Fine Structure and Iteration Trees

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Lecture Notes in Logic William J. Mitchell John R. Steel

Fine Structure and Iteration Trees

M-- - tl

f

l

u

Springer-Verlag

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Lecture Notes in Logic Editors: K. Fine, Los Angeles J.-Y. Girard, Marseille A. Lachlan, Burnaby T. Slaman, Chicago H. Woodin, Berkeley

William J. Mitchell John R. Steel

Fine Structure and Iteration Trees

Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest

Authors William J. Mitchell Department of Mathematics University of Florida Gainesville, FL 32611-2082, USA E-mail: [email protected] John R. Steel Department of Mathematics University of California, Los Angeles Los Angeles, CA 90024-1555, USA E-mail: [email protected]

Mathematics Subject Classification (1991): 03E45,03E55,03E10,04A10,04A15 ISBN 3-540-57494-8 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-57494-8 Springer-Verlag New York Berlin Heidelberg 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 1994 Printed in Germany SPIN: 10061642

46/3140-543210 - Printed on acid-free paper

CONTENTS

Section 0. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

Introduction Good Extender Sequences Fine Structure Squashed Mice Ultrapowers Iteration Trees Uniqueness of Wellfounded Branches The Comparison Process Solidity and Condensation Uniqueness of the Next Extender Closure under Initial Segment The Construction Iterability References INDEX OF DEFINITIONS INDEX

Page 1 5 10 28 34 47 58 69 74 89 96 99 108 125 126 128

§0. INTRODUCTION In these notes we construct an inner model with a Woodin cardinal, and develop fine structure theory for this model. Our model is of the form L[E], where E is a coherent sequence of extenders, and our work builds upon the existing theory of such models. In particular, we rely upon the fine structure theory of L[β] models with strong cardinals, which is due to Jensen, Solovay, Dodd-Jensen, and Mitchell, and upon the theory of iteration trees and "backgrounded" L[@] models with Woodin cardinals, which is due to Martin and Steel. Our work is what results when fine structure meets iteration trees. One of our motivations was the desire to remove the severe limitations on the theory developed in [MS] caused by its use of an external comparison process. Because of this defect, the internal theory of the model L[E] constructed in [MS] is to a large extent a mystery. For example it is open whether the L[B\ of [MS] satisfies GCH. Moreover, the use of an external comparison process blocks the natural generalization to models with infinitely many Woodin cardinals of even the result [MS] does prove about L[E], that L[E] \= CH + R has a definable wellordering. Our strategy for making the comparison process internal is due to Mitchell and actually predates [MS]. The strategy includes taking finely calibrated partial ultrapowers ("dropping to a mouse") at certain stages in the comparison process. Thus to define the internal comparison process and prove it succeeds one needs fine structure theory. Of course, fine structure theory requires a comparison process, but fortunately we are led not into a vicious circle, but into a benign helix: that is, an induction. The whole of what follows can be viewed as a long inductive proof that a certain construction yields a model L[E\ whose levels have certain fine structural properties. Among those properties is a strong local form of GCH. We have as a corollary that if ZFC 4- "There is a Woodin cardinal" is consistent, then so is ZFC + "There is a Woodin cardinal" + GCH. But our interest is not so much in this relative consistency result, which can probably be proved more easily using forcing, as in the inner model L[E] itself, and the fine structure techniques which should eventually decide many questions about L[E] and similar models containing more Woodin cardinals. The model L[E] and its fine structure theory are likely indispensable for proving certain relative consistency statements in which the theory hypothesized consistent does not directly assert the existence of large cardinals. For example the following conjecture is widely believed to be true:

Conjecture. If ZFC + "There is an ω^saturated ideal on ω\ * is consistent, so is ZFC + "there is a Woodin cardinal".

2

W. J. MITCHELL AND J. R. STEEL

Of course, the conjecture is really that the relative consistency is provable in Peano Arithmetic. Shelah has proven the converse relative consistency result. Mitchell ([M?]) has proved the conjecture with its conclusion weakened to "ZFC + Ξ/c(o(κ) = jc++) is consistent". The present paper is a step toward extending Mitchell's arguments so as to prove the full conjecture. What we lack at the moment is a method which does not use large cardinals in V for showing that a certain L[E] type model is sufficiently iterable. This "Core model iterability" problem is one of the key open problems in the area. Its solution should lead to a proof of the conjecture, and to much more. The notes are organized as follows. In §1 we introduce potential premice, which are structures having some of the first order properties of the levels of the model we eventually construct. Perhaps the most notable thing here is that the extender sequence EM of a potential premouse (ppm) Λ< may contain extenders which do not measure all sets in M. In general, an E on EM measures only subsets of crit E constructed in M before the stage at which E was added to EM. This tactic, which is due to S. Baldwin and Mitchell, greatly simplifies fine structure theory. Section §2 studies definability over potential premice. We introduce the rΣn hierarchy, a slight variant on the usual Levy hierarchy. We follow Magidor and Silver in introducing Skolem terms so as to avoid proving rΣn uniformization, and in working directly with rΣn formulae rather than master codes and iterated rΣi definability. We show that being a ppm is preserved under the appropriate embeddings. Finally, we introduce projecta, standard parameters, solidity and universality of parameters, cores, and soundness. These are standard fine structural notions, with the exception of solidity. The analysis of §2 is not appropriate for a certain sort of ppm, the "active type III" variety. In §3 we modify it slightly so that it suits these ppm. This leads to an annoying case split in the details of many arguments, a split which we have sometimes ignored. One important feature of the Baldwin-Mitchell tactic is that all levels of the model we build will be completely sound. Ultrapowers of sound structures can be unsound, but all proper initial segments of the ultrapower will be sound. So it suffices to consider only ppm all of whose proper initial segments are sound. These we call premice. In §4 we define the rΣn ultrapower Ult n (Λί, E) of a ppm M by an extender E measuring all sets in M and satisfying crit E < p^. We prove Los' theorem and show that the canonical embedding is rΣn+ι elementary if M is n-sound. We show that if p^l < crit E, M is n-sound, and E is "close to being a member of M", then the canonical embedding preserves the n + 1st standard parameter, provided this parameter is solid. This result explains the importance of solidity. Section §5 introduces iteration trees and n-iterability. It also proves the Dodd-

FINE STRUCTURE AND ITERATION TREES

3

Jensen lemma on the minimality of iteration maps, which is a key tool in our work. In Section §6 we investigate the uniqueness of wellfounded branches in iteration trees. Theorem 6.1 is a straightforward generalization of the uniqueness theorem of [MS]. Theorem 6.2 is a fine structural strengthening of theorem 6.1 which takes considerably more work to prove. Theorem 6.2 has the important consequence that all the iteration trees we care about have at most one cofinal wellfounded branch. Section §7 proves a comparison lemma for iterable premice. The lemma is never used in what follows, but the method of proof, the comparison process, is used throughout. In §8 we prove our main fine-structural result: the n + 1st standard parameter of an n-sound, n-iterable premouse is n + 1-solid and n + 1-universal. The method of proof traces back to Dodd's proof that GCH holds in the model of [D]. The method also gives a useful condensation result, Theorem 8.2. In §11 we finally construct (assuming there is a Woodin cardinal in V) some iterable premice. We in fact construct a model L[E] with a Woodin cardinal all of whose levels are ω-sound and u -iterable premice. §9 and §10 are devoted to some preliminary lemmas which guarantee that the construction of §11 puts enough extenders on E that we do indeed get a model with a Woodin cardinal. Section §12 shows that the construction of §11 produces an iterable structure L[E] by associating to any iteration tree on L[E] an iteration tree on V and then using the results of [MS]. We did the work described here during 1987-1989 and wrote it up in a set of notes which has been informally circulated since October 1989. This paper is essentially identical to that set of notes. We wish to thank Kai Hauser, Mitch Rudominer and Ernest Schimmerling for reading those notes carefully and bringing errors to our attention. Since 1989 the theory described here has advanced in several ways. In the spring of 1990, Steel found a solution to the core model iterability problem mentioned above, and with it was able to extend the work of [M?] to the level of a Woodin Cardinal [S?a]. He used this to show that if there is a saturated ideal on wi, together with a measurable cardinal, then there is an inner model with a Woodin cardinal. The measurable cardinal should not be necessary here and its use may indicate a weakness in the basic theory of [S?a]. As expected, other relative consistency results have come out of this work. Some of these use the weak covering lemma for the model of [S?a], which was proved in late 1990 by Mitchell [MSS?]. Schimmerling [Sch] has investigated the combinatorial set theory of the model L[E] described in this paper. He showed that Dωι holds in this model, and that weak DΛ holds for all K. It is open whether L[E] satisfies V/cD K . Schimmer-

4

W. J. MITCHELL AND J. R. STEEL

ling was able to combine his work on DΛ with the ideas of [S?a], [MSS?] and arguments of Todorcevik and Magidor in order to show that the proper forcing axiom implies that there is an inner model with a Woodin cardinal. Finally, Steel ([S?b], [S?c]) has extended the theory presented here to models having more Woodin cardinals.

§1. GOOD EXTENDER SEQUENCES DEFINITION 1.0.1. Let * < i/ and suppose that M is transitive and rudimentarily closed. We call E a (κ,v)- extender over M iff there is a nontrivial Σo-elementary embedding j : M —> N, with N transitive and rudimentarily closed, such that cήt(j) = /c, j(κ) > v, and E = {(α, x) I α 6 [v]<ω Λ x C [κ]carda Λ x G M Λ α G j(x)} . We write /c = crit JE, ϊ/ = Ih £. Remark. If the requirement that N be transitive is weakened to ι/-f 1 C wfp(W), where wfp(JV) is the wellfounded part of JV, then we call E a (/c, ι/) pre- extender over M. We are interested in this weakening for the purely technical reason that if v < ORM , then pre-extenderhood is expressible by a simple first order sentence about DEFINITION 1.0.2. Let S C OR and suppose E = (Ea | α G S) is a sequence of extenders (Ea over some Mα). Then

where

Remark. So J^ = Jjf ^ α . The ordinals in 5 are the stages at which extenders are activated. Let E = (Ea I α G 5) be given. Let α G 5 and Ea be a («,τ) pre-extender. We put (α, 6,6) G Ea

iff

[5 < T Λ 6 is a function

Λ dom 6 = K Λ 6 G j f

fα

Λ α = J?α Π ([£]<α; x ran 6)]. We then define

{

/ ΎE

It/

^

1

E

h t\ΐ\

if

« t* "-^ I C* I

/ T /? f- |τι h -. Iv ι ti "^ I C*

o \ *Jθc I

Λ/ ^ C^

11

C* \t *^

\f ll

-^ ^ C* *»* vZ *^

Remarks, (a) In the sequences ϊ? we shall consider, Ih Ea = Qf for all α G 5. (b) Of course if α G 5 then the sets first order definable over (jf, G, E \ α, £*«) are the same as those first order over j£, but the latter structure is a better starting point for fine structure.

6

W. J. MITCHELL AND J. R. STEEL

(c) We are interested only in the case that each Ea, (a £ S) is an extender over j£ f o r . Thus EQ is to measure no sets constructed after it is activated, a very useful idea due to Baldwin and Mitchell. As a consequence, the subsets of Jβ in J^+1 are just those first order definable over jf . In earlier setups one needed also "measure quantifiers" coming from the Ea, ot < /?, to define the new sets. This complicated the fine structure substantially. One consequence of the Baldwin-Mitchell idea is that we can work entirely with structures which are strongly acceptable in the sense of Dodd and Jensen [DJ1]. DEFINITION 1.0.3. A structure («7<ί, € , . . . ) is strongly acceptable iff whenever β < a and

P(*)n(Jf+1-JΪ)ϊ0

r

then J$+ι f= card(J^) < K. Notice that if j£ is strongly acceptable and j£ \= α/c+ exists" then j£ [= "P(κ) exists and P(κ) C J*+" In particular, GCH holds in strongly acceptable structures. It is a basic fact in the fine structure of L that ( J% , £) is always strongly acceptable. On the other hand, in the usual stratification of L[μ]t J%+2 l8 no^ strongly acceptable (for /c = crit μ). Let E be a (/c,ι/) extender over M. For K < ζ < v, we say ξ is a generator of E iff whenever α 6 [ξ]n and / E M and / : [κ]n -> /c, ζ / [α,/]^ (that is, {(uι t!n9t!n+ι) I /(ui Un) = ^n+ι] & Eau{ξ}) (Equivalently, ζ is a generator of E iff ζ is the critical point of the natural embedding from Ult(M, E \ Thus K is the least generator of E. All other generators are strictly greater than (κ+)M . Note that the property of being a generator of E depends only on E and P(κ)M , and E determines P(κ)M . Let η be the larger of (/c+)M and sup{£ +1 : ξ is a generator of E }. Then 77 < j/, since M ^= κ+ exists in the models of interest. We call η the natural length of E. Suppose Ult(M, E) is wellfounded, where the ultrapower is formed using functions in M. Regarding UltίM,^) as transitive, let i: M -*• Ult(M,£) be the canonical embedding. Then UltίM,^) ^ "η+ exists" since η < i(/c). We will use the ordinal (η+)υit(M>E) as the index of E in sequences of extenders. The trivial completion of E is the (/c, (r7+)ult(M'E)) pre-extender G consisting of pairs (α, x) such that Λβ

c [κ]cardβ Λ x G M Λ α G i(x) .

Then E \ a = G t α, where α = inf(lhJ5;,lhG). We now record some of the main properties of the extender sequences we shall consider:

FINE STRUCTURE AND ITERATION TREES

7

DEFINITION 1.0.4. A sequence (Eβ \ β E S) is good at a if it satisfies the following five clauses: (1) jf is strongly acceptable, and if a £ 5, then (2) Ea is a (/c, α) pre-extender over jf ία for some /c such that jf for |= /c+ exists, (3) (bounded generators) Ea is the trivial completion of Ea \ v, where i/ is the natural length of Ea. (4) (coherence) i(E \ a) \ a + I = E \ a where i : jf t* -+ Ult( jf '«, £α) is the canonical embedding, and (5) (closure under initial segment) Let i/ be the natural length of Ea. If η is an ordinal such that (κ+)Jaι < η < v and η is the natural length of Ea \ η, then one of (a) or (b) below holds: (a) There is 7 < α such that EΊ is the trivial completion of Ea \ η. (b) η E 5, and there is a 7 < α such that π(E \ η)Ί is the trivial completion of EQ \ η, where π: J^η -> \Jlt(J^η9Eη) is the canonical embedding. DEFINITION 1.0.5. A potential premouse (ppm) is a structure of the form Jβ , where E is good at all a < β. DEFINITION 1.0.6. A ppm Jβ is active if β £ dom E\ otherwise it is passive. DEFINITION 1.0.7. If M = jf

is active then ι/M is the natural length of Eβ.

Remarks, (a) Activity is determined by the similarity type of the ppm. (b) Condition 3 implies every β < a is represented mod Ea by a function with support C ι/M, so that Ea \ ι/M determines all of Ea. We include the extra coordinates just so that the functions witnessing coherence will be trivial (essentially projections on a coordinate) which helps show coherence is preserved by Σo ultrapowers. (Here M = J$ .) (c) The ultrapowers in the definition are "Σo ultrapowers" , that is, formed using a functions belonging to the model in question. Note JJ^ is always passive, hence amenable, so that we can move its predicate. Notice that since Ea is only a pre-extender, Ult(jf rα,£α) may not be wellfounded. However, α 4- 1 C wfp (Ult), which is enough to make sense of conditions (3) and (4). (d) Let E be good at α and i : jf rα -> Ult( j£ ία, Ea) the canonical embedding. (

rα)

Let z/ < a be the natural length of Ea. By coherence, Jα = jf . Since α = ϊ/+ in Ult(jf rα,£"α), which is strongly acceptable, there are no cardinals

8

W. J. MITCHELL AND J. R. STEEL

> i/ in J$ rα \ So there are no cardinals > v in jf . The ordinal i/ itself may be a successor ordinal. It is easy to see that if i/ is a limit ordinal, then in fact ι/ is a cardinal, both in jf and Ult(jf ίθf,£Λ). (e) Let K = crit EQ. By (3) there is a map of (P(/c) Π J f ) x [ι/]<w onto α, the map being in j£+ι Thus α is not a cardinal in (f) For the good E we construct, Ea is an extender over L[E \ α], which is strongly acceptable, and α = ι/+ in both L[£? f α] and Ult(L[J5? \ a],Ea) This in fact follows from the definition of goodness if we can iterate from J** via Ea and its images OR times (and preserve wellfoundedness). (h) It might be hoped that alternative (b) of the initial segment condition could be dropped, but we suspect that if L[E] is to have a Woodin cardinal, or even lots of strong cardinals, one cannot demand this stronger version of the initial segment condition. The initial segment condition is crucial in the proof that the comparison process terminates (cf. §7). We need some form of it as an axiom on our extender sequences in order to get a decent theory going. Sy Friedman has suggested that it might be possible to eliminate this clause if the sequences are indexed by letting an extender be Ea where a is the double successor of the natural length of E in the ultrapower by £?, instead of using the single successor as in this paper. We do not know whether this idea can be made to work. Notice passive ppm are amenable. For active ppm we have a weaker property, which we call weak amenability. DEFINITION 1.0.8. Let J* be an active ppm, and K = crit Ea. We say J* is weakly amenable iff whenever (Aβ \ β < /c) € jf and V/?3n < ω(Aβ C [κ]n) and η < α, then

The proof of the next lemma is well-known (due to K. Kunen?). Lemma 1.1. Every active ppm is weakly amenable. PROOF. Let (Aβ \ β < /c) be as in the definition of weak amenability, and »:jf ί o r ->Ult(jf '<«,£„) = Ult the canonical embedding. Let F = Ea Γ\ ([η]<ω x {Aβ \ β < K}) where η < a. Now (i(Aβ) I β < K) € Ult so as Aβ = i(Aβ) Π [«;]<", F = {(α, Aβ) I α e [η]
FINE STRUCTURE AND ITERATION TREES

By strong acceptability, since α is a cardinal in Ult, F E J^ FGjf.

9

By coherence, D

Remark. Let ϊ/ be the natural length of Ea. It is easy to see that if A C v and A € Ult(jf ,£α), then A can be computed from Ea Π (H<α; x {Aβ \ β < K}) for some sequence (Aβ \ β < K) £ J f of subsets of [/c]n, n < α;. Thus in fact a is the least ordinal 7 such that #α Π ([ι>]<w x {A0 | /? < K}) 6 J^ for all such (Aβ I /? < K) G J^. This is the motivation for condition (3) of good at α: we don't add an extender until we have weak amenability. Remark. For ξ < (/c+)J« define 7^ to be the least ordinal 7 such that Ea Π ([v]<ω x jf)

€ jf. The ordinals 7^8 are cofinal in a. To see this, let A G

Ult(j£, Ea) be any subset of the natural length ι/ of Ea, and let A = [α, /]E« Then A can be computed from (Ea)a^{ Aη : η < K} where A,, = { ΰ : i; E /(ϋ) }, so that if / € J^ for ί < K:+ of jf then Λ € /^+ι Since the sequence (7^ : ξ < (/c+)J* ) is in j£+l it follows that j£fl (= cf(α) =

§2. FINE STRUCTURE This and the next section explain some basic facts about the fine structure of definability over potential premice. Most of the notions and results here are due to Dodd and Jensen ([DJ4]). There is one important difference, however: because Ea measures only sets in JJ* ^α, the hydras of [DJ4] disappear; moreover, we never have to iterate in order to extend measures. This simplifies the theory considerably. Let M = Jβ . If VM is a successor ordinal, we let JM be the witness to 5(a) or 5(b) with respect to the trivial completion of Eβ \ VM — 1. That is, let G be the trivial completion of Eβ \ η, where η < VM — I is the natural length of Eβ\v". Remark. If η < ι/M - 1, then G = FM \ VM - 1. The proof uses the fact, which is not hard to prove, that if η is a limit of generators and is not itself a generator then η is not in 5. If 5 (a) of the definition of good at α holds, we set ΊM = Ih G = the unique ξ E dom EM such that G = E£* . If 5(b) holds, then η € dom EM and G is on the sequence of Ult(Λί , E*4). We set (fr,
DEFINITION 2.0.1. Let M be an active ppm, and /c = crit F, where F is the new extender introduced by M. Then (a) M is type I iff VM = (*+)" (b) M is type II iff VM is a successor ordinal (ι/M — 1 > (κ+)M follows from the definition of vM)> (c) M is type III iff VM is a limit ordinal and VM > (κ+)M. The definability hierarchy we study in this section is not appropriate for active type III ppm. The reason is explained at the beginning of §3, where we study a different hierarchy appropriate for such ppm. In this section we restrict our attention to ppm which are passive or active of types I or II. DEFINITION 2.0.2. £ is the language of set theory with additional constant symbols j>, 7, /t, additional 1-ary predicate symbol £", and 3-ary predicate symbol F.

FINE STRUCTURE AND ITERATION TREES

11

If M is a ppm, then we interpret C in M as follows: Suppose that the structure M = (jf, 6, E \ α, Ea) is active. Then = E\«, M

μ

= crit EQ

7

if M

is type II

0

if M

is type I

Suppose M = (Ja , E, £ f α) is passive. Then 1

^ = E \ a , F^ = 0

An active ppm is not amenable with respect to its interpretation of F. For this reason, the ultrapower of such a ppm via functions belonging to the ppm may not satisfy Los' theorem for ΣO formulae of £, and we must start our Levy-like hierarchy at a smaller class of formulae. DEFINITION 2.0.3. rΣ0 (or restricted Σ0) is the smallest class of £ formulae containing the atomic formulae of £, all ΣQ formulae of C — {F}, and closed under Λ, V,-*, and bounded quantification over finite sets. [That is, if 0(x,G, v) is rΣ0, so are ψ(G,υ) and φ(G,v), where ψ(G,ϋ) = "G is finite Λ 3x ζGΘ" and φ(G,v) = "G is finite Λ Vx 6 GP] DEFINITION 2.0.4. where φ is

Let θ(υ) be an C formula; then θ is rΣi iff θ =

We could continue and define rΣn, rΠπ for n > 1 by counting quantifier alterα nations. However, we shall reserve "rΣn" and rΠn" for n > 1, for different, more useful classes of formulae. For M a ppm, rΣjJ* is the class of relations on the universe of M definable over M by a rΣn formula (n = 0, 1). If M is passive, then rΣ^1 = Σ^1, the usual Levy class. The relativised classes rΣ^(X), for X C |Λ<|, and ), are as usual. Notice that rΣ^1 is closed under 3 , Λ, V, and bounded quantification over finite sets. The following normal form theorem makes clearer what an rΣi formulae can assert of an active ppm. Lemma 2.1. Ifφ(v) is any rΣi formula then there is a ΣI formula ψ(a,b,δ,υ) of C — {F} such that for all active ppm M (*)

M N Vΰ (φ(v) o 3α, 6, δ[F(a, 6, 6) Λ φ(a, 6, ί, δ)]) .

12

W. J. MITCHELL AND J. R. STEEL

Remark. That is, an rΣi formula asserts there is a "small chunk" of the new extender having a ΣI in (C — {F}) property. PROOF. We show first that if θ(v) is a rΣo formula and φ = θ or φ = -*θ then there is ψ as in (*). This is by induction on θ. E.g. for φ = -*F(x, j/, z), we have

Λ (z is not an ordinal V y is not a function V V d o m y φ d o m δ V ( y = bΛz=δΛa^ E. g. for the "Vx G G" step, let ψ(a> 6, δ, x, G, v) be ΣI in C - {F}; then we can rewrite G finite ΛVxEG3α,&,δ[F(α,6,δ)ΛV<αΛM,G,t>)]

as (**)

3α, 6, δ [F(α, 6, δ) Λ 3fc < ω 3 sequences x, ά, 6, δ of length k

where ^* is the formula Vi < fc(δ, is a function Λ dom&, = domt Λ ran 6, C ran 6 <ω

Λ δi < δ Λ α, = α Π ([ίt ]

x ran δt ) Λ ^(α< , 6, , «, , x,, G, v)) .

Then the formula (**) is equivalent to 3α, 6, δ3fc < ω 3 sequences x, α, 6, δ of length k

Thus every rΣo formula φ can be put in the form given by (*). This easily implies the same is true of rΣi φ. D As a corollary, rΣi satisfaction is rΣi, uniformly over all active ppm: Corollary 2.2. active ppm

There is an rΣi formula 0(t>o, vi) such thai whenever M is an

M \= θ[i, (α0

at)]

iffi

is the Gδdel number of an rΣi formula φ

and M (= φ[α,Q -

α*] .

Of course, rΣi = ΣI satisfaction over passive ppm is also uniformly rΣi. Henceforth we will use the same letter for the Gόdel number of a formula as we use for the formula, so that the displayed line in the corollary would start "Λ4N%, {GO •

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13

Lemmas 2.1 and 2.2 hold also for type III active ppm, setting ΊM = 0 for M type III, but once again we have no use for this fact. Remark. Let M = J& be active. We can construct an amenable structure Λ4* as follows. Define #* to be the set of quadruples (γ,£,α,z) such that M

M

(α > 7 > V ) Λ (μ

<ξ<

Λ (Ea Π α^]«* x jf ) G jf ) Λ ((α, x) G (Ea Π ([γ]<« x jf ))) . Our remark on p. 10 shows that M* is amenable; moreover one can easily see that rΣ^ is the usual ΣI over M*. This fact (which we didn't notice until we had written much of these notes) doesn't seem to simplify the definability analysis to follow. It might be used to give the analysis a more conventional look. For any ppm M , let <Λ/I be the usual order of construction (so if M. = jj[ and Jf = J& where α < /?, then <* end-extends <M). Lemma 2.3. There are ΣI formulae of £ — {F} defining uniformly over all ppm Ja the functions

PROOF. See Dodd-Jensen [DJ1].

D

The formulae of the lemma are in C — {F} hence rΣi. To apply [DJ1], notice every ppm is amenable with respect to its interpretation of E (and in fact EM G

Skolem terms and projecta. Following Magidor and Silver, we shall introduce Skolem terms into our language as a convenience. The existence of the amenable structure Λ4*, which we described following 2.2, shows that rΣf* relations admit rΣi uniformizations, and this means that we could avoid the Skolem terms if we cared to do so. There seems to be no great advantage to either approach. We shall also define the classes rΣ^4 for n > 1, and introduce a predicate 7^ related to rΣn satisfaction. Like Magidor and Silver, we shall work directly with the classes rΣjJ*, rather than with ΣI definability over master code structures. However, T*4 is closely related to the nth master code of M , and its use in constructing ΓΣjJ^ relations makes this difference more apparent than real. DEFINITION 2.3.1. Tn for 1 < n < ω.

£"*" is C together with additional 2-ary predicate symbols

The interpretation T^1, for M a ppm, will be defined shortly.

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W. J. MITCHELL AND J. R. STEEL

DEFINITION 2.3.2. Let θ(ΰ) be a formula of £+. (a) θ is rΣi iff θ is a formula of C which is rΣi in our former sense. (b) θ is rΣn+ι (where n > 1) iff there is a rΣi formula ψ(a,b, v) of C such that DEFINITION 2.3.3. Let φ = φ(vQ,..., v*, Vjb+i) be a formula of £+. Then τφ(vQ,.. .vjk) is the iαsi'c Skolem term associated to y>. Given that we have interpreted φ in a ppm Λ< (which we have not as yet done in general), we interpret τφ as follows: ^M least 6 such that Λi (= φ[ά, b] if such 6 exists 1 otherwise DEFINITION 2.3.4. For n > 1, Skn (the class of level n Skolem terms) is the smallest class which contains rφ for each rΣn formula φ and is closed under composition. DEFINITION 2.3.5. A formula φ of £+ is generalized rΣn for n > I iff φ results from an rΣn formula ψ by substituting terms in Skn for free variables in ψ (The substitution of r into ψ must be such that no variable free in T becomes bound in the resulting φ.) We can now define the predicate T£* for M a ppm; simultaneously, we define the nth projectum p^ of M. DEFINITION 2.3.6.

Let M be a ppm and n > 1. Then

(a) Th^(X) = {(φ,a) \ a G X<ω and φ is a generalized rΣn formula and (b) p? is the least ordinal p C \M\ such that Th^(pU {q}) <£ \M\ for some (c) The predicate T**(a, b) holds if and only if α = (α, q) for some α < p** and g G Λi, and 6 = Th^(α U {q}). Of course, Tn is essentially 3-ary. It's present form is a relic of an earlier version of these notes, where we set T^(α, 6) b = Th^(α). That earlier version led to a problem (showing rΣn ultrapowers give rise to rΣn+ι elementary embeddings). Remark, p** = ORM is possible. The definition above is by induction on n, as (a) for n depends upon (c) for 1 < m < n. We define the classes rΣ^, rΠ£*, rΔ^, for M a ppm, in the obvious way. The relativised and boldface versions of these classes are also defined in the obvious way.

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15

Notice that if p** < ω, then in fact p%* = 0, and rΣ^\.l trivializes. We are not interested in rΣj^ when p^ = 0 (although we are definitely interested in rΣ£* in this case). We shall tacitly assume henceforth that, in any discussion of a class of the form rΣ*ly p£* > 0. It is easy to see that rΣ^ is closed under 5, Λ, and V, and that (rΣ^ U rϋ^) C rΣ^j, for any ppm M. Moreover, the closure and inclusion are uniform over all ppm (there is a recursive translation procedure acting on formulae of the appropriate type). It is clear that -^T*4 is rΣ;^ in the parameter pft* , uniformly over all M. It follows that the class of sets definable by rΣn+ι formulae would be unchanged if we modified definition 2.3.3 by allowing allowed any formula of the form Ξα,6(T n (α,6) Λ ψ(a,b, v)) where ψ is a boolean combination of rΣn formulae. A similar argument shows that we could also have restricted ψ to be ΣI in C-{F,E}.

