CARDINAL TRANSFER PROPERTIES IN EXTENDER MODELS

ERNEST SCHIMMERLING AND MARTIN ZEMAN

Abstract. We prove that if L[E] is a Jensen extender model, then L[E] satisfies the Gap-1 morass principle. As a corollary to this and a theorem of Jensen, the model L[E] satisfies the Gap-2 Cardinal Transfer Property (κ^{++}, κ) → (λ^{++}, λ) for all infinite cardinals κ and λ.

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

Jensen isolated ♢, □, morass and other combinatorial principles in his work on the fine structure of L and its applications. One of these applications was to the Gap-1 Transfer Property

(κ^{+}, κ) → (λ^{+}, λ).

The arrow notation means the following. Let U̇ be a unary predicate symbol of an arbitrary countable first order language. Suppose that 𝔄 is a structure of this language with card(A) = κ^{+} and card(U̇^{A}) = κ. Then there exists B ≡ A with card(B) = λ^{+} and card(U̇^{B}) = λ.

Vaught proved that (κ^{+}, κ) → (ω_{1}, ω) for all infinite cardinals κ. Chang proved that (ω_{1}, ω) → (λ^{+}, λ) when λ is a regular cardinal. These two are results of ZFC + GCH alone. Finally, Jensen added □_{λ} to the hypothesis and proved that (ω_{1}, ω) → (λ^{+}, λ) when λ is a singular cardinal. It is interesting to note that Shelah [17] proved later that for singular strong limit λ, there is a principle S_{λ} very similar to □_{λ} such that

• □_{λ} implies S_{λ},

• S_{λ} implies Jensen’s weak square principle □_{λ}^{∗}, and

• S_{λ} is equivalent to (ω_{1}, ω) → (λ^{+}, λ).

The first author was supported in part by NSF grants DMS-0400467 and DMS-0700047.

The second author was supported in part by NSF grants DMS-0204728 and DMS-0500799.

2 ERNEST SCHIMMERLING AND MARTIN ZEMAN

Jensen also proved that the Gap-2 Transfer Property

(κ^{++}, κ) ⟶ (λ^{++}, λ)

holds for all infinite cardinals κ and λ in L. For regular λ, what Jensen really used besides GCH is a λ^{+}-morass. A combination of □_{λ} and a λ^{+}-morass with built-in □_{λ^{+}} is what he used for singular λ. It is interesting that the “built-in □_{λ^{+}}” is actually obtained purely combinatorially from □_{λ} and □_{λ^{+}}, again by an argument of Jensen.

This paper is concerned with Jensen’s combinatorial principles in models constructed from sequences of extenders. That is, models of the form L[E]. In [15], the authors showed that in L[E], □_{λ} holds iff λ is not a subcompact cardinal. Thus, the Gap-1 Transfer Property holds in L[E]. It seems like the methods developed in [15] can be used to generalize Jensen’s theorems about the combinatorial structure of L to L[E]. Here we verify this feeling at least as far as what went into Jensen’s work on the Gap-2 Cardinal Transfer Properties is concerned. We will see that in L[E], there is a λ-morass regardless of whether λ is subcompact or not. By applying results of [15] to Jensen’s argument on morasses with built-in square sequences, we also get λ^{+}-morass with built-in □_{λ^{+}} when λ is not subcompact. This suffices to conclude that the Gap-2 cardinal transfer property holds in L[E] for the trivial reason that singular cardinals are not subcompact.

The fundamental reason that the fine structure of L[E] is trickier than that of L is the failure of a certain form of condensation. Very roughly, this problem was overcome in [15] by laying out the universe differently, using a sequence of extenders and their fragments, and by showing that condensation holds relative to this replacement sequence. Our replacement method failed exactly when we were working between a subcompact and its successor, which is what led to the characterization of when □_{λ} holds in L[E]. As the reader will see, we can avoid this sort of failure entirely in the construction of a λ^{+}-morass.

For the reader who is familiar with a morass construction in L and with extender models, let us say more to illustrate why an overly naive generalization does not work. Assume V = L[E]. Let μ be a measurable cardinal and α = μ^{+}. We discuss a putative construction of an α-morass. Suppose that τ < α^{+} is a local successor of α. This means that α is the largest cardinal in J_{τ}^{E}. Say ᾱ is a local successor of μ

CARDINAL TRANSFER PROPERTIES IN EXTENDER MODELS 3

and τ̄ is a local successor of ᾱ. We must decide whether or not τ̄ ≺ τ where ≺ is the morass order. Let M be the level of L[E] over which τ is seen to have cardinality α. By E_{top}^{M} we mean the top extender of M, which in this case is just E_{On∩M}. For the sake of this example, we assume that cr(E_{top}^{M}) = μ and that every element of M is Σ_{1}-definable in M from parameters < α. Let σ : M̄ → M be the inverse of the Mostowski collapse of the Σ_{1}-hull of ᾱ in M. Assume that σ(ᾱ) = α and σ(τ̄) = τ. Based on the proof in L, we are inclined to set τ̄ ≺ τ and the morass map σ_{τ̄,τ} equal to the restriction σ ↾ τ̄. Condensation does not allow us to conclude that M̄ is an initial segment of L[E] but this is not far that from the truth. A more substantial issue arises when we try to prove the morass continuity axioms. Let τ̃ = sup(σ′′τ̄). A morass axiom requires that τ̄ ≺ τ̃ and σ_{τ̄,τ̃} = σ_{τ̄,τ̃}. Let

N = ⋃ σ′′M

and

E_{top}^{N} = ⋃ σ′′E_{top}^{N}.

In order to satisfy the continuity axioms, our definitions of both {τ′; τ′ ≺ τ̃} and {σ_{τ′,τ̃}; τ′ ≺ τ̃} should be based on Σ_{1}-hulls in N by analogy with how our definitions of {τ′; τ′ ≺ τ} and {σ_{τ′,τ}; τ′ ≺ τ} were based on M. Let M̃ be the level of L[E] over which τ̃ is seen to have cardinality α. The main issue is that M̃ ≠ N. We can tell this because E_{top}^{N} is not an extender over N.

In the course of the proof, we will define the terms protomouse and pluripotent mouse. As a reference for the reader, the structure N above is an example of a protomouse. Premice that look in some ways like M but in other ways like N are called pluripotent.

Finally, let us point out that so far our ability to construct L[E] models from any hypothesis is rather limited; the best result to date, due to Neeman [11], is at the level of a Woodin limit of Woodin cardinals. On the other hand, it has long been known that a λ^{+}-morass can be added by λ-directed closed forcing. Thus it is relatively consistent that there is a supercompact cardinal λ and a λ^{+}-morass.

The definition of the morass we are going to construct is in the next section. Modulo this definition, we can state our main result.

4 ERNEST SCHIMMERLING AND MARTIN ZEMAN

Theorem 0.1 (Main Theorem). If L[E] is an extender model, then L[E] satisfies that for all infinite cardinals κ, there is a κ-morass.

By the theorem of Jensen mentioned before, we may conclude that L[E] satisfies (κ^{++}, κ) → (λ^{++}, λ) for all regular λ. But more is true. Theorem 0.2. If L[E] is an extender model, then L[E] satisfies that for all infinite cardinals κ, there is a κ^{+}-morass with built-in □_{κ^{+}}.

