /f's", however, we show that there may be nontrivial elementary j : K —» M which are not iteration maps. Among the applications of the theory developed in these notes are the following theorems. Theorem 0.1. Let Ω be measurable, and suppose there is a presaturated ideal on ω\\ then there is a transitive set M C VΩ such that M [= ZFC + "There is a Woodin cardinaΓ. Corollary 0.2. If Martin's Maximum holds, then there is a transitive set M such that M \= ZFC-}- "There is a Woodin cardinaΓ. (It is known from [FMS] that Martin's Maximum implies that there is an inner model with a measurable cardinal and a saturated ideal on ω\\ by applying 0.1 inside this model we get 0.2. H. Woodin pointed this out to the author.) Further work of Mitchell, Schimmerling, and the author on the weak covering property for K (cf. [WCP]) together with his work on Jensen's D principle in K (cf. [Sch]), led Schimmerling to the following improvement of 0.2.

J. R. Steel Theorem 0.3. (Schimmerling, cf. [Sch]) If PFA holds, then there is a transitive set M such that M |= ZFC+ "There is a Woodm cardinal". (Theorem 0.3 also relies on an improvement, due to Magidor, of Todorcevic's result that PFA implies V/c (DΛ fails). (See [To].) In a different vein, we have the following, more immediate applications of the theory presented here. Theorem 0.4. Suppose that every set of reals which is definable over L ω ι (M) is weakly homogeneous] then there is a transitive set M such that M \= ZFC + "There is a Woodin cardinal". (H. Woodin supplied a crucial step in the proof of 0.4.) Since Woodin (unpublished) has shown that the existence of a strongly compact cardinal implies the hypothesis of 0.4, we have Corollary 0.5. Suppose there is a strongly compact cardinal] then there is a transitive set M such that M \= ZFC + (£There is a Woodin cardinal". (We shall give a different proof of 0.5, one which avoids Woodin's unpublished work, in §8.) We can also improve the lower bound of [IT] on the strength of the failure ofUBH. Theorem 0.6. Let Ω be measurable, and suppose there is an iteration tree Ί on Va such that Ί £ VΩ and T has distinct cofinal wellfounded branches] then there is a transitive set M C VΩ such that M (= ZFC+ "There are two Woodin cardinals". We can use the methods presented here to re-prove Woodin's result that Vz Gω ω(z fl exists) + Δ\ determinacy implies that there is an inner model with a Woodin cardinal. Finally, using an idea of G. Hjorth, the author has recently (at least partially) generalized Jensen's Σ\ correctness theorem to the core model constructed here. This yields a positive answer to a conjecture of A. S. Kechris. ω

Theorem 0.7. Assume Vx £ ω (x^ exists and Σ%(x) has the separation property)', then there is a transitive set M such that ί(

M \= ZFC + There is a Woodin cardinal".

4

The Core Model Iterability Problem

Each of the hypotheses of the theorems above is known to be consistent under some large cardinal hypothesis or other. We shall not attempt a scholarly discussion of the history of or context for these theorems, as our focus here is the basic theory which produces them. We shall prove 0.1, 0.4, 0.6, and 0.7 in §7. Historical note. We did most of the work described here in the Spring of 1990, and informally circulated it in a set of handwritten notes. Our main advance, which was isolating Kc and proving the results of §1 and §2 concerning it, was inspired in part by some ideas of Mitchell. The work in §8 was done somewhat later, and the Σ\ correctness theorem of §7D was not proved until the Spring of 1993.

The Core Model Iterability Problem

5

§1. The construction of K° Let us fix a measurable cardinal, which we call ί2, for the remainder of this paper. We shall sometimes think of Ω as the class of all ordinals; we could have worked in 3rd order set theory + "OR is measurable", but opted for a little more room. Fix also a normal measure μo on Ω. We now define by induction on ξ < Ω premice Λ/f. Having defined Λ/f, we set the ωth core of λίξ. The Mξ's will converge to the levels of Kc, the "background-certified" core model for one Woodin cardinal. That is, we shall define Kc by setting Jβ

— eventual value of J* * ,

as ξ —> Ω ,

for all β < Ω. The construction of the Λ^'s follows closely the construction in §11 of [FSIT]. In this section, we shall simply assume that the A f ' s are all "reliable", that is, that ^(Λfξ) exists and is fc-iterable for all k < ω. By Theorem 8.1 of [FSIT], this amounts to assuming that if <£fc(Λ/f) exists, then it is Ar-iterable (for all k < ω). We shall sketch a proof of this iterability assumption in §2, and give a full proof in §9. We shall also assume here that certain bicephali and psuedo-premice associated to (Λ/f | ζ < Ω) are sufficiently iterable. We prove this in §9. Iterability comes from the existence of background extenders. Definition 1.1. Let M be an active premouse, F the extender coded by FM (i.e. its last extender), K — crit(F), and v — v^ — sup of the generators of F. Let A C (Jn<ω P([κ]n)M. Then an A-certificate for M is a pair (TV, G) such that (a) N is a transitive, power admissible set, Vκ (JA C N, ωN C N, and G is an extender over N, (b) F Π ([ι/]«" x A) = G Π ([ι/]«* x A), (c) K+i C Ult(N,G), andωUlt(N,G) C Ult(N,G), (d) Vγ(u; 7 < ORM => jy = canonical embedding.

ri(J"}),

where i : N-+ Ult(N, G) is the

Definition 1.2. Let M be an active premouse, and K the critical point of its last extender. We say M is countably certified iff for every countable A Q \Jn<ω p([κ]n)M> ihere is an A-certificate for M. In the situation described in 1.2 we shall typically have \N\ = K, so that (OR Π TV) < IhG. We are therefore not thinking of (TV, G) as a structure to be iterated; TV simply provides a reasonably large collection of sets to be measured by G. The conditions Vκ C N and K+i C Ult(TV, G) are crucial; the closure of TV and Ult(TV, G) under ω-sequences could probably be dropped.

6

c

§1. The construction of K

It might seem that certificates (N,G) as in 1.1 are too much to ask for, and in particular condition 1.1 (c) might seem too strong. But notice that we get such pairs by taking Skolem hulls: if π : N =. H X Vη inverts the collapse of H, where Vκ C H but K £ H, then letting G be the length π(κ) extender derived from π, G is an extender over TV, Vκ C N, and Vπ(κ) C Ult(ΛΓ, G). We shall also see that the embedding associated to the measure μ 0 on Ω gives rise to certificates. Definitions 1.1 and 1.2 were inspired in part by earlier attempts by Mitchell to formulate a background condition along these lines. We proceed to the inductive definition of λίξ. As we define the Λ/^'s we verify: Aξ : Vα < ξ V/c (if pω( M ι) > κ for all 7 s.t. α < 7 < £, then letting η = («+)»••, Jf

=$?').

(Here we let ωη = ORΠ Ma in the case Ma \= «+ doesn't exist.) We begin by setting .Λ/o = (Vω, G). (So MQ = Λ/Ό ) Now suppose Λ/f , and hence Mξ, is given. Case 1. Mξ = (Jlf , G, E \ 7) is a passive premouse, and there is an extender F over Mξ such that (1) (J^,£,E \ 7, F) is a 1-small, countably certified premouse, and, letting K = crit(F), we have (2) K is inaccessible, (3)M (κ+)M* = κ+ => for stationary many β < K, β is inaccessible and

(β+) <=β+.

In this case, we choose an F as above with ^(F), the sup of the generators of F, as small as possible, and we set

As we mentioned above, the results of §9 and §8 of [FSIT] imply that Λ/ξ+i is reliable. Thus λiξ+i = <£u>(JVf+i) exists. We get Aξ+ι from (the proof of ) Theorem 8.1 of [FSIT]. Case 2. Otherwise. In this case, let ωj = OR Π A4f , and set

Again, Λ/^+i is reliable by §9, and 8.1 of [FSIT] yields Aξ+ι. So in Case 1 we add a countably certified extender to our extender sequence, while in Case 2 we take one step in the constructible closure of the sequence we have. In both cases we then form the ωth core of the structure we have. Now suppose we have defined Λfξ for ξ < λ, where λ < Ω is a limit ordinal. Set η = \immf( ξ—> λ

The Core Model Iterability Problem

7

(where ρω(Mξ)+ * = ORΓ\Mξ is possible. We set Λ/λ = unique passive premouse P s.t. (a) Vβ < η(jj = eventual value of jf4* as ξ -* λ) and (b) jf = P. Such a premouse exists as Aξ holds for all £ < λ. It is easy to verify A\. This completes the inductive definition of the Λ/f , for ξ < Ω. Theorem 8.1 of [FSIT] actually gives the following strengthening of our induction hypothesis Aη. (Cf. 11.2 of [FSIT].) Lemma 1.3. Suppose K < ρω(Mξ) for allξ > α 0 > o,nd ktξ > aΌ be such that K = pω(-Mξ). Then Mξ is an initial segment of Mη, for all η > ξ] moreover -Mf+i satisfies "every set has cardinality < κn . Lemma 1.3 implies liminf^^ Pω(Mξ) = β, and therefore we can define a premouse Kc of ordinal height Ω by jf° — eventual value of

Jβ * ,

as

ξ —> Ω ,

for all β<Ω. The following is a cheapo form of weak covering. It is crucial in what follows; it tells us that we haven't been too miserly about putting extenders on the K° sequence. Theorem 1.4. Exactly one of the following holds: (a) Kc \= There is a Woodin cardinal, (b)for μ 0 - a.e. a < Ω, (α+)*c = α+. Proof, (a) => -i (b): Every jjfc is 1-small, so if Kc \= 6 is Woodin, then Eίf° = 0 if 7 > δ. As Ω is measurable, there is a club class of indiscernibles for Kc , or equivalently, a countable mouse λί which is not 1-small. Comparing λί with Kc , we get that for μ$- a.e. α < β, (α+)^° has cofinality cj in V. Remark. We have no use for this direction in what follows. - (b) =* (a): Let j : V —*• M = Ult(F, μo) be the canonical embedding. We are assuming then +M

c

(Of course, β = β+.) Let ^ = P(Ω) Γ\ j(K ), so that A G M and M [= |^4 1 = ί?. Let i?y be the ( Ω , j ( Ω ) ) extender derived from j. By an ancient argument due to Kunen, whenever \B\ = β, E j Γ \ ( [ j ( Ω ) ] < ω xβ) E M. (Proof: if β = {5α | α < β}, then notice (j(5α) | α < β) = j((BQ \ a < β» Γ β, so O'(flo) I α < β) G M. Also, Ba G (£j)c iff c E j( In particular, setting

we have that F E M. Now it cannot be that every proper initial segment F \ is, v < j(β), of F belongs to j(Kc), as otherwise these initial segments

§1. The construction of Kc

8

witness that Ω is Shelah in j(Kc). But if Kc b There are no Woodins, then as ι7β = Kc, 1-smallness is not a barrier to adding these F \ v to the j(K°) sequence. The following claim asserts that there are no other barriers. Its statement and proof run parallel to those of Lemma 11.4 of [FSIT]. • / iv- C \

Claim. Suppose Kc \= There are no Woodin cardinals. Let (Ω+)^κ^ < p < κ j(Ω) and suppose that either p = (Ω^)^ °\ or p— I exists and is a generator of F, or p is a limit of generators of F. Let G be the trivial completion of F ί p, and j = IhG. Let E be the extender sequence of j(Ke). Then either (a) p = (β+y(*c), 7 G dom £?, and F7 = G = F \ 7, or (b) p — I exists, 7 G dom E, and £"7 = G — F \ 7, or (c) p is a limit ordinal > (β"*"}7'^0), p is not a generator of F, and 7 6 dom E and £"7 = G = F \ 7, or (d) p is a limit ordinal > (Ω+)^κ^ p is itself a generator ofF,p£ dom JS, and 7 G dom E and EΊ = G (but FT ^ F f 7), or (e) p is a limit ordinal > (Ω+)^κ°\ p is itself a generator of F, p £ dom E, and π(E)Ί = G, where π is the canonical embedding from J^ to Proof. By induction on p. As the proof is rather convoluted and follows closely the proof of Lemma 11.4 of [FSIT], we shall not give it here. The idea is as follows: we show that G satisfies, in M, all the conditions for being added to the j(Kc) sequence as EΊ. We have 1-smallness by hypothesis, and coherence because F is a restriction of E j . (If p is a generator of F, then we have to appeal to the condensation Theorem 8.2 of [FSIT] here.) We have the initial segment condition by induction. For our background certificates we can take (N, H), where VΩUACN, \N\ = β, and H = Ej Π ( [ j ( Ω ) ] < ω x N). (As we remarked earlier, H G M.) Since G can be added to j(Kc) in M as Fj, the construction of j(Kc) guarantees 7 G dom E. A "bicephalus" or "Doddage" argument guarantees G = EΊ . The foregoing is more or less a proof in the case (a) or (c) of the conclusion holds. If (b) holds, we run into technical problems with mixed type bicephali. • / τs C \ If (d) or (e) holds, we also run into the problem that Jj ' may not be a stage Mξ of the construction of j(Kc) done within M. We refer the reader to §11 of [FSIT] for a full proof. We should note that this argument requires the iterability of the bicephali and psuedo-premice which arise. (See §11 of [FSIT] for more detail on how they arise.) We shall prove this in §9. D We define Λ 0 C Ω by α G AQ <3> a. is inaccessible and (&*}K = α"1" and {β < a I β is inaccessible and (β+)K° = β+} is not stationary in a . One can easily check that if Kc \= there are no Woodin cardinals, then (i)

ΛO is stationary in β,

The Core Model Iterability Problem

(ii) (iii) (iv)

AO has a £ AQ α G AO c the K

9

μ0-measure 0, + c + => ((α )^ = α Λ a is inaccessible), c =^ α is not the critical point of any total-on-K extender from sequence.

It was in order to insure the existence of a set with the properties of AQ c that we included condition (3) in Case 1 of the construction of K . (Condition (2) could be dropped, but it does no harm.) The definition of Kc which we have given in this section is unnatural in one respect: its requirement that the Λ^'s, and hence Kc itself, be 1-small. We believe that, were this restriction simply dropped, the resulting Λ7's would converge to a model (Kc)' of height β, and one could show that either (α+)(^c) — α+ for μo a.e. a < Ω, or (Kc)f \= there is a superstrong cardinal. What is missing at the moment is a proof that if M is a countable elementary submodel of one of the Λ^'s or their associated psuedo-premice and bicephali, then M is ω\ + 1 iterable (in the sense of definition 2.8 of this paper). At the moment we can only prove this iterability result for mice which are "tame" (do not have extenders overlapping Woodin cardinals), and thus it is only to such mice that we can extend the theory presented here. (See [CMWC] and [TM].)

10

§2. Iterability

§2. Iterability In this section we shall sketch a proof that if T is a ά-maximal iteration tree on <£jb(Λf), and T \ X is simple for all limit λ < IhΊ ', and IhT is a limit ordinal, then for all sufficiently large «, ΐ/ColO,*) |= 7" has a cofinal, wellfounded branch. There are several other iter ability facts we shall need, and actually we shall not prove even this one in this section, since we shall make some simplifying assumptions on T. The reader seeking full detail and generality will find it in §9. The reader who would like to see the main ideas in our iterability proof, while avoiding full detail and generality, can content himself with this section. In this paper, we shall diverge slightly from the terminology of [FSIT] regarding iteration trees. By an iteration tree we mean a system T obeying all the conditions required in the definition of §5 of [FSIT] except possibly the increasing length condition. That is, we do not require a < β => Ih E% < Ih ET . Iteration trees in the sense of [FSIT] we call normal. (We note that even in a normal tree T, E% may not be applied to the earliest possible model. This last requirement is part of fc-maximality.) Although the trees which arise in comparison processes are all normal and fc-maximal for some k < ω, we must cover more than such trees in our proof that Kc is iterable. This is because the proof of the Dodd-Jensen lemma (5.3 of [FSIT]) involves non-normal trees. We shall say that T is simple iff for all sufficiently large /c, γCo\(ω,κ) μΊ has at most one cofinal wellfounded branch. (This diverges slightly from the terminology of [FSIT].) We shall need a relative of this notion. Definition 2.1. Let T be a tree on M. of limit length, and a £ OR. We say T is Oί-short iff for all sufficiently large K yCol(ωtκ)

μ

Ί

βnal

has no CQ

OL

branch

b

such ihat

is isomorphic to an initial segment of

ORMb .

The next two lemmas come from the uniqueness theorem of §2 of [IT], See also Theorem 6.1 of [FSIT]. Their formulation also owes a lot to work of Woodin, and to the Π\ mouse condition of §5 of [IT]. Lemma 2.2. Let M be l-small and T an iteration tree on M, and let λ < Ih T. Then for some a £ OR, T \ λ is a-short. Proof. Assume not. Let δ

=

E

=

sup{//ι Ej I β < λ} ,

The Core Model Iterability Problem

11

M

(so that E= common value of E *> \ <5, for all cofinal, possibly generic, branches bofT \ λ). Our hypothesis implies that Vα G OR, there are possibly generic cofinal branches 6 φ c of T \ λ such that α is in the wellfounded parts of both Mb and Mc. Hence so is La[E\. By §2 of [IT] or 6.1 of [FSIT], M

L[E] t= 6 is Woodin. But now E= E * \ <5, and El[ has length > δ. It follows that M\ is not 1-small, a contradiction. D We do not get from 2.2 that Ί itself is α-short for some a G OR. But the proof of 2.2 gives at once: Lemma 2.3. Let T be an iteration tree of limit length on a premouse M. Then either T is a-short for some a G OR, or there is a proper class inner model with a Woodin cardinal. The next lemma explains the importance of these uniqueness facts in our proof of iterability. It shows that the existence of a "bad" tree on M reflects to the existence of a bad countable tree on a countable λf X M.. Part (a) of the lemma is due to Woodin and the author independently; part (b) is due essentially to Woodin. Let us use "putative iteration tree" for a system having all the properties of an iteration tree, except that its last model, if it has one, may be illfounded. Lemma 2.4. Let T be a putative iteration tree on a 1-small premouse M such that T \ λ is simple for all λ < Ih T. Suppose that either (a) (T \M)^ exists, or (b) There is no proper class inner model with a Woodin cardinal. Suppose also that either T has a last, illfounded model, or T has limit length and for all sufficiently large K, γ^°\ω^) ]p T has no cofinal wellfounded branch. Then there is a countable λf ^ M and a countable putative iteration tree U on λf such that II \ λ is simple for all λ < Ih U and either U has a last, illfounded model, or U has limit length but no cofinal wellfounded branch. Proof. We give the proof under hypothesis (b). We also assume Ih T is a limit ordinal, and leave the contrary case to the reader. By 2.3 we can fix a G OR such that for all λ < Ih T, T \ X is α-short. Let θ be large enough that T,ΛΊ, a G VQ and Ve satisfies a reasonable fragment of ZFC. Lowenheim-Skolem gives us a countable transitive H and embedding

such that for some

So λf X M via 7Γ \ λf. Also, U is a countable iteration tree on λf of limit length. Now H \= U \ X is a short, for all λ < Ih U. But this notion is sufficiently absolute that U \ λ truly is α short, for all λ < Ih U. [Let x be a

12

§2. Iterability

real which is a member of H ^°^(ω^)t κ sufficiently large, and codes (M,U, ct). Being a short is a Π\ property of x, and so absolute to H^°^ω>κ\] Now H \= U \ λ is simple, Vλ < Ih U. Since U \ \ is ά short and ά £ #, this is absolute, and thus U \ \ truly is simple for all λ < Ih U. Similarly, if U has a cofinal wellfounded branch 6, then OR/^6 < ά, and thus U has a cofinal wellfounded branch in #^°l(ω,«) where K = (card(Λf)) card(W) δί)H . So U has no cofinal wellfounded branch. The proof under hypothesis (a) is similar. Where in case (b) we used αshortness and Π\ absoluteness, we use (T^M)^ and Π\ absoluteness. [We have π : (U.λT)^ -> (T,Λ4)«, and this guarantees (U,λf)*H = (ZΛ-Λf) 1 . That in turn implies H[G] is correct for Π\ statements about x, where x is a real coding (U,λf) in H[G], and G is generic /H for Col(ω,/c), K: = (card (U) caxd(λί))H .] D Remark. If /c = card(T) card(A^), then V0 > K, T has a cofinal wellfounded branch in FCo1^'^ iff T has a cofinal wellfounded branch in yCo!^,*). In particular, if ω = card(T) - card(ΛΊ), Ί has such a branch in γC°^ω^ iff T has such a branch in V. We are ready to state the main result of this section, which concerns the iterability of countable elementary submodels of Kc and its levels. Although we could prove that such structures are iterable with respect to arbitrary trees, to do so would add a layer of notational fog to the proof for normal trees. We shall therefore prove just the iterability we need, which is iterability with respect to linear compositions of normal trees. We call these trees almost normal. More precisely, suppose (Ta | α < /?) is a sequence of normal trees such that TO is on ΛΊ, Ta+ι is on the last model Λ4^ of Ta for all a + 1 < /?, and Tχ is on the direct limit of the Λ4^, for α < λ , i f λ < / ? i s a limit. We can form an iteration tree U by "laying the 7^'s end-to-end". We say U is generated by (Ta \ a < /?), and call a tree U generated in this way almost normal. Such a composition U will generally not be maximal, even if the Tα's are maximal, since maximality requires going back to the earliest possible model. We say U is almost k-maximal iff TO is ά-maximal, and Vγ < β (TΊ is .;' -maximal, where j = degTa(M^) for all sufficiently large α < 7. Theorem 2.5. Let P X ί*(Λ/i) for some k,θ, and P be countable. Let T be a countable, almost normal, almost k-maximal putative iteration tree on P such that T \ X is simple for all λ < Ih T . Then either Ί has successor length, and its last model is wellfounded, or T has limit length, and T has a cofinal wellfounded branch. Sketch of Proof. We shall give the proof in a special case which highlights the new ideas. The simplifying assumptions we make are: T is normal and ^-maximal, and (1) T has length ω,

The Core Model Iterability Problem

13

(2) k = ω\ moreover P is passive and P |= ZF~, and there is no dropping onT, (3) Letting Pi be the ith model of T, and V{ the sup of the generators of

Eΐ>

Pi |=

strength(£/ ) > ι/<,

moreover, z/t is a limit ordinal. As z/i is a limit ordinal, jjl is either type I or type III, where β = //ι E1?". This implies that ι/t is a cardinal of J^1, and therefore, by our strength assumption in (3), V{ is a cardinal of Pi. We have made assumptions (2) and (3) in part to avoid any need for "resurrection" (cf. §12 of [FSIT]) in the construction to follow. By (2), pω(Pi) — ORPt for all i £ ω, all ultrapowers on T are Σω (satisfy the full Los theorem), yet are formed using functions which belong to the model in question. It may seem that these assumptions just resurrect the "coarse structure" setting of [IT], but in fact they do not. For one thing, we don't have Ί G PQ. Because ρω(P) = ORP, £ω(λfθ) = λfβ Fix an elementary π : P -> Λ/0. We shall show that there is a cofinal branch 6 of T and elementary σ : Pb —»• λίe such that

commutes. Let U be the tree of attempts to build such a branch b and embedding σ. More precisely, let τ : P —» Q be elementary; we shall define a tree U = £/(r, Q) which tries to build (6, σ) such that σ :Pι> -+ Q and r = σ o i^"6. Fix an enumeration .

onto

such that ^((βjz)) is infinite for all (e,z) such that x G Pe We then put (a) (b) (Pβfc, a?0, . . . , xjb) Ξ (Q, j/o, - - , y*), where V i < *

(i) t(t) = (e, x) for β <E [0, et ]τ implies x f = i£ βfc (ίr), (ii) t(i) = (e, x) for e jί [0, e, ]τ implies Xj = 0, (iii) t(i) = (0, x) implies y, = r(x). Remark. (x 0 ) - - , #*} in (b) is determined by (e0, . . . , e*).

14

§2. Iterability

Let U = U(τ^Q) and i G ω, and suppose we are given an embedding σ : Pi —» Q such that r — σ o fζ^ . From σ we get an initial segment of a branch ofW(r,Q); let

P(», σ, r, Q) = ({e0, . - . , ejb), (y0, . . . , y*)) where (eo, . . . , ejt) is the increasing enumeration of [0, z]τ and letting (x 0 ) - - , £k come from (eo, . . . , βk) as in the definition of £/(τ, Q), we have yj = σ(xj) for all j < k. Theorem 2.5 is proved if we show that Lt(π,Afe) has an infinite branch. Let us assume otherwise toward a contradiction. By "coarse premouse" we mean a premouse in the sense of [IT]; that is, a structure M = (M, G,5), where M is transitive which is power-admissible and satisfies choice, the full collection schema for domains C Vjf* , and the full separation schema. We also require that ωM C M, and that δ be inaccessible in M. Write δM = δ. Let β

C = (Λ/H £ < 7) (7 < ) be the construction done in §1. Notice that C is definable from no parameters over VΩ (Here 7 is the first place < Ω where the construction breaks down, if any, and 7 = Ω otherwise. Thus θ < 7.) It follows that if Έ, = (R, G, δ) is any coarse premouse, then Cί^ makes sense: we interpret the definition of C inside V™. If Ti is a coarse premouse, then a cutoff point of Tl is an ordinal ξ such that <5π < ξ < ORπ and (V^π, G, <5π) is a coarse premouse. We now define by induction on i triples (^iyQi,Tli) with the following properties: (1) Hi is a coarse premouse, (2) Qi is an "N-modeP of the construction C^1 , moreover πt : Pi —» Qt elementarily, (3) for all j < i, Tlj agrees with 7^-, through TTJ(Z/J), (Recall that z/j = v(E?) = strict sup of the generators of Ej ' . We say coarse premice Έ, and S agree through 77 iff V™ = Vf.) (4) for all j < i, π< f ι/j = π; f Vj\ moreover Qj agrees with Qi through πXi/ί), (Ordinary, "fine" premice Q and 71 agree through η iff (5) Let W = Ufa o f^,., Qt ) and

(So W is a order type (6) i f i Clause

tree in 72.,- and p is a node of ZY.) Then ZY is wellfounded, and the of the set of cutoff points of 7£t is at least \p\Ui > 0, thenT^ € f t , -ι. (6) gives us the desired contradiction.

Base step: Set

The Core Model Iterability Problem

TΓo

=

7Γ,

QQ

=

λfβ

15

By assumption, U — W(πo, Qo) is wellfounded. Set

where £ is the |W|th ordinal a> Ω such that (Vα, E, ί?) is a coarse premouse. Our applicable induction hypotheses, namely (1), (2), and (5), clearly hold. Inductive step. We are given ((τrj,Qj,ftj) \ j < i) Let j = T-pred (i + 1), and set Q; |= ZF~ , so the ultrapower is formed using functions belonging to Qj. Notice that the ultrapower makes sense. For set k — crit Ej and K = πi(κ). Let E — πi(El). The rules for iteration trees guarantee k < ι/j. Induction hypothesis (4) states that πz \ ι/3 = πj \ ι/j] thus K < TTJ(Z/J). But 7Γj(i/j) is a cardinal of Qj, and Qi agrees with Qj through ?TJ(I/J). Thus P(κ)Qι C Qi, and the ultrapower makes sense. (We may have subsets of K in Qi but not Qj] 7Γj(i/j) may not be a cardinal of Qt . So E1 may measure more sets than necessary.) Let σ : Pi+i —»• Q^+1 be given by the shift lemma:

We have that σ is well defined and elementary, that Q(+1 agrees with through all η < Ih E, that σ f //ι #?" = πt f //ι £??", and that σoi^i where i^ : Qj —> Q(+1 is the canonical embedding. The following little lemma will be useful. Lemma 2.6. Suppose J~ V/c < ωβ[(Afη

\= K

η

is an initial segment of Mη such that

is a cardinal) Ό (Jβ

then there is a ξ < η such that J~

η

η

\= K

is a cardinal)]

= Λ/^.

Proof. We may assume ωβ < OR Π λfη . Let ξ < η be least such that Ja limit then Jβ

η

also suppose Jβ of MT- Since Jβ

n

is an initial segment ofλfξ. Clearly, if ξ is

= Λ/ξ, and we are done. Therefore we suppose ξ = τ+l. We η

φ Λ/V+i , and this implies that Jβ η

η

is an initial segment

is not an initial segment of Λ/V, and Mr = ίw(^/τ), we

have from the proof of Theorem 8.1 of [FSIT] that pω(λfτ) < OR** and J\fτ \= pω(λfτ)+ exists, moreover ωβ is strictly larger than (pω( Now let δ = mf{pω(λfθ) \ r < θ < η}, so that 6 < ρω(λfτ) and (pw(^r)+)Vr < ωβ. Then (<5+)^ is a cardinal of Mτ (by §8 of [FSIT]), and

§2. Iterability

16

thus it is a cardinal of Jβ η . On the other hand, the defining property of δ guarantees (6+)^r is not a cardinal of λίη. This contradicts the hypothesis of 2.6. D Since E is on the Q; sequence, E = E$l where β = Ih E. Let Qt = Λ/^'. Lemma 2.6 then gives us a ξ < η such that J?1 = Λ/^* . (We apply 2.6 within 7£;. Our simplifying assumptions tell us that 7Γi(ι/t ) is the largest cardinal of Jβ η , and πt (z/i) remains a cardinal in Λ^. Thus 2.6 applies.) Remark. Without our simplifying assumptions we don't get that 7Γ;(z/t ) is a cardinal of Q, , and therefore there may be no ξ such that j9l = J\Λπ' . At this point in the general argument we need to resurrect a background extender for E by inverting certain collapses. If we were in the situation of [FSIT] we would now have a "background extender" F for E such that Ult(7£, ,F) makes sense. We would let 7£ +1 = Ult(7£j,F), and then take TJ +i to be the collapse of a suitable hull of a suitable cutoff point of 7£[+1 . Qt+ι would be the image of Q(+1 under collapse. Now, however, we have no such F (after all, Vκ_^l — Vκ+\ , so F would be a full extender in Ίli). So instead we get a suitable background certificate (AT, F) for E in Tί{. Since N is large enough, and in particular V* 3 — V™1 C N, we can take an analogue of the hull producing 7£, +ι and Q, +ι "almost everywhere" below K. We get 7£i+ι(ύ), Qt +ι(ϋ) for ί& are ϋ, for a suitable 6. We then let fci+i = [&,λϋfti+ι(ϋ)]£ and g<+1 = [ Let

Since A is a countable subset of Hi , .4 G 7£i and is countable in 7£t . Also 1

where

§

E is the last

-4 C Un<ω ^(W )^' » -Λ/f = Jβ' extender of Λ/f , so we can let Hi |= (N, F) is an ^-certificate for tff1 . Since "Ult(7V,F) C Ult(^V,F), πt f ι/< G Ult(7V,F). Let us pick an F support 6 and functions \ΰ - 7rj(u), \ΰ z/(w) such that

We may assume that 7Γj(ΰ), z/(u) G V^ for all w. Claim. For F& a.e. ΰ, there are in V** \ a coarse premouse 7^, an "ΛΓ model" Q of C^ , and an elementary embedding π : PΪ+I —* Q such that (1) % = %, and J«fl) = J«ij (2) 7Γ ί Vi = Ti(«),

and

The Core Model Iterability Problem

17

(3) letting U - W ( π o t ^ i + 1 , Q ) and p = p(i + l , π , π o i£ ί+1 ,Q), we have: U is wellfounded, and there are in order type at least \p\u cutoff points of 7£. Proof. Fit measures the set of such ϋ, as the quantifiers in its definition range over V? , and J$ G N. Let ΰ G X if and only if ϋ G M | δ | and the claim fails for ΰ, and suppose toward a contradiction that X G F&. Fix an enumeration 7^ +ι = {xn | ft < ω} of P +I. For n < ω let σ(a? n ) = [cn,/n]^J , where cn G [^*(^t)]<α; an(^ Λ» £ Or If χn < Vi, so that σ(xn) = τr;(z n ), then we choose cn = {?Γi(xn)} and fn = identity function. Subclaim A. There is a set Yn G F Coϋ ... ϋCn order preserving, and ί" \Ji then

sucn

that if t : (Ji
κ ιs

Proof. Note (P< + ι,xo *n) Ξ (Q{ + ι,σ(x 0 ), ,σ(x n )). Now let Q{+1 μ ^>[σ(xo)) ) σ(#n)] By Los' theorem there is a set Y^ G -f7c0u uc n such that if ί : (Jt[/o(<"co), . .T, /n(t/;cn)]. We can choose Y* G ran π; l)ecause {/0, . . . , fn} C ran π, . Thus Y* G >t, and hence Y* G F C o U .. U C n . Let Yn = Π^,^; this works because F CoU ... UCn is countably complete. D Subclaim B. If xn < z/, , then there is a set Zn G ίc n ufc such that Ίft : c n U6 —*• K is order preserving and t"(cn\Jb) G ^n, then ττ ί (ί // 6)(x n ) = fn(t"cn) < v(t"b). Proof. Ult(ΛΓ, F) N [6, λδ π, (ti)]?(xn) - [cn> /n]£ < [6, λfi - ι/(ύ)]£ (noting that [cn,/n]^ = [{ττi(x n )}, identity]^ = πt (x n ) since xn < ι/t ). The subclaim now follows from Los' theorem for Ult(7V, F). D Subclaim C. lϊxn = «Ji+ι(2/)> then there is a set Wn G FCn such that whenever t"cneWn,fn(t"cn) = ^(y). Proof.