Hulls. DEFINITION 2.3.7. Let M be a ppm, n > 1, and X C \M\. Then

= (^(δ) I r € Skn Λα 6 X<»} , = τr"S^(X)

where it is the transitive collapse,

(The last predicate occurs only if M is active). We shall show n£*(X) is a ppm. To this end, recall the Q formulae of [DJ4]. One virtue of these formulae is that they go down under ΣI embeddings and up under cofinal ΣQ embeddings. We now define the appropriate analog in our situation. DEFINITION 2.3.8. Let M be a ppm, and π : M —> P be an rΣ0 embedding, with P transitive. We say π is cofinal iff (a) V ye\P\3x(yCπ(x)), and M

(b) *"(£+)" is cofinal in π((μ+) ). Recall here μM = 0 if M is passive, so that (b) is trivially true then. If M is active, μM = crit F**. DEFINITION 2.3.9. An rQ formula is one of the form: VzV0 < μ+ 5y3ι/ (x C y Λ (θ < v < μ+) Λ φ(y, v, ΰ)) where φ is rΣi and does not have x or θ free. Interpreted in a ppm Λί, an rQ formula asserts that, in the product order on (μ*)M x \M\ determined by the inclusion order on the factors, there are

16

W. J. MITCHELL AND J. R. STEEL

cofinally many pairs (ί/, y) with an rΣ^ property. (If M is passive, this reduces to asserting that, under inclusion, there are cofinally many y G \M\ with an 1 ^ property.) So we have clearly Lemma 2.4. Let φ(v) be an rQ formula, and π : M —* P. Then (a) 7/π is an rΣi embedding and P \= φ[π(ά)], then M \= φ[a\. (b) If π is a cofinal rΣo embedding and M [= ^[α], then P \= ^

The preservation properties given in 2.4 are interesting because one can say with an rQ sentence: "I am a (passive/active) ppm" . Lemma 2.5. There are rQ sentences φ\, φi, φ$ such that if M is a transitive £-structure, then (a) M ^= φ\ holds if and only if M is passive and (μM , ί / M , jM) = (b) M \=- φz holds if and only if M is active type I and (μM , VM,JM) = (μM,vM,ΊM);

(c) M [= φ% holds if and only if M is active type II and (μM , i/M > jM) =

PROOF. We construct ^3, the other constructions being slightly simpler. We shall use the fact that every rΠi formula can be put in rQ form; we leave it to the reader to check this. By Dodd-Jensen [DJ4] there is an rQ sentence θ\ of C — {F} whose transitive models M are precisely those of the form (jf M ,^EM,FM). We add to 0ι the rΠi sentence of C — {F} stating that M is strongly acceptable and that M

Now we define a rQ sentence #2 asserting that F codes a pre-extender over M M. The pre-extender coded is U{α | 36, ί F (a,b,6)}. The formula 02 is the conjunction of the formulas (i-vi) below: (i) "there are cofinally many (#,7) in the product order on μ+ x OR such that 3α 36(F(α, 6, 7) Λ 6 is a function from μ onto P(μ) Π Jf )." (ii) Vα, 6, 7, α x , 6;, j1 (if F(α, 6, 7) Λ j' < 7 Λ V is a function with dom V = i and ran ό; C ran 6 Λ α7 = α Π ([7/]<ω x ran ό x )> then F(α;, V, 7')). (iii) F(α, 6, 7) => 7 G OR and b : μ -> P(μ) and α C [7]<ω x ran 6 and, letting EC = {x I (c, x) G α}, the E^'s are compatible /i-complete measures on [/ί]cardc "as far as sets in ran 6 are concerned". (iv) F(α > 6,γ)Λήα' > 6, T )=».α = β'. (v) (Normality) (V/ : \μ}n -> /i)(V6 : /i -» (P([μ]n) U P([A]n+1))) pf 6 « /-closed then Vα, ί(F(α, 6, δ) =>• α = (Ee \ c € [ί]<ω) is normal with

FINE STRUCTURE AND ITERATION TREES

17

respect to /)] (where "6 is /-closed" stands for the formula: V A G ran δ Π P([/ι]n)Vι < n{{c*ι . . .αt ,/?,αt>1 . . .αn} | (αι...α n ) G A Λ β = /(αι...α n )} G ran 6). So far, condition (i) is rQ while (ii)-(v) are actually rΠi, and we have asserted enough to ensure that Ult(Λί,F) makes sense whenever M (= (i)-(v), where F = Uία I FM(a,b,6) for some δ,ί}. Normality guarantees ORΛ* C ω/p(Ult), but we must have OR ^ G wfp(Ult) for pre-extenderhood. From condition 3 of goodness at a (a = OR^), we know that we want to assert that [{i'}^, /]£* = α where /(/?) = (β+)M for β < μM. The next clauses in Θ2 do this. (vi) V ordinals 6 > is Vγ >δ Vα, 6 (if F(α, 6, 7) and {(α, 0) | /j? |= card /? < a] = x is in ran 6, then ({ι>, ί}, x) G α). We have to say finally that there is no function "between" /(/?) = /?+ on the z>th coordinate and the projection functions on arbitrary coordinates. (vii) For cofinally many pairs (θ, 7) in the product order on μ+ x OR there are α, 6 and 6 such that F(α, 6, 6) Λ δ > 7 Λ Vn < ω (P([μ]n) Π jf C ran ό) and for all functions / G Jf such that / : [μ]n —*• /i, and for all c G [τ]<ω such that c = {r/i r;n} for some ordinals ?/ι < < ηn with ιjt = i>, and (c, {(ax... α n ) I /(αi ... «„) < (at )J/ }) e a there is an ordinal £ such that 7 < ζ < δ and (cU {£} , {(αi -

αn+ι) | /(«ι

α n ) < «n4-i}) G α .

The formula in (vii) is rQ. To see that if M satisfies (i-vii) then /(/?) = (β+)M, on the ί/M coordinate, represents OR^ in Ult, notice that as μM is a cardinal of M, strong acceptability implies ((α+)J* )M = ((*t)M for αf < μM. We leave to the reader the not entirely trivial fact that any active ppm satisfies (vii). Let 02 be the conjunction of (i)-(vii). If M satisfies θ\ Λ #2, then M satisfies conditions 1, 2, and part of 3 of good at α, for α = OR^. We capture the rest of condition 3 with #3: #3: There are cofinally many 7 G OR such that 3α, 6, δ(F(ay 6, δ) Λ δ > 7 Λ 3f : [μ]n -> μ 3c G [i>]n such that ί> - 1 G c and (cU {7}, {(αi ...*„,/?) I /(αi . . .α n ) = £ Λ jf |= card(^) < αn}) G α . Moreover, /> — 1 is a generator of F; that is Vα, 6, δ V/ : [/i]n —>• /i Vc C z> - 1 (cU {j> - 1}, {(αi . . .α n ,/J) | /(ttl ...<*„) = /?}) g α .

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W. J. MITCHELL AND J. R. STEEL

The formula θ$ is the conjunction of an rQ sentence and an rΠi sentence, so #3 in rQ. One can check that if M \= #3, i>M - 1 is the largest generator of FM. Notice here that if 7 > j> satisfies the displayed clause of #3, then there are no generators between 7 and ι>. Recall that we are working with a type II ppm Λί, so that ύM — 1 exists. We can capture coherence, which is condition 4 of good at α, with an rΠi sentence 04: θ\ just says Vα,fr,ί (F(α,6, δ) =» "α is coherent as far as sets in 6 go"). We omit further detail. Condition 5 is a disjunction of two possibilities, (a) and (b), and we accordingly set 05 = ψι V Vj2 The formula VΊ, asserting that clause 5a holds, is "7 > ι> — 1 and 7 G dom £" and Vα, b(F(a, 6, ι> - 1) =>> a C E^) and Vξ < 7 (ζ a generator of E*.γ =Φ> £ < ί> — 1)." The formula V>ι is rΠi (its third conjunct is the only one not Σo in £ - {F}). The formula ^2, asserting that clause 5b holds, says that 7 = (τy,6,y), where if we set G = [6, g]f. then 77 is the natural length of G and is in dom(£*), the conjunction of the following three formulas holds:

g(u) is on E for (Eη)ι> a.e. ύ < Ih G(ί a generator of G => ζ < v - 1) and finally G ^ [α, /]^ whenever (α, /) is constructed before (6, g). We leave it to the reader to see that the formula V>2 is also The formula θ$ = ψ\ V V>2 captures (5) for the "last" proper initial segment of FM. Together with the Π0 in £ - {F} assertion that EM is good at all β < α, #5 captures (5). M

M

Let φ be the HI assertion that E is good at all β < QR . Then φ Λ Λ, <5 ^ is the desired rQ sentence. This completes the proof of 2.5. ~ D Corollary 2.6. Let M be a ppm which is passive or active of types I or II. Then (a) if π : ft —+ Λί is an rΣi embedding, then Tϊ is a ppm of the same type as M and ^μ*) = μM, π(i/w) = ι/M, and π(7W) = 7^, (b) if π : Λ4 —+ P is either a cofinal rΣo embedding or a *Σι over rΣi * embedding (see the proof of 2.7 for the definition) then P is a ppm of the same type as M, and π(μM) = μτ , π(ι/M) = vτ , and π(yM) = 7^.

The natural embedding π : Ή^(X) -* M is clearly rΣι, so it follows that H^(X) is a ppm of the same type as M. The next lemma shows that in certain circumstances π in fact preserves generalized rΣn formulae.

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Lemma 2.7. Let M be a ppm which is passive or active type I or II. Let U = Ή**(X\ whcre X C\M\ andn>l. Suppose that if n > 2, then

/tfίi < OR" ^3qεx (Th^o^ u {?}) ί and i f n > 3, Men

=» /rfia € X Λ 3g € X (Th£2(/£2 U Let π : 7ί —*• M be the inverse of the collapse. Then (a) H \= φ[ά] iff M )p φ[π(a)] for φ generalized rΣn and a G H (b) for 1 p^l

I

OR™ if no such a exists.

PROOF. For i > 0 and k > 1, we say a formula φ is Σ* over (generalized) rΣ, iff

φ= where φ is a Boolean combination of (generalized) rΣ, formulae and Q* = 3 or V as appropriate. (Here generalized rΣo = We show by induction on i that for 0 < i < n — 1 (i) If φ is Σ n _, over generalized rΣ*, then for all ά € H H\=φ(a}<#M\=φ[κ(a)},

(ii) for 1 < i < n - 1 and α, 6 G H

(iii) for 1 < i < n — 2, (b) of the statement of the lemma holds, and for i = n — 1, (c) holds (iv) if φ is generalized rΣ, +ι, then for all a E H

Proof of (i). If θ is Σ n _, over generalized rΣ, , then the translation procedure mentioned earlier gives us an rΣn formula θ* which is equivalent to θ over all ppm. As ranπ is closed under r£*, we see that if 3x(M ^= θ[xy

20

W. J. MITCHELL AND J. R. STEEL

then 3x G ranπ(Λί [= 0[z,π(6)]). But now π is elementary with respect to all generalized rΣ, formulae (trivially if i = 0, or by the induction hypothesis (iv) if i > 0). So the usual induction on the length of the quantifier prefix in φ gives (i) Proof of (ii). First we observe that for any i > I there is a HI over rΣ, formula 0"(vo> vι) such that for any ppm P

To see this, notice first that there is a recursive function associating to each term τ G Skt a ΣI over rΣ. formula ΘT such that rv[ά] = b iff P |= 0r[ά,6], for all ppm P. For basic r, say T = τφ, let 0 r (u, v) be the formula (^?(ϋ, v) Λ Vu; < t; -ιy>(ϋ, w)) V (v = 0 Λ Vw -^>(ϋ, u>)) . In this case θτ is a Boolean combination of rΣ, formulae. The extension of r ι-> θr to all of Sk, is obvious. Notice second that rΣ, satisfaction is uniformly rΣ, over all ppm. It then follows that generalized rΣ, satisfaction is uniformly ΣI over rΣ, , as well as uniformly Πi over rΣ, , over all ppm. This gives us the desired formula σ. Clause (ii) follows easily from (i) and the existence of σ. proof of (Hi). We first prove clause (b) for t < n — 3. Consider for example the first equivalence. The statement "p^ = ORM" can be expressed M \= Vα G OR VgΞfr σ(α U {?}, 6) where σ is the formula asserting that 6 = Thf(α) from part (ii). This sentence is Πa over rΣ, , so true in M iff true in W as i < n — 3 and we have induction hypothesis (i) at i. A similar calculation gives the second equivalence. Clause (b) for i = n — 2 comes from a similar calculation. If pf* = ORM then as we have just seen this is expressible by a Πa over rΣ, sentence which, since true in M, must go down to Ή by induction hypothesis (i). If pf* < ORM, then by hypothesis pf* and a suitable parameter p are in ran(π). We get M ^= Mb-^σ(ρ^ U {p},fr), which is U2 over rΣ, and thus goes down to ?ί, showing p? < π~l(ρf*). The second implication comes from a similar calculation. Finally, in the case i = n — 1 we must prove (c). Let τr(α) < pftiύ we claim α < p*_lβ For let q€\Ή\. Then τh

£-ι W α ) U {*($)}) = unique c such that 3α, 6(1^1 ^α, 6)Λ α = π(α) U {π(ςf)} Λ b = c)

so we can find b G \H\ such that

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But then Th^.^α U {q}) = 6 by (ii), so Th^α U {q}) G |7ί|, and as q was arbitrary, a < pjlj. On the other hand, if π(α) > p^, then by hypothesis we have a p G ran π such that Let π(g) = p. Then ThJ.^α U {q}) (g |7ί|, so α > p*^. Finally, we prove (iv) at t. Notice first that

6

iff

For suppose T^(a, 6). Then a = (a, q) where a < p?, and 6 = Th*(αU{g}). By (ii), π(6) = τh?(π(a)U{π(q)}), and by (iii) »(<*) < p? . Thus 7jW(»(a),ir(6)). The converse is equally easy. It follows at once that π is rΣn+ι elementary. Suppose for example that M. \= 3α, 6(7i(α, 6) Λ φ(a^ 6, τr(c))). Then applying the proper term in Skn to π(c) we get α, 6 G ran π such that Tf*(a, 6) Λ ^(α, 6, π(c)), and we're done. As the graph of any basic r G Sk, +ι is definable by a boolean combination of rΣt +ι formulae, uniformly over all ppm, we see now that π is generalized rΣ, +ι elementary. This completes the proof of lemma 2.7. Standard parameters and cores. DEFINITION 2.7.1. A parameter of M is a sequence (αo, . . . , α*) of ordinals of M such that αo > c*ι > > αjb DEFINITION 2.7.2. <ιex is the lexicographic wellordering of all parameters (i.e. of all descending sequences of ordinals). DEFINITION 2.7.3. Given a ppm M with p%* < ORM and given q G |Λί|, the kth standard parameter of (Λi, q) is the <ιex least parameter p of M such that We now define two useful properties a parameter might possess, solidity and universality. We shall eventually show that the appropriate standard parameters associated to the levels of the model we shall construct are solid and universal. Solid parameters. DEFINITION 2.7.4. Let r = {αo at) be a descending sequence of ordinals, M a ppm which is passive or active of types I or II, and q G |Λf |, and 1 < k < ω. We say r is k solid over (Λ4, q) iff for all i < ί

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W. J. MITCHELL AND J. R. STEEL

We are interested in the case that r is the kth standard parameter of (M,q). Notice that in this case α^ > p£*, and for any finite s C α, , Th£*(p£< U s U {{αo αt _ι),g}) G |Λ4| simply by the <ιex minimality of r. Solidity is the uniform version of this closure property of M : (αo . . .07} is Jb-solid over (M,q) iff π G M, where π(s, i) = Th^(/?^ U 5 U {(α0 . . .α, _ι}, ?}), for all i < ί and finite s C αf . Solidity is useful because it is easier to show solidity is preserved by the appropriate embeddings than to show standardness is. Universal parameters. DEFINITION 2.7.5. Let M be a ppm, q G \M\, r a parameter of M, and 1 < k < ω. We say that r is k-universal over (M,q) iff whenever A G \M\ and ω A C ρ%* , there is some term τ G Sk* and a G p% such that

Again, we are interested in the case r is the kth standard parameter of (Λ4, q). The Jk-universality of r will be used to show r remains the standard parameter in a certain hull of ΛΊ the argument is given in the next lemma. Lemma 2.8. Let π : Ή, —> M be generalized rE* elementary , where M is a ppm and I < k < ω. Suppose p%* C OR7* and π \ p^4 = identity. Suppose also that π(r) is the kth standard parameter (Λί,π(ςr)), and π(r) is k- solid and k-universal over (M, tf(tf)). Then ρ£ = pήf , r is the kth standard parameter of (W, q), and r is k-universal over (W, q). PROOF. For a < p£* we have Th£(α U {s}) = Th^(α U MS)}), moreover the theory in question can be regarded as a subset of α. If α < />£*, then as p%* is a cardinal of M and Λί is strongly acceptable, Th£*(α U {^(«)}) belongs to U. This shows that p? < p% But Th^(/?^ U {(r,q)}) £ |W|, as otherwise, letting A C pζ* code it, we have A = π(A) Π p%* G |ΛΊ|, and so U {^(r),π(ς)}) G |Λ1|, a contradiction. Thus p? = p% . We have Th^(p^ U {r, q}) £ |7ί|, so to see that r is the kih standard parameter of (W,ςr), suppose s <ιex r. So π(s) <ιex π(r), and we have A C p^, A G |Λ<|, such that A codes Th^(p^ U {π(s),π(ί)}). The k universality of π(r) over (M, π(q)) easily implies A G W, and so Th^(/# U {s, q}) G |W|, as desired. It is routine to check that r is fc-universal over (Ή,,q). We now define by induction on k > 0
the Jbth core parameter of M .

D

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We shall assume that certain parameters we encounter in the course of the definition are solid and universal; otherwise we stop the induction. This assumption may not be necessary for a sensible definition, but it is true of the ppm we are interested in, as we shall show later. DEFINITION 2.8.1. Let Λ4 be a ppm which is passive or active of types I or II. We define <£t(.M), pt(Af), and pk(M) by induction on k. k = 0: Let Po(M) = ORM,
be the inverse of the collapse. If ^-1(r) is not 1-solid over (Hi*(p\* then stop the induction, and otherwise set

u

M)>0)

Notice that pι(Λί) = τr((s,g}), where s is the first standard parameter of (<£ι(Λ4),g), and s is 1-solid and 1-universal over ( 1: Suppose we are given

= (s,q) where s is the Jfe-lst standard parameter of (£fc-ι(Λ<), q) and s is k - 1 solid and k — 1 universal over (ίjk-ι(Λί),g). Let « = (α0 •••£*/)

and bt = T h ^ ' α . U {α0, .... α,-ι, «}) for 0
r <.,,.*» 6/> =u=teϊ.-2ι(Λ<)

. . l( W£l-2 ^) otherwise.

Let r be the Hh standard parameter of (Cjfc-ι(Λί),u). If r is not Jb-solid and Jfc-universal over (£*-ι(Λ4),u), then stop the induction. If it is, consider * : K?-l(M) (p£k-l(m) U {r,ιι}) - Ck-

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W. J. MITCHELL AND J. R. STEEL

the inverse of the collapse. Suppose that π~l(r) is fc-solid over

if not then we stop the induction. Set then

and ΛO and k < ω. We call π a k-embedding iff (a) M and M are fc-sound, (b) π is rΣjb+i elementary, (c) *(pi(M)) = pί(W) for all i < Jb, (d) π(pi(M)) = Pi(tf) for all i < k - 1, and ^(jV) = sup *"ph(M). (We adopt the convention that π(ORM) = OR ^ in the previous definition.) Lemma 2.9. Let M be a k + 1 so/id ppm, αnrf π : <£jb+ι(Λ<) —*• <£fc(Λ<) 6e λί is a i-embedding, then ?r(iίt(Λί)) = Appendix to §2

We close this section by relating the rΣn hierarchy to the more traditional hierarchy involving master codes and iterated ΣI definability. First, the use of generalized rΣn+ι formulae rather than just pure rΣn+ι formulae in defining Th^pQ does not affect the value of />£+χ, at least if M is n sound.

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Lemma 2.10. Lei M be a ppm and n > 0. Let q £ \M\, and suppose in the case that n > 1 that M = ^(p*? U {q}). Suppose that α U {q}) Π {(φ, a) \ φ is pure rΣn+ι} is a member of\M\. Then in fact ThJ^α U {q}) G |Λί |. PROOF. We give the proof in full only for n = 0 and M passive and ORM = ωλ, for λ a limit. So let us make those assumptions. Let Λ4 = (jf,G,£)

(A

limit)

and

for β < λ. For τ G Ski, β < λ, and ΰ G |Λ^ί/?|<w, we say that r(u) changes value at /Jiff Notice that if τφ G Ski is a basic term, then τφ(u) changes value finitely often, since the new value precedes the old in the order of construction (unless the old is 0). It follows that for any T G Ski, τ(ΰ) changes value finitely often. Notice also that there is a recursive map (r, n) ι—* θT)n associating to each r G Ski and n < ω an rΣi formula 0r>n such that M [= 0τ,n[ΰ, α] if and only if τ(u) changes value at least n times and a is the nth value of τ(ΰ). Now, letting P = Th^(α U {q}) Π {(φ, ά> | φ is pure rΣn+ι} we can compute Th^(α U {q}) from P inside M as follows: Given a potential member (y>, a) of Thf*(α U {q}), which we write as ψ(τι(ΰ) τjb(ϋ)) where ψ is pure rΣi and τ\ r* G Ski and ΰ G (<*U{
and *<*

Then (^>, ά) G Thf^α U {?} iff aαι

' «<*

is a member of P. If M = (jf+ 1, G, ϊ?), then we can use the Sua+n, for n < ω, as we used the Mβ's of the previous argument.

26

W. J. MITCHELL AND J. R. STEEL

If M is active, then we can use the fact that rΣ™ = ΣI over M*, where M* is the amenable structure associated to M (cf. the remark following Corollary 2.2.) We can then ramify ΣI over M* as above to carry out the proof. This finishes the case n = 0. Now let n > 1. If r G Skn and β G (ftf)<ω , then we call the triple (r, β, q) a name of τM[β, q]. We are assuming every member of \M\ has a name. If θ is rΣn+ι and αi α* are names and β < p%* , then we let "Th^βU {q}) witnesses that θ(a\ α*)" have the obvious meaning. Namely, if θ = 3α, 6(Tn(α, 6) Λ ψ(a, 6, υi - - υk)) and

then Th£*(/? U {q}) witnesses θ(ctι α t ) iff there is a 7 < β and there are names (σι,ήι,q) and (0"2,rJ2»i) such that, first, the following sentences are in (a) σ\(ήι , q) = (7, x) for some x such that σϊ(ή 2} q) is a complete generalized rΣn theory of parameters in 7 U {x}. (b) ψ(σι(ηι,q), σ2(τ?2, ^), n(Λ, ί) Tk(fa, q)). (implicit here is that βi,ήi € /?<ω), and second, if we let σj [ήι,q] = second coordinate of σ^1 [ήi , q] , and we let f(φ, 6) =_a canonical name for (φ, 6* σ{ [ήι , q]) for each generalized rΣn formula ^> and ί G 7<ω , then ) € σ2(ij2> β)» € TC(0 U {«})iίr V(ϊ, σ*(ή!, ϊ))" € Th^(^3 U Remark. We have taken some liberties above, as Th£*(/? U {g}) is not literally speaking a set of sentences. Now for T G Skn+ι and αi τβ(aι

α fc ) = 6 iff

α*, 6 names and β < p** let Th^(/?U {q}) witnesses τ(aλ - -α k) = 6

where the right hand side is interpreted in the spirit above. We can use the r^'s to carry out the argument given in the case n = 0. D The calculations just indicated also give Lemma 2.11. Lei M be a ppm, q G \M\9 and M = Ή*?{ρ*? U {q}) where n>l. Lei A = Th^(p^ U {«}) Π {(φ, c) I φ

is pure rΣn} ,

coded in a natural way as a subsei of ρ£*. Then, leiiing

FINE STRUCTURE AND ITERATION TREES

λf is amenable, and for all B C p%* , B is rE^ iffB

27

is ΣI overλf.

PROOF. From A Π a we can compute Th^(α U {q}) in a simple way, as in the preceding lemma. Thus we may as well assume A = Th^(p^ U {q}). Now suppose η G B & M |= φ[η, x] and let x = σ[β, q], σ G Skn, β 6 (p^ί)<α>, where ^> is rΣn4.ι. Then iϊ€B«»3/?
witnesses

where ;;* is a canonical name for η. This shows B is ΣI over .ΛΛ The converse is easy. D

§3. SQUASHED MICE Let M be an active type HI ppm. Let E be an extender over M with K = crit E < VM. Even if wellfounded, Ult(ΛΊ, E) may not be a ppm. The trouble is in the initial segment condition: if i'f>ι^M is not cofinal in iε(^M)ί then this condition will fail in Ult(ΛΊ, E). The problem seems to be that we are using too many functions in forming Ult(Λί,#); we'd like to use only functions in J^M in order to get continuity of IE at VM . Lemma 9.1 and the remarks following it give a fuller explanation. This leads to DEFINITION 3.0.1. (Λi-squash) Let M be an active type III ppm. Let F be the extender coded by FM and v = VM . Then

The symbol ΛΊ8q stands for "Λί-squash" . The term "squashed mouse" was invented by Dodd for use in a similar, but more complicated, context. Recall that VM is a cardinal of M in the type III case, so that M*q includes all sets which have hereditarily cardinality < i/ ** in M . Our next lemma shows that M*g is amenable. Lemma 3.1. Let M be an active type III ppm. Then there are cofinally many 7 < VM such that ί™ = F \ 7 where F is the extender coded by FM. PROOF. Let /c = crit F, and let η — I be a generator of F. By the initial segment condition, there is a 7 < OR^ such that E** exists and is the trivial completion of F \ η. (Alternative (b) of the initial segment condition cannot hold as 77 is a successor ordinal.) Now the natural map π from Ult(Λ4,.F \ η) into Ult(Λf,F) has critical point > η, and hence crit(π) > j since 7 = (^+)Uit(M f Frf?) This implies that F \ 7 is the trivial completion of F \ ηy which is E** . To see this let G be the trivial completion of F \ η. We have M —^->

Ult(M,F)

K ΐ-

V\t(M,F\η)

and for α € [γ]<ω, x appropriate,

0*(β)€»(i G (*))

FINE STRUCTURE AND ITERATION TREES

29

Since there are arbitrarily large η < ι/M such that η- 1 is a generator of F, this completes the proof. Q So M*q is amenable. Moreover, the definition of VM guarantees that the rest of M can be recovered from Maq by taking an ultrapower. If E is an extender over M with crit E < ι/M, we'll have M

->

-I and Ult(Aί8ί,£) = tf*q for some tf C Ult(Aί,£). But λf φ Ult(Λί,#) is possible, and this is what leads us to iterate on the squashed level. As we shall iterate MBq and not Λί, the appropriate definability hierarchy is based on Λί8ί, not M as in §2. Note every ΛΊ-definable subset of VM is definable over M*q. DEFINITION 3.1.1. M.

N is an sppm iff λf = M*q for some active type III ppm

We now introduce a language appropriate for sppm. DEFINITION 3.1.2. £* is the language of set theory with additional 1- place predicate symbol E, 2-place predicate symbol F, and constant symbol μ. We interpret £* in an sppm

by setting EM = E, F* = F, and μ* = crit F. As sppm are amenable with respect to their predicates, we can work with the usual notions of ΣQ and ΣI. DEFINITION 3.1.3. (a) A formula of £* is ΣO iff it is built up from atomic formulae using Λ, V, -ι, 3x G y, and Vx G y. (b) The Σn and Π« formulae of £* are also as usual. We want now to say "I am an sppm" with a simple formula. DEFINITION 3.1 .4. A P formula is a formula of £* of the form θ(ϋ) = Vz3j/(z C y Λ ψ(y) Λ Vα 6 x 36 e y ¥>(α, 6, v) , where V' is ΣI without x free in it, and φ is ΣQ without x or y free in it.

30

W. J. MITCHELL AND J. R. STEEL

Thus a P formula can say a little more than that there are cofinally many y (under C) with a ΣI property. We aren't sure how necessary the little more is, but as the preservation lemma still goes through, there's no harm in it. Lemma 3.2. Lei M and M be transitive £* structures, and π : M —* λf, and ψ be a P formula. (a) Ifπ is a ΣI embedding and M |= Ψ[v(ά)}, then M ^= φ[a]. (b) Ifπ is a cofinal (i.e. |Λ/"| = U ranπ) ΣO embedding and M \= ψ[ά], then

λf

One can't quite say "I am an sppm" with a P sentence, since the decoding of M from MBq requires taking an ultrapower, and we can't capture the wellfoundedness of this ultrapower. We do get Lemma 3.3. There is a P sentence φ of C* such that (a) If M is an sppm, then N f= \l>. (b) If M is transitive and M \=. *φ, then F* is a pre-extender overM; moreover, if Ult(W, F*Γ) is wellfounded then J\f is an sppm or N is uof super strong type", that is i$(crit F) = length F = OK*). PROOF (Sketch). By Dodd-Jensen we have a P sentence θ\ whose transitive models M are those of the form λί = ( J f , . . . ) > "a limit ordinal. Let 02 be the HI sentence of £* asserting that E^ is good at all a < OR^. Let 03 be the Π x sentence: VαVx(F(α, x) => α E [OR]<ω Λ x C [μ]CΛrda) Let 04 be the P sentence: There are cofinally many ordinals 7 such that 7 E dom E and jf Π EΊ = F \ 7 Π J&. It may seem that "μ+ exists" is Σ2, but we can say with θ$: 3 ordinal α such that μ < a and {(/?o,/?ι) | tf |= βι = βo] € ?{*,<*} We claim ^ = Λ xs ft" is as desired. Clearly, if Λf is an sppm, then M \= /\ί<5 0, . Now suppose N is a transitive £* structure such that M f= /\ί<5 0, . As M ^= 04, we see that F^ = F is a pre-extender over ΛΓ. Suppose that Ult(^,F) is wellfounded, and that i^Γ(crit F) > OR^. Let α= ,

G = the (μ*, a) extender derived from i$ : tf -* Ult(^,

FINE STRUCTURE AND ITERATION TREES

31

Note that α exists since ip(μ^) > ι/. We claim M is an active type III ppm, and v = VM . For this, note iF(E^) \ a is good at all β < α, since E* is good at all β < μ* . So it is enough to check \ α~G is good at α. Clearly Λί is strongly acceptable. G is a pre-extender over M as G is a preextender over Λf and ) Π Ult(JV, F) = P(μ") Π M , The ordinal t/ satisfies condition 3 of good at α since P(μ^) Π \M\ C J as Λf f= 05- Since G is derived from #, Ult(^,G) = Ult( /i^ such that / G -V" we have

so so

This is enough to give 3(b). Finally, ϊ/ is the least ordinals satisfying clause 3 since if 7 < y, then G f 7 = F f 7 G Λ< by the fact that jV f= ί4 It is easy to see the coherence condition 4 is satisfied. The initial segment condition (only 5(a) is relevant) is satisfied as λf \= 04 and i^(E^) \ v = E* . Thus M is an active type III ppm with i/ = VM . Clearly N = Λ<βg .

D

Remark. It is annoying that we must include the possibility that λf \= φ be "of superstrong type" , but our attempts to strengthen V> so as to exclude this have not succeeded. Notice that if M \= ψ is of superstrong type, then a standard argument gives (JfV , G, E") \= ZFC + μv is a Shelah limit of Shelah cardinals . (ί/ = OR^). So λί is far above any mice our theory can handle anyway. The rest of this section is an obvious parallel to §2. Because sppm are amenable, we could adopt a very literal version of the Dodd-Jensen approach here (in particular, we could stick to the usual Σn hierarchy); however, for the sake of internal consistency, we shall adopt the approach of §2.

32

W. J. MITCHELL AND J. R. STEEL

Skolem terms and projecta. DEFINITION 3.3.1. 1 < n <ω.

£** is £* together with binary relation symbols Tn for

We define the quasi-Σn formulae for n > 1.

DEFINITION 3.3.2. (a) The gΣi formulae of £** are precisely the ΣI formulae of £*. (b) A formula θ(ϋ) of £** is gΣn+ι, where n > 1, iff

where φ is qΣ\. DEFINITION 3.3.3. For Ψ(VQ vt+i) an £** formulae, rφ(vQ υ*) is the basic Skolem term associated to φ. Having interpreted φ in an sppm Λf, we set :^ least b such that N [= φ[a, b] i

if no such b exists.

DEFINITION 3.3.4. SKn (for n > 1) is the smallest class of terms containing all τφ for φ qΣn and closed under composition. DEFINITION 3.3.5. A £** formula is generalized qΣn iff it results from substituting terms in SKn for free variables of a qΣn formula. (The substitution must be such that no free variable of a term becomes bound in the result.) DEFINITION 3.3.6. Let M be an sppm. Then for n > 1 (a) Ύh^(X) = {(φ,a) \ φ is generalized gΣn and ά 6 X<ω and M h= ^[α]} (b) p*? = least α < OR^ such that for somep € |Λί|, Th^(αU{p}) $\M\. (c) T^(α, b) iff α = {α, q) for some α < p* such that 6 = Th^(α U {q}). We define the classes of relations ^Σ^1, etc., in the obvious way. It is easy to see that qΣ** is closed under 3, Λ, and V, and that (gΣj^ U ίΠ^) ^ ^^ίίl-i) uniformly over all sppm. One can also show -»T^ is a qΣ™+x relation (uniformly) in parameter p**. Hulls. For M a sppm and X C \M\ and n > 1 let = the transitive collapse of {^[ά] | α G X
where π is the collapse map.