By Jensen, Theorem 0.2 is a direct consequence of Theorem 0.1 and the following theorem. Theorem 0.3 ([15]). If L[E] is an extender model, then L[E] satisfies that for all infinite cardinals κ that are not subcompact, the principle □_{κ} holds.

Immediately from the theorems above and and Jensen’s argument for obtaining cardinal transfer property we may conclude: Corollary 0.4. The Gap-2 Cardinal Transfer Property holds in every extender model.

The paper is organized as follows. In Section 1, we recall some definitions, useful notions and and properties of extender models. The account here will be, however, very brief and for the technical details we refer the reader to [15], [21] and [22].

[21] contains a detailed description of the fine structure theory and the models we are using in this paper. Moreover, our construction heavily relies on techniques developed in [15] that are not included in [21]. Finally, our notation is identical with that in [15], and thus slightly diverges from that in [21]. The reason for choosing this amended notation is given by the fact that we are dealing with a certain specific situation which allows such amendments and significantly simplifies combinatorial constructions. Finally we note that a construction of Gap-1 Morass in L based on Jensen’s Σ^{∗} fine structure can be found in [20]. This paper may be considered as a warm-up for a reader not familiar with basic aspects of the construction.

1. Preliminaries

1.1. Morasses. We begin with the definition of a Gap-1 Morass, which we state for the reader’s convenience. Our notion of a Gap-1 Morass is – up to some minor cosmetic amendments that we adopted in order make certain fine structural arguments tidy – identical with the standard definition, see e.g. [4] for a global version

CARDINAL TRANSFER PROPERTIES IN EXTENDER MODELS 5

or [2] for a local version. Our approach will be closer to that in [4] in that we construct a global version of Gap-1 Morass. Definition 1.1. A Gap-1 Morass is a structure characterized by the following data.

• A class of ordinals S(0) containing all uncountable cardinals such that S(0) ∩ κ is a closed unbounded subset of κ whenever κ is an uncountable cardinal;

• a sequence ⟨S_{α}; α ∈ S(0)⟩ of closed subsets of (α, α^{+}) satisfying: - if α is regular then S_{α} = S(0) ∩ (α, α^{+}),

- if α is a singular cardinal then S_{α} is a proper initial segment of S(0) ∩ (α, α^{+}), and

- if α fails to be a cardinal then for any β ∈ S(0), β > α we have

max(S_{α}) < β;

• a tree ordering ≺ on the set

S(1) = ⋃{S_{α}; α ∈ S(0)};

• a system of maps ⟨σ_{τ̄,τ}; τ̄, τ ∈ S(0) & τ̄ ≺ τ⟩. The system of maps ⟨σ_{τ̄,τ} ⟩ satisfies the set of axioms stated below. Before stating the axioms, we fix some notation. Given α ∈ S(0) and τ, τ′ ∈ S(1) set:

• α(τ) = the unique α such that τ ∈ S_{α};

• τ <_{0} τ′ iff α(τ) = α(τ′) and τ < τ′;

• τ(α) = max(S_{α});

• τ_{+} = the <_{0}-successor of τ. Note that the uniqueness of α(τ) follows from the description of S_{α} above. The following axioms are called morass axioms. (M0) If τ̄ ≺ τ then α(τ̄) < α(τ) and τ̄ is <_{0}-minimal, successor, limit iff τ is

<_{0}-minimal, successor, limit. (M1) If τ̄ ≺ τ then σ_{τ̄,τ} : τ̄ → τ is an order-preserving map such that

• σ_{τ̄,τ} ↾ α(τ̄) = id ↾ τ̄;

• σ_{τ̄,τ} (α(τ̄)) = α(τ). Moreover, these maps commute along the branches in ≺, i.e. τ^{∗} ≺ τ̄ ≺ τ implies that σ_{τ̄,τ} ∘ σ_{τ^{∗},τ̄} = σ_{τ^{∗},τ}.

(M2) Assume τ̄ ≺ τ and τ̄′ < τ̄.

6 ERNEST SCHIMMERLING AND MARTIN ZEMAN

(a) τ̄′ ∈ S_{α(τ̄)} ⟷ σ_{τ̄,τ} (τ̄′) ∈ S_{α(τ)}.

(b) If τ̄′_{+} <_{0} τ̄ then σ_{τ̄,τ} (τ̄′_{+}) = σ_{τ̄,τ} (τ̄′)_{+}.

(c) If τ̄′ <_{0} τ̄ and τ′ = σ_{τ̄,τ} (τ̄′), then τ̄′ ≺ τ′ and σ_{τ̄′,τ′} = σ_{τ̄,τ} ↾ τ̄′. (M3) If τ ∈ S(1) then

C(τ) = {α(τ̄); τ̄ ≺ τ}

is closed in α(τ) and if τ is distinct from the largest element of S_{α(τ)} then C(τ) is unbounded in α(τ). If C(τ) is unbounded in α(τ) then

τ = ⋃{rng(σ_{τ̄,τ}); τ̄ ≺ τ}.

(M4) If τ̄ is a <_{0}-limit and τ̃ = sup(rng(σ_{τ̄,τ})), then τ̄ ≺ τ̃ and σ_{τ̄,τ̃} = σ_{τ̄,τ}. (M5) Assume τ̄ is a limit point of S_{α(τ̄)}, τ̄ ≺ τ and the map σ_{τ̄,τ} is cofinal. Let

α(τ̄) < α̂ ∈ S(0) satisfy the following condition:

(1)  For each τ̄′ <_{0} τ̄ there is a τ̂′ ≺ σ_{τ̄,τ} (τ̄′) such that α̂ = α(τ̂′).

Then there is a τ̂ ≺ τ with α(τ̂) = α̂. The above definition describes a global version of the Gap-1 Morass. Jensen

proved that such a morass exists in L. The restriction of a Gap-1 Morass to {τ < κ^{+} | τ ≼ τ′ for some τ′ ∈ S_{κ}} is called a κ-morass. We can also “localize” the morass by restricting the class of cardinals in S(0). Notice that if we merely require that κ ∈ S(0) we get a κ-morass. Clearly, a κ-morass is interesting primarily when κ is regular. Axioms (M4) and (M5) are called continuity axioms. Without them what we would have is a coarse morass. Our main result is the following theorem. Theorem 1.2. Any extender model L[E] satisfies that there is a Gap-1 Morass.

This theorem obviously implies Theorem 0.1, and thereby guarantees the Gap-2 cardinal transfer property as explained in the introduction.

1.2. Extender Models. We will find a realization of a global Gap-1 Morass in an extender model with Jensen’s λ-indexing of extenders; the details regarding this indexing can be found in [21].

