Reconstructing copying and condensation Farmer Schlutzenberg September 30, 2015 Abstract We identify and correct some gaps in the inner model theory literature. These regard (i) the Shift Lemma for weak n-embeddings, (ii) the copying construction, for premice with Mitchell-Steel indexing, and (iii) the proofs of condensation, solidity and Dodd-solidity, for premice without extenders of superstrong type, and the ISC and weak ISC, for pseudo-premice without extenders of superstrong type.

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Introduction

We will identify some flaws in the inner model theory literature, and provide corrections. The first flaw we consider, noticed by Steve Jackson, lies in the definition of weak n-embedding, as defined in [1] or [9]. When one uses this definition, it seems unclear that the Shift Lemma, [1], actually holds for weak n-embeddings. (The problem here is just apparent, in that we do not have a specific counterexample.) We will describe two alternate ways of dealing with this problem. The first of these is due to Steve Jackson, and should be considered as a correction to the definition of weak n-embedding. This approach is very minimally invasive; it seems likely that most, if not all, uses of weak n-embeddings in the literature should go through with this change. The second approach is to replace the notion of weak n-embedding with a somewhat different (n) class of embeddings, the n-lifting embeddings (see 2.2), analogous to the Σ0 preserving embeddings of [10], and is somewhat more invasive. However, there are certain advantages to the latter approach, because it simplifies some issues. In particular, it leads to a simpler resolution to the third flaw we consider in the paper, described below. The second flaw, which only arises for premice with Mitchell-Steel indexing, lies in the copying construction, as presented in [1]. Recall that this is the algorithm for copying or lifting an iteration tree T on a premouse M to a tree πT on a premouse N , given some partially elementary π : M → N . There is a straightforward correction to this flaw. Part of this correction was presented in [9]. We give the full correction here. Fortunately, much of the time such corrections are not actually necessary; we also examine carefully the precise conditions under which they are. For example, we will show that if T is kmaximal and π is a near k-embedding, where either k ≥ 1 or M is not type 3, then the usual coyping construction algorithm works correctly. The third flaw lies in the proofs of condensation, solidity, Dodd-solidity, the ISC (initial segment condition) and weak ISC. (For condensation, solidity and

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the (weak) ISC see [1], [9, §5] and [5]; for Dodd-solidity, see [3, §3] and [5].) We refer to these properties collectively as condensation etc. In our discussion of condensation etc we only consider premice without extenders of superstrong type. The third flaw is independent of indexing, but only arises with certain formulations of the weak Dodd-Jensen property, such as that in [9]; we discuss the latter here. In the proof of condensation etc, a phalanx P and a premouse M are compared, producing trees T on P and U on M . At the same time, T is lifted to a tree V on M . The tree V is defined to be k-maximal, for a certain k ≤ ω. Ostensibly, this can have the consequence that for some α < lh(T ), the lifting map πα is only a weak degT (α)-embedding, not a near degT (α)-embedding (notwithstanding the fact that the copying construction usually propagates near embeddings, [4, 1.3]). Moreover, it seems this can occur in a context in which the standard proof invokes the weak Dodd-Jensen property with respect to an embedding of the form σ = πα ◦ iU : M → MαV , which is likely not a near embedding if πα is only weak; in fact it is not obvious that σ is even a weak embedding. But the weak Dodd-Jensen property of [2] is only shown to hold with respect to near embeddings. We give three different approaches to fill this gap. The first two stay closer to the original proofs, as they continue to deal with weak n-embeddings. The first of these, which is the simpler of the two, is to weaken the requirement that V be k-maximal. The second approach, which we just sketch, is to use a version of the weak Dodd-Jensen property which holds with respect to weak embeddings, given that there is a drop in model involved1 . This is a fairly straightforward adaptation of [2], but as mentioned above, it is not obvious that πα ◦iU is a weak embedding. It takes an extra argument to show that it is, making this approach a little more involved. The third approach, which results in the simplest proof overall, is to replace the use of weak n-embeddings with n-lifting embeddings (see 2.2). The issues mentioned in the last paragraph also relate to a fourth problem, which comes up in the same proof, and which we also address. By [4, 1.3], the copying construction propagates near embeddings. But this does not literally apply to the copying from T to V. Indeed, as mentioned above, some of our embeddings may fail to be near embeddings. So in our proof of condensation etc, we show that the proof of [4, 1.3] does adapt sufficiently well to give us near embeddings where we need them. This, and some further details, require that we analyse the extent to which extenders used in T are close to their target models. This was ignored in [1], [9] and [5]. It was partly addressed in [10], but there is also a gap in the calculations made there; indeed, [10, Lemma 9.1.11] is false, as we show at the beginning of §4. Conventions & Notation. Premice: Except where indicated otherwise, we deal with premice with Mitchell-Steel indexing, as in [9]. Let P be a premouse. We write F P = F (P ) for the active extender of P (possibly F P = ∅), EP = E(P ) P P for the extender sequence of P , excluding F P , and EP + = E+ (P ) = E b F . 1 It

is not relevant here whether one can prove a fuller version of weak Dodd-Jensen for weak embeddings.

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Given a structure M = (N, R0 , . . .) with universe N and predicates R0 , . . ., we write bM c = N . Given α ≤ ORP , we write P |α for the Q E P such that ORQ = α, and write P ||α = (bQc , EQ , ∅). If P has a largest cardinal δ, lgcd(P ) denotes δ. ISC stands for “initial segment condition”. L− denotes the language ˙ F˙ } (the premouse language, but excluding all constant symbols). We ˙ E, {∈, write P sq for the squash of P and (P sq )unsq = P . We also adopt the conventions described above and below for structures related to premice (such as pseudopremice). Fine structure: We use Mitchell-Steel fine structure. Let n < ω and let N be an n-sound premouse. Given Y ⊆ C0 (N ) we write Def N n+1 (Y ) for the set of all x ∈ C0 (N ) such that for some rΣn+1 formula ϕ and ~y ∈ Y <ω , x is the unique x0 ∈ N such that N |= ϕ(x0 , ~y ). See [6, §1.1] for the definition of Hulln+1 , cHulln+1 and Thn+1 . Let q ∈ (ORN )<ω and σ ∈ ORN . We say that (q, σ) is (n + 1)-solid for N iff for each α ∈ q, ThN n+1 (α ∪ (q\(α + 1))) ∈ N, and for each α < σ, ThN n+1 (α ∪ q) ∈ N. We say that q is (n + 1)-solid for N iff (q, 0) is (n + 1)-solid for N . Ultrapowers: Let E be a short extender over an m-sound premouse M with cr(E) < ρM m . We write Ultm (M, E) for the degree m ultrapower; recall that Ultm (M, E) = Ultm (M sq , E)unsq if M is type 3. We write iM,m for the corE responding ultrapower map, and given a ∈ lh(E)<ω and an rΣM m -function f f M,m M (where an rΣ0 -function is just a function in M ), we write [a, f ]E for the f object represented by [a, f ] in this ultrapower. We also drop the superscript m, and possibly M , where these are clear from context. Iteration trees: See [9, Definition 3.4] for the definition of k-maximal, and [9, paragraph preceding Definition 4.4] for k-bounded. If T is an iteration tree T = MαT , degT (∞) = degT (α), and of successor length α + 1, we write M∞ iTβ,∞ = iTβ,α for β ≤T α. Embeddings: Given premice M, N , fine structural embeddings from M to N will usually be of the form π : C0 (M ) → C0 (N ), and the elementarity of π will be between these structures. We sometimes explicitly write π : C0 (M ) → C0 (N ) to emphasize this, but usually just write π : M → N , with the convention that literally, dom(π) = C0 (M ) and cod(π) = C0 (N ). When M, N are structures related to premice, but either of M or N fails to be a premouse, we will literally have dom(π) = M , and the elementarity of π is literally between M, N . In special circumstances, both M, N will be type 3 premice, but dom(π) = M ; hopefully this will be clear in context.