FINE STRUCTURE AND ITERATION TREES

33

Lemma 3.4. Lei M be an sppm such that Vι/ € ORM (j£M \= there are no Shelah cardinals). Then for any X C \M\ and n > 1, Ή.%*(X) is an sppm. PROOF. By Lemmas 3.2 and 3.3. Strictly speaking, we haven't packed enough wellfoundedness of ultrapowers into being an sppm to be able just to quote 3.3(b), but the proof of 3.3(b) requires only the wellfoundedness we have. Lemma 2.7 carries over verbatim. Lemma 3.5. Lemma 2.7 remains true if one replaces appm which is passive or active of types I orlΓ by "sppm" and VΣn" by agEn". Standard parameters, solid parameters, and Cores. The definitions and results of §2 carry over verbatim. (The only "results" here are Lemmas 2.8, 2.9.) We shall say no more. Premice. DEFINITION 3.5.1. all β < α,

Let M .= jj; be a ppm. We say M is a premouse iff for

(1) Jβ is passive or active of types I or II =^ jf

is α -sound, and

8q

(2) jf is active type III => ( J/) is ω-sound. Notice that a premouse need not itself be u -sound. We shall eventually build an E such that every J™ is a premouse.

§4. ULTRAPOWERS Let M be either a ppm or an sppm, and K < ρ%*. Let E be a («, λ) pre-extender over M. (We are interested in the case that for some ppm or sppm tf such that P(κ)* = P(κ)M we have E = F* or Ξγ^ = JK^). It is easy to check that E is a pre-extender over M in this case.) We wish to define Ultn(Λf , E). We begin with the universe of Ult n (Λΐ, E) and the G relation on it. If n = 0, then the elements of Ult0(Λ4, E) are equivalence classes [α, /]£*, where α C λ is finite and / G |Λί| has domain [/c]cardα. The equivalence relation is as usual: (α, /) ~ (6, g) iff for Ea^ a.e. x, /(x) = ί/(x) where / and g come from / and g by adding the appropriate dummy variables. E measures enough sets that the definition makes sense. The G relation on equivalence classes is as usual. If n > 0, then let r = T(VQ vf ) be a term in Skn if M is a ppm or in SKn if M is an sppm. Let q G |Λί|. Then for a G [/c]* The elements of Ultn(Λ< , E) are equivalence classes [α, f]^ where α C λ is finite and / = fT)q for some q G \M\ and r G Skn (resp. SKΠ). The equivalence relation is as usual. E measures enough sets that the definition makes sense because K < p**. Again, the G relation is as usual. Ult n (Λΐ, E) may be illfounded; however, if it is wellfounded we shall identify it with the transitive set to which it is isomorphic. We must define £UM*<.£) and JJ UMΛΊ.E) to complete the definition of the structure Ultn(Λf ,£). Let M

^ iff { ά I f(a)

It is easy to see that Ea measures the set in question, using the amenability of M with resp. to EM in case n = 0. In case M is squashed or n > 0 we can set [{α, />]£* € F™*M& iff { β I /(α) € FM } £ EΛ , using amenability in the squashed n = 0 case. We are left with the case M is active and n = 0. Let μ = crit FM. Let also η = [{6, /}]£* G OR Π Ult0(Λί, £"), and Λ = [{6,
FINE STRUCTURE AND ITERATION TREES

35

Thus k G \M\. We then put

Case 2. /c < μ. Let t be a function in \M\ with domain = μ and

=

(J

(We may assume dom0(z) = μ all x.) Let 7 = sup (ran/ Π OR/'1). Let c be such that FM(cyt,i}. Using c we can compute in M k(x) = the unique d such that FM(d, g(x), f(x)) . So ke \M\. We then put

This completes the definition of Ultn(Λί,^). Notice the definition guarantees Los' Theorem holds for atomic formulae of C — {t), 7} (resp. £*). Theorem 4.1. (Los' Theorem). Let n > 0, lei M be a ppm or sppm, and lei E be a («,λ) pre-exiender over M, where K < p** . Let [α, , /,-]£* be an element o/Ult n (Λί, E) for each i < k, and let b = \Jk α, . Then , E) N p[[αo, /o]^, . , [α*f Λ]^]

iff 5B G ^Vϋ € B M for any generalized rΣn(resp. ^Σn) formula φ. Here /t comes from /,- by adding the appropriate dummy variables. Remark. Assume Λί, etc., are as in the hypotheses. If n > 0, then A = {« € M""14 I Λ< |= ¥»[/o(«) ../*(«)]} is in M as /c < p^ .

If n = 0, then A £ \M\ is possible. However, our proof will show there is a B G Eb (so B G |Λ4|) such that £ C A o r £ Π A = 0 . PROOF. We consider only the case that M is a ppm (passive or active type I or II) as sppm behave exactly like passive ppm here. Suppose first that φ is rΣo If n > 0 we get the desired conclusion easily as there are enough functions defined by terms in Skn. So suppose n = 0.

36

W. J. MITCHELL AND J. R. STEEL

For any rΣo formula φ = φ(vQ v*) and functions /o •/* G \M\ such that dom/t = [/c]card6 for all i < k (where 6 C λ is finite), we let

We show by induction on φ that there is a set B G EI> (so B G \M\) such that

J? C Aφj or B Π A^j = 0 and iff

B C Aφj

Ulto(Λ<, E) \= φ[[(b, /„)]#

((b, fk)]&] .

For formulas φ which are ΣQ in £, a subinduction on ΣQ in £ formulas (using amenability) gives the result as usual. For φ = F(VQ, υ\, ^2), the construction of ^uit0(Af,£?) guarantees the desired result. If φ is built from simpler rΣo formulae by Λ, V, or -i the inductive step is easy. Suppose φ(v$ Vk) = αvo is finite Λ w (Bϋjb+i G VQ) 0(t>o Vfc+ι) We may assume /o(ΰ) is finite £"5 a.e. as otherwise 5 = {ΰ I /o(ϋ) infinite} does the job. But then we can fix £ G ω such that card/b(ϋ) = £ for EI a.e. ϋ, and functions gi gt with dom = [/c]card6 such Γ ae that /o(ΰ) = {, or 7 now follows by the usual argument as there are enough functions defined by terms in Skn. But then 2.6 (b) implies that P = Ultn(Λί , E) is a ppm of the same type as Λ4, and that iε(μM) = μτ ', iE(vM) = ^P, i^ίr^) = ΊP This gives 4.1 for arbitrary formulae φ. So now let i > I . Notice first that as the relation Th^^α) = 6 is Πi over rΣ, _ι definable over Q, uniformly over all ppm Q, and as we have 4.1 for Πi over rΣ, _ι formulae by induction hypothesis and the fact that there are enough functions given by terms in Skn, we have (for Ult = Ultn(Λί, E).), (*)

ThJ^dα, /]) = [6, g] iff for £αυt a.e. x, Th^^/ί*)) = g(x) .

Let π : M —> Ultn(Λί, E) be the canonical embedding. It follows that

u,t

O

Pi-l —

/ Λ4 \

xi. otherwise.

We prove the case pV±\ = π(p^). Suppose p^l < ORM. We show first that *(p£ι) < P™\. For let a = [α,/]^ < π^J, and let q = [a,g]%. We may assume f(x) < ρ^l for all x. Define

= least 6 such that 7£Ί ({/(*),

FINE STRUCTURE AND ITERATION TREES

37

Then h is one of the functions used in forming Ultn(Λ<, E), and as we observed above in (*) Thus a < pYl\. Thus π(p^) < pj™.

On the other hand, pick q € |Λί| such that Th^p^ U {«}) g \M\. Then by (*) Th^OrO^) U Me)}) * Ult. Thus p?l\ < Putting (*) and (**) together, we have T?[([a, f], [b,
a.e. x, ϊ£ (/(*), g(x)) .

Now suppose where ^ is rΣi. We check one direction of the conclusion of 4.1. Suppose that for Eb a.e. ΰ, M \= φ[fa(ΰ) Λ(S)] Let /,- = fTiqt where r, G Skn and gt G |Λί|. We can translate "ψ(a, 6, v)Λ7i_ι(α, 6)" into an rΣn formula; this gives us terms σ0 and σ\ in Skn which Skolemize the result, i.e., such that for EI a.e. ΰ M \= φ(σQ(ΰ, q), σι(ϋ, g), r0(ϋ, go) where q = (gi

T A (U, ςf λ )) Λ 7i-ι(σ0(ϋ, g), σι(ΰ, g)) ,

g*}. But then, letting 00 = /<τ0,ί and gι = /σι>ί,

Ultn(M, ίO N Ώ.ι([6, yo], [6,
[6, Λ])

as desired. Finally, we prove 4.1 in the case φ is generalized rΣn with n > 0. Notice first that if T(VQ Vk) G Skn, then rϋlt[[α, /o], . . . , [α, /J] = [α, Afi - r^[/0(n)

Λ(ϋ)]]^

for any [α, /0] [α, /*] G Ult = Ult n (Λ<, E). To see this, it is enough to consider the basic terms τ& G Skn. But the graph of such a term is definable by a Boolean combination of rΣn formulae, uniformly over all ppm, so we can use the term-free case of 4.1 just proved. But now if φ(v) is rΣn and τ(v) G Skn then Ult |= rtr(tO)[[α,/]#] iff Ult f= ^[rult[[α,/]^]] iff Ult|= V [[α,Atι.r A4 [/(ii)]]^] iff for Ea a.e. ΰ , M \= φ[τm [/(«)]]

iff for Ea a.e. δ ,Λf as desired. Of course, the case φ or T having more variables involves only more notation. Q In the course of proving 4.1 we have shown

38

W. J. MITCHELL AND J. R. STEEL

Corollary 4.2. Lei M, etc., be as in the hypotheses 0/4.1, and let π : M Ultn(Λί , E) be the canonical embedding. Then for i
and

(where \nt = \J\tn(M,E)). We would like to show that under the hypotheses of 4.1, the canonical π : M —*• Ultn(Λί,£?) is generalized rΣn+ι elementary. For this we seem to need (essentially) that M be n-sound. Fortunately, we shall never want to form Vltn(M,E) unless M is n-sound. Corollary 4.3. Lei M, etc.f be as in the hypotheses of 4.1, and let π: M —> Vltn(M,E) be the canonical embedding. Suppose that for some p G \M\, M = Ήrf&n* U {p}) Then π is generalized rΣn+ι (resp. qΣn+ι) elementary; moreover PROOF. Let Ult = Ultn(Λί,J£). We show first that supπ"/?^ > p™1; for this it is enough to show that if K^dftf U {p}) = Λί, then

(For then Th^lt:(sup π"p** U (π(p)}) £ Ult by a diagonal argument.) So let ^(ffi* u {P}} = A<,_and let [α, /] 6 Ult. Then there is a term τ G Skn_(resp. SKn) and parameters_6 G [p** U {p}]<ω such that for all w, /(ϋ) = τM[ΰ, b]. Let id(ΰ) = ΰ , cj(ϋ) = 6. By the Los Theorem, [α, f]# = rult[[α, id]^ , [α, crf^] = rult[α,π(fr)]. Since α G [π(/c)]<α; and /c < />£<, and since π(δ) G [sup π;//>^ U , we're done. We claim next that p™ > sup π" p%* . For by the Los Theorem we have easily that for α, 6 G \M\ Th^(α) = 6 iff

Th^(π(α)) = π(6)

[For the "only if direction, let c 6 π(α)
FINE STRUCTURE AND ITERATION TREES

39

Thus pj^1* = sup π" p*f. Now let [α, /]£* = (α, q), where a < sup π'Vί^ We claim that for any g ΓnUlt([α, /], [α, g])

iff for Ea a.e. ΰ , f^(/(ϋ), y(ϋ)) .

<= is easy since Th^(α) = 6 is uniformly Πi over rΣ£, and we have the Los Theorem for rΣn formulae. So suppose T^lt:([α,/], [α,
Then we have (1)

(φ,c)€(a,g]iS φ is generalized rΣn and c e [α U {ί}]<ω and (φ* , c*) G

where ^>*(c*) is the obvious way of rewriting φ(c) so that the parameters c* come from π(j) U {ίr(p)}. _Thus the map (^>, c) •-* (y?*,c*) is rΔi over Ult in the parameters α, qy and β. Let /? = [α,Λ]^, where we assume for notational convenience that the support is α (otherwise enlarge all supports). Then the fact that (1) holds in Ult is a rΠi fact about [α,/], [α,0], and [α,Λ]. It follows that for Ea a.e. ϋ,

, φ is generalized rΣn and c G [c*o U {
for £^β a.e. ΰ, by the Los Theorem. As 6 = Th^(γ U {p}), we see

for J?α a.e. ή. As αfl < p** a.e., we get

for £"α a.e. ϋ, as desired. Finally, let

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W. J. MITCHELL AND J. R. STEEL

be an rΣn+ι formulae. If M )p φ[x], then we have α, 6 such that T^(α,6) Λ M Ult φ (a9b9x)9 so Γ™(π(α),τr(6)) and ^ (π(α),π(6),x), so Ult μ φ[π(x)]. On the other hand if Ult (== ^>[π(z)], then we have α, /, g such that [α,/] = (α,g) for some α ί^ , and Ult |= Tn((a, /], [α, g]) Λ tf([a, /], [α, ,], »(x)) . By our claim, for JPα a.e. ΰ Λί (= fn(/(ϋ), ί(«)) Λ Thus M ^= y>[x], as desired. M

ult

We can now show π(τ (x)) = r (π(a?)) for all r € Skn+ι, since the graphs of basic terms in Skn+ι are definable by boolean combinations of rΣn+ι formulae. It follows that π is generalized rΣn+ι elementary. Relations to Dodd-Jensen. It is easy to see that if M is n-sound, Ultn(Λ^,^) is exactly what is obtained by the Dodd-Jensen procedure of coding M onto p£* , taking a ΣQ ultrapower of the coded structure, and then decoding. For let M be a ppm or sppm, n > 1, and M = Ή^(p^ U {?}). Let be the canonical embedding. Now let AM = Th^(p^ U {g}), coded as a subset of p%* , A* = Th^(^ U {»(«)}), similarly coded .

Let

be the master code structures associated to M and M. Then π:P^Q cofinally; this is contained in 4.3. Note also that if [α, /]£* € |Q|, then 3β < ρ%* such that f(u) < β Ea a.e., so since / is given by a term in Skn, in fact / G \M\ and hence / E l^j. So in fact and π \ \P\ is the canonical embedding for this ΣO ultrapower. Notice finally that all of λί can be decoded from Q, since λί = Ή*[(f% U Although we can make sense of Ultn(M,E) in the case M is not n-sound, in practice we shall never need to form such an ultrapower. Thus our construction of Ult n (Λf, E) does not go beyond Dodd-Jensen in any important way. We describe now the preservation of the core parameters pι(Λί), for i < n, in the case that M is n-sound.

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Lemma 4.4. Let M be n-sound, lei E be an extender over M with cήt(E) < p**, and let π : M —> Ult n (Λί, E) be the canonical embedding. Then (a) Ultn(M,E)

is n-sound, and

(b) π 15 an n-embedding. PROOF. Let J\f = Ultn(Λ<, E). It is enough to show that for all i < n

and P,

= τp.

.

For then by soundness Pi(λf) = p? for all i < n, and similarly for Λ4, so that π maps the core projecta properly by 4.2 and 4.3. We proceed by induction on i < n. For i = 0 there is nothing to prove. Now let i = 1 and Let r be the first standard parameter of M. Thus as λi is 1-sound, pι(M) = (r, 0) and r is 1-solid and 1-universal over M.

Let r = (c*o * * *<*/ bj = ThfVj U {αo

<*j-ι}) ,

0 < j <ί

so that bj E \M\ by solidity. By 4.3, π is at least rΣ2 elementary, so »φ ) = Thf(»(βj) U {π(α0)

»(£»,_!)}) .

It follows that no s <\ex π(r) can serve as the 1st standard parameter of λf. On the other hand, Thf (>f U {π(r)}) ^ |Λ/"| and in fact Ή$(tf U {π(r)}) = |AΊ. [If n = 1 this is implicit in the proof of 4.3. Suppose n > 1. Let [α,/] by an arbitrary element of \Af\. Notice that if we let, for x G |ΛΊ|, h(x) = 1 st (in order of construction) (r, /?) such that r e S k x Λ/9e(^)

<ω

Λ4

Λr [^,r] =a:

M

then Λ(x) = σ [x, r] for some term σ 6 Sk2. So if we let

then y is one of the functions used to form tf, and if [α, g] = (r, /?), then r G Ski <α; and β 6 (/^) and r^[/?, π(r)] = [α, /], as desired.] So τr(r) is the 1st standard parameter of Λ/", is 1-solid and 1-universal and N is 1-sound. As tf is 1-sound, p\(N) = {*(**), 0) = π(pι(M))> as desired. The case i > 1 of the induction involves a bit more notation but no new ideas, so we omit it. D

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The next question, clearly, is how π moves ρ^l and the n-h 1st standard parameter of (Mypn(M)). We answer this under a solidity hypothesis. Our proof is in essence drawn from Mitchell [M?]; it was recently re-discovered by S. Friedman. We must also, it seems, impose an additional condition on E. DEFINITION 4.4.1. Let M be a ppm or an sppm, and E a («, λ) extender over M. Then E is close to M iff for every α G [\}<ω (1) Ea is rΣf1 (resp. qΣ?) and (2) if A G \M\ and M |= card A < K, then Ea ft A G \M\. The purpose of this restriction on E is explained by the following lemma. Lemma 4.5. Suppose E is a (K, λ) extender which is close to Λί, and K < p£*. Then

If, in addition, M = Ή^(ρ^ U {q}) for some q and pf^l < «, then M

O

_

UMΛ4.J5)

Pn+l — Aι+1

PROOF. The nontrivial part of the first sentence is the assertion that P(/c) Π I Ult n (Λ4, E)\ C \M\. So let [α, f]# C *, where / : [*]•' -* P(/c). Since /c < p? and / is defined from a parameter by a term in Skn, in fact / G |Λf |. For α < AC, let A a = {ϋ G [«]' I α G f(ΰ)} From J£β Π {Aα | α < /c} we can compute [α,/]£*. Since E is close to X, we get [α,/]£* G |-M|. For the second assertion it is convenient to use the master code structures. Let λf = Ult n (Af , E) and fix q such that M = W^(p^ U {q}). Set ^ U {q}) , coded as a subset of p A" = Th^O^ U {»(«)}) , coded as a subset o

so that Q = UltoίP, J5) with canonical embedding π, which is cofinal and ΣI elementary. If n = 0 then we take P = M and Q = N. By Lemmas 2.10 and 2.11, p^j is the least α such that some ΣI over P set B C α is such that B £ \P\, and similarly for pff+1 and Q. To see that /^ < p£l9 let β C ^ be Σj over P and B £ \P\. Since #H-ι < « and 7Γ is ΣI elementary, B is ΣI over Q. But P(/c) = ^(/c)^ = P(κ)^ = P(/c)^ using the first assertion of 4.5 and strong acceptability. Thus Bί\Q\.

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43

To see that p^j-j < ptf+l9 let a < pQi and B C a be ΣI over Q. It is enough to show that B is ΣI over P. Let

where ψ is ΣI in the language of Q. For 6 < OR^ let Q6 = (J?",^E«\ωδ,A and similarly define Pδ for δ < ORΛ So

so ,w \= φ[η, [α, P

36 < OR 3X E Ea Vϋ Now as E is close to Λ4, the Ea is an rΣ^ subset of \P\. By Lemma 2.11, Ea is ΣI over P. Thus 5 is ΣI over P, as desired. D We now consider preservation of the n + 1st standard parameter. Lemma 4.6. Let M be a ppm or sppm, n > 0, and M = U^((ρ^{ U {q}) if n>l. Let E be an extender close to M such that p^ < crit E < p*? . Let

be the canonical embedding. Suppose that r is the n + 1st standard parameter of ) and that r is n + 1-solid over Then π(r) is the n + 1st standard parameter o/(^, ^"(ί)), and π(r) is n-h l-solid

PROOF. We will give the proof for the case n = 0 with a passive premouse of limit length. The general proof is the same as this, using the fact that rΣn+ι is equivalent to ΣI over the appropriate master code structure. See lemma 2.11 for the case of n > 0 and the remark following corollary 2.2 for the case of n = 0 with an active premouse. For successor ordinals λ = 7 -f 1 write M\ = Un€ω $£y+n» where (5^λ : i/ < ωλ) is Jensen's S sequence, and use the same proof as below. Let us consider first the case n = 0, Λ< is passive, and M = (J?M,£,EM)

(λ limit).

Now by 4.5, p^4 = ptf and

U {π(σ), π(q)}) = Th^(p^ U {r, q}) £

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W. J. MITCHELL AND J. R. STEEL

so it is enough to show that π(r) is 1-solid over (Λf, τr(<j)). Let r = (α0 •••<*/) bi = ThfVi U {α0

α, _ι, q}) Π {(φ, c)\φ

d = Thf (τ(αrj) U {π(c*0)

is pure rΣi}

*(α. -ι), »(«)}) Π {(p, c) | ^> is pure rΣi

for 0 < ί < £ By 2.10, it will be enough to show ct G |,Λ/] for 0 < i < L So fix i such that 0 < i < L For γ < λ let Rv = bi

so that 6, = U < λ Ay Similarly, let

so that c, = Uu;7an(l thus

1. 3γ < λ(6, = Λy); i.e. Rγ is eventually constant as 7 —* λ.

PROOF. Let 6f = Ay = UKA^ Thus c, G H.

Then c

« = U
Cα^e 2. Otherwise. PROOF. For x G 6f, let 7^ = least 7 such that x G Aγ and if y G 6<, x < y iff Tar < 7y . Thus < is a prewellorder of 6, of limit order type. Notice < is computable within M from 6, , so that our solidity hypothesis on r means bi G \M\ and < G \M\. Now clearly, for x G fc.

so

where <* = ττ(<). By case hypothesis c, = {y € *(bi) I 3x €

so that c, is an <* initial segment of ττ(6t ). Subcase 2A. «>£**(<) ^ crit(£l). In this case, ran π is <* cofinal in π(6, ), so that ct = τr(ti), and c, G |Λ/Ί

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45

Subcase 2B. cofM(<) = crit(£). Let /c = cήt(E), f G \M\, f : /c -» δf such that ran / is <-cofinal. Then y G c< O 3α < ιc(y <* π(/(α))) ^ 3α < *(y <* π(/)(α)) so that c» G |Λ/Ί, as desired. D

Of course, we shall need to know that fine structure "up to level n" is preserved not just under passage to Ult n , but under iteration of this process. The following lemma summarizes the important facts. Lemma 4.7. Lei M =• MQ be n-sound, where n <ω. Suppose that for a < θ,

where EQ is close to ΛΊ α ; o>nd

M\ — dir lim Me β<\

for X < θ a limit. (We assume each Ma is wellfounded.) Let KM : M —>• MB be the canonical embedding. Then (a) 7Γo0 is an n-embedding. If, in addition, M is n + 1-sound (so n < ω) and p%\.\ < crit πo^, then = Pn+l(Mf

(d) Λίί is n + 1-solid, and in fact ίn+ι(Aί«) = M, and letting

be the inverse of the collapse, σ =

PROOF. This is a fairly routine induction on 0, using Lemmas 4.4, 4.5, and 4.6. The successor case is immediate from these lemmas, so let θ be a limit. Then (a), (b) are obvious, and (d) follows easily from (c). We sketch a proof of (c): let pn+ι(M) = (r, ή}, where f = (α0 at). For 7 < θ let 67 = Th£i (πθ7(α, ) U {^7(aO, . . . , πθ7(a, -ι), »o y(δ)}) Part of our induction hypothesis should be that b] G \MΊ\ for 0 < ϊ < £ This follows from 4.6 for successor 0, and for limit θ, our current case, from the proof of 4.6. For that proof shows that for each fixed i there are most finitely many

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W. J. MITCHELL AND J. R. STEEL

7 < θ such that ττ7+ι(&7) Φ ^7+1> since this can occur only when the ordering
REMARK. Under the hypotheses of 4.7 (including that M is n 4- 1-sound and pQt < crit(πo^)), we see that Me is not n + 1-sound. If crit(t'of ) Φ TM'(a,*oe(pn+i(M))] then Vά € [p^]<ω and r G Skn+1.

§5. ITERATION TREES We generalize the key tool of Martin-Steel [MS] to the fine structure context. DEFINITION 5.0.1. A tree order on a (for α £ OR) is a strict partial order T of a such that (1) (2) βT7=>β< 7, (3) {β I βTj} is wellordered by T, (4) 7 limit => {β | βTj] is cofinal in 7 (i.e. £ cofinal) and (5) 7 successor ^ 7 is a T-successor. DEFINITION 5.0.2. If T is a tree order then [0, T]T = {T? I η = /? V /?Tr;T7 V r; = 7} and similarly for (/?,γ]τ, [/?,τ)τ, and (/?,7)τ DEFINITION 5.0.3. T-Pred(7 + 1) is the unique ordinal ηTj such that (η, 7)7 = 0. DEFINITION 5.0.4. Let X = jf 7 > /?, J*4 is undefined.

be a ppm. Then for 7 < /?, 3** = J7^. For

DEFINITION 5.0.5. Let M and W be ppm's. Then M is an initial segment of Λ/' iff 3j(M = »7^) -M is a proper initial segment of ΛΓ iff Λ4 is an initial segment of .Λf and A is not an initial segment of M . Notice that if β £ dom E, then (Jβ , £, J? f /?) is not an initial segment of jf according to our definition, although we might reasonably have regarded it as such. DEFINITION 5.0.6. Let M and tf be ppm's. Then M and tf agree below 7 iff Jj* = J* for all β < 7. (In particular, j£* is defined iff J^ is defined, for all

β<τ) If Λί is a ppm then a iteration tree of length θ on M is a 4-tuple T = (T, deg, £>, (Ea, M'a+1 I α + 1< θ)),

where T is a tree order, which satisfies conditions (1-8) below. We write pa for the natural length of Ea. We will also define ppm Ma for α < θ and embeddings iaj : Ma -» Λ^ί^ for ordinals α and β less than 0 such that aTβ and £>n(α/?]τ = 0. (1) ΛΊo = My and each Λ^α is a ppm. (2) £*α is the extender coded by F^ t for some active ppm N which is an initial segment of Ma.

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W. J. MITCHELL AND J. R. STEEL

(3) α < β =» \h(Ea) < \h(Eβ). (4) If T-Pred(α + 1) = β then /c = crit Ea < Pβ> and Λΐ£ +1 is an initial segment JΊ+*β oΐMβ such that P(κ)ΠΛίJ + 1 = P(κ)Γ\λf. Moreover Λ4 A

α + 1 E .D <=> »/7

is a proper initial segment of Mβ .

If we take n = deg(α + 1) then /c < pί^"1"1 and

and if a + 1 §ϋ Z), then */?,<*+ 1 = canonical embedding of Mβ into Ult n (Ai0, J£α) , and i7,α+ι = */?,α+ι ° *y,β f°Γ a^ 7^ suc^ ^na^ (7ι )^]τ Π D = 0. (5) If λ < θ is a limit, then Z) Π [0, λ)χ is finite, and letting 7 be the largest element of DΠ[0,λ) τ , = direct limit of Λί α , α 6 [7, λ)τ, under the iα^'s fjyλ = canonical embedding of Mη into M\ , for 77 £ [7, (6) ΛC+i is deg(α + l)-sound. (7) If 7 + IΓα + 1 and D Π (7 + 1, α + l]τ = 0, then deg(7 + 1) > deg(α + 1). (8) For λ < θ a limit, deg(λ) = deg(α + 1), for all sufficiently large α 4- 1 Tλ. Notice that T determines the ordinals pα's, the embeddings t'α^X and the ppm Ma.

Conditions (6-8) can be dropped in some contexts. Condition (6) guarantees that i*+1 is a deg(α + l)-embedding. Condition (7) says that the ultrapowers taken along branches of T are of decreasing element arity; it allows us to "copy T" via certain embeddings. Lemma 5.1. Let Ί = (Γ, deg, D, (^α,Λ<*+1 | α + 1 < θ)) be an iteration tree. Then ifa<β<θ (1) Maf and Mβ agree below Ih Eat and (2) lh(^α) is a cardinal ofMβ, and in particular Ma and Mβ do not agree below lh(Ea) + l.

PROOF. By induction on β. Let β = 7 + 1. Since a <j =>lhEa < lh# 7 , it is enough for (1) to show that MΊ+\ and MΊ agree below lh£*7. Let EΊ =

FINE STRUCTURE AND ITERATION TREES

49

where Λf is an initial segment of MΊ. Now MΊ agrees with Ult0(W, EΊ) below lh£"7 by coherence. But M7+ι = Ultn(Λ<7+1,.E7), where M*+1 is an initial segment of Ms, some 6 < 7, with crit EΊ < mm(ORMl+l,lhEs). By induction, Ms agrees with MΊ below IhEs, hence below crit EΊ. Thus Λί*+1 agrees with MΊ below crit F7. So Λί7+ι agrees with Ult0(W, EΊ) below IhE^, hence with MΊ below lh£"7. (Notice here that if η < lh£"7, then the function representing jf*y+l is in both ΛΊ +i and tf. In fact, P(crit F7) ΠΛi +i = P(crit £7) Π.ΛΛ For C is true by fiat and D by our induction hypotheses.) For the second assertion it is enough to show Ih EΊ is a cardinal in MΊ+\ (using (1) and strong acceptability). Let us adopt the notation of the last paragraph. The definition of good extender sequence guarantees Ih EΊ is a cardinal in Ult0(JV, EΊ). But if A C Ih EΊ and A G MΊ+\ then A = [α, /] for some function / : [crit(£7)]" -+ j"$\) in M;+I. But then / G λf, so Λ G Ult0(^, £7), so Λ doesn't collapse lhF 7 . We leave the case β is a limit to the reader.