By Los' theorem for Ult(Qj, E), there is a set Wn G ^Cn as desired. But we can assume Wn G ran πy, so that Wn G >!, and thus Wn £ FCn. Ώ Since F is countably complete, we can find

18

§2. Iterability t : [J cn U 6 —»• K n<ω

order preserving such that t"b G ^, and t" |Jί
V> : Pi+i —>• QJ elementarily, and ^ ί ι>< = ^(^'fr), and πj = ψ o i?i+1. Let

and

so that by induction 7£j has at least \p\u many cutoff points. Let

One can check easily that g is a proper extension of p in ZY; this is true because jTi + I and πj = ψ o «J}+ι Thus there is an η G OR^J such that η = \q\uih

cutoff point of

Hj.

Set

72.

=

transitive collapse of closure of Vv^,,^ U {Qj} under Skolem functions for V^3 and

Q

=

image of

Qj

under collapse,

7Γ

=

image of

φ

under collapse .

ω-sequences

(Since ω*R, C 72, T and ψ belong to the uncollapsed hull. So also do U, p, and
Clearly (Q,7e,τr) G V*J = V* . Moreover, (Q,ft,π) witnesses the truth of the claim for ΰ = t"b G X. For (2): ψ \ V{ — π \ V{ because we put all of z/(ί"&) into the hull collapsing to π, and ψ(xn) < ι/(t/;6) for x n < i/j by β. But ^ f i/. = π t (t /x 6) by 5, as we observed earlier. For (3): Since KJ = ψ o fT i + 1 , π,- o ^ - φ o i£ί+1, and

moreover V^ J has at least \q\u cutoff points. This is first order, so 7£ has at least \q\if cutoff points, where q = collapse of q and U = collapse of U. Since U = U(π o ij£ί+1, Q) and q = p(i + 1, π, TT o i^f+1, Q), we are done. This proves the claim. Π

The Core Model Iterability Problem

19

By AC in TV, we have a function

denned F>, a.e., said function in N, picking witnesses to the claim. Let

nί+ι

= [b, λΰ fc(δ)]£ ,

Qί+1

= [b,\ϋ

Q(ΰ)]$,

and τrx +ι = [6, λϋ τr(ϋ)]£ . By (1) of the claim, V%% = V™^'F) = V^y So Ki+l agrees with Kk, k < 2, as desired. By (2), τrt +ι ί z/, = π, f z/z . To show that <3ί+ι agrees with Qt below τrt (z/t ), we argue as follows. Let 7 < ττt (ι/, ). Since Qi agrees with Q(u) below z'(ϋ), for F& a.e. ϋ, we easily get jQt+ι = [{τ}} flN?

where

j(α) = jg. for all α < « .

Now clearly / 6 Q», and the coherence condition on E, which is on the Qi sequence, gives

But then the agreement between E and F guarantees J\Λ t + 1 = J^1. (It is enough to see that if βo < βι < ωγ and φ(vQ,v\) is a formula, then jf'+l \= βι], because there is a uniformly definable +1

t surjection of ωj onto jQ . Let us assume /?0 < βi < 7 and Then for ^{^0,^,7} a.e. w, J^1 ^= ^[^o, ^ι] The set of such ΰ is in .4, and so is measured the same way by E{β0tβίtΊ}.) The remaining induction hypotheses for our construction are easy to check. This completes our proof of 2.5 under the simplifying assumptions (l)-(3) above. We now sketch how to do without our first simplifying assumption, that lhT = ω. ' We call a sequence ((TTJ, Qj,7£j) | j < i) satisfying our inductive hypotheses (l)-(4) an enlargement of T. Suppose we are given a simple T of length ω 4- 1 and want to construct an enlargement ((^j^Qj^j) \ j < i) of T. Let us assume that our simplifying assumptions (2) and (3) hold of T. Let Pj be the jth model of T. We are given by hypothesis

τro : 7>o -> Qo where Qo = .Λ/i ,

the flth model of Cv

and TΓo is elementary. For any r : P —»• Q let

20

§2. Iterability

U(τ,Q)

tree of attempts to build a pair (c, σ), where c is

=

a cofinal branch of T f ω, c φ [0, ω]τ , and r = σ oιζc. Note because T is simple, U(τ, Q) is wellfounded for all τ, Q. For j £ ω such that j ^ [0,u/|τ, and ψ : Pj —»• Q such that

commutes, set P(j) V 3 ) r> Q)

=

canonical initial segment of length \{k | kTjk-ιkTω}\ of a branch of U(τ} Q) given by (j, φ) .

We define by induction on i < ω enlargements ε% of T \ i + 1. There are two cases. Case l.i+lφ [0,ω]τ. In this case we proceed exactly as we did in the construction given in the length ω case. Let Letj = Γ-pred(i+l). LetW = ZY(fljθ^,<3 ; ) andp = p(j, ?TJ , TT, f z/j = πj(ΰ) f i/f. (Here [6, λΰ τr;(ϋ)]£ = π, f ι/, .) We then set g = p(i -f 1, V 5 , ?TJ o ij^ , Qj).) Since i -f 1 §ί [0,ω]τ, ί extends p in W. Thus, for F& a.e. ϋ, we are given a cutoff point of 7£j at which to take a hull. As before, we do this and collapse, producing 7£(ϋ), Q(ΰ), π(ύ). We then take Λi+ι = [b,Xm(ΰ)]$, Q, +ι - [6,AuQ(iZ)]£, πi+ι - [6, λϋ - π<(iϊ)]^. Set 2. f + 1 G [0,u;]τ. Again, let Γ = ((^,Qj,^) | j < i), and let j = Γ-pred(ί + 1). Arguing as before, we get "measure one many" Ψ : P<+ι -* Qj such that 7Γj = ψ o ij*t+ι. In this case, we are not given an ordinal at which to take a hull; we're on the wellfounded branch of T and so don't expect to

The Core Model Iterability Problem

21

get such ordinals. Along this branch, we shall realize the models of Ί back in V; that is, we take

= QJ, 7Γ;+1

=

Ψ,

for a ^ chosen to meet certain "measure one" conditions. (Thus by induction, ftf+ι = (V£,e,β),Q. +ι =-Λ/i, and π f+ ι o i£ί+1 = π0.) In order to do this, we must redefine (πk>Qk,'R>k) f°Γ j < fc < i, as otherwise our inductive hypotheses on agreement will fail. (After all, if k = crit Ej } then KJ(K) = ψ o ij^t +1 (/c) > Ψ(k) ) First, we find new K,Q' fc ,ft' fc ) for j < k < i such that K,^,^) G Ult(JV, F). (As in case 1, ( N , F ) is a ran π;- certificate for J^' where β — lh(πi(E?)).) For this, we must suppose that our induction hypothesis on the number of cutoff points gives us, for each k s.t. j < k < i, a cutoff point ηk of 7£fc which we can now afford to drop to. Let G be the finite set of relevant parameters, and Tl'k

=

collapse of Skolem closure of G U V^k) inside

Q'k

=

image of

TrJ.

= collapse o πfc ,

Qk

under collapse,

Now (πj., Q^H'jg) is coded by a subset of V^ϊ add to our inductive agreement hypotheses

x belonging to Hk Let us

(As our background extenders are "i/ + 1" strong, this is consistent with the construction in Case 1.) It follows that V^(fcί/fc)+1 C Ult(7V,F), where (N,F) is the background certificate in 7£f for πi(E^). Let for j < k < i,

Then using {(?ri(u),7Jj.(u)) | j < k < i) in the same way that we used 7r, (t/) in Case 1, we can define additional measure one sets for F so that by meeting them we guarantee that

is an enlargement with the desired properties. (Notice that if k < j, then ί/fc < k since j = T-pτed(i + 1) and k = crit E1^. But then «J"<+1 ί ^fc = identity, so V Γ ^ = (Φ ° *^<+i) ί ^ - π j Γ ^ = ^k \ "k- This is why we do not need to re-define Sl \ j.)

22

§2. Iterability

The existence of the cutoff point ηk of 7£fc, for j < k < i, is not a problem because for each k £ u>, only one such cutoff point is used. (It is used at stage i + 1, where i is least such that k < i and i + 1 £ [0,ω]τ.) Let now, for fc E ω fjj? — eventual value of

£].

as

i —> ω .

The eventual value exists since in fact Slk changes value at most once. Set Hωω - K%

=

= Qo

=

common value of Tl] , for

j G [0, ω]τ

and

i >j ,

common value of Q^ , for j £ [0, ω]τ ,

(Here £? = (π ω. The construction of [IT] did not have this property, and that led to complications. The techniques of §12 of [FSIT] allow one to drop our simplifying assumptions (2) and (3). This completes our sketch of the proof of Theorem 2.5. D Putting together Lemma 2.4 and Theorem 2.5, we get Corollary 2.7. Suppose there is no proper class inner model with a Woodin cardinal; then for all η < Ω and k < ω, ίjb(Λ/^) exists and is k-iterable. Proof. The reader can find Ar-iter ability defined in 5.1.4 of [FSIT]. Roughly speaking, it means: iterable with respect to simple, ^-bounded iteration trees. The existence of €jb+ι(Λ/"^) comes from the fc-iter ability of ^(Λ/i;) via Theorem 8.1 of [FSIT]. (The Strong Uniqueness theorem, 6.2 of [FSIT], is used here to show that the iteration trees arising in the proof of 8.1 are simple.) So we need only show that if (£fc(-Vrj) exists, then it is fc-iterable. Let T on (tjg(λίη) be almost normal and fe-maximal, and T \ λ simple for all λ < Ih T. Suppose Ih Ί is a limit; the case Ih Ί is a successor is similar. We want a cofinal wellfounded branch of T, and from 2.4 and 2.5 we get such a branch in v^ o K ω » Λ ) for all sufficiently large K. So if T is simple, we are done. If not, then letting M — ^(Λ/^), we have δ such that M \= δ is Woodin, and pk+ι(Λ4) > δ. Let (τ,e} be lexicographically least above (r?,&) such that
The Core Model Iterability Problem

23

tree T* on (ω, θ)). Definition 2.9. A premouse M is θ-iterable iff II has a winning strategy in G*(M,Θ). A winning strategy for II in ζj*(M,θ) is a θ-iteration strategy for M. Similarly, M is (ω,θ)-iterable iff II has a winning strategy in (/*(ΛΊ,(u>,0)) and we call such a strategy an (ω,θ)-iteration strategy. An obvious copying construction gives Lemma 2.10. If M -< λί and λί is θ-iterable, then M is θ-zterable. Similarly, ^λί and Λf is (ω, θ) iterable, then so is M. From 2.7 we get

24

§2. Iterability

Theorem 2.11. Suppose there is no proper class model with a Woodin c cardinal; then K is (ω, θ)-iterable for all θ. c

We shall only make use of the (ω, Ω -f l)-iterability of K . We can prove this without assuming there are no proper class models with a Woodin cardic nal, but using instead the measurability of Ω and assuming K \= there are no Woodin cardinals. More precisely, we use that A* exists for all sets A £ VΩ in order to see that Kc is well-behaved with respect to trees T £ VΩ (using 2.4 (a)), and then the weak compactness of Ω to see that K° is well behaved with respect to trees of length Ω. We use that Kc \= there is no Woodin cardinal to show that the appropriate trees are simple, and thus have not just generic branches, but branches in V. In a similar vein, one can omit the hypothesis "there is no proper class model with a Woodin cardinal" in 2.8, by using the measurability of Ω. We have stated 2.8 and 2.11 as we have in order to point out what can be proved without using the measurability of Ω. Most of the rest of this paper makes heavy use of Theorem 1.4, and we certainly do not know how to avoid' the measurability of Ω as a hypothesis in that theorem. So we shall take "Kc \= there is no Woodin cardinal" as our non-large-cardinal hypothesis, when we need one, instead of "there is no proper class model with a Woodin cardinal". We shall use: Thorem 2.12. Suppose Kc \= there is no Woodin cardinal] then Kc is (ω, Ω-\1)- iterable.

The Core Model Iterability Problem

25

§3. Thick classes and universal weasels We shall adapt Mitchell's notion of a thick class of ordinals to the present context. Definition 3.1. Let M and λί be premice. A coiteration of M with λί is a pair (TjU) of normal, ω-maximal iteration trees such that T is on M, U is on λf, and successor steps in the formation of T and U are determined by iterating the least disagreement and the rules for normal, ω-maximal trees. A coiteration (T,U) is terminal iff there is no extension of(T,U) to a properly longer coiteration. A terminal coiteration (T,U) is successful iffT and U have last models P and Q, and P
26

§3. Thick classes and universal weasels

Definition 3.4. A weasel M is universal iffM is Ω + l-iterable, and whenever (T,M) is a coiieration of M with some proper premouse M such that 1 Ih U = Ω + 1, then λί is a weasel, IΫ Π [0, Ω]u = 0, Vα < Ω(%Ω(a) < Ω), A weasel is universal just in case it is maximal in the mouse prewellorder <* on Ω + 1 iterable proper premice. (Roughly, M <* λί iff M iterates to an initial segment of an iterate of ΛΛ In order to give a precise definition and prove that <* is a prewellorder, one must impose a bound on the large cardinals in the proper premice being ordered. This is because one needs a simplicity hypothesis in the Dodd-Jensen lemma. Here we can assume that there is no proper class model with a Woodin cardinal.) Any two universal weasels are =*, that is, they have a common iterate. By 3.3, this common iterate is itself a universal weasel. If M is universal, then any Ω -f 1 iterable set premouse λί is strictly below Λί in the mouse order, and in fact a coiteration determined by Ω + I iteration strategies for M. and λf must terminate successfully at some stage α < Ω. If there is no inner model with a strong cardinal, then any Ω + 1 iterable weasel which is strictly above all Ω+l iterable set premice in the mouse order is universal. (This fact was noticed by Jensen, who made it his definition of universality.) If our weasels can have strong cardinals, however, then this condition no longer suffices for universality. For suppose P is universal, and P ^= K is strong. Let Q come from hitting some image of each extender from P with critical point some image of « once. Let M = JΩ . Then every Ω + I iterable set premouse is <* M, but M is not universal. We now show Kc is universal. Lemma 3.5. Let W be an ί?-fl iterable weasel such that for stationary many regular cardinals a < Ω, a+ = a+ then W is universal. Proof. Suppose that M is a proper premouse which is a counterexample to the universality of W. Let T on W and U on M result from the comparison process, and fix a £ [0, Ω]u such that i^Ω is defined (that is, there is no further dropping along [0, Ω]u) and fix K; such that z^ Ω(κ) = Ω. (We have Ih U — Ω +1, because otherwise, Ih U = 7 +1 < Ω for some 7, in which case Ύ OR^ < β, so that MΊ is an initial segment of the last model of T, contrary to hypothesis.) These are club many β £ [a,Ω]u such that ^^(K) = β Moreover, by 3.3 we have that iζ Ω is defined, and ΪQ Ω(Ω) = Ω. Thus there are club many β € [0, Ω]τ such that ίξ"ββ C β. Now let β be a regular cardinal in both the clubs of the last paragraph, and β+W = β+. As β is regular and i^β C β, ίξβ(β) = β, and thus (ξβ(β+) = β+ = (β+ )*β. On the other hand, (β+ )^> = ^ (κ+Vβ), and so (/?+ )^ < β+ as (*+)** < β+. But then, as β = crit iuβΩ and β < crit iJ Λ , (β+)λίn =

The Core Model Iterability Problem Wa

Wβ

0

27

+ Wn

and (β+) = (β+) . It follows that (β*)" < (β ) , and this contradicts the fact that WΩ is an initial segment of Λ/j? Π +

+

Remark. Below a strong cardinal, if (a )^ = α for cofinally many a < Ω, then W is universal. This is no longer true past a strong cardinal, as a modification of the previous example shows. Corollary 3.6. If Kc (= there is no Woodin cardinal, then Kc is universal. Proof. By Theorem 1.4, (a+)κ° = α+ for μ- a.e. a < Ω, and by 2.12 Kc is Ω 4- 1 iterable, so Lemma 3.5 does the job. D We can also prove a converse to 3.5, under the assumption that K° \= there are no Woodin cardinals. Theorem 3.7. (1) Let R and W be weasels having a common iterate, i.e., a successful coiteration (T,U) such that T andU have the same last model. Then for all but nonstationary many regular a < Ω, (α~*~)Λ — α+ iff(a+)w = α+. (2) Suppose Kc \= there are no Woodin cardinals; then for any Ω + 1 iterable weasel W, W is universal iff(a+)w = a+ for stationary many regular a< Ω. Proof. (1) Assume Ih T = Ih U — Ω + 1 for notational simplicity. Then for all but nonstationary many regular a < Ω, a < min(crit(zj β ), crit(^Λ)), and thus (α+X"" = (a+)Mϊ = (α+)Q, where Q = M% = M%. But also io"α(α) = a — ή)a(<x) for all but nonstationary many regular a. For such α, (α+)Λ = α+ iff (<*+)-"£ = α+, and (a+)w = α+ iff (a+)Mΐ = α+. This gives (1). (2) One direction is 3.5, and does not require the smallness hypothesis c c + κ + on K . Conversely, if K has no Woodin cardinals, then (a ) ° = α for stationary many α < Ω, and if W is universal then it has a common iterate with Kc. Thus we can apply (1). D For weasels small enough that linear iteration suffices for comparison (e.g. weasels no initial segment of which has a measurable limit of strong cardinals), the conclusion of (1) can be strengthened to: (a+)R = (a+)w for all but nonstationary many regular α < Ω. Moreover, there is a converse to this strengthening, for such "very small" weasels: if R and W are Ω + 1 iterable and (α+)β = (a+)w for stationary (equivalently, all but nonstationary) many regular α < Ω, then R and W have a common iterate. (In fact, R <* W iff (α+)Λ < (a~*~)w for stationary, or equivalently all but nonstationary, many regular a < Ω. The proof is based on the fact that if « < α, where α is inaccessible, and i comes from a linear iteration of length α of Λ, then |ί(«+) \R< (a+)R. This is true even if i £ R, since we can embed i into a "universal" linear iteration which is in R. We do not know whether this strengthening of (1), or its converse, hold for arbitrary 1-small weasels,

28

§3. Thick classes and universal weasels c

even assuming K has no Woodin cardinals. We conjecture that if R and W are Ω + 1 iterable weasels, then R <* W iff for club many a < Ω, α regular R w => (a+) < (a+) . The following definition adapts Mitchell's notion of a thick class to our context. The notion is useful in showing that certain hulls and iterations preserve universality. Definition 3.8. Let M be a weasel, and S, Γ C M. We say that Γ is S-thick in M iff (1) (2)

S C Ω and S is stationary in Ω, and for all but nonstationary many a £ 5: (a) α is inaccessible, (α+)M = a*, and a is not the critical point of a total-on-M extender from the M-sequence, and (b) ΓΠα" 1 " contains an a-club, and a. £ Γ.

Notice that if Ω is S-thick in M, and M is Ω -f 1 - iterable, then M is universal. Notice also that Ω is Ao-thick in Kc, provided that Kc satisfies "There are no Woodin cardinals" (cf.§l). We care most about Ao-thick sets in what follows. Lemma 3.9. Suppose Ω is S-thick in M. Then the class of sets which are S-thick in M is an Ω-complete filter. Lemma 3.10. Let π : H —+ M be elementary, where H and M are weasels, and suppose ran π is S-thick in M. Then {a \ ττ(α) = a} is S-thick in both H and M. Lemma 3.11. Let Ί be an iteration tree on the weasel M, and Ω be S-thick in M. Let λ < Ω, and suppose there is no dropping along [0, X]τ (and \ a. Thus α+ < crit ij β , and {7 | i%Ω(j) = 7} contains a and an α-club subset of α"1". From Theorem 1.4 we get immediately Theorem 3.12. Suppose Kc \= there are no Woodin cardinals; then Ω is Ao-thick in Kc.

The Core Model Iterability Problem

29

§4. The hull and definability properties Definition 4.1. Let M be a premouse and X C J\Λ. Then M

a E H (X)

<ω

& for some s £ X

and formula φ ,

α = unique υ such that M \= φ[v, s]. M

Notice here that H (X) in the uncollapsed hull of X inside M. Definition 4.2. Suppose Ω is S-thick in M, and let a < Ω. We say that M has the S-hull property at a iff whenever Γ is S-thick in M. M

P(ot)

C transitive collapse of

M

H (a U Γ).

In his work on the core model for sequences of measures, Mitchell makes heavy use of a lemma which states (translated into our context) that if Ω is 5-thick in M then M has the S-hull property at all α < Ω. This will fail as soon as we get past sequences of measures, as the following example shows. Example 4.3. Suppose Ω is S-thick in M. Let E be an extender from the M sequence which is total on Λί, and K = crit E. Suppose E has a generator > AC, and let ξ be the least such. (So κ+M < £, and E \ ζ = Eg4.) Now let jV = Ult 0 (Λί, E) = \Jltω(M, E). Then Ω is S-thick in ΛΛ We claim that λί fails to have the hull property at ξ. For let i : M —»• λί be the canonical embedding and Γ = ran i. Thus Γ is 5-thick in λί. Moreover we can factor i as follows:

M

^ Ult0(-M,ί?)

where t([α, /]) = i(/)(α). We have ζ = crit *, and ran(fc) = HλΓ(ξ U Γ), and so Ult(M,E \ ξ) is the transitive collapse of Hλί(ζt U Γ). On the other hand, by coherence έf = E£* = E \ ξ, so E \ ζ E -ΛΛ As E \ ζ is essentially a subset of £ (in fact, oΐ (*+)*) and E \ ξ £ Ult(ΛΊ, E \ ξ), we are done. Remark. If ί? is 5-thick in Λi, and ^ = M^ where T is an iteration tree on M and there is no dropping along [0, a]τ and a < β, and Λ^ has the S-hull property at all £, then P has the 5- hull property at ξ iff for no β+ 1 G [0, a]τ do we have (κ+)M? < ζ < v, where i/ = ^(£"J) and K = crit(ί?J). So we can recover from P, using the hull property, the pairs (AC, z/) such that some extender with critical point AC and sup of generators = z/ > (κ*)p is used on the branch from M to P. Notice also that P will have the 5-hull property at club many ξ < Ω.

30

§4. The hull and definability properties

Definition 4.4. Let Ω be S -thick in M, and a < Ω. We say M has the SM definability property at a iff whenever Γ is S-thick in M, a £ H (a U Γ). Even at the sequences of measures level, it is possible that Ω is S-thick in ΛΊ , but M fails to have the S-definability property at some α. For let Ω be S-thick in 7>, and M = Ult(7>,W) where W is total on P with critical point α. In view of the previous examples, we cannot expect that Kc will have the Ao-hull or definability properties at all α < Ω. We shall show, however, that Kc has these properties at many α < Ω. Lemma 4.5. Let W be an Ω -\- l-iterable weasel, and let Ω be S-thick in W; then there is an elementary π : M —* W such that ran π is S-thick in W, and M has the S-hull property at all a < Ω. Proof. Let us use "thick" to mean "S-thick" , and "hull property" for "S-hull property" . We shall define by induction on a < Ω classes Na -< W such that Na is thick in W. We shall have Na+\ C Na for all α, and N\ = Π/?<λ Nβ if λ is a limit. We then take ran π to be NΩ. In order to avoid dealing with collapse maps, let us say that a class N -< W which is thick in W has the hull property at AC, where K £ N, iff N has the hull property at σ(/c), where σ : TV = N is the transitive collapse. Equivalently, N has the hull property at K iff whenever Γ C N is thick in W, and Λ C AC and A £ TV, then there is a set B £ HW((N Π AC) U Γ) such that J5 Π AC = A. As we define the JV α 's we define κa for a < Ω. κa will be the αth infinite cardinal of NΩ . We shall have Na Π (AC Λ + 1) = Nβ Π (Acα + 1) for all β > a. We also maintain inductively that Nβ has the hull property at κ α , for all β > α. Base step:

No

=

W,

KQ

=

ω.

Kχ

= least AC £ AΓ^ such that

Limit step:

Kβ < K for all /? < λ .

(By induction, AC^ is a cardinal of N\ for all β < λ. So ACΛ is a cardinal of NX.) Successor step: Suppose we are given Na and Ac α , where Na \= κa is a cardinal. For each A C κa such that A £ Na and Λ is a counterexample to Na having the hull property at Ac α , pick a thick class ΓA witnessing this. Let Γ =

{ΓA I A C Acα Λ A £ Na Λ ΓA exists} ,

The Core Model Iterability Problem

31

where we set Γ = Na if no ΓΛ'S exist, i.e. if Na has the hull property at κa. Set (Each ΓACNa, so 7Vα+ι C Na.) This finishes the construction. It is clear that if a < β < Ω, then Na Π (κa + 1) = Nβ Π («α -f 1), «<* is a cardinal of Nβ , and Λ^ has the hull property at κa. Moreover, («7 | 7 < β) is an initial segment of the cardinals of Nβ. Moreover, Nβ is thick. Set NΩ = Πα<β -^α The assertions of the last paragraph are also obvious for β = β, except that TV/? is not obviously thick in W '. The following claim is the key to showing this. Claim. Let λ < Ω be a limit; then N\ has the hull property at /CA, and therefore Λ/A+I = N\. Proof. Let M be the transitive collapse of N\, and K the image of κ\ under collapse. So /c is a limit cardinal of M, and M has the hull property at all a < K. We want to show that M has the hull property at K. Notice Ω is thick in M. Let Γ be thick in M, and H = transitive collapse of HM(κ U Γ). We are to show P(κ)Γ\M C H. Let T on H and W on M be the iteration trees resulting from a coiteration of H with M determined by Ω + I iteration strategies. (Notice that H and M are Ω + 1-iterable because they are embeddable in W ', and by 3.3 the comparison ends as a stage < Ω.) Let Ih T — 7 + 1 and /Λ U — θ + 1, where 7,0< ί2by 3.3. Since H and M are both universal, HΊ = Me (where these are the final models or the two trees), and i^Ί and %e are both defined. It is enough to see that crit i^ θ > /c, as then P(κ) Π M — P(κ) Π Mg = P(κ) Π HΊ C H . So suppose that crit i^θ = μ < K. Notice (μ+)M < /c. 1 Let E be the first extender used along [0,0]c/; that is, E = E^ where η + I E [0,0]t/ and U-pτeά(η + 1) = 0. So crit £ = μ and /Λ E > K. The M argument of example 4.3 shows that E \ (μ+) witnesses that Me doesn't + M + M have the hull property at (μ ) = (μ ) * . On the other hand, M and hence + M Me has the hull property at all ordinals < (μ ) . M H If crit x£7 > (μ+) = (μ+) , then Hθ = M0 has the hull property at (μ+)M. Thus crit i£7 = crit i^θ = μ. Now let A C μ and A € M. Let Γ = {α | ij[7(α) = t^(α) = α}. By 3.9 and 3.11, Γ is thick (in H, M, and #7 = Me}. So we can find a term r such that where ^ G μ<ω and c G Γ <ω , using the hull property at μ in M. But then

32

§4. The hull and definability properties H

H

Now τ [β, c] Π μ = τ ^[β, c] Π μ = %j(A) Π μ = A. Thus

It follows that the 1st extenders used along [Q,θ]u and [0, j]τ agree up to the inf of the sups of their generators. This is a contradiction, as in the proof of the comparison lemma. This proves the claim. D The claim implies NΩ is thick in W. For by Fόdor's theorem, for all but nonstationary many a G 5, α = κa and for all β < a there is an α-club Cβ <Ξ Nβ C\a+. Fix such an a. Let C = [\β and that crit(ij β ) > α and crit(i^ β ) > α. One can easily show that for any β £ [0,α]τ, Mβ has the hull property at η whenever sup{/Λ E* \ j + I G [0,/?]τ} < 17. (Proof: let Γ be thick in Mβ and let A C 77, A £ M/j. Let 77* < 77 be least such that 77 < ϊo/?(ϊ7*) There is a function / E M , / : [τ/*]<ω x 17* -> {0,1}, and an α G [η]<ω , such that the characteristic function %A of A is given by: for ξ < η, %A(£) = «o/j(/)(α,ί) ^y ^ne ^u^ property in M we can find ξ G Γ <ω such that ίξβ(ζ) = ξ,be [77*]<-, and a term r such that / = τM[b,ξ] \ ([η*]<« x 77*). So ••£(/) - τM^β(b\ξ\ \ ([%β(η*)]<» x [io/ϊί ?*)]. So for 7 < ,, χ Λ ( 7 ) ^^[•o^WίίK0^)- Since z^(6) G [η] <ω by the leastness of 77*, A is in the collapse of HM?(η(JΓ).) Thus Mα has the hull property at α. Since α < crit tJ Λ , Mβ = Wβ has the hull property at α. Since α < crit i^ Ω, Wa has the hull property at α.

The Core Model Iterability Problem

33

Now let Γ be thick _in W^and A C a, A G W. We can find ξ G Γ<ω and 6 E ot<ω such that %a(ξ) = ξ and

for some term τ. Letting 6 be least which works for £, ^α(A), and α = ^α(«), 6 is definable from elements of ran z^α, so 6 = i^a(c) where c £ α < ω . Then A = τ^[c,£] Πα, as desired. Thus W has the hull property at all but nonstationary many inaccessible a< Ω. D Corollary 4.7. Suppose Kc |= Mere are wo Woodin cardinals; then K° has the Ao-hull property at μo- & e. a < ί?. Proof. This is immediate from 2.12, 3.12, and 4.6.

D

One should not expect that 4.6 will hold in full generality for the definability property. For suppose that for μo-a.e. α < ί?, a is measurable in Kc. Let W be the iterate of Kc obtained by using one total-on- Kc order zero measure from each measurable cardinal of Kc once. Then Ω is Ao-thick in W, and W is Ω + l iterable, but W does not have the A 0 -definability property at μo-a.e. α. Nevertheless, one can get a positive result in the case W — A' c , and this result will be important in the construction of "true K" . Lemma 4.8. Suppose Kc |= there are no Woodin cardinals; then for μo-α.e. α < Ω} Kc has the AQ- definability property at α. Proof. Assume the lemma fails, and for μ0-a.e. α pick Γa thick in Kc such that We can also arrange that α < β => Γa D Γβ . Let Vι = Ult(7, μo), and j : V —> V\ be the canonical embedding. Let V2 = Ult(VΊ ) t ;(μo)), and j\ : V\ —»• V^ the canonical embedding. Let Ω\ — c j(Ω) and ί?2 = ji(Ωι). Let ^ = j(K ) and ίf2 = jι(ffι). In V2, we consider the map

K

2

which inverts the collapse. Since Ω £ H *(Ω U Γ^ ), crit π = Ω. Since #2 is satisfied to have the hull property at Ω in 1/2, P(Ω)K<2 C if. Let £"w be the length τr(ί?) extender derived from π. So EV G V j, and measures all sets in P(Ω)K2. Not every Eπ \ z/, z/ X ττ(β), belongs to A'2, as otherwise β is Shelahin K2. <ω

2

Claim. Eπ = Ej Π ([π(Ω)] x P(ί2)^ ), where £; is the extender derived from j. Granted this claim, we can just repeat the proof of the main claim in the proof of Theorem 1.4 to get a contradiction. The point is that V2 has suitable

34

§4. The hull and definability properties

"background certificates" for the relevant fragments of Ej , so working in V^ we get that every E* \ v is "on" the K^ sequence in the right sense of "on" . (1-smallness in no barrier to putting them on, as K% \= there are no Woodins.) <ω

Aside. Why isn't this an outright contradiction? We don't get EjΓ\([j(Ω]] 2 ) as member of VΊ without our false hypotheses.

x

Proof of Claim. Let AC Ω and A G K
for some c G Ω<ω and d G (Γ^l)<ω . It follows that

Here we use that j o j = j± o j, so that j(K\) = K^. This also implies that vl} = Γ v 2 j so that ^v.j= Γv, τhus j(J) 6 (Γv,)<ω

j(Γ

On the other hand Γ%\ C Γ^2, and ;(.4) Π β = A, so

where c € β<ω and j(J) e (-Γ^2)<ω. Moreover, from the definition of π, π(A) =

τκ*(c,j(d)]nπ(Ω).