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Weak n-embeddings & n-lifting embeddings

Let n < ω. The first (apparent) problem is with the definition of weak nembedding π : M → N between n-sound premice M, N , as stated in [1, §5] or [9, §4], and its interaction with the Shift Lemma, as stated in [1, Lemma 5.2]. For this paragraph let us take weak n-embedding to be defined as in [9], except

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2 that we also require that pM However, we will n ∈ X (where X is as in [9]). adopt a modification of this definition shortly. It seems that there is a problem with the proof of the Shift Lemma in the case that n > 0. This was noticed by Steve Jackson. For consider the map σ defined in the proof of [1, Lemma 5.2]. It is unclear that σ is rΣn -elementary. (For the obvious proof to work, it seems that one would like to have π“TnM ⊆ TnN . (1)

It might seem that line (1) follows from the rΣn+1 -elementarity of π over X. In 3 fact, if π is in fact rΣn+1 -elementary over H = Def M n+1 (X), then line (1) holds. But it is not clear that π is indeed this elementary. However, the author does not actually know of a counterexample.) Steve Jackson suggested that the following is the correct notion. From now on we take weak n-embedding to be defined as follows: 2.1 Definition. Let M, N be n-sound premice and let π : C0 (M ) → C0 (N ). Then we say that π is a weak n-embedding iff the requirements stated in [9, M Definition 4.1] hold, and also, there is a set Y such that Y ∩ ρM n is cofinal in ρn and M Def M n+1 (Y ∪ {pn }) ⊆ Y a

and π is rΣn+1 -elementary on parameters in Y .

It was observed by Steve Jackson that the Shift Lemma goes through using the above definition. However, it turns out that the Shift Lemma goes through for the following, wider class of embeddings, also motivated by the preceding discussion. 2.2 Definition. Let M, N be n-sound premice and let L be the language of M ˙ ˙ E}). (for example, if M is passive then L = {∈, Let π : C0 (M ) → C0 (N ). We say that π is n-lifting iff π is rΣ0 -elementary with respect to L, and if n > 0 then π“TnM ⊆ TnN . We say that π is c-preserving iff for all α < ρM 0 , α is a cardinal of M iff π(α) is a cardinal of N . a (n)

The n-lifting embeddings are similar to the Σ0 -preserving embeddings of [10], and for n ≥ 1, they are a slight generalization of the n-apt embeddings of [4]. See [7, §2] for some basic facts regarding n-lifting embeddings. The main one of interest here is 2.3 below. Given a class X of embeddings, the Shift Lemma for X is the modification of [1, Lemma 5.2] given by replacing (i) “ψ be a weak 0-embedding” with “ψ be Σ0 -elementary”, (ii) “π be an weak n-embedding” with “π ∈ X”, and (iii) “σ is an weak n-embedding” with “σ ∈ X”. 2 In [9], it is not required that pM ∈ X. In [1], it is required that {pM , ρM } ⊆ X. The n n n author believes that in this regard, [9] was too weak and [1] too strong. 3 Letting D = Def M (∅), we have uM n+1 n−1 ∈ D. For example suppose that n = 2 and m m = lh(pM 1 ) > 0. The statement ϕ in variables u, p, w asserting “u = (p, w) and p ∈ OR and V = Def V (p ∪ min(p)) and p is 1-solid for V , as witnessed by w” is rΠ . But it is easy 2 1 M to see that for any u, p, w ∈ M such that M |= ϕ(u, p, w), in fact u = uM 1 . So u1 ∈ D. M }), we have D 4 Further, for any Y ⊆ M , letting D = Def M (Y ∪ {p rΣn+1 M . For n n+1 M ∪ {uM , pM }), so the claim follows from calculations suppose n ≥ 1. Then M = Def M (ρ n n n−1 n M in the proof of [1, Lemma 2.10], with q = (uM n−1 , pn ) as there (consider the stratification of M rΣn+1 ({q}) truth used in that proof).

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The Shift Lemma for n-embeddings holds, as proved in [1]. Moreover (see [7] for some details, though it is straightforward): 2.3 Lemma. Let n ≤ ω and let X be either the class of either weak nembeddings, or n-lifting embeddings, or c-preserving n-lifting embeddings. Then the Shift Lemma for X holds.

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Copying

In this section we identify and correct flaws in the copying construction, and examine the conditions under which such corrections are necessary. There are many variants on this construction which arise in the literature. We do not attempt to state a completely general version. We just give the details for a case in which the flaws show up, and through which we can also introduce notions relevant to our fix of the proof of condensation etc. We deal explicitly with weak n-embeddings, but the same proofs give the same results for c-preserving n-lifting embeddings. (For n ≥ 1, every n-lifting embedding is c-preserving. One can also prove related results for 0-lifting embeddings, but the copying process must be modified, as described in [7].) 3.1 Definition. Let P, Q be premice and π : P → Q be Σ0 -elementary. We define the map ψπ as follows. Let ψπ = π if P is passive, and let ψπ : Ult(P, F P ) → Ult(Q, F Q ) be the map determined by π and the Shift Lemma otherwise. We say that π is ν-preserving iff ψπ (ν P ) = ν Q ; that π is ν-low iff ψπ (ν P ) < ν Q ; and that π is ν-high iff ψπ (ν P ) > ν Q . a If π, as above, fails to be ν-preserving then P, Q are type 3. We will deal with copying of trees in which the copy maps might be any of these three kinds. (Moreover, any of these kinds can arise in the proof of condensation etc.) A weak 0-embedding can be ν-preserving, -high or -low. A 0-embedding can be ν-preserving or -high. An example of a ν-high 0-embedding is given as follows. Let P be a type 3 premouse, such that κ = cof P (ν(F P )) = cr(E) for some P -total extender E. Let Q = Ult0 (P, E) and let π = iE . Then π is ν-high. In contrast, we have the following lemma; we leave the proof to the reader. 3.2 Lemma. A near 1-embedding is ν-preserving. A near 0-embedding is either ν-preserving or -high. It is the presence non ν-preserving copy maps, and particularly ν-high copy maps, which lead to the flaws in the copying construction. The process for dealing appropriately with ν-high copy maps was explained in [8, §7] and [9]. Suppose M is type 3, π : M → N is a weak 0-embedding, T is a tree on / dom(π), it does not make M , and ν(F M ) < lh(E0T ) < ORM . Since E0T ∈ sense to define E0πT in the usual manner. A natural attempt to fix this is to define E0πT = ψπ (E0T ). This makes sense unless π is ν-high; in the latter case ψπ (E0T ) ∈ / EN + . So in this case, it is natural to insert an extra extender into πT , setting E0πT = F N , and then E1πT = ψπ (E0T ). This is how we proceed. Note then that the tree order of πT can differ from that of T . There is a technicality here. If α + 1 < lh(T ) is such that predT (α + 1) = 0 and ν(F M ) ≤ cr(EαT ) 5

∗T (and therefore T drops in model at α + 1, with Mα+1 / MαT ), then we will have πT πT pred (α + 1) = 1 instead of pred (α + 1) = 0. This is slightly irritating notationally. Because of this, we break the copying process into two steps. Given T on M , we define a partially adjusted tree S on M , which is essentially equivalent to T (but possibly with superfluous nodes inserted), and also a copied tree V = πS on N , such that the copying from S to V proceeds smoothly, preserving all tree structure. We then define πT to be πS. Before proceeding with the details, we introduce trees we will consider in the proof of condensation etc.

