DODD PARAMETERS AND λ-INDEXING OF EXTENDERS

MARTIN ZEMAN

Abstract. We study generalizations of Dodd parameters and establish their fine structural properties in Jensen extender models with λ-indexing. These properties are one of the key tools in various combinatorial constructions, such as constructions of square sequences and morasses.

AMS Subject Classification: 03E05, 03E45, 03E55 Keywords: Extender model, Fine Structure

Typical combinatorial constructions in extender models, such as constructions of square sequences, build on the analysis of Dodd parameters of certain active levels of the models. Dodd parameters come into play whenever we try to construct combinatorial objects that involve some form of “coherency”. The reason why they have to be considered is that the standard fine structural characteristics of premice are typically computed relative to a certain additional constant. This impairs the uniformity of the description of the combinatorial objects we are constructing, and thereby complicates the verification of the “coherency” conditions.

In this paper we will deal with Jensen extender models with the so-called λindexing introduced in [2]. The relevant background and notation can also be found in [17]. We establish the relationship between the Dodd parameter and the standard parameter for type B levels of extender models and prove the solidity theorem for the Dodd parameters. These results are formulated in Theorems 1.1 and 1.2. The actual formulations of both theorems are adjusted so that they can be immediately applied in [12] and [13], where the facts about the Dodd paremeters are used as a black box. Dodd parameters were originally introduced in [1] in connection with constructions of models for strong cardinals. For extender models of Mitchell-Steel type [7], the Dodd solidity was established by Steel [9, 15], and was used in a substantial way in combinatorial constructions in [9, 10] and in the

Research partially supported by the NSF grant DMS-0204728.

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proof of the weak covering lemma for Steel’s core model [5, 6]. Dodd parameters were also considered in [4] in constructions of extender models. The results of the current paper differ from those in [9] and [15] in several respects. The most obvious difference is that, here we deal with models with λ-indexing, which means that the relationship between the standard paramter and the Dodd parameter is different from that in models with Mitchell-Steel indexing. In models with λindexing, the fact that the Dodd parameter is distinct from the standard parameter is already a large cardinal axiom beyond a superstrong cardinal. In models with Mitchell-Steel indexing, it is a relatively modest large cardinal axiom well below one strong cardinal. We show that, in the models with λ-indexing, there is a canonical uniform way of conversion between the standard parameter and the Dodd parameter. Moreover, Proposition 1.3 shows that, under a certain relatively modest smallness condition, this conversion is particularly simple (although Proposition 1.3 does not seem to be relevant if we are interested in combinatorial constructions in their full generality). Secondly, in this paper we are considering slightly generalized versions of Dodd parameters; such generalizations are not needed for the basic construction of a square sequence, but they occur in more delicate constructions of square sequences with some amount of condensational coherency, and even in the construction of an ordinary Gap-1 Morass [13].

The only “smallnes condition” used in the fine structure theory of models with λ-indexing is iterability. This means that, modulo iterability, our fine structure theory is developed for models satisfying any large cardinal axiom that is witnessed by extenders in the sense of [17]. (That is, extenders whose sup of generators does not exceed the image of the critical point.) Let us also mention that the strategy of the proof of our main results is completely different from that used in the proof of Dodd solidity in Mitchell-Steel models in [15]. This was caused by our effort to avoid any direct reference to iterability in our arguments by substituting comparison arguments by applications of the condensation lemma and the solidity theorem for the standard parameters. However, we did not succeed completely in this effort and the paper does contain a version of a comparison argument after all.

We briefly recall some notation and terminology from [17]. Given two primitive recursively closed ordinals κ < λ, an extender at (κ, λ) is a map F : V → 𝒫(λ)

DODD PARAMETERS AND λ-INDEXING OF EXTENDERS 3

with V ⊂ 𝒫(κ) which preserves primitive recursive definitions; we of course require that V is closed under such definitions. The ordinal κ is called the critical point of F and denoted by cr(F), and λ is called the length of F and denoted by λ(F). If λ̄ < λ is primitive recursively closed, we define F | λ̄ : V → 𝒫(λ̄) to be the map x ↦ F (x) ∩ λ̄; this map is obviously an extender. Typically, V is a set of the form 𝒫(κ) ∩ M where M is some acceptable J-structure. So we can form an ultrapower Ult(M, F) of M by F. An ordinal λ̄ ∈ (κ, λ) is a cutpoint of F just in case that for every x ∈ dom(F) that codes a function from κ into κ (in the obvious way based on Gödel pairing ≺, ≻), F (x) ∩ λ̄ codes a total function from λ̄ into λ̄. More precisely, ≺α, ζ≻ ∈ F (x) implies ζ < λ̄ for every α < λ̄. In the language of ultrapowers this says that π(f)(α) < λ̄ for every function f : κ → κ from the model F is applied to and every α < λ̄; here π is the associated ultrapower map. A J-structure M is coherent just in case that M is of the form ⟨J_{ν}^{A}, F ⟩ and J_{ν}^{A} = Ult(J_{ϑ}^{A}, F) where κ = cr(F) is the largest cardinal in J_{ϑ}^{A} and λ(F) is the image of κ under the associated ultrapower map, and thus the largest cardinal in J_{ν}^{A}. The extender F measures all subsets of κ that are in J_{ϑ}^{A}, but not necessarily all subsets in 𝒫(κ)∩M, since the option ϑ < κ^{+M} is allowed. Thus, F need not be weakly amenable with respect to M, but M is always an amenable structure. If M is a coherent structue, we often write λ(M) for λ(F).

If M = ⟨J_{ν}^{E}, F ⟩ is a potential premouse in the sense of [17], Chapter 9 (see also [12]), M is of type A if F has no cutpoints, of type B if the set of all cutpoints of F is nonempty and bounded in λ(M) =^{def} λ(F), and of type C if the set of all cutpoints of F is unbounded in λ(M). A premouse is a potential premouse that satisfies the initial segment condition, which will be discussed in a little while. If M is a type B potential premice, the largest cutpoint of F is denoted by λ_{M}^{∗}. Letting κ = cr(F) and τ = κ^{+M}, the height of Ult(J_{τ}^{E}, F | λ_{M}^{∗}) is denoted by γ_{M}. In the genaral fine structure theory for premice, the language for coherent structures contains just predicate symbols denoting E and F, and the language for premice contains an additional constant symbol denoting F | λ_{M}^{∗}. In the case of type A or C premice, there is no difference between the two languages. The initial segment condition requires that F | λ̄ ∈ M whenever λ̄ is a cutpoint of F. In the case of type B premice, this is equivalent to the condition F | λ_{M}^{∗} ∈ M. This is in general

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weaker than the requirement that F | λ_{M}^{∗} is on the E-sequence, which considerably simplifies proofs of iterability in the constructions of extender models. However, if M is weakly iterable in the sense of [17] (see below), then γ_{M} is the index of F | λ_{M}^{∗}. For the purpose of doing combinatorics, it is more convenient to use γ_{M} instead of F | λ_{M}^{∗} as the additional parameter, so our expanded language of premice will contain a constant symbol γ̇ denoting γ_{M}. This makes no difference from the fine structural point of view, since F | λ_{M}^{∗} and γ_{M} are lightface Σ_{1}(M)-definable from each other in the language for coherent structures. For type B premice M, we shall stick to the following convention. The Σ_{1}-Skolem function computed in the language for premice will be denoted by h_{M}. The Σ_{1}-Skolem function computed in the language for coherent structures, i.e. in the language with no additional constants, will be denoted by h_{M}^{∗}. For type A and C premice, there is obviously no difference between h_{M} and h_{M}^{∗}, since the corresponding language for premice does not have any additional constant symbol. (Or, alternatively, if we want to have a uniform definition of the language for premice, we might introduce an additional constant symbol γ̇ and interpret it always as 0. Obviously, the difference between the two languages will be merely cosmetic.) Any cutpoint of F is a limit cardinal in M; since J_{θ}^{E} = V_{θ}^{M} for any such cardinal θ, the coherency condition combined with the initial segment condition guarantee that that F | λ is a superstrong extender in M whenever λ is a cutpoint of F. Thus, the presence of cutpoints is a very strong large cardinal hypothesis.

Let us now state the Condensation Lemma from [17] we will refer to throughout the paper. This lemma is true for weakly iterable premice. Recall that a premouse M is weakly iterable just in case every countable premouse that is elementarily embeddable into M is (ω_{1} + 1, ω_{1})-iterable in the sense of [17], Chapter 9. Such premice are automatically solid.

Lemma 0.1. Let M̄ and M be premice of the same type where M is weakly iterable and let σ : M̄ → M be an embedding which is both cardinal preserving and Σ_{0}^{(n)}- preserving (with respect to the language of premice!), and such that σ ↾ ωϱn+1 ; M̄ = id.

Then M̄ is weakly iterable and, consequently, solid. Furthermore, if σ ↾ ν = id and M̄ is sound above ν, then one of the following holds:

(a) M̄ is the core of M above ν and σ is the associated core map.

DODD PARAMETERS AND λ-INDEXING OF EXTENDERS 5

(b) M̄ is a proper initial segment of M.

(c) M̄ = Ult^{∗}(M || ζ, E_{α}^{M}) for some ζ and α such that ν ≤ ζ < ht(M), α ≤ ωζ and ν = κ^{+M || ζ} where κ = cr(E_{α}^{M}); moreover, ζ is maximal with these properties. Also, E_{α}^{M} has a single generator κ.

(d) M̄ is a proper initial segment of Ult(M, E_{ν}^{M}).

If M satisfies the assumptions of the Condensation Lemma and is additionally sound above ν, then (a) can be reformulated as follows:

(a′) M̄ = M and σ = id.

Throughout the paper, we will try to avoid any direct reference to iterability whenever possible and substitute it by an application of the Condensation Lemma. This approach is useful in many situations, as unlike the weak iterability, solidity is a first order property of premice.

1. Dodd Parameters

We first introduce the parameters d_{M}^{α}, which are generalizations of the Dodd parameters. Our context is adjusted to the needs of combinatorial constructions of e.g. [12, 13, 14] and [16]. Thus, the traditional notions of the Dodd projectum and Dodd parameter (for the definitios, see e.g. [12] or [9]) will actually not be considered. Recall that the canonical well-ordering of all finite sets of ordinals is denoted by <^{∗}. Definition. Let M be an active premouse and κ = cr(E_{top}^{M}) and let α be an ordinal satisfying

κ^{+M} ≤ α.

The parameter d_{M}^{α} is the <^{∗}-least finite set of ordinals d such that M = h_{M}^{∗}(α∪{d}), if defined. We write d_{M} for d_{M}^{ωϱ_{M}^{1}} whenever κ < ωϱ_{M}^{1}.

Obviously, d_{M}^{α} need not be defined for each M and α, and its existence guarantees that ωϱ_{M}^{1} ≤ α. However, d_{M}^{α} is defined, granting that ωϱ_{M}^{1} ≤ α and M is 1-sound above α, which is precisely the situation we are interested in. It is easy to see that if ᾱ ≤ α and dᾱ ; M is defined then so is d_{M}^{α} and d_{M}^{α} = dᾱ ; M − α. For 1-sound M with κ < ωϱ_{M}^{1}, the Dodd parameter in the traditional sense is just d_{M}. Clearly, d_{M}^{α} = ∅ if M is a type C premouse. If M is a 1-sound type A premouse then easily

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d_{M}^{α} = p_{M}^{1} − α, and the notions of Dodd solidity and solidity coincide. Thus, the notion is non-trivial only for type B premice.

Dodd solidity witnesses are defined in the obvious way. Given an ordinal β ≥ α and a parameter p ∈ [On ∩ M]^{<ω}, the standard Dodd solidity witness ^{∗}W _{M}^{β,p} for β with respect to M and p is the transitive collapse of the hull h_{M}^{∗}(β ∪ {p − (β + 1)}). Thus, the only difference between the standard solidity witness in the usual sense W_{M}^{β,p} and the standard Dodd solidity witness ^{∗}W _{M}^{β,p} is that ^{∗}W _{M}^{β,p} is computed with no reference to γ_{M}. Similarly, a pair ⟨Q, r⟩ is a generalized Dodd solidity witness for β with respect to M and p just in case that Q is transitive, r ∈ Q, and for every Σ_{1}-formula in the language for coherent structures φ(x, v_{1}, …, v_{ℓ}) and every ξ_{1}, …, ξ_{ℓ} < β we have

M ⊧ φ(p − (β + 1), ξ_{1}, …, ξ_{ℓ}) ⟶ Q ⊧ φ(r, ξ_{1}, …, ξ_{ℓ})

The property of being a generalized Dodd solidity witness is Π_{1} in the language for coherent structures. We will often make use of the following fact:

(1)

The existence of generalized witnesses is equivalent to the existence of standard witnesses.

This is expressed in a somewhat sloppy way, but the meaning of (1) is obvious. It says that if β ∈ On ∩ M and p ∈ [On ∩ M]^{<ω} are arbitrary, then W_{M}^{β,p} ∈ M if and only if M contains some generalized witness for β with respect to M and p, and the corresponding fact is true for Dodd solidity witnesses as well. In [17], this is formulated as Lemma 1.12.3. The lemma assumes that p is a good parameter and β ∈ p. However, it is easy to see that this assumption is superfluous: Recall that the heart of the proof of Lemma 1.12.3 is to show the “if” implication. In that proof, the assumptions on β and p are used to show that M has a cardinal larger than β. It is argued that β is the critical point of the witness map and, consequently, the image of β under the witness map is a cardinal in M. But if β and p are arbitrary, then either the witness map has a critical point ≥ β, or it does not have a critical point at all. In the former case, we still get a cardinal in M above β, and exactly as in the proof of Lemma 1.12.3 conclude that W_{M}^{β,p} ∈ M. In the latter case, W_{M}^{β,p} is an initial segment of M. But it must be a proper initial segment, as W_{M}^{β,p} is embeddable into some generalized witness that is an element of M.

DODD PARAMETERS AND λ-INDEXING OF EXTENDERS 7

Definition. We say that M is Dodd solid above α just in case d_{M}^{α} is defined and ∗W _{M}^{β,d_{M}^{α}} ∈ M for every β ∈ d_{M}^{α}. This is equivalent to the requirement that each β ∈ d_{M}^{α} has a generalized Dodd solidity witness with respect to M and d_{M}^{α} that is an element of M.

The reader familiar with the notion of Dodd solidity in the traditional sense will immediately notice that the above definition does not contain any requirement corresponding to the requirement in the traditional definition that the restriction of E_{top}^{M} to (ᾱ ∪ d_{M}^{α}) − (ᾱ + 1) is an element of M. for any ᾱ strictly smaller than the Dodd projectum. The reason why we do not need this requirement is that we only need to consider situations where the Dodd projectum agrees with the first projectum, and so the corresponding clause is automatically satisfied, as follows from the elementary properties of projecta.

In all relevant cases, the Dodd parameter d_{M}^{α} can be computed from the standard parameter of M and an additional finite set of ordinals, which we denote by e_{M}^{α}. This is, however, not immediately clear from the definition of e_{M}^{α}. Definition. Let M be a type B potential premouse and α ∈ On. The <^{∗}-least finite set of ordinals e such that γ_{M} ∈ h_{M}^{∗}(α ∪ {p_{M} − α, e}) is denoted by e_{M}^{α}.

Obviously, e_{M}^{α} is always defined and e_{M}^{α} ⊂ γ_{M} +1. The main result of this paper is summarized in the following two theorems. Theorem 1.1. Let M be a weakly iterable type B premouse and let κ = cr(E_{top}^{M}). Assume further that M is sound above α where α ≥ max{κ^{+M}, ωϱ_{M}^{1}}. Then d_{M}^{α} is defined and:

(a) d_{M}^{α} = (p_{M} − α) ∪ e_{M}^{α}.

(b) M is Dodd solid above α. Theorem 1.2. Let M be a weakly iterable type B premouse and let α ≥ κ^{+M} where κ = cr(E_{top}^{M}). Assume that d is a parameter satisfying:

• d ∩ α = ∅.

• M = h_{M}^{∗}(α ∪ {d}).

