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Reconstructing resurrection

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Farmer Schlutzenberg

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April 8, 2016

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Abstract Let R be an iterable weak coarse premouse and let N be a premouse with Mitchell-Steel indexing, produced by a fully backgrounded L[E]-construction of R. We identify and correct a problem with the process of resurrection used in the proof of iterability of N .

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Introduction

The purpose of this note is to fill a small gap in the proof of iterability of inner models with Mitchell-Steel indexing built by L[E]-construction with full background extenders (see [3, §12]). Such constructions are used to build canonical inner models of set theory having large cardinals. They are important, for example, in calibrating the strength of determinacy axioms against large cardinal axioms. Such constructions are ubiquitous in inner model theory. Iterability requires roughly that natural ultrapowers of the model must be wellfounded. It is essential to know that L[E]-constructions produce iterable inner models, when they indeed do; iterability helps ensure the canonicity of the inner models constructed, and it implies that they possess basic fine structural properties (such as condensation), which are needed for the general theory. The inner models under consideration are of the form L[E], where E is a sequence of partial extenders E. Consider a model L[E] built by a full background extender construction. The iterability proof of [3] relies on the resurrection process described there, which lifts elements E of E to extenders E ∗ of V . The (main) gap we discuss lies in this process: the process can fail, and in fact, for some E it seems there is no obvious candidate for E ∗ , if one requires that E ∗ be an extender of V .1 In Example 2.4 we provide a specific example in which the resurrection process fails. The problem is closely related to the problem with the copying construction described in [4].2 The main content of the paper is the proof of Theorem 2.6, an iterability proof for a model built by a full background extender construction, assuming iterability of the background universe, which uses a correct resurrection process. The fix to resurrection is similar in nature to the fix to the copying construction given in [4], but there are more details. In the correct resurrection process, resurrection can itself involve taking (finitely many) ultrapowers, and E ∗ can be an extender of an ultrapower of V , instead of V itself. Our modified resurrection process will produce a candidate E ∗ , and this may or may not be an extender of V , but even if it is, it may not be produced in the manner described in [3]. 2 In fact, because the iterability proof uses the copying construction, the problem with the copying construction itself also arises, so we incorporate the corrections described in [4] here. 1

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The problem with resurrection only arises with premice with Mitchell-Steel indexing (as opposed to Jensen indexing).3 Aside from the main gap, there also appears to be a small problem with the definition of weak n-embedding (see [3, ?] and [7, ?]); these embeddings arise naturally in the iterability proof (of [3, §12]). This problem was noticed by Steve Jackson, and is explained in [4, §?]. The problem is just apparent, in that we do not have an explicit example of it. To deal with this, we take weak n-embedding to be defined as in [4, §?]. We won’t discuss this issue any further here.

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Conventions and Notation

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Given some transitive structure R = (M, . . .) with universe M , we write bRc for M , and write J (R) for the rud closure of R ∪ {R}. If κ < ORR and cardR (κ) is the largest cardinal of R, (κ+ )R denotes ORR . All (fine) premice in this paper have Mitchell-Steel indexing. Let P be a premouse. Given α ≤ ORP , we write P |α for the initial segment of P of ordinal height α, and P ||α for its passive counterpart. We write F P for the active extender of P , EP = E(P ) for the extender sequence of P , excluding F P , and EP+ = E+ (P ) for EP b F P . Given a short extender E, we write cr(E) for the critical point of E, ν(E) for the natural length of E, lh(E) for the length of the trivial completion of E and str(E) for the strength of E. T Given an iteration tree T of length λ + 1, we write bT for [0, λ]T and iT for iT0,λ and M∞ for MλT . Given premice P, Q, and a fine structural embedding π : P → Q, the phrase “π : P → Q” conventionally indicates that, literally, dom(π) = C0 (P ). Recall that for type 3 premice P , P sq denotes the squash of P , and has universe P |ν(F P ) ∈ P (and a predicate coding F P  ν(F P )); see [3, §3]. When P is type 3, C0 (P ) = P sq , so π does not act, at least not directly, on elements of P \C0 (P ). It seems that this convention probably helped to disguise one of the problems we deal with here. From now on in this paper we display all domains and codomains literally, writing, for example, π : C0 (P ) → C0 (Q), so as to keep the true domain of π in mind. (However, we do use the convention that fine structural notions such as ρP1 , and fine structural ultrapowers, are literally computed over C0 (P ).) We take weak n-embedding to be defined as in [4, ?]. Other notation and terminology is standard and mostly follows [7].