3.3 Definition. Let M be a k-sound premouse. Say an iteration tree T on M is k-m-maximal4 iff it is formed according to the rules for k-maximal trees, except that we do not require that for each α < lh(T ), degT (α) is as large as possible.5 Let G m (k, θ) be the iteration game corresponding to k-m-maximal trees (analogous to G(k, θ)). Let G m (k, α, θ) then be the natural analogue of G(k, α, θ). A (k, θ)-m-iteration strategy is a winning strategy for player II in G m (k, θ). Likewise (k, α, θ)-m-iteration strategy. The terminology -m-iterable is then self-explanitory. A k-m-bounded tree is a partial run of G m (k, ∞, ∞) in which neither player has lost. a Our illustration of the copying construction will provide a proof of: 3.4 Lemma. Let N be (k, ω1 , ω1 + 1)-iterable and let π : M → N be a weak k-embedding. Then M is (k, ω1 , ω1 + 1)-m-iterable. A consequence is that M is (k, ω1 , ω1 + 1)-iterable iff M is (k, ω1 , ω1 + 1)m-iterable.6 Thus, m-iterability is not really any more interesting than regular iterability. However, in §4 we will be interested in m-iteration strategies. 3.5 Definition. An iteration tree T has increasing extender lengths iff for each α + 1 < β + 1 < lh(T ), we have lh(EαT ) < lh(EβT ). A tree has essentially increasing extender lengths iff for each α+1 < lh(T ), if η ≤ α and η is a limit T ) then then lh(EαT ) > δ(T  η); and for each α + 2 < lh(T ), if lh(EαT ) > lh(Eα+1 T T T T Mα is active type 3 and ν(F (Mα )) is a limit cardinal of Mα and Eα = F (MαT ) T and ν(F (MαT )) < lh(Eα+1 ). An iteration tree T is partially adjusted k-m-maximal iff it satisfies the requirements of a k-m-maximal tree, except that we replace the requirement of increasing extender lengths with essentially increasing, and whenever lh(EαT ) > T ), we have degT (α+1) = 0. We define partially adjusted k-maximal lh(Eα+1 similarly, but with all degrees dictated by k-maximality. For T a partially adjusted tree and α + 1 < lh(T ), we say that α + 1 is T ). (No limit α is superfluous.) a superfluous (for T ) iff lh(EαT ) > lh(Eα+1 3.6 Remark. Let T be partially adjusted and α + 1 < lh(T ) and P = MαT |lh(EαT ). If α + 1 is not superfluous then lh(EαT ) < lh(EβT ) for all β > α. Therefore, whether or not α + 1 is superfluous, if κ < lgcd(P ) then (κ++ )P < 4 The

“m” is for model. usual, we still require that (a) if α
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T lh(EβT ) for all β > α. If α + 1 is superfluous then Eα+1 ∈ E(MαT ) and if T ∗T pred (β + 1) = α + 1 then T drops in model at β + 1, and Mβ+1 / MαT .

The term superfluous is justified by the following lemma. It says that we can convert a partially adjusted tree into one with increasing extender lengths, by doing away with superfluous nodes, cutting any branches above superfluous nodes α+1 and grafting them above α. (When α+1 is superfluous in the formal T sense, EαT is superfluous in the intuitive sense, whereas Eα+1 is important to T .) We omit the proof of the lemma. Regarding padding, if U is a padded tree and EγU = ∅, then for all α, we set predU (α) = γ iff α = γ + 1. If there is β ≥ γ such that EβU 6= ∅ then let exitU (γ) denote the least such β. 3.7 Lemma. Let T be a partially adjusted k-m-maximal (k-maximal ) tree of length λ, on M . Let S be the set of superfluous nodes of T . Then there is a unique padded k-m-maximal (k-maximal ) tree U on M such that for all γ < λ: 1. lh(U) = λ, 2. degU (γ) = degT (γ), 3. if γ + 1 < λ, then EγU = ∅ iff γ + 1 ∈ S, 4. if EγU 6= ∅ then EγU = EγT , 5. if γ is a limit then there is α
∗k-embedding.  ~π ∗ = πα+1 are the corresponding copy maps, iff there is n ∈ [k, ω] such α+1<λ that: 7

1. M0V is n-sound and V is n-m-maximal. 2. The tree structures of T , V are identical. 3. For each α < λ, πα : MαT → MαV is a weak degT (α)-embedding. 4. For each α < λ, degT (α) ≤ degV (α). 5. The copy and iteration maps commute in the usual manner, and for each limit η < λ, πη is the usual direct limit. 6. For each α + 1 < λ, letting δ = predT (α + 1): (a) If EαT ∈ C0 (MαT ) then EαV = πα (EαT ). (b) If EαT = F (MαT ) and πα is not ν-low then EαV = F (MαV ). (c) If EαT = F (MαT ) and πα is ν-low then letting ν = ψπα (ν(EαT )), EαV is the trivial completion of F (MαV )  ν. (Note that ν is a cardinal of C0 (MαV ) and so EαV is on E(MαV ).)7 (d) If EαT ∈ MαT \C0 (MαT ) then πα is not ν-high and EαV = ψπα (EαT ). (e) If α + 1 is superfluous for T then πα is ν-high. (f) T drops in model at α + 1 iff U does, and if they drop in model, ∗V ∗T Mα+1 = ψπδ (Mα+1 ).

(g) The map ∗ ∗T ∗V πα+1 : Mα+1 → Mα+1

is the restriction of ψπδ . Let σα : MαT |lh(EαT ) → MαV |lh(EαV ) be the restriction of ψπα . Then the hypotheses for the Shift Lemma ∗ attain (regarding EαT , σα and πα+1 ). Let m = degT (α + 1) and U ∗V n = deg (α + 1) (so m ≤ n). Let M ∗ = Mα+1 and E = EαV and define T V ¯ α+1 π ¯α+1 : Mα+1 →M = Ultm (M ∗ , E) as in the proof of the Shift Lemma. Let ¯ V → MV τα+1 : M α+1 α+1 be the natural map, i.e.   ∗ ∗ ,m ,n τα+1 [a, f ]M = [a, f ]M . E E Then πα+1 = τα+1 ◦ π ¯α+1 . 7 One can show that because π is ν-low and a weak 0-embedding, ν(E T ) is a limit cardinal α α T ) is also a limit cardinal of M V . So a natural alternative of MαT . Therefore ν 0 = sup πα “ν(Eα α V = F (M V )  ν 0 , as is done in [8]. For our purposes it doesn’t matter whether we is to set Eα α use ν or ν 0 .

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Clearly some of the requirements are redundant. For example, we have required that V has increasing extender lengths (as it is k-m-maximal), but this actually follows from the other requirements. If degV (α) = degT (α) for all α ∈ λ\{0} then V is a minimal π0 -neat copy of T . The definitions above generalize in the obvious manner to trees on phalanxes (where some of the requirements might be less redundant, depending on what we start with). a 3.10 Lemma. Let M be an m-sound premouse and N an n-sound premouse, where m ≤ n. Let π : M → N be a weak m-embedding. Let T be an m-mmaximal tree on M . Then there is a unique (λ, S, V, ~π , ~π ∗ ) such that π0 = π and λ ≤ lh(T ) and S is a putative partially adjusted m-m-maximal tree on M such that Sˆ = T  λ, and V is a putative minimal (alternatively, n-maximal ) π-neat copy of S, with copy maps ~π , ~π ∗ , and either λ = lh(T ) or V has an illfounded last model. Proof Sketch. We first consider the case of arranging that V be minimal. We define S, V, ~π , along with a continuous, increasing sequence ~γ = hγξ iξ∈[1,λ] of ordinals, such that γξ = ξ for all limits ξ. We define ~γ  ξ + 1, S  γξ , V  γξ , ~π  γξ and ~π ∗  γξ by induction on ξ, maintaining that, with γ = γξ : (a) Sd  γ = T  ξ. (b) V  γ is a putative minimal neat copy of S  γ, with copy maps ~π  γ, ~π ∗  γ. (c) For each α < β < γ, ν(EαS ) is the least ν such that πβ (ν) ≥ ν(EαV ). (d) For each α < β < γ, πβ  ν(EαS ) ⊆ ψπα , and if EαS is type 1 or 2, then πβ  (lh(EαS ) + 1) ⊆ ψπα . We set γ1 = 1 and π0 = π. Suppose ξ ≥ 1 and we have defined γ = γξ , S  γ, V  γ, ~π  γ and ~π ∗  γ. Suppose that all models of V  γ (and therefore likewise S  γ) are wellfounded and ξ < lh(T ). If ξ = γ is a limit then we set γξ+1 = γ + 1 and produce S  γ + 1, etc, in the obvious manner. Suppose γ = β + 1 is a successor, and therefore so is ξ = ζ + 1. So we have defined MβS = MζT , etc. Case 1. πβ is ν-high and ν(F (MζT )) < lh(EζT ) < OR(MζT ). Then we set γξ+1 = γ + 2. We want S \  γ + 2 = T  ξ + 1; so we must set S EβS = F (MβS ) and degS (β + 1) = 0 and Eβ+1 = EζT . Case 2. Otherwise. Then we set γξ+1 = γ + 1 and must set EβS = EζT . Everything else is determined by our various requirements, but we highlight some points. The fact that