• Every β ∈ d has a generalized Dodd solidity witness with respect to M and

d that is an element of M. Then

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(a) d = d_{M}^{α};

(b) d_{M}^{α} = (p_{M} − α) ∪ e_{M}^{α};

(c) M is sound above α.

Under a certain smallness condition, the conversion between d_{M}^{α} and p_{M} − α simplifies as follows.

Proposition 1.3. Assume there is no inner model with a cardinal which is both subcompact and superstrong. Then (a) in Theorem 1.1 can be reformulated as follows: d_{M}^{α} = p_{M} − α or d_{M}^{α} = (p_{M} ∪ {λ_{M}^{∗}}) − α or d_{M}^{α} = (p_{M} ∪ {γ_{M}}) − α. Moreover, the second possibility can only occur if λ_{M}^{∗} = max(d_{M}^{α}).

Subcompactness is a large cardinal property slightly stronger than 1-extendibility. More precisely, if κ is subcompact then there are many 1-extendible cardinals below κ, see [11] or [12] or for a definition. It can be shown [18] that either of the three possibilities named in the above proposition can occur, and actually well below that smallness condition. We don’t know whether the smallness condition can be omitted, but we believe it cannot. Let us also note that there are obvious reformulations for type A premice. As has been mentioned above, Theorem 1.1 does not say anyghing new in this case. In Theorem 1.2, only (a) needs an argument; (b) and (c) then follow directly from (a). But (a) is nothing else but Lemma 1.12.5 from [17] in this case.

We now turn to the proof of the two theorems; as a by-product we will get a proof of Proposition 1.3. Throughout the proof of Theorem 1.1, we will assume that α < λ_{M}^{∗}. Roughly speaking, we can do this without loss of generality, since there is less work to be done for larger α. A closer examination of the entire situation reveals that the only more serious issue connected with this assumption occurs when γ_{M} ∈ e_{M}^{α}, but we will explain at the beginning of the proof of Lemma 1.6 why the assumption is harmless.

We fix the following notation. E will be the extender sequence of M, F will be the top extender of M, κ = cr(F) and τ = κ^{+M}. Furthermore, if π is the ultrapower map associated with Ult(J_{τ}^{E}, F) and f : ^{n}κ → κ or f : ^{n}κ → 𝒫(κ) is in M (here n is a natural number), we will often write F (f) for π(f). This is possible, since f is uniformly recursively encodable into a subset of κ in either of these cases.

DODD PARAMETERS AND λ-INDEXING OF EXTENDERS 9

Both theorems are proved by induction on the height of M. Thus, our argument will make use of the following induction hypothesis:

(2)

If M^{∗} is a weakly iterable premouse with ht(M^{∗}) < ht(M)

then Theorems 1.1 and 1.2 hold of M^{∗}.

Proof of Theorem 1.1. Since M is sound above α, the following holds of (p_{M} − α) ∪ e_{M}^{α}:

h_{M}^{∗}(α ∪ {p_{M} − α, e_{M}^{α}}) ⊃ h_{M}^{∗}(α ∪ {p_{M} − α, γ_{M}}) = h_{M}(α ∪ {p_{M} − α}) = M.

This computation does not make use of the minimality of e_{M}^{α}. It follows that d_{M}^{α} is defined and d_{M}^{α} ≤^{∗} (p_{M} − α) ∪ e_{M}^{α}. To see the equality, it suffices to verify:

(3)  ^{∗}W _{M}^{β,(p_{M} −α)∪e_{M}^{α}} ∈ M whenever β ∈ (p_{M} − α) ∪ e_{M}^{α}.

Indeed, the assumption d_{M}^{α} <^{∗} (p_{M} − α) ∪ e_{M}^{α} would lead to a contradiction as follows: Fix β ∈ [(p_{M}−α)∪e_{M}^{α}]−d_{M}^{α} such that [(p_{M}−α)∪e_{M}^{α}]−(β+1) = d_{M}^{α}−(β+1). Let W = ^{∗}W _{M}^{β,(p_{M} −α)∪e_{M}^{α}} and σ : W → M be the canonical map; this map is Σ_{1}- preserving in the language for coherent structures. Obviously α ∪ {d_{M}^{α}} ⊂ rng(σ), so rng(σ) = M. Consequently, W = M and σ = id. But (3) guarantees that W ∈ M, a contradiction.

The verification of (3) will take the most part of the paper. As usual, we will start with the largest β ∈ (p_{M} − α) ∪ e_{M}^{α} and gradually move downward. The assumptions of Theorem 1.1 require merely that M is sound above α. However, we can without loss of generality assume:

(4)  M is fully sound.

Although most of the argument will go through without this assumption, the two comparison arguments we describe below do make use of it.

To see (4), notice that if M^{∗} = core(M), σ_{c} : M^{∗} → M is the core map and α^{∗} = (σ_{c}^{−1})′′α, then σ_{c}(γ_{M^{∗}}) = γ_{M} and σ(e_{M^{∗}}^{α^{∗}}) = e_{M}^{α}. This follows from the preservation properties of σ_{c} by a straightforward reflection argument: Obviously γ_{M} ∈ h_{M}^{∗}(α ∪ {p_{M}, σ_{c}(e_{M^{∗}}^{α^{∗}})}), as γ_{M^{∗}} ∈ h_{M}^{∗}(α^{∗} ∪ {p_{M^{∗}}, e_{M^{∗}}^{α^{∗}}}). So e_{M}^{α} ≤^{∗} σ_{c}(e_{M^{∗}}^{α^{∗}}). On the other hand, if this inequality were strict, M would satisfy the statement

(∃e <^{∗} σ_{c}(e_{M^{∗}}^{α^{∗}}))(∃x ⊂ σ(α^{∗}))(x is finite & γ_{M} = h_{M}^{∗}(ξ, p_{M} ∪ e)).

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The preservation properties of σ_{c} would then yield γ_{M^{∗}} = h_{M^{∗}}^{∗}(x, p_{M^{∗}} ∪ e^{∗}) for some e^{∗} <^{∗} e_{M^{∗}}^{α^{∗}} and some finite x ⊂ α^{∗}, a contradiction. Once we have proved that ^{∗}W _{M^{∗}}^{β,(p_{M^{∗}} −α^{∗})∪e_{M^{∗}}^{α^{∗}}} ∈ M^{∗} for β ∈ p_{M^{∗}} ∪ e_{M^{∗}}^{α^{∗}}, the preservation properties of σ_{c} will imply that σ_{c}(^{∗}W _{M^{∗}}^{β,(p_{M^{∗}} −α^{∗})∪e_{M^{∗}}^{α^{∗}}}) is a generalized Dodd solidity witness for σ_{c}(β) with respect to M and p_{M} ∪ e_{M}^{α}, and thereby prove Theorem 1.1. Lemma 1.4. ^{∗}W _{M}^{β,(p_{M} −α)∪e_{M}^{α}} ∈ M whenever β ∈ (p_{M} − α) ∪ e_{M}^{α} and β > γ_{M}. Proof. Since e_{M}^{α} ⊂ γ_{M} + 1, we have

(p_{M} − α) ∪ e_{M}^{α} − (γ_{M} + 1) = p_{M} − (γ_{M} + 1).

Then easily ^{∗}W _{M}^{β,(p_{M} −α)∪e_{M}^{α}} = W_{M}^{β,(p_{M} −α)∪e_{M}^{α}}, and this object is an element of M by the solidity of M. ⊓ ⊔(Lemma 1.4)

Notice that the above lemma in fact establishes the agreement between d_{M}^{α} and p_{M} above γ_{M}. We now focus on the interval [λ_{M}^{∗}, γ_{M}]. Our next observation is: Lemma 1.5. If λ_{M}^{∗} ≤ β < λ(M) then λ_{M}^{∗} ∈ h_{M}^{∗}(α ∪ {β}) and h_{M}^{∗}(α ∪ {β}) ∩ γ_{M} is cofinal in γ_{M}. Proof. Let M̄ be the transitive collapse of h_{M}^{∗}(α ∪ {β}) and σ : M̄ → M be the uncollapsing map. Then β̄ =^{def} σ^{−1}(β) < λ(M̄) is an upper bound for (σ^{−1})′′λ_{M}^{∗}, so λ̄^{∗} =^{def}(σ^{−1})′′λ_{M}^{∗} ≤ β̄ < λ(M̄). Now λ̄^{∗} is a cutpoint of F̄, the top extender of M̄. Indeed, if f : κ → κ is in M̄ and δ < λ̄^{∗} then σ(δ) < λ_{M}^{∗}, so σ(F̄(f)(δ)) = F (f)(σ(δ)) < λ_{M}^{∗}, and, consequently, F̄(f)(δ) is an element of (σ^{−1})′′λ_{M}^{∗} = λ̄^{∗}. The way we defined λ̄^{∗} also guarantees that λ_{M}^{∗} ≤ σ(λ̄^{∗}) ≤ β. The latter inequality here ensures that σ(λ̄^{∗}) < λ(M). So σ(λ̄^{∗}) is a cutpoint of M, since being a cutpoint is a Π_{1}-property. But it cannot be strictly larger than λ_{M}^{∗}, as λ_{M}^{∗} is the largest cutpoint of M. So necessarily σ(λ̄^{∗}) = λ_{M}^{∗}, which proves the first part of the lemma.

Towards the second part of the lemma, recall that the ultrapower embedding π^{∗} : J_{τ}^{E} → J_{γ_{M}}^{E} associated with Ult(J_{τ}^{E}, F | λ_{M}^{∗}) is cofinal and π^{∗}(x) = F (x) ∩ λ_{M}^{∗} for every x ⊂ κ. Thus, the map

x ↦ the least ζ such that F (x) ∩ λ_{M}^{∗} ∈ S_{ζ}^{E}

maps 𝒫(κ) ∩ M cofinally into γ_{M}, and is easily seen to be Σ_{1}(M) in the parameter λ_{M}^{∗}. The rest follows from the facts that α ≥ τ and λ_{M}^{∗} ∈ rng(σ). ⊓ ⊔(Lemma 1.5)

DODD PARAMETERS AND λ-INDEXING OF EXTENDERS 11

The above lemma easily yields:

(5)  If (p_{M} ∪ e_{M}^{α}) − (λ_{M}^{∗} + 1) ≠ ∅ then e_{M}^{α} ∩ [λ_{M}^{∗}, γ_{M}) = ∅.

To see this, assume for a contradiction that e_{M}^{α} has an element in the interval [λ_{M}^{∗}, γ_{M}). Let β be the largest one. Then β > λ_{M}^{∗}. To see this, notice that the hypothesis in (5) implies that, assuming β = λ_{M}^{∗}, there is some β′ ∈ p_{M} ∪ e_{M}^{α} that is strictly larger than λ_{M}^{∗}. By Lemma 1.5, β = λ_{M}^{∗} must be then in the hull h_{M}^{∗}(α ∪ {β′}), so h_{M}^{∗}(α ∪ {p_{M} ∪ e_{M}^{α}}) = h_{M}^{∗}(α ∪ {p_{M} ∪ (e_{M}^{α} − {β})}), and we have a contradiction with the minimality of e_{M}^{α}. Since β > λ_{M}^{∗}, we can find an ordinal ζ ∈ h_{M}^{∗}(α ∪ {λ_{M}^{∗}}) that is larger than β. Obviously, h_{M}^{∗}(α ∪ {λ_{M}^{∗}}) contains a surjective map f : λ_{M}^{∗} → ζ, so we can fix a ξ < λ_{M}^{∗} with f(ξ) = β. Thus, β ∈ h_{M}^{∗}(α ∪ {λ_{M}^{∗}, ξ}). Let e = (e_{M}^{α} − {β}) ∪ {λ_{M}^{∗}, ξ}. Clearly, e <^{∗} e_{M}^{α} and h_{M}^{∗}(α ∪ {p_{M} − α, e}) ⊃ h_{M}^{∗}(α ∪ {p_{M} − α, e_{M}^{α}}) ∋ γ_{M}. This again contradicts the minimality of e_{M}^{α}.

Notice also that

(6)  If (p_{M} ∪ e_{M}^{α}) − (λ_{M}^{∗} + 1) = ∅ then λ_{M}^{∗} = max(p_{M} ∪ e_{M}^{α}).

Since λ_{M}^{∗} is a cutpoint of F, the map k : M || γ_{M} → M, defined by k(π^{∗}(f)(β)) = π(f)(β) for f ∈ ^{κ}κ ∩ M and β < λ_{M}^{∗} (here π^{∗} is as in the proof of Lemma 1.5), has critical point λ_{M}^{∗} and k(λ_{M}^{∗}) = λ(M). This map is Σ_{1}-preserving (with respect to the language for coherent structures), so γ_{M} ∉ h_{M}^{∗}(α ∪ {d}) for any finite d ⊂ λ_{M}^{∗}. Thus, (p_{M} − α) ∪ e_{M}^{α} ⊄ λ_{M}^{∗}, which immediately yields (6).

So far we have characterized d_{M}^{α} − (γ_{M} + 1) and established the Dodd solidity for this top part of d_{M}^{α}. By the above discussion, the next possible element of d_{M}^{α} is γ_{M}. This happens precisely when γ_{M} ∈ e_{M}^{α}. Notice that

(7)  e_{M}^{α} = {γ_{M}}

in this case, and we have to establish (3) for β = γ_{M}. As this requires a nontrivial amount of effort, we will formulate it as a lemma. Lemma 1.6. Assume that γ_{M} ∈ e_{M}^{α}. Then ^{∗}W _{M}^{γ_{M} ,p_{M}} ∈ M.

Obviously, we can write p_{M} instead of (p_{M} −α)∪e_{M}^{α} in the above lemma. Before going into the proof of this lemma, let us discuss the consequences. For every

12 MARTIN ZEMAN

β ∈ (p_{M} − α) ∩ γ_{M} we have ^{∗}W _{M}^{β,(p_{M} −α)∪{γ_{M} }} = W_{M}^{β,p_{M}} ∈ M; the membership to M follows from the solidity of M. Due to (7), we can then conclude:

Lemma 1.7. If γ_{M} ∈ e_{M}^{α} then d_{M}^{α} = (p_{M} − α) ∪ {γ_{M}} and M is Dodd solid.

Proof of Lemma 1.6. The proof is based on a comparison argument. This paper builds on the theory presented in [17], which uses fully Σ^{∗}-iterations in comparison arguments, and in this respect it differs from that in [7]. The use of fully Σ^{∗}-iterations has an advantage that the general theory becomes very “clean”. Unfortunately, there is also one drawback of this approach, namely, that the comparison argument breaks down in certain special cases when we apply a superstrong extender at the very first step on the “winning” side of the coiteration. Since the comparison described below does apply a superstrong extender at the first step on the “winning” side, it is convenient to treat certain situations independently. We will split the entire argument into two cases. Case 1 will deal with the situations which give rise to pathologies in the comparison argument; it will turn out that no comparison argument is needed in this case. In Case 2, we present the comparison argument itself and no pathologies will occur here. Our strategy is analogous to that in the proof of the solidity theorem in [17], Section 9.3. However, as the current situation is somewhat specific, the argument in Case 1 will be shorter than that in [17].

A simple reflection argument guarantees that we can without loss of generality assume that M is countable and (ω_{1} + 1, ω_{1})-iterable. We fix an enumeration e⃗ = ⟨e_{i}; i ∈ ω⟩ of M and the e⃗-minimal (ω_{1} + 1, ω_{1})-iteration strategy 𝕊 guaranteed by Neeman-Steel [8] (for the version for λ-indexing, see also [17], Lemma 9.2.11).

At this point, we should explain one issue. Recall that just before we went into the proof of Theorem 1.1, we made an assumption α < λ_{M}^{∗}. We argued that this assumption is harmless, since there is less work to do for larger α. This is really the case, since the same argument we used to identify d_{M}^{α} −(γ_{M} +1) and prove the Dodd solidity will also work for α > λ_{M}^{∗}, granting that γ_{M} ∉ d_{M}^{α}. Indeed, the argument did not require any knowledge of ωϱ_{M}^{1} other than the inequality ωϱ_{M}^{1} ≤ α, which is always true if d_{M}^{α} is defined. However, if γ_{M} ∈ d_{M}^{α} and α ≥ λ_{M}^{∗}, that argument is not going to work, and the comparison argument we are going to give will make a substantial use of the fact that ωϱ_{M}^{1} ≤ λ_{M}^{∗}. But this is always the case, as γ_{M} ∈ d_{M}^{α}

DODD PARAMETERS AND λ-INDEXING OF EXTENDERS 13

necessarily implies α ≤ γ_{M}, and there are no M-cardinals in the interval [λ_{M}^{∗}, γ_{M}] other than λ_{M}^{∗}.

