Fine Structure and Iteration Trees

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

Fine Structure and Iteration Trees

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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

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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

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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".

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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-

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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] 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 '. 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 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]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], 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 ...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, 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 < i < n - 2

1

~~ 1 π(p?) < ORM

ifp?

< OR*

f

the least α such that π(α) > 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} 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

FINE STRUCTURE AND ITERATION TREES

21

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) »(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] "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] 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.

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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(Λo 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) = { 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)[π(z)], then we have α, /, g such that [α,/] = (α,g) for some α < sup π"/>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.

FINE STRUCTURE AND ITERATION TREES

41

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 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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W. J. MITCHELL AND J. R. STEEL

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 [\} 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 ••• 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