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Resurrection

We first define a fairly general kind of full background extender construction (nice construction), which includes typical full background extender constructions in the literature. Then in Example 2.4 below, we give a specific example of the problem with resurrection. After this we will sketch the fix to this problem, and then, in (the proof of) Theorem 2.6, give a complete iterability proof incorporating the fix. 3

The same problem arises in the iterability proof of [6], i.e. the proof of [6, Theorem 9.14]. The author believes that the fix we describe here can be adapted to that context. However, in that context there is more work to do, particularly because the statement of [6, Theorem 9.14] itself depends on the notion of resurrection, and so as we will see, does not literally make sense. The author intends to provide a fix for its statement and proof in a future article.

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Definition 2.1. A weak coarse premouse (wcpm) is a premouse as in [2, Definition 1.1].4 Suppose V = (bV c , δ) is a wcpm.5 For λ ≤ δ + 1, a nice construction (of length λ) is a sequence hNα iα<λ such that (i) for each α < λ, Nα is a premouse, (ii) N0 = Vω ,6 (iii) for each limit γ < λ, Nγ = lim inf α<γ Nα , and (iv) for each α + 1 < λ, Nα is ω-solid and either Nα+1 = J (Cω (Nα )) or there are E, E ∗ such that

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(a) Nα+1 = (Nα , E),

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(b) E ∗ ∈ Vδ is an extender,

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(c) E  ν(E) ⊆ E ∗ , (d) for each κ < ν(E), if there is G ∈ E+ (Ult(Nα , E)) with G total over Nα and cr(G) = κ then κ < str(E ∗ ). a

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Remark 2.2. Suppose V is a wcpm and let C = hNα iα<λ be a nice construction. Recall the following basic facts from [3], which we will use freely. Let α < λ. Let ρ ≤ γ < OR(Nα ) be such that ρ is a cardinal of Nα and ρω (Nα |γ) = ρ. Then there is a unique ξ < α such that C0 (Nα |γ) = Cω (Nξ ). Let E ∈ E+ (Nα ) be such that E is total over Nα . Then cr(E) is measurable (in V ). Definition 2.3. Let M, N be premice of the same type and let π : C0 (M ) → C0 (N ) be an Σ0 -elementary embedding. We define the embedding ψπ as follows. If M is passive then ψπ = π. Otherwise, ψπ : Ult(M, F M ) → Ult(N, F N )

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is the embedding induced by the Shift Lemma. Note that in all cases, π ⊆ ψπ .

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Example 2.4. We now give an explicit example of the problem with resurrection, and sketch the fix we will use. Suppose V is a wcpm and C = hNα iα<λ is a nice construction, every Cn (Nα ) is fully iterable and there is α such that Nα has a type 3 proper segment M such that M |=“cof(ν(F M )) is measurable”. Let α be least such and M the least such proper segment of Nα . M M We claim that ρM that since M 1 = ω and p1 = ∅. For let H = Hull1 (∅) (recall sq M sq is type 3, this means that H is the premouse such that H = Hull1 (∅); cf. the convention established toward the end of §??). Then ρH 1 = ω, H is 1-sound, and 0iterable, so a comparison shows that H E M . So it suffices to see that H |= “cof(ν(F H )) is measurable”. Note that M |=“ν(F M ) = κ+µ ”, where κ = cr(F M ) and µ is the least 4

We do not use the term coarse premouse because this is used differently in [6]. This hypothesis just means that we work inside some wcpm. It is not intended to imply that V |= ZFC. 6 Although we restrict to pure premice Nα here, this is not important; everything in the paper relativizes immediately to premice above some fixed set. 5