As π(Ω) < ί?ι, π(A) = j(A) Π π(β), as desired.

The Core Model Iterability Problem

35

§5. The construction of true K c

The model K constructed in §1 depends too heavily on the universe within which it is constructed to serve our purposes. In this section we isolate c v v a certain Skolem hull K of K , and prove that K = K ^ whenever G is generic over V for a poset P £ VΩ . The uniqueness result underlying this fact descends ultimately from Kunen's proof of the uniqueness of L[μ] ([Kul]), and is based on the following lemma. Lemma 5.1. Let M and N be weasels which have the S-hull and Sdefinability properties at all β < a. Let (T,U) be a successful coiteration of M with N, let W be the common last model ofT and U, and letι:M-+W and j : N —> W be the iteration maps. Then i \ a = j \ a — identity. Proof. Suppose not, and let AC = inf(crit(z), crit(j)). Without loss of generality, let AC — crit(i). We claim first that AC = crit^'). For let Δ = {7 < Ω I f(γ) = j(γ) = 7} ,

and recall that Δ is 5-thick in M and TV. Now AC ^ HW(Δ), since otherwise AC is the range of i. On the other hand, TV has the 5-definability property at AC since AC < α. Thus AC £ HN(A), and if AC < crit(j), then AC £ HW(Δ). So /c = crit(j). We can now finish the proof as in 4.5. Let A C AC and A £ M; we claim that A £ TV and i(A) Π z/ = j(A) Π z/, where z/ = mί(i(κ)J(κ)). For by the 5-hull property of M at AC, we can find β £ Δ<ω and a Skolem term r such that A = τM(β) Π AC. (Notice that AC C Δ.) But then i(A) = τw(β) Π i(/c), so A = τw(β) Π AC = J(τN(β)) Π AC. Since crit(j) _= AC, this implies that A = τN(β) Π AC, so that A £ N. Also j(A) = rw(β) Π j ( κ ) , and therefore i(A) Π ι/ = j(A) Π z/ where z/ = inf(f(/c), j ( κ ) ) . A symmetric proof shows that if A C AC and A £ TV, then A £ M and i(A) Π z/ = j(A) Π z/. Let E and F be the first extenders used on the branches M'to W and N-to-W of Ί and U respectively, and let θ = mf(v(E), ι/(F)), so that θ < v. Then iE(A) Π θ = ί(A) Π 0 = j(Λ) Π ff = ι>(^) Π fl for A in M P(κ) . It follows that E \ θ = F \ θ\ on the other hand, since (T,W) is a coiteration, no extender used in Ί is compatible with any extender used in U. This contradiction completes the proof. D Corollary 5.2. Let M be an Ω+l iterable weasel which has the S-definability property at all β < a; then M has the S-hull property at a. Proof. By induction we may suppose M has the 5-hull property at all β < a. Let A C α, let Γ be 5-thick in M, and let TV be the transitive collapse of HM(a U Γ). We must show that A £ N. Now N is Ω + 1 iterable since it embeds in M, and Ω is 5-thick in TV. Also, TV has the 5 hull and definability properties at all β < a. Let (T,U) be a successful coiteration of M with TV, with iteration maps i : M -+ W and j : TV —> W. By 5.1, i\a = j\a =

36

§5. The construction of true K

identity. Then A = 2(A)Πα, so A G W . Since crit(j) > α, A G TV, as desired. D

Definition 5.3. Let M be a set premouse, and let S C Ω. We say that M is S-sound iff there is an Ω + 1 iterable weasel W such that (1) M < W, (2) Ω is S-thick in W, and (3) W has the S- definability property at all β G ORΓ(M. Condition (3) of 5.3 is equivalent to: for every 5-thick Γ, OR Π M C W W H (Γ). This is simply because if β is least such that β £ H (Γ), then w β i H (β(jΓ). Also, by 5.2, condition (3) implies that W has the 5-hull property at all β < OR Π M. Corollary 5.4. Let M and λί be S-sound; then either M. < λί or λί < M. Proof. Let W and R be weasels witnessing the 5-soundness of M and λί respectively. Let i :W —>T and j : R —» T be the iteration maps coming from a coiteration using Ω + I iteration strategies. Then if a = infiΌR^OR^), Lemma 5.1 implies i\a = j \ a = identity. This means that M < λί or λf<M. D Let S C Ω be such that, for some Ω + 1 iterable weasel W, Ω is 5-thick in W. Clearly, there are many 5-sound premice: J™ is an example, and J™ for a = ω^ is a slightly less trivial one. By 5.4 there is a proper premouse 11 such that the 5-sound mice are precisely the proper initial segments of K. We now give an alternative construction of 7£, one which shows that it is embeddable in W. Definition 5.5. Suppose Ω is S-thick in W . Then we put x G DeJ(W, S) <* VΓ(Γ is S-thick in W => x G HW(Γ)) . Clearly, Def(VP, 5) -< W. (More precisely, Def(W, 5) is the universe of an elementary substructure of W. Recall here that the language of W includes a predicate E for its extender sequence. Thus a more careful statement would be that (Def( W, 5), G \ Def(VF, 5), Ew nDef(W, 5)) is an elementary submodel of W.) We now show that, up to isomorphism, Def(H^, S) is independent of W. Lemma 5.6. Let Ω be S-thick in W, an let i : W —* Q be the iteration map coming from an iteration tree on W; then i" Def(W^S} = Proof. Let Δ = {7 < Ω \ 1(7) = 7}, so that Δ is 5-thick in both W and Q. Suppose first x _G Def(W, 5)_. Let Γ be 5-thick in Q; then ΓΠ Δ is 5-thick _in W, so x = τw(β) for some β € (ΓΓ\Δ)<ω and term r. But then i(x) - τQ(β), so i(x) € ff^(Γ). As Γ was arbitrary, i(x) G Def(Q, 5). Suppose next that y G Def(Q, 5). Since Δ is 5-thick in Q, we can find β G Δ<ω so that y = τ^(β) for some term r. Then t/ = i ( x ) , where x = τw (β).

The Core Model Iterability Problem

37 Q

Now let Γ be 5-thick in W. Then ΓΓ\Δ is 5-thick in Q, and so i(x) = τ (ά) <ω w for some term r and α £ (Γ Π A) . But then x = τ (a), and since Γ was arbitrary, we have x £ Όeί(W, 5). D Corollary 5.7. Let P and Q be Ω + I iterable weasels such that Ω is S-thick in each. Then Def(P,S} = De](Q,S). Proof. Once again, we are identifying Def(P, S) with the elementary submodel of P having universe Def(P, 5). To prove 5.7, let f : P -> W and j : Q -» W be given by coiteration; then by 5.6 Def(P, 5) = Όef(W, S) = Def(Q, S). D Definition 5.8. Suppose there is an Ω + 1 iterable weasel W such that Ω is S-thick in W; then K(S) is the common transitive collapse of De}(W, S) for all such weasels W. If there is no Ω + 1 iterable weasel W such that Ω is 5-thick in W, then K(S) is undefined. Lemma 5.9. Suppose K(S) is defined; then for any set premouse M, M is S-sound iffM
is

R

Then Γ\a<e <* S-ihick in W, so Def(Λ,5) C H (f}a<θ Γα), while Γα) Π fl = Def(Λ, 5) Π ff by construction. Thus if we set W = transitive collapse of HR then W is an Ω + 1 iterable weasel with Ω 5-thick in W, and M < W. It is easy to see that W has the 5-definability property at all β £ OR^: if not, then letting σ : W —» R invert the collapse, we have that R fails to have the 5 definability property at σ(β). Since β £ OR^, σ(β) = ττ(/?), and since π(β) £ Def(β, 5), this is a contradiction. Thus W witnesses that M is 5-sound. D As far as we know, it could happen that K(S) is defined (that is, there is an Ω + I iterable weasel W such that Ω is 5-thick in W) and yet K(S) is a set premouse, and hence not universal. We now show that if Kc satisfies "there are no Woodin cardinals", then K(Av), which exists by 2.12 and 3.12, is a universal weasel.

38

§5. The construction of true K

Theorem 5.10. Suppose that Kc \= there are no Woodin cardinals; then K(AQ) is a weasel, and moreover (a+)κ(A°) = α+ for μΌ — a.e. a < Ω, so that K(AQ) is universal. Proof. We first show that K(AQ) is a weasel, or equivalently, that Def(#c, AO) is unbounded in Ω. So suppose otherwise toward a contradiction. It is easy then to see that there are A0-thick classes Γξ, for ξ < Ω, such that ξ < δ =» Γδ C Γξ ,

and letting 6^ = least ordinal ι/ £ (Hκ°(Γξ) - Όef(Kc, AQ)) , we have that (Def(tf c , AO) U Ω) C 60 and ξ < δ => δξ < bδ . By Lemma 4.8, we can fix v such that 0 < v < Ω, v — sup{6ξ | ξ < ί/}, and Kc has the Ao-definability property at v. Let c £ v<ω and d £ /V+i and r a term be such that

v = τκc(c,d\. Fix ξ < v such that c € t " 1 , so that

This is an assertion about bξ,d, and &„+!, all of which belong to Hκc(Γξ). Thus we can find c* £ (6e Π Hκ°(Γξ))<ω such that

But 6e Π Hκc(Γξ) = Def(X c , AO) Π ί2, so c* £ Def(/ίc, A0). This implies rκc[c*,d] £ Hκe(Γ,,+ι), and since Def(/ίc, A 0 ) C 60, and 60 < rκc[c\d\ < fr^+i, this contradicts the definition of δ^+i. Thus Όef(Kc, AQ) is unbounded in Ω. We claim that, in fact, Def(ff c , AQ)Γ\Ω has ^ίo- measure one. For this it is enough to show that if v < Ω is regular, Def(JCc, AQ) is unbounded in z/, and Kc has the AQ- definability property at ι/, then v £ Def(/iΓ c ,Λo). So suppose v is a counterexample to the last sentence. For each η £ (z/ + 1) — Def(/£c, AQ), pick an Ao-thick class Γ^ such that η i Hκ\Γη), and let Γ = f)^. Let 6 be the least ordinal in HK°(Γ) which is strictly greater than v. Fix ξ £ Def(ίΓc, AQ) Π v and d £ Γ <ω such that for some c £ ζ<ω and term r, z/ = τκ°[c,d\. Then, as in the proof that Όef(Kc,Ao) is unbounded, for each η £ Def(/£c, AO) Π ^ we can find Cη £ £<ω Π Def(/f c , AQ) such that 77 < τκ°[cη, d\ < b. As v is regular, we can fix c* so that Cη == c* for arbitrarily large η < v. But then v < τκ°[c* , cf) < 6. Since c* £ Def(/^c, AO) C HK°(Γ), this contradicts the definition of 6.

The Core Model Iterability Problem

39

Finally, we show that for μo-a.e. z/, Def(Kc1Ao) is unbounded in z/+'. This clearly implies that (V+}K(AQ) = ι/+ for μo-a.e. v, and so completes the proof of 5.10. So suppose not; then we can fix v £ Def(ίfc, AO) such that (v+)κ° = ι/+, A'c has the A 0 -hull property at z/, and Def(tf c , A 0 ) Π z/+ is bounded in z/+ . We have then an ^o-thick class Γ1 such that HK°(Γ) is bounded in z/+, say by <5 < ι/+ . By the hull property we have a term τ and d £ Γ <ω such that for some

But now, set V, |P|; moreover, if C £ V[G] and C is club in some regular i/ > |P|, then 3D £ V (D C C and D is club in z/). It follows that for any class Γ C Ω in V[G] V[G] [= Γ is S thick in WΪS3ΔC Γ(V \= Δ is S -thick in W) . This implies that Ω is S-thick in W in V[G\, and that Def(W, v Όeΐ(W,S) . Since W is β + 1 iterable in V[G] by hypothesis, we get that V V V K(S) W exists and K(S) W = K(S) . Π We doubt that one can show that Ω + 1-iterability of W is absolute for "set" forcing in the abstract, although we have no counterexample here. It seems likely that one must appeal to the existence of a definable Ω + 1 iteration strategy for W. This will come from a simplicity restriction on the iteration trees on W, which in turn will come from a smallness condition on W. At the one Woodin cardinal level, we can use the following lemma, whose proof is a slight extension of that of 2.4(a). Lemma 5.12. Let W proper premouse such be V -generic over P, (respectively, (ω, Ω +

be an Ω + l-iterable (respectively, (ω,β+ l)-iterable) that W \= there are no Woodin cardinals, and let G where P £ VΩ Then V[G] (= W is Ω + I iterable l)-iterable).

40

§5. The construction of true K

Proof. We give the proof for Ω + 1-iterability. Using the weak compactness of Ω in V[G], it is enough to show that V[G] satisfies: whenever T is a putative normal, ω-maximal iteration tree on J^f ', for some VF-cardinal α < ί?, and Ih T < Ω, then either T has a last, wellfounded model, or T has a cofinal wellfounded branch. So suppose T is a tree on J™ which is a counterexample to this assertion, and let T,J™ G Vη[G\, where η < Ω is an inaccessible cardinal, and IP G Vη. By the Lόwenheim-Skolem theorem, we have in V a countable transitive M and elementary π : M -» Vη such that J™ ^ P G ran π. Let TΓ^W^P}) = (J<^,P}; then_M thinks that P has a condition forcing the existence of a "bad"_tree on W. Since M_is countable, we can find in V on M-generic filter G on P such that M[G\ \= T is a "bad" tree on W. Notice that since W satisfies "There are no Woodin cardinals" , T is simple; moreover, since π : W —»• J^f is elementary, T is "good" in V. Thus T cannot have a last, illfounded model, and T has a unique cofinal wellfounded branch 6 in V. It is enough for a contradiction to show that 6 G M[G], and for this it is_enough to_show 6 G M[Gj[7/J, where H is M[G\ generic for Col(u>, _max(|T|, |WΊ)M^). But now in M[0\[H] there is a real x which codes (T, W). Also, x* G M [Gf\[H], since M is closed under the sharp function on arbitrary sets because it embeds elementarily in Vη . It is a Σ\ assertion about x that T has a cofinal wellfounded branch, this assertion is true in V, and x^ G M[G\[H],so this assertion is true in M[G][/Γ]. As b is unique, this means that 6 G M[G\[H]. D Putting together 5.11 and 5.12, we get Theorem 5.13. Suppose K(S) is defined, as witnessed by a weasel W such that W |= there are no Woodin cardinals. Let G be V- generic for HP, where P G VΩ. Then V[G] \= Ί<(S) is defined, as witnessed by W* , and K(S)VW = V K(S) . Corollary 5.14. Suppose Kc \= there are no Woodin cardinals, and let G be V -generic over IP G VΩ. Then V[G] \= "K(Ao) is defined, as witnessed by c v κ A (K ) \ moreover (a^} ( ^ = a+ for μ 0 - a.e. a < Ω" . Let us observe in passing that if there is an Ω + 1 iterable weasel W such that Ω is 5-thick in W, for some 5, and W \= there are no Woodin cardinals, then in fact Kc |= there are no Woodin cardinals. [Sketch: If Kc \= there is a Woodin cardinal, then its coherent sequence is of size < Ω. Let (T,£/) be a terminal coiteration of Kc with W, using an iteration strategy on the W side and picking unique cofinal branches on the Kc side. (T,U) cannot be successful, since otherwise the Kc side would have iterated past W ', contrary + w + to (a ) = α for stationary many a. Thus it must be that T has no cofinal wellfounded branch. The existence of generic branches for trees on Kc then implies δ(T), the sup of the lengths of the extenders used in T, is Woodin in an iterate of W, a contradiction.] Thus we can add to the conclusion of 5.14: (Kc)ylG1 \= there are no Woodin cardinals. We are not sure whether

The Core Model Iterability Problem

41

Ω is (j4o)F-thick in (KC)V^G^ however. We now show that, if there is an (ω, Ω+ l)-iterable weasel, then there is at most one weasel of the form K(S}. First, let us note: Lemma 5.15. If there is an (ω,β + Inalterable universal weasel, then every Ω + l-iterable proper premouse is (ω, Ω + l)-iterable. Proof. Let W be universal and Σ an (ω, Ω + l)-iteration strategy for W. Let M be an Ω-\-l iterable premouse. By coiteration, we obtain a normal iteration tree T on W which is a play of round 1 of G*(Wy (ω, Ω + 1)) according to Σ, with last model P, and an elementary π : M -+P. But then P is (w, β + 1)iterable, and so by 2.9, so is Λi. D The next lemma says that, except possibly for its ordinal height, K(S) is independent of S. Lemma 5.16. Suppose there is an (ω,Ω + l)-iterable universal weasel, and that S and T are stationary sets such that K(S) and K(T) exist. Then K(S) < K(T) orK(T) < K(S). In particular, i f K ( S ) and K(T) are weasels, then K(S) = K(T). Proof. Let M be S-sound, as witnessed by W, and let λί be T-sound, as witnessed by R. We assume without loss of generality that OR/"1 < OR . W and R are (ω, Ω + l)-iterable by Lemma 5.15. By Theorem 3.7 (1), for all but non-stationary many α G SUT, (α+)Λ = (a+)w = α+. Now let W* be the (linear) iterate of W obtained by taking an ultrapower by the order zero total measure on α from W, for each a G T -ORM such that W \= a is measurable. Similarly, let R* be obtained from R by taking an ultrapower by the order zero measure on a at each a G S— OR^ such that R |= a is measurable. Then W* and R* still witness the S and T soundness of M and λf, respectively. Moreover, Ω is 5 U T thick in each of W* and#*. Let i : W* —> Q and j : R* —> Q come from coiteration. Let AC = min(crit(z),crit(a;)). It is enough to show that OR^ < AC, for then M < λf as desired, so assume that /c < OR^. Suppose that K = crit(i) < crit(.;). Since Ω is T-thick in Λ* and W*, and AC G Def(Λ*, T), we can find a term r and common fixed points c*ι ak of i and j so that AC = τR*[δί\. But then AC = j ( κ ) = r^[ά] = t(rw*[a]), so AC G ran(i), a contradiction. Similarly, we get crit(i) < crit(j), so crit(j) = crit(i) = AC. A similar argument with the hull property gives the usual contradiction. let A C AC and A G W* . We have a term r and common fixed points ά of i and j such that A = r^* [ά] Π AC, using here that W* has the 5-hull property as AC and Ω is 5-thick in R*. Then i(A) = r^[ά]nί(/c), so r^[α]ΠAC = r Λ *[ά]ΠAc = A, and j(A) = τQ[ά}Πj(κ). Thus i(A) and j(yl) agree below mm(i(κ)J(κ)). This implies that the extenders used first on the branches of the two trees

42

§5. The construction of true K

in our coiteration which produced i and j are compatible with one another. This is a contradiction. D Definition 5.17. Suppose there is an (ω, β+1) iterable universal weasel, and that K(S) exists for some S] then we say that K exists, and define K to be the unique proper premouse M such that \/P, S (P is S-sound Ό> P < M). We do know whether it is consistent with the definitions we have given that K exists, but is only a set premouse or a non-universal weasel. If we assume that Kc \= there are no Woodin cardinals, then K exists by 2.12, 3.6, and 3.12; moreover K is universal by 5.10. We summarize what we have proved about K under this "no Woodin cardinals" assumption: Theorem 5.18. Suppose K° \= there are no Woodin cardinals] then (1) K exists, and is (ω, Ω + 1) iterable, (2) (a+)κ = α+ for μo- Λ.C. a < Ω, and (3) if G is V-generic/\P, for some P G VΩ, then V[G] \= "K exists, is (ω,β+l) iterable, and (a+)κ = α+' for μ 0 - a.e. α < β"; moreover Kv.

The Core Model Iterability Problem

43

§6. An inductive definition of K The definition of K given in 5.17 is Σω(Vn+ι), and therefore much too complicated for some purposes. In this section we shall give an inductive c definition of K whose logical form is as simple as possible. Assuming that K has no Woodin cardinals, we shall show that K Π HC is ΣΊ(L W l (M)) in the codes; Woodin has shown that in general no simpler definition is possible. The following notion is central to our inductive definition of K. Definition 6.1. Lei M be a proper premouse such that M. \= ZF — {Powerset} and j£* is S-sound. We say M is (a^S)-strong iff there is an (ω, f?+1) iterable weasel which witnesses that j£* is S-sound, and whenever W is a weasel which witnesses that j£* is S-sound} and Σ is an (ω,Ω+ 1) iteration strategy for W, then there is a length 0 + 1 iteration tree T on W which is a play by Σ and such that V γ < θ(ι/(E^) > α), and a Q < Wj, and a fully elementary π : M —»• Q such that π \ a — identity. We shall see that it is possible to define "(α, S)-strong" by induction on α. First, let us notice: Lemma 6.2. Let W be an (ω,β+ 1) iterable weasel which witnesses that J^f is S-sound] then W is (α, S) strong. Proof. Let R be a weasel which witnesses J™ is S-sound, and let Σ be an β+1 iteration strategy for R. Let Γ be an β+1 iteration strategy for W, and let (T,l() be the successful coiteration of R with W determined by (Σ,Γ). Let Q be the common last model of T and ZY, and let π : W —» Q be the iteration map given by U. By Lemma 5.1, TT \ a = identity. D Lemma 6.2 admits the following slight improvement. Let W witness that J^f is S-sound, and let Σ be an (ω, Ω + 1) iteration strategy for W. Let Ί be an iteration tree played by Σ such that Vγ < θ(v(Ej ) > α), where θ+l = lhT; then Wj is (α,5) strong. [Proof: Let R be any weasel witnessing J™ is 5sound. Comparing R with W, we get an iteration tree U on R and a map π:W-+O%, where η = lhU-l.By 5.1, crit(ττ) > α. Let σ : Wj -* (Λ^)JT be the copy map. Then σ and IC^πT are as required in 6.1 for R.] This shows that we obtain a definition of (α, 5) strength equivalent to 6.1 if we replace "whenever W is a weasel" by "there is a weasel W" in 6.1. It also shows that there are (α,5) strong weasels other than those described in 6.2. For example, suppose W witnesses that J™ is S-sound, and E is an extender on the W sequence which is total on W and such that cήt(E) < a < v(E). Setting R = Ult(W9E)t we have that R is (a,S) strong, but R does not witness that j£ is S-sound. In view of the fact that K(S) is independent of S, one might expect the same to be true of (α, S)-strength. This is indeed the case.

44

§6. An inductive definition of K

Lemma 6.3. Suppose K(S) and K(T) exist, and a < ORΓ\K(S) Π K(T); then for any M, M is (α,5) strong iffM is (α,T) strong. Proof. Suppose M is (α,S)-strong. Let Ίl witness that j£* is S-sound, and 4 W witness that J* is T-sound. Let Σ be an (ω, Ω + 1) iteration strategy for Wj and Γ an (ω, Ω + 1) iteration strategy for R. From the proof of 5.16, we get iteration trees T and U on W and R which are plays of two rounds of g*(W, (ω, β + 1)) and Q*(R, (ω,Ω+ 1)) according to Σ and Γ respectively, and such that T and U have a common last model Q. The proof of 5.16 also shows that the iteration maps σ : W —»• Q and r : R —>• Q satisfy α < min(crit(σ), crit(r)). Since α < crit(σ), v(E%) > a for all 7+ 1 < lh Ί. 1 Now Σ yields an (ω, β -f l)-iteration strategy JC* for Q, and the strategy of copying via τ and using 17* on the copied tree is an (ω, Ω + 1 ^iteration strategy for R] call it Σ**. According to 6.1, there is an iteration tree V on R having last model P which is a play by Σ1**, and such that 77(7 + 1 < lh V => ι>(j£Jf) > α), and an embedding π : M —> P' for some P' < P such that TT f α = identity. Let r* : P —» £, where £ is the last model of the copied tree rV on Q, be the copy map; thus τ* \ a = T \ a = identity. Let £' < C correspond to P1. Then £' is an initial segment of the last model of T~rV, which is a play by Σ] moreover r* o π maps M into £' and (T* o TT) \ a = identity. This shows that M is (α,T)-strong, as desired. D Definition 6.4. Let M be a proper premouse, and let a < Ω. We say M is a-strong iff for some S, M is (a,S)-strong. We proceed to the inductive definition of "a-strong". The definition is based on a certain iterability property: roughly speaking, M is a-strong just in case M is jointly iterable with any λf which is /?-strong for all β < a. In order to describe this iterability property we must introduce iteration trees whose "base" is not a single model, but rather a family of models. Such systems were called "psuedo-iteration trees" in [FSIT]. Here we shall simply call them iteration trees, and distinguish them from the iteration trees considered so far by means of their bases. Definition 6.5. A simple phalanx is a pair ((Mβ | / ? < α ) , (λ/? | β < α}) such that for all β < a, Mβ is an ω-sound proper premouse, and (1) β < 7 < <* => (MΊ \= "\β is a cardinal" and pω(MΊ] > λ/j), (2) β < 7 < a => Mβ agrees with MΊ below λβ, and (3) β < 7 < α => A/? < λ 7 . We have added the qualifier "simple" in 6.5 because we shall introduce a more general kind of phalanx in §9. Since we shall consider only simple phalanxes in this section, we shall drop the "simple" when referring to them. If B = ((Mβ I β < α), (λβ I β < α}) is a phalanx, then we set lh B = α+1, Mf = Mβ for β < α, and λ(/?,B) = λ^ for β < α.

The Core Model Iterability Problem

45

A phalanx of length 1 is just a premouse. Iteration trees on phalanxes are the obvious generalization of iteration trees on premice; the main point is that we use λ(/?, β) to tell us when to apply an extender to M®, just as we used v(Ej) in the special case of a tree on a premouse. We shall have βTj for β < 7 < Ih B, but this is only a notational convenience, and it would be more natural to think of a tree with Ih B many roots. Since we only need normal, ω-maximal trees, we shall only define these. Definition 6.6. Let B be a phalanx of length a + 1, and θ > a + 1. An (ωmaximal, normal) iteration tree of length θ on B is a system T = (Eβ \ α-fl < β + I < θ) with associated tree order T, models Mβ for β < θ, and D C θ and embeddings iηβ : Mη —»• Mβ defined for ηTβ with (a U D) Π (77, β]τ — 0, such that (1) Mβ = M^ for all β
where M^ is the longest initial segment of MΊ containing only subsets of K measured by Eβ, and k is largest such that K < pk(M^). Also, β + 1 £ D iff MΊ φ Mj, and if β + 1 ^ D then i-γβ+i is the canonical embedding from MΊ into Ultk(MΊ,Eβ), and iηtβ+\ = ij,β+ι ° ^7 for ηTj such that (4)ifa<β<θ and β is a limit, then D Π [0,/?)τ w finite, [Q,β)τ is cofinal in β, and Mβ is the direct limit of the MΊ for 7 £ [0,/?)τ such that 7 > α Usup(Z)). Moreover, iΊβ : MΊ -* Mβ is the direct limit map for all 7 > αUsup(D). In the situation of 6.6, we set θ — Ih T, Mβ = ΛίJ, Eβ — Ej , and so forth. For β < θ, we let rootr(/?) be the largest 7 < Ih B such that jTβ. If β is a phalanx, then (?*(#, θ) is the obvious generalization of the length θ normal iteration game on premice: I and II build an iteration tree on β, with I extending the tree at successor steps and II at limit steps. If at some move a < θ, I produces an illfounded ultrapower or II does not play a cofinal wellfounded branch, then I wins, and otherwise II wins. A winning strategy for II in G*(B, θ) is a θ-iteration strategy for B, and B is θ-iterable just in case there is such a strategy. We wish to state an iterability theorem for phalanxes which are generated from iterates of Kc. Definition 6.7. Let It be a proper premouse and Σ an (ω,Ω + 1) iteration strategy for H. We say that a phalanx B is (Σ, IV) -generated iff for

46

§6. An inductive definition of K

all β < Ih B, there is an almost normal iteration a play according to Σ such that Mβ < P, where T, and such that (i) if β + 1 < Ih B, then X(β, B) Vγ (7 + 1 < Ih T => ι/(JE^) > λ(/?,β)), and (11) Vγ Vα < β(j + K Ih T => v(f%) > λ(α, B)).

tree T on ΊZ which is P is the last model of is a cardinal of U and if β + 1 = Ih B, then

Recall that ifKc has no Woodin cardinals, then there is a unique (ω, β-fl) iteration strategy for Kc (namely, choosing the unique cofinal wellfounded branch). Definition 6.8. Suppose Kc (= "There are no Woodin cardinals"] then a phalanx B is Kc -generated iff B is (Σ,KC) generated, where Σ is the unique (ω, β + 1) iteration strategy for Kc. Our iter ability proof for Kc in §9 will actually show: Theorem 6.9. Suppose Kc \= "There are no Woodin cardinals"] then every Kc-generated phalanx B such that Ih B < Ω is Ω + l-iterable. Proof. Deferred to §9.