satisfied 3.9(6e). In the situation of 3.9(6c), we have rg(πβ ) ⊆ MβV |ν, and σβ : MβS → MβV |lh(EβV ) is a weak 0-embedding, where σβ has the same graph as πβ . So it makes sense to apply the Shift Lemma with σβ here. Also here, if β = predS (α + 1) for some α ≥ β and S does not drop in model at α + 1, then neither does V, since ν is a cardinal of MβV . The remaining properties are established as usual. Clearly the construction was unique.8 This completes our sketch in the case that we want V to be minimal. Now consider arranging that V be n-maximal. We proceed mostly as above. The only difference is that degrees in V are determined by n-maximality, and that we define πβ+1 as in 3.9(6g). With notation as there, note that cr(τβ+1 ) > lh(EβV ), so we still get that πβ+1 agrees sufficiently with ψπβ to maintain hypotheses (c) and (d). 3.11 Definition (Copying). Let M, m, N, n, π, T , S, V be as in 3.10, with V a minimal (respectively, n-maximal) neat copy of S. Then we write πT = V and padj(π, T ) = S (respectively, π n−max T = V and padjn−max (π, T ) = S). (Note that S, V depend on m. We have not made m explicit in our notation, as we take it for granted that m is explicit in the metadata of T .) For notational convenience, we often denote the superfluous indices of S with half-ordinals; that is, with α + 12 , for some α ∈ OR. Thus (assuming πT has wellfounded models), for each α < lh(T ) the copy map πα : MαT → MαπT . Clearly there are many generalizations of these definitions. In §4 we will need to consider the case that T is a tree on a phalanx. a We now establish some restrictions on how copy maps which fail to be νpreserving can arise in copying. Part of the following lemma and its proof is taken from [8]. 3.12 Lemma. Let π : M → M 0 be a weak k-embedding and let σ : P → P 0 be Σ0 -elementary between premice (or squashes), with P, P 0 active, and cr(F P ) < ρM k , and such that the hypotheses for the Shift Lemma hold regarding π, σ (for 0 0 forming Ultk (M, F P ), etc). Let n ≥ k be such that cr(F P ) < ρM n . Let 0

τ : Ultk (M, F P ) → Ultn (M 0 , F P ) be defined as in 3.9(6g). Suppose n = k. Then τ is ν-preserving (-high, -low ) iff π is ν-preserving (-high, -low ). Suppose n > k. Then τ is ν-high iff π is ν-high, and if π is ν-low then τ is ν-low. Proof. We just give the proof assuming n = k, and leave the remaining case to the reader. We may assume that M is type 3, and also that k = 0, as near 0 1-embeddings are ν-preserving. Let E = F P , F = F M , ν = ν(F ), E 0 = F P , 0 F 0 = F M , ν 0 = ν(F 0 ), ψ = ψπ , κ = cr(E), µ = cr(F ) and let a, f ∈ M be such that ν = [a, f ]M F . Now iE is a 0-embedding, so is not ν-low. Using this, it is easy to see that if π is ν-low then so is τ . So suppose π is not ν-low. 8 It was in order to secure this uniqueness that we required the degrees of superfluous nodes be 0 in partially adjusted k-m-maximal trees.

10

Let j : M → U = Ult(M, E), where the ultrapower is computed at the unsquashed level. Then by the argument in [1, §9], U j(ν) = [j(a), j(f )]U F U = [iE (a), iE (f )]F U ,

so iE is ν-preserving iff cof M (ν) 6= κ. Likewise for M 0 . So if cof M (ν) 6= κ, then 0 we are done (if π is ν-preserving then also cof M (ν 0 ) 6= cr(E 0 ); if π is ν-high then we get that τ is ν-high by commutativity). So suppose that cof M (ν) = κ. Let g : κ → ν be strictly increasing, continuous, cofinal in ν, with g ∈ M . So ν = sup(g“κ) and since κ = cr(E) = cr(ψiE ), we have ν(F U ) = sup(ψiE (g)“κ). Now since iE 0 ◦ π = τ ◦ iE , we have ψiE0 ◦ ψπ = ψτ ◦ ψiE . Moreover, ψτ (κ) = τ (κ) = π(κ) = ψπ (κ). Therefore
ψτ (ν(F U )) = sup ψiE0 (ψπ (g))“π(κ) . (2) Now if π is ν-preserving then sup(ψπ (g)“π(κ)) = ν 0 , 0

and since cr(E 0 ) = π(κ), therefore line (2) shows that ψτ (ν(F U )) = ν(F U ), so τ is ν-preserving. It is similar if π is ν-high. We can now deduce that the copying construction as described in [1] often suffices: 3.13 Proposition. Let M, N be k-sound premice and π : M → N a weak kembedding. Let T be a k-m-maximal tree on M . Suppose πT has wellfounded models. Let πα : MαT → MαπT be the αth copy map, where we use half-ordinals as described in 3.11. 1. If β ≤T γ and (β, γ]T does not drop in model, then πγ is ν-preserving (-high, -low ) iff πβ is ν-preserving (-high, -low ), and if β is a successor, this is iff πβ∗ is ν-preserving (-high, -low ). 2. If either π0 = π is ν-preserving or there is a drop in model in (0, α]T , then πα is ν-preserving. Therefore if π is ν-preserving then πT is just as defined in [1, p. 55]. 3. If π is a near k-embedding then every πα is a near degT (α)-embedding, and if also T is k-maximal then πT is k-maximal. Proof. Item 1 follows from 3.12 by an easy induction on α < lh(T ). Item 2 also follows from 3.12 because fully elementary embeddings are ν-preserving. Item 3 follows from (the proof of) [4, Lemma 1.3], since if σ : P → Q is a near P Q n-embedding and ρP n ∈ dom(σ) then σ(ρn ) ≥ ρn . The proof of the following proposition is similar to the preceding one. 3.14 Proposition. Let M be an m-sound premouse and N be an n-sound premouse, where m ≤ n. Let π : M → N be a weak m-embedding. Let T be an m-m-maximal tree on M . Suppose V = π n−max T has wellfounded models. Let πα : MαT → MαV be the αth copy map, where we use half-ordinals as described in 3.11. 11

1. πβ is ν-high iff [0, β]T does not drop in model and π is ν-high. 2. Let β ≤T γ be such that (β, γ]T does not drop in model. If πβ is ν-low then πγ is ν-low. If πβ is ν-preserving then πγ is either ν-preserving or ν-low.

4

Condensation etc

In this section we identify and correct some problems with the proofs of condensation, solidity and Dodd-solidity, for premice without extenders of superstrong type, and the ISC and weak ISC for pseudo-premice without extenders of superstrong type. (The problems are less involved in the proof of universality, so we omit discussion of this.) We give three different approaches to correct the problems. The first two are closer to the original proofs from [1] and [5], in that they use weak n-embeddings; we give all the details in these cases. In the third, which we just sketch, we use n-lifting embeddings instead of weak n-embeddings. The latter approach works out somewhat simpler overall.9 The nature of the problems was sketched in the introduction; we will state exactly what they are after some preparation. However, let us immediately point out a problem with [10, Lemma 9.1.11]. That lemma is false, as the following example shows. In the following example we deal with premice with λ-indexing, as in [10]. Let M0 be an active premouse such that E = F M0 has only one generator κ = cr(E). Then E{κ} is not close to N = M0 ||ORM0 . Let M−1 = N and λ = (κ+ )N . Then P = (M−1 , M0 , λ) is a good phalanx (see [10, pp. 292, 296]). Let T be the normal tree on P, of length 2, with E0T = E. Then predT (1) = −1 and M1∗ = M−1 and 1 is not an anomaly for T (see [10, p. 293]). So the hypothesis of [10, Lemma 9.1.11] holds with respect to T , but its conclusion does not. In the above example, M−1 does not code as much information as does M0 . In the context in which we would like to use something like [10, Lemma 9.1.11], this will not be the case. For example, in the proof of condensation, we will have an embedding M0 → M−1 . In the arguments to follow we will show that a variant of [10, Lemma 9.1.11] does hold where we need it.