Let us now make some settings and also recall some notation that will be used throughout the rest of the proof. We set

(8)  W = ^{∗}W _{M}^{γ_{M} ,p_{M}} ,

so

(9)  cr(E_{top}^{W}) = cr(E_{top}^{M}) = κ and κ^{+W} = κ^{+M} = τ.

Let

m = \{ \begin{matrix}the least i ∈ ω such that ωϱ_{M}^{i+1} ≤ τ, & if defined, \\ ω & otherwise; \end{matrix}
M′ = Ult^{∗}(M, E_{γ_{M}}) and π : M → M′ is the associated ultrapower embedding;
M̃ = Ult^{m}(M, E_{γ_{M}}) and π̃ : M → M̃ is the associated ultrapower embedding.

For basic information on m-ultrapowers see e.g. [17], Section 3.5 or [12]. Denote the collection of all functions used to build Ult^{m}(M, E_{γ_{M}}) by Γ^{m}(κ, M). Working in the language for premice, π is fully Σ^{∗}-preserving and π̃ is Σ_{0}^{(m)}-preserving and cofinal at the level m. We also have the Łoś theorem for Σ_{0}^{(m)}-formulae for both ultrapowers. Moreover, π is Σ_{2}^{(i)}-preserving whenever ωϱ_{M}^{i+1} > κ.

Case 1: 𝒫(γ_{M}) ∩ Ult^{∗}(M, E_{γ_{M}}) ≠ 𝒫(γ_{M}) ∩ Ult(M, E_{γ_{M}}).

This happens precisely when m ∈ ω and ωϱ_{M}^{m+1} = τ. We only show that these two clauses necessarily follow, as only this half of the equivalence is relevant for the argument below. Notice that if m = ω (that is, ωϱ_{M}^{ω} > τ) or ωϱ_{M}^{m+1} ≤ κ then Ult(H_{M}^{m}, E_{γ_{M}}) = H_{M′}^{m} where H_{M}^{m} and H_{M′}^{m} is the domain of the m-th reduct of M and M′, respectively. So Ult(M, E_{γ_{M}}) and M′ agree up to a common cardinal that is larger than γ_{M} and, consequently, Ult(M, E_{γ_{M}}) and M′ contain the same subsets of γ_{M}.

As an immediate consequence we have ωϱ_{M}^{1} > κ, so

(10) π : M → M′ is Σ_{2}-preserving with respect to the language of premice.

Let σ̃ : M̃ → M′ be the canonical embedding defined by

σ̃ : π̃(f)(ξ) ↦ π(f)(ξ)

for f ∈ Γ^{m}(κ, M) and ξ < λ_{M}^{∗}. Recall that f need not be an element of M, but it has a functionally absolute Σ_{1}^{(m−1)}-definition in some parameter from M. More precisely, there is a Σ_{1}^{(m−1)}-formula φ(v_{0}, v_{1}, v_{2}) such that φ defines a partial function by (a_{1}, a_{2}) ↦ b iff Q ⊧ φ(b, a_{1}, a_{2}) whenever Q is an acceptable J-structure for the language of premice, and y = f(ξ) just in case that M ⊧ φ(y, ξ, p) for some parameter p ∈ M. Then π(f) is the function defined by φ over M′ in the parameter π(p). Standard arguments then yield:

• σ̃ : M̃ → M′ is Σ_{0}^{(m)}-preserving with respect to the language for premice;

• cr(σ̃) = γ_{M}^{+M̃};

• ωϱm+1 ; M̃ ≤ γ_{M} and M̃ is sound above γ_{M}. The first clause follows from the Łoś theorem for Σ_{0}^{(m)}-formulae by the standard argument. To see the second clause, notice that σ̃ is the identity up to γ_{M} + 1 by definition, so cr(σ̃) ≥ γ_{M}^{+M̃}. Our assumption guarantees that Ult(M, E_{γ_{M}}) and M′ have different power sets of γ_{M}. Since 𝒫(γ_{M}) ∩ M̃ = 𝒫(γ_{M}) ∩ Ult(M, E_{γ_{M}}), we see that M′ has more subsets of γ_{M} than M̃. Consequently, γ_{M}^{+M̃} < γ_{M}^{+M′}. Since σ̃(γ_{M}^{+M̃}) = γ_{M}^{+M′}, the conslusion follows. The third clause is a standard consequence of the soundness of M, the two facts that ωϱ_{M}^{m+1} = τ and π̃(τ) = γ_{M}, and general properties of m-ultrapowers.

The three clauses established in the previous paragraph enable us to apply the Condensation Lemma to σ̃ : M̃ → M′. This is possible, as M′, being a Σ^{∗}- ultrapower of M by an internal extender, is itself (ω_{1} + 1, ω_{1})-iterable. We want to infer that the conclusion (b) is the case, that is, we want to show:

(11)  M̃ is a proper initial segment of M′.

We first focus on ruling out (a). Notice that ωϱ_{M′}^{m+1} ≤ γ_{M} and M′ is sound above γ_{M}; this follows similarly as in the case of M̃ above. (a′) would then imply that M̃ = M′, which contradicts the assumption that M̃ and M′ have distinct power sets of γ_{M}. To see that (c) and (d) in the Condensation Lemma fail is easy, as both (c) and (d) would imply that the cardinal predecessor of cr(σ̃) in M̃ is a limit

DODD PARAMETERS AND λ-INDEXING OF EXTENDERS 15

cardinal in M̃. In the present situation, however, this cardinal predecessor is equal to γ_{M}, which is a successor cardinal in M̃. ⊓ ⊔(11)

The induction hypothesis (2) guarantees that the following statement is true:

(12)

Let R be a type B proper initial segment of M and α_{R} be an ordinal such that α_{R} ≥ max{(cr(E_{top}^{R})^{+R}, ωϱ_{R}^{1}}. Then

• R is Dodd solid above α_{R};

• d_{R}^{α} = (p_{R} − α_{R}) ∪ e_{R}^{α}.

By the preservation properties of π, this statement is true with M′ in place of M. This combined with (11) implies that M̃ is Dodd solid above α̃ = π̃(α). Letting α′ = π(α), general properties of fine ultrapowers easily imply that both M̃ and M′ are type B premice, π̃(p_{M} − α) = pM̃ − α̃ and π(p_{M} − α) = p_{M′} − α′. Furthermore, π̃(γ_{M}) = γM̃ and π(γ_{M}) = γ_{M′}; as a consequence we obtain that eα̃ ; M̃ = {γM̃} and e_{M′}^{α′} = {γ_{M′}}. Finally (12) implies that W̃ =^{def} ^{∗}W γM̃ ,dα̃ ; M̃ ; M̃ ∈ M̃.

Notice also that σ̃(pM̃ − α̃) = p_{M′} − α′ and σ̃(γM̃) = γ_{M′}; this is an immediate consequence of the above calculations and the definition of σ̃. Our next step is to show that:

(13)

In M′, there is a generalized Dodd solidity witness for γ_{M′} with respect to M′ and p_{M′}.

For this, we have to refer to the preservation properties of σ̃. We show:

σ̃ : M̃ → M′ is Σ_{1}-preserving with respect to the language of premice.

This is obvious if m > 0, so it suffices to focus on the case where m = 0. In this case, M̃ is a result of a coarse ultrapower, so π̃ is Σ_{0}-preserving and cofinal. M′ is not a result of a coarse ultrapower, but it is still true that π′ is cofinal. To see this, recall that the assignment

x ↦ the least ζ such that F (x) ∈ J_{ζ}^{E}

is Σ_{1}(M) and maps 𝒫(κ) ∩ M cofinally into ht(M), and similarly

x ↦ the least ζ such that F′(x) ∈ J_{ζ}^{E′}

is Σ_{1}(M′) and maps 𝒫(κ′) ∩ M′ cofinally into ht(M′) where F′ is the top extender of M′ and κ′ = cr(F′). Now notice that π′ maps τ cofinally into its image γ_{M},

16 MARTIN ZEMAN

which easily yields the cofinality of π′. Once we have established the cofinality of both π and π′, the cofinality of σ̃ follows immediately.

The calculations following (12), together with the preservation properties of σ̃ guarantee that W′ = σ̃(W̃) is a generalized Dodd solidity witness for γ_{M′} with respect to M′ and p_{M′} − α′, thus proving (13). Now the statement

(∃W′)(W′ is a generalized Dodd solidity witness for γ_{M′} with respect to M′ and p_{M′} and M′)

is easily Σ_{2}, so (10) then immediately yields that M contains some generalized Dodd solidity witness for γ_{M} with respect to p_{M} and M. This completes the discussion in Case 1. ⊓ ⊔(Case 1)

Case 2: 𝒫(γ_{M}) ∩ Ult^{∗}(M, E_{γ_{M}}) = 𝒫(γ_{M}) ∩ Ult(M, E_{γ_{M}}).

Let σ : W → M be the canonical witness map, that is, the uncollapsing map associated with h_{M}^{∗}(γ_{M} ∪ {p_{M} − (γ_{M} + 1)}). Then γ_{M} = cr(σ), so γ_{M} = (λ_{M}^{∗})^{+W} and W is a potential premouse which fails to satisfy the initial segment condition at λ_{M}^{∗} = λ_{W}^{∗} =^{def} λ^{∗}. Notice that the coherent structure determined by E_{top}^{W} | λ_{W}^{∗} is just M || γ_{M}; this follows from the fact that σ ↾ λ^{∗} = id. Fix the following data:

• π_{0} : J_{τ}^{E} → J_{γ_{M}}^{E} is the canonical ultrapower embedding by E_{top}^{W} | λ^{∗};

• π_{W} : J_{τ}^{E} → J_{ν_{W}}^{E} is the canonical ultrapower embedding by E_{top}^{W};
• σ_{W} : M || γ_{M} → W is the canonical map defined by
σ_{W} : π_{0}(f)(ζ) ↦ π_{W} (f)(ζ)
for f ∈ ^{κ}J_{τ}^{E} ∩ J_{τ}^{E} and ζ < λ^{∗}.

Obviously, σ_{W} maps J_{γ_{M}}^{E} cofinally into J_{ν_{W}}^{E}. As λ^{∗} is a cutpoint of E_{top}^{W}, we see that cr(σ_{W}) = λ^{∗} and σ_{W} (λ^{∗}) = λ(W). Notice also that each ζ < ν_{W} is of the form σ_{W} (f)(ξ) for some f : λ^{∗} → γ_{M} that is an element of J_{γ_{M}}^{E}, and some ξ < λ(W). (By the cofinality of π_{W}, choose ζ̄ so that σ_{W} (ζ̄) > ζ and some f̄ ∈ J_{τ}^{E} which maps κ surjectively onto ζ̄. Letting f = π_{0}(f̄), we see that σ_{W} (f) = π_{W} (f̄) is a surjection of λ(W) onto σ_{W} (ζ̄), so one can find some ξ as required above.) Letting H̄ be the extender at (λ^{∗}, λ(W)) derived from σ_{W}, the above observations allow us to conclude that

(14) J_{ν_{W}}^{E^{W}} = Ult(J_{γ_{M}}^{E} , H̄) and σ_{W} is the associated ultrapower embedding,

DODD PARAMETERS AND λ-INDEXING OF EXTENDERS 17

so Q̄ = ⟨J_{ν_{W}}^{E} , H̄⟩ is a coherent structure (see the introduction). Since γ_{M} = (λ^{∗})^{+W}, the extender H̄ is weakly amenable with respect to W and it is then easy to verify that Q̄ is actually a potential premouse. Moreover, Q̄ is a type A potential premouse: Indeed, the fact that λ^{∗} is the largest cutpoint of E_{top}^{W} guarantees that there are cofinally many ordinals below λ(W) that are of the form π_{W} (f)(λ^{∗}) for some f ∈ ^{κ}κ ∩ J_{τ}^{E}. Thus, the ordinals of the form σ_{W} (g)(λ^{∗}) where g ∈ ^{λ^{∗}}λ^{∗} ∩ J_{γ_{M}}^{E} also constitute a cofinal subset of λ(W), since the functions g include all functions of the form π_{0}(f) where f is as above. This allows us to conclude:

Q̄ = ⟨J_{ν_{W}}^{E} , H̄⟩ is a type A premouse.

The amenability of both structures ⟨J_{γ_{M}}^{E} , E_{top}^{W} | λ^{∗}⟩ and W together with (14) also yield that W = Ult(⟨J_{γ_{M}}^{E} , E_{top}^{W} | λ^{∗}⟩, H̄) and σ_{W} is the associated ultrapower embedding. So W is fully determined by ⟨J_{γ_{M}}^{E} , E_{top}^{W} | λ^{∗}⟩ and H̄. Also, ⟨J_{γ_{M}}^{E} , E_{top}^{W} | λ^{∗}⟩, being a coherent structure, is fully determined by J_{τ}^{E} and E_{top}^{W} | λ^{∗} = E_{top}^{M} | λ^{∗}. Thus, to prove that W ∈ M, it suffices to verify that both E_{top}^{M} | λ^{∗} and H̄ are elements of M. That E_{top}^{M} | λ^{∗} ∈ M is an immediate consequence of the initial segment condition for M. To see that H̄ ∈ M, we show

(15)  Q̄ ∈ M.

Let • π_{M} : J_{τ}^{E} → J_{ν_{M}}^{E} be the canonical ultrapower embedding by E_{top}^{M};

• σ_{M} : M || γ_{M} → M be the canonical map defined by

σ_{M} : π_{0}(f)(ζ) ↦ π_{M}(f)(ζ)
for f ∈ ^{κ}J_{τ}^{E} ∩ J_{τ}^{E} and ζ < λ^{∗}.

• H be the extender at (λ^{∗}, λ(M)) derived from σ_{M}. A discussion similar to that above shows that

(16)  Q = ⟨J_{ν_{M}}^{E} , H⟩ is a coherent structure

and

(17) M = Ult(M || γ_{M}, H) and σ_{M} is the associated ultrapower embedding.

Since γ_{M} < (λ^{∗})^{+M}, the extender H is not weakly amenable with respect to M. This means that Q is not a potential premouse. Instead, Q is a protomouse in the

18 MARTIN ZEMAN

sense of [17, 12] or [13]. The extenders H̄ and H are extracted from E_{top}^{W} and E_{top}^{M} respectively, so the structures Q̄ and Q cannot carry more information than W and M. (Actually, Q and M carry the same amount of information.) As a consequence we get the following, with respect to the language for coherent structures:

(18)  σ : Q̄ → Q is Σ_{0}-preserving and cofinal.

Indeed, σ can be viewed as a map from Q̄ to Q, since Q̄ has the same domain as W and Q has the same domain as M. This also guarantees the cofinality of σ. Thus, to see that σ is Σ_{0}-preserving, it suffices to show that σ(H̄ ∩ x) = H ∩ σ(x) for all sufficiently large x ∈ Q̄. By the cofinality of π_{W}, the set x can be chosen of the form J_{ζ}^{E^{W}} where ζ = π_{W} (ζ̄) for some ζ̄ < τ. In this case H̄ ∩ x is of the form {⟨σ_{W} (g)(ξ) ∩ λ^{∗}, σ_{W} (g)(ξ)⟩; ξ < λ^{∗}} for any surjection g : λ^{∗} → 𝒫(λ^{∗}) ∩ J_{ζ_{0}}^{E} that is an element of M || γ_{M}; here ζ_{0} = π_{0}(ζ̄). In particular, g can be chosen of the form π_{0}(f) where f : κ → 𝒫(κ) ∩ J E ; ζ̄ is any surjection that is an element of J_{τ}^{E}. Then σ_{W} (g) = π_{W} (f), and we have

(19)  H̄ ∩ x = {⟨π_{W} (f)(ξ) ∩ λ^{∗}, π_{W} (f)(ξ)⟩; ξ < λ^{∗}}.