D

We shall actually only characterize a strength inductively in the case α is a cardinal of K. In this case we have the following little lemma. Lemma 6.10. Suppose Kc \=" There are no Woodin cardinals" , and let a be a cardinal of K. Suppose a. < ORM , and M is a strong. Then a is a cardinal ofM. Proof. There is a weasel W which witnesses that J™ — j£ is 5-sound, and an elementary π : K —> W with crit(ττ) > α. Since a is a cardinal of K, a, is a cardinal of W. But then α is a cardinal of P, whenever P is an initial segment of a model on an iteration tree T on W such that Ih(E^) > a for all 7 + 1 < Ih T . We have σ : M —> V with crit(σ) > α, for some such P, and this implies that α is a cardinal of M . D We can now prove the main result of this section. Theorem 6.11. Suppose Kc has no Woodin cardinals. Let M be a proper premouse, and let a < ORM be such that a is a cardinal of K and J^ — j£ ] then the following are equivalent: (1) M is a strong, (2) if ((λf,λ4)> (α)) is a phalanx such that M is β strong for all Kcardinals β < a, then ((ΛΓ, Λ4), {«}) is Ω -\- 1 iterable. Proof. We show first (2)=>(1). Let W witness that J^ is 5-sound, and let Σ be an β 4- 1 iteration strategy for W. By 6.2, W is β strong for all β < α, and so our hypothesis (2) gives us an β + 1 iteration strategy Γ for the phalanx ((VFjΛί), (α)). We now compare M with W, using Σ to form an iteration tree T on W and Γ to form an iteration tree U on ((W, M), (a)). The trees T

The Core Model Iterability Problem

47

and U are determined by iterating the least disagreement, starting from M vs. VF, as well as by the rules for iteration trees and the iteration strategies. (See 8.1 of [FSIT] for an example of such a coiteration.) Let Ih U = θ + 1 and Ih Ί = 7 + 1. We claim that rootw(0) = 1. For otherwise τootu(θ) = 0, and the universality of W implies that Mfg — Λ4^, and that i^θ and ijγ exist. Moreover, the rules for U guarantee that crit(^) < a. Since W has the S-hull and 5-definability properties at all β < α, we then get the usual contradiction involving the common fixed points of i^θ and z^. Thus rootw(0) = 1. Since W is universal, i^ θ exists, and maps M into some initial segment of M^ . By the rules for ZY, crit(i^) > α. Thus T and Z

M witness that -M is (<*,S) strong. We now prove (1)^(2). Let us consider first the case α is a successor cardinal of K, say a = (β+)κ = (β+)M where β is a cardinal of K. Let ({.V, Λ4), α) be a phalanx such that λί is /?-strong. We shall show ({Λ/*, M), α) is β+ 1 iterable by embedding it into a A'c-generated phalanx, and then using 6.9. Note that M and λf agree below α, and since M is α- strong, J"^ is Ao-sound. Let W be a weasel which witnesses that j£* is Ao-sound. By Definition 6.1, there are (finite compositions of normal) iteration trees Ί~Q and TI on W, having last models PQ and PI respectively, such that Vγ[(γ + 1 < Ih TO => ι/(JE7^°) > /?) and (7 + 1 < Ih T0 =^ i/ί^1) > α)], and there are fully elementary embeddings TO and TI such that r0 : Λf -» J^°

and

TO f /? = identity ,

TΊ : Λί —> J^1

and

TI f α = identity .

and The proof of 5.10 shows that we may assume our Ao-soundness witness W is chosen so that there is an elementary σ : W —> Kc. Since α is a cardinal of A", we may also assume that α is a cardinal of W. Let σTo and σΊ\ be the copied versions of To and TI on Kc. Since V7 has no Woodin cardinals (because Kc has none), the trees TO and TI are simple. This implies that the copying construction does not break down, and that σTo and σΊ\ are according to the unique (ω, Ω + 1) iteration strategy for Kc. If E is an extender used in σT0, then v(E) > σ(β), and if E is used in σTi, then ι/(£) > σ(α). Let ΨQ'.PQ^ Qo

and

ψi PI ~> Qι

be the copy maps, where Q0 and Qι are the last models of σT0 and σΊ\ respectively. We have ψQ \ β = σ \ β and ψι \ a = σ \ a. Let, for i G {0, 1}, otherwise . We claim that ((7£0,7£ι), (^(»)}) is a /ic-generated phalanx, the trees by which it is generated being σT0 and σΊ\. For this, we must look more closely

48

§6. An inductive definition of K

at the extenders used in T0. We claim that if E is used in T0, then Ih E > a. For if some E such that Ih E < a is used in T0, then there is a B C β such that B G J™ and B £ PQ. Since M,λf, and W agree below α, 5 G -V, so τ0(B) G PO, so r0(5) Γ\ β = 5 G PO, a contradiction. Also, Ih E ^ a for all E1 on the W sequence, since α is a cardinal of W. Thus Ih E > a for all # used in TQ. Since α is a cardinal of W, this means ι/(£") > α for all E used in T0. That implies that v(E) > σ(α) for all E used in σT0. The remaining clauses in the definition of "/^-generated phalanx" hold obviously By 6.9 we have an Ω + I iteration strategy Σ for ({7^0,^1}, (σ(°t))) We can use Σ and a simple copying construction to get an Ω+l iteration strategy Γ for ((Λf, Λ4), (α)). We shall describe this construction now; it involves a small wrinkle on the usual copying procedure, and it shows why it is necessary that M. be α-strong, and not just /J-strong. Our strategy Γ is to insure that if T is the iteration tree on ({Λf, ΛΊ), (a)) representing the current position in G*(((λf, Λί), {α}), β+1), then as we built T we constructed an iteration tree U on ((72,o>7£ι}5 (σ(α))) such that U is a play by Σ and has the same tree order as T, together with embeddings r

U

πΊ : M Ί -* M

Ί

defined for all 7 < Ih T, satisfying: (a) for η < 7 < Ih T, πη \ vη = πΊ \ vη, where

β _ ί/(£,r) "" * Λ(ί?^)

if 17 = 0, if η > 0 and ^ is of type III, otherwise;

moreover, E^ = ^(E^) (b) for all 7 < Ih Ί such that 7 > 2, ττ7 is a (degr(7),X) embedding, where X = (ί^Ί o ^)"(Λ<*)r , for 77 the least ordinal such that i*Ί o ij exists; for 7 G {0, 1}, π7 is fully elementary; (c) for η < 7 < Ih T, if i^ exists, then i^7 exists and πΊ o i^ = i^ o πη. These are just the usual copying conditions, except that the agreementof-embeddings ordinal z/o is /?, rather than α. We have ΛίJ' = λf, M\ = M, M% = 7e0, and M^ = K\ to begin with, and we set r τr0 = -00 ° τ"o and ττι = -01 ° ι Since TΓQ Γ β = ?TI f /? and TTO, ττι are fully elementary, our induction hypotheses (a) - (d) hold. [To see πo and ?TI are fully elementary, notice that M and λί satisfy ZFPowerset, and TO and τ\ are fully elementary according to 6.1. If J^ = Pj, Tt this means Pi (= ZF-Powerset, so deg (^ ) = ω, where ^ = -ΛΛ^, and thus t/>t is fully elementary (i G {0, 1}). On the other hand, if j£* is a proper

The Core Model Iterability Problem

49

initial segment of Pi , then ψi \ J^v is obviously fully elementary. So in any case TΓo and ?TI are fully elementary.] Now suppose we are at a limit step λ in the construction of T and U. Σ chooses a cofinal wellfounded branch 6 of U \ λ, and we let Γ choose 6 as its cofinal wellfounded branch of T f λ. It is cofinal because T and U have the same tree order, and wellfounded because we have an embedding π : Ml —> M^ given by

defined for 7 E b sufficiently large. Setting TΓΛ = π, we can easily check (a) -

(d).

Now suppose we are at step η -f 1 in the construction of T and U. Player I in G * ( ( λ f ) M ) , {α}) has just played E^ and thereby determined T \ η + 2. We must determine U \ η + 2 together with TT^+I. In the case that Tpred(/7 + 1) φ 0, we can simply quote the shift lemma, Lemma 5.2 of [FSIT], and obtain the desired M^+ι and TT^+I. We omit further detail, and go on to the case T-pred(τ7 + 1) = 0. [Unfortunately, the agreement-of-embeddings hypothesis for the copying construction was mis-stated in [FSIT], because squashed ultrapowers were overlooked. We only get πη \ v(E^} — πΊ \ v(f%), for η < 7, in the case E% is type III, rather than πη \ (lh(E%) + 1) = π7 ί (Ih E* + 1) as claimed in [FSIT] (after 5.2, in the definition of πT). This weaker agreement causes no new problems, however.] Let K = crit(E^), so that K < a. and hence K < β. To simply quote the Shift lemma we would need that TΓQ \ (κ+)Mr> = πη \ (κ+)Mr> , and that is more than we know. Still, the proof of the Shift lemma works: set

From 6.6 (2), we get v\ > α, and our agreement hypothesis (a) then gives τrι \ a = πη \ a. Thus ^(/c) = ττι(/c) < ττι(α). Also, ττι(α) = σ(α). (Since Π \ a. = identity and ψ\ \ a = σ \ α, ττι(/?) = <τ(β). But ττι(α) is the li\successor cardinal of ττι(/?), and σ(α) is the / σ(α), these are the same.) Since M% agrees with 7£ι, and hence T^o, through σ(α) = τrι(α), M^ and 7£o have the same subsets of τr^(κ), and the ultrapower defining .M^+i makes sense. We can now define πη+ι : M^^ —> M^+l by:

The shift lemma argument shows that πη+ι is well defined, fully elementary, and has the desired agreement with πη. To see this, recall that v(E} > a for all E used in σTό. This implies that ΨQ \ a = σ \ α, and thus T/ΌJ ψι> ^ πη all agree with σ on α. Now K < /?, and for any A C β in Λf,

50

§6. An inductive definition of K

a

Thus, for example, if / = g on A C [κ]\ \ with A E (E*)a, then π 0 (/) = 1

7Γ0(0) on τr 0 (A), and hence π 0 (/) = τr0(#) on ττ 0 (A) Π [^(/c)]'" - But then π r o(/) = tfo(0) on τr^(A), and ^(A) G (πί?(£ ^))πj?(α). This shows that πη+ι is well defined, and the other conditions on it can also be checked easily. This completes the proof of (1)=>(2) in the case that a is a successor cardinal of K. It is worth noting that we really used that M was α-strong, and not just /?-strong. This guaranteed τ\ \ a = id, and thus ?TI \ a — σ \ a. That in turn gave πη \ a. = ψo \ α, which was crucial. It is not true that if M is /?-strong, where β is a cardinal of K, and j£* = J* for α = (β+)κ , then M is α-strong. The case α is a limit cardinal of K is similar. Let J\f be /?-strong for all A-cardinals β < α, and J^ = J^. Let H^ witness that J^ - J* is Ao-sound, and let σ : W —»• A c . For each A'-cardinal /? < α let Tβ be an iteration tree on W with last model Pβ, and let 7/3 : λf —> J^/ with r/j f /? = id for β < α. Let τa : M -+ J^« with τa \ a = id. For β < α, let σ7^ be the copied tree on Kc , Qβ its last model, ψβ : Pβ —>• Q/^ the copy map, and 7£/3 = J^ ^/

. oτ Ίlβ = Qβ as appropriate. Then ((Hβ \ β <

a f\β a cardinal of A"), (<τ(/?) | β < a /\β a cardinal of A")) is a Arc-generated phalanx, and therefore Ω + 1 iterable. But then we can win the iteration game (/"(((ΛΛΛ'ί), (»}), β -h 1) just as before; letting TT^ : λf —> Tβ be given by πβ — ψβ o Tβ, for β < α, and defining the remaining TT'S inductively, we copy the evolving T on ({Λ/", Λi), {α}) by applying πη(E%) to the model required by the rules for trees on ((Tlβ \ β < α Λ /? a A-cardinal), (<τ(^) | β < a /\β 3, A-cardinal)). Since for β < α, π/? \ β = ψβ \ β = σ \ /?, we have enough agreement to simply quote the shift lemma. Although T and its copy £/ have slightly different tree orders, this causes no problems. This completes the proof of 6.11. D To see that 6.11 gives on inductive definition of K, assuming Kc has no Woodin cardinals, suppose that α is a cardinal of K and we know which premice are α-strong. Then

3β < (a+)K(P = Jf} & 3M(M is α -strong Λ 3β < (a+)M(P = jj4)) . (We get =ϊ from 6.2. We get <= easily from the definition of "α-strong".) We can determine (a+)κ and J^+^K using this equivalence. Using 6.11, we can then determine which premice are (α+)κ-strong. The limit steps in

The Core Model Iterability Problem

51

the inductive definition of "α is a cardinal of K" and "M is α-strong" are trivial modulo 6.11. This definition still involves quantification over VΩ+I. In order to avoid that, we must show that if M is of size α, and 6.11 (2) fails, then there is an J\f of size a and an iteration tree T of size α on ({Λ/*, M), {α}) witnessing the failure of iterability. (We shall actually get a countable T.) This is a reflection result much like lemma 2.4. Definition 6.12. A premouse M is properly small iffM |= "There are no Woodin cardinals Λ there is a largest cardinal". A phalanx B is properly small ijffVα < lh(B) (Ma is properly small). The uniqueness results of §6 of [FSIT] easily yield the following. Lemma 6.13. Let B be a properly small phalanx, and let T be an iteration tree on B] then T is simple. Proof (Sketch). Suppose 6 and c are distinct branches of T with sup(6) = λ = sup(c), 6 and c existing in some generic extension of V. If 6 and c do not drop, then δ(Ί \ λ) < ORM* and δ(Ί \ λ) < ORM° because Mξ and MΊC have a largest cardinal. (This is why we included this condition.) From 6.1 of [FSIT] we get that 6(T \ λ) is Woodin in Mξ if OΈiM> < ORM*, and Woodin in M^ otherwise. This contradicts the proper smallness of the premice in B. If one of 6 and c drops, then we can argue to a contradiction as in the proof of 6.2 of [FSIT]. D We thank Kai Hauser for pointing out that our original version of 6.13 was false. (We had omitted having a largest cardinal from the definition of properly small.) By 6.13, a properly small phalanx can have at most one Ω + 1 iteration strategy, that strategy being to choose the unique cofinal wellfounded branch. Lemma 6.14. Suppose Kc has no Woodin cardinals, and that a. is a cardinal of K. Let M be a properly small premouse of cardinality a such that j£* — j£ but M is not ot-strong. Then there is a properly small premouse λί of cardinality a such that J*f = J* and λί a-strong, and a countable putative iteration tree T on ( ( λ f , M ) , ( a ) ) such that either T has a last, illfounded model, or T has limit length but no cofinal wellfounded branch. Proof. Let W be a weasel which witnesses that J*f is A0-sound. By 6.2, W is α-strong. From the proof of (2)=»(1) in 6.11, we have that ((W,M), (a)) is not Ω + 1 iterable. It follows that there is a putative iteration tree U of length < Ω on ((W, M), (a)) which is bad] i.e., has a last, illfounded model or is of limit length and has no cofinal wellfounded branch. Since Ω is weakly compact, Ih U < Ω. This means that for all sufficiently large successor cardinals μ of VF, we can associate to U a tree Uμ on ((Jμ , M), {«)). Uμ has the same tree order and uses the same extenders as

52

§6. An inductive definition of K

U\ the models on lίμ are initial segments of the models on U. We claim that there is a μ such that Uμ is bad. If U has successor length this is obvious, as the last model of U is the union over μ of the last models of the lίμ. Suppose U has limit length, and bμ is a cofinal wellfounded branch of Uμ, for all μ < Ω such that μ is a successor cardinal of W. Notice that if μ < η, then bη is a cofinal wellfounded branch of Uμ, and thus by 6.13, bη = bμ. Letting 6 be the common value of bμ for all appropriate μ, we then have that 6 is a cofinal wellfounded branch of ZY, a contradiction. Let V = Uμ and P = J™, where μ is a successor cardinal of W large enough that V is a bad tree on ((P,M), (α)). Note that P is α-strong, and ((P, Λί), {α}) is properly small. Let X x Vη, for some η, with V, P, M,a £ X, and X countable. Let π : R = X be the transitive collapse, and ττ(V) = V, etc. Let λ £ X Π Ω be such that V,P,M,a eVχ then V J £ X, and thus Λ |= 1/J exists. Because TT embeds (V*)R into F λ B , we have (V^)R = (V^)B, and so R[x] is correct for Π\ assertions about z, whenever x is an Λ-generic real coding V^. But now R satisfies "V is a bad tree on ((P,M), (α))", and because V is simple by 6.13, R[x] must satisfy the same. Thus V is indeed a bad tree on ((P,M), (a)). Now let X -4 Y -< Vη, where (α + l)uM C Y and \Y\ < a. Let σ : 5 S Y be the transitive collapse, and ψ : R —> S be such that π = σ o ψ. Notice that ψ(M) = M and ψ(ά) — a. Let λί = ψ(P). Using ψ we can copy V as a tree ψV on ({Λ/*,Λ^),α), noting that because V is simple, ψV can never have a wellfounded maximal branch. It follows that *φ V is a bad tree on ((N,M), (α)). Since σ : N -+P and σ f (α + 1) = identity, W is α-strong. This completes the proof of 6.14. D Clearly, if a is a cardinal of K and /? < (α+)x, then there is a properly + <>M small, α-strong M such that Jβ* = J^ and /? < (α ) . So in our inductive definition of K we need only consider properly small mice. Thus 6.11 and 6.14 together yield: Theorem 6.15. Suppose Kc has no Woodin cardinals; then there are formulae V'(^θ)^ι), ^(^Oj ^i) in the language of set theory such that whenever G is V-generic/ P, where P £ Vfa, then V[G] satisfies the following sentences: ω (1) Vz ; y £ ω Vα < ωι [(Lα+ι(ffi) |= [x,y]) & 3δ < a (x codes δ Λ y codes a δ-strong} properly small premouse)]; (2) Vx, y e ωω Vα < ωι [(Lα+ι(M) |= ^[«,y]) <ί=> 36 < a(x codes δ Λ y codes Jδκ)}.

The Core Model Iterability Problem

53

§7. Some applications In this section, we use the theory developed in §1 - §6 to show that various propositions imply the existence of an inner model with a Woodin cardinal. A. Saturated ideals

Shelah has shown, in unpublished work, that Con (ZFC + There is a Woodin cardinal) implies Con (ZFC -f There is an u^-saturated ideal on u>ι). (For earlier results in this direction, see [Ku2], [SVW], [W], and [FMS].) Here we shall prove what is very nearly a converse to Shelah 's result. We shall show that Con (ZFC + There is an α;2-saturated ideal on ωι + There is a measurable cardinal) implies Con (ZFC -f There is a Woodin cardinal). The best lower bound on the consistency strength of the existence of an ^-saturated ideal on ω\ known before our work is due to Mitchell ([M?]). He obtained Con (ZFC + 3κ(o(κ) = ft"1"1")), which of course is as far as the models studied in [M?] could go. Actually, our proof does not require that the given ideal be on α>ι, nor does it require u^-saturation in full. A generic almost-huge embedding will suffice. Theorem 7.1. Lei Ω be measurable, and let G be V-generic/ P for some IP £ VΩ. Suppose that in V[G] there is a transitive class M and an elementary embedding j : V -+ M C V[G] with critical point K such that V α < j(κ)(aM Π V[G] C M) . Then Kc |= There is a Woodin cardinal. Proof. Suppose toward contradiction that K° has no Woodin cardinals. This supposition puts the theory of §1- §6 at our disposal. In particular, by 5.18 we have that Kv = KV^G1. Moreover, by 6.15, the agreement between M and V[G] implies that if P is a properly small premouse of cardinality < j ( κ ) in V[G], and a < j ( κ ) , then (M \= P is α-strong) & (V[G\ |= P is α-strong) . It follows that KM agrees with KVW below j ( κ ) . That is, J*M = for all a < j ( κ ) . Since K — crit(jf), K is a regular cardinal in V, and thus j ( κ ) is a regular cardinal in M. Since P(a)M = P(a)vW for all a < j(/c), j ( κ ) is a cardinal of V[G\. Thus j(/c) is a cardinal of both KM and KVW. We claim K is inaccessible in Kv . For otherwise, we have β < K such that /c = (β+)κV. This means j ( κ ) = (β+)RM = (β+)κV[G] = (β+)κV , a contradiction. So AC is inaccessible in Kv . But then j ( κ ) is inaccessible in

54

§7. Some applications

KM , and j(κ) is a limit cardinal in KVW. Also, since Kv and A'M agree below j ( κ ) , K is inaccessible in KM . Let Ey be the extender over V derived from j. We shall show that for all a < j(κ): Ejn([a]<ω xKv)eKV . This is a contradiction, as then these fragments of Ej witness that K is Shelah in Kv. (That is so because j(Kv) = KM agrees with Kv below j ( κ ) . ) So fix α < j ( κ ) , and take α large enough that (κ+ )κ < a. Let W £ V be a weasel which witnesses that J% is ylo-sound. We may assume a is chosen to be a cardinal of Kv and W. It will be enough to find an extender F on the W sequence such that crit(F) = «, v(F) > α, and for all A £ P(κ)Π Kv ,

Working in V[G], we shall compare W with j(W). Notice that by 5.12, there is in V[G] a (unique) Ω + 1 iteration strategy Σ* for IV. We shall show that there is a (unique) β+1 iteration strategy Γ for the phalanx ((W,j(W)), {α}). Let us assume for now that such a Γ exists, and complete the proof. Let T on W and U on ({VF, j(W)), {α}) be the iteration trees resulting from a (Σ, Γ) coiteration. (Coiteration was defined only for premice, but it makes obvious sense for phalanxes. Here we start out comparing j(W) with W, iterating the least disagreement, but the tree U, which begins on j(W), goes back to W whenever it uses an extender with critical point < α.) Let Ma be the αth model of T and Λfa the αth model of U. In order to save a little notation, let us assume T and U are "padded", so that Ih T = Ih U. Let Ih Ί = Ih U = θ + 1, where θ < Ω. We claim that rootw(0) = 1. For otherwise, rootw(0) = 0; that is, λfθ is above W = Λ/o in U. Now W is universal, and therefore there is no dropping on [0,0] c; or [0,0]τ, so that i% θ and ι£0 are defined; moreover, MΘ = MeLet ^ = { 7 < « l ^ ( 7 ) = i^(7)=7>, so that Δ is thick in W and Me- The construction of U guarantees crit i% θ < Oί. It follows that crit i^ θ is the least 7 such that 7 ^ H^Θ(Δ). From this we get crit IQΘ = crit i^θ. Using the hull property for W at crit i%θ, we proceed to the standard contradiction. So λίβ is above j(W) = Λ/Ί on W. Now j(lV) is universal (in V[G\) since the class of fixed points of j is α-club in Ω for all sufficiently large regular α, so that j(W) computes α+ correctly for stationary many α < Ω. Thus Λ/i = Λί^j and i^e and ιζe and defined. Let so that Γ is thick in W and Λ40 = Λ/0. Now « = cτit(i^θ o j), so K = least r y s . t . r?

The Core Model Iterability Problem

55

It follows that /c = crit ιζe. Similarly, using the hull property for W at K,

for all A C K s.t. A G W. Let 77 + I E [0, θ]τ be such that T-pτed(η + 1) = 0. Now all extenders used in T or U have length > α, and sup of generators > α. So crit i^+i)θ > &. w Also, crit i^ θ > a by construction. So for all A £ P(κ) ,

Let F be the trivial completion of E^ \ a. Then F is on the sequence o It follows (using coherence if η > 0) that F is on the sequence of W = Moreover, for A C K s.t. AζW,

So F is as desired. It remains to show that the phalanx ( ( W , j ( W ) ) , (a)) is Ω -f 1 iterable in F[G]. We claim that the strategy of choosing the unique cofinal wellfounded branch is winning in the length Ω -f 1 iteration game. If not, then as in 6.14 there are properly small Ίl < W and S < j(W) such that α G ORπ Π OR5, and a putative iteration tree on ((7£, <S), (a)) which is bad; that is, which has a last, illfounded model, or is of limit length but has no cofinal wellfounded branch. Since Ω is weakly compact, our bad tree has length < β, so that its sharp exists. Using this for absoluteness purposes, as in the proof of 6.14, we can find in V[G] σ:P

-+U

(where ft < W) ,

r:Q

-> S

(where S < j(W)) ,

such that V and Q are of cardinality < a and σ \ a = r \ a, = identity , together with a countable bad tree on ({^,(5), (α)). Now V is α-strong in F[G], as witnessed by σ. Also, Q is α-strong in M, as witnessed by r; note that T e M as M<jW C M in V[G\. Since M and V[G\ have the same subsets of α, 6.11 and 6.14 imply that Q is α-strong in V[G\. But then the (l)->(2) direction of 6.11 implies that ({P, Q), (α)) is Ω + 1 iterable, a contradiction. D

Corollary 7.2. Xe< Ω be measurable, and suppose there is a pre-saturated ideal on ω\\ then Kc \= There is a Woodin cardinal. We conjecture that the measurable cardinal is not needed in the hypotheses of 7.1 and 7.2.

56

§7. Some applications

B. Generic absoluteness One of the most important consequences of the existence of large cardinals is that the truth values of sufficiently simple statements about the reals cannot be changed by forcing. For example, if there are arbitrarily large Woodin cardinals, then L(R)V = ^(M)^^ for all G set-generic over V. (This result is due to Hugh Woodin.) We shall show that this generic absoluteness implies that there are inner models with Woodin cardinals. Hugh Woodin pointed out this application of §1 - §6. The key is the following lemma. Lemma 7.3. (Woodin) Let Ω be measurable, and suppose Kc has no Woodin cardinals. Then there is a sentence φ in the language of set theory, and a partial order P £ Vfa, such that whenever G is V-generic over P

Proof. Here (Lωι(R))vW = Lωv[G](Rv^G]). Using the formula ψ described in 6.15 (2) which defines (jj^ \ δ < ωi) in all generic extensions of V by posets IP G VΩ , we can construct a sentence φ such that (provably in ZFC -f "ί? is measurable" + "Kc \= There are no Woodin cardinals") we have Lωι(R)\= φ iff

ωι is a successor cardinal of K .

Our hypotheses guarantee (a ) — α+ for some a. If (Lωι(Sk))v ¥ φ, then take P = Col(u>,α); letting G be ^-generic / P, we have (Lωι(M))v[σl \= φ by 5.18 (3). On the other hand, if (Lωι(R))v [= φ, then take P = Col(ω, < a) where α < Ω is inaccessible; letting G be F-generic/P, we have * φ. D + κ

Theorem 7.4. (Woodin) Suppose that Ω is measurable, and that whenever v v c G is V-generic/^ for some P G VΩ, (Lωι(R)) = (Lωι(R)) W . Then K \= There is a Woodin cardinal. It is well known that weak homogeneity can be used to obtain generic absoluteness. We can therefore use 7.4, together with standard arguments, to show Theorem 7.5. // every set of reals definable over L ωι (M) is weakly homogec neous, then letting K be the model constructed in §1 below Ω, where Ω is the least measurable cardinal, we have Kc \= There is a Woodin cardinal. Proof. If any set is weakly homogeneous, then there is a measurable cardinal. Let Ω be the least measurable cardinal. For any weakly homogeneous tree T, let T* be the tree for the complement coming from the Martin-Solovay construction. (The notation assumes the homogeneity measures for T are given somehow.) So

The Core Model Iterability Problem

57

is true in V[G] whenever G is generic for P with card (P) < additivity of homogeneity measures for T, hence whenever G is generic for P £ VΩ - Let Sn

=

universal Σn(Lωι(lΆ)) set of reals

Pn

=

universal 77n(Lωι(lR)) set of reals

for 1 < n < ω. Pick weakly homogeneous trees Un such that Pn = p[Un] and let Tn+ι be the canonical weakly homogeneous tree which projects to &p[Un] in all V[G] p[Γn+ι] = ΈPp[Un] in all V[G] . (Thusin V Γ ,p[Γ n+ ι] = 5n+ι.) Claim. If G is P-generic, where P 6 Vh, then for all n > 2

Prw/. Fix F[G]. Since p[Un] Πp[Tn] = 0 in 7, this remains true in V[G] by absoluteness of wellfoundedness. On the other hand, if x £ R F t G J and x £ (p[Un] Up[Tn]), then z G p[l/*] Πp[Γ^] because £/"* and Γ* project absolutely to the complements of the projections of {7n, Tn. But then p[U*] Πp[T^] ^έ 0 in V by absoluteness of wellfoundedness. This is a contradiction as p[Un] = inF. D It follows that in all V[G], G generic for P G VΩ, p[Un+ι] — universal 77^(^4) set of reals, where A = p[Uι] . But now the fact that A is the universal Πι(Lωι(R)) set of reals is a TZ^o fact about A. So in all such V[G] p[Un+ι] = universal 77n+ι(Lωι(M)) set of reals. Thus for any sentence φ of the language of set theory

By Theorem 7.4, /f c [= There is a Woodin cardinal, where Kc is constructed below Ω. D Woodin has shown (unpublished) that if there is a strongly compact cardinal, then all sets of reals in L(IR) are weakly homogeneous. So we have at once: Theorem 7.6. Suppose there is a strongly compact cardinal, and let Kc be the model o f § l constructed below f2, where Ω is the least measurable cardinal. Then Kc \= There is a Woodin cardinal.

58

§7. Some applications

We shall give a more direct proof of Theorem 7.6 in §8, a proof which does not rely on Woodin's work deriving weak homogeneity from strong compactness. We conclude this section on generic absoluteness by re-proving a slightly weaker version of the following theorem due to Woodin: Con (ZFC + ^-determinacy) =» Con (ZFC + There is a Woodin cardinal). Because the theory of §1 - §6 relies on the measurable cardinal cardinal Ω, we do not see how to use it to prove Woodin's theorem in full, although we believe that should be possible. We can, however, prove the theorem with its hypothesis strengthened to: Con (ZFC + Δ^-άeteτmmacy+Vx G M (x* exists)). Modulo the theory of §1 - §6, our proof is simpler than Woodin's. Our proof relies on the observation that the theory of §1 - §6 uses somewhat less than a measurable cardinal. Namely, suppose A is a set of ordinals and A^ exists. Let CQ be an indiscernible of L[A], let j : L[A] —>• L[A] have critical point CQ, and let U be the L[A]-ultrafilter on CQ given by: X G U & CQ G j(X). Working in L[A], we can construct (KC)L^ below CQ just as we constructed Kc below Ω in §1. Let us assume that L[A] satisfies: There is no proper class inner model with a Woodin cardinal. We can then conduct our proof of iterability within L[A] (using 2.4 (b) rather than 2.4 (a)), and we have that indeed (KC)LM exists and (by 2.10) is (ω,θ) iterable for all θ. Further, the proof of 1.4 shows that for U a.e. α < CQ, L[A] \= (a+)κ° = a+. (The main point here is that we don't really need U G L[A] to carry out the proof; it is enough that if Ej is the (CQ, J(CQ)) extender over L[A] derived from j, and A G L[A] and |^4|L[Λ1 < CQ, then Ej Π ([j(cQ)]<ω x A) G L[A]. That these fragments of Ej are in L[A] is well known.) This implies that L[A] [= "CQ is Ao-thick in Kc". We can therefore carry out the arguments of §3 - §6 within L[A], and we get that KL^ exists, is absolute for forcing over L[A] with posets P G V^o , and inductively definable over L[A] as in §6. (The only serious use of the measurable cardinal Ω in these sections occurs in the proof of 4.8. Once again, it is clear from that proof that we only need the fragments Ej Π ([j(co)]<ω x .4), for \A\L[A^ < c 0 , to be in L[A].) We also have that for U a.e. α < c 0 , L[A] \= (α+)K = α+. Theorem 7.7. (Woodin) //Var G ωω (x^ exists) and all A\ games are determined, then there is a proper class inner model with a Woodin cardinal. Proof. According to a theorem of Kechris and Solovay (cf. [KS]), A\ determinacy implies that there is a real x such that for all reals y >τ x, L[y] f= "All ordinal-definable games are determined". Fix such a real x, and let CQ be the least indiscernible of L[x]. We may suppose that L[x] \= "There is no proper class inner model with a Woodin cardinal". As we have observed, this means that KL^ exists and is absolute for size < CQ forcing over L[a?], and that for W-a.e. α < c0, L[x] \= (&+}κ = α"1", where U is the L[x]-ultrafilter on c0 given by x f l . Let α < c0 be such that L[x] \= (a+)κ = α + , and let

The Core Model Iterability Problem

59

y = ( x , z ) where z is (a real) generic over L[x] for the poset Col(u;, a) collapsing OL to be countable. Then in L[y], K exists and is inductively definable as in §6, and ω\ is a successor cardinal of K. Moreover, OD determinacy holds in L[y]. Let us work in L[y]. Now OD determinacy implies that every OD set A C ω\ either contains or is disjoint from a club, and therefore that ω\ is measurable in HOD. On the other hand, K C HOD, so since u% = (a+)κ, ω\ = (α+)HOD. But HOD \= AC, so HOD |= all measurable cardinals are inaccessible. This contradiction completes the proof. D C. Unique branches The Unique Branches Hypothesis, or UBH, is the assertion that if T is an iteration tree on V, then T has at most one cofinal wellfounded branch. Martin and the author showed that the negation of UBH has some logical strength, in that it implies the existence of an inner model with a Woodin cardinal and a measurable above. (Cf. [IT], §5.) Woodin then showed, in unpublished work, that if there is a nontrivial elementary j : V\+ι —> VA+I, for some λ, then UBH fails. The gap between these two bounds on the consistency strength of -UBH is, of course, enormous. Here we shall improve the lower bound to two Woodin cardinals. (However, we must add "There is a measurable cardinal" to -UBH because the basic theory demands it.) We conjecture that -UBH is equiconsistent with the existence of two Woodin cardinals. Theorem 7.8. Let Ω be measurable, and suppose there is a normal iteration tree T on V such that T £ VΩ and T has distinct cofinal wellfounded branches. Then there is a proper class inner model satisfying "There are two Woodin cardinals". Proof. Assume toward contradiction than there is no such model. We shall need a slight generalization of the K° construction in §1. Let X C be any transitive set, X 6 Vfo where Ω is measurable. We can form K (X) by relativizing the construction of §1. So Λ/Ό = X, and all hulls used in forming Gu(tft(X)) = Mξ(X) contain X U {X}, so that X € λΓξ(X) for all ξ. We C require that all extenders added to the K (X) sequence have critical point > ORΠ X. We require that the levels λίξ(X) of the construction be "1-small above X", that is, if K is a critical point of an extender from the Λfξ(X) ίW sequence, then for no δ > OR Π X do we have J^ |= δ is Woodin. By C K (X) we mean the limit as ξ —*• Ω of the Mξ(X). Let us call a structure with the appropriate first order properties of the Mξ(X) an X-premouse. C If there is no δ > (ORΠX) such that K (X] |= δ is Woodin, then as in §2 C we get that K (X) is (ω,Ω+ΐ) iterable "above X", that is, via extenders on its sequence and the images thereof. (All such extenders have critical point > OR Π X, so none of the embeddings move X.) Of course, any two Ω + I iterable-above-X X-premice have a successful coiteration. As in 1.4, we also

60

§7. Some applications

have (a+)κc(χ) = α+ for μ 0 a.e. α < β, where μ 0 is a normal measure on Ω. The rest of §3 - §6 adapts in an obvious way. (We shall not need §6.) Now let Ί be our iteration tree on V having distinct cofinal wellfounded branches 6 and c. We have T G VΩ, where Ω is measurable. Let 6 = δ(T) = sup{//ι E%\a+l δ and / G Mj Π M* , then ΛΛf |= "6 is Woodin with respect to /". (Equivalently, Mrc satisfies this.) Notice that %b(Ω) = ϊξc(Ω) = β. Working in Mξ and Λίf, let us form the models

Notice here that V* = V 5 are 1-small above X.