D

Let H\ be the set of sets hereditarily of cardinality < λ. From 5.1 we get, using the notation there, that if a < β and λ = lhEa, then H^fi = \Jχ*a\. A few miscellaneous remarks on the definition of an iteration tree: (a) It is easy to see from the above that if T is an iteration tree of length θ , α < β < 0, and F is an extender from the Mβ sequence (i.e. F on EMfl or F = FM*), then Ea \ pa + F \ pa. For suppose Ea \ Pa = F \ pa If F is on i^*, this implies Ea \ pa 6 Λ40, and therefore that lh£"α is not a cardinal of Λί/?, contrary to 5.1. If F = F^*, then ί/^" = v > lhEa since Ih Ea is a cardinal of Λί/j, and pα < i/. By the initial segment condition on good extender sequences, F \ ρa G Mβ. Since Ea \ pa collapses lhEa, we again have a contradiction. (b) The demand in (4) that crit Ea < pβ, rather than just crit Ea < r makes a difference only when Eβ = F for some P of type III, so that pβ = i/^, p and crit Ea = pβ = v . In this case our official definition won't allow us to apply EQ to an initial segment of Mβ to form Λf β+ι (c) Suppose we have an iteration tree T = (Γ,deg, D, (^Λς+x I τ + K *+!)) , so that the last model Λί Λ of T is determined. Suppose F = F^ for some initial segment P of MQ. How may we extend T one step further so that F = £"α? Let us assume all ultrapowers to follow are wellfounded. Assume also that Ih F > Ih EΊ for all 7 < α. Let /c = crit F. (i) We may set αTα -f 1 and take ΛΊ*+! to be any initial segment of MQ such that P is an initial segment of Λί^+1 and P(/c) Π |P| = P(/c) Π |>ί;+1|. Notice

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W. J. MITCHELL AND J. R. STEEL

that if Q is a type III initial segment of Ma, P an initial segment of Q, and P(/c)^ = P(κ)S, then /c < v<* since (*+)*> = (*+) IhEβ and P(/c) Π \J^β \ = P(/c) Π |J^ |. Any of these candidates will do for ΛΊ*+1. Notice again that if Q = JΊ ft for such a 7, then K < ι/Q as (/c+)^ is a cardinal of Q. So we can squash Q if necessary and still apply F. In almost all of the iteration trees used in this paper, the extension of T to α + 1 will be determined by the choice of Ea We take Γ-Pred(α + 1) to be the least ordinal α*, if there is one, such that pa > crit(#α) and α* = T-Pred(α + 1) = α otherwise. Then we take Λί^+i to be the largest initial segment of Λία which does not contain any subset of crit(J£α) other than those measured by Ea. Finally we take deg(α + 1) to be the the largest ordinal such that Λ4 * crit(J£α) < ρn 0f"M. See the definition of n-maximal, definition 6.1.2, for details. Iterability. If T is a tree order on 0, then a branch of T is a set 6 C θ such that b is wellordered by T with limit order type, and Vα 6 bVβ(βTa = > / ? € & ) . We call b cofinal iff sup 6 = 0. We call b maximal iff 6 φ [0, λ)τ for all λ < θ. If T = (T, deg, D, (Ea, Λ^+i | « -h 1 < 0)) is an iteration tree, then a (maximal, cofinal) branch of T is a (maximal, cofinal) branch of T. If 6 is a branch of T such that D Π 6 is finite, with largest element 7, then we set = direct limit of Λί α , <* G b — 7,

under the

iα^'s .

We say a branch bofT is wellfounded iff D Π b is finite and Λίj is wellfounded. We now state the iterability property which qualifies premice having no more than one Woodin cardinal as mice. We shall eventually show that all levels of the model we construct have this property by quoting results of Martin-Steel

[MS]. DEFINITION 5.1.1. VT = (T,deg,D9(Ea,M*a+i \a+ 1 < 0)} then for β< θ

DEFINITION 5.1.2. Let T be an iteration tree of length 0. T is simple if and only if every maximal wellfounded branch of T is cofinal in 0, and T has at most one cofinal in 0 wellfounded branch. Notice that by definition 5.0.1(4) it follows that T is simple iff for every limit λ < 0, T \ X has at most one cofinal wellfounded branch. We shall deal almost exclusively with simple iteration trees. The fact that it suffices to do so is one of the key things we must prove, (c.f. Theorem 6.2.)

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51

DEFINITION 5.1.3. Let /c < ω. Then an iteration tree T is k-bounded iff degr (α + 1) < k whenever α is such that [0, α + l]τ Π Dτ = 0. Notice that by clause (7) in the definition of "iteration tree", if deg(α + 1) < k whenever α -f 1 £ D and T-pred(α -f 1) = 0, then Ύ is Jb-bounded. DEFINITION 5.1.4. Let M be a ppm, and let k < ω. (1) M is singly k-iterable if any Jfc-bounded iteration tree

Ί = (T, deg, D, (Ea, M*a+l \a+KΘ)) such that T \ X is simple for all A < θ satisfies conditions (a) and (b) below: (a) If θ is a limit ordinal, then T has a cofinal wellfounded branch. (b) Suppose α < θ = β + I and M is an active initial segment of Λί/j, such that crit(F^) < pα, and suppose that T> = jf*<* for some 7 > lhEa, with /c = crit(F^) < p* and P(κ) Π \P\ C tf. Then is wellfounded (provided also n < k when [0, α]τ Π D φ 0 and P = ΛΊ α ) (2)We say M is k-iterable if it is singly fc-iterable and satisfies conditions (a) and (b) below: (a) If n < ω, and (7J : i < n) is a sequence of iteration trees such that TQ is a Jk-bounded simple iteration tree on Λ4, and for i > 0 7; is a simple iteration tree on the last model M^.'1 of 7ί_ι, and 7ί is Jb-bounded whenever Dr* Π [0,fy]Tj = 0 for all j < i, then the last model Λί£ of Tn is singly fc-iterable. (b) Suppose that (T{ : i < ω) is as in (a). Then [0,0, ]τ, Π A = 0 for all but finitely many i, so that we have a canonical embedding r, : MQ —> Λ^ό"1"1 = jA/fJ defined for sufficiently large i < ω. Moreover, the direct limit of the Λ^ό's under the TV'S is wellfounded. It is easy to see that if M is fc-iterable, T is a Jb-bounded simple tree on Λί, and P is a model on T, then P is Jb-iterable. It may seem that we can derive (2) and (3) from (1). Given Tί's as in (2) or (3), we can lay the Tfs "end-to-end" and produce a tree S to which we can then apply (1). The problem is that S may not be, formally speaking, an iteration tree: we may have a < β such that lh£"f ff IhEβ. This can definitely occur in the proof of the Dodd-Jensen lemma on the minimality of iteration maps, which is our application of (2) and (3). Rather than generalize the definition of "iteration tree" we prefer to complicate the definition of iterability. The fc-iterability of M allows us to build it-bounded iteration trees on M freely as long as the tree built so far is simple. For then (lb) guarantees we can proceed at successor steps without fear of illfoundedness. Clause (la) guarantees that at

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a limit ordinal λ we have a cofinal in A wellfounded branch. Thus we can choose this branch to be [0, \)τ It should be remarked that a theorem of Woodin asserts that the model L[S\ which we are constructing is not fully iterable, in the sense that there is a tree which is a member of L[E] but which has no well founded branch which is a member of L[E]. If we make the additional assumption that every set has a sharp then we can prove that V ^= L[E\ is iterable: that is, every tree on L[E] has a well founded branch, with both the tree and the branch being in V. It is a theorem of ZFC that every iteration tree which involves only extenders from a proper initial segment of the sequence E has a well founded branch, so that this much iterablity is true in both V and L[E]. The proof that our construction works will depend on this iterability in V of initial segments of E. It is important for this that L[E] has no more than the one Woodin cardinal, which is the supremum of dom(E). DEFINITION 5.1.5. Let M be a ppm. Then M is 1-smα// iff whenever K = crit F* for some initial segment M of M, then J** (= "There are no Woodin cardinals". It is possible for a 1-small ppm M to satisfy "there is a Woodin cardinal"; however, such an M cannot satisfy "there is a sharp for an inner model with a Woodin cardinal". DEFINITION 5.1.6. A 1-smα// mouse is a 1-small, u -iterable premouse. DEFINITION 5.1.7. A 1-small coremouse is a 1-small mouse which is completely sound. In general (for models with more than a Woodin cardinal) ω-iterability will not convert a premouse into a mouse. Since all the mice we shall deal with in the moderately near future will be 1-small, we make the temporary convention: mouse = 1- small mouse coremouse = 1- small coremouse Embed dings of Iteration Trees. We now head toward the Dodd-Jensen lemma on the minimality of iteration maps. For that we must show, given an embedding π: M —* λf and a iteration tree T on ΛΊ, how to extend π to an embedding from T into an iteration tree U on N. Since not all of the embeddings involved will be full n-embeddings we need a new definition: DEFINITION. We say π: M —*• λf is a weak n-embedding if M and M are premice of types I or II or sppm's, and there is a set X C M such that the following four conditions hold: (i) The models M and λf are n-sound, and X is a subset of M such that

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X, and X is cofinal in />£*. (ii) 7Γ is rΣn (respectively gΣ n ) elementary, and π is rΣn+ι (respectively q Σn+ι) elementary on parameters from X. (iii) π(Pi(M))=Pi(tf)foτi Q is a weak n-embedding and K is an ordinal in ORP. Then P ^= K is a cardinal if and only ifQ \= π(κ) is a cardinal. PROOF. This a is obvious if n > 1, so let n = 0. Recall po(P) = OR7*, so that the set X on which π is rΣi elementary is cofinal in OR^. Fix /c s.t. P ^= K, is a cardinal, and let μ G X be such that /c < μ. Let ξ £ X, μ < ζ, be such that

where "ST" refers to the £th level of the Jensen 5-hierarchy. Then P \= card5* (μ) is a cardinal and as £, μ € X

Q \= card s^»«) (ττ(μ)) is a cardinal. So, setting i/ = card5* (μ), we know that /c < ι/ and π(ι/) is a cardinal of Q. If K = v we're done. If /c < i/, then jf \= K is a cardinal, so since the relation R(z, x) O "x is a cardinal relative to z" is Σo-in-£ \ {F} we know that J%v\ (= τr(/c) is a cardinal, and hence Q \= π(/c) is a cardinal. D Lemma 5.2 (Shift lemma). let π : M —> M ψ \ λf —> N

Let M and N be ppm's, let k = crit(F
be a weak Q-embedding

such that M and N agree below (κ+Y* < (/c"1")^, while M and M agree below (κ+)M < (/c+)^, and π \ (*+)* = <φ \ (κ+)*. Suppose k < p*, so

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that Ult^M^F*) makes sense, as does \Jltn(M9FM), and that both of these ultrapowers are wellfounded. Then there is an embedding σ: Ult n (Λ4,F ^) — > Ultn(Λ< , F^) satisfying the following four conditions: (a) The map σ is an weak n-embedding, and ifπ is an n-embedding then so is σ. (b) Ultn(A4,F^) agrees with jf below Ih(F^), while Ultn(Ai,F^) agrees with M below Ih(F^). (c) σ t Ih(F^) + 1 = V r Ih(F^) + 1(d) The diagram

M

——>

M

l commutes, wiere i and j are the canonical n-embeddings. Remark. We want to allow the possibility (κ+)M = OR^. In this case, we make our standard convention: π(ORΛ) = OR^. We allow Ih(F^) = OR^ as well, and make a similar convention in (c) of the conclusion. PROOF. The map σ is defined by

W(α), /r,»(f )]/ir

if n > 0.

If X is the set used to show that π is a weak n-embedding then the set i"X will show that σ is a weak n-embedding. It is straightforward to verify that this works. D DEFINITION. If Ύ and £/ are iteration trees then we say that π = (ττ α : a < lh(T) ) is a weak n-embedding from T to W if the following 6 conditions are satisfied. (1) Tr = 1", degr = deg" and Dτ = I? . (2) TΓo : Mo —* Λ/o is a weak n-embedding. (3) For each ordinal α with 0 < a < IhT there is a set Y such that fl"α* Λ^α —>• Λ/"α is a (degr(α), y)-embedding, where Λ4 α and Λ^r are the αth models of T and Z/ respectively. (4) πa \ \hEa + l = * f \ \hEa + 1 whenever α < δ < θ' . (5) π7 o iJ7 = i^7 o πα whenever αTγ and (α, γ]χ Π D = 0. DEFINITION. We say that TT is a
Suppose that T = (T, deg, D, (Eα, M*α+1 I α + 1< *))

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is a n-maximal, n-bounded iteration tree on M, where n < ω, and that π: M —> J\f is an weak n-embedding, where N is a n-iterable premouse. Then there is a tree πT on M and a tree embedding π: T —> πT such that πo = σ. PROOF. We define πT \ α + 1 and πα by recursion on α < θ. For β = 0 we have Λ/o = λf and TΓQ = π. Now suppose we have defined πT \ β, together with sets Yα such that πα is a (degr(α),yα)-embedding for each ordinal α < β. If β > 0 is a limit ordinal then we set λfβ = dirlim{,Vαr : αTβ and D Π [α, β]τ = 0 }, where the direct limit is taken along the maps jαΊ> and we define π/j by setting πβ(i<*β(x)) = jα,^(^"α(ίp)) for αT/? such that [α,/?)τ Π D = 0. Finally we set yα = iβt0t"Yβ for any /?Tα large enough that ij α is defined. For successor ordinals β = 6 + 1, let £$ = Fp, where 7> = Jj* . Set Q = J*δ(η), (with the usual convention if η = dom π$) and f j = ί1^. Let Γ-Pred(6+1) = α, let Λ<J+1 = J^** i and set .Λ/£+1 = J*J(Ίγ again with the usual convention if 7 = dom πα. We will use the shift lemma, to define vt+\. Let σ be the natural embedding AΊJ +1 into Λ/J +i Let /c = crit ^7. Then (/c+)Mi+i < lhjE;α (possibly with (R+)M +ι = ORM +ι), so σ and π7 agree up to and at (ϊi+)M +ι. Thus we can apply the shift lemma to get π&+ι: MS+I —* Λ/i+i satisfying our inductive hypotheses on commutativity and agreement. If M J+j = Mα and degr(ί+l) = degr(α) then set Yw = <£,+1"yβ. Otherwise take Yw = i*βτ"M*β. To see π^^i is a deg(ί + l,y^^ι)-embedding when T-Pred(ί + 1) = 0 , use nboundedness, and for Γ-Pred(6 + 1) > 0. Note that σ is fully elementary if •MJ+i Φ Mα> and that degr(ί + 1) < degτ(α) if the degrees are not equal.

of

This finishes the recursive definition of πT, and it only remains to verify that each Mβ is well founded. Suppose that it is not. Since M is n-iterable, it follows that there is another branch 6 in T, cofinal in /?, such that if M is the limit along the branch 6 in U then λfb is well founded. This is impossible since there is an embedding π^: Mb —* λfbi where Mb is the limit in T along the branch 6, and Mb is ill founded since T is simple and Mβ is well founded. G The Dodd-Jensen Lemma. We are now ready to prove the Dodd-Jensen lemma on the minimality of iteration maps. This is a powerful tool which will be crucial in what follows. We shall call it simply the Dodd-Jensen lemma, though without meaning to suggest that this is the most important of the lemmas which they have proved. Our proof is just the obvious generalization of the original proof of Dodd and Jensen. Lemma 5.3 (Dodd-Jensen Lemma). Let T = (Γ, deg, D, (Eα, M*α+1 | α + 1 < ϋ + 1}) be an n-bounded, simple iteration tree of length θ + 1 on a n-iterable

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premouse MQ. Suppose σ:MQ-+Q is an weak n-embedding, where n <ω and Q is an initial segment ofM$. Then (1) Q = M9. Moreover, if there is an ordinal 7 G [0, θ]τ such that deg(γ') > n whenever J1 G [7, θ]τ then the following two clauses hold in addition: (2) D Π [0, θ]τ = 0, so that deg(γ) = n for all j G [0, θ]τ, (3) »o,*(n) < σ(η), for all n G OΛ Π M Q .

Remark. Notice that the additional precondition for clauses (2) and (3) is equivalent to the condition that MB is n-sound. This equivalence will be used in many of our applications: We will know from the construction of σ that Q is n-sound, so that clause (1) implies that MΘ = Q and hence Me is n-sound so that that clauses (2) and (3) of the lemma must are valid as well. PROOF. We will define a sequence ( 7ί : t < ω ) of iteration trees as in clause (b) of the definition of Jb-iterable, together with maps σ, : MQ —* M19 where M1Ί is the γth model of 7i. For each integer t the pair (7ί,σ, ) will satisfy the same conditions as the pair (To, σ0) = (T, σ), and it will follow that any failure of the lemma will imply that ( 7J : ί < ω ) violates condition (b) of the definition of Jb-iterable. We first give the definition under the assumption Q = M$. We will then modify the definition slightly to prove that Q = M$. We have TQ = T and σo = σ. Now suppose we are given a simple, n-bounded tree Ti on the n-iterable model A/ό, together with a (n,X t )-embedding σ, : MQ —> Λ ί . Let Q is n-iterable and 7ί is simple, M\ = M^1 is n-iterable. Thus 7ί+ι has length θ + 1 and is simple and n-bounded. Let π1 : 7ί —» σ, 7ί = 7ί+ι be the tree embedding given by the copying procedure, and set

Since deg(γ + 1) > n for all sufficiently large 7 + 1 G [0, 0]τ, 0"t+i is a (n, embedding, where Xf +ι is given by the copying procedure. Thus we are ready for the next stage of the construction. r

This completes the definition of the T^s and σ, 's. We must have jD(Ί[0, θ] ± 0, since otherwise Di Π [0,0]τ; -φ 0 for all i < ω, contradicting clause (b) of the definition of Jb-iterable. Thus there are canonical n-embeddings

given by composing the embeddings along the branch [0, 0]τ of 7i. We have the commutative diagram

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__ T2

σQ I A jQ

T

°

Suppose toward a contradiction that »'o,0(*?o) = ro(»/o) > σ(^o) Set ι;t +ι = <Ti(ηi). It is routine to check that rf (f7$ ) > σ, (ι/, ) = ι/,+ι for all ί and it follows that dirlim(Λit : i <ω) is not well founded, contradicting clause (b) in the definition of n-iterability To show Q = Me we proceed essentially as above. If Q φ MB we will have σ, : MI —> <2, , with Q0 = Q and Q, a proper initial segment of M* . In this case σt 7ί is a tree on Q, rather than Λ4J, but it can be modified slightly to make it a tree on Me which immediately drops to Q, at all TQ successors of 0. That is, 7}+i is the same as σ, 7ί except that we put γ f 1 into D\ whenever T-pred(7+ 1) = 0, and we set (Λ^+1)* = Qi or the appropriate initial segment thereof. With this modification the construction works as before, giving a sequence of trees ( T{ : i < ω ) such that Z>, Π [0, θ]^ Φ 0 for every i > 0 and thus contradicting clause (b) of the definition of n-iterability. Notice that in this case we don't need the hypothesis that deg(γ + 1) > n for all sufficiently large 7 + 1 G [0, 0], since for example it is not σt but σ, f Qi which will be used to produce 7<+ι, and σ, f Qi is fully elementary. D

§6. UNIQUENESS OF WELLFOUNDED BRANCHES We shall show that, roughly speaking, all iteration trees which are important for the comparison of 1-small mice are simple. Let T = (T,deg,D, (Ea.M^ \ a + 1 < 0}} be an iteration tree of length θ. We set £(T)= \J(EM~

\lhEa)

a<θ

δ(T) =

\JlhEa a<θ

By 5.1, EM* \ lh£α = EM> \ lh£α for all β > α, so that E(T) is a good extender sequence with domain included in ί(T). Notice that if b is a cofinal wellfounded branch of T, then E(T) = EM* \ 6(T). Theorem 6.1 (Uniqueness Theorem). Let T be an iteration tree of limit length θ, and b and c be distinct cofinal wellfounded branches of T. Let a = ORMb Π ORMc, so that a > 6(T), and suppose that a > δ(T). Then i« Woodin . PROOF. Just as in [MS]. Here is a slightly cleaner presentation of that argument, adapted to our context. Let δ = ί(T), E = £?(T), and let / : 6 -> δ with / G jf . Let β < θ be large enough that

D Π (6 U c) C β and bΠβ^cΠβ

and τ

€ 6 - β => /, ^, ί E ran i7» ,

τ

G c - )3 => /, E, δ G ran t7C ,

and α G ran t'7& if α ^ OR^6, and a E ran i;-c if α ,έ ORMc. CLAIM 1. If 7 e 6 - /? and η e c - )9, then (ran iγi Π ran iηc Π jf ) X Σ l ^f

PROOF. Straightforward. The restriction to ΣI is due to the limited elementarity of the maps iΊb,iηc

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CLAIM 2. Let 7 + 1 6 6 with Γ-pred(γ + 1) = ξ > β, and let η be a member of c such that β < c < 7 -f 1 such that if c < ξ then η is the largest member of c such that η < 7 + 1. Then ran iξi flran iηc Π δ = inf{crit i£&, crit i^c} . PROOF. D is obvious. Let us define

To = 7 + 1 τ/n = least ordinal in c — γn 7n_l_1 = least ordinal in 6 — τ/n for all n < ω. The 7n's and rjn's are all successor ordinals. Also we have supn ξ in which case this may fail), and T-pred(7o) = ξ. Now suppose μ G ran i^ Π ran iηc Π δ. As μ < δ, we have an n < ω such that

Since μ € ran z^ and ξTjn+ι, μ < crit £'7Λ^1 . By clauses (3) and (4) on iteration trees, μ < lh£?τ.pred(7n+1) Since μ 6 ran iηc and ηTηn, μ < crit Eη By clauses (3) and (4) on iteration trees

So we may repeat the cycle until we get μ < lh£"7o-ι. Then applying the argument again we get μ < crit£7o_ι ξ (under either hypothesis on 77) so that μ < Ih Ev , so μ < crit Ev. Thus μ < crit ι^c and μ < crit ίξδ. CLAIM 3. Claim 2 holds with the roles of 6 and c reversed. PROOF. The proof is the same as that of claim 2.

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Now fix /?' > β such that 6 Π (β1 - β) ^ 0 and c Π (/?' - β) φ 0. Let /c = least i/ such that v = crit #7 for some 7 + 1 £ (6 U c) — /?' . Let 7 be largest such that /c = crit EΊ and 7 + 1 £ (6 U c) — /?', and suppose without loss of generality that 7 -f 1 G 6. Let r; be the largest element of c which is < 7 -f 1. Notice crit iηc = crit £"„ for some i/ 4- 1 € c such that 7 + 1 < v + 1; thus crit iηc> K. So /c = ran iηc Π ran iξi Π 6 where £ = T-pred(7 -f 1), and it follows by Claim 1 that /c is closed under /. Now let v = inf{crit iηc, crit i7+ι,δ} Claim 3 implies that z/ = ran iηc Π ran i y+ι,& Π 8 so that i/ is closed under /. Note also that K < v. We claim that i/ < p7. (Recall that pΊ is the sup of the generators for EΊ.) Let T € c and T-pred(r) = η. Then ι/ < crit iηc < crit #τ_ι < pη. So if 77 = 7 we're done. Otherwise 77 < 7, so IhE^ is a cardinal of Λ<7, and as lhEη < lhEΊ, lhEη < ρΊ. As v < ρη, ι/ < ρΊ. Our initial segment condition on good extender sequences implies that EΊ \ v is an initial segment of some extender F which is on the sequence of Λ<7 before EΊ. By coherence we see that F is one of the extenders on E = E(T). So EΊ \ v G Jf . We leave it to the reader to check that v is an inaccessible cardinal of J f . By strong acceptability and the fact that F coheres with E,

Finally, suppose i^(/) = /. Then / f K, = / \ /c, and

so But

«'e,7+ι(/ Γ «) t ^ = •Xr 'ί/ t «) Γ " as computed in jj^. Thus EΊ \ v witnesses that δ is Woodin with respect to /

in For the purpose of comparison we are only interested in iteration trees in which each Ea is applied to the earliest model to which it can be.

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DEFINITION 6.1.1. Ί = (T,deg,D, (EaίMa+1 \a+l < θ)) is non-overlapping iff whenever Γ-pred(7 + 1) = β, then ρη < crit EΊ for all 77 < β. Here />,, is the sup of the generators for Eη, so that crit EΊ < pβ. Clearly, generators are not moved along the branches of a nonoverlapping tree, and in fact not moving generators is equivalent to being non-overlapping. We want also to restrict ourselves to trees in which Λ4 *+1 and deg(7 + 1) are as large as possible, subject perhaps to an n-boundedness requirement. DEFINITION 6.1.2. Let T = (T,deg, D, (Ea,M*a+ι \ α + 1 < θ)) be an iteration tree, and n < ω. We say T is n- maximal iff T is non-overlapping, and whenever T-pred(7 + 1) = β, EΊ = F^ where λί is an initial segment of Λ<7, and K = crit EΊ, then (a) ΛΊ^+i is the longest initial segment T> of Mβ such that P(«) Π \P\ = P(/c) Π μ/1, and (b) if D Π [0, 7 + l]τ = 0 then deg(γ + 1) is the largest integer k < n such AΊ*

that K < ρk Ύ+1 , and (c) if D Π [0,7 + l]τ τ£ 0, then deg(γ + 1) is the largest k G ω such that Notice that in (a) of the definition P is the longest initial segment Q of Mβ such that

Since Jlh ί = Jlh j, it follows that if β φ 7 then P is the longest initial segment Q of Λ40 such that P(/c) Π Q = P(/c) Π |Λί7|. The iteration trees for which we have any practical use are all n-maximal for some n < ω. One important elementary property of such trees is the following. Lemma 6.1.5. Let Ί = (T,deg, D, (Ea,M*a+ι \ a + 1 < θ)) be an n-maximal iteration tree, where n < ω; then for any a + 1 < θ, Ea is close to ΛΊ £+1PROOF. By induction on α. Let β = T-pred(α + 1). We may assume β < α; otherwise Ea is on the Mβ sequence, and so by the restrictions on how far Λf £ can drop in Mβ> on the Λ<* +1 sequence. Thus Ea is close indeed to Let α C lhEa be finite. We wish to verify the two conditions in closeness to jVi* + 1 for (Ea)a. We begin with the second. Let /c = crit Ea and T = IhEβ. As β = T-pred(α + 1), K < τ, and as τ is a cardinal of Ma, (κ+)M* < τ. Let A C P([/c]cardα), A e |Λίi+1|, be such that M*Q+! (= card(Λ) < K. We want to see that (Ea)a Π A e |X«+1|. Now P(/c) Π |Λία| = P(«) Π |Λi* +1 |, so A has cardinality < K in Λi α . But then (^α)α Π A is in Ma and has cardinality < /c there, by weak amenability. But then (Ea)a Π A G |Λ<* +1 |, as desired.

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It remains to show (Ea)a is ΣI over M«+I. The following claim is useful; notice that Jτ β is an initial segment CLAIM 1. If A C r and A G \MΊ\ for some 7 > β, then A is ΣI over jf*β. PROOF. By 5.1, A G |Λί/j+ι| Let A = [α,/]^, where Q = Λί£+1. Since A C r, we can take / to map [μ]cardα into J^, where μ = crit Eβ. We can therefore assume / G |Q|, as μ < ρ% where Mβ+i = UItm(Q,^). But also, Mβ agrees with Q below r, and / G J^* = P. Moreover, A = [α,/]^ = [α,/]^ . It is easy, then, to define A in a ΣI way over P from the parameters α and /. D It follows that if (Ea)a G |ΛΊ β |, then since (Ea)a is coded by a subset of r, (Ea)a is ΣI over ^r *, hence ΣI over Λί^i, as required. Thus we may assume that (Ea)a & l Mαl, and hence Ea is on the Ma sequence, Ma is active and Ea= ' CLAIM 2. Let 7 G [0,α]τ be such that 7 > /? and L> Π (τ,α]τ = 0- Then crit(i7α) > /c, and (#<,)<, is ΣI over MΊ. If, in addition, j > β and 7 is a successor ordinal, then crit(z'7>a o i* ) > « and (Ea)a is ΣI over Λ47 PROOF. Since K = crit Ea and £"α = F^0, K G ran t"7α. On the other hand, every extender used in i7β has length at least IhEβ, since 7 > β. It follows that K < crit(i7βr). By our induction hypothesis, #,, is close to Mη+ι for all r; < a. Thus the preservation facts recorded in 4.5, 4.6, and 4.7 hold for the embeddings of T \ (a + 1). Now pf1" < T = (κ+)Λ*0' since (^α)α ^ |Λίβ|, and r < crit ί τα , so deg(ry) = 0 for all η G (7, α]τ The proofs of 4.5 and 4.7 (see especially 4.5) show that every Σ^* subset of crit(i7α) is Σ^Ύ. Thus (Ea)a is Σ^ Ύ , as desired. Suppose finally that 7 > β and 7 is a successor ordinal. The extenders used in ι*7α o i* are just those used in i7α together with EΊ-\. Since 7 — 1 > /?, all these have length at least lh£"^, hence > K. The argument of the previous paragraph now shows crit(t'7α o *'7) > « and (Ea)a is ΣI over Λ4 7 . D Now let η G [0, α]τ be least such that β < η. Suppose first that DΓ\ (r/, α]τ ^ 0. Let 7 be largest in £) Π (TJ, α]τ, and ^ = T-pred(7). Since 7 > β, Claim 2 implies that (Ea)a is ΣI over >ί*. Since 7 G D, Λί} G |Λίf|, so (J^αjα G \Mξ\. Since ί > /?, Claim 1 implies that (£α)α is ΣI over Λί£+n ω desired. So we may assume D Π (TJ, α]τ = 0. We claim that η = β. For if η > /?, then the least ness of 77 implies that η is not a limit, so let ί = T-pred(τ ). Since η is least, 6 < β. By Claim 2 with 7 = η, crit(i*) = crit (£?,,. i) > /c. But crit(^«ι) < ps, so K < pt. But the rules for non-overlapping trees then require that T-pred(α + 1) < δ, a contradiction. So η = β. Also, by Claim 2, crit i/jα > /c, and (£"α)α is ΣI over Λί/j. But then

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63

P(/c) Π \Mβ\ = P(κ) Π |ΛΊ α |, and since Ί is n-maximal, Mβ = Λί* +1 . Thus (Ea)a is ΣI over M^+ι, as desired. D Lemma 6.1.5 has the important consequence that the preservation facts listed in 4.5, 4.6, and 4.7 apply to the embeddings along the branches of an n-maximal tree. We shall use this repeatedly and without explicit mention in the future. The following is a crucial strengthening of the uniqueness theorem (6.1). It will imply that only simple iteration trees arise in our proof that 1-small, fc-iterable premice are fc-solid for all k. This is important because our proof of that fact uses heavily the Dodd-Jensen lemma, which requires a simplicity hypothesis. If M is a ppm, an "extender from the Λ4-sequence" is an extender E such that E = FM or E is on the sequence EM . Theorem 6.2 (Strong uniqueness). Lei M be an n-sound, l-small n-iterable premouse and p£+ι < lh E for some extender E from the M-sequence and some integer n. Let T be an n-maximal iteration tree on M. Then T is simple. PROOF. Assume toward a contradiction that 6 and c are distinct cofinal wellfounded branches of T with OR^6 < ORMc. Let 6 = δ(T). CLAIM 1. IhF < 6 for all extenders F from the Mb sequence. PROOF. Let F be the first extender on the Mb sequence such that IhF > δ. Notice δ is a limit of Mb cardinals, as crit iab is an Mb cardinal whenever iab is defined. Thus IhF > ί, as 3i/ < IhFVγ < IhF (Mb \= card 7 < i/). Let 7 = IhF. By Theorem 6.1, j 7(r)

δ is Woodin

SO

G, E(T), F) \= δ is Woodin . Now let λί = Ult0( J**\ F). As F is a pre-extender over J**\ 7 E wfp(jV). By coherence and strong acceptability and the fact that 7 is a cardinal of Afy Af \= δ

is Woodin.