The abuse of notation we introduced immediately before the beginning of the proof of Theorem 1.1 allows us to write E_{top}^{W}(f) instead of π_{W} (f), so (19) can be rewritten as H̄ ∩ x = {⟨E_{top}^{W}(f)(ξ) ∩ λ^{∗}, E_{top}^{W}(f)(ξ)⟩; ξ < λ^{∗}}, and the following calculation can be easily verified:

σ(H̄ ∩ x) = σ({⟨E_{top}^{W}(f)(ξ) ∩ λ^{∗}, E_{top}^{W}(f)(ξ)⟩; ξ < λ^{∗}})
= {⟨σ(E_{top}^{W}(f))(ξ) ∩ λ^{∗}, σ(E_{top}^{W}(f))(ξ)⟩; ξ < λ^{∗}}
= {⟨E_{top}^{M}(f)(ξ) ∩ λ^{∗}, E_{top}^{M}(f)(ξ)⟩; ξ < λ^{∗}}
= H ∩ σ(x).

The second equality here follows from the preservation properties of σ and the fact that cr(σ) = γ_{M}. To see the third equality, notice that σ(E_{top}^{W}(f)) = E_{top}^{M}(f). The last clause is a consequence of the fact that E_{top}^{M}(f) = π_{M}(f) is a surjection of λ(M) onto 𝒫(λ(M)) ∩ J_{σ(ζ)}^{E} = 𝒫(λ(M)) ∩ σ(x), so it enables us to isolate all x ⊂ λ(M) from σ(x) which are in the rangle of H in the same way we did it for H̄ in (19).

DODD PARAMETERS AND λ-INDEXING OF EXTENDERS 19

For the argument we are going to do below, we will need the following fine structural fact about Q̄. Let p̄ = σ^{−1}(p_{M} − (γ_{M} + 1)). Then

(20)  hQ̄(γ_{M} ∪ {p̄}) = Q̄, so ωϱ1 ; Q̄ ≤ γ_{M}.

The proof of (20) contains an ingredient from the proof of (30), namely a construction of an auxiliary funcion f. Although that proof comes later in this paper, we still prefer to give the details of the construction of f there rather than here. Recall that h∗ ; Q̄ is the same as hQ̄ since Q̄ is a type A premouse and W = h_{W}^{∗} (γ_{M} ∪{p̄}). Fix an ordinal ζ < ν_{W}. As σ_{W} is cofinal, the argument from the proof of (30) yields a function f : ^{1+|p̄|}λ^{∗} → γ_{M} that is an element of J_{γ_{M}}^{E} and such that ζ = σ_{W} (f)(ξ, p̄) for some ξ < γ_{M}. Now f(ξ) = otp(g(ξ)) for a suitable g : λ^{∗} → 𝒫(λ^{∗}) that is an element of J_{γ_{M}}^{E}, so our abuse of notation (see the previous paragraph) enables us to write ζ = otp(H̄(g)(ξ, p̄)). This tells us that ν_{W} = On ∩ Q̄ ⊂ hQ̄(γ_{M} ∪ {p̄}), so hQ̄(γ_{M} ∪ {p̄}) = Q̄. ⊓ ⊔(20)

The conclusion (18) combined with the fact that Q and M have the same domain imply that ⟨M, Q̄, γ_{M}⟩ is a good phalanx in the sense of [17], Section 9.1. Our strategy now is to compare this phalanx with M and use standard comparison techniques to infer that Q̄ ∈ M, and thereby complete the proof of Lemma 1.6. This is not possible verbatim, as it is not clear that the phalanx is embeddedable into a sufficiently iterable premouse in the manner described in [17], Section 9.2. For this reason, the entire comparison argument needs certain amendments which are described below. The first step in our argument is to show that the phalanx is coiterable with M. In fact, we show that the phalanx is normally (ω_{1} + 1)-iterable. We show this by embedding iterations of ⟨M, Q̄, γ_{M}⟩ into slight modifications of iterations of M. In order to describe how these modifications will be formed, we need the following notation.

Given a type B premouse P, we define a protomouse Q(P) in the same manner we defined Q in (16). More precisely, letting σ_{P} : P || γ_{P} → P be the canonical map defined in an analogous way as σ_{M}, the structure Q(P) is of the form ⟨J_{ν_{P}}^{E^{P}} , H_{P} ⟩ where ν_{P} = ht(P) and H_{P} is a (λ_{P}^{∗} , λ(P))-extender derived from σ_{P}. The structures P and Q(P) have the same domain.

20 MARTIN ZEMAN

Lemma 1.8. Let P be a type B premouse and F_{P} be a total extender on P such that P′ = Ult(P, F_{P}) exists (that is, the corresponding ultrapower is well-founded; P′ is transitive). Let π : P → P′ be the associated ultrapower embedding. Then Q(P′) = Ult(Q(P), F_{P}) and π : Q(P) → Q(P′) is the associated ultrapower embedding. Proof. Recall also P and Q(P) have the same domain, and the same applies to P′ and Q(P′). So π is clearly an ultrapower map when viewed as a map between the domains of Q(P) and Q(P′). Thus, it suffices to show that π retains the preservation properties when viewed as a map between Q(P) and Q(P′). This amounts to showing that π(H_{P} ∩ x) = H_{P′} ∩ σ(x) for all x ∈ P and can be proved the same way as (18). ⊓ ⊔(Lemma 1.8)

Write (Q̄_{−1}, Q̄_{0}) for (M, Q̄), so from now on we can write ⟨Q̄_{−1}, Q̄_{0}, γ_{M}⟩ instead of ⟨M, Q̄, γ_{M}⟩. We have seen that Q̄ is embedded into Q(M) via σ. Now if Q̄_{i} is a model of some iteration of ⟨Q̄_{−1}, Q̄_{0}, γ_{M}⟩ that is on the same branch as Q̄_{0} = Q̄ and there is no truncation point on this branch, the critical point of the iteration map π̄_{0,i} : Q̄ → Q̄_{i} is strictly above γ_{M}, so λ^{∗}^{+Q̄_{i}} = γ_{M}. If M̃_{i} is the corresponding model of the iteration of M that arises in the course of the copying construction then the critical point of the iteration map π̃_{0,i} : M → M̃_{i} is also above γ_{M}, but γ_{M} is not a cardinal in M and M̃_{i}. The relationship between Q̄_{i} and M̃_{i} resembles to that between Q̄ and M, and we will embed Q̄_{i} into Q(M̃_{i}). In all other cases we can imitate the usual copying construction; the treatment of anomalies, however requires a bit of extra care, too.

Let us now proceed with the construction. We shall give a recursive definition of a normal iteration strategy 𝕊̄ for ⟨Q̄_{−1}, Q̄_{0}, γ_{M}⟩. Following the standard approach, this iteration strategy will be a pullback of 𝕊 under σ, but we have to be a bit careful about the copy maps (recall that 𝕊 was fixed at the beginning of this section). Let ℑ̄ be a normal iteration of ⟨Q̄_{−1}, Q̄_{0}, γ_{M}⟩ according to 𝕊̄ with iteration indices ν̄_{i}, critical points κ̄_{i}, premice Q̄_{i}, iteration maps π̄_{ij} and the associated tree ordering <ℑ̄. Starting from ℑ̄ and the embeddings id : Q̄_{−1} → M and σ : Q̄ → Q, we define a normal iteration ℑ̃ of M according to 𝕊 with iteration indices ν̃_{i}, critical points κ̃_{i}, premice M̃_{i}, iteration maps π̃_{ij} (so M̃_{0} = M) and the associated tree ordering <ℑ̃ together with “copy” maps σ_{i} such that:

(a) σ_{−1} = id : Q̄_{−1} → M̃_{0} and σ_{0} = σ : Q̄_{0} → Q(M̃_{0});

DODD PARAMETERS AND λ-INDEXING OF EXTENDERS 21

(b) σ_{i} : Q̄_{i} → Q(M̃_{i}) is Σ_{0}-preserving and cofinal whenever 0 ≤ℑ̄ i and there is no truncation point on the branch [0, i]ℑ̄;

(c) σ_{i} : Q̄_{i} → M̃_{i} if at least one of the conditions in (b) fails and κ̄_{i−1} ≠ λ^{∗}; in this case, σ_{i} is fully Σ^{∗}-preserving;

(d) σ_{i} : Q̄_{i} → M̃_{i} = Ult(M̃_{0} || γM̃_{0}, E_{top}^{Q(M̃_{i−1})}) = M̃_{i−1} if

(∗) 0 ≤ℑ̄ i − 1, there is no truncation point on the branch [0, i − 1]ℑ̄ and

ν̄_{i−1} = ht(Q̄_{i−1}); in this case κ̄_{i−1} = λ^{∗} as π̄_{0,i−1} ↾ (γ_{M} + 1) = id, and i is a strong anomaly in the sense of [17], Section 9.1; it follows that {−1, i} is a maximal branch in ℑ̄. Regarding the equality M̃_{i} = M̃_{i−1}, see below.

(e) σ_{i} : Q̄_{i} → π̃_{0,i}(M || γ_{M}) is Σ_{0}-preserving and cofinal if the following two

conditions are satisfied: – κ̄_{i−1} = λ^{∗} and

– the condition (∗) from (d) fails; as in (d), i is a strong anomaly, so {−1, i} is a maximal branch in ℑ̄.

Clauses (a), (c) and (e) can be verified similarly as in the case where we have a standard embedding of a phalanx into a premouse (see [17], Lemma 9.2.9). Clause (b) is a consequence of Lemma 1.8. Let ξ_{i−1} be the immediate <ℑ̄-predecessor of i. If σ_{ξ_{i−1}} : Q̄_{ξ_{i−1}} → Q(M̃_{ξ_{i−1}}) and σ_{i−1} are already given, the lemma allows us to define the embedding σ_{i} : Q̃_{i} → Q(M̃_{i}) in the standard way. To see (d), notice that (∗) implies that the extender EQ(M̃_{i−1}) ; ν̃_{i−1} = E_{top}^{Q(M̃_{i−1})} is not total on M_{0}, and the result of corresponding ultrapower is just M̃_{i−1}; this follows from the fact that M̃_{i−1} agrees with M̃_{0} up to γM̃_{0} + 1. Strictly speaking, ℑ̃ is not a normal iteration, as it may contain branches of the form {0, i} whenever EQ(M̃_{i−1}) ; ν̃_{i−1} is a top extender of Q(M̃_{i−1}). But obviously, by removing such branches from ℑ̃, we obtain a normal iteration. All the remaining issues can be handled in the standard way.

The copying construction described above provides us with the obvious definition of 𝕊̄, namely 𝕊̄(ℑ̄) = 𝕊(ℑ̃) whenever ℑ̄ is an iteration of limit length. This gives the normal iterability of the phalanx ⟨M, Q̄, γ_{M}⟩, and thereby its coiterability with M. The coiteration of the phalanx with M terminates in, say, θ + 1 < ω_{1} steps. To keep the notation visually consistent with that in the previous paragraph, let ℑ̄ be the iteration on the phalanx side, let ℑ̃ be its copy as described above and let ℑ be

22 MARTIN ZEMAN

the iteration on the M-side. Denote the premice in ℑ by M_{i} and the iteration maps in ℑ by π_{ij}. Thus, the last models in ℑ̄, ℑ̃ and ℑ are Q̄_{θ}, Q̃_{θ} and M_{θ}, respectively. Also, ν̄_{i} are the indices of the coiteration.

The application of the Neeman-Steel Lemma as described in [17], Lemma 9.2.12, goes literally through even in the present context; the details are left to the reader. We thus obtain:

• 0 ≤ℑ̄ θ;

• Q̄_{θ} is an initial segment of M_{θ};

• [0, θ]ℑ̄ has no truncation points.

Notice that if Q̄_{θ} = M_{θ}, then [0, θ]_{ℑ} necessarily contains a truncation point. This follows from the fact that if the iteration maps π̄_{0,θ} : Q̄_{0} → Q̄_{θ} and π : M_{0} → M_{θ} are total, then they preserve the premice type. Now M is a type B premouse, so M_{θ} would be of type B as well. On the other hand, Q̄_{0} = M is a type A premouse, so Q̄_{θ} is of type A. So we have:

Either Q̄_{θ} is a proper initial segment of M_{θ} or else there is a truncation point on the main branch of ℑ.

Now we can proceed in the usual way. If Q̄_{θ} is a proper initial segment of M_{θ}, then Q̄_{θ} is sound. So the phalanx side of the coiteration is not moved, as otherwise (20) would imply that Q̄_{θ} fails to be sound. This means that Q̄_{θ} = Q̄_{0} = Q̄. Notice that ν̄_{0} = γ_{M}, as E_{γ_{M}}^{M} ≠ ∅ = EQ̄ ; γ_{M}. Assume there is a disagreement between the extender sequences of M_{1} and Q̄. Then ν̄_{1} is strictly larger than γ_{M} and is a cardinal in M_{θ}. Since Q̄ projects to γ_{M}, we see that Q̄ is actually a proper inital segment of M_{θ} || ν̄_{1} = M_{1} || ν̄_{1}, a contradiction. So Q̄ must be a proper initial segment of M_{1} and θ = 1. By (20), Q̄ can be encoded into a subset b of γ_{M} that is Σ_{1}(Q̄), and therefore into an element of M_{1} = Ult^{∗}(M, E_{γ_{M}}). Now recall that we are in Case 2, so b ∈ Ult(M, E_{γ_{M}}). We want to see that b ∈ M, as this will enable us to decode b inside M and conclude that Q̄ ∈ M. The argument is described in [17], proof of Theorem 9.3.1, Case 1. The point is that in M, there are sequences ⟨g_{ξ}; ξ < γ_{M}⟩ and ⟨α_{ξ}; ξ < γ_{M}⟩ such that g_{ξ} : κ → τ, α_{ξ} < λ_{M}^{∗} and ξ = π̂(g_{ξ})(α_{ξ}) where π̂ is the ultrapower embedding associated with Ult(M, E_{γ_{M}}). Since E_{γ_{M}} ∈ M and b = π̂(f)(α̂) for some f and α̂ that also are elements of M, the statement “ξ ∈ b”

DODD PARAMETERS AND λ-INDEXING OF EXTENDERS 23

can be expressed as π̂(g_{ξ})(α_{ξ}) ∈ π̂(f)(α̂). Using the Łoś Theorem, this can be expressed internally in M.

Now assume that Q̄_{θ} = M_{θ}. Let i > 0 be the least such that EM_{i} ; ν̄_{i} is an extender. Notice that i exists, as there must be a truncation point on the main branch of ℑ (so θ > 1). Obviously, M_{i} = M_{1} so ν̄_{i} indexes an extender on the M_{1}-sequence (possibly the top one). Also, let j + 1 be the last truncation point on that branch. The set b encoding Q̄ is Σ_{1}(Q̄), so by standard arguments, b is Σ_{1}(M_{j}^{∗}) where M_{j}^{∗} is the result of the last truncation on the main branch in ℑ. So b ∈ M_{ξ_{j}} where ξ_{j} is the immediate <_{ℑ}-predecessor of j + 1. Now either ξ_{j} = 1 or else ξ_{j} > 1, in which case ν̄_{i} is a cardinal in M_{ξ_{j}}. In either case, b is an element of M_{1}, due to the agreement between M_{1} and M_{ξ_{j}}. Now apply the assumption of Case 2, and as in the previous case conclude that b ∈ M and Q̄ ∈ M. ⊓ ⊔(Lemma 1.6)

From now on we can assume

(21)  e_{M}^{α} ≠ {γ_{M}}.