R

=

Kc(Vδ}Mΐ

5

=

Kc(Vδ)M

; setting X = V

, w e have that both R a n d

Claim 1. Let α > 6 be a successor cardinal of R such that J* ¥ 3κ(δ < K Λ K is Woodin); then J^ is Ω + 1 iterable above X. Similarly for 5. Proof. Our "proper smallness above X" requirement on a guarantees, as in §6, that no iteration tree on J^ which is above X can have distinct cofinal wellfounded branches. Our standard reflection argument (cf. 2.4 (a)) shows that it is enough to prove the following. Subclaim. Let π : P —»• J^ be elementary, with V countable, and let ττ(X) = X. Let U be a countable putative iteration tree on P\ then either W has a last, wellfounded model, or U has a cofinal wellfounded branch. Proof. Since 7> is countable, P G ΛΊf , and of course J* G Λ J* such that σ G Λf f . But also W G Λίf . We can therefore carry out the iterability proof of Theorem 2.5 within M^ using the background extenders given by the construction of R = Kc(Vs)Mb . D Claim 2. P(δ) Γ\R= P(δ) Π 5. Proof. Let α = (δ+)R and /? = (5+ )5. By Claim 1, both J* and J^5 are ί?+l iterable above X. It follows that they have a successful coiteration above X, and since neither can move without dropping, we get j£ < jj or Jjj < j£. Suppose without loss of generality that j£ < Jjj . It follows that P(6) Π R C 5, so P(δ) Π R C Λίf Π Λίf, so that R\=6 is Woodin. We are done if R has another Woodin cardinal above 6, so we assume otherwise. But then, whenever 7 is a successor cardinal of R above δ, then J* P 3κ(δ < K Λ K is Woodin). Claim 1 then shows jf is Ω + I iterable, and since this is true for all 7, R itself is Ω + 1 iterable above X.

The Core Model Iterability Problem

61

Theorem 1.4, applied within M^ to R = KC(X), implies that M^ satisfies "for %b(μo) a.e. α < β, α+ - (<*+)*"(*>. But for μ 0 a.e. a < β, *oδ(α) = α an(^ z oft( a + ) —α+ Thus it is true in V that for μo a.e. α < β, Since R and J^f are β + 1 iterable above X, they have a successful coiteration above X. Since R computes successor cardinals correctly μo a.e., and jj cannot move without dropping, jj < R. This completes the proof of the claim. D Inspecting the proof of claim 2, we have: Claim 3. δ is Woodin in both R and S. Both R and S are β + 1 iterable. Finally, (a+)R = (a+)s = α+ for μo a.e. α. Let us emphasize that R and 5 are (ω} Ω + 1) iterable in V, not just in the models M\ or Λί£". We wish to compare R with 5, but first we must pass to models for which the comparison will have a large set of fixed points. Working in Λ4jf, let R* come from R by taking ultrapowers by the order zero total measure at each measurable cardinal of R. Thus R* is M% definable (from δ and β), R* is a linear iterate of β, and if δ < K < Ω and K, is strongly inaccessible in Λ4j", then K is not the critical point of a total extender on the R* sequence. Let 5* be obtained from 5, working inside MJ, in a similar fashion. Now let (W, V) be a successful coiteration of R* with 5*, according to their unique β + 1 iteration strategies. Since R* and S* compute α+ correctly for a.e. a < β, U and V have a common last model Q. Let j : R* -+ Q and fc : 5* —»• Q be the iteration maps. Let

be the set of common fixed points of j o ιζb and k o ί^e. We have then that + + + μo(Z) = 1, and for μo a.e. α, Z is cofinal in α and a = (a )Q. Now let c*o G 6 — c, and define /?n αn+ι

=

least 7 G (c-α n ),

= least 7 G (δ-/?„).

Let us assume that αo is chosen large enough that δ G ran i^ 6 Π ran ^c . It follows, of course, that R* G ran i^ 6 and 5* G ran ij^ c. Set /c = crit i£ι>6

and

τ

# = transitive collapse of HullQ(v"b (J Z \J {δ}). The next claim comes directly from the proof of the uniqueness theorem of §2 of [IT] (see also 6.1 of [FSIT]). Claim 4. EullQ(V^ (J Z U {δ}) Π vf4* =

62

§7. Some applications

Proof. (Sketch) For all i > 1, on and $ are successor ordinals, and

and

Here stιM(E) is the strength of E in the model M. Now suppose t is a ΛΊb

sequence of parameters from Vκ

T

U Z U {<5}, and

Let /c, = crit(£'Jt_1) and ι/t = crit(£?Ji_1). Since sup{/c λ \ i £ ω} = sup{ί/, | i G ω} = 6, we can let i be least such that for some x £ VKt ΛΊb

b

, Q ^= φ[x,i\.

T

Fix such an x in VKl . We claim i = 1; since «ι = /c this will complete the proof of claim 4. Suppose then i = e + 1. b U Z U {<$}, we have fe((x,ί)) = (x,t), so Since fc is the identity on Vδ 5* (= y>[x,*]. Let /? = Γ-pred (/?e), and let

Now z/e = crit iJ j C , and κe+ι < i^c(ve). Since i^c is elementary and x G Vίί^, we have x 7 G VΪf* such that 5 |= φ[x'tΐ\. But then 5* |= φ[x' >t], and hence Q \= φ[xf,t\. We can now go apply the argument of the last paragraph to i J b , where T !M α = T-pred(αe), using R* and j instead of 5* and k. We get x" G V Γ Λe b such that Q |=
b

-premice which are Ω + 1 iterable above Vκ

ful coiteration above Vκ

b

b

, so they have a success-

. Since J*f \= K is not Woodin, there is a subset

The Core Model Iterability Problem

63

of K which is in J*f but not H. This means J*f must iterate past H. On the other hand, H computes α"1" correctly for μoa e. α < ί2, so J*f cannot iterate past H. D Now K < δ, δ is Woodin in R, and M = KC(VK)R. A standard argument shows that for some v such that /c < z/ < <$, M [= ι/ is Woodin. (See the proof of 11.3 of [FSIT]. Thus M \= There are two Woodin cardinals, and the proof of 7.8 is complete. D D. Σg correctness and the size of UΊ We say that a transitive model M is Σ$ correct iff whenever x £ M Γ\ωω and P is a nonempty Π^x) set of reals, then P Π M -φ. 0. The proof of the following theorem was inspired by, and relies quite heavily upon, an idea due to G. Hjorth. Theorem 7.9. Suppose Kc [= "There are no Woodin cardinals", and suppose there is a measurable cardinal μ < Ω; then K° (or equivalently, K) is Σ% correct. The remarkable insight that there are theorems along the lines of 7.9, and the proof of the first of them, are due to Jensen (cf. [D]). Jensen's work was later extended by Mitchell ([M2]), and by Steel and Welch ([SW]). The smallness hypotheses on K in these works are, respectively: no inner model with a measurable cardinal, no inner model with a cardinal /c such that o(κ) = K"1"1", and no inner model with a strong cardinal. The smallness hypothesis on K in Theorem 7.9 is necessary. For if Kc \= "There is a Woodin cardinal", then Kc is not Σ% correct. [Let P = {x £ ω ω I x codes a countable, Π\ -iterable premouse which is not 1-small}. The existence of the measurable cardinal Ω gives P / 0. On the other hand C PΓ\K = 0, since if M is coded by a real in P, then J*° < M for a = ωf °. c (Cf. [PW], 3.1.)] However, if we liberalize our definition of K so as to allow levels which are not 1-small (but still retain some weaker smallness condition, e.g. tameness, which suffices to develop the basic theory of A'c), then we can simply drop the hypothesis that Kc satisfies "There are no Woodin cardinals" from 7.9. This is because if there are arbitrarily large α < ω^c such that j£° is not 1-small, then Kc is Σ\ correct. (In fact, if x is a real coding a countable, ω\ + 1-iterable, non-1-small mouse M such that y £ M, and P is nonempty and Π\(y), then 3z £ P(z <τ x). This result is due to Woodin; cf. [PW], §4.) Where we have assumed in 7.9 that there are two measurable cardinals, [D] requires only that every real has a sharp, and [M2] and [SW] require only the sharps of certain reals. We believe that it should be possible to eliminate the hypothesis that there is a measurable cardinal < Ω from 7.9. Of course, the need for Ω itself is also problematic, here and elsewhere.

64

§7. Some applications

Proof of 7.9. Our proof descends from a proof of Jensen's Σ\ correctness theorem which is much simpler than Jensen's original proof. That simpler proof is due to Magidor. c

Suppose that K |= "There are no Woodin cardinals", and let μ < Ω be measurable. For a > 1, we let ua be the αth uniform indiscernible relative to parameters in Vμ, that is ua

=

αth ordinal β such that Vx G Vμ (β is an indiscernible of L[x]).

Thus iίι = μ. Magidor's argument is based on the following lemma. Lemma 7.10. (Magidor) Suppose u% = 1/2 >' then there is a tree T% £ K such that p[T2\ is the universal Π\ set of reals, and thus K is Σ\ correct. Proof. (Sketch) We first show that for all α, u% = ua. The proof is by induction on α; the cases a = I and α is a limit are trivial. Let α = β + I. Let 71(7, x) = least indiscernible of L[x] which is > 7 . We have Uβ+ι = sup{n(tι/j, x) I x G Vμ] , and

ί/2

=

sup{n(tiι, x) I x G Vμ}

=

sup{n(uι,a?) | x G V^} ,

since 1/2 = t/^ But then for any x £ Vμί we can find t; £ V*f so that n(ιtι,x) < n(uι,y), and thus 71(11/3, x) < n(uβ,y) by the uniform indiscernibility of the it^'s. It follows that ti/j+i = sup{n(tι/3, x) I x G ΐ^f} , as desired. It is well known that for any ordinal 77, these are an x G V^ and a term r and uniform indiscernibles w α o < < uan < η such that η = τL^(uao - - -uan). (This result is due to Solovay; the proof is an easy induction on η.) Since ua = u# for all α, we can take x G K in the above. By Γ2, we mean the Mart in-Solovay tree for U\ constructed as follows. Let L = (Jί ^M Ix ^ ^μ} ^t S on ω x ω x μ be the Shoenfield tree for a Buniversal Σ\ set. For w, υ G ω<ω such that dom(iί) = dom(t ), let S(u,v) = {w I (u, υ, w) G 5}. We define an ultrafilter on P(S(UfV)) Π L as follows. For X C μ and n < ω, let [X]n = {(α0 «n-ι) I «o < «ι < < <*n-ι A Vi < n(θίi £ X)}. Letting n = dom(u) = dom(υ), there is a unique permutation (f'o,...,i n -ι) of n such that 5(u>v) = {(αίo - - α» n-1 } | {α0 α n _ι} G [μ]n}. For A C 5(W)t;) with A G L, we put

The Core Model Iterability Problem μ(n,v)(Λ) = 1 «•

3C(C is club in μ Λ V{α0 (K

65

α n _ι) € [C]n

•<*._,}

Since Vz £ Vμ(x* exists), μ(U)υ) is an ultrafilter on P ( S ( U y V ) ) Γ \ L . If tί C r and v C Sj then μ(u,v) is compatible with μ(r,*)> so we nave a natural embedding * (u,t,),(r,5) : Ult(L,/l( t t > t ,)) -> Ult(L,μ ( Γ ) S )) .

The ultrapowers here are formed using functions in L. The result of Solovay mentioned above yields {«<0+ι, . . . , tι,n _ 1 + ι) = [identity]^ v) , where (io, . . . , i n -ι) is the permutation of n = dom(iί) used to define μ(u,v)ι and the uz 's are the uniform indiscernibles. By convention, μ(0,0) is principal and Ult(L,μ(0 } 0)) = L. We then have: μ(Utυ)(A) = 1

iff

(ns0 + ι,...,tii n - 1 +ι)

£π(V>j),(u,v)(A).

Except for the fact that they are not total on V, the measures μ(u,υ) witness the weak homogeneity of S. In particular x £ p[S\ iff By Gωω (the direct limit of the Ult(L, μ(x\nty\n)) under the *(s\niy\n)t(x\n+i,y\n+i) is wellfounded). The tree TI builds a real x on one coordinate, and proves x £ p[S] on the other by showing continuously that all associated direct limits are illfounded. More precisely, let (r, | i G ω) enumerate ω<ω so that ΓQ = 0 and =n (ti, {α0, . . . , α n _ι» G T2 iff α 0 = μ Λ Vi < j < n (r< C r; => τr Since if (= Var £ V^(x* exists), we can form T£ inside /C. In order to see that T£ — TV, we must see that for any iί, υ £ ω <ω with dom(iί) = dom v

and if iί C r and v C 5 and dom(r) = dom s, ^Kt;),^,,) ί / = Titi.t/Mr.O ί /

for μ* = / 7Γ(0,0) ) (r ) 5)(A<). Now clearly, μfu^ = μ(u,υ) Π X for all (ti, v). We are sucn done, then, if we show that for any (u, v) and / : S(u,v) —* μ that / £ L, [/]μ(u υ) has a representative in A'. We may assume that for some x £ Vμ and term r, f ( w ) = TL^[W] for all w £ S(M>t,). Let ιy* = [identity]^ ^ < ίίί7n+ι , where n = dom(υ), so by the result of Solovay mentioned above, applied inside K, we can find a y £ V* and a term σ such that σL[yl[υ;*] = rLW[w;*]. It follows that for μ(u>v) a.e. tu, σLW[u;] = TL^[W]. Letting flf(ιy) = σLW[w] for all ιy £ 5(tt>f;), we have 5r £ A and [0]μ(U)V) = [/]μ (U(V) .

66

§7. Some applications

This completes the proof of 7.10

D

Now let

T — {M G Vμ I M is Ω + I iterable and properly small} . Recall from 6.12 that a premouse is properly small just in case it satisfies "There are no Woodin cardinals" and "There is a largest cardinal". There can be at most one cofinal wellfounded branch in an iteration tree based on a properly small premouse, so any M £ F has a unique Ω + 1 iteration Bstrategy ΣM For M, λί G T , let P and Q be the last models of Ί and U, where (T,Z/) is the unique successful (Σju,Σj\r) coiteration of M with λί. We define M <* λί iff P < Q. Thus <* is just the usual mouse order, restricted to f '. The Dodd-Jensen lemma implies that <* is a prewellorder. Set δ - order type of (T , <*) . Also, for M £ F, let |Λί|<* be the rank of M in the prewellorder <*. The following lemma is part of the folklore. Lemma 7.11. δ < u% . Proof. It is easy to see that if U is any normal ultrafilter on μ, then (θί+}κ = α+ for U a.e. a < μ. B(We prove this as part of the proof of lemma 8.15 in the next section.) It follows that K Γ\f is <*-cofinal in T . For let M £ F, and let (T,U) be the successful coiteration of M with jff determined by Ω+ I iteration strategies for the two mice. Since J^ computes successor cardinals correctly almost everywhere, max(//ι T, Ih U) < μ, and the last model P of Ί is an initial segment of the last model of U. Let a < μ be a successor cardinal of K and such that Ih E^ < a for all ξ + 1 < Ih U] then we can regard U as tree on J*, so that (T,U) demonstrates that M <* J* . It suffices then to show that if Λί £ KΓ\f, then |ΛΊ|<* < uf. Fix Λί, and let G be F-generic for Col(u>, < μ), and let z0 be a real coding M in V[G]. L χ L M Choose xQ to be generic over L[Λί], so that (μ+) l °l = (μ+) ί l < u$ . For x and y reals in V[G], let Λ(x, y) iff

(x and y code properly small premiceΛία; and Λί y , and there is a successful coiteration (T,U) of Λί^with My such that T and U are simple, and the last model of Ί is a proper initial segment of that of U ),

and let

It is easy to check that 5 is a Σ\(x§) relation on the reals in V[G\. Claim. S is wellfounded.

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67

Proof. Suppose not, and let S = SG, where 5 represents the natural definition of S from M over V[G\. Working in V, we can construct a countable, transitive P and an elementary π : P -» VΩ with π(M) = M and π(μ) = μ for some M, μ. Let G in V be P-generic for Col(α;,< μ). Then Sg is illfounded, and this implies that the mouse order below M is illfounded. Since π \ M : M —»• M, and Λ4 G J7, this is a contradiction. Since 5 is wellfounded and Σ\(XQ), its rank is < (μ+)Llχ°1 by the KunenMartin theorem. Clearly, (|Λ4|<*) V ' is less than or equal to the rank of 5. This proves 7.11. G In view of 7.10 and 7.11, we would like to show that 8 — u^. The key idea for doing this is due to Greg Hjorth. Lemma 7.13. (Hjorth) Suppose δ < u^; then there is a set M G Vμ such that F Γ ( L [ M ] is <*-cofinal in T. Proof. Since δ < 1/2, we have an x G Vμ and a term r such that δ = τL\x*[x} μ ] . Now let x G Vη where η < μ, and let (Z, e) < (Vfo, e) be such that card(Z) < Pi Vη Q Z> and A* £ Z. Let M be the transitive collapse of Z, π : M —»• VΩ the collapse map, and ττ(μ) = μ. Let 17 be such that M [= ί7 is a normal ultrafilter on μ, let TV be the μth (linear) iterate of M by U and its images, and let i : M —>• TV be the iteration map. By an argument due to Jensen, there is an embedding σ : N —* VΩ such that π = σ o i. Now i(μ) = μ, so σ(μ) = σ(i(μ)) = ττ(μ) = μ. We also have π \ Vη = i ί V« = identity, so σ ί Vη = identity, so σ(x) = x. Thus σ(τLW[x,μ]) - rLM[z,μ]; that is, σ(6) = 5. It follows that (FN, <* N ) has order type 6. Now if P G ^, then σ f P : T> : P ->• σ(P) and σ(P) G 7", and thus P is β + 1-iterable. Thus TN C T. It is easy to see that (<*)N =<* Π7V. Since N G L[M], J7^ C T Π i[M], and we are done. D We will actually use the proof of 7.12, rather than the lemma itself. So far we haven't worked with K above μ, and indeed Hjorth formulated his lemma with μ = Ω. But now let M and N be as in the proof of 7.12. N We would be done if we could find V G T such that V<5 G T (Q <* P). M M There is a natural candidate for such a P, namely K . (K is not actually properly small, but this problem is easily finessed.) Of course, the iteration map i : KM -» KN comes from an "external" iteration of all of M, but suppose we could absorb its action into an internal iteration of KM. We'd be done. Since crit(i) = μ, we must use the part of KM above μ to do this. So we must work with μ < Ω. The following lemma is the key to absorbing the map from KM to KN into an iteration of KM. Its proof borrows Lemma 8.2 from §8, a lemma we originally proved as part of the proof of 7.9. Lemma 7.13. Let j : V -» Ult(V,U), where U is a normal ultrafilter on μ; then there are almost normal iteration trees T on K and U on j(K),

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§7. Some applications

having common last model Q and associated embeddings k t: j(K) -> Q, such that k = £oj.

K

Q and

Proof. Let W be a weasel such that Ω is thick in W, and W has the hull property at all α < Ω. Lemma 4.5 shows that such a weasel exists. By Lemma 8.2, there is an iteration tree TO on K having last model W whose associated embedding to ' K —> W satisfies BDef(H^) = t^K. In fact, TO is a linear iteration by normal measures. Notice that j(To) is an iteration tree on j(K) with last model j(W) and associated embedding j(ίo) Since the class of fixed points of j is thick in W, Ω is thick in j(W) and Όeΐ(j(W)) = j" Όef(W). Now let (Tι,Uι) be the successful coiteration of W with j(W), using their unique Ω + I iteration strategies, and let Q be the common last model of TI and U\. BLet t\ : W —* Q and u : j(W) —» Q be the associated iteration maps. We have the diagram:

K

J(K)

The bottom rectangle commutes: j o^ 0 = j ( t $ ) o j because j is elementary on V. The upper "triangle" may not commute, but it commutes on ran(tfo), since = Def(Q) (

=

u" Def(j(W)) = u"(j

=

«''(J''(ΦO)

It follows that, setting T = T TV is the iteration map coming from hitting a normal meaAsure of M on μ repeatedly, μ times in all. LeAt iaβ : Ma —> Mβ be the natural map, where Mα and Mβ are the αth and βih iterates of M. So i = io«.

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We define premice Qα, for α < μ, by induction on α. We shall have that Qα+ι is the last model on an almost normal iteration tree Ta on Qa, with an associated iteration map fcα,α+ι : Qa —» Qα+i We shall simultaneously 0 define embeddings -ία : A'*' ' —>• Qa so that for α < β < μ

Qβ

commutes. (Here we are setting Ar Λ > 7 + ι = &7)7+i ° & ατ , and fcαλ : Qa —> Qλ to be the canonical embedding into Q\ = dir lima<χQa for λ limit.) Set Qo = AM° and t = identity. Now, given Qα and 4, we apply 7.13 inside the model Ma to the ultrapower which produces Mα+ι. This gives an almost normal iteration tree T on KMa with last model Q and iteration map k : KM« -> Q, and an embedding I : KM*+l -> Q such that Jb = t o f α > α + ι. Note T E Mα. Let 7^ = £aT be the result of copying T to a tree on Q α , and let Qα+ι be the last model of Tα. (KM<* is a model of ZFC, Ί doesn't drop on its main branch, and k and ί are fully elementary. So, by induction, all Qj are ZFC models, no 7^ drops on its main branch, and all kηΊ and £Ί are fully elementary. So we can copy.) Let u : Q —* Qa+ι be given by the copy construction, and £a+ι = uol. The commutative diagram below summarizes the construction of Qα+ι and

For λ a limit < μ, let l χ ( i a χ ( x ) ) = &αλ(4*(#)) whenever α < λ and x E KMot. This completes the inductive definition of Qa and ίa. The Qα's are not properly small, but we can easily finesse this problem. Let Ψ : V —+ Ult(V, U) be the canonical embedding, where U is a normal

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ultrafilter on Ω. Let T> = J$(κ\ where a = Ω+ = (β+)^(*). Clearly, P is properly small, its largest cardinal being Ω, and K = jj. Also, 7> is β + 1 iterable in V, since 7> is ^(β + 1) iterable in Ult(F, £7) and Vlt(V, U) is closed under ί?-sequences. It follows that any iteration tree on K of length < Ω + 1 which is built according to the unique Ω+l iteration strategy for K can be regarded as an iteration tree on P. Now we can assume that the hull Z -< Va collapsing to M is such that Z — Y Π VΩ for some Y -< V# (for 0 > ί2 large) with f?, "P £ y. Let M' be the transitive collapse of y and Q'Q be the image of Ql

Ί> under this collapse. Thus Q'Q £ .T7 and Q0 = t7α °, where a is the collapse of Ω. We can interpret TO as a tree on Q'Q according to its unique Ω + 1 iteration strategy, and let Q( be the last model of TO, so interpreted. Then Ql

Qi = Jal , where α is the largest cardinal of Q(, and Q{ £ J7. Proceeding similarly by induction, we define Q'a for a < μ so that Qa = Jβ" for β the largest cardinal of Q'a . Now let UeFN. Working in TV, we see that Ίl <* j£N for some /? < μ. Since A"^ is elementarily embedded into Qμ by ίμι and Q^ is an almost normal iterate of Q'0 by its unique Ω + 1 iteration strategy, 7£ <* QQ. Thus ^"^ is not <*-cofinal in T\ Q'Q is an upper bound. The argument in the proof of 7.12 now yields a contradiction. D Ql

We can use our Σ% correctness theorem to show that certain apparently weak consequences of A\ determinacy actually imply Δ\ determinacy. The ideas here are due to A. S. Kechris; what we have contributed is just Theorem 7.9. Corollary 7.14. Suppose Vx € ωω (a?* exists), and Vx £ ωω (the class of subsets ofω has the separation property). Then Λ\ determinacy holds. Proof. We show that A\ determinacy holds; the proof relativizes routinely to an arbitrary real. By a theorem of Woodin, it is enough to show that there is a transitive proper class model M and an ordinal δ such that M |= δ is Woodin, and V^ is countable. Let x be a real which codes up witnesses to all true Σ\ sentences; that is, let x be such that whenever P is a nonempty Σ^ set of reals, then 3y £ P (y <τ x)- Using the Jensen-Mitchell Σ$ correctness theorem, we get a proper class model N such that x £ N and TV |= "There is are two measurable cardinals". For if there is no such TV, then KDJ(X) is Σ% correct, where KDJ(%) is the Dodd- Jensen-Mitchell core model for two measurable cardinals, relativised to x. Now KDJ(X) \=" There is a Z\g(x)-good wellorder of M", and thus KDj(x) \= "There are Σ£(X) sets A, B C ω such that AΓ\B = 0 and for all Δ^(x) sets C, A C C =» B Π C φ 0". Since KDJ(x) is Σ^ correct, there really are such sets A and 5, and thus Σ$(x) separation fails. Now let N be as described in the previous paragraph, and let TV \= "μ and Ω are measurable", where μ < Ω. If (KC)N \=" there is a Woodin cardinal", then we get the desired proper class model M with one Woodin

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cardinal δ such that V^ is countable. (Let P be the transitive collapse of a countable elementary submodel of Vn+ω, and i : P —» P^ the result of iterating a normal measure on the image under collapse of Ω through OR, and c p C N let M = i((K ) ).) But if (K ) satisfies that there are no Woodin cardinals, N then K is Σ% correct in TV by 7.9. The choice of x guarantees that, since Σ*3 ΠP(u ) has the separation property in V, it has the separation property in N. The correctness of KN implies that Σ^Γ\P(ω) has the separation property in KN. But KN \= "M has a Λ^-good wellorder", so KN f= "Σ* ΠP(ω) does not have the separation property". D If Π^Γ\P(ωω) has the reduction property, then for all x Gωω of sufficiently large Turing degree, ΣQ(X) Π P(ω) has the separation property. [Let (A, B) reduce a universal pair of Π^ subsets of ωω x ωω. Then whenever A and B are Π^(x)9 Σ^(x)Γ\P(ω) has the separation property.] Thus the proof of 7.14 shows that Vx Gωω (x* exists) + "11$ Π P(ωω) has the reduction property" implies Δ\ determinacy. We do not know whether Vx G ωω (x fl exists) + "£3 Π P(ωω) has the separation property" implies A\ determinacy. We conjecture that Vx G ωω (x" exists) plus "Σ\ Π P(ω) has the separation property" implies Δ\ determinacy. If one tries to prove this lightface refinement of 7.14 by the method of 7.14, then the fact that our Σ\ correctness theorem required two measurable cardinals, (rather than none) becomes a problem. Another application of our Σ$ correctness theorem in "reverse descriptive set theory" can be found in [Hj], where Hjorth uses it to show that Π\ Wadge determinacy implies Π\ determinacy. A problem which is closely related to the Σ% correctness problem is: what is the consistency strength of ZFC + Vx Gωω(x^exists) + #2 — ^2? Woodin has shown that the strength of ZFC-f "there is a Woodin cardinal with a measurable cardinal above it" is an upper bound. It is shown in [SW] that ZFC + "There is a strong cardinal" is a lower bound. We conjecture that the lower bound can be improved to ZFC + "There is a Woodin cardinal". Unfortunately, our proof of 7.9 does not seem to help with this conjecture, because of our use of the measurable cardinals μ and Ω. One wants to replace μ with ω\ (and Vμ with HC), and avoid Ω altogether, and we don't see how to do this. However, our proof of 7.9 does give the consistency strength lower bound ZFC H- "There is a Woodin cardinal" for a certain variant of ZFC + "Vx Gωω(x* exists) + δ\ = u>2" which we now explain. Let μ < Ω be measurable, and let ua be the αth uniform indiscernible relative to elements of Vμ, as in the proof of 7.9. Notice that in γc°^ω'<μ\ μa is the αth uniform indiscernible relative to reals, and so u^ = (δ\)v ° " ><M . One can ask whether γc°i(ω><μ) |= £* = ω 2 ; we do not know whether it is consistent relative to any large cardinal hypothesis that this be true. But if we replace vrCol(ω'<^) by its L(R), then the resulting proposition follows from ADL(^) in v°ol(ω><μ\ which of course holds if there are enough Woodin

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cardinals in V. We now show that «vCol(ω>2)" is at least as strong as the existence of one Woodin cardinal. Theorem 7.15. Let μ < Ω be measurable, and suppose vCol(ω
\ a contradiction.