But then tf is not 1-small, so that Mb is not 1-small and hence M is not 1-small, which is a contradiction. D CLAIM 2. Mb is an initial segment of Mc. PROOF. Otherwise Mc is not 1-small. For let F be the first extender from the Me sequence with IhF > ί; if none exists Claim 2 is obvious from Lemma 5.1. So IhF > δ as in Claim 1. If Mb is not an initial segment of MC1 IhF < OR^6. But now we can show Mc is not 1-small as in Claim 1. D

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CLAIM 3. If ORMb < ORMc, then there is no dropping of any kind along 6; that is, Dr Π 6 = 0 and degr (α + 1) = n for all α + 1 € 6. PROOF. If ORMb < ORMc, then Mb is a proper initial segment of Λi c , and hence Mb is ω-sound since Mc is a premouse. But now suppose the last drop of any kind along 6 occurs at α + 1. Then α + 1 £ 6, and k = deg(α + 1) = deg(γ) for all 7 G 6 — (a + 1). Also, Λi* +1 is Jb + 1 sound and crit(iα+ιt& o t'^+i) = crit(ί*+1) > p^i*1 From Lemma 4.7 it follows that Mb is not k + 1-sound, a contradiction. D CLAIM 4. If ORΛ** = ORMc, then on one of b and c there's no dropping of any kind. PROOF. Suppose the last drop along 6 occurs at η -f 1, and the last drop along c at γ + 1. Since Mb = ΛΊ C , deg(τj -f 1) = deg(γ -f 1) = fc, where k < ω is least such that Mb — Me is not k -f 1-sound. But then

This implies that T-pred(ry + 1) = T-pred(γ -f 1). For let β = T-pred(r/ + 1); then E1^ is on the ΛΊJJ+x sequence, so Eβ is on the Aί^+1 sequence, so ^ is on the Λίξ-sequence where ξ = Γ-pred(7 + 1). Thus ζ < β by remark (a) following 5.1. That β < ζ is proved symmetrically. Now then «Vn.» o ij+i = «7+ι,c o «7+ι , since by lemma 4.7 each side is the natural embedding from &k+ι(Mb) to &k(Mb) = Λ4δ inverting the collapse. Since T is non-overlapping, crit ifj+ι,6 > Pr; and crit i7+i,6 > />7 So letting ί/ = inf(/?^,/?7), we have crit Eη = crit EΊ
then where for any ζ G b U c such that f > 17 -f 1 «

But then

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65

as iβtb \ p = iβtC \ p = id, and iβ,b(qβ) = iβtc(Vβ) = ( r » w )» where Γ is the Jb -f 1st standard parameter of (ΛΊδ, u) and u is as in the definition of pk+ι(Mb) (cf. Lemma 4.7). Let σ+1 e 6, r+ 1 G c, and T-pred(σ+l) = Γ-pred(r+l) = /?. As iβte = i^j, we see that crit J5?σ = crit Eτ, and E^ f ι/ = Eτ f i/, where i/ = inf(p σ ,p r ). This implies σ = r, a contradiction. D In view of Claims 3 and 4, we may assume there's no dropping of any kind along 6 (perhaps by exchanging 6 for c). The proof of the following claim will take several pages and will nearly finish the proof of theorem 6.2. CLAIM 5. p^ < δ. PROOF. We show by induction on η £ 6, that if αTr;, or if η = b and a £ 6, then

and

(**)

If /#! = *«!(/#.!) and Th£ϊ(/#ϊ U {?} * Ma) then Th^p^ U {«„,(«)}) * ΛV

By (*) for TJ = 6 and α = 0 we have ρ£J.\ < iQb(p'%+ι)
u

ί

For this, suppose Th^^ίp^'i) U {iΊη(q)}) = »e»?(x)' where we may assume ^TξTη. As iξ^ is generalized rΣn^i elementary, we see x = Thn+\(i7^(pn+r1 U {iΊξ(q)}). This contradicts (**) at £. Now let ?; = ζ + 1 and set β = T-pred(^). If (*) or (**) fails at η we must have q G \Mβ I such that but

Fix such a q. Let p = p^ , i = iβη , E = We may assume f ( ΰ ) C p for all ΰ E dom /. Also p < p^* by (*) and the fact that p^J0! < p^°. If we let .4 = {(ΰ, ι/) | i/ € /(δ)}, then A is (generalized) rΣn, !. Thus/€|Λί/j|.

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W. J. MITCHELL AND J. R. STEEL

Now (t)

* € TO, U {?}) O i(x) € [α, f}%>

since t is generalized rΣn+ι elementary. This gives an rΔft definition of Th^^U {q}) since Ea is rΣ^. This is a contradiction if n > 0, so we now assume n = 0. Let /c = crit E. We have /c ^' = (/C We will now complete the proof of claim 5 by showing that there is a rΣx β function t : /c —> p such that ran(t) is cofinal in p. To see that this proves claim 5, we let S be the set of triples (α,7, z/) such that 7 -<<(<*) *Λ where -
For any tf and X C |Λ/], let Thf (X) = Thf (X) Π {(y>, ά) I φ is pure rΣi} . Using the proof of Lemma 2.10 we see that Thx ft(p(J {q}) $. \Mβ\ implies that TΪh^(/>Ll{?}) ^ \Mp\, so we can use Th?ft(p(j{q}) n

instead

Let / be the function representing Thx (i(p) U {i(q)})> two cases:

ofTh?"(pU{q}).

We need to consider

Case 1. There is a total, continuous, order-preserving, rΣx ORMfi such that g"κ is cofinal in

β

function g : K —+

In this case, we set for ΰ G dom(/)

so that Λ is rΣβ . Notice that if A G Ea, then Ξϋ G Ah(ΰ) ± /(ϋ), as otherwise h\ At \Mβ\, so that Th^^^U {q}) G |Λί0|, a contradiction. Now set, for all ΰ G dom(/) Γ least α \ 0

such that (/(fi)Δ A(δ)) Π (w x (α U {ί})<α;) Φ 0 if no such α exists .

So t is total and rΣx ". It is enough to see ran(t) is unbounded in /?. Fix any ordinal θ < p. We will complete the proof of case 1 by finding a ΰ such that t(ΰ) > θ. Define a function k by

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67

Then k G \Mβ\ since it can be computed from Thf "(0 U {g,r}), where r is a parameter chosen so that the function g is Σl fl({r}). Moreover

(tt)

[a,fc]£> = ™"'(;(0)u {%)}).

One direction, D, of equation (ff) is easy. To prove C, let [6,^]^ β G [<*,&]#*, where we may assume α C 6. We may assume that for all t; G dom J

where v* is the appropriate subsequence of v. For v G dom I such that v0 is a limit, let

_ j"β s(ϋ) = least α < VQ such that I(v) G Thx g(θf) (0 U {q}) . Then 5 is a rΣx * map from /cn to K, so 5 G |Λi^|. By normality, fix c*o such that s(v) = c*o for EI a.e. v, and let ^ =
= τhίί«'(i(ί) u {»(,)}) c τEf'(i(β) u {»(?)}) , as desired. This completes the proof of equation (ft) It follows that there is an A G Ea such that for all ΰ G A, f(ΰ) Π (ω x (0 U {g})<«) = ft(δ) Π (ω X (0 U {,})<«) .

Let ΰ G A be such that Λ(ϋ) ^ /(ϋ); then t(ϋ) > 0. This completes the proof of case 1 of claim 5. 2. There is no function g as in case 1. In this case, define the function ί(ϋ), where ΰ G dom(/), by t(ΰ) = least α such that (/(ΰ) Δ Th?β(p U {g})) Π (ω x (α U {?})<ω) φ 0 . Thus t is total rΣx *. To see that rant is unbounded in /?, note that for θ < p Thf'W) U {%)}) = i(Thf ^(fl U {q}))

as

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W. J. MITCHELL AND J. R. STEEL

for some ζ < OR^* by case hypothesis. This completes the proof of case 2, and hence of Claim 5.

D

Fix now p G \Mβ\ and p < δ such that Th^(p(J {p}) <£ \Mb\. We obtain a contradiction via an easy generalization of the proof of 6.1. Fix β < length of T so large that (1) 6 Π β φ c Π β, and there's no dropping on 6 U c above β. (2) 7 G b - β => crit iΊb > p and p G ran i7& and (δ < ORMb => δ G ran a'7&). (3) 7 G c — β => crit ι'7C > p and p G ran ι'7C and (δ < OR^6 => δ G ran t'7C) and G ran ι7C). As in Claim 2 of the proof of 6.1, we can find 7 G b — /? and η £ c — β such that

ran ι"7j Π ran i^c Π δ = /c where p < K < δ. Let where X = ran i'7& Π ran ι,,c and π is the inverse of the collapse. Then π is generalized rΣn+ι elementary. This follows from the fact that both i7& and ιΊC are generalized rΣn+ι elementary. To see that iΊC is generalized rΣn^ι elementary, note that if Mb = Mc, then deg(f -f 1) > n for all sufficiently large ( + 1 6 c, so ί»;c is generalized rΣn+ι elementary. If Mb is a proper initial segment of Mc, then i nc f «7c1( ^*) *s ^n ^act ^u^v elementary. Notice that crit π = /c, and tf = J*(r)riί for some α > «. Also Th£j!ί(p U {p}) is definable over Λ/', and hence is a member of L[E(T) \ /c]. As J?(T) Γ « ^ 1-^6 1 and Mb has an internally iterable extender on its sequence with critical point greater than /c, we get Th£J.\(pU {p}) G \Mb\, a contradiction. This completes the proof of theorem 6.2. Π

§7. THE COMPARISON PROCESS We prove in this section a comparison lemma for 1-small mice. Our interest is not so much in the lemma itself, but in the method by which it is proved. We shall use that method in a much more important way in the next section. For bookkeeping purposes we shall use "padded iteration trees" . These are just like ordinary iteration trees except that we modify the successor clause in the definition of "iteration tree" so as to allow αT(α+l), Ma = Λf α+ι, and *α,α+ι = identity, and then require that aTβ => β = a + 1 or (α -f 1) Tβ. So a padded tree is essentially an ordinary tree with the indexing of the models slowed down by repetition. We shall no doubt often fail to distinguish between iteration trees and their padded counterparts. Theorem 7.1 (The comparison lemma). Lei M and N be n-sound, l-small, n-iterable premice, where n < ω. Then there are n-maximal padded iteration trees T on M and U on N such that either (1) T and U have successor length θ + 1, and either (a) Me is an initial segment ofNe and Dr ΓΊ[0,0]τ = 0 and deg(a + l) = n for alla + le [0, θ]τ, or (b) MB is an initial segment of Me and ΣP Γ\[Q,θ]u = 0 and deg(a + l) = n for alla + le [0,%,

or (2) T andlί have limit length, one of the two is not simple, and in some yc°l(**ω) there are wellfounded cofinal branches bofT and c ofU such that either (a) Mb is an initial segment ofAfc, a + 1 E bf or

Dτ Γ)6 = 0, and deg(a + 1) = n for all

(b) Mc is an initial segment of Mb, D^ C\c = 0, and deg(a+ 1) = n for all a+lζc.

PROOF. We define by induction on 7 T ί T = (ΓΠ (7 x 7), Dτ Πτ,deg r \ 7, {££,ΛCn I * + K 7» and

together with the associated Ma, Ma for α < 7, pζ and fa for a + 1 < 7, and embeddings ij^, %β (for (α,/?) as appropriate). The method for defining T and U is the standard one of "iterating the least disagreement" .

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W. J. MITCHELL AND J. R. STEEL

We begin with 7 = 1. In this case we need only define Mo and Λ/Ό, which we do by setting

Now consider the case 7 is a limit ordinal. Then

β<Ί

where the union is taken along each of the 4 coordinates. Now suppose 7 = λ + 1, where λ is a limit ordinal. If Ύ \ λ or U \ λ is not simple, we stop our induction. Suppose T \ λ and U \ λ are simple. As M and N are n-iterable we have wellfounded branches b of T \ λ and c of If \ λ which are cofinal in λ. Set

τn(τ x 7) = (rn(λ x λ)) u{(α,λ) I α e 6} [7Π(7 x 7) = (tf Π(λ x λ))U{(α,λ) | a € c} ,

r

for α E 6 - sup(D Π 6) ,

*'L = %b

=

for α G c

The rest of T \ 7 and U \ 7 is determined by this. Finally, we have the case 7 = τ?+2. Here we must define E%, -M*^, £)rn{τ7+l}, r deg (η+1), T-pred(τ7+ 1), and similarly for the U side. We are given the models Mη and J\fη. If Mη is an initial segment of A f η , or vice-versa, then we stop our inductive definition. Otherwise we have a least ordinal 7 such that JΊ n φ JΊ n . Set Γ 0

r

,

y Jl**

*?

—

n

if »77

η

is passive,

^ F&ι "

if JΊ

f 0

if JΊ " is passive,

I

if JV ' is active.

X

* Jy /r-^n

^ F

J\Γ

is active,

FINE STRUCTURE AND ITERATION TREES

71

The rest is determined by the rules for forming non-overlapping, n-maximal, padded iteration trees. So, on the Ύ side: If El = 0, then T-pred(i7 + !) = ?, M^ = M,+i = M,, *£„+! = identity. In this case, we also set p% = 0. 1

Now suppose E ^ ^ 0. Let K = crit £"^, and let β < η be least such that K < prβ. Then we set T-pred(τ7 + 1) = β. Let M *+! = longest initial segment V of Λί/j such that

p(/c)n|7>|= POOnlJ^'l = longest initial segment 7* of Λi/j such that

We let

τj+leI>r^Λί;+ Let Λ4*

Ar = largest m <ω such that /c < pm *+ί and T

£> Π [0, η + l]τ = 0 =* m < n .

We let degr(τj + 1) = Jb, and

and let ι*+1 : Λ^*+1 —*• Mη+\ be the canonical embedding, and for &T(η + 1) such that £>r Π (α, η + l]τ = 0, let iJ|JJ+1 = ij+1 o ίj^. This completes the definition of T f ?/ + 2. We obtain W f η + 2 from E^ in a similar fashion. This completes the definitions of T and U. It is easy to see they are iteration trees. CLAIM. If α < β, then max(/?J,/#) < min(/> PROOF. 7 = IhE^ is a cardinal of Mβ, and hence a cardinal of Jγ^βE*r

As

7 is a cardinal of Jlh J τ , 7 < />J. As 7 is a cardinal of «/ τ , T < /^ So lh ^τ, l h <7 < min(/?J,/^). Symmetrically, /# < mi Lemma 7.2. ίeί α -h 1, β + 1 < IhT. Suppose E% ± 0 and £^ ^ 0. Suppose crit E% = crit E% = K. Then there is a parameter a £ [p« Π />^]
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W. J. MITCHELL AND J. R. STEEL

PROOF. We may as well assume a < β. Notice then pζ < ffjf, and

P(«) n j

= P(«) n j£

= p(κ) n

by 5.1 and the fact that Mβ and tfβ agree below lhE%. So E% and Έty are defined on the same subsets of /c, and it will suffice to show

Suppose E% \ Pa = E% \ pζ. Now E^ is the trivial completion of E% \ /£, and so the initial segment condition on Hβ gives us two possibilities. We may have E% on the sequence of Λ//? at or before the position of Ejf . But E% is not on the Ma sequence because it is part of the least disagreement at α, and by coherence then EB is not on the λfβ sequence for all β > a. Thus the second possibility from the initial segment condition is realized: p% £ dom E^ and E% is on the sequence of Ult0(P, F), where F = (E^)pr and P = J^f . In this case, F is on the sequence of Λ^, and hence of Mβ as pζ < IhEβ. Thus F is on the sequence of MQ as p^ < InFj. Also P = J^}a . By coherence, F is on the sequence of Ult0(P, J5£) and F is not on the sequence of Ult0 (Ulto(P,F),f7j). This is a contradiction, as these ultrapowers agree past IhE'J. D CLAIM. The inductive definitions of T and U halt at some ordinal 7 such that 7 < max(card Λ<, card λί)+. PROOF. Let θ = max(card Λ(, card Λ/*)"*". If the claim is false, then Me and M$ are defined. Let 6 = [0, θ)τ and c = [0, 0)c/. So b and c are club in θ. By the standard closure argument we can find a club d C 6 U c such that

(i) Dr C\d= (ii) α E d => α = crit i^b = crit (iii) a,βedAa<β=>(%p(a) (iv) (α E d Λ A C [α]Λ Λ A E |Λ<β| Π \λfa\) =» t^(A) = Now let d satisfy (i)-(iv) and take a £ d. Let /? 4- 1 and 7 -I- 1 be the successor of α in 6 and c respectively, so that T-Pred(/? + 1) = U Pτed(y + 1) = a. Since T and ί/ are non-overlapping, critίj +ljk > pj and crit ^+ι>c > /^. By (iv) we see that for all A C [α]n, A E |Λίβ| Π |Vα|,

where /? = pj Π (% . It follows that Ej \ p = E^ \ p, contradicting the lemma. This proves our claim. Π There are two ways the construction ofT and U can halt. Suppose first we reach θ + 1 such that Me is an initial segment of Λ* or vice- versa. If Me is a proper

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73

initial segment of Λ/i, then there's no dropping along [0,0]τ because .Λ/i is a premouse so that Me is ω-sound. So we have (1) (a) of our desired conclusion. If Me is a proper initial segment of Me we have (1) (b) of our desired conclusion. Finally, if Me = Me then on one of [0,0]τ and [0,0]c; there's no dropping; the proof is just like that of Claim 4 in the proof of 6.2, so we omit it. Suppose next the construction halts because we reach a limit ordinal θ such that one of T \ θ and U \ θ is not simple. Say T \ θ is not simple, so there are distinct wellfounded cofinal branches ofT = T \ θ. Just as in the proof of the first 4 claims of 6.2, we can find a cofinal wellfounded branch 6 of T such that Dr Π 6 = 0 and deg(α + 1) = n for all α + 1 G 6, and Mb has no extenders with length > 6 = δ(T). Let c be any cofinal, wellfounded branch of U. If Mb is an initial segment ofAfc, we are done. If Afc is a proper initial segment of Mb then Λ/"c is ω-sound, so there's no dropping along c and we're done. The remaining possibility (since Mb and Nc agree below ί) is that λfe has an extender F such that δ < IhF < ORMb. But this contradicts the 1-smallness of λΓc. D Remark. We haven't ruled out the possibility that (1) of our Theorem 7.1 holds, that Me = λί$9 and that one (but not both!) of [0,0]τ and [0,0]t/ has a drop. Nor have we ruled out the analogous situation in case (2) of 7.1. One can show that this cannot happen in the case n = ω, but for n < ω we don't know.

§8. SOLIDITY AND CONDENSATION In this section we prove the central fine structural result of the theory we are developing, namely that every 1-small mouse is Jb-solid for all Jb. We also derive, by the same method, some condensation results we shall need later. Our proofs of these facts trace back to Dodd's proof that the models of [D] satisfy the GCH. For mice M up to a strong cardinal (that is, for mice M such that J*4 (= "There are no strong cardinals" whenever K = cήtE for some extender E on the M sequence), our proof actually shows that pk+ι(M). That is, every "very small" mouse is an iterate of its core. We suspect that this is not true for arbitrary 1-small mice. Recall that uQ(M) = 0, and that uk(M) = (ph(M), to, , bs, p^) for t > 1, where to *δs are the solidity witnesses for pk(M) and the last coordinate p^l occurs only if it is defined and is smaller than OR/'*. Thus pk+ι( M) is the appropriate collapse of {r, tijk(Λί)), where r is the fc + 1st standard parameter of Recall that if π : M —* M is a Ar-embedding, then π(uk(M)) = Uk(Λf). Theorem 8.1. Lei M be a k- sound, \ small, k-iterable premouse, where k < ω. Lei r be the k + 1st standard parameter of (Λί,tijb(Λί)) Then r is k + l-solid and k + 1 universal over (Λ4, PROOF. Let u = Uk(λ4) and r = (αo> , Qfs}, with the ordinals α, in decreasing order. Let α$+ι = p%+ι Let s < 5-f 1 be least such that T h > , U {αo^

, α,-ι,

Such an s certainly exists, since 5 + 1 will do. Let , «*-ι, tι}) ,

let π : Ή, —>• M be the inverse of the collapse (so that π is a Jfe-embedding), and let ΰ = π~ 1 (w) and δtj = π~1(αj ) for j < s. Our strategy is to compare Ή, with Λ4, using fc-maximal trees. Suppose that P is the model produced at the end on the W side, and Q the model produced on the M side. Suppose the branches Ή, to P and M to Q involve no dropping of any kind, so that we have generalized rΣ*+ι maps »: H -» P and j : M —> Q. Suppose criti > α, and critj > /?£+!. Then

f

U {i(

FINE STRUCTURE AND ITERATION TREES

75

and u

so that neither of T> and Q is a proper initial segment of the other, and hence P = Q. Now if M is not k + 1-solid then s < S -f 1 and hence p^ < p?^ because we didn't throw α, as a member into the hull collapsing to Ή,. But we can show PfcVi < PfcYi, so p$+l < p^ < p%+l < p£+1 contradicting the fact that P = Q. Thus M is k -f 1-solid. It follows that s = S + 1, and critj > ρ^\l so we have pM (Pk[ι) Q |W|. Thus Λ< is Jb + 1-universal. There are many problems in completing this sketch, but the main one is arranging that critί > α,. Our strategy will be to modify the comparison. Instead of comparing the models M and 7ί by iteration trees U on M and T on Ή, we will use a iteration tree U on the model M and a pseudo-iteration tree T on the pair of models (Λί,W). The situation can be represented by the following diagram:

M = PQ MΓO I -

_ T

*• J

M = Qo -The horizontal lines in this diagram indicate that the corresponding models are in the same tree, so that there is an embedding between them just in case they are on the same branch of the tree and there is no dropping on the branch between them. The comparison takes place between U, which is a genuine iteration tree, and the pseudo-iteration tree f. The thing which makes T_a pseudo- iteration tree, rather than a real one, is that its underlying tree T has two separate roots, —1 and 0, corresponding to the models P-\ = M and PQ =_W. We take p_ι = α β , and then we continue the comparison exactly as if T were a real iteration tree. This means that whenever an extender Ev appears in the pseudo- tree f such that cήt(Ev) < α, then T-Pred(ι/ + 1) = -1, so that the ι/ + 1st model T>v+\ of T is equal to Ult(^, Ev) for some initial segment P* of M. Since f is not a genuine iteration tree, we don't know directly that it has well founded branches. For this we use the iteration tree T and embeddings πα, which are defined by setting π_ι = id, letting πo be the inverse of the collapse map, and then using the shift lemma to copy T. Since T is a genuine iteration tree, theorem 6.2 implies that it is simpje. Thus it has well founded branches at every stage, and the embeddings πv : Pv -+ Pv ensure that the corresponding branches of T are also well founded. We will show that QTΘ and that there is no dropping along the main branch of θ either tree. Thus the maps Ϊ0>0 : Ή -» P and ijf^ : M —> Qe are defined. In

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W. J. MITCHELL AND J. R. STEEL

addition we show that Pg = Qβ and finally that Ϊ0,0 = 3£g o fl"o We now begin the actual proof of theorem 8.1. Notice first that if ρQl >lhE for all extenders E from the Λf-sequence, then H is already an initial segment of M (since α, > p£+ι) ^n ^is case, no iteration is necessary, and we have that Ji = ΛΊ, which easily gives the desired results. Thus we may and do assume that /?£{.! < Ih E for some extender E on the Λf-sequence. According to the strong uniqueness theorem then, every fc-maximal iteration tree on M is simple. This fact will make the Dodd-Jensen lemma applicable in what follows. We now define by induction on length: (1) a_"psuedo iteration tree" T on the pair (H, M), (2) a tree Ί on M "enlarging" T, and (3) a tree U on M. We use Pa, Pa, and Qa for the_αth models of T, T, and U respectively. We use T for the tree ordering of T,T for that of T, and U for the tree ordering ofli. The rest we indicate with superscripts; e.g., p%, pα, and f% or i^, ϊα0, and ^. The systems T and £/ will literally be a padded iteration trees on M\ they will be Jk-maximal and non-overlapping. T will not literally be a tree ordering in our sense, as it will have two roots, but will agree with T on OR — {0, —1}. Simultaneously with T, T, and U we define πa : Pa -+ Pa such that the map τrα is a weak deg(α)-embedding. We begin by setting

P-! = ΛΊ, PQ = W, 7>o = Λ<, Qo = M and π_ι = identity, TΓQ = inverse of collapse. Notice that π_ι and πo are t-embeddings. Now suppose that we have defined T \ θ, f \ θ, and U \ θ. (This means we have defined the models Pa^Pa, and Qa for α < 0, together with the extenders Ea, E0, and £*f, for α -f 1 < 0, etc.) Suppose we have also defined πa : Pa ~* Pa for α < θ with the following commutativity and agreement properties. (i) If afβ and DΓ\(a,β\f = 0 then (*β oπa = πβoιaβ. (ii) If 0 < a < β < θ, then PQ agrees with Pβ below lh^α; moreover letting 7 = lh^α and N = /f« = jf", we have πα f ΛΓ = π^ f N. Remark. Some simple observations about H. (1) We may assume a, £ |W|. For otherwise 7ί is an initial segment of M (if Λί and hence 7ί is active, then the initial segment condition on good extender sequences implies FH = FM \ ORW is on the Λί-sequence) but then W = Λί, and we are done. (2) W (= α, is a cardinal, since α, = crit π0 if s < 5 -f 1 and, α, = π(α,) = p^j

FINE STRUCTURE AND ITERATION TREES

77

(3) For β > 0 and K < α,, P(/c) Π \Pβ\ = P(/c) Π |W| = P(/c) Π |J^|. However, it seems possible at this point that P(κ) Π \Mj might be larger than P(/c) Π \H\. We now define Ί \ θ + 1, T f θ +1 and W \ θ + 1. CASE 1. 0 is a limit ordinal. In this case, we have only to pick cofinal wellfounded branches in each of our trees. As T \ θ is fc-maximal and p^l < Ih E for some extender E from the M sequence, T \ θ is simple. As M is fc-iterable, there is a cofinal wellfounded branch 6 of T. Similarly, there is a cofinal wellfounded branch c of U. Finally, let 6 = 6 or 6 = (6 - {0}) U {-!}, whichever is a branch of Ί'. Set Pθ = direct limit of Pa , a E 6 - sup Dτ PO = direct limit of Pa , a e b - sup D Qe = direct limit of Qa , a E c — sup ΣP and extend T, f and U to 0-f 1 correspondingly. For α £ 6 —sup ί) and x G \PQ\ we can set ^(ϊα,β(x)) = ίί,(?(?Γα(«))

(where of course ϊα,^ = « α j j, etc.), and by induction hypotheses (i) and (ii) this gives a well-defined π$ : P» —> Pe. Clearly πe is a deg (ί)-embedding and (i) and (ii) continue to hold. CASE 2. θ = η -f 1. In this case we "iterate the least disagreement" between Pη and Qη, as in the proof of the comparison lemma. Let 7 be least < OR*5* Λ ORQ* such that

if no such 7 exists then we stop the construction of T, T, and ZΛ Set f FJ** , if jf* is active

* η

~~

^ 0 I

~

ιτ# _ j ^^ " > ^ 0

otherwise At

^ ^r^11 ιs actiγe otherwise.

On the U side the rest is determined by the demands of a ύ-maximal iteration tree. So W-pred(τ; + 1) = ζ, where ί = least α such that crit £^ < Jί .

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(Assuming now E% φ 0; if E% = 0 we just pad for one step.) Let K = crit £^, let Q*+ ! be the longest initial segment M of Qξ such that P(κ) Π \λί\ = P(κ) Π \Qη I and let where n = largest s such that /c < />,

* and 5 < fc if Et4 Π [0, 77 + !](/ = 0 .

On the T side we proceed similarly. We assume Eη φ 0; otherwise we pad for one step. Set K = cτit(Eη) and let let T-pred(τ; -f 1) = £, where ξ = least α such that /c < />£ , so that in particular £ = — 1 if /c < αrs = p_ι. Now set 7^+ι equal to the longest initial segment tf of Pe such that P(κ) Π [.Λ/Ί = P(/c) Π |P,,| and

Q* where n is the largest integer s such that /c < p9 n"M and such that s < k if D Π {α I af(η -f 1) V α = η -f 1} = 0.

That 7*t7+ι agrees with P^ below lh^ is proved as usual. Notice that if ξ = — 1, then as P(/c) Π 1*7^1 = P(κ) Π \pη\, there is an Λ/" as called for in the definition Finally, we extend T by "copying" what we just did with f. Let 7 be least such that JΊ " φ Jιη . Assume that JΊ n is active; otherwise we just pad T for one step. Let where ^ = J , where as usual we let SUBCASE A. T-pred(»;-f 1) = -1. Let T-pred(fj -f 1) = 0,

Pη+i = Ult«(^;+1 , ^) ,

where n = de^ (9 + 1) .

We get TΓ^+i : Pη+ι —> ^+ι by the shift lemma, lemma 5.2 which implies that π,j+ι is a deg (η+ l)-embedding with the required commutativity and agreement properties (i) and (ii). SUBCASE B. T-pred(τ; -f 1) = ζ > 0. Let T-predfa +!) = {. Let P*+1 = then

ffi\

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where πξ(ORp*) = ORP< . Let n = deg (77 + 1), then

Finally, we get the desired π^+i by the shift lemma. This completes the construction of T, T, and II. We leave it to the reader to check the many details we ought to have verified in the course of the construction. (In particular, that T is a i-maximal iteration tree, and that the π^'s have the required commutativity and agreement properties.) Now because T and U are simple we must reach an ordinal Θ such that Pe is an initial segment of Qe or vice- versa. The proof is exactly the same as the proof in §7 that the comparison process stops. We shall say that a branch 6 of U drops if either ΣPr\b φ 0 or Ξα £ 6 (deg (a) ^ k), and similarly for branches of T or T . We need to verify that, just as with the comparison in section 7, at most one side of the comparison drops, and that the side which drops is the longer. That is, if the main branch { β : βTθ } of T drops then the main branch [0, θ]u of U does not drop and Pe is not a proper initial segment of Qe , while if the main branch of U drops then the main branch of T does not drop and Qe is not a proper initial segment of Pelt is immediate that if either branch drops then its final model is not u -sound, and hence cannot be a proper initial segment of the final model of the other branch. If follows that if both branches dropped then we would have Pe = Qe> This implies that if the last drop on { β : βfθ } occurs at α + 1 and the last drop on [0,0]u at /?+!, then + 1) = the least n such that Qe is not n + 1 sound = the least n such that Pe is not n + 1 sound

Moreover, if n = deg^(/? + 1),

Also crit

o,

and CΓit ta+l,t O ?a+l >

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Also crit E% = least /c such that K φ τQβ[δi,pn+ι(Qθ)} for any rGSk n +! and ά G (p^)<ω = least /c such that /c ^ ^*[ά,pn+ι(^)] for any r G Skn+χ and ά G (p?;^" = crit £α . Finally, if A G QJ+1 and ,4 C [crit JSj^]n, then letting

where

It follows that one of £^ and J5α is an initial segment of the other, and this is a contradiction as in the proof of the comparison lemma. Thus at most one of the trees Ύ and U can have a drop along its main branch. CLAIM 1. OT0, that is, Pθ lies above PQ = Ή in the f system. PROOF. Assume not; that is, assume that — 1T0, so that P$ lies above *P-ι = M. We know that at least one of the branches [— l,0]y and [0,0]t/ does not drop. CASE 1. [-1,% drops. Then P0 is not ω-sound, so is not a proper initial segment of Qg. Suppose first Qβ is a proper initial segment of ϊ>g\ say Qe = jf9. Let σ = π$ \ J^9 , so that σ : Qo —f Jf/~\ is fully elementary. Then the map σoi^θ is a weak Ar-embedding from M to a proper initial segment of Tg. As T is Jb-bounded and simple, this contradicts the Dodd-Jensen lemma. Suppose next that Qe =Pe Then as Qe is fc-sound, deg(0) > k, so that πe is a weak Jb-embedding. Thus πe o i£9 is a weak Jk-embedding from Λf to Pβ But by case hypothesis, [0,0]τ drops. This contradicts the Dodd-Jensen lemma. (As (θ) > ίr, we must have Dr Π [0, θ]τ φ 0.) CASE 2. [0, θ]u drops. In this case, [-1,0]? doesn't drop and Pe is an initial segment of Qe. If proper, then ϊ_ι ( 0 is a Jb-embedding of M to a proper initial segment of Qe, which lies

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on a fc-bounded, simple iteration tree with base model M. This contradicts Dodd-Jensen. If Pθ = Qβ, then as Pθ is Jb-sound, de^(θ) > k. But then the fact that [0,0]c; drops contradicts Dodd-Jensen. CASE 3. Neither [-1,0]? nor [0,0]c/ drops. If Pe is a proper initial segment of Qg, then t_ι f f contradicts Dodd-Jensen. If Qe is proper initial segment of P$, then π$ o i^ θ contradicts Dodd-Jensen. So we must have P$ = Qe . We now use the minimality of the iteration maps x^, t"0"0 given by the Dodd-Jensen lemma. Let
since i^ is afc-boundediteration map and i^oτr^ is a weakfc-embedding.Then

so so But if f-i^ = i^, then the first extender used on [— l,β]^» is compatible with the first extender used on [0,0]t;, which is impossible. D This proves Claim 1, and it follows that [0, θ]f is the main branch of T. Again, we know that at most one of the branches [0, θ]f and [0, θ]u drops.