We will work in a fixed extender model L[E]. Thus, whenever we say “extender model” or “L[E]”, we mean this fixed model. The following lemmata list three basic properties of premice that will be used in the course of our construction. Recall that if E_{ν} is on the extender sequence of a premouse M with λ-indexing then the

CARDINAL TRANSFER PROPERTIES IN EXTENDER MODELS 7

initial segment M||ν is a coherent structure whose top extender is weakly amenable, that is, M||ν = ⟨J_{ν}^{E}, E_{ν}⟩ where, letting μ = cr(E_{ν}) and ϑ = μ^{+M||ν}, we have J_{ν}^{E} = Ult(J_{ϑ}^{E}, E_{ν}). We will also need to consider certain coherent structures whose top extender is not weakly amenable; these will be discussed in the next section. The largest cardinal in J_{ν}^{E}, i.e. the image of μ under the ultrapower embedding, is denoted by λ(E_{ν}). Furthermore, we denote the standard parameter of M by p_{M} and the standard solidity witness for an ordinal α with respect to some parameter p ∈ M and M is by W_{M}^{α,p}. Recall that if α is such that ωϱ_{M}^{n+1} ≤ α < ωϱ_{M}^{n} then W_{M}^{α,p} is the transitive collapse of the Σ_{1}^{(n)}-hull h̃_{M}^{n+1}(α ∪ {p − (α + 1)}) where h̃_{M}^{n+1} is the Σ_{1}^{(n)}-Skolem function for M. We will also consider the so-called generalized witnesses. A generalized witness for α with respect to p ∈ M and M is a pair ⟨Q, r⟩ where Q is a J-structure of the same type as M and r ∈ Q is a finite set of ordinals of the same size as p − (α + 1) such that for every Σ_{1}^{(n)} formula φ(v_{0}, v_{1}, …, v_{ℓ}) and any ordinals ξ_{0}, ξ_{1}, …, ξ_{ℓ−1} < α we have

M ⊧ φ(ξ_{0}, ξ_{1}, …, ξ_{ℓ−1}, p − (α + 1)) ⟺ Q ⊧ φ(ξ_{0}, ξ_{1}, …, ξ_{ℓ−1}, r).

Although the standard solidity witness for α with respect to p and M is unique, there may be many generalized witnesses. If we work with a fixed generalized witness ⟨Q, r⟩, we usually denote it by Q_{M}^{α,p} to indicate that it is a generalized witness for α and it was computed with respect to p and M. Two crucial properties of levels of extender models is the solidity (S) and the universality (U) of the standard parameter.

(S) Any level of an extender model M is solid, i.e. if W_{M}^{β,p_{M}} ∈ M whenever β ∈ p_{M}. Equivalently, there is a generalized witness ⟨Q, r⟩ ∈ M for β with respect to p_{M} and M.

(U) For any level of an extender model M and any n ∈ ω, the standard parameter p_{M} is n-universal. That is, if M̄ is the transitive collapse of the Σ_{1}^{(n)}-hull h̃_{M}^{n+1}(ωϱ_{M}^{n+1} ∪ {p_{M}}) then 𝒫(ωϱ_{M}^{n+1}) ∩ M̄ = 𝒫(ωϱ_{M}^{n+1}) ∩ M.

Recall that we have three types of active premice, depending on the distribution of cutpoints of the top extender. Type B premice are those premice whose set of cutpoints is nonempty, but bounded in λ(E_{top}^{M}). The largest element of this set is denoted by λ_{M}^{∗} and the index of E_{top}^{M}|λ_{M}^{∗} is γ_{M}. In the case of type B premice,

8 ERNEST SCHIMMERLING AND MARTIN ZEMAN

we extend the language for premice by a constant denoting γ_{M}, which has the consequence that all fine structural characteristics are computed relative to γ_{M}. In the case of type A and C premice, there is no additional parameter used in the computation of these characteristics. To distinguish between the two languages, we will call the language with the additional constant for γ_{M} the language for premice (to have a uniform notation, we can view the additional constant as a singleton predicate in the case of type B premice and the empty predicate for type A and C premice) and the language without the constant the language for coherent structures. Obviously, the language for coherent structures is the same as the language for type A and C premice.

For for active premice M we will need to consider the Σ_{1}-Skolem functions that are computed in the language for coherent structures. To distinguish between the usual Σ_{1}-Skolem functions (i.e. those computed in the language for premice), we will denote the Σ_{1}-Skolem function for M computed in the language for coherent structures by h_{M}^{∗}. In many arguments, it will be crucial to distinguish between h_{M} and h_{M}^{∗}. This, of course is non-void only in the case of type B premice.

The techniques used in combinatorial constructions also require to consider the so-called Dodd parameters for premice. The issue arises when dealing with type B premice. Unlike the construction of square sequences in [15], the construction of a Gap-1 Morass requires an elaboration on the notion of a Dodd parameter. This is caused by the fact that in certain situations, we have now less control on projecta than we had in the construction of square sequences. We state the crucial definitions and facts below and refer the reader to [22] for the details. Recall that <^{∗} is the canonical well-ordering of finite sets of ordinals which can be viewed as the lexicographic ordering of finite descending sequences of ordinals.

Definition 1.3. Let M be an active premouse and μ = cr(E_{top}^{M}). Assume that α is an ordinal satisfying max{μ^{+M}, ϱ_{M}^{1}} ≤ α. The parameter d_{M}^{α} is the <^{∗}-least finite set of ordinals d such that M = h_{M}^{∗}(α ∪ {d}). We write d_{M} for d_{M}^{α} if α = ωϱ_{M}^{1}.

Obviously, d_{M}^{α} need not be defined for each M and α. However, it is defined granting that M is sound, which is precisely the situation we are interested in. It is easy to see that if ᾱ ≤ α and both d_{M}^{α} and dᾱ ; M are defined then d_{M}^{α} = dᾱ ; M − α. For sound M with ωϱ_{M}^{1} ≥ μ^{+M}, the Dodd parameter in the usual sense is just d_{M}.

CARDINAL TRANSFER PROPERTIES IN EXTENDER MODELS 9

For an ordinal β ≥ α and a parameter p ∈ M, the standard Dodd solidity witness ^{∗}W _{M}^{β,p} for α with respect p and M is the transitive collapse of the hull h_{M}^{∗}(β∪{p−(β+1)}). This is a direct analogue to W_{M}^{β,p} with the only difference that there is no reference to γ_{M}. Similarly, a pair ⟨Q, r⟩ is a generalized Dodd solidity witness for β with respect to p and M just in case that Q is transitive, r ∈ Q is a finite set of ordinals of the same size as p − (β + 1) and for every Σ_{1}-formula in the language for coherent structures φ(v_{0}, v_{1}, …, v_{ℓ}) and every ξ_{0}, ξ_{1}, …, ξ_{ℓ−1} < β we have

M ⊧ φ(ξ_{0}, …, ξ_{ℓ−1}, p − (β + 1)) ⟷ Q ⊧ φ(ξ_{0}, …, ξ_{ℓ−1}, r)

M is Dodd solid above α just in case that d_{M}^{α} is defined and ^{∗}W _{M}^{β,d_{M}^{α}} ∈ M for every β ∈ d_{M}^{α}. This is equivalent to demanding that for each β ∈ d_{M}^{α} there is a generalized Dodd solidity witness ⟨Q, r⟩ ∈ M with respect to d_{M}^{α} and M. Although the standard Dodd solidity witness for β with respect to p and M is unique, there may be many generalized Dodd solidity witnesses. Analogously to the case of solidity witnesses, if we work with a specific generalized Dodd solidity witness ⟨Q, r⟩, we usually denote it by ^{∗}Q_{M}^{β,p} to indicate that it is a generalized witness for β and it is computed with respect to p and M. The following facts summarize the properties of d_{M}^{α}.