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measurable of M , and that if π : C0 (H) → C0 (M ) is the uncollapse map, then µ, κ ∈ rg(π). Moreover, for each α ∈ µ ∩ rg(π), we have (κ+α )M ∈ rg(π), since the identity of (κ+α )M is coded into a segment of F M in a Σ0 way. Therefore H |=“ν(F H ) = κ ¯ +¯µ ”, where π(¯ µ, κ ¯ ) = µ, κ, which suffices. It follows that α = ξ + 1 for some ξ, and letting N = Nξ , C0 (M ) = C1 (N ) = Cω (N ), and N is active and type 3. Note that all 0-maximal iteration trees on M are linear. T Now there is a successor length 0-maximal tree T on M such that N = M∞ and bT does not drop in model. This can be seen in two ways: either because M is below 0¶ , or by the stationarity of L[E]-constructions [5, §4]. Moreover, the core embedding % : C0 (M ) → C0 (N ) is % = iT . Let κ = cr(F M ), ν = ν(F M ) = OR(M sq ) and let µ be the least measurable of M . So µ < κ and cof M (ν) = µ and iT (κ) = cr(F N ) and iT restricts to a fully elementary map M |κ → N |cr(F N ). So iT (µ) is the least measurable of N . Therefore iT (µ) is measurable, so µ < iT (µ). It follows that µ = cr(iT ). Let P = M1T . By the preceding remarks, P = Ult0 (M, U ) (recall that this means that P sq = Ult(M sq , U )) where U ∈ EM is the normal measure on µ, and 1 ∈ bT and degT (1) = 0. We claim that ν(F N ) < ψiT (ν) = ψ% (ν). This follows from [4, Lemma 2.11], but here things are simpler, so we include the proof for self-containment. In M , let f : κ → κ be sq T → P sq ). Then f (α) = α+µ . Then ν = [{κ}, f ]M F M . Let j = i0,1 (so j : M ν(F P ) = sup j“ν = (j(κ)+µ )P < (j(κ)+j(µ) )Ult(P,F

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= [{j(κ)}, j(f )]PF P .

Since also ν(F N ) = sup iT “ν, the claim follows easily. Therefore, since ν is a limit cardinal of M , we have (†) N ||ORN / ψ% (M |ν). Now the resurrection maps of [3] are formed by composing core embeddings. In particular, if M |λ / M is active, and we wish to resurrect this to find some backgrounded ancestral extender, then according to [3], we should consider %(M |λ), then resurrect this structure with further core embeddings, as needed. But if ν < λ < ORM , the first step here does not make literal sense, since M |λ ∈ / dom(%). Moreover, we can’t correct this by lifting M |λ with ψ% , since by (†) we have N ||OR(N ) / Q where Q = ψ% (M |λ), and so standard facts about nice constructions show that Q was never constructed by C. So the usual resurrection process, applied to M |λ, breaks down. To solve this problem, in the proof of 2.6 below, we will use, approximately, the following approach. It is similar to the correction to the copying construction given in [4]. We continue in the scenario above. Let E ∗ be a background extender for N . Then (see the proof to follow) there is Q∗ in Ult(V, E ∗ ) and an elementary embedding Q → Q∗ , such that Q∗ is constructed by iE ∗ (C). Thus, we can move into Ult(V, E ∗ ) and continue the resurrection process there. In the example above, the same issue will not arise again, but in the more general case it could. In the latter case we can take another ultrapower, and so on. This procedure will terminate in finitely many steps, yielding a successful resurrection. We next give a detailed iterability proof incorporating the fix to resurrection sketched here (or a slight variant).7 7

We will not actually define an explicit resurrection process, but instead fold the details directly into the iterability proof. Moreover, because we need to use background extenders E ∗ as above in order to produce some form of resurrection, and there need not be a canonical choice of such E ∗ , there need not be a canonical resurrection for a structure such as M |λ above. However, using the methods to follow, it is easy enough to formulate an abstract notion of a resurrection (of some initial segment of a model produced by a nice construction computed in a wcpm R), in terms of finite iteration trees on R.