CLAIM 2. [0,% doesn't drop. PROOF. Suppose it did drop. Then [0, θ]u does not drop and P$ is not a proper initial segment of Q$. Suppose Qβ is a proper initial segment of P$. Then π^oi^tf is a weak t-embedding from M to a proper initial segment of P$ , contrary to Dodd-Jensen. Suppose Q$ = Pφ. Then as Qe is fc-sound, deg(0) > fc, so that π $ o f β θ is a weak Jb-embedding from M to Pg . As [0, θ]τ drops and deg (θ) > fc, we have Dr Π [0, θ]τ ^ 0. This contradicts Dodd-Jensen. D CLAIM 3. Qe is an initial segment ofP$. PROOF. Claims 1 and 2 together imply that

U

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Moreover, as ThJ^1(α, U {{c*o, , d,_ι, ϋ}}) is essentially a subset of α,, and is not in Λf, it is not in Qθ. (Note here that P(a3)Ql C |Q0|, and that if ζ > 1 and E% φ 0, then lh£^ > α,, so that P(α,) Π IQ^+il <Ξ ^(α*)n \Qtl with equality holding after the least such ζ.) It follows that Pe, over which the subset of α, in question is definable, is not a proper initial segment of Qg. D CLAIM 4. [Q,θ]u doesn't drop. Suppose otherwise. Then QQ is not ω-sound, so Qβ = Pe Jb-sound, so that deg"(0) > k.

But then Q# is

Let 7 4- 1 be the largest member of D" Π [0,^]^. Thus deg^^) > k for all ί > Ί + 1 such that £ € [0, 0]c;.

For any X C |Λ/Ί, any j, let Thf (X) = {(φ, ά) G Th/(X) I φ is pure rΣ,-} . Then set

Thus ^ is rΣf^, and by Lemma 2.10, A$\P9\. CASE 1. crit(^+1tθ o i*^) > α,. As in the proof of Lemma 4.5, we can show by induction on β £ [7 + 1, θ]u that any set X C α, which is rΣJJ^ is in fact rE^JJ1. Thus A is rEJ^J1. Thus Λ 6 Qξ, where ζ = tf-pred(7 -f 1). But then A G \M\ = |Qo|, since A C α,. But then the proof of 2.10 shows that Thf ^α, U {ϊo,*({c*o,

, «,-ι,

a contradiction. CASE 2. Otherwise. Let

Since * = crit(i*Z^1) = crit E*f ', and 7 -f 1 £ Z/', we have

Let ί be least such that E% ± 0; thus ^ < 7 and φ = Q0 = M. Now Λ< agrees with Q*+1 below Ih £^, and lh£^ > α,; thus there must be a subset of K in M but not in j£ . So Λ< (= card(α,) < /c.

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Thus α, φ /?£+! and 8 < S + 1 and a, = crit π0 where πo : H —> M is the inverse of the collapse. But then α, =

Thus P(κ) Π \Ή\ = P(«) Π J* = P(κ) Π |^|, all 17 > 0. Now since Qξ agrees with Pξ below lh£^, and lh£^ is a cardinal of Qη for 17 > ζ, we have

(all ,,>£ and Qη \= a, = κ+

(all η>ξ

It follows, since γ + 1 € Σf4 , that CA-pred(γ + 1) = ξ. Also,

We can then show by an induction using the proof of 4.5 that

Say A is rΣ^Jl in the parameter p, where p = [α, f]Eu*1 . It will be enough to show that E*j \ a, U α is a member of M, for then, since Q^+χ G Qξ = Λί, we get that A G |ΛΊ|, a contradiction. Suppose first 7 = £. Since 7 + 1 £ Z^, £*f ^ F^, and thus £^ G Qξ, as desired. Now let 7 > {. Since ^r; for all η > ξ, ζUj. If £% φ F^, then ί% € |Q7|, and since E^ f α, U α is a subset of α,, £^ f α, U α £ |Q^|, as desired. So we may assume that E^ = F^Ύ. Now J^ Π [ί,7]w φ 0, as otherwise since crit F^ = /c, crit ι*£Ί > K and So let η + 1 be largest in EM Π [£,7]^. So F^**1 has critical point /c, and *»H- 1,7 ° lί-hi ^^ CΓ^ical point > /c, hence > α,. But now F^y \ (α, U α) is an rE?^ subset of α,, and hence (as in the proof of 4.5) FQ^ \ a, U α is rΣ^ ^1 . Since η + 1 G D", we get F^ f (α, U α) G Q^* ω desired. D CLAIM 5. Pθ = QΘ. PROOF. Otherwise Q$ is a proper initial segment of TV But then 7Γ$ o i^tf is a weak fc-embedding from M to a proper initial segment of Pβ , which is on a fc-bounded simple iteration tree based on M. This contradicts Dodd- Jensen. D

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CLAIM 6. ϊo,*(ϋ) = 'M(U), and for

< s - 1, ϊb,*(αj) = #,*(<*,-)•

PROOF. ΪQt$(ΰ) = uk(Pθ) = uk(Q$) = ^(u), since ϊ0tθ and iote(u) are kembed dings. We show the second assertion by induction on j. Assume it for p < j. As i%θ is a fc-embedding, the proofs of 4.6 and 4.7 show that •Γh&ι(,

, α, -!, «»}) €\Q,\.

On the other hand Thf ^ϊoXαy -f 1) U {tb,*((ao,

, α,_ι, δ))} £ ft .

So our induction hypothesis implies that i%Θ(GJ) < %Qt$(άj). On the other hand, since the iteration map ϊζθ is minimal and π$ o ήf 9 is a Jk-embedding of M into Pe , we have or

so that io,0(c*j) < ί?ί(θfj)ι

anc

^ ^us *o,*(άj ) = l^(αj

desired.

CLAIM 7. crit i^ > p^lβ PROOF. Assume not, and let /c = crit i%θ = crit £ where β + 1 € [0,0]# is such that [7-pred(/? -f 1) = 0. Then

It follows as in the proof of 4.6 that

But now α, < lh£^ < t'o f /j+i(tf), so ,

, α.-i, «))}) € \Qβ+ί\ .

So, again using the proof of 4.6 if crit i^+1 ^ < α, (which seems possible; we may have ff£ < or,), ThftΛ*, U {^({α0, -

, α,_!, «))}) e KM .

This contradicts the conjunction of our previous claims. CLAIM 8. 5 = 5 -f 1; that is, (c*o,

, <*5) is t + 1-solid over (Λί , u).

D

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PROOF. Let A C p^l be rΣ^ but not a member of \M\. Then A is hence rΣf^, hence rΣ^+1. But if s < 5+1, this means A is rE£J.j in a parameter from (α, U {α0, ,α,_ι,u})<α;, hence in u and a parameter <ιex (c*o, , QS) This contradicts the minimality of (αo, , (*s) Π CLAIM 9. P(p%+\)M = ^(Pfc+i)7*; that *s> r is Jb + 1-universal over (M,u). PROOF. This follows easily from the facts that Pe = Qe and crit t0|0 > />£+ι, crit i£, >p*+ι. Π This completes the proof of Theorem 8.1.

Π

The method by which 8.1 was proved gives some condensation results for 1-small coremice. One which will be of use to us is the following. Theorem 8.2. Let U and M be l small coremice, and suppose there is a nontrivial fully elementary π : Ή —*• M such that crit (π) = p%. Then either (a) Ή is a proper initial segment of M

or (b) There is an extender E on the M sequence such that Ih E = p* and 7ί is a proper initial segment 0/Ulto(Λ4, E). Remark. In case (b), H is not an initial segment of M. The following example shows that case (b) can occur. Suppose P is an active 1-small coremouse, K = crit Fp, and Fp \ a is on the P sequence for some α > (κ+)P. (We shall later construct such a "P.) Let σ : Ult0(7>, Fφ \ α) -> Ult0(7>, Fr) be the natural embedding. It is easy to see α = crit (σ). Let

and

M = σ(H),

π = σ \ Ή,.

Clearly α = crit(π) = p™, π is fully elementary, and 7ί is not an initial segment ofM. PROOF OF 8.2. Suppose first that IhE < p* for all extenders E from the U sequence. Then either Ti is an initial segment of Λί, so that (a) holds, or we have a first E from the M sequence such that p™ < IhE < OR7*. As M is internally iterable, lh£" is a cardinal of L[EM \ p%]. But cardiOR7*) < p% in L[EM \ /#], so IhE = p%. Moreover, U is an initial segment of Ult 0 (Λi,£ ) as otherwise again we have a cardinal of L[EM \ ρ%] strictly between p™ and OR7*. So we have alternative (b).

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So we may assume p* < IhE for some E from the Ή. sequence, and hence PΪ? < lh # for some E from the M sequence. The next section of the the proof will be almost the same as the start of the proof of theorem 8.1. We will compare Jί with M as in in theorem 8.1, with p* in the place of α, and ω in the place of Ar, noticing that the proof of the strong uniqueness theorem gives easily that every u -maximal iteration tree on Ή, or M is simple. Everything will go through almost exactly as before until the point where we used the fact that there was a subset A of α, which is definable in Ή, and not in M. Thus we will conclude that OT0, that there is no dropping along [ϋyθ]f , and that Pg < Qθ It will follow immediately that ϊo,0 is the identity, since the use of any extender with critical point greater than or equal to ρ% would cause a drop. We now continue with the detailed proof. As before, we define three ω-maximal trees by induction on length: (1) a"psuedo iteration tree" T on the pair (W,Λ<), with models Pa\ (2) an iteration tree T on M enlarging T, with models Paί and (3) an iteration tree U on M with models Qa. We also have embeddings TΓαr

i or ~"~^ ' o r

such that πα is a deg(α) embedding. The πα's have the natural commutativity and agreement properties they had in 8.1.

Set

P0 = K, P-! = M,P0 = Q0 = M and τr0 = π, π_ι = identity. The remainder of T, T, and U is defined by induction just as in 8.1: we get Pa+i and Qa+ι by "iterating the least disagreement" between Pa and Qa, as in the comparison process. We get πα+χ and Pa+ι by Copying. The role of α, in the proof of 8.1 is played here by ρ*\ that is, if crit Ea < ρ%, then -lΓ(α + 1). As before, we get θ such that P$ is an initial segment of Qe or vice- versa. We say a branch 6 of U drops if either ΣP Π 6 / 0 or deg^α) < ω for some α G b. Similarly for branches of T and f. Since we are dealing with α -maximal trees on fully sound mice, we have that (a) if {/? I βfθ} drops, then Qe is a proper initial segment of P$ and [0, θ]u doesn't drop and (b) if [0, θ\u drops, then Pe is a proper initial segment of Qe and {β \ βfθ} doesn't drop.

CLAIM 1. {/? | βfθ] doesn't drop.

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PROOF. By (a) above, if {β \ βfθ} drops then -KB o fy$ is a fully elementary embedding from M to a proper initial segment of Pg , which lies on a simple iteration tree based on M. This contradicts the Dodd- Jensen lemma. CLAIM 2. OT0. PROOF. Suppose -ITΘ. CASE 1. [0, θ]u drops. Then ϊ_ι,0 is a fully elementary embedding from M to a proper initial segment of Qg. This contradicts the Dodd- Jensen lemma. CASE 2. [0,0]c; doesn't drop. If one of Pg and Qg is a proper initial segment of the other, then we have a contradiction to the Dodd-Jensen lemma. So suppose Pθ = Qθ. Then as in Case 3 of the proof of Claim 1 of 8.1, t-i,* = i",. This means that the first extender used along [— l,θ]τ is compatible with the first extender used along [0,0]i/, which is impossible. D CLAIM 3. ϊo,0 = identity. PROOF. Otherwise, since ρ%° < crit ϊo,0, [0, θ]f drops. This contradicts Claim 1. D CLAIM 4. Pg = Ή, is a proper initial segment of Qg. PROOF. If [Q,θ]u drops, then in fact Pg must be a_proper initial segment of Q0, as Pg is ω-sound. If [0,0]# doesn't drop, then Pg is an initial segment of Qg as otherwise π$ o i%θ contradicts the Dodd-Jensen lemma. But p% < ρ£* < %,i(PΪ?) = Pω$ , so ^ = W = <2* is impossible.

D

Our proof now deviates from that of theorem 8.1. In order to show that U is the desired tree we must verify that either (a) or (b) of the statement of 8.2 holds. Suppose (a) fails, that is, U is nontrivial. So Έ% φ 0. Now p% < IhJE^ since W crit(τr) = p* , and Ih EQ < OR , since otherwise W would be an initial segment of M. But now lh£^ is a cardinal of Qg , and H is a proper initial segment of Qθί so that cardtOR*) < p£ in Qθ. So we must have lh£# = p£. Similarly, if W E% exists, then OR < lh£^. So in fact ί% doesn't exist, that is, θ = 1 and U is a proper initial segment of Q\ = Ult*(.M,E^), where fc = degw(l). We can take k = 0 because Ulto(Λ<, E^) and Ultjk(Λί, £^) agree to their common value for (p^)"*~ and beyond. D Remark. The hypothesis that crit(τr) = ρ% is necessary in 8.2. For notice that crit(π) > p% is impossible since π is fully elementary. (That is, this case is vacuous.) On the other hand, crit(π) < ρ% can occur while conclusions (a) and (b) of 8.2 fail: e.g., let M = Ult^T^i;) where E is on the Ή. sequence and crit(£") < pj, and let π be the canonical embedding. One can also derive a version of 8.2 with p**^ replacing p* . Namely, suppose W and M are 1-small, n -f 1 sound mice, and π : Ή —* M is rΣn+ι elementary with crit(π) > p?+1. Then either

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(a) 7ί is a proper initial segment of Λί, or (b) />^+1 = Ih E for some E from the M sequence, and W is a proper initial segment of Ult0(Λί, E). The example following the statement of 8.2 shows alternative (b) is necessary. The proof of this version is almost the same as that of 8.2. We use n-maximal trees in the comparison and modify the uses of Dodd-Jensen slightly to accommodate this_change. Note that in this case we don't know that if e.g. [0, β\u drops then P$ is a proper initial segment of Qe. Also notice that we can assume that there is an extender E from M sequence with lh(Λί) > ρM, since the result is trivial otherwise. Notice that alternative (b) of 8.2 (or its "n + 1 version") cannot arise when p* (respectively p%+ι) is a cardinal of Λ4, simply because IhE is never a cardinal of M when E is on the M sequence and Ih E < ORM. As a sample application of the n +1-version: let Λί be a 1-small, 1-sound mouse, and let a < rf4, a a cardinal of M. Let p = pι(Λί), and K = H^(a U {p}). Let π : H —* Λί be the inverse of the collapse. Clearly α = p^ < crit(π), and π is rΣi elementary. Suppose a is large enough that the solidity witnesses for p are all of the form τM\β,p] for some β 6 <*<ω and r £ Ski. This guarantees that π~l(p) is the first standard parameter of Ίi, and that ft is 1-sound. We can then conclude that W is a proper initial segment of Λί. We don't know whether the assumption that Ή. is n + 1 sound can be reduced to n soundness. If this can be done, then in the application just mentioned we needn't assume p = pι(Λ<) or make the largeness assumption about α.

§9. UNIQUENESS OF THE NEXT EXTENDER In §11 we shall construct an extender sequence E such that L[E] ^= "there is a Woodin cardinal, and every level Jg of L[E] is a 1-small coremouse". The sequence E will be defined by recursion. The recursion is substantially more subtle than it is for sequences of measures, but the basic idea is still to define EΊ by recursion on 7, by making EΊ be the least extender which can be added to the sequence E \ 7 so that the extender sequence remains good. Part of the strategy will be to pick EΊ without regard to the initial segment condition and then prove that in fact it does satisfy the initial segment condition as well. We would like to show that there is always only one possible choice of EΊ for each 7, so that if p is the natural length of EΊ \ p and G of length 7' is its trivial extension then G, being a legal choice for £7y, must in fact be EΊ>. Of course this ignores the second alternative in the initial segment condition, but more important we are unable to prove this uniqueness: so far as we know there could be one choice of types I or III and a second of type II. In this section we will prove uniqueness for types I or III, and in section 11 this will be used for the case when p is a cardinal in L[S\. In section 10 we will prove a related result which will apply in the cases when p is not a cardinal in L[E]. The standard method for showing uniqueness of the next extender on the sequence involves Doddages and comparison of a Doddage with itself. The method originates in Mitchell's [M74R], see also [D]. We need only a simple sort of Doddage, dubbed by Jensen a bicephalus. A bicephalus is like an active premouse, except that it has two predicates corresponding to two candidates for a last extender. By comparing bicephali with themselves we show that in sufficiently iterable bicephali, these candidates are not distinct. Unfortunately, when we want to form an ultrapower of a bicephalus whose last extenders differ in type, we have a problem. We may want to squash for the sake of one extender, but if we do so it is not clear how to carry along the other. This is the reason we will also need the alternative technique from section 10. The first problem in dealing with bicephali will be to verify that when we form the ultrapower of a bicephalus both of whose last extenders are of type III, the squashing procedures in the two cases are consistent with one another. We shall verify this now, in Lemma 9.1. If M is an active ppm then VM is just the the natural length of the extender coded by FM, that is if M is of type II or III then VM is the strict sup of its generators, while if M is type I, then ι/M = (κ+)M.) Lemma 9.1. Let M be a type HI ppm, and G an extender over M with crit G = K, < VM . Let P be the uttrapower of M via G, where functions in \M\ are used, and let i : M —>P be the canonical embedding. Assume P is well founded. Let v* = supi"vM. Then

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(a) ι/* = sup of generators of Fφ'. (b) Let 7 = (I'*)*", or 7 = OR** i/P f= (ί/*)+ rfoesn'* exist Lei Then Q is a type III ppm. and Qs« = Ult0(Λίβί,G).

Remarks. (1) P is defined more carefully at the beginning of §4. It is to be constructed without squashing. (2) According to (b), the structure Q is equal to the structure Ult0(Λί,G). (3) If i/* = i(vM), then 7 = ORP, so that P = Q. (4) If i/* < ifi/*1), then 7 < OR7* since 1(1^) is a cardinal of P. Part (a) of 9.1 then tells us that P is not a ppm, as it violates the bounded generators clause of "good at ORP". (It also violates the initial segment clause.) Roughly speaking, its last extender Fr was added too late; it should have been added at j. Replacing P by Q, which is the net effect of squashing, amounts to adding Fφ at 7. PROOF OF 9.1. (a) Notice that for ζ < ι/M, ξ is a generator of FM iff i(ζ) is a generator of Fr. The reason is that FM Γ (( + 1) € |Λ4|, and the fact that ζ is a generator is a ΣO fact about FM \ (ζ + 1). Thus i/* is a sup of generators of Fpy and we need only show that no η > v* is a generator of Fp. So let v* < η < ORP. We want to find α C i/*, a finite, and ft G |P| such that Let 77 = [6, /]£*, where 6 is a size n set of coordinates of G and / € |Λ4| and dom / = [/c]n. It will be enough to find maps ΰ »-> αc and ΰ >-*• Λ f i , both in \M\, such that (i) for Gb a.e. ϋ, /(ΰ) = [a0,fto]^ and

(ii) \J{ αδ : ΰ G [/c]n } is bounded in VM . (We can then take α = [6, Aϋ αo]^1 and ft = [6, Aΰ-fto]^. Because FM is weakly amenable over M we have enough of Los' theorem to show this works. We omit the details.) Now as / G \M |, the coherence condition on FM implies that / G Ult(Λί, FM). Let /=[e,^ where c C ι/M is finite and g G \M\. For any ϋ = {t/i

u n } € [/c]n, set

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So the map ΰ H + αc is in Λί, and condition (ii) above holds. Let αfi = {<*ι αj} in increasing order. Let c = {ak0 c*te} and ϋ = {<*mo * αm,} in increasing order. If ϋ = {vi wj} is any sequence, given in increasing order, then we will write v? = {vjeQ v*.} and VQ = {vmo vm,}. Note that the map (v, ΰ) ι-+ (v§ , v£) is in M. Now set, for ΰ £ [/c]n and v in the space of (Take ΛO(V) = 0 if this doesn't make sense; it makes sense F£ a.e.) Clearly the map ϋ^ hΰia in \M\. We need only verify condition (i). Let j : M -+ Ult(Λ<, FM) be the canonical embedding. Then

= j(ί)(c)(δ) = /(«), as desired. (b) Clearly, Ult 0 (ΛΊ 8ί ,G) = (J^,^Er \ v* ,Fr \ i/*). This structure is for some type III ppm ft, by results of §3. We decode ftp from ftsί by taking Ult0(ft8ί,F^ \ z/*) and cutting off at its (ι>*)+. By (a), this is the same as taking Ulto(Λ8ί, Fτ) and cutting off. As Fτ coheres with the T> sequence, this gives us Q. D DEFINITION 9.1.1. A bicephαlus is a structure

such that Bo = ( jf , €, ^ f α, F) and »ι = (jf , G, ^ f α, G) are both premice, and either (a) both ®o and 58ι are of type II or

(b) neither 95o nor 9$ι is of type II. Remark. As the reader may have noticed, the distinction between types I and II is not very important elsewhere - here it is. If both Q5o and 2$ι are of type II we say C has type II. Otherwise 95 has type III. We let FO* and F? be the two last extenders of ». Certain notions appropriate for premice - e.g. J*, agreement below 7- extend to bicephali in an obvious way.

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Suppose 05 and A are bicephali, and G is an extender from the ^-sequence such that crit G = K. Suppose P(κ) Π |03| = P(κ) Π \A\. Suppose also that if 03 is type III, then /c < i/®0. (Notice that ι/®° = ϊ/®1 = largest cardinal of 03, in the case 03 is of type III.) Suppose Ult0(030,G) and Ult0(03ι,G) are wellfounded, and hence premice of the same type as 2$o and 03ι. We claim there is a unique bicephalus C such that CQ = Ult0(03o,G) and C\ - Ult0(Si,G). If 03 is of type II this is obvious, so suppose 03 is of type III. Suppose one of 03o and 03ι in of type I. If both are type I, there is no problem. Suppose e.g. 03o is type I while 03ι is type III. Then j/®1 = (/c"*")95, where /c is the critical point of the last extender of 2Jo, i.e. of F$;. (For v*1 is the largest cardinal of OSi in the type III case, and (/c+)®° is the largest cardinal of 03o in the type I case.) But then if ί is the canonical embedding from the full ultrapower of 03ι by G, using all functions in |03ι|, then i is continuous at i/®1. So by Lemma 9.1, the squashed and unsquashed ultrapowers of 03ι coincide. This gives us the desired <£ι at once. Now suppose both 03o and 03ι are of type III. Recall that if M is a type III premouse, then Ult 0 (Λ4,G) is the unique Q such that Q** = Ult0(Λί8ί,G). It will be enough to show that OR Π Ult0(B0,G) = OR Π Ult0(#i,G), and that Ulto(030,G) agrees with Ult0(03ι, G) below ORΠUlt 0 (β 0 ,G). But now let T> be the full ultrapower via GofB formed using all functions in |β|, and i: B —*• V the canonical embedding. By Lemmaθ.l we see ORnUlt0(#0, G) = (sup i"i/e°)+p = ORnUlto(/?ι,G), and that the necessary agreement holds. So we may define DEFINITION 9.1.2. In the situation described above, Ult0(03,G) is the unique bicephalus £ such that <£o = Ult0(030,G) and <£ι = Ult0(03ι,G). Notice that if B is type II, we have a canonical i : 03 —> Ulto(03,G) which is rΣi elementary (in the obvious sense.) If 03 is type III we get an embedding i : 038ί —» Ult0(03,G)8ί - which is qΣi elementary. We get an embedding i : 03 -> Ult0(03, G) in the case Ult0(03, G) happens to be the full ultrapower of 03 by G, using all functions in (031. This happens when the canonical embedding of 03 into the full ultrapower, call it i, is continuous at ι/®°. That is, this happens when 03 \= cof (i/®0) φ /c, where K = crit G. Notice in this regard that if 03 |= cof (ι/®°) = «, and £ = Ult0(03, G), then £ |= cof (ι/e°) = /c, since ί/€° = sup i" v*°. This implies that along any branch of an iteration tree on 03, the natural embeddings map 03 into Ulto(03, G) in all but at most one instance. This is because we can hit a given /c at most once along any branch. The notion of a 0-maximal iteration tree generalizes in an obvious way to trees on bicephali, so we shall just mention a few points. Let 03 be a bicephalus; a 0-maximal iteration tree on 03 is a system Ί = (T, deg, D, (EQy
together with associated models 03α and embeddings iafi : 03α —* 030 defined

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whenever aTβ and [c*,/?]τ Π D = 0. We have Q50 = 55. One can build T freely at successor steps except for the following restrictions: Let T-pred(α + 1) = /?, and crit J£α = /c. Then we must have lh#α > Ih-B,,, all 17 < α, and /? = least £ < a such that /c < sup of generators of Eξ .

Let 7 =largest η such that ^ * exists and P(«)n|J,T'|C|» e |; then α -f 1 G D O ι77

ft

is a proper initial segment of 930, and

ίf de (a l ϊ - ί ° l°' α + 1 ί τ Π J 5 =0 ' [ largest Jfe s.t. /c < p$f, for Af = JΊ

β

, otherwise

and Finally, if n = deg(α -1-1), then

and if a + 1 £ D we have a canonical embedding *0,α+ι : ®/J ~~* ®α+ι This last statement is not true in the case when 930 is of type III and i/®* has cofinality K in 930. In this case we let t0,α+ι be the canonical embedding of We also have an embedding i*+1 —* ®α+ι, again with a possible exception in 8ί the type III case, when we may only have i*+1' : (2J£+ι) * -* (93α+ι) . If λ < θ is a limit, then D Π [0, λ]τ must be finite. Moreover the special case referred to above will only occur finitely often, so that domi*+1 = |®α+ιl f°Γ all but finitely many a + I G [0,λ)τ». Thus the direct limit of the models 930 under the maps i0ι7 for /?, 7 in [/?o, λ) for some /?o € [0, λ) exists; and we require that 93λ be this direct limit. Remarks. 1. If [0, a + l]τ Π D -φ 0, then 2Jα+ι is a premouse rather than a bicephalus. Moreover, one can see by an easy induction that 93£+1 is deg(α + 1) sound, whenever [0, α + l]τ Π D φ 0. Also, if 7 + ITα + 1 andD Π (7 + 1, α + l]τ = 0, then deg(7 + 1) > deg(α -h 1). 2. By coherence and the fact that Ih Ea increases with α, we get the counterpart of Lemma 5.1; 930 agrees with 93α below Ih Ea, for all β > α, and lhEa is a cardinal of 930 for all β > a.

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The notion of a simple iteration tree generalizes in an obvious way to trees on bicephali. We can then define k-iterability for bicephali just as we did for premice. The notion of 1-smallness generalizes in the obvious way: a bicephalus 05 is 1-small iff both 05o and 05ι are 1-small. The uniqueness theorem 6.1 generalizes in an obvious way. (We have no analog of 6.2, strong uniqueness. In general, we don't care about the Levy hierarchy over bicephali.) The main theorem about bicephali is that there aren't any interesting ones. Theorem 9.2. Let 05 be a 1-small, l-iterable bicephalus. Then F

a, built by the method of "iterating the least disagreement". Notice that if F* φ Fj* then it is guaranteed that there will be such a disagreement, since even if the bicephalus <£tt = (J.~, £, E, F, G) is matched to Va except for the final extenders F and G then each of F and G would have to agree with whatever is on the Ί)Q sequence at 7 (or possibly to both of the final extenders of "Pα), and if F and G are different then they cannot both agree with the same extender. At limit steps λ, we stop the construction unless T \ X and U \ X are simple. In the latter case, we let [0, X)τ and [0, X)u be the unique cofinal wellfounded branches of their respective trees, and continue. Suppose the construction never stops because we reach a A such that one of T \ X and U \ λ is not simple. Then the proof of 7.1 shows that we must reach a θ such that $, or vice-versa. (So the construction stops at θ.) Say <£$ is an initial segment of IV By the proof of 7.1, there's no dropping on [0,0]τ [Otherwise <£$ is unsound, so <£$ = ίV So ίu (^), the core of £0, is ££+ι f°Γ some α +1 G [0, θ)u. This is too much agreement at an earlier stage.] Thus <£* is a bicephalus. If Fζ* φ Ff*, then <£* cannot be on initial segment of T>β\ at worst, one of FQ° and Ff* will participate in a disagreement with Ί>θ. So Fξ = Ff«, so FO* = Ff. Suppose we reach a λ s.t. e.g. T \ X is not simple. Let 6 and c be distinct 6 C cofinal wellfounded branches of T \ X. Suppose e.g. OR* < OR5 . Let 6 = δ(Ί \ λ) = sup {lh££ | a G λ}. The proof of Claim 1 in the proof of 6.2 shows IhF < δ for all extenders F from the (£5 sequence. Clearly, then, δ = sup {IhF : F from the (£5 sequence}. As Co = 05 has a maximum length r realized by extenders on its sequence, D Π 6 φ 0. Thus £& is unsound. On the other hand, the proof of Claim 2 in/a the proof of 6.2 shows <£& is an initial segment of <£c Thus <£& = <£c, and Dτ Π c •£ 0. But then the proof of Claim 4

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in the proof of 6.2 yields a contradiction. Thus we can never reach a λ such that one of T \ λ and U \ λ is not simple. This completes the proof. D

§10. CLOSURE UNDER INITIAL SEGMENT We prove a result, theorem 10.1, which implies that certain structures arising in the construction done in §11 satisfy the initial segment condition on premice. As was pointed out in the last section, this will be used when bicephali cannot be used because one of the extenders being compared is of type II and one is not. DEFINITION 10.0.1. A psuedo-premouse is a structure M = (J^f, 6, E> F) such that (1) (J^ , G, E) is a passive premouse, (2) F satisfies conditions 1 through 4 in the definition of "good at a" (i.e. everything but the initial segment condition), and (3) There is a δ < a s.t. (i) M \= 6 is the largest cardinal and (ii) for some 7 s.t. £ < 7 < α, 7 G dom E and EΊ = trivial completion of F \ δ. Any psuedo-premouse Λί is weakly amenable with respect to its predicate FM for the last extender. Consequently, if E is an extender from the sequence of some psuedo-premouse Ulto(ΛΊ, E) is the canonical embedding, i is rΣi elementary. The calculations of §2 show that, if transitive, Ulto(Λί, E) is a psuedo-premouse. (If 6,7 witness 3 for M, then ι(ί), 1(7) witness 3 for Ulto(Λί ,£").) We can thus construct 0-maximal iteration trees on a psuedo-premouse Λf. We define the notions of simplicity and iterability for psuedo-premice just as for premice. (We only consider 0-maximal trees.) The notion of 1-smallness also generalizes in an obvious way. Theorem 10.1. premouse.