(DP1) If M is a level of an extender model and α is as in Definition 1.3 then

(a) d_{M}^{α} = (p_{M} − α) ∪ e_{M}^{α} where e_{M}^{α} is the <^{∗}-least finite set of ordinals e such that γ_{M} ∈ h_{M}^{∗}(α ∪ {p_{M} ∪ e}).

(b) M is Dodd solid above α. (DP2) Let M and α be as in Definition 1.3. Assume that for some n, there is

a Σ_{0}^{(n)}-preserving (with respect to the language for coherent structures!) embedding σ of M into an L[E]-level such that σ ↾ ωϱ_{M}^{n+1} = id. Let d be a parameter satisfying

• M = h_{M}^{∗}(α ∪ {d});

• for every β ∈ d there is a generalized Dodd solidity witness ⟨Q, r⟩ ∈ M

with respect to d and M. Then

(a) d = d_{M}^{α};

10 ERNEST SCHIMMERLING AND MARTIN ZEMAN

(b) d_{M}^{α} = (p_{M} − α) ∪ e_{M}^{α} where e_{M}^{α} is the <^{∗}-least finite set of ordinals e such that γ_{M} ∈ h_{M}^{∗}(α ∪ {p_{M} ∪ e}).

(c) M is sound above α. Finally we state the Condensation Lemma, that will be used many times through-

out the entire construction. Lemma 1.4 (Condensation Lemma). Let M̄ and M be premice of the same type where M is a level of an extender model and let σ : M̄ → M be an embedding which is cardinal preserving, Σ_{0}^{(n)}-preserving and such that σ ↾ ωϱn+1 ; M̄ = id. Then M̄ is solid and pM̄ is k-universal for every k ∈ ω. Furthermore, if M̄ is sound above ν = cr(σ) then one of the following holds:

a) M̄ = M and σ = id.

b) M̄ is a proper initial segment of M.

c) M̄ = Ult^{∗}(M||η, E_{α}^{M}) where α ≤ ωη and letting κ = cr(E_{α}^{M}), the ordinal η is largest with the property ν = κ^{+M||η}. Additionally, E_{α}^{M} has a single generator κ. (Notice that ωϱ_{M||η}^{ω} = κ in this case.)

d) M̄ is a proper initial segment of Ult(M, E_{ν}^{M}). The construction of a morass is analogous to that of a square sequence in that

to each ordinal of interest (this ordinal is usually a local successor cardinal) we assign a unique collapsing structure. In L, such a collapsing structure is just the corresponding collapsing level. In an extender model, however, we have to pick the collapsing structure from a broader class of structures, but in a way that makes our choice unique. The new kind of structures we have to consider are the so-called protomice [15].

1.3. Protomice. We need to adapt the account on protomice from [15] (in that paper, we used the term “fragment” instead of “protomouse”) for the present purposes. The amendments we will make are needed because the collapsing structures we consider here are of a slightly different kind than those considered in [15]. We will also discuss properties of protomice and divisors which were irrelevant there, but will be of use now. The notion of a protomouse is defined relative to a fixed extender model W = L[E].

Recall that M = ⟨J_{ν}^{E}, F ⟩ is a coherent structure just in case that F is an extender such that J_{ν}^{E} = Ult(J E ; ν̄, F) where, letting μ = cr(F), we have 𝒫(μ) ∩ J E ; ν̄ = dom(F)

CARDINAL TRANSFER PROPERTIES IN EXTENDER MODELS 11

and ν̄ is the least ordinal with this property. Clearly ν̄ ≤ μ^{+M} and μ is the largest cardinal in J E ; ν̄. The ordinal ν̄ is denoted by ϑ(M).

In this section we shall consider a coherent structure of the form M = ⟨J_{ν}^{E}, F ⟩ whose top extender F is not weakly amenable. So F measures only subsets of μ = cr(F) from some initial segment J_{ϑ(M)}^{E} where ϑ(M) < μ^{+M} . We denote the collapsing level for ϑ(M) in M by N^{∗}(M). We say that M is a protomouse just in case that N(M) = Ult^{∗}(N^{∗}(M), F) is a level of W or Ult(W, E_{β}) for some β.

We will primarily deal with collapsing structures for local successor cardinals of a certain special kind. Recall that an ordinal α is cardinal-correct in W just in case every β < α is a cardinal in J_{α}^{E} iff β is a cardinal in W. Definition 1.5. Let τ be an ordinal. We say that N is the collapsing structure for τ just in case the following conditions are satisfied:

• α is cardinal-correct in W where α is the cardinal predecessor of τ in N (the option τ = On ∩ N is allowed);

• N is the collapsing level for τ in W or in Ult(W, E_{α}). For τ as in the above definition, the collapsing structure N is uniquely deter-

mined. To see this, notice that if E_{α} ≠ ∅ then W ||α is the collapsing level for α in W, so α cannot be a cardinal in any level of W properly extending W ||α. Thus, the “or” in the above definition is actually an “exclusive or”. One crucial property of the collapsing structure N is that ωϱ_{N}^{ω} ≤ α. In our applications, however, we will often have equality here. We will also consider collapsing protomice. Unlike collapsing structures, these need not be unique; therefore we talk about “a” collapsing protomouse in the next definition. Definition 1.6. Let M = ⟨J_{ν}^{E}, F ⟩ be a protomouse and τ = α^{+M}. We say that the protomouse M is a collapsing protomouse for τ iff α is cardinal-correct in W, μ < α < λ(F) and N(M) is the collapsing structure for τ.

Protomice might emerge in condensation-interpolation arguments, if these arguments involve premice whose top extender overlap α. In many cases, the situation will require to consider fine structure of these premice with respect to the language for coherent structures, even if they are type B premice. We will call such premice pluripotent.

12 ERNEST SCHIMMERLING AND MARTIN ZEMAN

Definition 1.7. Let α be cardinal-correct in W and let N be a level of either W or else Ult(W, E_{α}). We say that N is pluripotent at α just in case the following three conditions are satisfied:

(a) E_{top}^{N} ≠ ∅;

(b) μ < α < λ(E_{top}^{N}) where μ = cr(E_{top}^{N});

(c) ωϱ_{N}^{1} ≤ α.

We will frequently omit the reference to α, as α will be clear from the context. Thus, we will briefly write “pluripotent” instead of “pluripotent at α”. The main objects of our interest will be collapsing levels of extender models, in which case α will be implicitly determined. Definition 1.8. Let N be the collapsing structure for τ. N is pluripotent just in case that N is pluripotent at α where α is the cardinal predecessor of τ in N.

In the above definition, τ is a suppressed parameter. In the construction of a morass each suitable ordinal τ uniquely determines its collapsing structure N. The construction then depends on the pluripotency of N at the cardinal predecessor of α in N.

Under the above settings we define the notions of a divisor and strong divisor for N. We adopt the following conventions. Let N be the collapsing structure for τ.