D

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73

5. Embeddings of K In this section we prove some general theorems concerning embeddings of K. We also use these theorems and the ideas behind them to give another proof that if there is a strongly compact cardinal, then there is an inner model with a Woodin cardinal. In his work on the core model for sequences of measures ([Ml] , [M ?]), Mitchell has shown that if there is no inner model satisfying 3κ(o(κ) = K"1"1"), then for any universal weasel M there is an elementary j : K —» M; moreover, for any weasel M and elementary j : K —»• M, j is the iteration map associated to some (linear) iteration of K. Thus the class of embeddings of K is precisely the class of iteration maps, and the class of range models for such embeddings is precisely the class of universal weasels. It follows at once that if there is no inner model satisfying 3κ(o(κ) = /c++), then any j : K —* K is the identity; that is, K is "rigid". Mitchell's results extend the original Dodd-Jensen theorem ([DJ1]) that if there is no inner model with a measurable cardinal, then whenever j : K —> M is elementary, M = K and j = identity. The authors of [DJKM] strengthen Mitchell's results by weakening their non-large-cardinal hypothesis to "There is no inner model with a strong cardinal". We shall also prove such a strengthening of Mitchell's results, in Theorem 8.13 below. The situation becomes more complicated once one gets past strong cardinals. We shall see that it is consistent with "There is no inner model having two strong cardinals" that there is a universal weasel which is not an iterate of K, and an elementary j : K —> M which is not an iteration map. Assuming only that there is no inner model with a Woodin cardinal, however, we can still show that K is rigid. Using this fact, we can characterize K as the unique universal weasel which is elementarily embeddable in all universal weasels. We shall also show that if j : K —> M, where M is β+1 iterable, and μ = crit(.;), then P(μ)κ = P(μ)M. We shall assume throughout this section that Kc satisfies "There are no Woodin cardinals", so that Ω is Λo-thick in Kc and K = Def(/f c ,AO). Since we need only consider 5-thick sets for S = ΛO, we make the following definition. Definition 8.1. We say Γ is thick in W iff Γ is Ao-thick in W. Similarly, W has the hull (resp. definability) property at a iffW has the A^-hull (resp. definability) property at a. Finally, Def(W) = Def(W, AQ). We begin by showing that for any α < ί2, one can generate a witness that Ja is Ao-sound from K by taking ultrapowers by the order zero measures at each measurable cardinal /c of K such that α < K < Ω. The key to this result is the following. Lemma 8.2. Let W be an Ω+l iterable weasel which has the hull property at all a < Ω; then there is an iteration tree T on K with last model Mj = W, and such that

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(1) Vαfα+1 < θ => E% is a normal measure (i.e., has only one generator), so that T is linear, Proof. Let T on K and U on W be the iteration trees resulting from a successful coiteration of K with VF determined by the unique Ω + 1 iteration strategies on the two weasels. We show first that W never moves; i.e., U is trivial. Claim I . Ih U = 1. Proo/. Assume not, so that E% exists. Let α < Ω be inaccessible and such that Ih(E^) < a. Let R be an f? + 1 iterable weasel which witnesses that J^ is Ao-sound. Let S on R and V on W be the iteration trees resulting from a successful coiteration of R with VF. Let Q = ΛΊ^ = A4]f be the common last model of the two trees, and i = if7 and j = ϊ%6 be the iteration maps. Let μ = crit(j). Since J* = J* and lh(f%) < α, we have £# = £#. But then μ < V(EQ) < a. Since j : VF —+ Q is an iteration map and W has the hull property everywhere, Q has the hull property at all ξ < μ. Let θ = least η £ [0, j]s such that 77 = 7 or crit(i^7) > μ . From example 4.3 and the remark following it, we see that whenever η + 1 £ [0,0]s, then E^ is a normal measure, that is, has crit(i^) as its only generator. (Otherwise, Q would fail to have the hull property at (κ+)Q , where K = cτit(Eη) < μ. Since (κ+)Q < μ, this is impossible.) But now in any normal iteration tree, a normal measure can only be applied to the model from which it is taken. It follows that S \ θ -f 1 is just a linear iteration of the normal measures E^ for η 4- 1 < θ. We claim that Λif has the definability property at μ. For let Γ be thick <ω M in Λίf we want a G (μ U Γ) and a term τ such that μ = τ [a]. Let 4 be the thick class of fixed points of i%θ. Suppose first ifi θ(μ) = μ; then since .β witnesses that j£ is Ao-sound, and μ < α, we can find α £ (Γ Γ\ Λ)<ω R Mθ and r such that τ [a] = μ. But then r [α] = μ, as desired. Suppose next that i Q θ ( μ ) > μ. The usual representation of iterated ultrapowers gives us <ω <ω a function / : [μ] —» μ and an α £ [μ] such that if fl(/)(α) = μ. Since Λ witnesses that J* is A0-sound, and / £ J* , we have 6 £ (Γ Π Zi)<α; and r such that / = τΛ[6]. But then μ = TM(> [δ](α), which gives the desired definition of μ. Since μ = crit(j), Q does not have the definability property at μ. It follows that θ < 7 and crit(if ^) = μ. But now the ancient Kunen argument yields a contradiction: let A C μ and A £ Λ4f . Since Λίf has the hull property at μ, we can write A Π μ = r^ [α], where if,0(α) = α ~ ^(α) Ii; f°H°ws tnat ^M(^) n ^ ~ ^(^) n ^ where i/ = inf (if^(μ), j(μ)) This implies that the

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first extender used in S on [0,7]s and the first extender used in V on [0,0] v are compatible, which is a contradiction. This proves claim 1. D The proof of claim 1 also gives: Claim 2. T is a linear iteration of normal measures. Proof. If not, then we can find a weasel R which witnesses that j£ is AQsound, where α is inaccessible and large enough that some extender with more than one generator used on T has length < α. Again, let S on R and V on W come from coiteration, with M^ = M^. Since j£ = j£, S and Ί have the same initial segment using extenders of length < α. It follows that there is an η -f 1 Sj such that cnt(E^) < a and E% has more than one generator. Thus there is ξ < a such that M^ fails to have the hull property at ξ. On the other hand, the proof of claim 1 shows that cτit(iQ δ) > α, and since W has the hull property everywhere, this means M^ has the hull property at all ξ < a. This is a contradiction. D

Let Ih T = θ + 1, so that W = Mj. Claim 3. Όef(W) = ίξJ'K. Proof. Let a < Ω be inaccessible and ΪQθ"a C α; we shall show that Όef(W)Γ\ a = I^Q'OL. Let R witness that j£ is Ao-sound, and let S on R and V on W come from coiteration, with M^ — M^'. Let η be least such that η = θ or crit(t^) > a. Since J* = J*, S \ η + I = T \ η + I. We claim that η -f 1 C [0,7]$; that is, the linear iteration giving us T \ η + 1 is an initial segment of the branch of S leading to M^. For otherwise, letting β+1 G [0, j]s be least such that Ih E% > α, we have crit Eβ < α. This implies that M* has the hull property at all ξ < crit(£?!), but not at (cnt(E$)+)M$. But then we get crit(^) = crit(£'|), and that Eβ is compatible with the first extender used in [0,<5]\/, as in the ancient Kunen argument. This is a contradiction. Thus η + 1 C [0,7J5, and the argument also shows crit(ί^7) > α. So *M Γ <* = *ltη \<* = ^η \ <* = *0 >7 Γ <*

Finally, since M^ has the hull property everywhere below α, crit(ij^) > α. This gives

= Clearly, the claims yield 8.2.

io/

/α

D

D

Lemma 8.3. Suppose Kc |= "There are no Woodin cardinals". Let a be a cardinal of K, and let W be the iterate of K obtained by taking ultrapowers by

76

§8. Embeddings of K

the order zero total measure at each measurable cardinal of K which is > a. Then W witnesses that j£ is AQ-sound] moreover, W has the hull property at all β <Ω. Proof. Clearly W is universal, and no K £ AQ is measurable in W. It follows from 3.7(1) that Ω is thick in W', and it remains only to show that a C Def(W) and that W has the hull property everywhere. Let i : K —> W be the iteration map. Lemma 4.5 gives us an β+1-iterable weasel M which has the hull property at all β < Ω. Let α* be the αth ordinal in Def(M), and for each β < α* such that β £ Def(M), pick a thick class Γβ such that β g HM(Γβ). Let R = transitive collapse of HM(f]{Γβ

\ β e a* - Def(M)}).

Clearly, a C Def(Λ), so R witnesses that j£ is Λo-sound. It is easy to see that R has the hull property at all β < Ω. [Let A C /?, A £ R, and let Γ be thick in R. Let TT : R —»• M be the collapse map. Since π" Γ is thick in M, and M has the hull property at ττ(/?), there is a term τ and b £ Γ<ω and α £ ττ(/?)<α; such that π(A) = rM[α, τr(6)] Π π(β). The least α with this property is of the form ττ(ά) for some a £ /?<ω, and then A = rΛ[α, 6] Π /?.] Let j : /f —> Λ be the iteration map associated to the linear iteration of normal measures given by 8.2. We have j"K = Def(jR), and hence α < crit(j). Let T on R and U on W be the iteration trees associated to a successful coiteration of R with W, and let Λ4^ = ΛΊ^ be the common last model. Since ij^ o j and ίβδ o i are iteration maps, the Dodd-Jensen lemma gives fζ o j = i^j o i. From this we get that crit(z'o^) > a and crit(i^) > α. For otherwise, since i and j have critical point > α, we get /c < α such that /c = crit(tjy) = crit(i^). Also, for any A C K such that A £ K, i(A) = j(A) = A, so io γ(A) = i^(-A). This leads to the usual contradiction that the first extenders used in [0,γ]τ and [0,<5]c/ are compatible. Now by 5.5, iJ/'Def(Λ) = Όeΐ(M^) = Όeΐ(M^) = ^/'De^W). Since α C Def(Λ) and crit(tjy) > α, a C Όef(W). It remains to show that W has the hull property everywhere. Let β < Ω be a cardinal of /C, with a < β. Let V^* be the iterate of K obtained by hitting the order zero measure on each measurable cardinal /c > β of K exactly once, so that W* witnesses that j£ is Ao-sound by what we have just shown. In particular, W* has the hull property at all 7 < β. Clearly, W is a linear iterate of W* by normal measures (i.e., those of order zero on cardinals K such that a < K < /?), and therefore W has the hull property at all 7 < β. Since β was arbitrary, W has the hull property everywhere. D The reader may have noticed that 4.5 and 8.2 gave us a linear iterate by normal measures W of K satisfying the conclusion of 8.3, without much effort. (See paragraph 2 of the proof of 8.3.) What 8.3 gives, beyond this, is an iteration leading from K to W which is definable over K.

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Our next lemma expresses a maximality property of K and its iterates. Definition 8.4. Let M be a premouse. We say thai an extender F coheres M with M just in case (J£* , £, E \ α, F) is a premouse, where a = Ih F. Of course, any extender on the Λ4-sequence coheres with M. As another example: if (T,U) is a coiteration in which, at some intermediate stage, the current models are M% and Λi^, and E^ is part of the least disagreement at this stage, then E^ coheres with -Mj. We now show that if an extender E coheres with the last model M of an iteration tree T on K, and a certain iterability condition is satisfied, then E is on the .M -sequence. The iterability condition is that we can extend Ί one step using E as if it came from the .M-sequence, and then continue iterating in the normal fashion. The following definition enables us to make this condition precise. The reader should see 9.6 and 9.7 for the general notion of a phalanx, and the definition of Φ(T), the phalanx derived from an iteration tree T. Definition 8.5. Let T be an iteration tree with last model M%, let E cohere with M^, and suppose Ih E > Ih ET for all β < a. Let 7 be least such that j = a or crit(E) < v(E^), and letP be the longest initial segment Ίi of such that P(κ)Γ}K = P(κ) Π J^j1 , where K = crit(E). Let k < ω be least such that pk+ι(P) < K, if there is such a k < ω, and let k = ω otherwise. Suppose M = Ultk(P, E) is wellfounded. Then the E- extension ofΦ(T) is the phalanx Φ(T)~(λf, fc, ί/, λ), where v — v(E) and λ is the least cardinal of M which is > v. Theorem 8.6. Suppose Kc \= There are no Woodin cardinals". Let T be a normal iteration tree on W of length a. + 1 < β, and suppose E coheres with Mra and Ih E > IhE β for all β < a. Suppose that W witnesses that J*f is Ao-sound, where μ is inaccessible and large enough that a < μ and E £ Vμ and V/? < ct(Ej £ Vμ). The following are equivalent: (a) E is on the M% sequence, (b) the E-extension ofΦ(T) is Ω + 1 iterable. Proof, (a) => (b) is just a re-statement of the fact that W is Ω + 1 iterable. Now suppose B = Φ(T~(λf, fc, v, λ) is the ^-extension of Φ(T), and that B is J?+l iterable. Let us form the natural coiteration of β with Φ(T): at successor steps we iterate the least disagreement, beginning with the least disagreement between the last models N of B and M ra of Φ(T); the rules for iteration trees on phalanxes determine the models to which we apply the extenders from the least disagreement. At limit steps we use the (unique) Ω + 1 iteration strategies for B and Φ(T) to pick branches. The usual argument shows that this coiteration terminates successfully at some stage < Ω. The iteration tree on Φ(T) which is produced can clearly be regarded as an extension of T; let us call this extension U. We note that U is normal. [This is clear, except perhaps for the increasing-length condition on its extenders; for that we need

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only show Ih E% > Ih E% for all β < α. But since E coheres with λί agrees with M% below lh(E), and the disagreement of which E% is a part must occur at a length > lh(E}] The iteration tree on B which is produced can be regarded as a system S extending Ύ which has all the properties of a normal iteration tree except that, since E% = E, it may not be true that E% is on the Λ4f sequence. We shall use iteration tree terminology in connection with 5; its meaning should be clear. Claim 1. If E% is compatible with E% , then η = ξ < a and E* = E% . Proof. If V is a normal iteration tree, then Ih E% is a cardinal in all models M^ for r > σ, so that E% is incompatible with any extender on the M^sequence, for r > σ, by the initial segment condition and the fact that E% collapses its length. The system S has this property of normal trees because E% coheres with M% . Since S \ a = U \ a = Ί , this gives the claim immediately if η < a or ξ < a. It cannot happen that η > a and ξ > a. For then E^ and E¥ are each part of a disagreement in the coiteration of B with Φ(T). They cannot be part of the same disagreement since they are compatible. Thus Ih(E^) φ Ih(E^). Suppose that lh(E*) < lh(E^)] the other case leads to a similar contradiction. By the inital segment condition on premice, E^ is on the M^ sequence or an ultrapower thereof. Letting Λif be the model on S with which M^ is being compared, we have η < r and E^ on the M? sequence or an ultrapower thereof. This means lh(Eη) is not a cardinal of -A/if, a contradiction. We are left with the possibility that η = a and ξ > a. Thus E^ — E. If Ih(E^) > lh(E], we get that E is on the M2^ sequence or an ultrapower thereof, and hence on the Λf-sequence or an ultrapower thereof, which is impossible since E coheres with -M J. Thus E^ = E^ . D Now let M^ and M^ be the last models of S and U respectively. Claim 2. M* = Mub '. Proof. If [0,γ]s Π D8 = φ, then since Ω is thick in W = MQ, Ω is thick in M*. But then [0, % Π D" = φ and hence Muδ = Λίf . If [0, 7] 5 Π D8 φ φ, then M*ξ < M^ because M^ is not ω-sound. But then [0,<5]c/ Π if* φ φ, as otherwise Ω is thick in M^ while M^ computes /c+ incorrectly for a.e. K G AQ. Thus M^ < M*f, as M$ is not α -sound. D Now let θ < a be largest such that 067 and ΘUδ. Since Λ4f+1 = λί exists, θ < 7, so we can set η + 1= unique β £ [0, 7] 5 such that S-pτed(β) = θ. Claim 3.θ<δ. Proof. If not, then since θ < a < <5, we must have θ = a = δ.

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Suppose first that Dr Π [0, a]τ = Φ Since then M J = Λί5 is a universal weasel, [0,7)5 Π D8 - φ. Let K = crit(E5). Since K < v(Eθ) < μ, and μ C Def(H^), Λίf has the definability property at K. Since K = crit if 7, Λ45 does not have the definability property at /c. But Λίf = M% = M^ = Λίf, a contradiction. By claim 3, we can let ξ + 1 £ [0,<5]c/ be the unique β such that U-pτed(β) = θ. Claim 4. E5 is compatible with E^. Proof. We claim first that Ds Π (77 + 1,7)5 = 0, and deg5 (77 + 1) = deg5 (7). For otherwise, let σ + 1 be the site of the last drop in model or degree along (η + 1,7]5, and let k = deg (σ + 1) By standard arguments, (Λ^^+1)5 = <£jfe+ι(Λί5), k is least such that Λ45 is not k + l sound, and if+ι )7 °(C+ι)5 ^s the canonical core embedding from £jb+ι(.M5) into .Λ/f5. Since M8 = M^, there is a last drop r+1 in model or degree along [0, δ]t/, and deg w (r-fl) = ky (Λ4*+1)w = <£fc+ι(Λ4^), and i^+1 δ °(i* +1 ) w is the core embedding. This gives Ef compatible with E^, so that σ = r < a by claim 1. But then σ < θ by the definition of θ, while 0 < 77 < σ by the definition of σ. A similar argument shows that D^ Π (£ + 1> <$)?/ = <^ and deg (£ + 1) = deg^γ). Next, suppose 77 + 1 £ Ds or deg (η + 1) ^ deg5(0). Arguing as above, we get that E5 is compatible with E^, where r-f 1 is the site of the last drop in model or degree along [0, <$][/. We cannot have r + 1 £ [0,0]c;, since then E^ = Ef, so Ef is compatible with E^, while r φ η because η + 1 ^ [0, θ]u The only other possibility i s r + l = £ + l, which gives E5 compatible with E^, as desired. Similarly, if ξ + 1 £ D" or degw(ξ + 1) $ degw(0), then E5 is compatible with E^. So we may assume that if and ήf δ exist, and each is a degr(0) embedding. Suppose that Dr Π [0, θ]τ = 0, and let ιx = sup{z/(Ej) | σ -f 1 £ [0,0]τ}. Since μ C Def(W), we have μ C HM?(v U Γ) for all thick Γ. Taking Γ to be the class of common fixed points of if 7 and i^ δ, and noting that v < crit(if 7 ) < μ and v < crit(i^) < μ, we get that if >7 (A) = %tδ(A) for all A C K, where K is the common critical point of the two embeddings. This implies othat E5 is compatible with E^. Finally, suppose that Dr Π [0,0]τ Φ φ, and again let v — sup{z/(Ej) | σ + 1 € [0,%}. Let fc = degr(<9). Then Λtf = Jϊj^{(ι/U p), where p = pk+ι(Mj). Since if j7 (p) = pjk + ι(Λ^ 5 ) = ^^(p), we again get ίf >7 (A) = i*ott(A) for all A contained in the common critical point of the two embeddings, and thus that E5 is compatible with E%. This proves claim 4. D

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By claims 1 and 4, η = ζ < a and Ef* = E% . We cannot have η < α, for then 7 7 + 1 = £+1 < α, which gives 77 + 1 < 0, contrary to the definition of η. Thus η - £ = α, so that E = E* = E%. Since E£ is on the Λ4£ sequence, we are done. Π Remark 8.7. (a) We believe that one can prove 8.6 with K replacing W '. c That is, suppose K \= "There are no Woodin cardinals", and let T be a normal iteration tree of length α + 1 < Ω. suppose E coheres with M J, lh(E) > Ih(E^) for all β < α, and the ^-extension ofΦ(T) is Ω + 1-iterable. Then E is on the Λ4 J sequence. For let μ be inaccessible and T, E £ Vμ, and let W come from K by iterating normal order zero measures above μ, as in 8.3. Let π : K —>• W be the iteration map, and let πT on W be the copied tree; it is enough to show E is on the M*J sequence. By 8.6, it is enough for this to show that the £"-extension of Φ(τrT) is Ω + 1-iterable. We believe that one can do this by chasing the proper diagrams, but haven't gone through the details. (b) We believe, but haven't checked carefully, that the methods of §9 show that if T, E, and a are as described in the hypothesis of 8.6, and if (M ^ , E) is countably certified (in the sense of 1.2), then the E'-extension of Φ(T) is iterable. Taking T = 0, this means that every countably certified extender which coheres with K is on the /^-sequence. We now show that K is rigid. Theorem 8.8. Suppose that Kc |= "There are no Woodin cardinals" , and let j : K —» K be elementary] then j = identity. Proof. Suppose otherwise, and let K = crit(j). Let μ < Ω be inaccessible and such that j ( κ ) < μ. Let W be the result of hitting each order zero measure of K with critical point > μ exactly once (in increasing order) as in 8.3. Thus W witnesses that J*f is Ao-sound. Let F be the length j ( κ ) extender over K, or equivalently W , derived from j. It will be enough to show F \ p 6 W for all p < j ( κ ) , since then these initial segments of F witness that /c is Shelah in W. The proof of this is an induction on p organized as is the proof of Lemma 11. 4 of [FSIT]. Lemma 8.9. Let (κ+)w < p < j ( κ ) , and suppose p is the sup of the generators of F \ p. Let G be the trivial completion of F \ p, and 7 = Ih G. Then E™ = G = F ί 7 unless p is a limit ordinal greater than (κ+)w , and is itself a generator of F. In this case EΊ i(Ew)Ί

if p i dom Ew , if pedomEw ,

where i : J™ -+ UltQ(J™ , E™) is the canonical embedding. Proof. By induction on p. Suppose first that p is not a generator of F which is > (κ+)w . It follows that the natural embedding from Vlt(W,F \ p) into

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Ult(W, F) has critical point at least 7 = ( / ? + ) ( ^ , F t p ) From this weget that G coheres with W\ (J^, £, Ew , G) satisfies the initial segment condition on premice by our induction hypothesis. We now apply 8.6 to the trivial iteration tree on W whose only model is W'. By 8.6, we are done if we show that the phalanx B = (((W,ω), (Ult(VP, G),ω)), {/?, 7)) is Ω+ I -iterable, for then G is on the W sequence. In order to show that B is β+1 iterable, it is enough to find an elementary π : Ult(VF, G) —>• W such that π \ p = identity, since then we can copy iteration trees on B as ordinary iteration trees on W. Now W is definable over K from μ, hence j : W —»• j(W) is elementary, where j(W) is defined over K from j ( μ ) as W was from μ. Since G is an initial segment of the extender derived from j, we have the diagram

Ult(WζG) where ψ \ p = identity. So it is enough to find an embedding σ : j(W) -» W which is the identity up to /?. Let U be the linear iteration leading from K to W, so that /ί = M% and W = Λ4/7. Let 5 = ,;'(£/) be the linear iteration leading from K to j(W), so that K = Λif and j(W) = Λ<£. Let α < Ω be least so that crit(^ Ω) > j(μ). We now define by induction on β > 0 maps Tβ : M^ —*• -^^+0 such that T 7 ° ^7 ~ *α+/3,α+7 for ^ < 7 We begin with r0 = 2^α. Given r/j, set where if = E%\ notice here that Tβ(H) = ί^+« so that this works out. [Let Z be the a + /?th order zero total measure of K above μ. By definition of ct, Z is the βth order zero total measure of K above j(μ). Thus £%+β = %a+p(Z) and JSf = iξβ(Z). But then r^JEf ) - rβ(^(Z)) = z

'α,α+/?(ro(^)) = #>β+/j(^) = ^+/j ]We define rλ usinβ commutativity, for λ a limit. Clearly Tfi : j(W) —>• W, and it is easy to see that r/j f μ is the identity. This finishes the proof of 8.9 in the case p = (κ+)w or p is not a generator of F. If p > (κ*)w and p is a generator of F, then the natural embedding ψ : Ult(ΐ^, G) —>• j( W) has critical point p. It is therefore not obvious that G coheres with W. Nevertheless, we can apply the condensation Theorem 8.2 of [FSIT]. For notice that φ(p) < j ( κ ) < μ, and that W and j(W) agree below μ, and that p is a cardinal of Ult(VP, G). Further, 7 = (p+) ult W G ), so there are arbitrarily large η < 7 such that pω(Jη

^ '

) = p and ψ maps

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/T UltfWGO

4/T7

'

!

.

xTTirr i

i

i

i

elementarily into j •ψyη) ,, ^ which is a level of W. If M

i

then 8.2 of [FSIT] gives jfli(w'G) = J™ for all η < 7, so that G coheres with W. (Again, the initial segment condition is our induction hypothesis.) We can then finish the proof just as in the case p is not a generator of F. So assume p £ dom Ew. In this case, 8.2 of [FSIT] implies that G coheres with Ult(W, E™). (Notice that Ult(W, E™) makes sense: letting λ = crit E™, we have (λ+)J™ = ^χ+^(j[t(W,G) because p is a cardinal of Ult(H^, G), which agrees with W below p. Since p = crit(VO, (λ+)ult<w σ> = (λ+y'W = (λ+) w . Thus E™ measures all sets in W.) The proof of 8.9 will be complete if we show that G is on the Ult(V7, E™) sequence. To this end, we apply 8.6 to the iteration tree T on W of length 2 such that E*ζ = E™. We are done if we can show that the G-extension of Φ(T) is Ω+1 iterable. Using the natural map from Ult(P^, G) into j(W) we can copy iteration trees on the G-extension of Φ(T) as iteration trees on B, where B is the phalanx with models (W,Vlt(W,E™),j(W)) and "exchange ordinals" (v(E™)>p}. So it is enough to see that B is Ω+ 1-iterable. But we can embed B into a Xc-generated phalanx as follows: let π : K -+ Kc with ran(τr) = Όef(Kc). Since W is obtained from K by a K-definable iteration process with critical points above μ, we have π : W —*• Λ, where R is obtained in the same way from Kc, the critical points being above ττ(μ). Similarly, π : j(W) —* 5 where 5 is obtained from Kc by iterating above π(j(μ)). Finally, let σ : Ult(V7, £?J^) -» Ult(Λ, Jr(£^)) be the shift map induced by π. We have σ f z/ = π f z/, where v =• v(E^). Now Ult(JR, π(E^)) is not quite the last model of an iteration tree on Kc, since the ultrapowers do not come in the increasing length order. But this is easy to fix: we can find an embedding ψ : Ult(β, π(E™)) —»• Q, where Q comes from forming Ult(X c , π(E™)), c and then doing the images of the ultrapowers in the iteration from K to R. We have φ \ ττ(μ) = identity. We have then that the phalanx T> with models (Λ, Q,5) and exchange ordinals {ττ(i/), π(/?)) is /^-generated, and therefore is Ω + 1 iterable. (cf. 6.9) We can use the maps {π, ψ o σ, π) to reduce trees on B to trees on D, and hence /? is Ω + 1 iterable. This completes the proof of Lemma 8.9, and hence of Theorem 8.8. Ώ Theorem 8.8 leads to the following characterization of K. Theorem 8.10. Suppose Kc \= "There are no Woodin cardinals"} then K is the unique universal weasel which elementarily embeds into all universal weasels. Proof. Let M be a universal weasel. For a < Ω, we construct a linear iterate Pa of M by hitting each total order zero measure above α once. (More precisely, let Uv be the ί/th total-on-M measure of order zero on the M-sequence which has critical point > α. Let T be the linear iteration tree on M such that Ef = ιξtV(Uv) for all i/, and let Ih Ί - 7+1 < β+1; then we set Pa = M*.)

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It is not hard to see that if a < β < β, then Pa is a linear iterate of Pβ. For let Uv be the z/th total measure of order zero on the M-sequence with critical point between a and β, and let j : M —> Pβ be the iteration map. We define a linear iteration tree Ί on Pβ by: Ej = io^/C; (£/„)). Letting 7 +1 = Ih T (so that 7 is the order type of {z/ | Uv exists}), we have Pa = M%. (This comes down to the fact that if i and k are the embeddings associated to normal measures on distinct measurable cardinals, then i(k) = k.) Since M is universal, (α+)M = α+ for all but nonstationary many α £ AQ, by 3.7 (1). The construction of P^ guarantees that no 7 £ Λo—β is the critical point of a total-on-P0 extender from the Pβ sequence. Therefore Ω is Ao-thick in Pβj for all β < Ω. From 5.7 we then have that K = Def(P/j), for all β < Ω. Let πβ : K —> Pβ have range Όef(Pβ). Let i^ : Pβ —+ PQ be the linear iteration map described above, so that i"β Όeΐ(Pβ) = Def(P0) by 5.6. Since iβ(πβ(μ)) = π0(μ), we have ^(μ) < 7r0(μ) for all β and μ. Now let us define σ : /£ —•» M as follows: for x £ K, pick any μ such that x £ J^, and let /? be inaccessible and such that ττo(μ) < /?; then we set σ(x) = ττ^(x). Note here that πβ(x) £ ^ ^ s C Jββ = jjf, so indeed σ(x) £ M. Also, if/? < 7, then the iteration from P7 to P0 has critical point > /?, so that Def(PΊ) Π J^ = Def(P/j) Π J^β. This implies that σ is well-defined. Finally, if K |= y?[ar], then P^ |= ^[^(x)], so M |= φ[σ(x)] because the iteration from M to Pβ has critical point > β and 7Γ0(x) £ Jβ1. This shows that σ is elementary. Finally, we show uniqueness. Let j : M —* K be elementary, where M is a universal weasel. Let i : K —»• M be elementary. Then j o i : K —* K elementarily, so j o i = identity by 8.8. It follows that M = K. D We now consider the situation "below O p ". Definition 8.11. A proper premouse M is below Op iff whenever E is an extender on the M-sequence, and K = crit(E), then j£* |= ((There are no strong cardinals". One important way in which a premouse M below Or is simple is that every normal iteration tree Ύ on M is "almost linear". More precisely: if T is a normal tree on M and Γ-pred(/? + 1) = α, then for some n £ α ; , / ? = α - h n and crit(£'J) = crit(£'J+A.) for all k < n. Thus T misses being linear only in that it may hit the same critical point finitely many times in immediate succession (and thus branch finitely) before going on to critical points larger than the lengths of all preceding extenders. [Proof: It is enough to show that whenever a < β and crit(Ej) < ι/(i£), then crit(#J) = crit(^). So let a < β, K = cήt(Ea), μ - criK-Ej)) and suppose μ < v(E^γ If K < μ, then there are arbitrarily large λ < μ such that E% \ X is on the Mj sequence, so MTβ is not below Or. If μ < /c, then there are arbitrarily large λ < K such

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that Ej \ X is on the M J sequence, so M J is not below O . Both statements follow at once from the inital segment condition and the normality of T.] c

p

Lemma 8.12. Suppose K is below O . Lei W and M be universal weasels, with Ω Ao-thick in M and W, and μ C Def(W}. Let i : W -> Q and j : M -+ Q come from coiteration, and π : K —» M with ran π = Def(M). Then j \ sup(π"μ) = identity. Proof. Let T be the iteration tree on W producing i = ιζδί and U the iteration tree on M producing j. Since we are below Op, T and U are almost linear. Suppose toward contradiction that crit(j) < ^(7), where 7 < μ. Let K = crit(j) = crit(£#). Since i"Def(HO = Def(Q) = j"Def(M) by 5.6, we have 2(7) = j(7r(7)), and thus K < sup(2/; μ). But now for any η < sup(i" μ), either Q has the definability property at η or crit(jE^) < η < ι^(E^) for some αT<5. (If 77 < i(f) and the second disjunct fails, we can write η = i(f)(ά) where a G [η]<ω and / : ζ -> ί, so that / G Def(W) and hence t(/) G Def(Q).) Since « = crit( ), Q does not have the definability property at /c, and thus we can fix α-f 1 < Ih T such that crit(E£) < /c < i/CE*"). We claim that K = crit(E^). This is where we make real use of the fact that our mice are below Or. For otherwise, letting θ = crit(E'J), we have that J™ |= θ is a strong cardinal. This is because there are arbitrarily large λ < K such that E% \ X is on the M-sequence. (Proof: K is inaccessible in Q, and hence in J^ a, where ξ — Ih E%. The initial segment condition gives arbitrarily large λ < K such that E% \ X is on the M J sequence. But /rM _ ηM a \ Jκ - JK •) Let i^δ : M^ —» Q be the remainder of the iteration. Since M^ has the hull property at /c and K = crit(^), Q has the hull property at /c, and M hence so does M. But then i^δ(A) Π v = j(A) Π z/ for all A G P(κ) , where 1 ι/ = inf^E'J), V(EQ)). It follows that E£ is compatible with E ^, the usual contradiction. D Theorem 8.13. Suppose Kc is below Op; then any universal weasel is a normal iterate of K, and any j : K —*• M is the iteration map associated to a normal iteration of K. Proof. The second statement actually implies the first, via 8.10, but we give an independent proof. Let (T,£Y) be the coiteration of K with M. Since Kc is below Op, both T and U are normal, "almost linear" iteration trees. We wish to see Ih U = 1. If not, then EQ exists; let μ < Ω be inaccessible and such that Ih E% < μ. Let K* and M* come from K and M, respectively, by hitting each total order zero measure above μ. So Ω is Ao-thick in K* and M*, and μ C Όef(K*). If (T*,W*) is the coiteration of K* with M*, then E* — E. This contradicts Lemma 8.12.