By (5), e_{M}^{α} ⊂ λ_{M}^{∗} + 1. Lemma 1.9. Assume that λ_{M}^{∗} ∈ e_{M}^{α}. Then ^{∗}W _{M}^{λ_{M}^{∗} ,(p_{M} −α)∪e_{M}^{α}} ∈ M. Proof. By (6), λ_{M}^{∗} ∈ e_{M}^{α} just in case that λ_{M}^{∗} = max(p_{M} ∪ e_{M}^{α}). Letting W = ∗W _{M}^{λ_{M}^{∗} ,(p_{M} −α)∪e_{M}^{α}} and σ : W → M be the canonical witness map, we see that cr(σ) = λ_{M}^{∗} and σ(λ_{M}^{∗}) = λ(M). This follows from the fact that λ_{M}^{∗} is the largest cutpoint of E_{top}^{M}. So W is a coherent structure with λ(W) = λ_{M}^{∗} and E_{top}^{W} = F | λ_{M}^{∗}. By the initial segment condition, F | λ_{M}^{∗} = E_{γ_{M}}, so W = M || γ_{M} ∈ M.

⊓ ⊔(Lemma 1.9)

This takes care of the top part of the Dodd parameter above λ_{M}^{∗}, and we have:

d_{M}^{α} − λ_{M}^{∗} = (p_{M} ∪ e_{M}^{α}) − λ_{M}^{∗} and ^{∗}W _{M}^{β,d_{M}^{α}} ∈ M whenever β ∈ d_{M}^{α} − λ_{M}^{∗}.

Recall that we are in the process of verifying (3) and that we have already done the job for β ≥ λ_{M}^{∗}. From now on we will assume:

β < λ_{M}^{∗} and β ∈ (p_{M} − α) ∪ e_{M}^{α} (22) d_{M}^{α} − (β + 1) = (p_{M} ∪ e_{M}^{α}) − (β + 1) (23)

(24) ^{∗}W _{M}^{δ,d_{M}^{α}} ∈ M whenever δ ∈ d_{M}^{α} − (β + 1).

24 MARTIN ZEMAN

We will first discuss the case where β ∈ p_{M} − α. Lemma 1.10. Assume β ∈ p_{M} − α and (22) – (24) hold. Then ^{∗}W _{M}^{β,p_{M} ∪e_{M}^{α}} ∈ M. Proof. Let W = W_{M}^{β,p_{M}} and let σ : W → M be the canonical witness map. Let further

e_{1} = e_{M}^{α} − β and e_{2} = e_{M}^{α} ∩ β.

Then σ(β) > β. By the definition of e_{M}^{α}, there is some finite a ⊂ α such that

(25) (∃e)(∃β̄)(e ∈ [On]^{<ω} & β̄ < σ(β) & γ_{M} = h_{M}^{∗}(a, {p_{M} − {β}, β̄, e, e_{2}});

this is witnessed by e_{1} and β. Since σ is Σ_{1}-preserving with respect to the language for coherent structures (in fact, with respect to the language for premice) and the objects γ_{M}, p_{M} − {β}, a and e_{2} are in the range of σ, there is an e ∈ rng(σ) witnessing (25). Assume e is the <^{∗}-least such.

Claim. e = e_{1}, so e_{M}^{α} ∈ rng(σ). Proof. Suppose this is false. There are two possibilities. First consider the possibility e_{1} <^{∗} e. If this happens then (25) can be rewritten as

(∃ē)(∃β̄)(ē <^{∗} e & β̄ < σ(β) & γ_{M} = h_{M}^{∗}(a, {p_{M} − {β}, β̄, ē, e_{2}}).

It would follow that rng(σ) contains some finite set of ordinals ē <^{∗} e witnessing (25), which contradicts the minimality of e.

Now consider the case where e <^{∗} e_{1}. Fix δ ∈ e_{1} − e such that e_{1} − (δ + 1) = e − (δ + 1). Let ^{∗}W = ^{∗}W _{M}^{δ,p_{M} ∪e_{M}^{α}} and σ^{∗} : ^{∗}W → M be the canonical witness map. Since δ > β, we can apply (23) and (24) and conclude that W^{∗} ∈ M. On the other hand, the definition of δ guarantees that p_{M} ∪ e ⊂ rng(σ^{∗}). Since σ^{∗} is Σ_{1}-preserving with respect to the language for coherent structures, we see that both p_{M} and γ_{M} = h_{M}^{∗}(a, {p_{M} − {β}, β̄, e, e_{2}}) are in rng(σ^{∗}); here β̄ < β comes again from (25). But this means that rng(σ^{∗}) ⊃ h_{M}^{∗}(α, p_{M} ∪ {γ_{M}}) = M. So ∗W = M, a contradiction again. ⊓ ⊔(Claim)

Let q be such that σ(q) = (p_{M} ∪ e_{M}^{α}) − (β + 1). The Dodd solidity witness ∗W _{M}^{β,p_{M} ∪e_{M}^{α}} is the transitive collapse of

Y = h_{M}^{∗}(β ∪ {(p_{M} ∪ e_{M}^{α}) − (β + 1)}).

DODD PARAMETERS AND λ-INDEXING OF EXTENDERS 25

By the preservation properties of σ, the structures (Y, ∈) and (h_{W}^{∗} (β ∪ {q}), ∈) are isomorphic (this notation suppresses the predicates), so^{∗}W _{M}^{β,p_{M} ∪e_{M}^{α}} is the transitive collapse of h_{W}^{∗} (β ∪ {q}). Since M is solid, we know that W ∈ M. Now M is active, so its domain is a ZFC^{−}-model. It follows immediately that both h_{W}^{∗} (β ∪ {q}) and ∗W _{M}^{β,p_{M} ∪e_{M}^{α}} are in M. ⊓ ⊔(Lemma 1.10)

It remains to discuss the case where β ∈ e_{M}^{α}. This will require some effort. In a series of lemmata, we first show that the top extender of ^{∗}W _{M}^{β,p_{M} ∪e_{M}^{α}} can be factored into two extenders; one of them will be an element of M and the other one will satisfy the initial segment condition. We then apply a comparison argument of the same kind we did in Case 2 in the proof of Lemma 1.6 to conclude that this extender is in M. We will also see that such a factoring implies the existence of a cardinal in M that is both subcompact and superstrong. Thus, under a suitable (and relatively modest) anti-large cardinal assumption, we will be able to conclude that e_{M}^{α} ∩ λ_{M}^{∗} = ∅, which will prove Proposition 1.3.

In addition to (21) – (24), from now on we assume:

β ∈ e_{M}^{α} (26)

(27) W = ^{∗}W _{M}^{β,p_{M} ∪e_{M}^{α}} and σ : W → M is the canonical witness map γ = (σ^{−1})′′γ_{M} (28)

Obviously, W is a type B potential premouse and σ(λ_{W}^{∗}) = λ_{M}^{∗}. Lemma 1.11. γ < (λ_{W}^{∗})^{+W}. Proof. Obviously, γ ≤ (λ_{W}^{∗})^{+W}. Assume for a contradiction that γ = (λ_{W}^{∗})^{+W}. Since σ maps γ cofinally into γ_{M}, we can apply the Interpolation Lemma ([17], Lemma 3.6.10) to the embedding σ : W → M and find a coherent structure M̃, and maps σ̃ and σ′ such that:

• σ̃ : W → M̃ is Σ_{0}-preserving with respect to the language for coherent structures and cofinal;

• σ′ : M̃ → M is Σ_{0}-preserving with respect to the language for coherent structures;

• σ̃ ↾ J_{γ}^{E^{W}} = σ ↾ J_{γ}^{E^{W}};

• cr(σ′) = γ_{M} and σ′(γ_{M}) = (λ_{M}^{∗})^{+M};

26 MARTIN ZEMAN

• σ′ ∘ σ̃ = σ.

The definition of σ guarantees that (p_{M} ∪ e_{M}^{α}) − {β} is contained in rng(σ), and therefore also in rng(σ′). But since cr(σ′) = γ_{M} > β ≥ α, we also have the inclusion α ∪ {β} ⊂ rng(σ′). It follows that α ∪ {p_{M} ∪ e_{M}^{α}} ⊂ rng(σ′). Now the fact that cr(σ′) > τ (recall that τ = κ^{+M} where κ = cr(F)) implies that σ′ : M̃ → M is cofinal, and therefore Σ_{1}-preserving with respect to the language for coherent structures. From this we immediately infer that rng(σ′) = h_{M}^{∗}(α∪{p_{M} ∪e_{M}^{α}}) = M, and obtain a contradiction with the next-to-last clause above, which tells us that rng(σ′) has a critical point. ⊓ ⊔(Lemma 1.11)

Before proceeding further, we observe:

(29)  γ_{M} ∉ rng(σ).

Otherwise γ_{M} ∈ h_{M}^{∗}(α ∪ {p_{M} ∪ (e_{M}^{α} − {β}) ∪ e}) for some finite e ⊂ β. But (e_{M}^{α} − {β}) ∪ e <^{∗} e_{M}^{α}, which contradicts the minimality of e_{M}^{α}. ⊓ ⊔(29)

Let us also point out that (29) implies the failure of the Initial Segment Condition for W. If W satisfied the Initial Segment Condition then E_{top}^{W} | λ_{W}^{∗} ∈ W, and the preservation properties of σ would yield σ(E_{top}^{W} | λ_{W}^{∗}) = F | λ_{M}^{∗}. Since J_{τ}^{E} is obviously in the range of σ, also γ_{M} = ht(Ult(J_{τ}^{E}, F | λ_{M}^{∗})) would end up in rng(σ), which is impossible.

Recall that λ_{W}^{∗} is the largest cardinal in J_{γ}^{E^{W}}. Let N be the level of W collapsing γ and n ∈ ω be such that ωϱ_{N}^{n+1} ≤ λ_{W}^{∗} < ωϱ_{N}^{n}; these objects exist by Lemma 1.11. Our observation (29) guarantees that σ(γ) > γ_{M}. Letting N′ = σ(N), we now apply the interpolation lemma to the embedding σ ↾ N : N → N′. We obtain a transitive structure Ñ together with maps σ̃ and σ′ satisfying:

• σ̃ : N → Ñ is Σ_{0}^{(n)}-preserving with respect to the language for coherent structures and cofinal at the level n;

• σ′ : Ñ → N′ is Σ_{0}^{(n)}-preserving with respect to the language for coherent structures;

• σ′ ∘ σ̃ = σ;

• σ̃ ↾ J_{γ}^{E^{N}} = σ ↾ J_{γ}^{E^{N}};

• γ_{M} = σ̃(γ) = (λ_{M}^{∗})^{+Ñ} (this has a meaning also when γ = ht(N));

• σ′ ↾ γ_{M} = id;

DODD PARAMETERS AND λ-INDEXING OF EXTENDERS 27

• cr(σ′) = γ_{M} and σ′(γ_{M}) = σ(γ) =^{def} γ′ whenever γ < ht(N).

For n > 0, the cofinality of σ̃ at the level n follows from the soundness of N (soundness with respect to the language for premice, of course), as it implies that R_{N}^{n}, when computed in the language for coherent structures, is nonempty ([17], Lemma 3.6.3 (h)). The second clause is then an immediate consequence of the cofinality of σ̃ at the level n ([17], Lemma 3.6.10). For n = 0, the cofinality of σ̃ follows directly from the pseudoultrapower construction.

Fix the following notation. Assuming that N is active, we let • μ = cr(E_{top}^{N}) and μ̃ = σ̃(μ);

• ϑ = μ^{+N}, ϑ′ = σ̃(ϑ) and ϑ̃ = sup(σ̃′′(ϑ));

• G = E_{top}^{N}, G̃ = EÑ ; top and G′ = E_{top}^{N′}. Lemma 1.12. N is active, ht(N) = γ, ϑ = β and E_{β} ≠ ∅. Proof. First observe that β = cr(σ). This is standard: If not, then β ∈ h_{M}^{∗}(β ∪ {p_{M} ∪ (e_{M}^{α} − {β})}), so h_{M}^{∗}(β ∪ {p_{M} ∪ (e_{M}^{α} − {β})}) ⊃ h_{M}^{∗}(β ∪ {p_{M} ∪ e_{M}^{α}}) = M. We then arrive at a contradiction exactly as in the proof of (29).

We will split the proof into a sequence of claims in which we rule out all possibilities that are incompatible with the statement of the lemma.

Claim 1. N is active, n = 0 and σ̃′′ϑ is bounded in ϑ′. Proof. Assume for a contradiction that this fails, so either N is passive or n > 0 or σ̃′′ϑ is cofinal in ϑ′. We first show that γ < ht(N). This is clear if N is passive, as J_{γ}^{E^{N}} is a ZFC^{−}-model. Now suppose that N is active and γ = ht(N), that is, E_{γ}^{W} ≠ ∅. In this case we have n = 0, so our assumption implies that σ′′ϑ = σ̃′′ϑ is cofinal in ϑ′. Recall that W ⊧ cf(γ) = cf(ϑ), that is, in W there is a strictly monotonic function f : ϑ → γ that is cofinal in γ. Then σ(f) : ϑ′ → γ′ is strictly monotonic and cofinal in γ′, so

γ′ = sup(σ(f)′′ϑ′) = sup(σ(f)′′(σ′′ϑ)) = sup(σ′′(f′′ϑ)) = sup(σ′′γ) = γ_{M}.

This contradicts the fact that γ′ > γ_{M}. The second equality here follows from the cofinality of σ′′ϑ in ϑ′. So indeed γ < ht(N) and, consequently, γ_{M} = cr(σ′).

As a next step we show Ñ is a premouse, and if N is active then Ñ is of the same type as N and N′. The key point here is to verify that G̃ is a total extender on Ñ. This is a Π_{2}-property. If n > 0 then G̃ is total on Ñ, as σ̃, being a fine pseudoultrapower embedding, is Σ_{2}-preserving ([17], Lemma 3.6.3 (d)). If σ̃′′ϑ is cofinal in ϑ′, a standard argument yields

dom(G̃) = ⋃{σ̃(x); x ∈ J_{ϑ}^{E^{N}} & x ⊂ 𝒫(μ)} = 𝒫(μ̃) ∩ Ñ.

So far we have seen that Ñ is a potential premouse. That Ñ is a premouse of the same type as N and N′ (this includes the verification of the initial segment condition) follows from the preservation properties of σ̃ with respect to the language for coherent structures established above and from [17], Lemma 9.1.7. It also follows that the preservation properties of maps σ̃ and σ′ stated above hold with respect to the language of premice.

The soundness of N and our choice of n guarantee that N = h̃_{N}^{n+1}(λ_{W}^{∗} ∪ {p_{N}}). This together with the solidity of N, the fact that Ñ is a result of a fine pseudoultrapower of N by σ ↾ J_{γ}^{E^{N}}, and the preservation properties of the maps σ̃ and σ′ yields:

• Ñ = h̃n+1 ; Ñ(λ_{M}^{∗} ∪ {p̃}) where p̃ = σ̃(p_{N});

• ωϱn+1 ; Ñ ≤ λ_{M}^{∗};

• Ñ is solid;

• each δ ∈ p̃ has a generalized witness with respect to p̃ and Ñ, that is an element of Ñ.

• p̃ = pÑ − λ_{M}^{∗}, so Ñ is sound above λ_{M}^{∗}. The first clause here follows from the construction of fine pseudoultrapowers (note that p_{N} ∩ λ_{W}^{∗} = ∅, as ωϱ_{N}^{ω} = λ_{W}^{∗}), and immediately yields the second clause. The third clause follows from the second clause and from the first part of the Condensation Lemma applied to the embedding σ′ : Ñ → N. The fourth clause is a consequence of the preservation properties of σ̃ and the solidity of N. Finally, the last clause is a consequence of the previous two ([17], Lemma 1.12.5).