• α(τ) denotes the cardinal predecessor of τ in N;

• n(τ) is the unique n ∈ ω satisfying ωϱ_{N}^{n+1} ≤ α < ωϱ_{N}^{n}. Definition 1.9. Let N be the collapsing structure for τ. Assume that μ ∈ On and q ∈ [On]^{<ω}. The pair (μ, q) is a divisor for N iff there is an ordinal λ = λ_{N}(μ, q) such that, setting α = α(τ), n = n(τ) and

r = r(μ, q) = \{ \begin{matrix}p_{N} − λ if N is not pluripotent \\ d_{N} − λ if N is pluripotent, \end{matrix}

the following clauses (F1) – (F4) hold.

(F1) μ ≤ ωϱ_{N}^{n+1} ≤ α < λ < ωϱ_{N}^{n} and μ < α.

(F2) q = \{ \begin{matrix}p_{N} ∩ λ, & if N is not pluripotent \\ d_{N} ∩ λ, & if N is pluripotent. \end{matrix}

(F3) h̃_{N}^{n+1}(μ ∪ {r}) ∩ ωϱ_{N}^{n} is cofinal in ωϱ_{N}^{n} if N fails to be pluripotent and h_{N}^{∗}(μ ∪ {r}) is cofinal in N if N is pluripotent.

CARDINAL TRANSFER PROPERTIES IN EXTENDER MODELS 13

(F4) (a) λ ∩ h̃_{N}^{n+1}(μ ∪ {r}) = μ and λ ∈ h̃_{N}^{n+1}(μ ∪ {r}) if N is not pluripotent;

(b) λ ∩ h_{N}^{∗}(μ ∪ {r}) = μ and λ ∈ h_{N}^{∗}(μ ∪ {r}) = μ if N is pluripotent. What makes the present situation different from that in [15] is that we have to

allow the possibility that ωϱ_{N}^{n+1} = α and ωϱ_{N}^{ω} < α.

Let us make some trivial remarks. First, observe that if (μ, q) is a divisor for N, then μ ≤ ωϱ_{N}^{n+1}. Indeed, (F1) together with (F4) imply that μ must be a limit cardinal in N^{∗}, and therefore in N. Hence μ is a cardinal in W. In our setting, ωϱ_{N}^{n+1} = α whenever τ < max(S_{α}). Finally notice two facts. First, r ∩ α = ∅. Second, no pair (μ, q) can be a divisor for N if N is active and (cr(E_{top}^{N}))^{+N} = α.

It is straightforward to verify that the arguments in Subsection 2.1 in [15] (subsection titled “Premice and Fragments”) go through when working with the above definitions. We can thus conclude that the transitive collapse N^{∗} of X = h̃_{N}^{n+1}(μ∪{r}) (resp. X = h_{N}^{∗}(μ ∪ {r})) is a premouse that is sound and solid, and in fact N^{∗} is an initial segment of N. For pluripotent N this requires an extra argument showing that γ_{N} is in X ([15], Lemma 2.4). If π is the associated uncollapsing map, then by [15], Lemma 2.5, we have:

(2) π(p_{N^{∗}}) = r whenever N is not pluripotent. For pluripotent N we have π(d_{N^{∗}}) = r and, moreover, if λ < λ(E_{top}^{N}) then q = p_{N} ∩ λ.

It follows that the description of q on the top line in (F2) also covers that on the bottom line of (F2) unless N is a type B premouse and λ = λ(E_{top}^{N}); in this case q = d_{N} and is in general different from p_{N}. The map π turns out to be the ultrapower embedding associated with the fine ultrapower Ult^{∗}(N^{∗}, F) where F = π ↾ (𝒫(μ) ∩ N^{∗}). Letting ϑ = μ^{+N^{∗}} and ν = π(ϑ), we see that M = ⟨J_{ν}^{E}, F ⟩ is a protomouse, N(M) = N and ϑ(M) = ϑ. Notation. Given N together with some divisor (μ, q) as in the above definition, we let

(3)  N(μ, q) = the protomouse determined by (μ, q).

In general, we will write N^{∗}(μ, q) for N^{∗}, ϑ(μ, q) for μ^{+N^{∗}} = ϑ(N(μ, q)), λ(N(μ, q)) for π(μ), F ((μ, q)) for π ↾ 𝒫(μ), i.e. for E_{top}^{M}, etc...

Our construction of a Gap-1 Morass will also require a generalization of the notion of a strong divisor. The reason for this is that if τ is the largest element

14 ERNEST SCHIMMERLING AND MARTIN ZEMAN

of S_{α} (see Definition 1.1) we do not have any control of the bottom part of the standard parameter p_{N} ∩ α where N is the singularizing structure for τ. Definition 1.10. Let N be the collapsing structure for τ and α = α(τ). A divisor (μ, q) for N is strong above β where α ≤ β < λ(N(μ, q)) just in case that for every ξ < μ and every x of the form h̃_{N}^{n+1}(ξ, p_{N} − β) (resp. h^{∗}(ξ, d_{N} − β), if N is pluripotent) we have x ∩ μ ∈ N^{∗}(μ, q).

A divisor (μ, q) for N is strong iff it is strong above ωϱ_{N}^{n+1}.

Of course, this notion trivializes if β > max(q). By (2), the characterization of strongness using h̃_{N}^{n+1}(ξ, p_{N} −β) also works for pluripotent N if λ < λ(E_{top}^{N}). (With n = 0, of course.) In the following we state some properties of strong divisors. We omit the proofs, as they are easy modifications of those in [15].

The property of being strong above β is equivalent to any of (S1) – (S3) below, unless N is pluripotent and λ = λ(E_{top}^{N}):

(S1) If x is of the form h̃_{N}^{n+1}(ξ, p_{N} − β) for some ξ < μ, then x ∩ μ = y ∩ μ for

some y ∈ H_{N}^{n} of the form h̃_{N}^{n+1}(ξ, r). (Here r = r(N(μ, q)).) (S2) 𝒫(μ) ∩ N^{∗}(μ, q) = 𝒫(μ) ∩ N′(μ, β) where N′(μ, β) is the transitive collapse

of h̃_{N}^{n+1}(μ ∪ {p_{N} − β}). (S3) |p_{N^{∗}(μ,q)}| = |p_{N′(μ,β)}|.

For pluripotent N with λ < λ(E_{top}^{N}) we can equivalently write h_{N}^{∗}(ξ, d_{N} − β) and h_{N}^{∗}(ξ, r) in (S1), and h_{N}^{∗}(μ∪{d_{N} −β}) in (S2). If N is pluripotent and λ = λ(E_{top}^{N}) then q = d_{N} ∩λ, and N′(μ, β) is defined using d_{N} −β instead of p_{N} −β. Again, the obvious reformulations of (S1) and (S2) will hold here. We also have the following characterization of strongness in this case, granting that d_{N} − β ≠ ∅:

(S4) (μ, d_{N}) is strong above β iff E_{top}^{N′(μ,β)} | μ ∉ N′(μ, β). Thus, the strongness of (μ, d_{N}) above β is equivalent to the failure of N′(μ, β) to satisfy the initial segment condition at μ.

The following is a refinement of two crucial properties of strong divisors ([15], Lemmata 2.24 and 2.26). For fixed q and β we have:

(S5) The set of all μ where (μ, q) is a divisor strong above β is closed in ωϱ_{N}^{n+1}

where n = n(τ). (S6) If (μ, q) is a divisor strong above β, μ̄ ≤ μ and q̄ is a proper bottom segment

of q such that q̄ ∩ β = p_{N} ∩ β, then (μ̄, q̄) is not a divisor.