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Usually one deals with k-maximal trees, or stacks thereof. However, it does not take much more work to give the iterability proof for the following more general class of trees (which includes stacks of k-maximal trees, and more), so it seems worthwhile doing so: Definition 2.5. Let M be a k-sound premouse. We say that T is a standard degree k iteration tree on M iff T satisfies the conditions described in [3, §5], except that we drop condition (3) (the condition “α < β =⇒ lh(Eα ) < lh(Eβ )”), and strengthen the other clauses as follows. Let Mα = MαT and Eα = EαT . For α ≤ β < lh(T ) and κ < OR(Mβ ) we say that [α, β) is κ-valid (for T ) iff κ < minγ∈[α,β) ν(EγT ) and if α < β then (κ+ )Mα |lh(Eα ) = (κ+ )Mβ . For E ∈ E+ (Mβ ) we say that [α, β) is E-valid (for T ) iff [α, β) is cr(E)-valid and if α < β then E is Mβ -total. We require that if predT (β + 1) = α ∗T and degT (β + 1) be chosen as for then [α, β) is Eβ -valid. We also require that Mβ+1 k-maximal trees. For an ordinal α, we say that M is standardly (k, α)-iterable iff there is a winning strategy for player II in the iteration game on M for standard degree k trees of length at most α. a We make a couple of remarks on standard iteration trees. See [1, pp.3–5] for more discussion; standard trees all meet the definition of iteration tree used in [1]. Let T be standard. If κ < OR(Mβ ) then [β, β) is trivially κ-valid. Let α < β and κ be such that [α, β) is κ-valid. Then either (κ+ )Mβ < lh(Eα ) or Eα is type 2 and (κ+ )Mβ = lh(Eα ). For all γ ∈ (α, β), (κ+ )Mγ = (κ+ )Mβ < lh(Eγ ). Suppose further that κ = cr(E) for some Mβ -total extender E ∈ E+ (Mβ ). Then the type 1 initial segment G of E is on E(Mγ ) for all γ ∈ (α, β], and also G is on E(Ult(Mα , Eα )). Theorem 2.6. Let θ ≥ ω1 be a cardinal. Let R be a wcpm8 and ΣR a (partial ) (θ + 1)iteration strategy for R. Let C ∈ R be such that R |=“C is a nice construction”. Let ζ < lh(C) and z ≤ ω. If ΣR is total then Nζ is z-solid and Cz (Nζ ) is standardly (z, θ + 1)-iterable. If ΣR is defined on all stacks of non-overlapping trees, then Nζ is z-solid and Cz (Nζ ) is (z, θ, θ + 1)-iterable. If ΣR is defined on all non-overlapping trees, and if Nζ is z-solid, then Cz (Nζ ) is (z, θ + 1)-iterable. Proof. We mostly follow the usual proof, lifting trees on N = Cz (Nζ ) to R via copying and resurrection, but make modifications to deal with the problem described in 2.4. Assuming that ΣR is defined on all trees of length ≤ θ, we will describe a strategy ΣN for player II in the standard (z, θ + 1)-iteration game on N . Let T be a standard degree z tree on N which is via ΣN . Then by induction, we can lift T to a tree U = πT on R (U is to be defined), via ΣR , and if T has limit length, use ΣR (U) to define ΣN (T ). At the end we will make some modifications to the construction which will ensure that U is non-overlapping if T is z-maximal. We will have lh(U) ≥ lh(T ), but in general may have lh(U) > lh(T ). For each U node α of T , (α, 0) will be a node of U, and the model Mα0 will correspond directly to T Mα . However, there may also be a further finite set of nodes (α, i) of U, and models U Mαi associated to proper segments of MαT . For indexing, let OR∗ = OR × ω; we order ∗ OR lexicographically. We index the nodes of U with elements of OR∗ . Note though 8

It is not particularly important that R be a wcpm. We just need that iteration maps on R for trees based on VδRR are sufficiently elementary, but we leave it to the reader to reduce the hypotheses.

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that if lh(T ) > 1 then the nodes of U will form a set not closed downward under <. U For notational convenience we allow U to use padding. If E = Eαi = ∅ we consider U U Mαi U U str (E) = OR(Mαi ). (And whether or not Eαi = ∅, we allow pred (β, j) = (α, i) for (β, j) > (α, i).) The following definition is a coarser variant of the notion of dropdown sequence used M |η M |η M |η in [3, §12], which also records the various projecta ρn i in the interval (ρω i , ρω i−1 ). At this stage we ignore these intermediate projecta. In the end we will index partial resurrection maps by potential critical points κ, not by projecta. Definition 2.7. Let M be a k-sound premouse and γ ≤ ORM . Let ρ(ORM ) = ρM k and for η < ORM let ρ(η) = ρω (M |η). The (γ, k)-model-dropdown sequence of M is the sequence σ = h(ηi , %i )ii≤n of maximum length such that η0 = γ, and for each M i ≤ n, %i = ρ(ηi ), and if i < n then ηi+1 is the least η ∈ (ηi , OR ] such that ρ(η)