We now apply the second part of the Condensation Lemma to the embedding σ′ : Ñ → N′. There are four possibilities to be discussed, and we show that each of them yields a contradiction. This will complete the proof of Claim 1. Clause (a′) in the Condensation Lemma is clearly false, as σ′ ≠ id. (We are allowed to use (a′), as N′ is sound.) To see that clause (b) fails, recall that γ_{M}, being the critical point of σ′, is a cardinal in Ñ, so EÑ ; γ_{M} = ∅. On the other hand, N′ is an initial

DODD PARAMETERS AND λ-INDEXING OF EXTENDERS 29

segment of M, so E_{γ_{M}}^{N′} = E_{γ_{M}} = F | λ_{M}^{∗} by the initial segment condition. Clause (c) is false, as Ñ is sound above λ_{M}^{∗}, but the fine ultrapower from (c) fails to be. Finally, clause (d) fails, as it would imply that ωϱω ; Ñ ≥ γ_{M}, and we have seen that Ñ projects to λ_{M}^{∗}. ⊓ ⊔(Claim 1)

Claim 2. μ < β. Proof. Notice first that μ < λ_{W}^{∗}. Otherwise ϑ > λ_{W}^{∗}, so σ̃′′ϑ would be cofinal in σ̃(ϑ), as follows from general properties of pseudoultrapowers (by Claim 1, Ñ is the result of a coarse ultrapower in this case). This would contradict Claim 1. Assume for a contradiction that μ ≥ β. Then β < ϑ < λ_{W}^{∗}. Since λ_{W}^{∗} is a cardinal in W, so is ϑ, as follows from the acceptability of W and the fact that ϑ is a cardinal in N. The agreement between σ and σ̃ below λ_{W}^{∗} together with Claim 1 guarantee that σ′′ϑ = σ̃′′ϑ is bounded in ϑ′ = σ(ϑ) = σ̃(ϑ).

Now argue as in the proof of Lemma 1.11. We apply the Interpolation Lemma to the embedding σ : W → M; this time we form a pseudoultrapower of W by σ ↾ J_{ϑ}^{E^{W}}. We obtain an acceptable structure W̃ together with maps σ_{0} and σ_{1} satisfying:

• σ_{0} : W → W̃ is Σ_{0}-preserving with respect to the language for coherent structures and cofinal;

• σ_{1} : W̃ → M is Σ_{0}-preserving with respect to the language for coherent structures;

• σ_{1} ∘ σ_{0} = σ;

• σ_{0} ↾ J_{ϑ}^{E^{W}} = σ ↾ J_{ϑ}^{E^{W}};

• cr(σ_{1}) = ϑ̃ and σ_{1}(ϑ̃) = ϑ′.

Exactly as in the proof of Lemma 1.11 we observe that σ_{1} is Σ_{1}-preserving with respect to the language for coherent structures and (p_{M} ∪ e_{M}^{α}) − {β} ∈ rng(σ_{1}). Since ϑ > β, we have cr(σ_{1}) = ϑ̃ > β. It follows that α ∪ {β} ⊂ rng(σ_{1}), so actually α ∪ {p_{M} ∪ e_{M}^{α}} ⊂ rng(σ_{1}). Thus, rng(σ_{1}) = M, which contradicts the fact that σ_{1} has a critical point. ⊓ ⊔(Claim 2)

Claim 3. ϑ = β.

30 MARTIN ZEMAN

Proof. By Claim 2, μ < β, so ϑ ≤ β (recall that β, being a critical point of σ, is a cardinal in W). If ϑ < β, we would have ϑ′ = σ(ϑ) = ϑ and σ̃′′ϑ = σ′′ϑ = ϑ, which would yield an immediate contradiction with Claim 1. ⊓ ⊔(Claim 3)

Claim 4. E_{β} ≠ ∅. Proof. Building on the previous claims, we conclude:

• σ̃ : N → Ñ and σ′ : Ñ → N are both Σ_{0}-preserving with respect to the language for coherent structures, and σ̃ is cofinal;

• cr(G̃) = μ;

• dom(G̃) = 𝒫(μ) ∩ J_{β}^{E} ⫋ 𝒫(μ) ∩ Ñ;

• Ñ is a coherent structure that is not a potential premouse. Let N^{∗} be the level of Ñ collapsing β. Obviously, β = μ^{+N^{∗}} and dom(G̃) =

𝒫(μ) ∩ N^{∗}. By the agreement between Ñ and M, the premouse N^{∗} is the level of M collapsing β, so N^{∗} is a proper initial segment of J_{β′}^{E} where β′ = σ(β). Let π′ be the ultrapower embedding associated with Ult(J_{β′}^{E}, G′) and let R′ = π(N^{∗}). Let further R̃ = Ult^{∗}(N^{∗}, G̃) with the associated map π̃. Then R′ is a level of N′, and thus a level of M. To see that R̃ exists (that is, the corresponding ultrapower is well-founded), notice that the assignment [δ, f] ↦ π′(f)(σ′(δ)) is Σ_{0}-preserving; here ⟨δ, f⟩ appears in the construction of the fine ultrapower, so δ < λ(Ñ) and f ∈ Γ(μ, N^{∗}). This assignment gives rise to the embedding k : R̃ → R′.

The map π′ ↾ N^{∗} : N^{∗} → R′ is fully elementary in the language for premice, and N^{∗} and R′ are of the same type. We would like to obtain an analogous conclusion with R̃ and π̃ in place of R′ and π′ ↾ N^{∗}. This is, however, not possible in general. Let m be such that ωϱ_{N^{∗}}^{m+1} ≤ μ < ωϱ_{N^{∗}}^{m}. Notice that, in fact, ωϱ_{N^{∗}}^{m+1} = μ = ωϱ_{N^{∗}}^{ω}.

Now we apply our induction hypothesis (2) to the premouse N′, which is a proper initial segment of M. We obtain that d_{N′} = d_{N′}^{λ_{M}^{∗}} is defined and N′ is Dodd solid. Since fine structural properties of N′ are first-order expressible over N′, they are preserved under σ. Thus, d_{N} = d_{N}^{λ_{W}^{∗}} is defined and N is Dodd solid. The general properties of fine pseudoultrapowers yield:

• hÑ(λ_{M}^{∗} ∪ {d̃}) = Ñ where d̃= σ̃(d_{N});

• W δ,d̃ ; Ñ ∈ Ñ whenever β ∈ d̃.

In the latter case, of course, we first conclude that each δ ∈ d̃ has a generalized Dodd solidity witness with respect to d̃ and Ñ that is in Ñ, and then use the

DODD PARAMETERS AND λ-INDEXING OF EXTENDERS 31

generalized witness to obtain the standard Dodd solidity witness by applying (1). (Recall: now we work in the language for coherent structures.)

Assume for a contradiction that E_{β} = ∅. Then obviously β < ht(N^{∗}), and, moreover,

λ(N^{∗}) > μ whenever N^{∗} is active.

It follows that R̃ is either passive or is a premouse of the same type as N^{∗}, and π̃ is Σ_{0}^{(m)}-preserving with respect to the language for premice and cofinal at the level m. The verification of this, as well as that of the following clauses, is standard:

• k(π̃(f)(δ)) = π′(f)(σ′(δ)) for f ∈ Γ(μ, N^{∗}) and δ < λ(Ñ);

• k ↾ J EÑ ; ν̃ = σ′ where ν̃ = ht(Ñ);

• k ↾ γ_{M} = id;

• k : R̃ → R′ is Σ_{0}^{(m)}-preserving with respect to the language for premice;

• h̃m+1 ; R̃(λ(Ñ) ∪ {π̃(p_{N^{∗}})}) = R̃;

• every δ ∈ π̃(p_{N^{∗}}) has a generalized witness with respect to R̃ and π̃(p_{N^{∗}}),

that is an element of R̃. Our next task is to show that ωϱm+1 ; R̃ ≤ λ_{M}^{∗} and R̃ is sound above λ_{M}^{∗}. This

together with the middle two clauses on the above list will enable us to apply the Condensation Lemma and derive a contradiction the same way we did in the proof of Claim 1. Toward the soundness of R̃ above λ_{M}^{∗}, we first show:

λ(G̃) ⊂ h̃m+1 ; R̃(λ_{M}^{∗} ∪ {d̃∪ π̃(p_{N^{∗}})}) (30)

(31) W δ,d̃∪π̃(p_{N^{∗}}) ; R̃ ∈ R̃ whenever δ ∈ d̃

The former together with the next-to-last clause on the list in the previous paragraph will imply

(32)  h̃m+1 ; R̃(λ_{M}^{∗} ∪ {d̃∪ π̃(p_{N^{∗}})}) = R̃,

and, consequently, that ωϱm+1 ; R̃ ≤ λ_{M}^{∗}. This inequality together with the middle two clauses from that list guarantee the solidity of R̃. Finally (32) combined with (31), the last clause on the above list, and the solidity of R̃ yield d̃∪ π̃(p_{N^{∗}}) = pR̃ − λ_{M}^{∗}, and thereby the soundness of Ñ above λ_{M}^{∗}. (That d̃∩ λ_{M}^{∗} = ∅, follows from the fact that d_{N} ∩ λ_{W}^{∗} = ∅.)

32 MARTIN ZEMAN

Clauses (30) and (31) follow from lemmata established in [12], Section 2, where a more general theory is developed. The proof of (31) makes use of the induction hypothesis (2). In order to show how (2) is used, and also to make this paper self-contained, we give the proof of both.

First focus on (30). Given a ζ < λ(Ñ), we have seen above (proof of the current claim, the fourth paragraph) that ζ = hÑ(ξ, d̃) for some ξ < λ_{M}^{∗}. Let H(z, x, y, w) be a Σ_{0}(Ñ)-relation that determines the Skolem function hÑ, i.e. such that for every x, y, w ∈ J EÑ ; ν̃ we have

y = hÑ(x, w) ⟷ (∃z)H(z, y, x, w).

We stress that now we work in the language for coherent structures, as Ñ is a coherent structure that is not a potential premouse. Fix β̄ < β such that, letting ν̄ = π̃(β̄), there is a z ∈ J ER̃ ; ν̄ = J EÑ ; ν̄ satisfying H(z, ζ, ξ, d̃). This is possible, since π̃ maps β cofinally into π̃(β) = ν̃. Define a function f̃: ^{1+|d̃|}λ(Ñ) → λ(Ñ) by

f̃(ξ′, w) = \{ \begin{matrix}the unique ζ′ < λ(Ñ) with (∃z ∈ J ER̃ ; ν̄)H(z, ζ′, ξ′, w), if defined; \\ 0, otherwise. \end{matrix}

Obviously, ζ = f̃(ξ, d̃). The function f̃ is Σ_{0}-definable over ⟨J ER̃ ; ν̄, G̃ ∩ J ER̃ ; ν̄⟩. Now J ER̃ ; ν̄ is in the range of π̃, and therefore is an element of h̃m+1 ; R̃(μ̃ ∪ {π̃(p_{N^{∗}})}). The predicate G̃∩J ER̃ ; ν̄ can be expressed as {⟨π̃(g)(η)∩μ, π̃(g)(η)⟩; η < μ} where g ∈ N^{∗} is a surjection of μ onto 𝒫(μ) ∩ J E ; β̄, so G̃ ∩ J ER̃ ; ν̄ ∈ h̃m+1 ; R̃(μ ∪ {π̃(p_{N^{∗}}), μ}). Taken

together, f̃ ∈ h̃m+1 ; R̃(μ ∪ {π̃(p_{N^{∗}}), μ}), so ζ ∈ h̃m+1 ; R̃(μ ∪ {μ, ξ, d̃∪ π̃(p_{N^{∗}})}). This

completes the proof of (30) ⊓ ⊔(30)

Now turn to the proof of (31). Let δ ∈ d̃, let W̃_{δ} = W δ,q̃ ; Ñ and let σ_{δ} : W̃_{δ} → Ñ be the canonical witness map. W̃_{δ} is computed in the language for coherent structures. (Obviously, W̃_{δ} fails to be a potential premouse for the same reason Ñ fails to be.) This map is Σ_{1}-preserving with respect to the language for coherent structures and its range is precisely hÑ(δ ∪ {d̃ − (δ + 1)}). Thus, W̃_{δ} = hW̃_{δ}(δ ∪ {d̃_{δ}}) where σ_{δ}(d̃_{δ}) = d̃− (δ + 1). Denote the top extender of W̃_{δ} by G̃_{δ}. Since σ_{δ} ↾ δ = id, the domain of G̃_{δ} is exactly 𝒫(μ) ∩ N^{∗} = dom(G̃). Let W_{δ} = Ult^{∗}(N^{∗}, G̃_{δ}) and let π̃_{δ} : N^{∗} → W_{δ} be the associated ultrapower map. The existence of W_{δ} follows from the preservation properties of the assignment [ξ, f] ↦ π̃(f)(σ_{δ}(ξ)) where ξ < λ(G̃_{δ})

DODD PARAMETERS AND λ-INDEXING OF EXTENDERS 33

and f ∈ Γ(μ, N^{∗}). This assignment gives rise to a map σ̃_{δ} : W_{δ} → R̃. We show:

(33)  W_{δ} = W δ,d̃∪π̃(p_{N^{∗}}) ; R̃.

We have seen above that R̃ is a premouse of the same type as N^{∗} and π̃ : N^{∗} → R̃ is Σ_{0}^{(m)}-preserving with respect to the language for premice and cofinal at the level m. The same argument yields the same conclusion for W_{δ} and π̃_{δ} : N^{∗} → W_{δ}. The map σ_{δ} : W_{δ} → R̃ is Σ_{0}^{(m)}-preserving with respect to the language for premice, as follows by the standard argument based on the Łoś Theorem. We also obtain the cofinality of σ̃_{δ} at the level m; this is a consequence of the cofinality of both π̃ and π̃_{δ} the level m. So σ̃_{δ} is Σ_{1}^{(m)}-preserving with respect to the language for premice. The definition of σ̃_{δ} guarantees that σ̃_{δ}(π̃_{δ}(p_{N^{∗}})) = π̃(p_{N^{∗}}) and σ̃_{δ} ↾ λ(G̃) = σ_{δ} ↾ λ(G̃); the latter implies that σ̃_{δ}(d̃_{δ}) = d̃ − (δ + 1). Now since W̃_{δ} = hW̃_{δ}(δ ∪ {d̃_{δ}}), we can imitate the proof of (32) and infer h̃_{W_{δ}}^{m+1}(δ ∪ {π̃_{δ}(p_{N^{∗}}), d̃_{δ}}) = W_{δ}. This together with the preservation properties of σ̃_{δ} just established guarantees that rng(σ̃_{δ}) = h̃m+1 ; R̃(δ ∪ {π̃(p_{N^{∗}}), d̃− (δ + 1)}), which completes the proof of (33).

Recall that we proved that W̃_{δ} ∈ Ñ. We also know that W̃_{δ} can be encoded into a Σ_{1}(Ñ) set b ⊂ δ. Since δ < λ̃ =^{def} λ(G̃), we see that b ∈ J EÑ ; λ̃ = J ER̃ ; λ̃. Now J ER̃ ; λ̃ is a ZFC^{−} model, so inside this structure we can both recover W̃_{δ} and form Ult(N^{∗}, G̃_{δ}). It follows that W δ,d̃∪π̃(p_{N^{∗}}) ; R̃ ∈ R̃. ⊓ ⊔(31)

Finally we obtained the soundness of R̃ above λ_{M}^{∗}, which allows us to turn to the application of the Condensation Lemma. The embedding in question is k : R̃ → R′ and, as we have already indicated above, we can then proceed exactly as in the proof of Claim 1, with k in place of σ′. ⊓ ⊔(Claim 4)

Claim 5. ht(N) = γ. Proof. Assume for a contradiction that the claim is false, that is, ht(N) > γ. Define R′, π′, R̃, π̃ and k as in the proof of Claim 4. The structure R̃ is an active potential premouse. The map k : R̃ → R′ is Σ_{0}-preserving with respect to the language for coherent structures and cofinal. The cofinality of k follows from the fact that E_{top}^{R′} and ER̃ ; top have the same critical point that is below μ, μ is a cardinal in both R′ and R̃ and k ↾ μ = id. Claim 4 implies that N^{∗} = Ñ || β = M || β, so λ(R̃) = π̃(μ) = λ(G̃). Since γ is the cardinal successor of λ_{W}^{∗} in the sense

34 MARTIN ZEMAN

of N, our assumption ht(N) > γ implies that λ(N) > λ_{W}^{∗}, and, consequently, λ(R̃) = λ(Ñ) > λ_{M}^{∗}.

Working in the language for coherent structures, we argue similarly as in the proof of Claim 4 to get the following analogues to (30) and (31).

λ(G̃) ⊂ h∗ ; R̃(λ_{M}^{∗} ∪ {d̃}) (34)

(35) ^{∗}W δ,d̃ ; R̃ ∈ R̃ whenever δ ∈ d̃.