CARDINAL TRANSFER PROPERTIES IN EXTENDER MODELS 15

Similarly as in the construction of a □_{κ}-sequence in [15] it will be crucial to identify certain divisors that would serve as a “canonical divisors”. The morass embeddings with target τ are determined by Skolem hulls over either the collapsing structure for τ or over a collapsing protomouse for τ. Each τ determines uniquely which of the two kinds of collapsing structures should be chosen; the “canonical divisor” guarantees the unique choice of the protomouse.

In the following we explain how to identify the “canonical divisor”. Fix a bottom segment q of p_{N} (resp. of d_{N}). If there were cofinally many μ < ωϱ_{N}^{n+1} such that (μ, q) is a strong divisor for N then (F1) – (F4) would also hold for μ = ωϱ_{N}^{n+1} with the exception of the requirement μ < α in (F1). The proof of (S5) would also work for μ = ωϱ_{N}^{n+1} and we could conclude that the pair (ωϱ_{N}^{n+1}, q) satisfy (S1). This is clearly impossible, as N is sound. We thus conclude that there are only boundedly many μ < ωϱ_{N}^{n+1} such that (μ, q) is strong. Consequently, there is always a largest such μ < ωϱ_{N}^{n+1}, granting that N admits at least one strong divisor of the form (μ, q). This is the key toward determining the “canonical divisor”.

In our setting, ωϱ_{N}^{n+1} will turn out to be either α or the cardinal predecessor of α in N. If (p_{N} − ωϱ_{N}^{n+1}) ∩ α = ∅ then “strong above α” has the same meaning as “strong”. It follows immediately from the previous paragraph that there is a divisor (μ, q) for N strong above α with largest possible μ, provided that there is at least one strong divisor. If (p_{N} − ωϱ_{N}^{n+1}) ∩ α ≠ ∅ then ωϱ_{N}^{n+1} is the cardinal predecessor of α in N. So again there is a divisor strong above α with the largest possible value of μ, this time for the trivial reason that ωϱ_{N}^{n+1} < α. (Notice that ωϱ_{N}^{n+1} may be the largest possible value of μ in this case.)

The above considerations suggest the following way of identifying the “canonical divisor”: If there are μ and q such that (μ, q) is strong above α then there is a largest such μ. The associated q is then uniquely determined as the shortest possible. That is, if (μ, q) and (μ̄, q̄) are divisors for N strong above α and μ is largest possible then q is a bottom segment of q̄.

Definition 1.11. (μ(N), q(N)) is the divisor (μ, q) for N strong above α with largest possible μ, granting that some such divisor for N exists.

16 ERNEST SCHIMMERLING AND MARTIN ZEMAN

Being a strong divisor is a Π_{2}^{(n)}-property. This fact was irrelevant in the proof of □_{κ}, but will be useful here. Given an ordinal α and a finite set of ordinals s, we let max^{α}(s) = max(s ∪ {α}). Lemma 1.12. Let N be the collapsing structure for τ, let α = α(τ) and let (μ, q) be such that μ < α and q is a bottom segment of p_{N} resp. of d_{N}, if N is pluripotent. Let further r = p_{N} −(max^{α}(q)+1) resp. r = d_{N} −(max^{α}(q)+1) if N is pluripotent.

(a) The statement “(μ, q) is a divisor for N” is uniformly Q^{(n)}. More precisely: (a1) There is a Q^{(n)}-formula ψ(v_{1}, v_{2}, v_{3}, v_{4}) in the language for premice

such that for every τ, N, α, (μ, q) and r as above we have: • (μ, q) is a divisor for N iff N ⊧ ψ(μ, max^{α}(q), r, α). assuming that either N is not pluripotent or λ(μ, q) < λ(E_{top}^{N}). (a2) There is a Π_{1}-formula ψ(v) in the language for coherent structures

such that for all τ, N and μ as above we have: • (μ, d_{N}) is a divisor for N iff N ⊧ ψ(μ). assuming that N is pluripotent and λ(μ, q) = λ(E_{top}^{N}).

(b) The statement “(μ, q) is a divisor for N strong above α” is uniformly Π_{2}^{(n)}. More precisely: (b1) There is a Π_{2}^{(n)}-formula ψ^{∗}(v_{1}, v_{2}, v_{3}, v_{4}) in the language for premice

such that for every τ, N, α, (μ, q) and r as above we have:

• (μ, q) is a divisor for N strong above α iff N ⊧ ψ^{∗}(μ, q −α, r, α) assuming that either N is not pluripotent or λ(μ, q) < λ(E_{top}^{N}). (b2) There is a Π_{2}-formula ψ^{∗}(v_{1}, v_{2}) in the language for coherent struc-

tures such that for every τ, N, α and μ as we have:

• (μ, d_{N}) is a divisor for N strong above α iff N ⊧ ψ^{∗}(μ, d_{N} − α) assuming that N is pluripotent and λ(μ, q) = λ(E_{top}^{N}).

Proof. For (a1), the formula ψ(μ, q − α, r, α) is the conjunction

(∀u^{n})(∃v^{n} ⊃ u^{n})(∃ξ^{n})(v^{n} = h̃_{N}^{n+1}(ξ^{n}, r)) & (∀ξ^{n})(∀ζ^{n})ψ̃(ξ^{n}, ζ^{n}, μ, q − α, r, α)

where ψ̃(ξ^{n}, ζ^{n}, μ, q − α, r, α) is the Σ_{1}^{(n)}-formula

[ξ^{n} < μ & ζ^{n} = h̃^{n+1}(ξ^{n}, r) & ζ^{n} < max^{α}(q − α) + 1] ⟶ ζ^{n} < μ.

The first formula in the above conjunction expresses (F3) and the second formula (F4) in the definition of a divisor. Notice that any reference to α is superfluous

CARDINAL TRANSFER PROPERTIES IN EXTENDER MODELS 17

unless q ⊆ α. The case (a2) is easy, as the formula ψ just says that μ is a cutpoint of E_{top}^{N}.

Regarding (b), we prove only the former part of the statement and leave the latter part where (μ, d_{N}) is a divisor to the reader. Notice first that in (S1), we can without loss of generality assume that x, y ∈ H_{M}^{n}. The formula ψ^{∗}(μ, q, r, α) is the conjunction of ψ(μ, q, r, α) and the formula

(∀x^{n})(∀ξ^{n} < μ)[x^{n} = h̃_{N}^{n+1}(ξ^{n}, ((q − α) ∪ r) ⟶ ψ̃^{∗}(x^{n}, μ, r)]

where ψ̃^{∗}(x^{n}, μ, r) is the Σ_{1}^{(n)}-formula

(∃η^{n} < μ)(∃y^{n})(y^{n} = h̃_{N}^{n+1}(η^{n}, r) & y^{n} ∩ μ = x^{n} ∩ μ)
⊓ ⊔(Lemma 1.12)

1.4. Protomice, Fine Structure and Condensation. We discuss fine structural lemmata for protomice relevant for the construction of a morass. The focus is on translating fine structural information between N and N(μ, q) where (μ, q) is a fixed divisor for N, as well as on condensation lemmata for protomice. We need two new condensation lemmata. One is a refinement of a lemma in [15] and will be applied to embeddings with critical points “at α^{+}”. The other deals with embeddings with critical points “at α” and will be applied to morass embeddings.