 9 < %i . M M M M If (OR , ρ(OR )) ∈ σ then let τ = ∅; otherwise let τ = (OR , ρ(OR )) . The extended (γ, k)-model-dropdown sequence of M is σ b τ . Let σ = (σ0 , . . . , σn−1 ) be a sequence. The reverse of σ is (σn−1 , . . . , σ0 ). If each σi = (ai , bi ) then p0 [σ] = (a0 , . . . , an−1 ). a We now fix some notation and state some intentions. Fix α + 1 < lh(T ). We write Mα = MαT , Eα = EαT and mα = degT (α). For now let M = Mα and E = Eα . Let σ be the extended (lh(E), mα )-model-dropdown sequence of M and let σ ∗ be its reverse. Let uα + 1 = lh(σ) and let hγαi ii≤uα = p0 [σ ∗ ]. Fix i ≤ uα . Let nαi = mα if i = 0 and nαi = ω otherwise. Let Cαi = iU00,αi (C) and ∆αi = lh(Cαi ). We will define ξαi < ∆αi , and letting Qαi = NξCαiαi , will define an nαi -lifting embedding παi : C0 (M |γαi ) → Cnαi (Qαi ).

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For m ≤ n ≤ nαi let nm ταi : Cn (Qαi ) → Cm (Qαi )

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be the core embedding. Let Q∗α = Qαuα and πα∗ : C0 (M |lh(E)) → C0 (Q∗α ),

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where letting n = nαuα ,

n0 ◦ παuα . πα∗ = ταu α

Let cα be the set of infinite cardinals κ < lh(E) of M |lh(E) such that if E is type 3 then κ < ν(E). Fix κ ∈ cα . Let (γ, nακ ) be the lexicographically least (γ, n) such that γ ≥ lh(E) and either (i) γ = ORM and n = mα or (ii) ρn+1 (M |γ) ≤ κ. Note that γ ∈ p0 [σ ∗ ]. Let i = iακ be such that γ = γαi . We also define the nακ -lifting embedding πακ : C0 (MαT |γαi ) → Cnακ (Qαi )

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nm by πακ = ταi ◦ παi , where n = nαi and m = nακ . If lh(T ) = β+1 then (β, 0) will be the last node in U, and we will also define ξβ0 < ∆β0 , C and letting Qβ0 = Nξβ0β0 , will define an mβ -lifting embedding

πβ0 : C0 (MβT ) → Cmβ (Qβ0 ). We will maintain the following conditions by induction on lh(T ):

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Note that these notions depend on k as ρ(ORM ) = ρM k .

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U 1. For all α + 1 < lh(T ), Eαu 6= ∅. α

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2. For all α + 1 < lh(T ) and all κ ∈ cα , πακ  (κ+ )Mα |lh(Eα ) ⊆ πα∗ ⊆ πα+1,0 ,

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and if α < β < lh(T ) and [α, β) is κ-valid then πακ  (κ+ )Mα |lh(Eα ) ⊆ πβ0 .

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3. Let α + 1 < lh(T ) and κ ∈ cα and i ∈ [iακ , uα ]. Suppose that if M 0 = Mα |γαi is active with extender F then there is E ∈ E+ (Ult(M 0 , F )) with E total over M 0 and U U ). (If M 0 is passive this will be trivial as we cr(E) = κ. Then πακ (κ) < strMαi (Eαi U = ∅.) will have Eαi 4. Suppose α = predT (β) and let i ≤ uα be such that Mβ∗ = Mα |γαi . Then (α, i) = predU (β, 0). (So T drops in model at β iff i 6= 0.) 5. Suppose (α, i) = predU (β, j). If j 6= 0 then ξβj < iUαi,βj (ξαi ). Suppose j = 0; so α = predT (β). Then ξβ0 = iUαi,β0 (ξαi ) and letting n = mβ , nαi n U πβ0 ◦ i∗T ◦ παi . β = iαi,β0 ◦ ταi