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Next, let j : K —>• M be elementary. By the Dodd-Jensen lemma, M is universal, and hence there is a normal iteration tree Ί on K such that M = M^, for some 7 < β. We wish to show that j = ίζy. So fix 77 < β; we shall show that j(η) = ιζΊ(η). Let β < Ω be such that j(η) < β and Vα (cήt(E%) < β => /ft (#£) < /?). 1 Let F be the length β extender derived from j, and let j : K —> M' be the 1 canonical embedding of AT into \]lt(K,F) = M . Clearly, M1 agrees with M below β, and j'(η) = j(ή). Let T be the normal iteration tree on K whose last model Mj' = M'. The agreement between M and M' implies that Ma[(lh(El] < β V /ή(#f) < / ? ) = > f£ = E*']. It follows that if 'oi'fa) = JΌ?)» then 'Jyfo) = ^'(|?)» and we are done Let K* come from K by hitting each total order zero measure with critical point above β once. Let M* come from M' via the same process, using critical points above j'(β). Clearly j1 : K* -> M*, Ω is A0-thick in K* and M*, and /? C Def(A*). Also, {α | /(α) = α} is AQ thick in K* and M*, which is why we switched from j to j1. Now let i : K* —» Q and k : M* —»• Q be the iteration maps coming from the coiteration of K* with M*. Since K and A"* agree below /?, as do M' and M*, it will be enough to show that i(η) = /(f?), for then i(ι/) = ϊζ6 (η) and we are done.But since j' has a thick class of fixed points, the proof of 5.6 gives i11 Όeΐ(K*) = Def(Q) = (k o j')" Def(/C). Thus i(η) = k(j'(η)). Since k \ β = identity by 8.12, i(ιy) = /(»/), as desired. D We have developed the theory of Kc and K under the assumption that there is a measurable cardinal β, and so this is a tacit hypothesis in 8.13. The measurable cardinal is not needed for the theory of K "below Or", however. (See [DJKM].) Thus 8.13 does not require this tacit hypothesis. We now sketch an argument which shows that the hypothesis of 8.13 that Kc is below Or cannot be substantially weakened. The reason is that the conclusion of 8.13 implies, via work of Jensen and Mitchell, that K Π EC is Σ\ in the codes. On the other hand, Woodin has shown that it is consistent c that K is "below two strong cardinals", and yet K Π EC is not Σ\ in the codes. p We sketch the definition of K "below O " due to Mitchell and Jensen. Let us call α a closure point of a premouse M iff a is a limit of Λί-cardinals and V/? < αΞγ < αV0 G (τ,α) (JΘM is active => cnt(E^) > β). Let us call a premouse M α-good just in case M is iterable, ρω(M) = ct, and either a is a closure point of M or there is a universal weasel W such that for some β: M = jjf, Pω(J?) > α for all 7 > /?, and crit(J^) > β for all 7 > β. Clearly, if W witnesses that M is α-good, then α is a cardinal of W. It can be shown that the relation R is 77^, where R(x, y) O (x, y £ ωω Λ x codes a premouse M Λ y codes an ordinal α Λ M is α-good). [If Λ4, α £ #C, and α is not a closure point of M, then M is α-good iff VAf G #C [3/? ( > β(mi(Eίf) > /?)Λ 'W is iterable via extenders with critical

86

§8. Embeddings of K

point > /?") =» λf is iterable and pu,(.V) > a.] This is 77^ in the codes; the antecedent in the bracketed conditional is Π\ because of the iterability assertion. The =>> direction of this equivalence comes from the coiteration of a weasel W witnessing M is α-good with a potential counterexample λί to the right hand side. For the <ί= direction, the weasel W which witnesses that M is α-good is "KC(M)"> the result of the construction of §1 modified so that it begins with Λo = M and only uses extenders with critical point > ORM . We use here that α is not a closure point of M. This, together with the other conditions, implies that /?+ 1 is contained in every hull formed in the KC(M) construction, so that this construction does indeed produce the desired W . If α is a closure point, a ^ some hull is possible.] Notice that if α is a cardinal of K, and pw(J^) = α, then jjf is α-good. This is clear if α is a closure point of K . If α is not a closure point of A", then the weasel W which witnesses that jjf is α-good is Ult(A, £!^), where 7 is least such that 7 > β and cήt(Eϊf) < /?, unless there is no such 7, in which case W = K is the witness. (If there is such a 7, then cήt(E^) < a for the least such 7. For if crit(£"^) =. α, then α is a limit cardinal of K , so since α is not a closure point, there is a K < a which is the critical point for extenders on the AΓ-sequence with indices unbounded in α. The initial segment condition gives v such that β < v < 7 and crit(^) = «, where P - \J\t(jf ,E*) But E% — Eff by coherence, and this contradicts the minimality of 7. Thus crit(J5'^) < α, and therefore Ult(/ί, Eff) makes sense.) Suppose that, conversely, whenever α is a cardinal of A, J*4 = jζ , and M is α-good, then M = jjf for some β. It would follow that

5 T < (α+)* (P

=

JK)

= J* ^M is α-good Λ 3Ί(P = jf)) . This easily implies that the function α i-» J*f , restricted to HC, is Π\ in the codes, and hence that K Π HC is Σ§ in the codes. It is easy to see that if M is α-good, j£* — j£ , and α is a closure point of Λί, then M = jf for some β. This is because all critical points in the coiteration of M with K are > α. So if K Π HC is not Σ\ in the codes, there is a cardinal α of K which is not a closure point of A r , and an α-good M such that j£* — j£ but M. is not an initial segment of K. Let M be α-good, as witnessed by W , and J^ — j£ If W is an iterate of Ky then the iteration must use extenders of length > α because J™ — j£ , and this implies M is an initial segment of K because ρω(M) = α. So if K Π HC is not Σ\ in the codes, then there is a universal weasel W which is not an iterate of A'. By 8.10, there is nevertheless an elementary j : K —»• W , and of course this j cannot be an iteration map.

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We shall not attempt to sketch Woodin's proof that it is consistent that Kc has no Woodin cardinals (and is in fact "below two strong cardinals") and yet K Π EC is not Σ\ in the codes. See [H]. Although we do not have the decisive characterization of embeddings of K given by 8.13, once we get past Op, we can prove the following consequence of the characterization. Theorem 8.14. Suppose Kc |= There are no Woodin cardinals, and let M be Ω + I iterable, j : K —> M be elementary, and K = crit(j). Then (1) P(K)M = P(κ)κ, (2) the trivial completion of the (/C,AC + 1) extender derived from j is on the K-sequence, (3) if K \= "j is regular but not measurable", then j ( j ) = sup(j"j). Proof. We simply trace through the proofs of 8.12 and 8.13, and see what we get. Let M, j, AC, and 7 be as in the statement of 8.14. Let β < Ω be inaccessible with /c, 7 < /?. Let F be the length β extender derived from j. Let A'* be the witness that jjf is A0-sound which is obtained from K by hitting each order zero measure above β. Let M* — Ult(/ί*, F). Since M is universal by Dodd-Jensen, (α+)M = α+ for all but nonstationary many a £ AQ by 3.7, and hence Ω is AQ thick in M*. Further, {a \ j*(α) = α} is Ao-thick in K*, where j* : K* —*• M* is the canonical embedding. It will be enough to show that P(AC)M* = P(/c)κ*, that the (/c, AC + 1) extender derived from j* is on the K* sequence, and that j * ( j ) — sup(.;* 7). Let i : K* —>• Q and k : M* —> Q be the iteration maps coming from a coiteration of K* with M*. We have β C Def(/f*), and since j* has a thick class of fixed points, K C Def(M*). Standard arguments then show that i \ K = (k of) \ K = identity, and therefore P(κ)κ* = P(κ)Q = P(/c)M*. Further, since K = cήt(k o j*), K £ Def(<3), and therefore K = crit(f). Since K* and M* have the hull property at AC, if crit(fc) = K then the usual argument shows that the first extenders used along the branches of the iteration trees producing i and k are compatible, which is a contradiction. Therefore crit(fc) > AC. But then the (AC, AC + 1) extender derived from i is the same as the (AC, AC + 1) extender derived from j*. Since i is an iteration map, the (AC, AC + 1) extender derived from i is on the /£"-sequence, and we have proved (1) and (2). Since K* \= "7 is regular but not measurable", and i is an iteration map, i(γ) = sup(i"7). Since β C Def(#*), i \ β = (k o j*) f /?, and therefore (* 0 ^*)(τ) = sup((fco j*)7/7). It follows that j*(y) — sup(.;'* 7), which proves (3). D Clearly, 8.14 can be pushed a little further, in that more of j is given by an iteration of K than its normal measure part. We conclude this section by giving another proof that if there is a strongly compact cardinal, then there is an inner model with a Woodin cardinal. The

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proof we gave in §7 required an excursion into descriptive set theory, whereas this proof does not. Lemma 8.15. Suppose Kc \= There are no Woodin cardinals, and lei μ < Ω be measurable] then (μ+)κ = μ+. Proof. Let j : V —> M with crit(j) = μ. Since μ is measurable, we can prove all the results of §1 - §6 with μ replacing Ω. (Let Kcμ be Kc as constructed in Vμ. If K* \= There is a Woodin cardinal, then there is a model of height Ω having a Woodin cardinal δ < Ω. As we remarked in §2, this is impossible if Kc |= There are no Woodin cardinals.) Let Kμ be the model constructed in §5, but "below μ". Let U = {X C μ \ μ G j(X)} be the normal ultrafilter generated by j. By 5.18, for U a.e. a < μ, (a+)Kμ = α + . But now the results of §6 give an inductive definition of Kμ which is precisely the same as that of JK, and therefore Kμ = J*. It follows that (α+)κ = α+ for U a.e. a < μ. But then (μ+)κM = (μ+)M = μ+. Since j : K -> KM, we have (μ+)κ = (μ+)κM by 8.14. So (μ+)κ = μ+, as desired. D Theorem 8.16. Let Ω be measurable, and let μ < Ω be μ+-strongly compact] then Kc \= There is a Woodin cardinal. Proof. Suppose otherwise. Let j : V —»• M come from the ultrapower of V by a fine, μ-complete ultrafilter on Pμ(μ+). It is well known that j(μ+) > sup(/'μ+). But μ+ = (μ+)κ by 8.15, and j is continuous at (μ+)κ by (3) of 8.14. This is a contradiction. D

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§9. A general iterability theorem In this section we give a full proof of the iterability facts we have used. The proof results from an amalgamation of §12 of [FSIT], §4 of [IT], and §2 of this paper. Given a premouse M of the construction C of [FSIT], and an iteration tree Ί on Λ4, §12 of [FSIT] shows how to use the background extenders of C to "enlarge T" to an iteration tree U on V. The good behavior of U guarantees that of T. That U is well-behaved is shown in §4 of [IT], by realizing in V the models ΛΊ^ occurring on countable elementary submodels U of U. However, since the construction C of the present paper does not involve background extenders over V, we cannot in the current situation enlarge T to a tree on V. Instead, we shall run the enlargement process of [FSIT] and the realization process of [IT] simultaneously, making do with the partial background extenders of C as in §2. We have also re-organized and streamlined the construction of §4 of [IT]. Moreover, in order to cover all our applications, we shall consider more than just iteration trees on premice. Definition 9.1. A creature is a structure which is either a premouse, a psuedo-premouse, or a bicephalus. Let C be the construction of §1, that is, C = (λfξ I ξ < ΩΛtft

is defined).

Definition 9.2. ΛΊ is a creature ofC just in case for some j,ξ (a)Λ4 = C, M)> or (b) M = (&w(Aft),F), M is a psuedo-premouse, and letting K = crit(F), MA C P(κ)M(\Λ\ < ω => F has a certificate on A), or (c) M = (i^(A^),Fo, FI), M is a bicephalus, and letting i £ {0,1} and Ki = crit(Fi), MA C P(κ,i)M (\A\ < ω => Ft has a certificate on A). We say M is C-exotic just in case condition (a) above fails to hold. If M is a creature of C which is not a premouse, then M must be Cexotic, but we do not know whether the converse is true. If M — £j(-A/f), then we call ( j , ξ ) an index (in C) for M\ a non-exotic M can have more than one such index, but all its indices have the same second coordinate. If M is C-exotic, it must be of the form (C(A/^),F) or (£u,(-Λ/e),ίo,ίι), and then we say (0,£) is an index (in C) for M. A C-exotic creature of C has exactly one index in C. By ind(Λ4) we mean the common second coordinate of all indices of M. Recall that a coarse premouse is a structure M = (M, £, ί) such that M is transitive, power admissible, satisfies choice, infinity, and the full separation schema, satisfies the full collection schema for domains contained in V§, and such that ωδ = δ and ωM C M.

§9. A general iterability theorem

90

M

Definition 9.3. // M is a coarse premouse, then C = (λf£* \ ξ < M 6 and λf£* exists) is the construction of §1 as done inside M, up to M stage 6 . M

Thus C = C^ for all coarse premice M such that 6 = Ω and V/f = VΩ. 4 Notice that for any coarse premouse Λ4, λfjf* £ V£M whenever Me exists. Further , there are in V^M certificates for all extenders put into models of C". By convention, all creatures are 0-sound. The notion of a weak 0-embedding extends in an obvious way to creatures which are not premice. If π : M. —> M and M and Λf are psuedo-premice, then π is a weak 0-embedding just in case M 7Γ is rΣo elementary, and for some cofinal X C OR , π is rΣ\ elementary ; on parameters from X. If π : M —»• λf where M = (Λί , FQ, FI) and Λf = (λfΊGQjGi) are bicephali, then π is a weak 0-embedding just in case it is a weak 0-embedding from (Λ4', FQ) to (ΛΛ, GO) and a weak 0-embedding from (M',Fι) to (ΛΛ,Gι). For k > 0, we shall consider fc-soundness and weak Ar-embeddings only as applied to premice. Definition 9.4. Let M be a creature and let k < ω; then (7£, <2, π) is a k-realization of M just in case Ίi is a coarse premouse and (a) Q is a creature of Cπ of the same type as M, and if k > 0 then M is premouse and Q — <£jb( Λ/jr)^ for some ξ, (b) π is a weak k-embedding from M. into Q, and (c) 7Γ, M G π. In the situation of 9.4, if M is a premouse then the ordinal ξ as in (a) is determined uniquely by Q and k. If M is a creature and ωβ = ORM , then we set J$* = M . If ωβ < ORM , then we let j£* be the unique premouse Q such that Q is an initial segment ofΛΊ Definition 9.5. Two creatures M and λί agree below 7 just in case for all

β<Ί, J^ = 3$ We wish to consider iteration trees whose base is a family of creatures. Definition 9.6. A phalanx of creatures is a pair (((Mβ,kβ)\β Vβ such that MΊ \= η is a cardinal, and moreover, pky( M.Ί) > \β', (4) λ^ < ORMβ; and

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(5) if β < j < a, then Mβ agrees with MΊ below Xβ. If βis a phalanx of creatures, say B = (((Mβ,kβ) \ β < a) , ((vβ,\β] \ β < a)) , then we set Λίf - Mβ, degβ(/?) = kβ, v(β,B) = vβ and λ(/?,β) = λβ. We also set lh(B) = a + 1. Notice that, because of (3), the λ(/?,β)'s are determined by the z/(/?, #)'s and the ΛΊ|'s. If B is a simple phalanx in the sense of §6, then B becomes a phalanx in the sense of 9.6 if we set kβ = ω for all β < /Λ(β), and ι/(/J, B) = X(β, B) for all β + 1 < lh(B). The notion of an iteration tree on a simple phalanx, as defined in §6, extends in an obvious way to phalanxes as defined in 9.6. Definition 9.7. Let T be an iteration tree of length θ + 1 on a phalanx B of length a + 1. Then: (i) (a) for β such that lh(D such that lh(D) = θ + l and (a) M$ = Mτβ and degp(/?) = degr(/?) for all β<θ, (b) ι/(/?,l>) = ι/()9,T) and X(β,Ί>) = X(β,T) for all β < θ. A realization of a phalanx B will be a family of realizations of the creatures occurring in B. We shall demand that these realizations agree with one another in a certain way. In order to explain this agreement condition, we now recall the terminology associated to "resurrection" in §12 of [FSIT]. Let M be a creature ωa = ORM , and t < ω. Suppose t = 0 if M is not a premouse. Let ωX < OR^ . Set

and

!, fci+i) = lexicographically least pair {/?, k). such that (\,0)<(βi,ki) 0. Also, letting ωa = ORM , {flkO^) I λ < /?Λ (/?,*)
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§9. A general iterability theorem

The following lemma is proved in §12 of [FSIT]. We gave its proof in a typical special case in Lemma 2.6 of these notes. Lemma 9.8. Let M be a creature of C with index (*,£), let ((βe,ke) \ e < i) be the (tf, λ) dropdown sequence of M. Then there is a unique 7 < ξ such that Jβ^ is a creature of C with index (^,7). Now let M be a creature of C with index (t,ξ), and let ωX < ORM . We define the (.M,tf,£) resurrection sequence for λ as follows. Let {/?, k) be the last term in the (ί, λ) dropdown sequence of M. If {/?, k) = (λ,0), then the (M,t,ξ) resurrection sequence for λ is empty. If (β,k) φ (λ,0) (so that k > 0), then let 7 < ξ be unique so that J$* = Cjk(Λ/*7), as given by 9.8. Let

be the canonical embedding. Then the (-M,i,£) resurrection sequence for λ is {/?, fc,7, π)~s, where s is the (<£fc_i(t), s° this is indeed a legitimate inductive definition. Now suppose M is a creature of C with index (t, f ), ωλ < ORM , ((βe,ke) \ e < i) is the (tf, λ) dropdown sequence of Λί, and {{<5e,4,7e, TΓe) | e < 5) is the (M,t,ξ) resurrection sequence for λ. As explained in §12 of [FSIT], we can find stages i < eι < e2 < < et _ι = s such that for 1 < j < i — 1, (δej , ίej) = π β j _ι o π β j _ 2 o - - o We set eo = 0, and interpret "7Γ eo _ι o o TΓQ" as standing for the identity embedding; this makes the equation just displayed true for j = 0 as well. Set σ

i-j

=π

e3 ° TΓβj-1 O

O 7ΓQ

so that is an 4j — 1 embedding, for 0 < j < i — 1. In order to simplify the indexing a bit, we set r, _j = jej for 0 < j < i — 1. Notice that &, _j = tej . Thus, setting p = i — j, we have that for 1 < p < i,

is afcp- 1 embedding. Let us set Resp = ίfc p -ι(Λ/V p ) Definition 9.9. In the situation described above, we call (σp,Resp) the pth partial resurrection of X from stage (/,£).

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The partial resurrections of λ from stage (t,ξ) agree with one another in the following way. For 1 < p < i, let

Then one can check without too much difficulty that KI > «2 >

> «t ,

p < q => σp \ κq_λ = σq \ κq-ι ,

and

p < q => Resp and Res? agree below sup(σqκq-ι) . Definition 9.10. In the situation described above, we call (σ, Res) the complete resurrection of X from (M,t,ξ) if and only if (a) the (M,t,ξ) resurrection sequence for X is empty, and (σ,Res) — (identity, M), or (b) the (M,t,ξ) resurrection sequence for X is nonempty, and (σ, Res) = Notice that in either case of 9.10, Res is a creature of C with index (k, 7), for some (7, k) <\ex (£,*)• ^ ^es ^s C-exotic, then 9.10 (a) must hold. Of course, the notions associated to resurrection can be interpreted in any coarse premouse 72., using C^ , and not just in V '. We shall do this in the following. Let (σ, Res) be the complete resurrection of A from (M,t,ξ). Suppose Jχ* is active, which is a case of particular interest. If ( t , ξ ) — (0,λ), then M = J*Λ = Res, and σ is the identity. Otherwise, (λ, 1} <jeχ ( ζ , t ) , so (βi , *ι) = (λ, 1). It follows that Res = Λ/"7 for some 7 < ξ, and σ : jf* -^ λfΊ is a 0-embedding. Definition 9.11. Let B be a phalanx of length a + 1. Then a realization of B is a sequence ((TlβyQβ,πβ) \ β < <*) such that β (1) for all β < a, (Kβ,Qβ,πβ) is a deg (β)- realization ofM%, and β (2) if β < 7 < α, and τ is the unique ordinal ζ such that (deg (/?),£) is πβ an index of Qp in C , and Xβ = X(β,B), and (σ^,Re/) is the complete resurrection of πβ(Xβ) from (Q β , degβ (β) , r) , and v$ — v(β,B), then (a) V*e = Vf\ and V^\ C V^\, for μ = σ" o *β(Vβ), (b) Resr agrees with QΊ below σ@ (c) (σ& o πβ) \ Xβ = πΊ \ Xβ, and If £ = ((Έ,β, Qβ, πβ) I β < a) is a realization of B, then we write li^ for Ίlβ, etc. Definition 9.12. Let B be a phalanx of length α + 1, S a realization of B, and T a putative iteration tree on B. Let a + 1 < 7 < Ih T, and let β < a + 1 and βTj. We call a pair ( P , σ ) an S-reahzation of M^ if and only if

§9. A general iterability theorem

94

(1) (Kεβ,P,σ) is a degτ(7) realization of M*, (2) if Qβ has index (degβ(/?),£) and P has index (degτ(j),θ) construction ofR,^, then θ < ξ, and

in the

(3) if Dr Π [/?,7]τ - Φ and degr(T) = degβ(/J), then P = Qβ and β =σoιβ,Ί

π

Definition 9.13. Let Ί be an iteration tree on B, and b a maximal branch of T such that DΊΠ& is finite. Then an ε-realization ofb is just an ε-realization of ΛΊ^, where 7 = sup 6 and S is the putative iteration tree of length 7 + 1 such that S \ 7 = T \ j and b — {η \ ηSj}. We say b is ε-realizable iff there is an ε-realization o f b . We can now state the main result of this section. Recall that a cutoff point of a coarse premouse (M, E, δ) is an ordinal Θ £ M such that (VθM, E, δ) is a coarse premouse. We say that M has a cutoff points if the order type of the set of cutoff points of M. is at least a. Theorem 9.14. Let B be a hereditarily countable phalanx, and let ε be a realization of B such that Vα < lh(β) (R,ea has δπ<* cutoff points). Let T be a countable putative normal iteration tree on B. Then either (1) T has a maximal, ε-realizable branch, or (2) T has a last model M^, and this model is ε-realizable. Proof. Fix £o, a realization of BQ as in the hypotheses, and T a putative normal iteration tree of countable length θ on BQ . We shall consider no iteration trees but T in the proof to follow, and so we set Mβ = M*β , Eβ = Ej, vβ = ι/(/?,T), λβ = λ(/?,T), and deg(/?) = degr(/?). Let n* : θ —> ω be one-one, and set n(α) = inf{π*(/?) | a - β

or aTβ} .

Clearly aTβ => n(α) < n(/?), and for λ a limit < 0, n(λ) is the eventual value of n(β) for all sufficiently large βT\. Notice that if n(a) = n(β), then aTβ or /?Tα or α = β. Also, for 6 a branch of T, 6 is maximal O> sup{n(α) | a £ 6} = ω .

For a,β < θ, we say α

survives at

β

& [a - β V (αT/2 Λ n(α) = n(β) Λ n

< 7 < β Λ 7 ^ (α, /?)τ) =^ (

c

It is easy to see that if a survives at β and β survives at 7, then α survives at 7. Also, if α survives at 7 and aTβTj, then α survives at β and /? survives

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at 7. One can also easily see that if λ is a limit, then all sufficiently large βT\ survive at λ, and that for 6 a branch of T, 6

is maximal

^ V α £ δ Ξ / ? E & (α < /? Λ α doesn't survive at /?).

Let Ih (Bo) = α 0 + 1. For each β < α 0 , letting A! = degβ(/?), choose a cofinal Yβ C pk(M®°) such that 7Γ0° is rΣfc+i elementary on parameters from Yβ. Next, for α 0 < /? < 0, we define y/? by induction. If T-pτed(β) = 7 and β £ Dτ and deg(/?) = deg(γ), then set Yβ = iΊβ"YΊ. If T-pτed(β) = j but β £ Dr or deg(β) < deg(7), then set Y^ = ^"(Λ^). Finally, if β is a limit ordinal < 0, let Y^ = common value of ^Y7 for all sufficiently large jTβ. The idea here is that in a copying construction beginning from £Q, Y/? is the subset of Mβ on which we expect rΣk+i elementarity of the copy map, for k = deg(β). We call a k realization (7£,Q,τr) of Mβ a (k,Y) realization just in case π is rΣfc+i elementary on Yβ. A realization Σ1 of Φ(T f α +1) is a Y realization just in case V/? < α((^, Q^, π|) is a (deg(/?), Y) realization of Mβ). All realizations we consider in the proof to follow will be Y-realizations. Let α < θ and let (72., (J, π) be a deg(α) realization of Λ4α We shall define a tree {/ = ί7(α,7J,Q,π). Roughly speaking, 17 tries to build a maximal branch 6 of T such that α £ 6, together with a realizing map σ for Λ4jf which extends TT. More precisely, we put a triple ({/?o,...,/?n},{^o,...,^n),(Qo,...,Qn}) into ί7 just in case (1) /?<> = α, 9?0 = π, and Q0 = Q, and for all i < n, (2) βiTβi+ι and /?,- does not survive at $+1, (3) indπ(Qί+ι) < indπ(Qa and Dr Π (A, β+ι]τ ^ 0 iff indπ(Qί+1) < ind^(Qi); moreover, if Dτ Π (/?,-, A +I]T = <^ and deg r (A) - degr(β+ι), then Qt = Q, +ι, and (4) (Λ,Q t+ι,^t+ ι) is a (deg(/?;+ι), Y) realization; moreover, if Dr Π (A, ft+ι]τ ^ Ψ and deg(β) = deg(^>ι), then φ{ - φi+ι o ίj t) p t+1 Suppose that ((/?; | i G ω), (^ | f E ω), (Q< | f 6 ω)) is an infinite branch of t/(α,7i,Q,π). Set 6 = {T? | 3i(τyT^)}; then (1) and (2) guarantee that r 6 is a maximal branch of T such that α £ 6. By condition (3), D Π 6 is finite, and Q, is eventually constant as i —* ω, say with value Q^. Condition (3) also guarantees Qoo = Qo = Q in the case £>r Π (6 — (a + 1)) — φ and deg(α) = deg (b) (i.e., deg(r ) = deg (α) for all η E 6 - (α + 1)). Finally, let y G M'ζ, and let & be large enough that Dτ Π (6 — βk) = ^, deg(/?fc) = deg (b), and y = ^βkb(x) ^OΓ some x E Mβk. We then set σ(y) = φk(x) By condition (4), σ is a well-defined weak deg (b) embedding from Mb into Qoo Moreover, if Dr Π (6 — (a + 1)) = φ and deg (α) = deg(6), then π = σ o ί£6. So if for some α < α 0 ) ί7(α,^β°,Oβ°ιίrS0) nas an infinite branch, then conclusion (1) of 9.14 holds. We therefore assume henceforth that for all a < α 0 , !7(α,ft£ 0 ,Q£ 0 ,π£ 0 ) is wellfounded. Notice that U(ajπεa°,QεQ0,πεaQ) belongs to 7έ£°, and has size < δn<*Q in ^°.

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§9. A general iterability theorem

Notice that if a < 0, then there are only finitely many 7 < θ such that α < 7 and T-pred(γ) < a. and T-pred(γ) survives at 7. [If not, then we can Γ fix k < n(θί) such that k — n(j) = n(T-pred(7)) f° infinitely many 7 such that T-pred(7) < α < 7. Fix two distinct such 7's, say 70 and 71. Then 70 and 71 are T-incomparable, yet 71(70) = ^(71). This contradicts the definition of n] For α < β < 0, we define c(α, β] = |{7 I β < 7 <θ Λ T-pred(7) < a Λ T-pred(7) survives at 7} | . Definition 9.15. Let 7 < θ = Ih Ί, and lei £ be a realization of Φ(T \ 7). We say S has enough room iff ^ ex. < 7 (a) ^(αsT^jQαj^α) f's wellfounded, and (b) ft£ Λ α s ω rvnk(U(a,'R,*ίJQ*l,π*)) + c(ct,Ί) cutoff points. Definition 9. 16. Let a <J<Θ; then a is a breakpoint atj iff whenever β is a successor ordinal such that a < β < 7 and T-pred(β) < a, then T-pred(β) does not survive at β. We can now prove our main lemma, which concerns the extendibility of realizations of the phalanxes determined by initial segments of T. Lemma 9.17. Let C*Q < a < η < θ, and let £ be a realization ofΦ(T \ α + 1) such that S has enough room. Then: (1) Suppose a. is a break point at η. Then there is a realization T of Φ(T Γ η + 1) such that T \ a + I - £ , F has enough room, and Ίlζ E #£. (2) Suppose that for some δ < a, δ survives at η, and let SQ be the largest such ordinal δ. Then there is a realization T of Φ(T \ η + 1) such that f \ <50 — £ \ όo, T has enough room, and (a) Ίlζ = ftfo and ιndπϊ (Qζ ) < ind*** (Q*a), (b) Dr Π (<*,η]τ φφ^ md^(Qζ) < ιndπ«(Qεa), and τ

τ

r

ε

(c) if D Π (<x,η]τ = Φ and deg (a) - deg (η), then Q Q = Qζ and Proof. By induction on η. First, supposing 9.17 known for η < 7, we prove it for 7 + 1. So let α 0 < α < 7 + 1, and let £ realize Φ(T \ a -h 1) and have enough room. Let β = T-pred(7 + 1). We shall ultimately consider two cases in the construction of the desired T realizing Φ(T \ 7 + 2): the case that for some δ < α, δ survives at 7 -f 1, and the case that α is a break point at 7 + 1 and β does not survive at 7 + 1. Ostensibly there is a third case, the case that α is a break point at 7 + 1 and β survives at 7 + 1, but this case reduces easily to case one. For in this third case, a < β < 7 + 1. Since a is a break point at β, induction hypothesis 9.17 (1) gives us a Q realizing Φ(T \ β+ 1), having enough room, and such that S — Q \ a + 1 and 7£^ E 1iεa. Now case one gives us an T realizing Φ(T \ 7 + 2), having enough room, and such that T \ β = Q \ β and nζ+l = *R?β. Clearly, T is as required in 9.17 (1) with 77 = 7 + 1.

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The desired F will come from a realization H of Φ(T \ j -f 1) which we now define. The definition depends on which of the two cases we are in. Case 1. For some δ < α, δ survives at 7 + 1. Let <50 be the largest such δ. Since β — T-pred(7-f 1), <5o = β or (δQTβ and δQ survives at β). Let G = £ if δ0 = /?, and otherwise let £7 be the realization of Φ(T ί /?+ 1) given by our induction hypothesis 9.17 (2), with η = β. Since /? survives at 7 + 1, either β = 7 or β is a break point at 7. [If Γ-pred(£) < β < ξ < 7 and T-pred(£) survives at ξ, then n(β) > n(ζ) = n(T-pred(£)), so β doesn't survive at 7 + 1.] Let H = Q if β — 7, and otherwise let Ή. be a realization of Φ(Ί \ 7 + 1) such that Ή. \ β + I = £ as given by our induction hypothesis 9.17 (1), with η — 7. Notice that in any case, TΪft = KQβ - K£a. Also, if Dr Π (<*,/%• = ^ and degr(α) = degr(/?), then Q* = Qεa and ττ£ = ττ£ o ί^. Finally, indπ?(Q£) < indπ*(Q£), and if Dτ Π (α,/?]τ ^ <£, then indπ?(g£) <

2. a is a break point at 7 + 1, and β does not survive at 7 -f 1. In this case, α is a break point at 7. If a = 7 we let Ή = S . If a < 7, we let 7ί be the realization of Φ(T f 7 -f 1) given by induction hypothesis 9.17 (1). In either case, we have 8 = Ή. \ a -f 1 and 7£^ C ft£. Now, using W, we produce the desired T realizing Φ(T \ j + 2). We shall have to consider the case split above again later, but for now we can run the two cases simultaneously. In order to clean up our notation a bit, we set (Qη,Kη,*η)

= (Q^i^i^) &Γ all η < J.