4.1

First approach, via weak n-embeddings

We will make use of a modification of the weak Dodd-Jensen property (see [2]), in which k-maximality is replaced by k-m-maximality. There is one point here we want to clarify, so we give the precise statement. 4.1 Definition. Let M be a k-sound premouse and let T be a k-m-bounded tree on M . We say that T is (M, k)-m-large iff T has a last model P , and there is Q E P such that if Q = P then degT (∞) ≥ k, and there is a near k-embedding π : M → Q.10 9 This third option was only noticed later by the author, after the original submission of the paper for publication. 10 Aside from allowing T to be k-m-bounded instead of k-bounded, the definition of (M, k)m-large differs from the definition of (M, k)-large (in [2]) in the requirement that “if Q = P then degT (∞) ≥ k”. The point of this difference is that, for instance, when verifying that a strategy Σ we construct has the weak Dodd-Jensen property, one wants to know that, given

12

The m-weak Dodd-Jensen property is the natural analogue of the weak Dodd-Jensen property, but with (M, k)-m-large replacing (M, k)-large. a 4.2 Lemma. For any countable, k-sound, (k, ω1 , ω1 + 1)-iterable premouse M , and enumeration e of M in ordertype ω, there is a (k, ω1 , ω1 + 1)-m-iteration strategy Σ for M which has the m-weak Dodd-Jensen property with respect to e. The proof of the lemma is a simple modification of the version for the regular weak Dodd-Jensen property. We leave to the reader the obvious adaptation of the definition and lemma for pseudo-premice. We now set up some terminology in order to handle the anomalous case of the proof of condensation etc. (In the anomalous case, the first gap in the proof does not arise; however, we do still need to adapt [4, 1.3] to apply here.) 4.3 Definition. A segmented-premouse is a structure P = (JαE , E, F ) such 6 ∅ then: that (JαE , E, ∅) is a passive premouse, and if F = 1. P has a largest cardinal δ, 2. there is an extender F 0 over P such that, letting U = Ult(P, F 0 ), then U |(δ + )U = P ||ORP , 3. iF 0 (κ) > δ, where κ = cr(F 0 ), 4. F is the P -amenable code for F 0 ; that is, it is defined as in [9, §2.3], but with ν(F 0 ) replaced by max(ν(F 0 ), δ). Note that a segmented-premouse can fail the ISC. A properly segmentedpremouse is a segmented-premouse which is not a premouse. We deal with at most Σ1 -elementarity for properly segmented-premice, literally over P (not P sq ), in the language L− (see §1). (Moreover, the embeddings of properly segmented-premice we encounter will be cofinal.) We take ultrapowers at the unsquashed level, shifting the predicate F P as with type 2 premice. For an active segmented-premouse P , let ιP = max(ν(F P ), δ), where δ = lgcd(P ). Thus, if P is an active premouse then ιP = ν P . Given a segmented-premouse P and an extender E over P , let Ult−1 (P, E) denote the 0-ultrapower of P , formed at the unsquashed level. For any segmentedP premouse P , let ρP −1 = OR , and for any properly segmented-premouse P , let 0 0 0 0 ρP F P  ν(F P ) = 0 = ρ0 (P 0 ), where P is the least active premouse such that 0 F P  ν(F P ) and the ISC fails for P with respect to F P . Generalizing the some tree T which is (M, k)-large, as witnessed by σ : M → Q, then T lifts to a tree πT on a certain iterate N of M , and that, letting π 0 be the final copy map, π 0 induces a near k-embedding π 00 : Q → Q∗ for some Q∗ . But if Q = P and degT (P ) < k then π 0 need not be sufficiently elementary for this. Note here that P might be k-sound even if degT (P ) < k, even for trees T which are a run of G(k, α, θ) (see [2]). For example, it might be that T is a stack of two normal trees T0 , T1 , and T0 is the trivial tree on M , and then player I drops to (M, k − 1), so T1 is (k − 1)-maximal on M , and T1 uses just one extender E, which is total over T1 T1 M , and cr(E) < ρM is k-sound. k . In this case, lh(T1 ) = 2, deg (1) = k − 1, but P = M1 T The requirement that deg (∞) ≥ k solves this problem. (The author believes that the same restriction should have been made in [2].)

13

usual terminology for iteration trees T , we allow degT (α) ∈ ω ∪ {−1}, with the obvious meaning. Let M be a properly segmented-premouse, N a segmented-premouse. A (weak, near) 0-embedding π : M → N is defined as usual, but using L− . And π : M → N is a (weak, near) −1-embedding iff π is a (weak, near) 0-embedding. (In both cases, dom(π) = M , not M sq , independent of whether N is a premouse or not.) We define ψπ as in 3.1. For such π, we have ψπ (ιM ) = π(ιM ) = ιN ; in fact, Σ0 -elementarity suffices for this, because the predicates for the active extenders F M and F N , as specified in 4.3, explicitly encode the identities of ιM and ιN . When using F M in an iteration tree, we will use ιM , not ν(F M ), as the corresponding exchange ordinal. Let π : M → N be a weak 0-embedding, where M is a segmented-premouse. We say that π is ι-preserving iff either M is not a premouse or π is ν-preserving. a We now fill a gap in the proof of the following: 4.4 Fact. Let M be a k-sound, (k, ω1 , ω1 + 1)-iterable premouse, where k ≤ ω. Then if k < ω: 1. Degree k + 1 condensation holds for M , 2. M is (k + 1)-solid, and 3. if M is 1-sound then M is Dodd-solid; and if k = ω: 4. Degree ω condensation holds for M . Moreover, every (0, ω1 , ω1 + 1)-iterable pseudo-premouse M satisfies the ISC and weak ISC.11 See [1, §8, §10], [9, §5] and [3, §3] for the definitions of the statements above (in particular, degree k + 1 condensation is defined in [1, pp. 87, 88]). Proof. We may assume that M is countable. Fix a (k, ω1 , ω1 + 1)-m-strategy Σ for M , with the m-weak Dodd-Jensen property (with respect to some enumeration of M ; here k = 0 in the proof of Dodd-solidity or the (weak) ISC). Initially we separately set up the proofs of each fact, and specify the context in which we will be working. After that we deal with all proofs simultaneously. Condensation. The degree ω version follows from the degree k + 1 < ω version, so we just consider the latter. Suppose that M, H are (k + 1)-sound and π : H → M is a near k-embedding with ρH k+1 ≤ cr(π). Consider the proof of condensation with respect to π. The gap in the standard proof occurs in the case that cr(π) = (γ + )H < ORH , where γ is a cardinal of M (and H), so assume H this is the case. In particular, H 6= M . Now H ∈ M . For if (π(pH k+1 ), ρk+1 ) is (k + 1)-solid for M , and therefore H M M (π(pH k+1 ), ρk+1 )
as certainly “≤lex ” holds, but if “=” holds then H = M , contradiction. 11 Recall that the weak ISC does not follow from the ISC, as the weak ISC deals with type Z extenders, whereas the ISC does not.

14

Solidity. Consider the proof that M is (k + 1)-solid. Let p = pM k+1 and let M ε ∈ p. Let q = p\(ε + 1) and H = cHullk+1 (ε ∪ q). We want to show that H ∈ M . Let π : H → M be the uncollapse. Then ρH k+1 ≤ ε = cr(π). The gap in the proof of solidity occurs in the case that ε = (γ + )H < ORH , where γ is a cardinal in M ; assume this is the case. We first dispense with some easy cases. + M Case 1. k > 0, M is active, cr(F M ) = γ, and ρM k = (γ ) . M + H M Then since cr(π) = (γ ) , clearly Hullk+1 (ε ∪ q) ∩ ρM k is bounded in ρk . But then H ∈ M , so we are done.