We will consider a coherent structure M = ⟨J_{ν}^{E}, F ⟩ together with a premouse N with the following properties:

(C1) J_{ν}^{E} is a premouse all of whose initial segments are sound and solid and F

is an extender at (μ, λ). (C2) dom(F) = 𝒫(μ) ∩ J_{ϑ}^{E}, ϑ = ϑ(M) < μ^{+M} and N^{∗} is the collapsing level for

ϑ in M. (C3) ωϱ_{N^{∗}}^{n+1} ≤ μ < ωϱ_{N^{∗}}^{n}.

(C4) π : N^{∗} ⟶_{F}^{∗} N.

Clauses (C1) – (C4) cover all cases where M is of the form N(μ, q) for some divisor (μ, q), and in fact are of higher generality. This higher generality will be needed in our arguments. In some cases we will consider M to be a pluripotent mouse instead of a structure as in (C1) – (C4); in all such cases we will explicitly mention this. Let us recall once more time that we adopt the following convention. Whenever we

18 ERNEST SCHIMMERLING AND MARTIN ZEMAN

deal with protomice we automatically use the language for coherent structures. So in (C1) – (C4), the language used for M is the language for coherent structures. On the other hand, for N^{∗} we use the language for premice. If μ < λ(E_{top}^{N^{∗}}) or N^{∗} is of type A or B then N^{∗} and N are premice of the same type, so we use the same kind of language for both N and N^{∗}. However, if μ = λ(E_{top}^{N^{∗}}) and N^{∗} is a type C premouse then the language for premice is the same as the language for coherent structures when dealing with N^{∗}. But the associated premouse N from (C4) may turn out to be a type B premouse, in which case the two languages differ. This kind situation will always be treated separately, if the circumstances require it. The issue here is important as it has a substantial effect on verifying coherency in combinatorial constructions. In general, to avoid any possible misunderstandings, we will explicitly say which language we use in all situations where the confusion is likely.

Given a finite s ⊂ λ, the standard witness W_{M}^{β,s} is the transitive collapse of the hull h_{M}(β ∪ {s}). Here h_{M} is the Σ_{1}-Skolem function with respect to the language for coherent structures. A generalized witness for β with respect to M and s is defined in the obvious way; see the beginning of Subsection 1.2.

For M, N as in (C1) – (C4) we obtain refined versions of results in [15], Section 2 summarized in (FS1) – (FS4). The refinement is needed, as in the construction of a morass we also have to consider the case ωϱ_{M}^{1} = μ < α(τ) where M is a collapsing protomouse for τ. (For comparison with the construction in [15], here α plays the same role as κ in [15].)

Notice first that the map π : N^{∗} → N is Σ_{0}^{(n)}-preserving and cofinal at the n-th level, as easily follows from the soundness of N^{∗} and general properties of fine ultrapowers. Also, ωϱ_{N}^{n+1} ≤ λ since N = h̃_{N}^{n+1}(λ ∪ {π(p_{N^{∗}})}). Let r = π(p_{N^{∗}}) − λ. If N is solid then r = p_{N} − λ, as follows by a standard argument. Recall that by the definition of R_{N}^{n+1}, a ∈ R_{N}^{n+1} iff h̃_{N}^{n+1}(ωϱ_{N}^{n+1} ∪ {a}) = N and similarly a ∈ R_{M}^{1} iff h_{M}(ωϱ_{M}^{1} ∪ {a}) = M.

(FS1) ωϱ_{M}^{1} = ωϱ_{N}^{n+1}.

(FS2) R_{N}^{n+1} ≠ ∅ just in case that R_{M}^{1} ≠ ∅. In fact, if q ⊆ λ is a finite set of ordinals then

r ∪ q ∈ R_{N}^{n+1} ⟹ q ∪ {ϑ(M)} ∈ R_{M}^{1} and
q ∈ R_{M}^{1} ⟹ r ∪ q ∪ {μ} ∈ R_{N}^{n+1},

granting that λ(E_{top}^{N^{∗}}) > μ. If λ(E_{top}^{N^{∗}}) = μ then R_{N}^{n+1} is computed in the language for coherent structures.

(FS3) If s ⊂ λ is finite and μ^{+M} ≤ β then

h̃_{N}^{n+1}(β ∪ {s ∪ r}) ∩ λ = h_{M}(β ∪ {s}) ∩ λ,

granting that λ(E_{top}^{N^{∗}}) > μ. If λ(E_{top}^{N^{∗}}) = μ then the equality holds with h_{N}^{∗}

in place of h̃_{N}^{n+1}; in this case r = ∅.

(FS4) If max{μ^{+M}, ωϱ_{M}^{1}} ≤ β < λ, s ⊆ λ is finite and W_{M}^{β,s} = ⟨J Ē ; ν̄, F̄⟩ then

W_{N}^{β,s∪r} = Ult^{∗}(N^{∗}, F̄) or ^{∗}W _{N}^{β,s} = Ult^{∗}(N^{∗}, F̄),

where the former holds if λ(E_{top}^{N^{∗}}) > μ and the latter otherwise. Recall also useful notation from [15]. If β < λ then β is of the form π(f)(s) where

f : ^{|q|+1}μ → μ is an element of N^{∗} and s is the decreasing sequence enumerating q ∪ {ξ} for some ξ < μ. We will write “π(f)(q, ξ)” in place of π(f)(s). The map f is an element of J_{ϑ}^{E} and is essentially a subset of ^{|q|+2}μ, so by a slight abuse of notation we can write F (f) instead of π(f). Thus, the equality

(4)  β = F (f)(q, ξ)

has the obvious meaning. We adopt similar notation when dealing with a subset x ⊆ λ in place of an ordinal β < λ. In this case f would be a function with rng(f) ⊂ 𝒫(μ), so f would be essentially a subset of ^{|q|+3}μ. Again, the equation x = F (f)(q, ξ) has the obvious meaning. The following useful proposition was essentially proved in [15]. It specifies how ordinals definable over N can be expressed in the form described in (4) and vice versa. Proposition 1.13. Assume (C1) – (C4). Let a ∈ N ∩ 𝒫(λ), r = π(p_{N^{∗}}) and x ∈ [λ]^{<ω}. Then a ∈ h̃_{N}^{n+1}(μ ∪ {r ∪ x}) just in case that there is a function f ∈ J_{ϑ}^{E} and ξ < μ such that a = F (f)(x, ξ).

20 ERNEST SCHIMMERLING AND MARTIN ZEMAN

Now turning to a different topic and no longer assuming (C1) – (C4), we state the two condensation lemmata. The lemmata refer to a fixed extender model W = L[E]. Recall the notion of collapsing structure from Definition 1.5. Recall also the notation introduced below Definition 1.8, in particular that α(τ) is the cardinal predecessor of τ in the collapsing structure for τ. The first condensation lemma is a slight refinement of Lemma 2.19 from [15].