Let j — deg(7), and let Q7 have index (j,ξ) in C7^. Let (σ7, Res7) be the complete resurrection of ττ 7 (λ 7 ) from (<27, j', ^), as computed in 7£7, of course. Since 7 > αo, λ7 = Ih E^ . If λ7 = OR/^, then as usual we set π 7 (A 7 ) = ORQ^. Claim 1. If η < 7, then σ7 f ^(λ^) = identity. Proof. Since Φ(T f 7 + 1) is a phalanx, definition 9.6 guarantees that cardinal of MΊ and pj(MΊ) > λη. Since π7 is a weak j-embedding, is a cardinal of QΊ and pj(QΊ) > πΊ(λη). Also, π 7 (λ^) < ττ 7 (A 7 ). It that all projecta associated to the (j, ττ 7 (λ 7 )) dropdown sequence of >π 7 (λ,). Set F — σΊ o πΊ(E^) — last extender of Res7 ,

λ^ is a ^(λ^) follows QΊ are D

where if ResΊ is a bicephalus we choose the extender interpreting the same predicate symbol that EΊ interprets in MΊ. We wish to consider Ult(Q* +1 ,F), where Q*+1 is the creature of C7^ we shall now define. Let n = deg(/?), and ((?7o> ^o), , (ηe, ke)) = the (n, \β) dropdown sequence of and set

§9. A general iterability theorem

98

for 0 < i < e. The following claim relates these to the (n, πβ(\β)) dropdown sequence of Qβ. The claim is slightly complicated by the fact that πβ is only a weak n-embedding. Claim 2. The (n, πβ(λβ)) dropdown sequence of Qβ is (a) {(^(770), fc0)> , (M7?*), ke))

if κ

e < Pn(Mβ),

(b) {(71^(770), fco),..., (πβ(ηe), ke))~u, where u = φ oτ u = (η,n) for ωη = M if κe = pn(Mβ) but (ωηe,ke) φ (OR *,n), and (c) {(π/jfao), *o), , (M»7e-ι), *e-ι))~t*, where w = <£ or tz = (πβ(ηe), ke) Q (ωιj,n), for ωij = OR * if (u;r/e, t e ) = Remark. Note that κe = pn(Mβ) in case (c). If e = 0, then n = 0 = &0 and ft ηQ = Xβ = ωXβ = OR . The (n, τr0(λ0)) dropdown sequence for Qβ is then Q/9 {(OR ,0)}, which falls under case (c). Remark. The u = φ case in (c) would not be necessary if πβ were a full n-embedding. The claim follows quite easily from the fact that πβ is a weak (n,Yβ)embedding. For (a), notice that π'βpn(Mβ) < pn(Qβ) Recall that πβ preserves cardinals, so that if for example ωηe < OR

β

then Mβ \= Vγ >

ηe(pω(Jf) > Λ.(J^)), and thus Qβ \= Vγ > πβ(ηe)(pω(Jf) Let μo = crit(.Z£^), and let

> *β(κe))

e+1 if μo < κe, least j s.t. KJ < //o > if κ e ^ A*o • Notice that since KQ — λ^ > //QJ ί' > 0. Because T is maximal,

if if

i < e, i = e + l,

and j / . -v ( ki — 1 deg(7+l)=( „

if if

{

t < e, -e+l

Let (σf, Resf) be the ith partial resurrection of λ/? from (Q/?, n, r), where Q0 has index (n, r) in C7^, if this resurrection is defined. (The resurrection is undefined if i = e + 1, and defined if i < e by claim 2. If i = e, (σf, Resf) is undefined just in case (ωηe,ke) — (OR^, n) and the conclusion of (c) of claim 2 holds with u = φ.) Now let

The Core Model Iterability Problem /n*

s

is defined , otherwise

_

7+1

99

I Qβ

{

σf identity

if Resf is defined , otherwise .

Thus, in any case, σo(πβ \ Λ4* + 1 ) is a weak deg(γ + 1) embedding from Mj+ι into <37+ι Moreover, σ o πβ is ^deg^+iHi elementary on Z, where Z = Yβ if 7 + 1 £ £>r and deg(γ + 1) = deg(£), and Z = universe of Λ<*+1 otherwise. Set t = deg(γ+l). C/αzm 2.5. σ o π/j is a weak ^-embedding which is rΣfc+i elementary on

Proof. Assume first that Resf is defined, so that i < e, deg(γ -f 1) = ki — 1, and σ = σf is a full fcj — 1 embedding. Looking at claim 2, we see that in all cases the domain of σ is J^β, ^ since we cannot have the situation in (c) with i = e and u = φ. But M.^+1 = Jηt embedding. In fact, if ωηi < OR

ft

β

, and πβ \ Λ"ί*+1 is a weak ki — 1

, then π^ f M*+1 is fully elementary,

and if ωηi = ORMβ , then fci < n, so π/j f -^7+1 is a weak fci embedding. It follows that σ o (π^ f -M*.^) is a weak Ar, — 1 embedding from M^+ι into <37+ι. Assume next that Resf is undefined. Then either i — e + 1, or we have the situation in (c) of claim 2 with u = φ. In either case, deg(γ + 1) < n. Also Λ4* +1 = Mβ, Q7+ι = Qβ, and σ=identity. Since πβ is a weak n-embedding, σ o πβ is a weak deg(γ H- 1) embedding from Λ47+1 into <37+ι Let (σ^, Res^) be the complete resurrection of π^(λ^) from (Q^, n, r). Let -0 be the complete resurrection embedding for σ(πβ(λβ)) from the appropriate tuple. (This tuple is (Q 7 +ι,ft,τ) if Resf is undefined, and (Q*+1, Art — 1,77) where Resf = (£fc t _ι(M/) π/? otherwise.) Then

and

7

C/αz'm 3. ψ \ (sup(σ o π^ /Ci_ι)) = identity. /. Suppose first that Resf exists, so that i < e and σ = σf . From claim 2 and the fact that πβ is a weak n-embedding we see that πβ(κi-ι) s is the projectum associated to the (i — l) t element of the (n, πβ(Xβ)) dropdown sequence of Qβ. As we remarked earlier, ψ is therefore the identity on sup(σf "πβ(κ,i-ι)}> and this implies the claim. Suppose next that Resf is undefined, so that either i = e + I or i = e and (c) of claim 2 holds with u = φ. In either case the projectum associated to the

100

§9. A general iterability theorem

last term of the (n, πβ(λβ)) dropdown sequence of Qβ is at least sup(τr^'/q_ι). Thus σ& \ sup(7Γ0/q_ι) = identity. But ψ = σ^ and σ = identity, so this implies the claim. D Now let

•{

Otf)"^ 1 OR^**

1

if >ς+ιN/4 otherwise.

Claim 4. μ\ < λ/?, and if μ\ = OR largest cardinal of M^+l.

Ύ+1

exists,

then Λi7_|_ι — J^

β

and μo is the

AΎ

Proof. If β = 7, then (μj) exists (is < λ 7 ) since EΊ has index A 7 on the MΊ sequence. Also, Λί7+1 is the shortest initial segment of Λi 7 over which a subset of μo not in Jχ 1

MΊ

is definable. Thus μ\ — (μo )

M* Ύ+1

= (^o )

M J λγ ^

< ^7,

1

and λ7 < OR^ ** , which yields the claim. Now let β < 7. We have μo < vp < \β, and Xβ is a cardinal of ΛΊ 7 . Also P(μ0) Π MΊ = P(μ0) Π J^Ί = P(μ0) Π j£ft - P(μ0) Π Λf* + 1 . It follows that μi < Xβ. If μi = OR^**1, then as λβ < OR^**1, Aί*+1 = J^* and μo is the largest cardinal of M^+I. Π My From the proof above we see that if β < 7, then μ\ — (μ^) . Also, claim 4 implies μi < /q_ι. If «, _ι = A/? this is obvious. Otherwise κt _ι is a cardinal of Jχ β, since it is a projectum of some Jη β with η > Xβ. Since μo < κ>i-ι by the choice of i, μi < /ct _ι. The next claim shows that Res7 and Q7+ι have the agreement required for an application of the shift lemma. Claim 5. (a) Res7 agrees with Q7_|_ι below sup(σ o π^'μi), 7 (b) σ o ττ7 \ μi = σ oπβ \ μ\. Proof. Subclaim A. Q7+ι and Res^ agree below sup(σ o πβ'1'μi), and σ o πβ \ μ\ = "0 o (7 o TΓβ f μi. Proof. This follows at once from claim 3 and the fact that μ\ < /q_ι. Subclaim A yields claim 5 at once in the case β = 7, so let us assume Subclaim B. If β < 7, then Res^ and QΊ agree below sup(σoττ^ / / μι), and /

h

K

Proof. Recall that *ψ o σ o πβ = σ^ o πβ. This sub claim therefore follows at once from the fact that Ή is a realization of Φ(T \ 7 + 1); see clause 2 of 9.11. Notice here that μ\ < Xβ by claim 4. Subclaim C. If β < 7, then QΊ and Res7 agree below sup(σoπ^ // μι), and 7 ττ7 \ μi = σ o π7 f μ\.

The Core Model Iterability Problem

101

Proof. μ\ <\β, and σ o πp \ μι = π7 \ μi, so sup(σ o πβ"μι) < πΊ(\β). By claim 1, QΊ and Res7 agree below π7(λ/3), and σ7 is the identity there. Together, A, B, and C yield claim 5. D Let us define K = σΊ o π7(μ0) = & ° Kβ(μo) = crit F . Thus («+)
Proof, μo < Vβ because Ύ is an iteration tree, so K = σ o πβ(μQ) = σβ o π^(μo) < σ^ o πβ(vβ). The claim now follows from the fact than Ή. is a realization; cf. 9.11 (2) (a). D Now Res7 is a creature of C?*Ύ with an index of the form (0, η) in C^Ύ . Therefore 7£7 has background certificates for the countable fragments of F. Let (TV, G) =

some (σ7 o ττ7(ί/7), ran(σ7 o ττ7)) — certificate for F, as computed in HΊ .

Since Ult(ΛΓ, G) is closed under α -sequences, σ7 o ττ7 f VΊ £ Ult(7V, G). Let us fix 6 G [lh G]<ω and a function ΰ •-»• π(w) mapping [/c]'6' into V^

Ύ

so that

Suppose for a moment that case 1 of 9.17 applies, that is, that OQ < a and <$o survives at 7 + 1. It follows that c(η, 7 + 2) < c(r/, 7 + 1) for all η such that β < η < 7. Therefore, for such 77, TLη has ω rank(ί/(r7,^,Q,7,πf;)) + c(η, 7 -f 2) + 1 cutoff points, because H has enough room. Let ξη be the last of these cutoff points, and set 7£*

=

transitive collapse of Eu\\v^(Vσ^η(l/η) U {6**, Qη, πη} U σ» o π,(A,)) , as computed in ΊZη ,

102

§9. A general iterability theorem

and (Qη> πη) = image of (Qη^η) under collapse. Notice that ft* is coded by an element of VσT)*π , s +1 , which is a subset of ft7 because Ή. is a realization. (Note here ση o ττ^(λ^) has cardinality ση o ^(i/,,) in ft,,.) So (ft*, Q*, TT*) G ft7 for all η such that /? < η < 7. Set

Then 7ί* G HΊ since 7£7 is closed under ω sequences. Clearly, Ή* is coded by 72. a member of Vσ^π , s+1 . It is easy to check that (H \ β)"W* is a realization of Φ(T f 7+ 1). It may not have enough room as a realization of Φ(T \ 7 + 1), of course, because we have dropped an ordinal on coordinates η such that β < η < 7 Since G is σ7 o π7(ι/7) + 1 strong in ft7 , W* G Ult(7V, G). We may suppose our finite support 6 was chosen so that for some function ΰ «—»• W*(ϋ) mapping [/c]W into V^,

If there is no δ < a such that δ survives at 7 + 1, then W* is undefined. Let k = deg(7 + 1), and Q'Ί+ι = Ultfc(Q* +1 , F). The ultrapower makes sense by claim 6, and it is wellfounded because F has background certificates in 7£7, and 7£7 is ω-closed. Let r : MΊ+ι —»• <97+ι be given by the shift lemma, that is,

(Here, if k > 0, then σ o π^(/ r>g ) = fr,σoπβ(q) for all terms r G Sfcfc and q G A4 7+1 . For simplicity, we shall use the k = 0 ultrapower notation.) By 7 the shift lemma, Q7+ι agrees with QΊ below σ o π7(λ7), and r \ \Ί = 7 σ o π7 ί A 7 . Also, r is a weakfc-embeddingwhich is r Σσ+ι elementary on YΊ+I. We now use the countable completeness of G to reflect τ below K. Let {xn I n < ω} be an enumeration of the universe of ,M7+ι , and let xn = — jwf * — [άn, /nlj^*1 where αn G [ι/7]<ω. Set an = σ7 o π τ (α n ) and /„ = σ o πβ(fn), so that r(xn) = [αnj/n]?;+1. For notational reasons, we shall sometimes regard the component measures EC of an extender E as concentrating on order-preserving t : c —* crit(E'), so that "for E a.e. t : c —* crit(E'), ί G X" means that there is a set Y G J^c such that whenever i : c —>• crit(E') is order preserving and ΐ"c G Y, then t G -X". Let us write /(/?, Q,σ) just in case σ is rΣfc elementary on its domain, σ is r Z^+i elementary on dom σ Π Y/j, V i < fc(p*(A ^(Λ(^ί/j)) = Λ(0))» and *"Pk(Mβ) C Λ(Q). Thus, if TT : Λί^ -. Q, then π is a weak k -embedding from Mβ into Q which is r Σk+i elementary on Yβ & (V finite F C Mβ)I(β, Q, π ί F).

The Core Model Iterability Problem For t : (b U αo

103

U α n ) —»• K order preserving, let


for all i < n. Claim 8. Let n< ω and let c = b U α0 U U α n . Then there is a set Wn £ Gc such that whenever t : c —> K is order preserving and £"c £ W n , (ii) if i;+1(y) € dom y>?, then y>?(ς+1(y)) = σ o (iii) if a?n < λγ, then #(*„) = τ(t"6)(arn), and (iv) if arn = λ 7 , then ¥>?(*«) > »(*"&)(*„). Proof. We first show that (ii) holds for G a.e. ί : c —»• K. Let i^+i(y) — %i = — JWf *

—

[άj, /j]^ "H"1, where i < n. Then /<(ΰ) = y for (-Ey)s, a.e. ϋ. Since σ o πβ and σ7 o π7 agree on P(μo), this means that /i(ΰ) = σ o τr/?(j/) for F0t a.e. ϋ. The set of such ΰ is in ran(σ7 o π7), so fi(t"a,i) = σ o π/j(2/) for G a.e. t : c —>• «. Since ^(ί^+ι(y)) = fi(t"a-i)ι we are done. Next, we show (i) holds G a.e. First, let p(vo - -vn) be an rΣk formula. Then

iff for F a.e. t : I) α,

iff for G a.e. t : c

Notice, for the third equivalence above, that the appropriate set of ΰ is in the range of στ o ττ7, so that Fι ι and G\ \ a give it the same measure. Second, we show φ" is rΣk+i elementary on YΊ+ι Π {XQ z n }, for G a.e. t : c —* K. Notice here that Y7+ι = i^+iZ, where σ o πβ is rZ"jk+i elementary on Z. [If 7 4- 1 g £>r and A: = deg(γ + 1) = deg(/?), then Z = Yjj, and σ o πβ = πβ is rΣΆ ^i elementary on y^ because W is a Y-realization. Otherwise, Z is the universe of Λ4* + 1 , σ is a full fc-embedding, and TT/? is at least rΣk+i as a map from Λ<*+1 to J^^, where ωη = πβ(OR Π Λί*+1).] Thus, if we set Y7+ι Π {XQ '"Xn} = {i^+ι(yo), »*7+ι(ym)}, then we have for all rΣk+ι formulae p •^7+1 N p[*7+ι(yo) σ

«7+ι(ym)] iff Λί* +1 |= p[y0

iff O^+i N /^[ ° τ/?(yo) iff for G a.e. t : c —» K

σo

ym]

104

§9. A general iterability theorem

This completes the proof of (i). We now prove (iii). Let xn < λ 7 , and assume first that λ7 = ι/7. Since xn < ι/7,

SO

because of the agreement between σ o πβ and σ7 o π7. Letting d = αn U {σ7 o π7(zn)}, this means that for F a.e. t : d —> «, fn(t"an) = ί(σ7 o τr 7 (z n )). Because the set of all ί"cί for which this equation holds is in ran(σ7 o π7), we get that /n(*"fln) = *( z/7. We then AΊ*

get a function g G Vrμo_j7j[1"1 such that

[αn, p]^ Ύ+1 = some wellorder of z/ of order type xn . Applying the shift lemma map τ to this fact, with g = σoπβ(g) — σ7 oττ 7 (y), Q*y+l α [ n, 9]p

= some

7

7

wellorder of σ o ττ7(z/) of order type σ o ττ7(a?n) .

But now Fd agrees with Gd on all sets in ran(σ7 o π7), whenever d G [σ7 o <ω π 7 (ι/+ l)] This implies

It follows that for G a.e. ί , g(t"an) is a wellorder of order type t(σΊ o π7(a?n)). We also have that for F a.e. t, hence for G a.e. ί, g(t"an) has order type f(t"an}. So we get that /(2 x/ α n ) = ί(σ7 o ττ 7 (x n )) for G a.e. tf. Now we can finish the proof of (iii) as in the first case. We leave the proof of (iv) to the reader. The main point is that [αn, /n]^ ^ σ7 o ττ7(λ7).

The Core Model Iterability Problem

105

— jw<* l — 7 7 (We may assume fn E VμQ+\ , so that fn = σ oπ 7 (/ n ) is in ran(σ oττ 7 ).) This follows from the agreement between F and G; the proof breaks into the cases VΊ — λ7 and VΊ < λ7 as did the proof of (iii). This completes the proof of claim 8. We can now finish the proof of 9.17 in case 1, the case that for some δ < α, a survives at 7 + 1. For β < η < 7, let σ* be the complete resurrection embedding in ft* for Q* from ττ*(λ^). Then σ* f λ,, = σΊ o π7 \ Xη, and σ η ° ^(^η) ^σ7 ° ^(^f?) for all 77 < 7; this one sees from the construction of H*. This agreement is a fact about H* and σ7 o π7 \ (A 7 + 1) in Ult(7V, G); by Los' theorem we get a set X G G& such that for all ΰ G X

and

for /? < η < j. Here (σj o π*)(ϋ) = σ;(ΰ) o ^(tϊ), where W;(δ) (7£*(u),Q*(u),7Γ*(iί)) and cτ*(iί) is the complete resurrection of π*(u)(Xη) from Qη(ΰ) in ft^(i/). We can also arrange that for ΰ G X, 7

σ o ττ7 Γ μo = ττ(^) Γ μo ,

because σ7 o ττ7" μo is just a countable subset of /c = crit G, and G is countably complete. Finally, we can arrange that for ΰ G X, ftJjW 'ιas ω - rank(W(77, ^(ϋ), g;(u), π;(δ)) + cfo, 7 + 2) cutoff points. Now let Wn be as in claim 8, for all n < ω, and let t : b U Un<ω an -^ K be order preserving and such that t"b E X and t"(b U αo U α n ) G Wn for all n. Such a t exists because G is countably complete. Set φ(xn) — fn(i//an) for all n < ω, and

It is easy to verify that f fulfills the requirements of 9.17 as a realizations of Φ(T \ 7 + 2) in case 1. Now let us prove 9.17 in case 2, the case that α is a break point at 7 -f 1 and β does not survive at 7 + !. From claim 8 and the countable completeness of G we get Claim 9. For G& a.e. ϋ, there is a ( deg(7 + 1), Y) embedding φ : Λί7+ι —»• <57+1 such that (a) φ \ λ7 = π(ϋ) \ A 7 , (b) φ(\Ί) > π(ϋ)(A 7 ), (c) ^ o ΐ * + 1 =σoπβ. Now if w and φ are as in claim 9, then (7 + 1,<^,Q7+1) is a node of the tree W(/?,ft*,Qjgf,π^). (The fact that β does not survive at 7 + 1 is relevant here.) Moreover, ^(7 + 1,7£?, Q 7 + χ, <£>) is isomorphic to the subtree

§9. A general iter ability theorem

106

of W(/?,ft^,<2^,τr^) consisting of nodes below (7 + l,p,Q 7 +ι). It follows that Tϊfi has an ω rank(ZY(7 + l,ft^,<2* + 1 , <£>)) + 0(7 + l,γ + 2) + 1 72.^ β

point 77. Working in ft^, we can form a Skolem hull of Vη 72, Ή

^r(o)(λ )

U

st

cutoff

containing

{Φy+ι^}> closed under ^-sequences and having size < K. The Ή

'R,β

collapse of this hull belongs to Vκ

= V**. This gives us

Claim 10. For G& a.e. ϋ, there is a triple (ft, Q, v?) such that (a) (ft, Q, ττ(ΰ)(λ 7 ). By the axiom of choice in N, there is in TV a function f ( ΰ ) = (ft(ϋ), Q(ϋ), <£>(ϋ)) which picks, for each ΰ in the relevant G^-measure one set, a triple satisfying claim 10. Let

It is easy to see that T is a realization of the phalanx Φ(T \ 7 + 2); the necessary agreement of models and embeddings comes from parts (d) and (e) of claim 10. Part (c) of claim 10 implies that T has enough room. As case 2 governed our definition of Ή, £ \ α+1 -U \ α + 1 and ft^ C ft£. It follows that T \ a + 1 = S [ α + 1, and since ft^+1 E Ult(ΛΓ,G) C ft*, we have 1lζ+1 E ft£ Thus f witnesses the truth of 9.17 (1). This finishes the successor step in the inductive proof of 9.17. Now let η be a limit ordinal, and αo < <* < η- Let βTη, where β is large enough that α < β, β survives at 77, Dr Π [/?, 77]^ — ψ, and deg(/J) = deg(τ7). Let (βn I n E ω) be such that /?o = β, and βnTβn+ιTη for all n, and 77 = sup{/?n I n E ω}. Let £ be our given realization of Φ(T \ a + 1). Suppose first that α is a break point at 77. Then α is a break point at /?, so by induction we have a realization ^Ό of Φ(T \ β + 1) which has enough room. We also get f§ \ a + 1 = 8, and ft^° E ft£. Now suppose fn realizing Φ(T) \ (βn + 1) is given. Since βn survives at βn+ι, and between βn and βn+i there is no dropping in model or degree, our induction hypothesis gives a realization Fn+ι of Φ(T \ /3n+1 -f 1) having enough room, and such that .Fn+l n+1

ί βn ~ ^n

TΓx, o z'T Λ + l . Let Pn-fl βnβn

\ βn, H'β"

= ^β^

an

d Q^ = Qβ**[ >

an<

^ ^Γ ~

The Core Model Iterability Problem

107

where for x £ Mη we define π(x) by

It is easy to check that f witnesses 9.17 (1) for a and η. Next, suppose <5o < α is largest such that <5o survives at η. Let (βn \ n < ω) be such that Γ-pred(/?0) = <$o, βnTβn+ιTη for all n, and sup n /3 n = r?. By induction hypothesis 9.17 (2) we get a realization FQ ofΦ(T \ β0 + 1) which has enough room, and such that T§ \ SQ = £ \ SQ and 7^° = 7£fo, and Q^o° is related as required to (?fo, and ττfo = *fa°iJ0jQ if this is required. We can then use induction hypothesis 9.17 (2) repeatedly as in the last paragraph, and we easily get 9.17 (2) at η. This completes the proof of Lemma 9.17. D We can now easily complete the proof of 9.14. Suppose first that Θ is a limit ordinal. For 0 < i < ω, let α,+ι be defined by

(Recall that c*o = lh(Bo) — l ) Clearly, α t < α t +ι and α, +ι is a break point at 0, for all i. We may suppose that n* was chosen so that n*(c*o) = 0, which means that α 0 is a break point at θ. But then 9.17 (1) gives us a sequence (Fi \ i £ ω) such that /Ό = £o> J7* is a realization of Φ(T f α, + 1), and ftαί+ϊ € 7£α,*> f°Γ aΉ 2 <ω This is a contradiction. Next, suppose 0 = 7+1. We may suppose n* is chosen so that n*(ηf) = 0, which implies that β survives at 7 whenever βTj. But then 9.17 (2) clearly implies that M^ is £o-realizable, as desired. D Theorem 2.5 obviously follows from 9.14. (While 9.14 was only proved for normal trees, whereas 2.5 was stated for linear compositions of normal trees, we can nevertheless take care of such "almost normal" trees by applying 9.14 (2) repeatedly to their normal components.) The iterability of the exotic creatures of C which we used in the proof of 1.4 also follows immediately. This represents all the iterability we used in §1 - §5. c It remains only to prove Theorem 6.9, which states that if K \= "There c are no Woodin cardinals", then every K generated phalanx B such that Ih B < Ω is Ω + 1-iterable. We shall now sketch the minor modifications of the proof of 9.14 which yield this result. First, the reflection arguments of §2 show that it is enough to prove the following: let π : M —»• Ke be elementary, where M is countable. Let B be a hereditarily countable phalanx which is (Σ1, Λ4)-generated, where Σ is the strategy of choosing unique cofinal wellfounded branches. Let T be a countable, normal, putative iteration tree on B. Then either T has a cofinal wellfounded branch or T has a last, wellfounded model. So fix π, ΛΊ, and B with these properties; we shall show that there is a realization S of B such that Vα < Ih (B) (Hεa has <5π« cutoff points). The desired conclusion concerning T then follows from 9.14.

108

§9. A general iterability theorem

Since Ω is measurable, we can find ξ < Ω such that π : Λ4 —-»• λίξ is elementary. Let η be such that (Vη, £, Ω) is a coarse premouse having Ω + Ω cutoff points, and let SQ = ((Vη , £, β), Λ/f , π). Thus £Q is a realization of Λ4. Now fix a. < lh(β), and let «S be a countable iteration tree on M such that M^ is an initial segment of M^, the last model of *S, and such that 5 has no maximal wellfounded branches. Such a tree S exists because B is (Σ,M) generated. We have by definition 6.7: if a + I < /Λ(β), then Vj λ(α, β)), and if α + 1 = /A(5), then

We now apply a slight variant of 9.17 to find the desired realization of ί,*ί) M% Let n* : 7 + 1 -» u; be 1-1 and n*(0) = 0. Let n : 7 + 1 —> ω be defined from n* as in the proof of 9.14, and interpret "survives" and "break point" relative to n as in 9.14. Thus 0 is a break point at 7. For β < 7, and (ft, Q, σ) a degs(β) realization of Λff , let £/(/?, ft, Q, σ) be defined as in the proof of 9.14: it is the tree of attempts to build a maximal branch b of S and realize M% appropriately. Since <S has no maximal wellfounded branches, U(β, ft,<2,σ) is always wellfounded. For f a realization of Φ(S \ £), let us say that f has more than enough room just in case Mβ < ζ(R,% has <5π? + ω - rank([/(/?, UT , Qβ, πj")) + c(β, 7) cutoff points), where of course c(/?, 7) is defined as in the proof of 9.14. So EQ has more than enough room. It is clear that the proof of 9.17 works equally well when "more than enough room" replaces "enough room" in its hypothesis and conclusion. Since 0 is a break point at 7, this version of 9.17 gives us a realization T of Φ(S) such that £0 = F \ 1 and ft^7 has δny cutoff points. Since F is a realization of Φ(5), Q^ agrees with QQ = Λ^ below σoπ(i/(E 1 ^)) and π^ f ι/(£?f ) = σ o π \ v(E$ ) - σ o π f ι/(i?o), where σ is the appropriate complete resurrection embedding (bringing K(EQ) back to life). Now whenever β < a and \(β,B) is defined (i.e. β+l< /Λ(β)), λ(/?,β) is a cardinal of M and ι/(ί^) > λ(/?,β). Thus π(λ(/?,β)) is a cardinal of A'c and σ f ττ(λ(/?, β)) + 1 is the identity. This implies that πζ \ (λ(β, B) + 1) = π f (λ(/3,β) + 1), whenever /? < α and λ(/?,β) exists. Let us set ft£ = ft^, π^ = π^ Γ ^α> and Q£ = Q^ if Ml = M* , and Q£ = <(Mα) otherwise. Doing this for all α < ίft(β), we obtain a realization S of B which has enough room; the agreement properties of £ follow from the fact that if α + 1 < Ih B, then ττ£ f (λ(α,β) + 1) = π f (λ(α,β) -f 1) and Qεa agrees with Λ/"ξ below ττ(λ(α,β) + 1), and from the corresponding facts when α + 1 = /ft(#). From 9.14 we get that the tree T on B is well-behaved, and this completes the proof of 6.9.

References

[D] [DJ1] [DJ2] [DJ3] [DJKM] [FMS] [H] [Hj] [KS] [Kul] [Ku2] [IT] [Ml] [M2] [M?] [FSIT] [WCP] [Sch] [TM] [CMWC] [PW]

A. J. Dodd, The core model, London Math. Soc. Lecture notes, vol. 61, 1982. A. J. Dodd and R. B. Jensen, The core model, Ann. Math. Logic 20 (1981), 43-75. A. J. Dodd and R. B. Jensen, The covering lemma for K, Ann. Math. Logic 22 (1982), 1-30. A. J. Dodd and R. B. Jensen, The covering lemma for L[U], Ann. Math. Logic 22 (1982), 127-155. A. J. Dodd, R. B. Jensen, P. Koepke, and W. J. Mitchell, The core model for nonoverlapping extender sequences, to appear. M. Foreman, M. Magidor, and S. Shelah, Martin's maximum, saturated ideals, and non-regular ultrafilters, Ann. of Math. (2) 127 (1988), 1-47. K. Hauser, The consistency strength of projective absoluteness, habilitationsschrift, Ruprecht-Karls-Universitat, Heidelberg, 1993. Greg Hjorth, 772 Wadge degrees, Ann. Pure and Applied Logic 77 (1996), no. 1, 53-74. A. S. Kechris and R. M. Solovay, On the relative consistency strength of determinacy hypotheses, TAMS, 290 (1), 179-211. K. Kunen, Some applications of iterated ultrapowers in set theory, Ann. Math. Logic 1 (1970), 179-227. K. Kunen, Saturated ideals, J. Symbolic Logic 43 (1978), 65-77. D. A. Martin and J. R. Steel, Iteration trees, JAMS 7 (1994), 1-73. W. J. Mitchell, The core model for sequences ofmeasuresl, Math. Proc. Cambridge Philos. Soc. 95 (1984), 228-260. W. J. Mitchell, Σ\ absoluteness for sequences of measures, in Set Theory of the Continuum, H. Judah, W. Just, H. Woodin eds., MSRI publications, no. 26, Springer-Verlag 1992. W. J. Mitchell, The core model for sequences of measures II, unpublished. W. J. Mitchell and J. R. Steel, Fine structure and iteration trees, Springer Lecture Notes in Logic 3 (1994). W. J. Mitchell, E. Schimmerling and J. R. Steel, The weak covering lemma up to a Woodin cardinal, to appear in Ann. Pure and Applied Logic. E. Schimmerling, Combinatorial principles in the core model for one Woodin cardinal, Ann. of Pure and Applied Logic, 74 (1995) 153-201. E. Schimmerling and J. R. Steel, Fine structure for tame inner models, J. Symbolic Logic, vol. 61 (1996), 621-639. J. R. Steel, Core models with more Woodin cardinals, to appear. J. R. Steel, Protectively wellordered inner models, Ann. of Pure and Applied Logic, vol. 74 (1995), 77-104.

110 [SVW] [SW] [To] [W]

References J. R. Steel and R. Van Wesep, Two consequence of determinacy consistent with choice, Trans, of AMS 272 (1982), 67-85. J. R. Steel and P. D. Welch, Σ\ absoluteness and the second uniform indiscernible, to appear. S. Todor£evic, A note on the proper forcing axiom, Contemporary Mathematics 95 (1984), 209-218. W. H. Woodin, Some consistency results in ZFC using AD, in Cabal Seminar 79-81, Springer Lecture Notes in Mathematics, vol. 1019 (1983), 172-198.

Index of Definitions Definitions not numbered in the text are indexed here by the number of the theorem, lemma, or definition immediately preceding; thus "1.4 if." indicates an unnumbered definition occuring in the body of the text after Theorem 1.4. 1.1 A certificate for M 1.2 M is countably certified 1.2 ff. Mt,λft,foτζ <Ω 1.3 ff. Kc 1.4 ff. the stationary class AQ 2.1 T is α-short 2.4 ff. almost normal iteration tree 2.4 ff. almost fc-maximal iteration tree 2.9 0-iterable, ^-iteration strategy, (ω,0)-iterable, (ω,0)-iteration strategy 3.1 coiteration of M with λf 3.1 ff. (Σ, Γ) coiteration of M with λf,P
5 5 6 7 8 10 12 12 23 25 25 25 26 28 29 29 30 36 36 37 42 43 44 44 44 45 45 45 46 51 73 77 77 83 89 89

112

Index of Definitions

9.2 ff. 9.2 if. 9.3 9.4 9.5 9.6 9.7 9.7 ff. 9.8 ff. 9.9 9.10 9.11 9.12 9.13 9.13 ff. 9.14 ff. 9.14 if. 9.15 9.16

(j, ξ) is an index of M in C, ind(ΛΌ coarse premouse M c C , the K construction of M (U, Q, π) is a ^-realization of M M and λί agree below j phalanx of creatures ΦOΠ, the phalanx derived from T the (tf, λ) dropdown sequence of M (M,t,ξ) resurrection sequence for λ the pth partial resurrection (^,Resp) the complete resurrection (σ, Res) realization 8 of a phalanx B ^-realization of M* ^-realization of a branch 6 of T cutoff point of a coarse premouse Oί survives at β Y-realization S has enough room α is a break point at 7

89 89 90 90 90 90 91 91 92 92 93 93 93 94 94 94 95 96 96

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