Case 2. k = 0, M is active type 1 or type 2, and cr(F M ) = γ. Then again H ∈ M .12 Case 3. Otherwise. In this case we need to do more work. Since ε = (γ + )H ∈ pM k+1 , if M is active type 1 or type 2 with cr(F M ) = γ then k > 0 and (γ + )M < ρM k . Therefore if H is active type 1 or type 2 and cr(F H ) = γ then k > 0 and (γ + )H < ρH k . Dodd-solidity. Consider the proof that M is Dodd-solid; here k = 0. We follow the notation of [5, §4], except that we use “H” in place of “N ” and “π” in place of “ψ”. (So H is the candidate premouse and π : H → M is cofinal and Σ1 -elementary in L− (see §1).) As in [5], by the weak ISC (see below), we may assume i > 0. The gap occurs in the case that ε = si = (γ + )H < ORH , for an M -cardinal γ. Note that cr(F H ) < γ. The ISC and weak ISC. Let M be a (0, ω1 , ω1 + 1)-iterable pseudo-premouse. We follow approximately the proof of [1, Theorem 10.1]. So M has largest cardinal γ and F M has largest generator ε > γ. Let G = F M  ε and H = Ult0 (M, G) and U = Ult0 (M, F M ) and π : H → U be the natural factor map. So cr(π) = ε = (γ + )H . Let ν = νG and λ = (ν + )H . We will prove that either: – λ = ε > ν and G = EM ε , or + H – λ = ε > ν = γ + 1 and G is type Z, F M  γ = EM ε , and letting η = (ε ) M W and W = Ult0 (M, Eε ) and E = Eη , then E is type 1 with cr(E) = γ and G = E ◦ EM ε , or

– λ > ε = ν and M |ε is passive and G = EM λ , or – λ > ε = ν and M |ε is active, and letting W = Ult0 (M, EM ε ), we have G = EW λ . This completes our preparation. We now continue with all cases simultaneously. We have that (γ + )H < ORH and if H is active and cr(F H ) = γ then either H ∈ M or we are in case 3 of the proof of solidity. In the cases of condensation, solidity and Dodd-solidity, let U = M . So π : H → U . Now cr(π) = (γ + )H < (γ + )M . Let J / M be such that H||(γ + )H = J||(γ + )J and ρJω = γ. Let j < ω be such that ρJj+1 = γ < ρJj , if there is such j. Otherwise J is type 3 and ν(F J ) = γ (i.e. the anomalous case); in this case we treat J as a segmented-premouse, and set j = −1, and so still ρJj+1 = γ < ρJj . 12 Although

in solidity cases 1 and 2 we are done, the proof for case 3 also applies to cases 1 and 2, giving the extra information about the nature of H which results from the proof of solidity.

15

Consider the phalanx P = ((M, < γ, k), (J, γ, j), (H, > (γ + )H , k)), where, for example, the tuple (J, γ, j) indicates that extenders with critical point γ are applied to J, with degree j. A (k, j, k)-maximal tree T on P is defined in the obvious manner, where if MαT is not a premouse and EαT = F (MαT ) then we set the exchange ordinal ναT = ι(MαT ), instead of ν(EαT ). Consider the phalanx ~ = ((M, < γ, k), (M, γ, k), (U, > γ, k)) M (which is equivalent to M if U = M ). ~ ”, where ~π = (π−2 , π−1 , π0 ) and We have base copy maps “~π : P → M π−2 = id : M → M, π−1 = id : J → J / M, π0 = π : H → U, where if j = −1 then literally, dom(π−1 ) = J, not C0 (J). If T is a tree on P and α < lh(T ), we write rootT (α) for the unique i ∈ {−2, −1, 0} such that i ≤T α. Likewise for other trees on phalanxes. Note that π might be ι-high. No πi is ι-low for i ∈ {−2, −1, 0}, but we still might end up with ι-low embeddings above J. Now P is (k, j, k)-maximally iterable, via copying a (k, j, k)-maximal tree T ~ , using Σ to form V. Here we copy as on P to a k-m-maximal tree V = ~π T on M in 3.11, making the obvious adaptations for phalanxes, and as follows. We write T T V V M−2 = M , M−1 = J, M0T = H, R−2 = M−2 = M , R−1 = J / M = M−1 and V ~ R0 = M0 = U . As we build T , we also build S on P and V on M , and we define Rα E MαV , and copy maps πα : MαS → Rα , such that for all α ∈ [−2, lh(T )): 1. Sˆ = T and V is a “~π -neat copy of S” (that is, most of the details of the copying are as in 3.9); we index S and V with ordinals (we do not use half-ordinals). 2.
∗V i. If S does not drop in model at ε + 1 then Mε+1 = Rβ . ∗V ∗S ii. If S drops in model at ε + 1 then Mε+1 = ψπβ (Mε+1 ).

For α < lh(T ) and the corresponding node α0 < lh(S), we can have rootT (α) 6= rootS (α0 ), but in this case, both [rootT (α), α]T and [rootS (α0 ), α0 ]S drop in model. As usual, we have MαS0 = MαT . Now let T , U, S, V be such that (T , U) is a comparison of (P, M ), U is kmaximal and via Σ, and T , S, V are as above. It is clear that because S is (k, j, k)-maximal, the only way that some MαS can be a properly segmented-premouse, or that we form a degree −1 ultrapower leading to MαS , is in the anomalous case, and where rootS (α) = −1 and (−1, α]S does not drop in model. We next show that most maps πα are near degS (α)-embeddings. We also extend 3.13 to our situation, describing when πα is ι-low, and when ι-high. ∗ For α + 1 < lh(S) we define πα+1 as in 3.9. We say that α < lh(S) is exceptional iff rootS (α) = −1 and (−1, α]S does not drop in model or degree (in this case there are no superfluous nodes in (−1, α)S ). Claim 1. Let α < lh(S). Then: 1. πα is ι-low iff α is exceptional, J is type 3, (γ + )H < ORJ , j = 0, and γ = cof J (ν(F J )) (in particular, MαS is a premouse). 2. πα is ι-high iff rootS (α) = 0, H is type 3, k = 0, π is ι-high, and (0, α]S does not drop in model. 3. πα is a weak degS (α)-embedding, and if α is not exceptional, then πα is a near degS (α)-embedding. 4. If α + 1 < lh(S) then: ∗S (a) if predS (α + 1) 6= −1 then EαS is close to Mα+1 , and S (b) if predS (α + 1) = −1 then ρj 0 +1 (Mα+1 ) = γ where j 0 = max(j, 0).

Proof. We first prove items 1 and 2. Since π−2 is ι-preserving, the proof of 3.13 shows that if rootS (α) = −2 then πα is ι-preserving. Now π is a near 0-embedding, so is not ι-low. Therefore if rootS (α) = 0 then the proof of 3.13 shows that πα is not ι-low, and establishes item 2 (for such α). Now suppose that rootS (α) = −1 and that πα is not ι-preserving, but for every δ ∈ [−1, α)S , πδ is ι-preserving. The proof of 3.13 shows that J is type 3 and [−1, α]S does not drop in model. We are not in the anomalous case, as in that case, J is type 3 and πα is ι-preserving. (Recall that in the latter case, we do not squash when forming the ultrapowers along (−1, α]S , and dom(πα ) = MαS , not (MαS )sq . As explained in [5], all generators of F (MαS ) are < iS0,α (γ).) V So (γ + )H < ORJ . For δ ∈ [−1, α]S let j−1,δ : J → Rδ be the natural map. Now π−1 is ι-preserving. Thus, 3.12 and an easy induction gives that if iS−1,δ is V ι-preserving then j−1,δ and πδ are ι-preserving. It follows that α = β + 1 for S some β, deg (β + 1) = 0, and letting δ = predS (β + 1) and κ = cr(EβS ), MδS |= “κ = cof(ν(F (MδS )))”. 17