As before, the proof of the latter uses the Dodd solidity of N, and thus makes use of the induction hypothesis (2). The inclusion (34) immediately implies

(36)  h∗ ; R̃(λ_{M}^{∗} ∪ {d̃}) = R̃,

which tells us that dλ_{M}^{∗} ; R̃ is defined and ωϱ1 ; R̃ ≤ λ_{M}^{∗} < λ(R̃). It follows that R̃ cannot be a type C potential premouse (see the introduction). In fact, R̃ must be a type B premouse, as μ is easily seen to be a cutpoint of ER̃ ; top.

We claim that R̃ is a premouse, that is, it satisfies the initial segment condition. Consider two cases. If d̃≠ ∅, let δ̃ = max(d̃). Then δ̃ ≥ λ∗ ; R̃ (to see this, recall that an argument similar to the proof of (30) shows that λ ∉ h∗ ; R̃(λ) for any cutpoint λ of ER̃ ; top). So σδ̃ ↾ λ∗ ; R̃ = id where σδ̃ : ^{∗}W δ̃,d̃ ; R̃ → R̃ is the canonical witness map. From this and (35) we immediately get

ER̃ ; top ↾ λ∗ ; R̃ = E_{top}^{^{∗}W δ̃,d̃ ; R̃} ↾ λ∗ ; R̃ ∈ R̃,

which tells us that R̃ satisfies the initial segment condition. Now assume d̃ = ∅. This means that no λ ≥ λ_{M}^{∗} can be a cutpoint of ER̃ ; top. Consequently, λ∗ ; R̃ < λ_{M}^{∗}. But we know that k ↾ γ_{M} = id, and using this it is easy to see that ER̃ ; top | λ∗ ; R̃ = E_{top}^{R′} | λ∗ ; R̃. By the initial segment condition for R′, the extender E_{top}^{R′} | λ∗ ; R̃ is an element of R′. But R′ agrees with R̃ up to λ_{M}^{∗} and λ_{M}^{∗} is a cardinal in both structures, so ER̃ ; top | λ∗ ; R̃ ∈ R̃. This proves that R̃ is a premouse.

Our aim is to apply the Condensation Lemma and obtain a contradiction in the same way as in the proof of Claim 4. Since the Condensation Lemma is formulated in the language for premice, we have to replace k by a suitable Σ_{1}-preserving map (with respect to the language for coherent structures) k̃ : R̃ → R̂ where where R̂ is a type B premouse and k̃(γR̃) = γR̂. The Condensation Lemma cannot be applied to k directly, as R′ is a type C premouse. This follows from the elementarity of

DODD PARAMETERS AND λ-INDEXING OF EXTENDERS 35

π′ and the fact that N^{∗} = M || β is a type C premouse; the latter, in turn, is a consequence of the fact that λ(E_{β}) = μ is inaccessible in M. Let λ^{∗} = k(λ∗ ; R̃) and λ̂ = sup(k′′λ(R̃)). We show:

(37)  λ̂ is the least cutpoint of E_{top}^{R′} larger than λ^{∗}.

To see that λ̂ is a cutpoint of E_{top}^{R′}, it suffices to show that E_{top}^{R′}(g)(δ) < λ̂ whenever δ < λ̂ and g : μ̄ → μ̄ is a monotonic function that is an element of N^{∗}; here μ̄ = cr(ER̃ ; top) = cr(E_{top}^{R′}). (It is enough to consider just monotonic functions, as every g′ : μ̄ → μ̄ is majorized by such a function.) Now if δ̃ < λ(R̃) is such that δ ≤ k(δ̃), we have:

E_{top}^{R′}(g)(δ) ≤ E_{top}^{R′}(g)(k(δ̃)) = k(ER̃ ; top(g)(δ̃)) < λ̂.

Now focus on the proof that no cutpoint of E_{top}^{R′} lies in the interval (λ^{∗}, λ̂). Since λ∗ ; R̃ is the largest cutpoint of ER̃ ; top, there are cofinally many ordinals in λ(R̃) that are of the form ER̃ ; top(g)(ξ) for some g as above and some ξ = ≺ξ_{1}, …, ξ_{ℓ}≻ where ξ_{1}, …, ξ_{ℓ} ≤ λ∗ ; R̃ (here ≺ξ_{1}, …, ξ_{ℓ}≻ is the Gödel ℓ-tuple). By the preservation properties of k, there are cofinally many ordinals in λ̂ that are of the form E_{top}^{R′}(g)(ξ′) where g is as above and ξ′ = ≺ξ′_{1}, …, ξ′_{ℓ}≻ where ξ′_{1}, …, ξ′_{ℓ} ≤ λ^{∗}. ⊓ ⊔(37)

Let R̂ be the initial segment of R′ whose top extender is E_{top}^{R′} | λ̂. By the above discussion, R̂ is a type B premouse and λ∗ ; R̂ = λ^{∗}. Let k̂ : R̂ → R′ be the canonical factor map, so k̂ : ER̂ ; top(g)(δ′) ↦ E_{top}^{R′}(g)(δ′) whenever g : μ̄ → 𝒫(μ̄) is an element of J_{τ}^{E} and δ′ < λ̂. (This again involves our abuse of notation.) We define the map k̃ : R̃ → R̂ by k̃ = k̂^{−1} ∘ k. It is easy to check that

k̃(ER̃ ; top(g)(δ)) = ER̂ ; top(g)(k(δ))

for g as above and δ < λ(R̃) = λ(G̃). That k̃ is Σ_{0}-preserving with respect to the language for coherent structures and cofinal follows from the fact that both k̂ and k have these properties. As an immediate consequence of the definition of k̃ we have

(38)  k̃ ↾ λ(R̃) = k ↾ λ(R̃) = σ′ ↾ λ(R̃),

so k̃(λ∗ ; R̃) = λ∗ ; R̂. Since R̃ satisfies the initial segment condition, ER̃ ; γR̃ = ER̃ ; top | λ∗ ; R̃. Using the preservation properties of k̃ we obtain ER̂ ; k̃(γR̃) = ER̂ ; top | λ∗ ; R̂, which proves

36 MARTIN ZEMAN

that k̃(γR̃) = γR̂. This means that k̃ : R̃ → R̂ is Σ_{1}-preserving with respect to the language for premice.

We know that ωϱ1 ; R̃ ≤ λ_{M}^{∗}, as this follws from (36). We also have k̃ ↾ γ_{M} = id, as this follows from (38). These two facts combined with the preservation properties of k̃ established in the previous paragraph and with the first part of the Condensation Lemma yield the weak iterability of R̃ (in fact they yield (ω_{1}, ω_{1} + 1)-iterability of R̃, as we are assuming that M is countable). In order to be able to apply the second part of the Condensation Lemma and obtain the desired contradiction, we need to show that R̃ is sound above λ_{M}^{∗}. Now R̂, being a proper initial segment of R′, is a proper initial segment of M. Since R̃ is embeddable into R̂, we have ht(R̃) ≤ ht(R̂) < ht(M). By the induction hypothesis (2), Theorem 1.2 holds of R̃. We have just shown that R̃ is weakly iterable; that the rest of the assumptions of Theorem 1.2 is satisfied follows from (35) and (36). This guarantees the soundness of R̃ above λ∗ ; R̃. Now we can proceed exactly as in the proof of Claim 4 and get a contradiction. ⊓ ⊔(Claim 5)

It is now obvious that the proof of Lemma 1.12 is merely a direct combination of the five claims we have just established. ⊓ ⊔(Lemma 1.12)

In view of Lemma 1.12, we can summarize the current state of affairs as follows.

• F = E_{top}^{M} and F̄ =^{def} E_{top}^{W};

• γ′ = σ(γ) and N′ = σ(N);

• G = E_{top}^{N} = E_{γ}^{W}, G′ = E_{top}^{N′} = E_{γ′} and G̃ = G′ ∩ J_{γ_{M}}^{E};

• μ = cr(G) = cr(G′), as σ(μ) = μ;

• β = μ^{+N} = μ^{+W} is the critical point of σ and β′ =^{def} σ(β).

Since E_{β} is a total extender in M and β = cr(σ), an easy argument shows that, in the sense of M, there are stationarily many ordinals β̄ ⊂ β′ with Eβ̄ ≠ ∅. This means that μ is subcompact in M (see [12]). Also, E_{γ_{M}} witnesses that μ is superstrong in M (see the introduction). These observations give a proof of Proposition 1.3: Under the assumption that no cardinal is both subcompact and superstrong in an inner model, we immediately obtain that β as above cannot exist, so e_{M}^{α} ∩ λ_{M}^{∗} = ∅.

DODD PARAMETERS AND λ-INDEXING OF EXTENDERS 37

Now focus on the general case. The structure ⟨J_{γ}^{E^{W}} , F̄ | λ_{W}^{∗} ⟩ is a potential premouse. If F̄ | λ_{W}^{∗} were an element of W, it would be equal to E_{γ}^{W}. Consequently, σ(F̄ | λ_{W}^{∗}) would be equal to F | λ_{W}^{∗} = E_{γ_{M}}. So γ_{M} ∈ rng(σ), contradicting (29). It follows that F̄ | λ_{W}^{∗} ∉ W, so W fails to satisfy the initial segment condition. The structure ⟨J_{γ}^{E^{W}} , E_{γ}^{W} , F̄ | λ_{W}^{∗} ⟩ looks like a bicephalus, and it is not difficult to prove that it is iterable. Because of this, one might attempt to apply the bicephalus lemma to conclude that F̄ | λ_{W}^{∗} = E_{γ}^{W}, and thereby obtain a contradiction. This contradiction would show that β as above does not exist, that is, it would enable us to remove the smallness condition from the assumptions of Proposition 1.3. However, the bicephalus argument is not going to work here, as we will see in a little while that ⟨J_{γ}^{E^{W}} , F̄ | λ_{W}^{∗} ⟩ fails to satisfy the initial segment condition.

Let ν̄ = ht(W) and • π̄ : J_{τ}^{E} → J E^{W} ; ν̄ be the ultrapower embedding associated with Ult(J_{τ}^{E}, F̄);

• π̄^{∗} : J_{τ}^{E} → J_{γ}^{E^{W}} be the ultrapower embedding coming from Ult(J_{τ}^{E}, F̄ | λ_{W}^{∗});

• k̄ : J_{γ}^{E^{W}} → J E^{W} ; ν̄ be the canonical map defined by k̄ : π̄^{∗}(f)(ξ) ↦ π̄(f)(ξ) for f ∈ ^{κ}κ ∩ J_{τ}^{E} and ξ < λ_{W}^{∗}; so k̄ ↾ λ_{W}^{∗} = id and π̄ = k̄ ∘ π̄^{∗};

• K̄ be the extender derived from k̄. Using standard arguments we infer that cr(k̄) = λ_{W}^{∗}, k̄(λ_{W}^{∗}) = λ(W) and K̄ has no cutpoints (see the proof of Lemma 1.6, the discussion at the beginning of Case 2). Since k̄ is a cofinal map, there is a unique predicate H̄ on J E^{W} ; ν̄ such that

k̄ : ⟨J_{γ}^{E^{W}} , G⟩ → ⟨J E^{W} ; ν̄ , H̄⟩ is Σ_{0}-preserving.

The cofinality of k̄ guarantees that

Q̄ = ⟨J E^{W} ; ν̄ , H̄⟩

is a coherent structure, and since cr(k̄) = λ_{W}^{∗}, H̄ is an extender with critical point μ, λ_{W}^{∗} is a cutpoint of H̄ and λ(H̄) = λ(W). It also follows that H̄ is a total extender on J E^{W} ; ν̄, so Q̄ is a potential premouse. And since K̄ has no cutpoints (any cutpoint of K̄ would be a cutpoint of F̄ larger than λ_{W}^{∗}), there are no cutpoints of H̄ above λ_{W}^{∗}, so λ_{W}^{∗} is the largest cutpoint of H̄. The definition of H̄ guarantees that H̄ | λ_{W}^{∗} = G = E_{γ}^{W}, so Q̄ satisfies the initial segment condition. It follows that Q̄ is a type B premouse and λ∗ ; Q̄ = λ_{W}^{∗}. Lemma 1.13. H̄ ∘ E_{β} = F̄.

38 MARTIN ZEMAN

Proof. We first show that F̄ | λ_{W}^{∗} factors in a similar way and then use this information to get the desired conslusion about F̄. Claim 1. G̃ ∘ E_{β} = E_{γ_{M}}. Hence cr(E_{β}) = cr(F) = κ < μ. Moreover, μ is a cutpoint of both F and F̄. Proof. We use the notation from the proof of Claim 4 in Lemma 1.12. Notice that N^{∗} = M || β in our case. We have the maps π̃ : N^{∗} → R̃, π′ ↾ N^{∗} : N^{∗} → R′ and k : R̃ → R′. Both R̃ and R′ are coherent structures whose top extenders have critical points equal to cr(E_{β}) < μ. Furthermore, λ(R̃) = λ(G̃) = λ(G′) = λ(R′) and this value is λ_{M}^{∗}. It is now easy to see that k ↾ (λ_{M}^{∗} + 1) = id, so in fact k = id and R′ = R̃. But ht(R̃) = ht(Ñ) = γ_{M} and R′ is an initial segment of M, so ER̃ ; top = E_{γ_{M}}. It follows that cr(E_{β}) = cr(ER̃ ; top) = cr(E_{γ_{M}}) = cr(F) = κ, which proves the second part of the lemma. It also follows that μ is a cutpoint of ER̃ ; top, which yields the last sentence in the lemma. Regarding the first part of the lemma, for x ∈ 𝒫(cr(E_{β})) ∩ N^{∗} we have:

(G̃ ∘ E_{β})(x) = G̃(E_{β}(x)) = π̃(E_{β}(x)) = ER̃ ; γ_{M} (π̃(x)) = E_{γ_{M}} (x),

which completes the proof. ⊓ ⊔(Claim 1)

Claim 2. G ∘ E_{β} = F̄ | λ_{W}^{∗}. Proof. Given any x ∈ 𝒫(κ) ∩ W, we have

σ((G ∘ E_{β})(x)) = σ(G(E_{β}(x))) = G′(E_{β}(x)) = G̃(E_{β}(x))
= (G̃ ∘ E_{β})(x) = E_{γ_{M}} (x) = F (x) ∩ λ_{M}^{∗} = σ(F̄(x) ∩ λ_{W}^{∗}).

The third equality here follows from the fact that G̃ and G′ agree on 𝒫(μ) ∩ J_{β}^{E} and the fifth one follows from Claim 1. ⊓ ⊔(Claim 2)

The following computation then completes the proof of the lemma:

F̄ = K̄ ∘ (F̄ | λ_{W}^{∗}) = K̄ ∘ G ∘ E_{β} = H̄ ∘ E_{β}.

The equality on the left follows from the fact that π̄ = k̄ ∘ π̄^{∗}; this fact also implies H̄(x) = k̄(G(x)) = K̄(G(x)) for all x ∈ 𝒫(κ) ∩ W, which yields the equality on the right. The middle equality comes from Claim 2. ⊓ ⊔(Lemma 1.13)

DODD PARAMETERS AND λ-INDEXING OF EXTENDERS 39

Our aim is to prove that W ∈ M, and since W is a coherent structure, this amounts to showing that F̄ ∈ M. By Lemma 1.13, it suffices to establish the following. Lemma 1.14. H̄ ∈ M. Proof. We generalize the argument from the proof of Lemma 1.6. As before, the argument splits into two cases. In the present situation, β plays the role γ_{M} played in that proof. The case where 𝒫(β) ∩ Ult^{∗}(M, E_{β}) ≠ 𝒫(β) ∩ Ult(M, E_{β}) can be handled in the same way as before, and we will leave the details to the reader. Thus, for the rest of the argument we will assume

(39)  𝒫(β) ∩ Ult^{∗}(M, E_{β}) = 𝒫(β) ∩ Ult(M, E_{β}).