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6. Let λ < lh(T ) be a limit and let α
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The inductive hypotheses are trivial for T  1 and U  (0, 1), with π00 = id. Now let λ be a limit ordinal and suppose that the inductive hypotheses hold of T  λ and U  (λ, 0); we will define U  (λ, 1) and show that they hold for T  λ + 1 and U  (λ, 1). Note that U  (λ, 0) has limit length and is cofinally non-padded. Let c = ΣR (U  (λ, 0)). Let b = ΣM (T  λ) be the unique branch such that for eventually all α ∈ b, we have (α, 0) ∈ c. By conditions 4–6, b is indeed a well-defined T  λ-cofinal branch, and there are only finitely many drops in model along b, and there is a unique choice for πλ0 maintaining the commutativity (and all other) requirements. Now let λ = δ + 1 and suppose that the inductive hypotheses hold for T  δ + 1 and U  (δ, 1). We will show that they hold for T  δ + 2 and U  (δ + 1, 1). Case 1. Eδ = F (MδT ). We just give a sketch in this case as the details are mostly standard here, and anyway U U they are simpler than the next case. We have uδ = 0. Set Eδ0 to be some E ∗ ∈ Mδ0 T U witnessing 2.1 with respect to Qδ0 in Mδ0 . Let κ = cr(Eδ ) and α = pred (δ + 1) and ∗ i = iακ . Note that Mδ+1 = Mα |γαi and nακ = mδ+1 . We claim that it is possible to U set pred (δ + 1, 0) = (α, i); and we do this. For suppose α < δ. Then κ < ρmδ (Mδ ) so πδ∗ (κ) = πδ0 (κ). Let G be the normal measure segment of Eδ and let (α, i) ≤ (ε, l) < (β, 0) be such that M 0 = Mε |γεl is active with extender F . Then G ∈ E(Ult(M 0 , F )), and so by conditions 2 and 3, U

πδ∗ (κ) = πδ0 (κ) = πακ (κ) = πεκ (κ) < strMεl (EεlU ), 254 255

which suffices. Now we can define πδ+1,0 as usual and standard arguments show that the inductive hypotheses are maintained. 7

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Case 2. Eδ 6= F (MδT ). In this case we must deal with the problem described in 2.4. Let σ, σ ∗ be defined as before, with α = δ. Let (γ, ρ) = (σ ∗ )1 (so γ is the largest element of p0 [σ] excluding OR(Mδ ) and ρ = ρω (Mδ |γ)). Then ρ is a cardinal of Mδ . Subcase 1. ρ ∈ C0 (Mδ ). mδ 0 U Set Eδ0 = ∅. Let ϕ : C0 (Mδ ) → C0 (Qδ0 ) be ϕ = τδ0 ◦ πδ0 . Then ϕ(ρ) is a cardinal of C0 (Qδ0 ) and ρω (ϕ(Mδ |γ)) = ϕ(ρ). So we can let ξ = ξδ1 be least such that C0 (ϕ(Mδ |γ)) = Cω (NξCδ1 ), and let πδ1 = ϕ  C0 (Mδ |γ). Subcase 2. ρ ∈ / C0 (Mδ ). So Mδ is active type 3 and ρ = ν(F (Mδ )) = ρ0 (Mδ ). Let % : C0 (Mδ ) → C0 (Qδ0 ) be mδ 0 % = τδ0 ◦ πδ0 . Let ψ = ψ% . Subsubcase 1. ψ(ρ) ≤ ν(F (Qδ0 )). U Proceed as in Subcase 1, with ϕ = ψ; in particular, Eδ0 = ∅. Subsubcase 2. ψ(ρ) > ν(F (Qδ0 )). Here we need to do something different because ψ(Mδ |γ) is not constructed in Cδ0 . U Let Eδ0 = E ∗ witness 2.1 with respect to Qδ0 . Let F = F (Mδ ) and κ = cr(F ) and let T 0 0 be the putative standard tree on M of the form (T  δ + 1) b F , with α = predT (δ + 1) as T0 small as possible. Then Mδ |γ / Mδ+1 . Let i = iακ and n = nακ . Let predU (δ, 1) = (α, i); ∗ as in Case 1 this is possible. Let Q = iE ∗ (Qαi ) and 0

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be given as usual. Then ϕ(ρ) < ρn (Cn (Q∗ )) and ϕ(ρ) is a cardinal of Cn (Q∗ ) and ϕ(ρ) = ρω (ϕ(Mδ |γ)). Let ξδ1 be the unique ξ such that C0 (ϕ(Mδ |γ)) = Cω (NξCδ1 ). (So ξδ1 < iUαi,δ1 (ξαi ).) Let πδ1 = ϕ  C0 (Mδ |γ). This completes the definition of U  (δ, 2) in all subcases. If lh(Eδ ) = γ, we have uδ = 1 U U and we set Eδ1 to be a background extender for Qδ1 , with minimal strength. If Eδ0 =∅ U then we can set pred (δ + 1, 0) as required by condition 4 as in Case 1, and because ω0 cr(τδ1 ) ≥ ρω (ϕ(Mδ |γ)) = ϕ(ρ),