Lei M be an iterable, l-small psuedo-premouse. Then M is a

PROOF. We must show that the initial segment condition holds. Let M = (J^ , G , J5, F), and suppose toward a contradiction that the initial segment condition fails for F \ p. Thus p is the natural length of F \ p and if G is the trivial completion of F \ p then G is not on the E G is not on the Eυ}t(M Ep ) sequence.

sequence, and if p G domE

then

Notice that if p is a successor ordinal then p — 1 is a generator of F, and if p is a limit ordinal then either p is a limit of generators of F or else p is equal to /c+ of M where /c = crit(F). Also, p is smaller than natural length of F and as M is a psuedo-premouse p is larger than any cardinal of Λί. We obtain a contradiction by comparing M with Ult0(Λί, G). That is, we define 0-maximal iteration trees T and U on M with models Pa and Qα respectively

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as follows: PQ = QQ = M

_ ( Ulto(Λi), E?

if p G dom

IΛ<

Thus E'ζ is always equal to G, and jE^ either does not exist or is equal to The remainder of the trees T and U is determined by the comparison process. At successor steps we pick an extender, or two extenders, representing the least disagreement, and apply these to possibly earlier models in their respective trees so as not to move generators along branches of T and U. At limit steps we use the unique cofinal wellfounded branches of T and U given by the iterability and 1-smallness of M . First we will verify that the iteration stops, that is, that there is an ordinal θ such that the 0th model Pβ ofT is an initial segment of the 0th model Qe oΐU or vice- versa. There is a slight wrinkle here because the proof that the comparison process terminates uses the initial segment condition on premice, and we don't yet know that this holds for the final extender F of M. Suppose that the iteration never stops. As in the proof of the comparison lemma (7.1) we have ordinals 1 < a < β such that E% is the trivial completion of Eβ ί Pa (where pζ is the sup of the generators of E%) or, symmetrically, E% is the trivial completion of Ej \ (%. We may as well assume the former. This is a contradiction as in the proof of the comparison lemma unless [0, β]u Π Ef* = 0 fi and Ί$β = F^ \ that is, E% is the unique extender from the Qβ sequence for which we don't have the initial segment condition. But then Qβ is a psuedopremouse, and thus obeys the initial segment condition on F^ft somewhere past its largest cardinal. It follows that pζ > largest cardinal of Qβ. Thus IhE'J is not a cardinal of Qβ. On the other hand, lh£"J is a cardinal ofPQ+ι, hence of ft Pβ. This contradicts the fact that F^ is part of the least disagreement between Pβ and Qβ, and hence the comparison must terminate. So let θ be such that Pβ is an initial segment of Qβ or vice- versa. The DoddJensen lemma, adapted to our present situation, implies that Pg = Qβ, Dr Π [0,% = 0 = Σf4 Π [0,0]c/, and ιξ$ = %θ. The trees involving the extender G must have well founded branches since they can be embedding into trees using F instead of G. Thus we can apply the Dodd-Jensen lemma to a tree involving G even though G is not a member of M. Now let α be least such that α + 1 £ (0, 0)τ, and β be least such that β + I G (0,0)c;. As t"2i = ι'(f $ we have that E% and E*jf are compatible up to inf(/>£,/)^), that is, either E% is the trivial completion of E% \ pζ or E^ is the trivial

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completion of E% \ ffβ. When we proved that this comparison terminates we derived a contradiction from this situation using the assumption that both of the ordinals α and β are greater than 0. It follows that at least one of the ordinals α and β must be equal to 0, but on the other hand α and β cannot both be 0, for E % is always G while if E*jf exists then it is equal to E*4 . If Eβ exists then p is a limit ordinal and p is a limit of generators of G, but not a limit of generators of E*4 and hence G and E** are not compatible with each other. Thus we have two cases, depending on which of a and β is equal to 0. Suppose first β = 0, so that a ± 0 and E% = E*4 (and p G dom EM). Then, p is a limit ordinal, hence a limit of generators of F, and hence a cardinal of Pl = Ult(Λί,G). So p is a cardinal of PQ. As Pa is a psuedo-premouse or a premouse, E% satisfies the initial segment condition in Pa somewhere past p. But E** is the trivial completion of E% \ v, where ι/ < p < p% is the sup of the generators of E*4 . Thus E*4 £Pa, so that p is not a cardinal ofPa. This is a contradiction, and hence β φ 0. Now suppose α = 0, so that E% = G. First suppose that p — 1 exists. Then p £ dom EM so Q\ = QQ. Also, letting 7 = Ih G, Ult0(Λ4, G) and Ult 0 (Λi, F) agree below 7, so that M and Ult0(Λ4 , G) agree below 7. That is, PI and Qi agree below 7. Since 7 is a cardinal of PI, 7 is a cardinal of Pβ. As Q0 is either a premouse or a psuedo-premouse, and E*jj is part of the least disagreement between Pβ and Qβ (so that 7 is a cardinal in Jη , where r/ = lh£^), E^ f 7 = G is on the sequence of Qβ. (One must also consider the η = 7 case. Then E^ = G, and we have the same contradiction.) Thus G is on the sequence of Qi = QQ = M. This contradicts our choice of G, so that p is not a successor ordinal. Suppose next p is a limit ordinal, but p ^ dom £7^ . If p is not itself a generator of F, then again Ult0(Λί,G) agrees with Ult0(.M,F) below 7 = IhG, and the argument from the last paragraph yields a contradiction. If p is a generator of F, then the natural embedding π : Ult(Λί, G) —> Ult(Λ<, F) has critical point p, so the agreement is not obvious. Nevertheless, Theorem 8.2 easily implies that Ult 0 (Λi,G) and Ult0(.M,F) do agree below 7 = IhG = (p+)ulto(Λ4,σ)β So again we reach a contradiction as in the last paragraph. Finally, suppose p is a limit ordinal and p G dom EM. From Theorem 8.2 we get that Ult0(Λί,G) agrees with Ulto(yito(M,F), J^4) below 7 = IhG, which implies PI agrees with Qι = Ult0(Λί, Ef*) below 7. As 7 is a cardinal of PI, 7 is a cardinal of Pβ, hence of Jηft where η = lh£^. Since p £ dom EQ*, and Qβ satisfies at worst the initial segment condition on psuedo-premice, G is on the Qβ sequence. So G is on the sequence of Qι = Ult0(.M, E**). This contradicts our choice of G. Π

§11. THE CONSTRUCTION At last we are in a position to construct our extender sequence E. We will construct the sequence E inside of Ve where θ is least such that L(Ve) satisfies that θ is Woodin. Note that every bounded subset of θ in L(E) is in L$[E] since θ is inaccessible. The construction of E will differ from that for sequences of measures in that we do not simply define Ea by induction on a. The reason is that we want the construction to provide each Ea with an ancestry tracing back (by inverting certain collapses) to an extender on V having a certain amount of strength. The illustrious ancestry of the extenders which lie on E guarantees that all levels of L[E] are u -iterable. Let us call a premouse M reliable iff for all k < ω, &k(M) exists and is kiterable. We shall simply assume in this section that the premice we produce in our construction are reliable, and discharge our obligation to show this in §12. We now define by induction on ζ a reliable coremouse Mξ. Simultaneously, we verify an induction hypothesis Aξ describing the agreement between Mξ and the Ma for α < ξ: A

( t) 3^" = J^ for all α < ξ and /c < uΛ{pω(Mv) : a < v < ξ }, where η = (*+)"<*. In the formulation of Aξ, we understand that ωη = ORM<* in the case that Ma f= K* doesn't exist. We begin by setting MQ = (Vω, 6, 0). Now suppose that Mξ is given and that Aξ holds. We define Mξ+ι and verify Aξ+i. Case 1. Mξ = (Ja,ε,E) is a passive premouse, and there are an extender F* over V, an extender F over Λίf, and an ordinal v < a such that

and

F r ^ = F*n([i/] and

jf)

-Λ/e+ι = (jf.e,£,F)

is a 1-small, reliable premouse, with ϊ/ = ι/^+l . In this case we choose F*, F, i/, and J\fξ+ι as above with i/, the natural length of F, minimal among all such F*. Let

Case 2. Otherwise.

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W. J. MITCHELL AND J. R. STEEL M

In this case, let ωa = OR * , and set

M

(Of course, F * = 0 is possible.) Thus Λ/^+i is a passive premouse. If Λ/^+i is not reliable, stop the construction. Otherwise,

We must verify Aξ+\. Now Theorem 8.1 tells us that Nξ+\ agrees with ία,(.Λ/ξ+ι) below (pi)^+l = (pί)M*+l The obvious agreement between and Afξ+ι, together with our induction hypothesis Aς, easily gives -Af+i Now suppose Λ is a limit ordinal. Let ^liminf (/>+)"< ς—*A

(where again we set (p*)M* = unique α s.t. ωa = ORΛ1* in case <Mf [= pω ί has no successor cardinal, or pω * = OR/'** .) Then we let λfχ be the passive premouse P = J^ , where for all β < η we set jj equal to the eventual value

Λ/λ exists since Aς holds for all ξ < X. Now suppose M\ is reliable; if not we stop the construction. Set It is easy, using 8.1 and the induction hypothesis, to verify A\. This completes the inductive definition of the Λ^^'s. For the moment, let us assume: Lemma 11.1. The construction above never stops; Mξ is defined for all ordinals ξ. PROMISE OF PROOF. We have to show Jfς is reliable for all ξ. We will prove as theorem 12.1 that ί*(Λ/Je) is Jfc-iterable, for all Jb < ω, provided that Cjb(Λ/ξ) exists. Given this it follows from theorem 8.1 that Nξ is reliable. D Lemma 11.2. Suppose QQ and ξ are ordinals such thai c*o < ζ and K = pω * < p^* for all α > <*o. Then Mξ is an initial segment of Mη, for all η > ξ. Moreover, Mξ+\ ^= every set has cardinality at most K. PROOF. We may assume K < ORM*. We claim Λΐξ+i is defined by Case 2. For suppose not; let Mξ = (/f , G, E) and let F be as in Case 1. Then Mξ is a

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proper initial segment of Ulto(Λίf,F), and Ulto(Λίξ,F) ^= a is a cardinal. As there is a map from AC onto a which is Σn * for some n, we have a contradiction. Let Mξ = (j , G, £, F), where F = 0 if Mξ is passive. Let Λ/Jt+i = (j+Γ/, € , E^F) be as in Case 2. Then the Σn * map from AC onto α guarantees .Λ/f+i ^= every set has card = AC, and pl *~M < AC. Thus pω**1 = /c. Theorem 8.1 implies that (^(.Λ/^+i) = Λ/ξ+i Thus Λ<£+ι = Λ/Je+i, and the claim holds for r; = ξ + 1. For η > ξ + 1, the claim follows easily from the induction hypothesis Aη. D The claim implies that lim inf^_,oκ pω * = OR. So we can define our desired E by jjp = eventual value of Jjf*6 , all sufficiently large ξ E OR . Clearly this determines E, and we have that every level Jβ sound, α -iterable 1-small mouse.

of L[E] is an ω-

We can think of the construction as producing, in increasing order, the cardinals of L[E] together with the levels of L[E] whose α th projectum is a cardinal of L[E]. Namely, let KQ = ω, ξo = 1 , and now suppose we have /c7 and ξΊ for 7 < α. Set KQ = inf {p^β I β > sup {ί7 I 7 < a}}

and

ζa = least β > sup {£7 | γ < α} such that pω

ft

= /cα .

One can check easily that {/cα | α £ OR) enumerates in non-decreasing order the cardinals of L[£\, that p^<0f = /cα, and that Mζa is a level of the eventual L[E]. In fact, for /c a cardinal of L[E], the Λί^tt for /cα = AC are precisely those levels J^ of L[^] whose u th projectum is AC. We now show that L[E] \= there is a Woodin cardinal. Once again, certain iterability assumptions will crop up during the proof. We shall verify these assumptions in §12. Theorem 11.3. Suppose there is a Woodin cardinal. Let E be the extender sequence constructed above. Then L[E] \= there is a Woodin cardinal. PROOF. Let θ be least such that L(VΘ) \= "θ is Woodin." We show that θ is Woodin in L[E]. So fix / : θ -> θ such that / € L[E\. Define g : θ -> θ in V by g(a) = 2nd strongly inaccessible (of V) > /(α) .

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(The strong inaccessible is just a security blanket.) As θ is Woodin in V there is an extender F* over V, F* G Vθ, crit F* = /c, such that if j* : V -> Ult(F,F*) is the canonical embedding, then g"κ C K and

and

^r

Let

F = F*Γι([lhF*]<ω Notice that L[E] agrees with Ult(£[J£], F) below j*(fif)(κ), where the ultrapower is computed using functions in L[E]. That is, F "coheres" with E sequence out to j * ( g ) ( κ ) . Notice j*(g)(κ) is a strongly inaccessible cardinal of V, hence of L[E]. We now show that for p < .;*(
Let

i:L[E]-+V\i(L[E],F\p) be the canonical embedding, and let

G = {(α, x) I a G [γ]** Λ x C [/c]card(α> Λ x G I[F] Λ α G i(x)} . Thus G is the trivial completion of F \ p. The generators of G are of course just those generators of F which are less than />, and G \ p = F \ p. Lemma 11.4. Let (/c+)L^l < p < j*(g)(κ), and suppose that p is the natural length of F \ p. Let G be the trivial completion of F \ p, and 7 = IhG. Then EΊ = G = F \ p unless p is a limit ordinal greater than (κ+)L^J, and is itself a generator of F. In this case \ EΊ

i/7 ^ dom£?

E

ι/7GdomF

[ (i p(E))Ί where iEf> : jf

—> Ulto(«7^ , Ep) is the canonical embedding.

PROOF (modulo §12). The proof proceeds by induction on p, and is divided into a number of cases. In those cases where p is not a cardinal we will apply theorem 10.1, and in the other cases we will be able to use bicephali. Case A. p is a successor. In this case p — 1 must be a generator of F. Let σ: Ult(L[F], F \ p) -+ Vlt(L[E], F)

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be the canonical embedding. From the case hypothesis we see that σ(p) = p and hence σ f 7 = id. Also, G = F f 7 as (α,x) G G 'ΦΦ α G ip\p(x) & a G )) <£* α G ι>(x) <=> (α, x) G F, for all α G [i\<ω and appropriate x. We claim there is a stage 17 of the construction such that

For let 6 be the largest cardinal of L[E] which is < p.

(So δ < p.) Now

*>tp(#) Γ Ί = *>(#) ί 7 = # ί 7, and by the definition of 7, we have j'ΊFtt>(β) |= every set has cardinality < p, so J^ [= every set has cardinality < p. On the other hand, 6 is the largest cardinal ί218

σ(γ) is a cardinal of L[£?]) hence in J^ . So J,- [= every set has cardinality ί . Let (£α j a < (δ+)LW) enumerate in increasing order those ordinals ζ such that Pω(Mξ) = ί and pω(Mβ) > δ for all β > ξ. We observed earlier that the Mξa are precisely those levels of L[E] whose ωth projectum is δ. It is clear that 7 is a limit of such levels. So letting η = sup {ζa \ OR Π Mξa < 7}, we have that η is a limit and Mη = (J^, G, E \ 7). This proves our claim. Now clearly («Λ~ , G, E \ 7, F \ 7) is a type II premouse. It is also 1-small, since otherwise J% satisfies that some ordinal α < /c is Woodin. But then since K is a cardinal of L[Ϊ3\, α is Woodin in L[E], and a < K < ί, contrary to our initial assumption that no ordinal α < θ is Woodin in L[E]. Let us assume until §12: Sublemma 11.4.1. (J-f , G, E \ 7, F \ 7) is reliable. It follows that Mη+\ is defined by Case 1 in our construction. That is, λfη+ι = But

δ since

(jf , G, E \ 7, #) for some H, and Λ4,,+ι = £ω( Λ/Ί?+ι) A*(-^+i) > δ is the largest cardinal of L[E] and hence Mη+\ = M -f i since J^ satisfies that every set has cardinality at most 6. Moreover Mη+\ is an initial segment of the eventual L[E]y and H = EΊ. Thus it is enough to show EΊ = F f 7. Notice that 7 is a generator of F, as otherwise σ(j) = 7, so that 7 is a cardinal of £[!£], contrary to 7 G dom jE?. Let G' be the trivial completion of F f 7 + 1. Arguing as above, with £ = Ih G;, we see that G1 = F \ ξ, and that E \ ζ~F \ ζ satisfies conditions 1-4 of "good at ζ" . We now show that (J^ , G, E \ ξ, F is a psuedo-premouse.

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Since it is easy to see that δ remains the largest cardinal of J?, as 7 is not a cardinal of J^ , we need to verify that the trivial completion of F \ δ is on E. Now either δ = (/c+)L^' or δ is a limit of generators of F. [Otherwise, let τ < δ be such that r = \J{ξ < δ \ ζ = (/c+)L^l or ζ is a generator of F}. By our inductive hypothesis the trivial completion of F \ τ is on E - it falls under either (b) or (e) of the lemma. But from F \ τ we easily construct a collapse of δ.] By our inductive hypothesis, as δ < />, the trivial completion of F \ 6 is on E. (Note here that clause (d) of the lemma cannot apply as δ ^ dom E as δ is a, cardinal of L[E].) That is, if β = (ί+jUitWή^W, then Eβ is the trivial completion of F \ δ. Clearly β < 7 < ξ. Thus ( j f , £,E \ξ,F \ ξ ) satisfies the initial segment condition on psuedo-premice, as desired. We now borrow from §12: Sublemma 11.4.2. (j£, e,&\ζ,F\ξ) is iterαble. Granted 11.4.2, Theorem 10.1 tells us that (J*, e,E\ξ,F\ζ) satisfies the full initial segment condition, so that F \ 7 = EΊ. Remark. We can't use bicephali here because EΊ might be of type III, while F f 7 is of type II. Case B. p is a limit of generators of F, but not itself a generator of F. Let σ : Ult(L[#],F \ p) -*> Ult(L[.E],F) by the canonical embedding. As p is not a generator of F, σ f 7 = id and G = F f 7. Note p is a cardinal of J~% hence of L[E\ because σ exists. Arguing exactly as in Case A we find a stage η of the construction such that

and

is a premouse of type III. In §12 we prove: Sublemma 11.4.3. (J^, 6, E \ 7, F f 7) is reliable. Thus Mη+\ is defined through Case 1 of our construction. Let H be the set such that λfη+ι = (jf, €, E \ 7, H). Now p is the largest cardinal of (jf, e, F Γ 7), and we chose H so as to minimize i/^»+l, the sup of the generators of H. Thus i/-^i+i = p and .Λ/^+i is of type III or type I. Drawing on §12, we get

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Sublemma 11.4.4. The structure (J^ ,G, E \ y,F \ 7,ff) is an iterable type HI bicephalus. It follows from Theorem 9.2 that H = F \ 7. Also, as in Case A, Mη+\ = &w(Nη) = Nηy and Mη+\ (note here p = pω(λίη) is the largest cardinal of tfη, and p is a cardinal of L[E]) is an initial segment of L[B\. Thus 7 G dom E and EΊ = F\Ί. Case C. p is a limit of generators of F} and is itself a generator of F, and p £ dom E. Again, let σ : Ult(L[£], F \ p) -* Ult(L[£], F) be the canonical embedding. This time we have p = crit σ, and thus it is not obvious that G "coheres" with E up to 7. Nevertheless, Theorem 8.2 implies that this is true. Claim 1. Ult(L[£], F \ p) agrees with L[E] below 7. Proof.

Let η be any ordinal such that p < η < 7 and p™ = p where Ή =

J™t(L[£],Ftp). Since 7 is the successor cardinal of p in \Jlt(L[S\,F \ p), there are arbitrarily large such ordinals η < 7. It will thus be enough to see that 7ί is an initial segment of L[E]. But now σ ί Ή is a fully elementary map from H into σ(H}\ moreover crit(σ \ Ή) = p* and /# £ dom J5 (so /# £ dom ί^W). Thus Theorem 8.2 implies that Ή is an initial segment of σ(H). But σ(H) agrees with L[E] below IhF, hence below 7, and thus 7ί is an initial segment of L[E]. Claim 2. (J^, G, 5 ί 7, G) is a 1-small type III premouse. Proof. For coherence, we use Claim 1. The initial segment condition follows from our induction hypothesis on p. We get 1-smallness as in Case A. We now consider two subcases. Subcase C I . p is a cardinal of L[E]. In this case we have, just as in Case A, that there is a stage η of our construction such that

(Here η is the sup of all ξa s.t. /cα = p and p < QRM** < 7.) Granted this, we will proceed just as in Case B: Sublemma 11.4.5. ( J^, G, E \ 7, G) is reliable.

PROOF. In §12. So Mη+i is defined via Case 1 in our construction. Let

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As we chose H to minimize v**+* , Afη+ι is either type I or type III, and P Sublemma 11.4.6. The structure (J^,G,£ \ 7,G,#) is an iterable type III bicephalus.

PROOF. In §12. From Theorem 9.2 we get that G = H. As in Case B, Theorem 8.1 and Lemma 11.2 guarantee that Mη+ι = tfη+ι and Mη+ι is an initial segment of L[E]. ThusG= Ey. Subcase C 2. p is not a cardinal of L[E\. We use the argument from Case A. Let 6 be the largest cardinal of L[E] which is < p. Let G' be the trivial completion of F \ p + 1, and ζ = IhG'. Thus G' = F \ ξ, and (J* ,G,£ \ ζ,F \ ζ) satisfies conditions 1 through 4 of goodness at ζ. Using σ we see that δ is the largest cardinal of J^ which is less than p. It follows that δ is the largest cardinal of J^ (note p is not a cardinal in J^ since the natural embedding from Ult(L[£],F \ p + 1) into Ult(L[£],F) fixes p, and p is not a cardinal of Ult(L[J5],F)). Our induction hypothesis guarantees that the trivial completion of F \ δ is on E, and hence on E \ ξ. Thus ( j f , G, E \ ζ, F \ ζ) is a psuedopremouse. Sublemma 11.4.7. (J^, e,E\ζ,F\ξ) is iterable.

PROOF. In §12. Theorem 10.1 implies that ( j f , €, E \ ξ, F \ ζ) satisfies the full initial segment condition on premice, so that 7 G dom E and EΊ = F \ 7, as desired. Remark. We do not seem to get that there is a stage η of the construction such that Mη = (jf , G, E \ 7) in Subcase C2. CASE D. p is a limit of generators of F, a generator of F itself, and /? E dom ^. As p G dom £?, p is not a cardinal of L[E]. We can now just repeat the argument from Subcase C2. Letting ξ be the length of the trivial completion of F f P+ 1, ( jf , G, ΐ? Γ ί, F ί ί) is a psuedo-premouse and, borrowing from §12, is iterable. By Theorem 10.1, (j£, €,E \ ξ,F \ ξ) satisfies the full initial segment condition on premice. As p G dom Ey this means G is on the sequence of Ult((J^, G, E \ p),Ef>)y as desired.

The proof is the same as that in Case B. We omit further detail.

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This completes the proof of Lemma 11.4.

D

We can now easily finish the proof of Theorem 11.3. Let p = least strongly inaccessible cardinal of L[E] > j*(f)(κ) . Let G be the trivial completion of F \ /?, and 7 = Ih G. By the choice of j* we know that p is definable in U\t(L[E],F) from j*(f)(κ) 

V\ί(L[E],F\j) where the upper ultrapower is computed using functions in V, and the lower using functions in L\Eί\. The function k is defined by k([a,h]^^) — [α,Λ]£.. Since k \ 7 = id, »(/)(«) < p. By coherence, jf = j™Wi*\fW

d thus

t an

L(E\\=Vpcm(L(β],F\j). So F \ 7 witnesses the Woodin property for the function /.

D

§12. ITERABILITY We now discharge our obligation to show that various of the structures we encountered in §11 are iterable. We shall concentrate on proving Lemma 11.1, which states, in the language of §11, that Nη is reliable for all η. The other iterability lemmas from §11 are proved in almost the same way. A complete proof of these lemmas will be given in the paper [S?a]. As we observed in §11, it is enough to show Theorem 12.1. Let λfη be the ηth W-model" of the construction of §11. Let 0 < k < ω and suppose &k(Nη) exists. Then ^(M'η) »'* k-iterable. PROOF. The proof of theorem 12.1 will take up all of this final section of the paper. Let

Ί = (T, deg, D, (El, P +I I « + K *» be a fc-bounded, Ar-maximal iteration tree on

The assumption of Jk-maximality is not necessary, but it simplifies the notation a bit, and we have never used non-maximal trees anyway. We let Pa be the αth model of T. Suppose that T \ λ is simple for all λ < 0, and that θ is a limit ordinal. We shall show that T has a cofinal wellfounded branch. For 7 < η such that Case 1 applied in the definition of MΊ+\ from Λ4 7 , that is, such that Λ/^y+i is equal to MΊ expanded by a new predicate for a last extender, we let F* be the background extender for the last extender of λfy+ι Thus F* is ι/ + ω strong, where v = ι/^+l . Set C = ((XΊ I 7 < ι/>, (F; I 7 < η and F; defined}) . Our strategy for the proof of theorem 12.1 is straightforward. We shall associate to T a tree U which will be an iteration tree on V in the sense of [MS]. As such the models of U will be well founded by results methods of [MS]. The tree ordering of U will be the same tree ordering, T, as T, and we will define embeddings πα from the models of U to those of U. Thus the models of the tree T will also be well founded, which is what we need to show. Since U is not a fine structure iteration tree it doesn't make sense to ask that π be a tree-embedding in the sense of section 5. However, if we let RQ be the αth model of U then the embeddings πa will be embeddings from the αth model Pa of T into Qa =
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We must also maintain a certain amount of agreement between πα and the τr0's for β < a. We now state some definitions which allow us to describe this agreement. DEFINITION 12.1.1. Let M be a premouse, and ω\ < ORΛ*. Then the \-dropdown sequence of M is the sequence ((/?o, &o)> , (A, *«)) defined as follows: (1) (&,to) = {λ,0). (2) {/?,-+!, fct +ι) is the lexicographically least pair (β, k) such that A < /?, ωβ < OR", and pk(jf) < Pk.(jtf) If there is no such pair (β, k) then {/?, +ι , fci+i) is undefined. Let i be the largest integer such that {/?,-, Jb, ) is defined. Notice that if ((/? e ,fc e ) I ^ < i) is the λ-dropdown sequence of Λί, then kt < ω for all e , /?w < OR^, and p < λ < /? is in the set { Pke(j£} \ e < i }. Now we prepare to define the (j,ξ)-resurrection sequence for an extender E, where E is on the extender sequence of M = <£j (wVξ), the jth core of one of the models of our construction C. We are allowing the possibility that E = FM . The idea is just to trace E back to its origin as the last extender of some NΊ with 7 < ξ. Let λ = Ih J£, and suppose that {{/?o, fco) {/%>*<)) is the initial segment of the λ-dropdown sequence of M consisting of those pairs {/?, Jb} on the sequence such that (/?, k) * whenever (τ» e ) ^lex ( r ) n )
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then Jβ* is a proper initial segment of ίc( A/^) PROOF. By the 1st claim pe+ι(-V7) > K. But # < (/C+^-H^) since j£ has projectum K. By 8.1, J^ is a proper initial segment of
It will be enough to see that p*' = p*, since this implies that (C^ (Λ^) = J^ , contrary to the minimality of (7, e}. So suppose we have t s.t. Jfe» < t < e and ρ\^\ < p\. We may assume t is the largest such. Now the reader can easily check1 that for any u, pJJ+1 = pJJ+J, and p«+1 < p« =» pftj < p^1. Thus we have p\\\ < p]*1 < p**1 . As Ct+i(^7) = Ce^) by the maximality o f / , Pe = Pill < Pit1 = Pfc. **ut ^(-^7) = *^' so thίs contradicts the fact that {/?,-, ki) is the last term of this restriction of the λ-dropdown sequence of Λi, so that ρe(Jβ?) < pk^Jβ*} is impossible if Jfc. <e<j. If jg = Λί then we must verify that e < j in order to apply this fact. Now if 7 = £, then β < j by the choice of {7, e), and if 7 < ξ, then ίβ(Λ^) = ίj( Λ/ξ) and it is easy to see that this is impossible. Next, suppose J^ is a proper initial segment of (CeCΛ/^)- From Claim 2, we see that e = 0, so that J^1 is a proper initial segment of If 7 is a limit, then the definition of NΊ guarantees that j£f is a proper initial segment of some (^(Λ/V) for τ < 7. But then Claim 2 implies J^ is a proper initial segment of λ f T j a contradiction. Thus 7 is a successor. Let 7 = r + 1. From the definition of λfτ+ι (either we add an extender predicate to Mr or extend the J-hierarchy for one more step), J^ is an initial segment of Mτ = £ω( Λ/V) This contradicts the minimality of (7,e). Thus J^ •=- ίjbiί^y) for some 7 < ζ. There is a unique such 7 by the following easy fact, whose proof we omit: if 7 / δ then CeCΛ/'-y) ^ ίjb(M), v*7, δ, e, fc. D We can now define the (j, ξ) resurrection sequence for E. CASE 1. i = 0. Notice that pι(Jχ*) < λ, since Jj^ is active. Since (/?ι,*ι) = (λ, 1} is not defined we must have Λ^ = M = j£* and j = 0. Then E is the ,*+! < S PU/.«+! the way, it is also true, though not at all obvious, that pJJ < pJJ_j => pίί

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last extender of λfξ, and the (j,ξ) resurrection sequence for E is defined to be the empty sequence. CASE 2. i > 0. Let 7 < ξ be such that j£ = C^ί-A/V). Notice that ikt > 1 as pki(Ni) < ^ and ω\ < OR^. Let π : &ki(Mj) -» tfc.-iί-Λ/γ) be the inverse of the collapse. Then the (j,ξ) resurrection sequence for E is {/?,-,fc,, 7, π)~S, where 5 is the (fc, — 1,7) resurrection sequence for π(E). (Here if E is the last extender of <£jb,( Λ/^), then by π(E) we mean the last extender of (tkt-ι(λfy).) This completes the recursive definition of the (j,ξ) resurrection sequence for E. For any premouse P with ωa = ORP and t <ω, and α λ < OR^, the (/, λ) down sequence of ^ is just that initial segment of the λ-dropdown sequence of P consisting of pairs (/?, k) such that (/?, k) <\ex (α,t). Now let us return to the situation of Case 2 of the definition of the (j, ζ) resurrection sequence for E, and adopt the notation there. Let us adopt our standard notational device by taking π(ORc* ^^) to be OR*"*-1^. One can easily see from our results on preservation of projecta that the (k{ — l,τr(λ)) dropdown sequence for 1tkt-ι(λfy), which is what we use to resurrect π(E), has the form

where tι = 0

or ti = {*(/%), t, -l).