But then, since iS−1,δ is ι-preserving, iS−1,δ (κ0 ) = κ where κ0 = cof J (ν(F J )). 0 V Now κ0 ≥ ρJ1 since κ ≥ ρ1 (MδS ). If δ 6= −1, we have iV −1,δ (κ ) ≥ ρ1 (Rδ ), V ∗V and therefore Mβ+1 = RδV and deg (β + 1) = 0. But then by 3.12, πβ+1 is ι-preserving, contradiction. So δ = −1 and j = 0 and J |=“cof(ν(F J )) = γ”. And now by the proof of 3.12, and since [0, β +1]V does not drop in model (since predS (β + 1) = −1) it follows that πβ+1 is ι-low. Also by the proof of 3.12 and induction, if β + 1
Therefore an induction shows that item 4 holds for α if β ≥ 0. Suppose β = −2. We proceed by induction. Let E = EαS . So cr(E) < γ and <ω E is total over M . Let b ∈ νE . Clearly if Eb ∈ MαS then Eb ∈ H|(γ + )H , so Eb ∈ M . So we may assume that E = F (MαS ) and Eb ∈ / MαS , so ρ1 (MαS ) ≤ γ. S Therefore letting i = root (α), [i, α]S does not drop in model, and cr(E) < cr(iSi,α ), and (i, α]S is all at degree 0 or −1. If rootS (α) = −2 then by induction, every extender used along [0, α]S is close to its target model, which suffices. If rootS (α) = 0 then E is close to H (by our earlier closeness results), and we are not in the ISC case (as there, cr(F H ) > γ). But then π : H → U = M is Σ1 -elementary and cr(π) = (γ + )H , so E is close to M . So suppose that rootS (α) = −1. Let −1 γ, and we know that E is close to Mξ+1 . Now γ is a limit cardinal S S of J since cr(Eξ ) = γ. So it suffices to see that ρ1 (Mξ+1 ) ≥ γ (this contradicts S the fact that Eb ∈ / Mα , because γ is a limit cardinal). Let F = EξS and let a ∈ νF<ω . Then it suffices to see that ρ1 (Ultj (J, Fa )) ≥ γ. But we claim that Fa ∈ M . Since J ∈ M and J|γ = M |γ, this suffices. So we show that Fa ∈ M . Since (γ + )H < ORH , if Fa ∈ MξS or [rootS (ξ), ξ]S drops in model then Fa ∈ H. But then Fa ∈ M by condensation (applied to π  H 0 for some H 0 / H). So suppose that Fa ∈ / MξS and [rootS (ξ), ξ]S does not drop in model. Since cr(F ) = γ, we must have rootS (ξ) = 0, and so F is close to H, and H is active, with cr(F H ) = γ, and M is active with cr(F M ) = γ. So we are not in the ISC or Dodd-solidity case, and so U = M . If H ∈ M or H is type 3 we are done, so suppose H ∈ / M and H is type 1 or 2. Then we are in solidity case 3, so k > 0 and ρM > (γ + )M . But F is close to H and π is 1 rΣ1 -elementary, and therefore Fa ∈ M . S The foregoing argument also shows that if β = −1 then ρj 0 +1 (Mα+1 ) = γ. This proves item 4. We say that an extender E is trivially close to a model M iff E is close to <ω M and for every a ∈ νE , Ea ∈ M . The definition of strong closeness at α 18

(see [4, 1.3]) generalizes in the obvious manner to our setting. Item 3 follows from: Subclaim. Let α + 1 < lh(S) be unexceptional. Then (a) strong closeness at α holds; and (b) πα+1 is a near degS (α + 1)-embedding. Proof. We proceed by induction.13 Given (a), we can deduce (b) using the ∗ argument of [4, 1.3]. This is because α + 1 is unexceptional, and so πα+1 is a S near deg (α + 1)-embedding. So it suffices to prove (a). Let δ = predS (α + 1). So δ 6= −1. If δ = α then (a) is easy, so suppose δ < α. If δ ≥ 0 then let E = EδS and κ = cr(E). First suppose that (i) either: – δ = −2 and E is trivially close to M ||(γ + )H , or – δ ≥ 0 and E is trivially close to MδS |lh(EδS ), or – δ ≥ 0 and κ = lgcd(MδS |lh(EδS )) and (ii) either: – E is type 3, or – πα or ψπα induces a Σ1 -elementary (with respect to L− ) embedding MαS |lh(E) → MαV |lh(EαV ). We outline the proof of (a) under these assumptions, letting the reader fill in the details from [4, 1.3]. First suppose that δ = −2 and E is trivially close to M ||(γ + )H . Then cr(E) < γ and E is trivially close to MαS |lh(EαS ), and πα  (γ + )H = id = π−2  (γ + )H . <ω Moreover, by (ii), for each a ∈ νE , we have

ψπα (Ea ) = (EαV )ψπα (a) . So (a) follows easily. Now suppose that δ ≥ 0 and κ 6= lgcd(MδS |lh(EδS )), so E is trivially close to S Mδ |lh(EδS ). Let S S η = (κ++ )Mδ |lh(Eδ ) . So η ≤ lh(EδS ). Moreover, by 3.6, η ≤ lh(EβS ) for all β ≥ δ. But E is trivially close to MδS ||η, so E is trivially close to MαS |lh(EαS ), and also ψπδ  η ⊆ πα , which gives (a). Finally, suppose that δ ≥ 0 and κ = lgcd(MδS |lh(EδS )). Then EδS is type 2, and arguing as in [4, 1.3] and using the induction hypothesis, one can show that S E is trivially close to Mδ+1 and (a) holds. (If E is trivially close to MαS this is S easy. Otherwise, E = F (Mα ) and S

ρ1 (MαS ) ≤ lh(EδS ) = (κ+ )Mα , will actually just use the following inductive assumption: Let δ = predS (α + 1). Then for all β < α, if β + 1 is unexceptional then πβ is Σ1 -elementary, and if also δ ≤S predS (β + 1) then we have strong closeness at β. 13 We

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and much as in [1, 6.1.5], we get that δ + 1
4.2

Second approach, via weak n-embeddings

It is natural to consider an alternate correction to the proof of condensation etc, using a strengthening of weak Dodd-Jensen, which applies to weak embeddings, at least in the context in which the difficulties arise. We now sketch such an alternate fix. This approach has the one advantage that we don’t need to consider k-m-maximality. For simplicity, consider just the case of solidity. So we have premice H, M and a k-embedding π : H → M . Also for simplicity, assume that π is not ι-high. We construct T , U, V mostly as before, but V is k-maximal, and so it is possible that for some α, Rα = MαV and degT (α) < degV (α). (We are yet to specify the strategy we use to form U and V.) Because π is not ι-high, the tree orders of T and V will be the same. Consider the case that b = bT is above J V T be the final copy → M∞ and drops in degree but not in model. Let π∞ : M∞ T map. Then π∞ is a weak deg (∞)-embedding, but it might fail to be a near degT (∞)-embedding, because of the k-maximality of V. Thus, it is not obvious that π∞ ◦ iU is a weak k-embedding. We will use a result from [6] to see that it is. This will contradict the modification of weak Dodd-Jensen we will be using. We first describe this modification. Let us say that a k-bounded tree T on M is weakly (M, k)-large iff T has a last model P and there is Q E P , with degT (∞) ≥ k if Q = P , and there is a weak k-embedding σ : M → Q. We say that a (k, ω1 , ω1 + 1)-strategy Σ for M has the weak+weak Dodd-Jensen property iff Σ has the weak DoddJensen property14 , and whenever T , via Σ, is weakly (M, k)-large, as witnessed by Q, P , then bT does not drop in model or degree, and Q = P . 4.5 Lemma. Let M be an m-sound premouse and N and n-sound premouse, where m ≤ n. Let π : M → N be a weak m-embedding. Let T be an m-maximal tree on M . Suppose that U = π n−max T has wellfounded models. For α < lh(T ) let πα be the αth copy map. Let α < lh(T ) be such that [0, α]T drops in model. Then degU (α) = degT (α) and πα is a near degT (α)-embedding. Proof. Argue as in the proof of item 3 of Claim 1 in the proof of 4.4. 4.6 Lemma. Let M be a k-sound, (k, ω1 , ω1 + 1)-iterable premouse. Then there is a strategy Σ for M with the weak+weak Dodd-Jensen property. Proof. This is an adaptation of the construction in [2]; we explain the differences. Let Σ0 be any (k, ω1 , ω1 + 1)-strategy for M . Let T0 be a weakly (M, k)-large tree on M , via Σ0 , as witnessed by Q0 , P0 and π0 : M → Q0 , such that, letting ΣQ0 be the tail of Σ0 for k-bounded trees on Q0 , then whenever T 0 is a weakly 0 (M, k)-large tree on Q0 via ΣQ0 , as witnessed by Q0 , P 0 , then Q0 = P 0 and bT does not drop in model or degree. Let Σ1 be the π0 -pullback of ΣQ0 . Now run the construction of [2], producing T1 an (M, k)-large tree T1 on M , via Σ1 , as witnessed by Q1 , P1 , π1 (so P1 = M∞ T1 and deg (∞) ≥ k if Q1 = P1 , and π1 : M → Q1 is a near k-embedding), 14 Actually, we modify the weak Dodd-Jensen property of [2] a little. For T a tree via Σ, we say T is (M, k)-large iff T has a last model P and there is Q E P such that, if Q = P then degT (P ) ≥ k, and there is a near k-embedding π : M → Q. We then formulate the weak Dodd-Jensen property using this to replace the notion of (M, k)-large used in [2]. The point of this modification is explained in footnote 10.