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U for each µ ≥ ω we get the right agreement between πδµ and πδ∗ . Suppose Eδ0 6= ∅ and U T cr(Eδ ) < ν(F (Mδ )), and so we want to set pred (δ + 1, 0) < (δ, 1). Let G be the normal measure segment of EδT . Then G ∈ E(C0 (Mδ )) and G is total over Mδ . So again as in ω0 Case 1, we can set predU (δ +1, 0) appropriately. Also in this case cr(τδ1 ) ≥ ν(F (Qδ0 )), so again each πδµ and πδ∗ agree appropriately. To see that condition 3 holds in this case, let µ ∈ cδ and suppose that 0 = iδµ and there is G ∈ E+ (Mδ , F (Mδ )) which is total over Mδ and cr(G) = µ. Then µ < ρ0 (Mδ ), so the normal measure segment of G is on E(Mδ |γ), U U so both Eδ0 and Eδ1 are strong beyond πδµ (µ). The other conditions are maintained as usual. Now suppose that lh(Eδ ) < γ. Let γ1 = γ and (γ2 , ρ2 ) = (σ ∗ )2 . Repeat the subcases, working with Mδ |γ1 , ρ2 , πδ1 , etc, in place of Mδ , ρ, πδ0 , etc. Continue in this manner until reaching some lift of Eδ . This completes the definition of U  (δ + 1, 1). Arguments like those above show that the inductive hypotheses are maintained.

This completes the proof for standard iterability. Now suppose that Nζ is z-solid and T on Cz (Nζ ) is z-maximal. We make the following adjustments to the preceding 8

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construction to ensure that U is non-overlapping. (The rest of the theorem follows as U usual.) Most things are as before. But suppose we need to set Eαi 6= ∅. Then Mα |γαi is active with some type 3 extender F and ν(F ) is a limit cardinal of Mα |γαi . Then let m0 ◦ παi “ν(F ), let F ∗ = F (Qαi ) and F 0 = F ∗  ξ, and let Q0 be the m = nαi , let ξ = sup ταi segment of Qαi with F (Q0 ) = F 0 . Because ξ is a cardinal of Qαi , Q0 = C0 (Nχ ) for some U U witness 2.1 with respect let Eαi χ ≤ ξαi . In all other cases let Q0 = Qαi . Working in Mαi U U to Qαi , chosen with strMαi (Eαi ) as small as possible. The foregoing proof still goes through with these changes.10 Moreover, we claim that U U is non-overlapping. For let (α, i) be such that Eαi 6= ∅ and let m = nαi . Our choice of U Eαi implies that U U m0 strMαi (Eαi ) ≤ sup ταi ◦ παi “ν(F ),

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and from this and the agreement condition 2, it is straightforward to prove the claim. This completes the proof.

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References

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[1] Steve Jackson, Richard Ketchersid, Farmer Schlutzenberg, and W. Hugh Woodin. Determinacy and J´onsson cardinals in L(R). To appear in Journal of Symbolic Logic, preprint available at https://sites.google.com/site/schlutzenberg/home1/research/papers-and-preprints. [2] Donald Martin and John R. Steel. Iteration trees. Journal of the American Mathematical Society, 7(1):1–73, January 1989. [3] William Mitchell and John R. Steel. Fine structure and iteration trees. Number 3 in Lectures Notes in Logic. Springer-Verlag, 1994. [4] Farmer Schlutzenberg. Reconstructing copying and condensation. Submitted, preprint available at https://sites.google.com/site/schlutzenberg/home1/research/papers-and-preprints. [5] Farmer Schlutzenberg and Nam Trang. Scales in hybrid mice over R. Submitted, preprint available at https://sites.google.com/site/schlutzenberg/home1/research/papers-and-preprints. [6] John R. Steel. The core model iterability problem. Number 8 in Lecture Notes in Logic. Springer-Verlag, 1996. [7] John R. Steel. An outline of inner model theory. In Matthew Foreman and Akihiro Kanamori, editors, Handbook of set theory, volume 3, chapter 19. Springer, first edition, 2010.

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If i < uα and ξ < ν(Qαi ) and ν = ν(F (Mα |γαi )) it seems that we might now have πα,i+1 (ν) = ξ (whereas in the former construction we would have had πα,i+1 (ν) > ν(Qαi )), but this is okay.

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