We have already constructed Q̄; this structure which will play an analogous role as Q̄ constructed in the proof of Lemma 1.6. One major difference between the two situations is that in the present case, Q̄ is a type B premouse. The next step is to define the structure Q. We have seen in Case 1 in the proof of Lemma 1.13 that μ is a cutpoint of F. This enables us to repeat the construction described in (16), this time with μ in place of λ_{M}^{∗} and set

Q = Q(M, μ)

where, given an active premouse P and a cutpoint υ of E_{top}^{P},

Q(P, υ) = ⟨J_{ν_{P}}^{E^{P}} , H_{P,υ}⟩.

Here ν_{P} = ht(P) and, letting ζ_{υ} be the index of E_{top}^{P} | υ and υ_{P,υ} : P || ζ_{υ} → P be the canonical factoring map, H_{P,υ} is the extender derived from υ_{P,υ}. (Precisely: υ_{P,υ} : π_{P,υ}(f)(ξ) ↦ π_{P} (f)(ξ) for f ∈ P with dom(f) = cr(E_{top}^{P}) and ξ < υ; here π_{P,υ} and π_{P} are the ultrapower maps associated with Ult(J_{τ_{P}}^{E^{P}} , E_{top}^{P} | υ) and Ult(J_{τ_{P}}^{E^{P}} , E_{top}^{P}), respectively, and τ_{P} = (cr(E_{top}^{P})^{+P}).) The proof of (18) can be literally repeated with present structures Q̄ and Q, and the ordinal μ in place of λ_{M}^{∗}, so we have

σ : Q̄ → Q is Σ_{0}-preserving and cofinal.

Again, here we work in the language for coherent structures. As before, we see that ⟨M, Q̄, β⟩ is a good phalanx and that the pair of maps ⟨id, σ⟩ is an embedding of this phalanx into the pair ⟨M, Q⟩ in the sense described in the proof of Lemma 1.6.

40 MARTIN ZEMAN

We again write Q̄_{0} for Q̄ and Q̄_{−1} for M and compare ⟨M, Q̄, β⟩ with M. This gives rise to iterations ℑ̄, ℑ̃ and ℑ where ℑ̄ and ℑ constitute the coiteration in question and ℑ̃ is the “copy” of ℑ̄ in the sense described in the proof of Lemma 1.6. This works in the present situation, since the obvious generalization of Lemma 1.8 goes through — this time for extenders with critical points above μ (actually, this is true for arbitrary extenders that are weakly amenable with respect to Q). More precisely, we have:

If F_{P} is a total extender on P with cr(F_{P}) > μ and π : P → P′ is the ultrapower map associated with Ult(P, F_{P}), then Q(P′, μ) = Ult(Q(P), F_{P}) and π : Q(P, μ) → Q(P′, μ) is the associated ultrapower map.

Clauses (a) – (e) concerning the “copying” construction from the proof of Lemma 1.6 hold with:

• the current versions of Q̄_{0} and Q̄_{−1};

• Q(M̃_{i}, μ) in place of Q(M̃_{i});
• β in place of γM̃_{0};
• μ in place of λ^{∗}.

Our notation is consistent with that in the proof of Lemma 1.6, so for instance the models of ℑ̄, ℑ̃ and ℑ are denoted by Q̄_{i}, M̃_{i} and M_{i}, respectively. The application of the Neeman-Steel Lemma can be carried out exactly as before, so we conclude:

• The coiteration of ⟨M, Q̄, β⟩ with M terminates at some θ + 1 < ω_{1};

• 0 ≤ℑ̄ θ;

• There is no truncation point on the branch [0, θ]ℑ̄;

• Q̄_{θ} is an initial segment of M_{θ}. The next step in the argument is showing that M wins the coiteration, which is formulated rigorously as the next claim. This was easy to see in the situation from the proof of Lemma 1.6, but requires an argument in the present case. Claim. Either there is a truncation on the main branch of ℑ or else Q̄_{θ} is a proper initial segment of M_{θ}. Proof. Assume this fails, so Q̄_{θ} = M_{θ} and both iteration maps π̄ : Q̄ → M_{θ} and π : M → M_{θ} along the main branches in ℑ̄ and ℑ are total. The existence of π̄ guarantees that Q̄ is (ω_{1} + 1, ω_{1})-iterable. Notice that the coiteration begins with no action on the phalanx side and with the ultrapower by E_{β} on the M-side.

DODD PARAMETERS AND λ-INDEXING OF EXTENDERS 41

Subclaim 1. h∗ ; Q̄(β ∪ {d̄}) = Q̄ where σ(d̄) = (p_{M} ∪ e_{M}^{α}) − (β + 1). Consequently,

ωϱ1 ; Q̄ ≤ β.

Proof. Choose any ordinal ζ < ν; we will show that ζ ∈ h∗ ; Q̄(β ∪ {d̄}). Since W = h_{W}^{∗} (β ∪ {d̄}), the proof of (30) gives us a function f : ^{1+|d̄|}κ → 𝒫(κ), f ∈ J_{τ}^{E} such that ζ = otp(F̄(f)(ξ, d̄)) for some ξ < β (again, with a slight abuse of notation). By Lemma 1.13, the right side here is equal to otp(H̄(f′)(ξ, d̄)) where f′ = E_{β}(f) ∈ J_{β}^{E}. So ζ is Σ_{1}(M)-definable from d̄ and elements of J_{β}^{E}. ⊓ ⊔(Subclaim 1)

Subclaim 2. ωϱ_{M}^{1} ≤ τ. Proof. Let i + 1 be the immediate successor of 0 on the main branch of ℑ. Recall the ν_{i} are the indices in the coiteration and κ_{i} are the critical points in ℑ. As we have mentioned above, ν_{0} = β, that is, the first extender applied on the M-side of the coiteration is E_{β}. It follows that ν_{i} ≥ β and κ_{i} < μ = λ(E_{β}).

Assume that ωϱ_{M}^{1} > τ. This means that ωϱ_{M_{i+1}}^{1} > π_{0,i+1}(τ) = β. Since the iteration map π_{i+1,θ} : M_{i+1} → M_{θ} is Σ^{∗}-preserving, ωϱ_{M_{θ}}^{1} ≥ ωϱ_{M_{i+1}}^{1} > β. On the other hand, all extenders applied on the main branch of the phalanx side of the coiteration have critical points at least β, so ωϱ_{M_{θ}}^{1} = ωϱ1 ; Q̄ ≤ β; the inequality on the right here comes from Subclaim 1. Contradiction. ⊓ ⊔(Subclaim 2)

Subclaim 3. The option ωϱ_{M}^{1} ≤ κ is impossible. Proof. Assume ωϱ_{M}^{1} ≤ κ. Then both sides of the coiteration are above the first projectum on their main branches. This is clear in the case of ℑ̄, as we have seen in the proof of Subclaim 2. Let i be as in that claim. If the main branch of ℑ failed to be above the first projectum, then necessarily κ_{i} < ωϱ_{M}^{1}, as the critical points ascend along the branches, and the projecta, that are below the critical points, are preserved. Consequently, κ_{i} < κ, so i > 0. But then ωϱ_{M_{i+1}}^{1} > λ(E_{ν_{i}}^{M_{i}}) > β, so ωϱ_{M_{θ}}^{1} > β. As in Subclaim 2, this yields a contradiction.

Since we are assuming that M is sound (see (4)), the conclusion fom the previous paragraph implies that M = core(Q̄). Let σ̄ : M → Q̄ be the associated core map, d = (p_{M} ∪ e_{M}^{α}) − (β + 1) and d′ = σ̄(d). Recall that σ(d̄) = d. We have already proved that ^{∗}W _{M}^{δ,d} ∈ M for all δ ∈ d. It follows that for every δ′ ∈ d′, there is a generalized Dodd solidity witness with respect to Q̄ and d′, that is an element of Q̄, namely σ̄(^{∗}W _{M}^{δ,d}) where δ′ = σ̄(δ). This together with Subclaim 1 yields the middle inequality in the following formula; the other two inequalities follow from the monotonicity of both σ̄ and σ:

d ≤^{∗} d′ ≤^{∗} d̄≤^{∗} d.

It follows that d′ = d̄, so h∗ ; Q̄(β ∪ {d′}) = Q̄. In particular, σ̄(γ_{M}) ∈ h∗ ; Q̄(β ∪ {d′}). Consequently, γ_{M} ∈ h_{M}^{∗}((σ̄^{−1})′′β ∪ {d}) ⊂ h_{M}^{∗}(β ∪ {d}). Thus, there is a finite e ⊂ β such that γ_{M} ∈ h_{M}^{∗}(α ∪ {p_{M} ∪ (e_{M}^{α} − (β + 1)) ∪ e}). This contradicts the minimality of e_{M}^{α}, as (e_{M}^{α} − (β + 1)) ∪ e <^{∗} e_{M}^{α}. ⊓ ⊔(Subclaim 3)

Subclaim 4. The option ωϱ_{M}^{1} = τ is impossible. Proof. The proof is an elaboration on that of Subclaim 3. Assume that ωϱ_{M}^{1} = τ and seek for a contradiction. Let i be as in Subclaim 2; we first observe that i = 0. Indeed, if i > 0, then

ωϱ_{M_{θ}}^{1} ≥ ωϱ_{M_{i+1}}^{1} = π_{0,i+1}(τ) > π_{0,i+1}(κ_{i}) = λ(E_{ν_{i}}^{M_{i}}) > β,

which yields a contradiction as in the proof of Subclaim 3. So M_{1} is on the main branch of ℑ and ωϱ_{M_{1}}^{1} = β. Also, M_{1} is sound above β, as follows from the soundness of M. Since both branches [1, θ]_{ℑ} and [0, θ]ℑ̄ are above β, we conclude that M_{1} = core_{β}(Q̄). Let σ̄ : M_{1} → Q̄ be the associated core map.

This time set d′ = σ̄ ∘ π_{0,1}(d) (d was defined in the proof of Subclaim 3). We now see that every δ′ ∈ d′ has a generalized Dodd solidity witness with respect to Q̄ and d′, that is an element of Q̄. Using Subclaim 1, we argue as in the proof of Subclaim 3 and conclude that d′ = d̄. So h∗ ; Q̄(β ∪ {d′}) = Q̄. In particular,

σ̄ ∘ π_{0,1}(γ_{M}) ∈ h∗ ; Q̄(β ∪ {d′}). It follows that

γ_{M} ∈ h_{M}^{∗}(((σ̄ ∘ π_{0,1})^{−1})′′β ∪ {d}) ⊂ h_{M}^{∗}(β ∪ {d}),

which yields a contradiction with the minimality of e_{M}^{α} as before. ⊓ ⊔(Subclaim 4)

By Subclaim 2, either ωϱ_{M}^{1} = τ or ωϱ_{M}^{1} ≤ κ. Subclaims 3 and 4 guarentee that neither of these options can occur, which yields a contradiction and thereby completes the proof of the Claim. ⊓ ⊔(Claim)

The rest of the proof of Lemma 1.14 is standard. By Subclaim 1 above, Q̄ can be encoded into some Σ_{1}(Q̄) subset b of β. As in the proof of Lemma 1.6 we show that b ∈ M_{1}, so b ∈ Ult(M, E_{β}) by (39). Then we apply the argument from the

DODD PARAMETERS AND λ-INDEXING OF EXTENDERS 43

end of the proof of Lemma 1.6 to conclude that b ∈ M. Using the fact that J_{ν}^{E} is a ZFC^{−}-model, we decode b inside M, so Q̄ ∈ M. ⊓ ⊔(Lemma 1.14)

This also completes the proof of Theorem 1.1. ⊓ ⊔(Theorem 1.1)

Proof of Theorem 1.2. We verify (a) first. The assumptions of the theorem imply that d_{M}^{α} is defined and d_{M}^{α} ≤^{∗} d. Assuming that d_{M}^{α} <^{∗} d, we have an ordinal β ∈ d − d_{M}^{α} such that d − (β + 1) = d_{M}^{α} − (β + 1). Then d_{M}^{α} ∈ rng(σ) where σ : ^{∗}W _{M}^{β,d} → M is the witness map. This implies that h_{M}^{∗}(α ∪ {d_{M}^{α}}) ⊂ rng(σ), so rng(σ) = M. On the other hand, we are assuming that M contains a generalized Dodd solidity witness for β with respect to M and d. Remark (1) guarantees that ∗W _{M}^{β,d} ∈ M, so rng(σ) ≠ M, a contradiction. It follows that d = d_{M}^{α}. ⊓ ⊔(a)

We next prove that M is sound above α, that is, h_{M}(α ∪ {p_{M}}) = M. Since h_{M}^{∗}(α ∪ {d}) = M, it suffices to show that d ∈ h_{M}^{∗}(α ∪ {γ_{M}, p_{M}}). Assume this is false and denote the largest element of d that is not in h_{M}^{∗}(α ∪ {γ_{M}, p_{M}}) by β. Let β′ be the least ordinal that is in h_{M}^{∗}(α ∪ {γ_{M}, p_{M}}) − β. Then β′ > β and d ∩ (β + 1) witnesses the following statement in M:

(∃x)(x ∈ [β′]^{<ω} & p_{M} ∪ {γ_{M}} = h_{M}^{∗}(y, (d − (β + 1)) ∪ x))

where y ∈ [α]^{<ω} is such that p_{M} ∪ {γ_{M}} = h_{M}^{∗}(y, d). It follows that this statement is witnessed by some d^{∗} ∈ h_{M}^{∗}(α ∪ {γ_{M}, p_{M}}); here d^{∗} is a finite subset of β. Thus, we have p_{M} ∪ {γ_{M}} = h_{M}^{∗}(y, (d − (β + 1)) ∪ d^{∗}). Let σ : ^{∗}W _{M}^{β,d} → M be the canonical witness map. Then {y, d′} ⊂ rng(σ) where d′ = (d − (β + 1)) ∪ d^{∗}, and therefore p_{M} ∪ {γ_{M}} ∈ rng(σ) as well. Let σ(p̄) = p_{M} ∪ {γ_{M}}. Since p_{M} is a good parameter for M with respect to the language for premice, we have a set A that is Σ_{1}(M) in p_{M} ∪ {γ_{M}} such that A ∩ ωϱ_{M}^{1} ∉ M; here Σ_{1}(M) is meant with respect to the language for coherent strutures. The map σ is Σ_{1}-preserving with respect to the language for coherent structures and σ ↾ ωϱ_{M}^{1} = id, so letting Ā be the Σ_{1}(^{∗}W _{M}^{β,d})-set by the same definition as A in the parameter p̄, we have Ā ∩ ωϱ_{M}^{1} = A ∩ ωϱ_{M}^{1} ∉ M. Now by the assumptions of the theorem, ^{∗}W _{M}^{β,d} ∈ M, so Ā ∩ ωϱ_{M}^{1} ∈ M after all. This contradiction shows that d ∈ h_{M}^{∗}(α ∪ {γ_{M}, p_{M}}) and thereby completes the proof. ⊓ ⊔(c)

Clauses (a) and (c) together with our assumption that M is weakly iterable now allow to apply Theorem 1.1 (actually, just (b) of Theorem 1.1) and obtain (b).

⊓ ⊔(Theorem 1.2)

Let us finally remark that although we did apply Theorem 1.1 to M at the end of the above proof, such an application does not make the proof of the two main theorems as a whole circular. This is the case, because the proof of Theorem 1.1 does not make any use of an application of Theorem 1.2 to M — the only application of Theorem 1.2 in that proof occurs at the end of the proof of Claim 5 in the proof of Lemma 1.12, where Theorem 1.2 is applied to a premouse of height strictly smaller than ht(M).

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[12] Schimmerling, E. and Zeman, M., Characterization of □_{κ} in core models, accepted for publication in Journal of Mathematical Logic.

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[14] Schimmerling, E. and Zeman, M., Cardinal transfer theorems in extender models II, in preparation [15] Schindler R. D., Steel, J. R. and Zeman, M. Deconstructing inner model theory, Journal

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[16] Zeman, M., Global □ in extender models, in preparation

[17] Zeman, M., Inner Models and Large Cardinals, de Gruyter Series in Logic and its Applications Vol. 5, de Gruyter, Berlin 2002

[18] Zeman, M., A note on Dodd parameters, unpublished, http://math.uci.edu/∼mzeman

Department of Mathematics, University of California at Irvine, Irvine, CA 92697

 E-mail address: mzeman@math.uci.edu