We do not know whether it is possible that u φ 0. In order for this to happen we would need to have (#-i,fc,--i) φ (ft, t< - 1), Pk.-itffr) = Λ-iί^Ji and Λ.-ι(ί*.-ι(-V7)) < '(P*.-ι W)) We only know that Pfci-iίίfc.-iί-Λ/V)) = sup π" Pki-ι(jβ*) It seems plausible that π preserves the Jfe, — 1st projectum, so that in fact u — 0 must hold. Notice that if u φ 0, then the last integer Jfc, in the dropdown sequence gets decreased by 1 at the next stage of resurrection. Thus there are cofinally many stages in the resurrection at which the u associated to the stage is 0. These stages are important, so we now give a formal definition. Let E be on the sequence of £>(.Λ/ξ), A = lh£", and let

be the (j, λ) dropdown sequence of €j(Λ/"^), and let

be the (j, ξ) resurrection sequence for E. (We suppose the resurrection to be nonempty. Thus (βι,kι) = (λ, 1) is defined.) We have at once from the definitions

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that («0,40 = (/%.*<),

and for 1 < e < ί,

(ίβ,^e) =last term in the ((e-ι — 1, Ke-ι o

o ττo(λ))

dropdown sequence of ί/ β _ l -ι(ΛΛy β _ I ),

and

From our earlier remarks on the new dropdown sequences, we can find stages 1 <eι < e2 < ••• < e, _ι = t

such that

o 7Γ0((#_2, ti-a

(<**,_! Λ.-J = »e,-i-l O

Here if ei = 1, the notation "π βl _ι o . o TΓQ" stands for π0. We also set eo = 0, and interpret απβo-ι o o TΓQ" to stand for the identity embedding. We then have for 0 < n et..-i(JVT.j

is an ^βn — 1 embedding, for 0 < n < i — 1. In order to simplify the indexing a bit set r, _ n = 7βw for 0 < n < i — 1. Notice also that fcj_n = ί eΛ Thus, setting p = f — n, we have that for 1 < p < i

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is a kp — 1 embedding. Let us set Resp = ί* F _ι(Λ/; F ) and call (σp,Resp) the pih partial resurrection of E from stage ( j , ζ ) . (Notice that if p < q, then Resp represents "more resurrection" than Res^ in the sense that it goes back to an earlier model Afη and hence nearer to the first appearance of the prototype of E. On the other hand, Resp resurrects less of Λi in the sense that the domain Jβ* of σp is smaller than that of σq . The partial resurrections of E agree with one another in the following way: For 1 KI > > /c, , and that if p < q then σp \ κ,q-\ = σq \ /c g _ι and the models Resp and Res^ agree below supσ^K^-i. For example, consider the case q = i. Then σ, = TΓQ : Jβf —> ίjb.-iίΛ/'-yo), and moreover, the last term of the (Jb, — 1, fl*o(λ)) dropdown sequence for (£t1-ι(^y0) corresponds to a projectum which is greater than or equal to sup(πό//c, _ι). This implies that πy \ supπό' /c, _ι is the identity, for all j > 0. So

σp \ /c, _ι = π e ,_ p o -

o πi o π0 f /c, _ι = π0 \ /c -i = σ, f /c, «ι.

See figure 1 for a diagram of some of the relationships above. Finally, the complete resurrection of E from (j*,f) is the pair (identity, λfς) if the (jyξ) resurrection sequence for E is 0 (so that j = 0 and E is the last extender of .Λ/^), and the pair (σι,Resι) if the (.;,£) resurrection sequence for E is nonempty. Notice that in any case, Res =
r

(commutativity) If βTa and (/?, a]τ Π D = 0 then πα o ίjα = t^

W. J. MITCHELL AND J. R. STEEL

114

Res3

Res2

Resi

FIGURE 1. To simplify matters, this diagram assumes that i = 3 and a so βι < βi < β3 ft l assumes that the new dropdown sequence is just the image of the old minus its last term, that is, that "u = 0" always holds. Thus t = 2, CQ = 0, e\ = 1, and 62 = 2. Also, σ$ = τr0, σ2 = πi o π0, and σ\ = π2 o π\ o KQ. Finally, we assume that *β(Λ.-ι(€e.(-Λf7.))) = p/ β -ι(ί/ β _ι(jVV.)) for e = 1,2, which, together with a similar assumption on TTQ, implies u = 0.

Next, we have some agreement of models and embeddings to maintain. For each ordinal β < IhT, let Vβ be the natural length of Ej and let l(σ^, Res^) be the complete resurrection of πβ(Ej) from stage (j, r), where j" = degr(/?) and H3.

For each β < α, if Res^ is type I or III then Qa agrees with Res^ below

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yRes ^ moreover ττα \ Vβ = cr o Kβ \ i/β

and

ίΓαv^) ^! ^

H4. For each /? < α, if Res^ is type II then Qa agrees with Res^ below and moreover Λ

T

= <τ o τr/3 t Ih Ea H5.

and

7

For each β < α, Ra agrees with Λ/? below z/Res/9 + ω, that is V** = V7Λ'

where 7 = i'1*68 4- ω In order to handle the limit case in the definition of W, we will require two final induction hypotheses. If Q = <£jb(.Λ£y) and Q' = &j(λfξ) where NΊ and Λ^ are two models of the construction C, then we write Q
Let β = T-Pred(α + 1) and C**1 = #,α+ι(C) Then

(a) Qα+i (b) if a + I G Dτ , then H7.

and

If λ is a limit ordinal then i^λ(Q0) = Qλ for all sufficiently large αΓλ.

We shall need to know that U is a tree in the "coarse structure" sense of [MS]. Set ffi = v***β . Then it will be obvious from the construction that Efβ is ff£ + ω strong in the model Rβ. We shall show in the remark following claim 1 below that f/β < $ whenever β < 6, and the agreement condition on the models Rβ follows at once from this. This guarantees that U is a normal iteration tree in the sense of [MS], provided that no illfounded model appears in U. Thus we need to know that we encounter no illfounded ultrapowers or direct limits in the formation of If. This follows from the following theorem, which is proved by the methods of [MS]. Theorem. If there is no ordinal 7 < ξ such that L(VΊ) ^ "7 is a Woodin cardinal" then every iteration tree on L(Vζ) has a unique coβnal wellfounded branch. Note that if theorem 12.1 holds for all η' < η, so that Mη exists, then Afη is constructed in Vξ for some ordinal ζ smaller than the least cardinal δ such that L[Vt] satisfies that δ is a Woodin cardinal. Thus we can apply this theorem to the trees derived from If. We now begin the recursive definition of the tree U and the embeddings πα. For α = 0 we take Q0 = PQ, RO = L(V$) where θ is the least ordinal 7 such that \= "7 is a Woodin cardinal", and π0 = identity.

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Definition oflί: The Successor Step. We assume that the tree has been defined through the αth model Λ α , and we have the embeddings πα mapping Pa into Qα, where j = degr(α) and QQ = (ίj(-A^))Λβr, and we have (in Ra) (<7a, Resα) = complete resurrection of πa(E%) from (j,ξ) , where πQ(E%) is the last extender predicate of QQ in case E% is the last extender predicate ofPa. CLAIM 1.

If 7 is strictly smaller than a then σα f πα(lh£!^) = identity.

PROOF. Fix 7 < α. Then lh£^ is a cardinal of Pa, so πα(lhί?^) is a cardinal of Qa. Thus ρω(J^a) > πα(lh£^) for all β such that πα(lh££) < ωβ < ORQ<*. We claim that also pj(Qa) > πα(lh£^). (Recall that j = deg(α).) Assume first that α is a successor ordinal. Then Pa = Ultj^^li^j), and so !!!££_! < pj(Pa) Thus lh£^ < pj(PQ), and as πα is a weak ./-embedding, πQ(lhE^) < pj(Qa) Now our claim for the case α is a limit ordinal follows from the successor case applied to sufficiently large α'Tα. Thus no projectum associated to a term in the (j, πa(lhE^)) dropdown sequence for Qct lies below πα(lh E^)y and it follows that σa is the identity below REMARK. The claim enables us to show that ^ >

for all β < α. For

But now, for β < α, lh#J is a cardinal of Pa and IhEj < va. Thus vβ < ι/α for β < a. So

< lhi£, so that

But Claim 1 tells us σα o πα(ί//j) = πα(ϊ/^), and our induction hypotheses on agreement of embeddings say τra(vβ) > v1168 . So

We can now define £^f and Λα^ι. Set F = σa o πα(i;J) = last extender of Resα . Now Resα is an "W model" in the universe Ra, so its last extender has a "background extender". Set £^ = F*9 the background extender for F in Ra. Let β = Γ-pred(α + 1) and set

FINE STRUCTURE AND ITERATION TREES

117

Notice that Ult0 = Ultα, since Rβ \= ZFC. Let us note that Ra and Rβ are in sufficient agreement that this ultrapower makes sense. This is clear if β = α, so we may suppose that β < a. By our induction hypotheses, Ra agrees with Rβ to v***β + ω. Now crit f% < ι/β because β = Γ-pred(α + 1). As σa is the identity on πa(lhEj), crit F* = crit F = crit σa(πa(E^)) ultrapower makes sense.

-

i ("e, *e)) be the (n, λ) dropdown sequence of Pβt

and set /c, = pkt(j£β) for 0 < i < e. The following claim relates these to the (n,π^(λ)) dropdown sequence of Qβ. The claim is slightly complicated by the fact that πβ is not a full n-embedding. Notice that κe < pn(Pβ) CLAIM 2. The (n,π^(λ))-dropdown sequence of Qβ is the sequence given by the appropriate clause below: (a) If κe < pn(Pβ) then the dropdown sequence is

(b) If κe = pn(Pβ) but (ωηe, ke) φ (ORP^, n) then the dropdown sequence is

where u = 0 or w = (17, n) for ωr; = T

(c) If (ωηe, ke) = (OR * , n) then the dropdown sequence is

where u = 0 or u = (π^Tje), ie) = (ωry, n), for ωη = REMARK. Note that κe = ρn(Pβ) in case (c). If e = 0, then n = 0 = fc0 and 170 = λ = α λ = ORp/>. The (n,^(A)) dropdown sequence for Qβ is then ((ORS',0)), which falls under case (c). REMARK. The u = 0 case in (c) would not be necessary if π^ were a full n-embedding. The claim follows easily from the fact that πβ is a weak n-embedding. For (a), notice that *β(κe) ηe(pω(J$) > thus Q^ f= V T > *β(η.)(fr(j)

> *β(*J).

D

118

W. J. MITCHELL AND J. R. STEEL

Let μ = crit(E£), and let j=

f β+1

\ least .; s.t. KJ < μ

if κe < μ.

Notice that i > 0 since KQ = λ > μ. Because T is maximal

»,.,+ !,

" «-\r,

and

d e g τ ( α + l ) =(f fnc < ~ 1

β ί ifίi~ = e'+ ,1.

Let (σf , Resf ) be the ith partial resurrection of πβ(E^) from stage (n, r), where Qβ = CnCΛ/τ) Λ *> if this resurrection is defined. The resurrection is undefined if i = e + 1 and defined if i < e by claim 2. If t = e then (σf , Resf ) is undefined just in case (ωηe , ke) = (OR^ , n) and the conclusion of (c) of claim 2 holds with u = 0.

Now let

_ f Resf

φ +1

ί

I Q/? σf . identity

if Resf is defined otherwise, if Resf is defined, otherwise.

Then σ o (πβ \ T^+i) is» in any case» a wea^ degr(α -f 1) embedding from ^α+i into Φα+i To see this, assume first that Resf is defined, so that t < e, degr(α -hi) = ki — 1, and σ = σf is a full (Jbt — 1) embedding. Looking at claim 2, we see that in all cases the domain of σ is J^* \ since we cannot have =

an(

the situation in (c) with i = e and u = 0. But P^i Λ?/> i is a weak (Jb, — 1) embedding. In fact, if ωηi < ORPβ then πβ \ fully elementary, and if ωηi = OΈίf>β then Jfc, < n, so π^ f Pa+i ιs a wea^ embedding. It follows that σo(π / u f ί^β+i) isa wea^ &t — 1-embedding from P^ into (?£+!• Assume next that Resf is undefined. Then either i = e+l or we have the situation in (c) of claim 2 with u = 0. In either case, degr(α + 1) < n. Also an( σ 8 ne Pa+i = Pβ, Qa+i = Q/?) l i ^ identity. Since πβ is a weak n-embedding, σ o 7Γ0 is a weak degr(α -H l)-embedding from Pa+ι into (JJ+1. Let Qβ = ίπί-A/i-)*' , so that (σ^, Re/) is the complete resurrection of πβ(Ej) from stage (n, r). Let V> be the complete resurrection embedding for σ o πβ(Ej) from the appropriate stage, which is (n, τ) if Resf is undefined and (fc, — 1, 77),

FINE STRUCTURE AND ITERATION TREES

where Resf = itj-i(X,), otherwise.

CLAIM 3.

119

Then ψ: J%°?f\χ) -" Res" and σ* =

φ \ (sup(σ o π/?" /ct -.ι)) = identity.

PROOF. Suppose first that Resf exists, so that i < e and σ = σf . From claim 2 and the fact that πp is a weak n-embedding we see that π^(/c, «ι) is the projectum associated to the (i - l)st element of the (n,π0(λ))-dropdown sequence of Qβ. As we remarked earlier, ψ is therefore the identity on sup(σf "πβ(κi-ι)), and this implies the claim. Suppose next that Resf is undefined, so that either i = e + 1 or else i = e and (c) of claim 2 holds with u = 0. In either case the projectum associated to the last term of the (n,π^(λ)) dropdown sequence of Qβ is at least sup(π0"/ct _ι). Thus σ& \ sup(π^///c, _ι) is the identity, but ψ = σ@ and σ is the identity, so this implies the claim. D We can now define Qα+ι = *J9,α+ι(Oα+ι) Before we define πα+ι and verify the induction hypotheses, however, we must describe the agreement between and Resα. Set exists ~~" CLAIM 4. is type II.

otherwise.

7 < λ = Ih(JSj), and if 7 = OR7""*1 then 7>£+1 = jf' and

PROOF. If β = α, then (μ+)^λ Λ exists and ^+1 is the shortest initial segment of Pa over which a subset of μ not in Jχ* is definable. Thus (μ+^+i = (μ+)J?~ < λ < ORP +S soT < λ < OR7'--". If β < a then the subsets of μ in Pa are just those in Jχ β and shortest initial segment of Pβ over which a subset of μ not in Jχ fi is definable, "P Λ

"P Λ

so if (μ*)^* exists then (μ+)p<*+1 = (μ+)^λ < λ. Otherwise /i is the largest cardinal of Jχ ft , so P^+i ^ Jχ* since λ is definably collapsed over the active ppm Jχ ft . In this case we see also that 7^+1 ls ^yPe H» since otherwise /i < 1/^3 < λ and ι//j is a cardinal of Jχ ft . D Claim 4 implies 7 < /ct _ι. If /c, _ι = λ then this is obvious. Otherwise /c, _ι is a cardinal of Jχ fi , since it is a projectum of some Jη β with 77 > λ. Since μ < /c, _ι by the choice of i, we have 7 < /c, _ι. The next claim shows that Resα and Q^+i have the agreement required for the use of the shift lemma. CLAIM 5. (a) Res" agrees with Q*+1 below sup(σoπ^ ;/ 7).

120

W. J. MITCHELL AND J. R. STEEL

(b) σa o πa ί 7 = σ o πβ \ 7. PROOF. The proof of claim 5 is divided up into three subclaims. Subclaim A. Qa+ι and Res^ agree below sup(σ o π^^γ), and σ o πβ \ 7 = V> o σ o πβ \ 7. This follows at once from claim 3 and the fact that 7 < κ, _ι. Subclaim B. If β < a then Res^ and Qa agree below sup(σoτr^;/7), and ψoσoπβ \ Ί = *a \ 7 Recall that ψ o σ o πβ = σ& oπβ. This subclaim is therefore just our induction hypotheses on agreement. If Res^ is type I or type III then claim 4 yields 7 < vp and we can apply H3. If Res^ is type II then 7 < IhE'J by claim 4 so we can apply H4. Subclaim C. If β < a then Qa and Resα agree below sup(σ o πβ'Ί) and πα f 7 =σ« o πa \ 7. We have 7 < λ, and σ o πβ = πα f 7, so sup(σ o πβ"j) < πQ(\). By claim 1, Qα and Resα agree below τrα(λ) and σa is the identity there. Together, subclaims A, B and C yield claim 5.

Π

Now define, for α G [i/α]<α; and appropriate / ([α, /]g+1) = [σα o πα(α), σ o If / = fτ|f then by "σ o π^(/)" we mean fτtσo*ft(q)> the later function being 1 defined over the ppm Qα+i I* order to see that πα+ι has the desired properties, it is useful to factor it. Let * = degr(α + 1) and Q'a+1 = Ult(Qi+1,F). Let ι: e e Φα+i "^ Qa+ι ^ the canonical embedding and let πj,+1: Pα+i —^ Qα+i ^ the weak i-embedding given by the shift lemma. Finally, let r: Q'α+1 —* Qα+i be the natural map given by r ί [α, 1\0Z** (πτ\\ = [°) ΛF* Then πα^ι = r o π^+1 and we have the commutative diagram

/ **1

}

Qα+l

•ί

r

~* β«+l

In order to verify HI we need to show that πα+ι is a weak fc-embedding, where k = deg(α -hi), which means that we have to find a witness set X on which

FINE STRUCTURE AND ITERATION TREES

121

πα+ι is rΣfc+i elementary. If k = deg(α + 1) = n and P^+i = *Pβ then we can take the witnessing set to be X = i*a+\"Xβy where Xβ is a set witnessing that 7Γ0 is a weak Jb-embedding. Otherwise take X = C+i" |^α+ι| *n e^her case the shift lemma implies that π^+1 is rΣt+i elementary on parameters from X. On the other hand the Los theorem 4.1 implies that r is rΣ*+ι elementary on parameters from i" |Q£+ι|, and since ic'a+^X C i" |φα+ι| ^ follows that πα+ι is rΣfc+i elementary on parameters from X. Thus X witnesses that πa+ι is a weak Jb-embedding and we have verified HI. Induction hypothesis H2 comes from the commutativity of the diagram above. We now verify H3 and H4. Let η < a -f 1. If Res'7 is type I or III then we must show that Qa+i agrees with Res'7 below i/1168* and moreover that πα+ι \ ι/η = ση o πη \ vη and πa+ι(vη) > i/***8 . If Res17 is of type II, on the other hand, then we must show that Qa+ι agrees with Res'7 below OR1168 and moreover that f Ά£ = σ" o », f &£ and We consider first the case η = α. Set ^' = *Ί(/J). By claim 3, so that

where the ultrapowers are computed using all functions which are members of o* Rβ> or equivalently of Λ α , and which map [μ']1 into J^**1 for some integer i. Now the canonical embedding ψ : Ult0(Resα, F) -+ Ult(Resα, F*) (where the first ultrapower uses all functions belonging to Resα, and the second uses all functions in Ra) has critical point > j/11*8* if Resα is type I or III, and > ORResOΓ = IhF if Resα is type II. Moreover, UltoίRes*, F) agrees with Resα below IhF = OR11680'. As i^a+1(^) > IhF, QQ+l agrees with Res0 below v***" 1168 in the type I or III case, and below OR * in the type II case. α

Next we consider the agreement of embeddings. Suppose first Res is type I or id 1 III, and ξ < va. Then ζ = [{£}» ]^τ* , where id = identity function, so = σ o

as desired. Also, let / E \Pa\ Π \PZ+ι\ and α € [v<*]<ω be such that z/α =

= [σ α oπ α (α), σα o >[σ α oπ α (α), σ Λ o

122

W. J. MITCHELL AND J. R. STEEL

But for (££)βu{*Όr} ae (ΰ,t>)> /(ΰ) = t>. Also σa o πα(ι/α) = ι/Resa , so σ α o = v for (F)σttoTtt(α)u{l/Re.«} a.e. (ϋ, v). Thus

and πα+i(ί/α) > i/1168*, as desired. These calculations carry over easily to the case Res" is type II to give the agreement of embeddings facts in part (b) of the claim. We omit further detail. We must now consider the case η < a. Let's just prove (a), the proof of (b) being similar. So assume Res'7 is type I or III. From the η = α case we know that Qα+ι agrees with Resα below ι/Reβ°' . But we showed in the proof of claim 5 that Resα agrees with Qa below πa(lbE%). Also, πa(lhE%) is a cardinal of Res", hence πα(lh£^) < i/**8*. Thus v****. Thus Qa+ι agrees with Res'7 below v***\ as desired. For agreement of embeddings, we argue similarly that πα+ι f va = σ α o πα f 2/α. Furthermore since IhE^ is a cardinal of Pa and lhE% < lh£"J, we know that lhE% < ι/α, and since σa is the identity on πα(lhJ?^) we get that πα+ι f lhE% = τrα f lh£^. But then since πα f ϊ/^ = σ*7 o πη \ vη by the induction hypothesis, πα+ι ί ι/η = σ17 o π^ \ ι/η, as desired. Notice also that we get πα+ι(i/^) = πa(vη) > v1168" by induction. This verifies H3 and H4. A much simpler coarse structural argument along the same lines gives H5. Finally, H6 is easy to check and H7 is vacuous in the successor case. Now let λ be a limit ordinal with λ < θ = IhT. We are given sequences U \ λ, (Qa \ Λ < λ), and (πa \ a < λ) satisfying our inductive hypothesis, and must define U f A + 1, Q\> and π\. Let c = [0,λ)τ = {α | αTλ}. We claim that lim α € c Λ α is wellfounded, where the limit is taken along the maps f£β for α, β G c. For this it suffices, using results of [MS] asserting that T has at least one well founded branch, to show that if 6 is a branch of T \ A which is cofinal in A, and b φ c, then lim^ Ra is illfounded. So let b be such a branch. We may assume t^(Qα) = Qβ f°r &U sufficiently large α and β in 6, α < /?, as otherwise our last induction hypothesis 6(a) implies that i^(k = liπiα€& Pα, and Qι = limα€6 Q α , which is the common value of iJk(Qe) for α 6 6 sufficiently large. Then P& exists as D T Π6 is finite, and Pi is illfounded as T f A is simple and 6 φ c. There is a natural π : P\> —> Qi given by our

FINE STRUCTURE AND ITERATION TREES

123

commutativity hypothesis: π(i^b(x)) = ^(iΓαOO), for α 6 6 sufficiently large. Thus Qb is illfounded, and hence Tib is illfounded since Tib = limα€δ RQ ^ "Qb is wellfounded" . We set R\ = limα €c Λ α , and this gives us U \ λ + 1. Notice that i^λ(Qα) is constant on all sufficiently large αTλ, as otherwise *
»λ(£λ(*)) = &(»«(*)) for α < λ sufficiently large, αTλ. Let n = degr(λ) = degr(α) for αTλ sufficiently large. It is easy to check that π\ is a weak n-embedding which is rΣn+ι elementary on the appropriate set, and that π\ commutes properly. Our last induction hypothesis is just the definition of Q\ so we need only check that Q\ and π\ agree properly with and σ^ o π^ for β < λ. Let β < λ. We have already shown that if 7 > /?, then ι/ReβΎ > ί/**8*. But v**y < Ihfiζf, and thus β < Ί =» v**** < #ι7+ι(crit JEξf) where 17 = T-pred (7+ 1). As R\ is wellfounded, we must have i/11®8 < crit £^, for all sufficiently large 7 + 1 Tλ. We can then find 7 + 1 Tλ sufficiently large that v***ft < crit i^+1 λ and i^+i.λίΦy+i) = Qλ

By induction, Q7+ι agrees with Res^ below

Qλ agrees with Q7+ι below crit &f+ιt\ So Q\ agrees with Res^ below v For the embeddings, notice that β < 7 =Φ- ι/β < ι/Ί < t^7+1(crit E%)9 where η = T pred(7 + 1). So we can assume the ordinal 7 + 1 of the last paragraph is such that iJf+ι \ is defined and ι/β < crit i^i \ But then, for α <

Since τr-y+ι f i/^ = σ^ o πβ \ ι/β by induction, ?TA \ vp = σ^ o π^ ί ι/^, as desired. This proves the agreement hypothesis in the case Res^ is type I or HI. The type II case is almost the same. We have completed the definition of U \ θ = U. Assuming that θ is a limit ordinal, methods of [MS] yield a cofinal, wellfounded branch 6 of U. It is easy to see (cf. the limit case above) that 6 is a wellfounded branch of T. This was what we needed. In the case θ is a successor, the fact that U can be extended freely one more step guarantees the same for T, as desired. The remaining clauses of fc-iterability can be proved similarly, using that the corresponding operations on L(Ve) yield wellfoundedness. This completes the proof of 12.1.

D

124

W. J. MITCHELL AND J. R. STEEL

The 0-iterability of the bicephali and psuedo-premice arising in the construction of § 11 can be proved similarly.

References [D] A. Dodd, Strong cardinals, unpublished notes. [DJ1] A. Dodd and R. B. Jensen, The core model, Annals of Math. Logic 20 (1981) 43-75. [DJ2] A. Dodd and R. B. Jensen, The Covering Lemma for K, Annals of Math. Logic 22 (1982), 1-30. [DJ3] The Covering Lemma for L[{7], Annals of Math. Logic 22 (1982) 127155. [DJ4] A. Dodd and R. B. Jensen, untitled notes on fine structure below a strong cardinal, unpublished. [M74R] W. Mitchell, Sets constructive from sequences of measures: revisited, JSL 48 (1983), 600-609. [M85] W. Mitchell, The core model for sequences of measures I, Math. Proc. of the Cambridge Phil. Soc. 95 (1984) 229-260. [M?] W. Mitchell, The core model for sequences of measures II, to appear. [MS] D. A. Martin and J. R. Steel, Iteration trees, to appear. [MSS] W. Mitchell, E. Schimmerling, and J. Steel, The Covering Lemma up to One Woodin Cardinal, 1992, preprint. [Sch] E. Schimmerling, Combinatorial Principles in the Core Model for One Woodin cardinal, 1992, preprint.o [S?a] J. R. Steel, The Core Model Iterability Problem, in preparation. [S?b] J. R. Steel, Projectively Wellordered Inner Models, 1993, submitted to Annals of Pure and Applied Logic. [S?c] J. R. Steel, Inner Models with Many Woodin Cardinals, 1993, Annals of Pure and Applies Logic, to appear.

INDEX OF DEFINITIONS Definitions not numbered in the text are indexed here by the number of the theorem, lemma, or definition immediately preceding; thus "1.0.3 ff." indicates an unnumbered definition occuring in the body of the text after definition 1.0.3.

1.0.1 1.0.2 1.0.3 l.O.Sff 1.0.3 l.O.Sff 1.0.4 1.0.5 1.0.5 1.0.5 1.0.8 2.0.1 2.0.2 2.0.2 ff. 2.0.3 2.0.4 2.3.1 2.3.2 2.23 2.3.4 2.3.5 2.3.6 2.3.6 2.3.7 2.3.8 2.3.9 2.7.2 2.7.3 2.7.4 2.7.5 2.8.1 2.8.1 2.8.1 2.8.2 2.8.3 2.8.4 3.0.1 3.1.1

(/c, ι/)-extender (/c, i/) pre-extender strongly acceptable generator of E natural length of E trivial completion good at α ppm (potential premouse) active passive weakly amenible types I, II, and III active ppm the language £ M Ί

rΣ0 rΣi the language £+ rΣx basic Skolem term Skn generalized rΣn Th^pO ?^(α,6) H**(X) cofinal rΣ0 embedding rQ the kth core parameter of M M is fc-solid M is i-sound Jb-embedding Λίsq squashed ppm (sppm)

5 5 6 6 6 6 7 7 7 7 8 10 10 11 11 11 13 14 14 14 14 14 14 15 15 15 21 21 21 22 23 23 23 24 24 24 28 29

FINE STRUCTURE AND ITERATION TREES

3.1.2 3.1.4 3.3.1 3.3.2 3.3.3 3.3.4 3.3.5 3.5.1 4.0 if. 4.4.1 5.0.1 5.0.2 5.0.3 5.0.4 5.0.5 5.0.6 5.0.6 ff. 5.0.6 if. 5.1 ff.

the language £* P-formula the language £** ςrΣn τφ(vQ •••Vk) SKn generalized $Σn premouse UltnίΛί,^) E is close to M tree order [/?,T]T Γ-Pred(7 +1) Jf* M is an initial segment of M M and W agree below 7 iteration tree on M DτίE^ίM^l maximal, cofinal, wellfounded branch of T wellfounded branch 5.25 simple 5.1.3 Jfc-bounded iteration tree 5.1.4 λ-iterable ppm 5.1.5 1-smallppm 5.1.7ff. weak n-embedding 5.2 ff. weak n-embedding from T to U 5.2 ff. tree embedding from Ί to U 5.2 ff the copying process: πT 6.13 n-maximal 7.0 ff. padded iteration tree 9.1.1 bicephalus 10.0.1 pseudo-premouse 11.Off. reliable premouse 12.1.1 λ-dropdown sequence of M 12.1.1 ff. (j, ^-resurrection sequence for E 12.1.1 ff. (J, λ)-dropdown sequence for M 12.1.1 ff. pth partial resurrection of E 12.1.1 ff. complete resurrection of E

127

29 29 32 32 32 32 32 33 34 42 47 47 47 47 47 47 47 47 50 50 50 51 51 52 52 54 54 54 61 69 91 96 99 109 109 109 113 113

INDEX

Q formula, 15 Los Theorem, 35 acceptable, 6 active, 7 ammenability, 8 Baldwin, S., 2, 6 bicephalus, 89, 91 bounded, 51 closure, 7 cofinal, 15, 50 coherence, 7 Comparison Lemma, 69 comparison process, 8, 69 complete resurrection, 113 completion, 6 condensation, 74, 85 copying, 78 core, 22, 24 coremouse, 52 Dodd, A., 1, 28 Dodd-Jensen Lemma, 55 Dodd-Jensen lemma, 63 Doddage, 89 dropdown sequence, 109, 111 drops, 79 extender, 5 Friedman, S., 8, 42 generators, 6, 7 good,7 hydras, 10 initial segment condition, 7, 8, iterability, 50, 115 iterablity, 51, 52 iteration tree, 47 iteration trees, 69

FINE STRUCTURE AND ITERATION TREES

Jensen, R., 1 Kunen, K., 8 Linus, 102 Los Theorem, 11 Magidor, M., 2, 13 Martin, A., 1 master code, 13, 24, 40, 43 maximal, 50, 61 Mitchell, W., 1, 2, 6 mouse, 52 natural length, 6 non-overlapping, 61 normal form theorem, 11 overlapping, 61 P-formula, 29 padded iteration trees, 69 parameter, 21, 22 passive, 7 potential premouse, 2, 7 ppm, 7 pre-extender, 5 premouse, 2 projectum, 22 pseudo-iteration tree, 75 pseudo-premouse, 96 Q formula, 15 quasi-Σn, 32 reliable, 99 resurrection, 109, 110 rQ formula, 15 saturated ideals, 1 security, 102 sharps, 52 Shelah, S., 2 shift lemma, 53 Silver, J., 2, 13 simple, 50, 58 simplicity, 63 Skolem hulls, 15

129

130

W. J. MITCHELL AND J. R. STEEL

Skolem terms, 13 small, 52 solidity, 21, 24, 74 Solovay, R., 1 soundness, 24 sppm, 29 squashed mouse, 28, 29 Steel, J., 1 strong uniqueness theorem, 63 superstrong, 30 tree embedding, 54 Uniqueness Theorem, 58 uniqueness theorem, 63 universality, 21, 74 weak n-embedding, 54 wellfounded branch, 50 Woodin Cardinal, 1 Woodin cardinal, 101 Woodin, H., 52

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