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such that, letting ΣQ1 be the tail of Σ1 for k-bounded trees on Q1 , then the π1 -pullback Σ2 of ΣQ1 has the weak Dodd-Jensen property. ·O o

%00 π10 T2

·O o

π10 T2

QO 2 o

T2

%

·O o

π00 T1

%0

%1

QO 1 o

QO 0 o

T0

M

π0 T1

M

π1

M

π2

M We claim that Σ2 has the weak+weak Dodd-Jensen property. (The diagram illustrates the following argument.) For let T2 be a weakly (M, k)-large tree on M via Σ2 , as witnessed by Q2 , P2 and π2 : M → Q2 . In the following let π 0 π0 T abbreviate π k−max . Let %0 : P1 → M∞0 1 be the final copy map. Our choice of T0 ensures that bT1 does not drop in model or degree, and that Q1 = P1 . For suppose that Q1 = P1 . If bT1 drops in degree but not model then 0 degT1 (∞) < k, contradiction. So bT1 drops in model, so bπ0 T1 does too, and 0 by 4.5, degπ0 T1 (∞) = degT1 (∞) ≥ k and %0 is a near k-embedding. So %0 ◦ π1 is a weak (in fact near) k-embedding. But π00 T1 is via ΣQ0 , contradicting the choice of T0 . So Q1 / P1 . If %0 is not ν-high we reach a similar contradiction.

 If %0 is ν-high then we get a contradiction by considering (π00 T1 ) b F R where π0 T

R = M∞0 1 . A slight elaboration of the argument in the previous paragraph shows that bT2 does not drop in model or degree, and Q2 = P2 , as required. For example, π0 T %0 π 0 T let P = M∞1 2 and P 0 = M∞0 1 2 and %1 : P2 → P and % : P → P 0 be the final copy maps. Consider the case that Q2 = P2 , and suppose that bT2 drops in some sense, and therefore in model, so 4.5 applies to π10 T2 and %00 π10 T2 . 0 0 Therefore deg%0 π1 T2 (∞) ≥ k and % ◦ %1 is a near k-embedding. So % ◦ %1 ◦ π2 is a weak k-embedding (though maybe not a near k-embedding), a contradiction. If Q2 / P2 , argue much as before. So we may assume that U, V, as described above, are via Σ, a strategy with the weak+weak Dodd-Jensen property. We say that α is weakly exceptional iff rootT (α) = −1 and (−1, α]T does not drop in model. Claim 1 of the proof of 4.4 goes through with exceptional replaced by weakly exceptional by almost the same proof, but one also shows that
M M latter. If k < l then this is immediate, so suppose k = l. Let (z, ζ) = (zk+1 , ζk+1 ) (see [6] for the definition). By [6] and standard fine structure, and since k+1 ≤ j, T T z 0 =def pk+1 (M∞ ) = zk+1 (M∞ ) = iU (z),

(3)

T T ζ 0 =def ρk+1 (M∞ ) = ζk+1 (M∞ ) = sup iU “ζ.

(4)

T T Now π∞ is rΣk+1 -elementary on K 0 = rg(σ) where σ : Ck+1 (M∞ ) → M∞ is the core map, and T M∞ K 0 = Hullk+1 (ζ 0 ∪ {z 0 }) = rg(σ).

Letting K = HullM k+1 (ζ ∪ {z}), M / M . By lines (3) and (4), iU “K ⊆ K 0 . K is cofinal in ρM k , because Thk+1 (ζ ∪z) ∈ U So π∞ ◦ i is rΣk+1 -elementary on K, so is a weak k-embedding, as required. The foregoing argument also rules out the case that k = j. (But the proof we used to discount this case in our first approach also still applies when k = j.)

4.3

Third approach, via n-lifting embeddings

We finish by sketching a third approach to filling the gaps in the proofs of condensation etc. 4.7 Definition. Let M be a k-sound premouse and let T be a k-bounded tree on M . We say that T is (M, k)-lifting-c-preserving iff T has a last model P , and there is Q E P such that if Q = P then degT (∞) ≥ k, and there is a k-lifting c-preserving embedding π : M → Q. The lifting-weak Dodd-Jensen property is the natural analogue of the weak Dodd-Jensen property, but with (M, k)-lifting-c-preserving replacing (M, k)large. a 4.8 Lemma. For any countable, k-sound, (k, ω1 , ω1 + 1)-iterable premouse M , and enumeration e of M in ordertype ω, there is a (k, ω1 , ω1 + 1)-iteration strategy Σ for M which has the lifting-weak Dodd-Jensen property with respect to e. Again, there are obvious modifications for pseudo-premice, and the proofs are routine modifications of the arguments in [2]. Now consider the proof of condensation etc, using the lifting-weak Dodd~ and a comparison (T , U) of (P, M ~ ) much as Jensen property. One defines P, M ~ before, but by lifting T to a k-maximal tree V on M , with copy maps πα : MαT → Rα E MαV (we may have degT (α) ≤ degV (α)), and V being via a strategy with the liftingweak Dodd-Jensen property. Then every πα is an n-lifting c-preserving embedding, where n = degT (α). Thus, the lifting-weak Dodd-Jensen property suffices. The remaining details are given by simplifications of the arguments in §4.1. (They are also closer to the arguments in [10], with [10, Lemma 9.1.11] replaced by some of the arguments in §4.1.)

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References [1] William Mitchell and John R. Steel. Fine structure and iteration trees. Number 3 in Lectures Notes in Logic. Springer-Verlag, 1994. [2] Itay Neeman and John Steel. A weak Dodd-Jensen lemma. Journal of Symbolic Logic, 64(3):1285–1294, 1999. [3] Ernest Schimmerling. Combinatorial principles in the core model for one Woodin cardinal. Annals of Pure and Applied Logic, 74(2):153–201, 1995. [4] Ernest Schimmerling and John R. Steel. Fine structure for tame inner models. Journal of Symbolic Logic, 61(2):621–639, 1996. [5] Ralf Schindler, John R. Steel, and Martin Zeman. Deconstructing inner model theory. Journal of Symbolic Logic, 67(2):721–736, 2002. [6] Farmer Schlutzenberg. The definability of E in self-iterable mice. Available at arxiv.org: 1412.0085. [7] Farmer Schlutzenberg. A premouse inheriting strong cardinals from V . Available at arxiv.org: 1506.04116. [8] Farmer Schlutzenberg. Measures in mice. PhD thesis, University of California, Berkeley, 2007. [9] 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. [10] Martin Zeman. Inner Models and Large Cardinals. Walter de Gruyter, 2002.

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