Lemma 1.14. Let α = α(τ). Let M̄ be a coherent structure and let M satisfy one of the following:

(i) M = N(μ, q) where N is the collapsing structure for τ and (μ, q) is a divisor for N;

(ii) M is the collapsing structure for τ pluripotent at α.

Let σ : M̄ → M be Σ_{0}-preserving with critical point τ̄ = α^{+M̄} and σ(τ̄) = τ. Assume there is a finite set of ordinals q̄ such that:

(a) hM̄(α ∪ {q̄}) = M̄.

(b) For every β ∈ q̄ there is a generalized witness Qβ,q̄ ; M̄ ∈ M̄.

(c) σ(q̄) = \{ \begin{matrix}q − α, & if (i) above applies \\ d_{M}^{α}, & if (ii) above applies; \end{matrix}

Then M̄ ∈ M, ωϱ1 ; M̄ = α and q̄ = p1 ; M̄. Also q̄ = pN̄ ∩ λ(M̄) if λ(EM̄ ; ϑ(M̄)) < μ and

q̄ = dα ; N̄ if λ(EM̄ ; ϑ(M̄)) = μ where N̄ is the collapsing structure for τ̄. Finally (μ, q̄) is a divisor for N̄ and M̄ = N̄(μ, q̄).

We omit the proof of this lemma, as it is – up to some minor modifications – the same as the proof of [15], Lemma 2.19.

In the same sense that our first condensation lemma deals with maps that reflect τ while keeping α fixed, our second condensation lemma deals with maps that reflect α.

Lemma 1.15. Let α = α(τ). Let M̄ and M be coherent structures satisfying:

• M = N(μ, q) where N is the collapsing structure for τ and (μ, q) is a divisor for N;

• M̄ is not a potential premouse, i.e. its top extender is not weakly amenable.

CARDINAL TRANSFER PROPERTIES IN EXTENDER MODELS 21

Assume σ : M̄ → M is a Σ_{1}-preserving embedding with critical point ᾱ satisfying σ(ᾱ) = α and q̄ ∈ M̄ is a finite set of ordinals such that the following conditions are met:

(a) hM̄(ᾱ ∪ {q̄}) = M̄.

(b) For every β ∈ q̄ there is a generalized witness Qβ,q̄ ; M̄ ∈ M̄.

(c) σ(q̄) = q − α. Let N̄ = N(M̄) and μ = cr(E_{top}^{M}). Then q̄ = p1 ; M̄ − ᾱ. Also q̄ = (pN̄ ∩ λ(M̄)) − ᾱ if λ(EM̄ ; ϑ(M̄)) > μ and q̄ = dᾱ ; N̄ if λ(EM̄ ; ϑ(M̄)) = μ. Furthermore, exactly one of the following two clauses holds:

(A) N̄ is sound. In this case N̄ is the collapsing structure for τ̄ = ᾱ^{+M̄} and

there is a divisor (μ, q̄^{∗}) for N̄ such that q̄ = q̄^{∗} − ᾱ and M̄ = N̄(μ, q̄^{∗}). (B) N̄ is not sound. Let N̄^{∗} = N^{∗}(M̄). In this case α is a successor cardinal

in N̄, so there is an ordinal κ such that ᾱ = κ^{+N̄}. Moreover, ωϱn+1 ; N̄ = κ where n = n(τ), so ωϱn+1 ; N̄^{∗} = κ < ωϱn ; N̄^{∗}. If λ(EN̄^{∗} ; top) > μ then

N̄ = h̃n+1 ; N̄(κ ∪ {r̄, q̄, κ})

where r̄ = π̄(p_{N^{∗}}) and π̄ is the ultrapower map coming from Ult^{∗}(N̄^{∗}, EM̄ ; top).

If λ(EN̄^{∗} ; top) = μ then N̄ = h∗ ; N̄(κ ∪ {q̄, κ}).

Proof. We first verify that N̄ is as in the statement of the lemma. Let F̄ and F be the top extenders of M̄ and M, respectively. Obviously, cr(F̄) = μ = cr(F). Because neither of F̄ and F is weakly amenable, we have ϑ(M) < α and ϑ(M̄) < ᾱ. Letting E = E^{W}, notice that 𝒫(μ) ∩ M̄ ⊂ J E ; ᾱ and 𝒫(μ) ∩ M ⊂ J_{α}^{E}. The fact that a ⊂ μ is measured by F̄ can be expressed by the Σ_{1}-formula (∃y)(⟨a, y⟩ ∈ F̄) and similarly for F. Now σ is Σ_{1}-preserving, so for a ∈ 𝒫(μ) ∩ M̄ we have

a ∈ dom(F̄) ⟷ a ∈ dom(F).

This immediately implies that ϑ(M̄) = ϑ(M).

By the previous paragraph, N^{∗}(M̄) = N^{∗}(M); we will write briefly N^{∗}. Let n be such that ωϱ_{N^{∗}}^{n+1} ≤ μ < ωϱ_{N^{∗}}^{n}. Let N̄ = N(M̄) = Ult^{∗}(N^{∗}, F̄) and N = N(M) = Ult^{∗}(N^{∗}, F) with the associated ultrapower maps π̄ and π. We define a map σ̃ : N̄ → N by

σ̃(π̄(f)(β)) = π(f)(σ(β))

22 ERNEST SCHIMMERLING AND MARTIN ZEMAN

for f ∈ Γ(μ, N^{∗}) and β < λ(F̄). The map σ̃ is easily seen to be Σ_{0}-preserving, which guarantees that N̄ is well-founded, and we will consider it to be transitive. Standard methods [21] then show that σ̃ is actually Σ_{0}^{(n)}-preserving. The definition of σ̃ immediately implies π = σ̃ ∘ π̄ and σ̃ ↾ λ(F̄) = σ ↾ λ(F̄). This means that ᾱ = cr(σ̃).

To see that N̄ is as in the conclusion of the lemma we apply the Condensation Lemma (Lemma 1.4). In the following we verify that N̄ satisfies the assumptions of the lemma. Assume first that λ(E_{top}^{N^{∗}}) > μ. Let r̄ = π̄(p_{N^{∗}}). Clause (a) in the statement of the current lemma together with (FS3) guarantee that

h̃n+1 ; N̄(ᾱ ∪ {r̄, q̄}) = h̃n+1 ; N̄(λ(M̄) ∪ {r̄}) = N̄.

This alone implies that ωϱn+1 ; N̄ ≤ ᾱ, so N̄ is solid by the first part of the Condensation Lemma. The solidity of N̄, the above fact on the hulls and (FS4) then altogether yield that N̄ is sound above ᾱ ([21], Lemma 1.12.5). This completes the verification in the case where λ(E_{top}^{N^{∗}}) > μ. Now assume λ(E_{top}^{N^{∗}}) = μ. In this case we apply (DP2) additionally to (FS4) to conclude that q̄ = dᾱ ; N̄ and N̄ is sound above ᾱ. However, we need to do some additional work to see that N̄ is a solid premouse. Notice that it is not automatically clear that N̄ satisfies the initial segment condition; the issue arises since N̄ is a type B potential premouse, whereas N^{∗} might have been a type C premouse. Here we use the latter part of (FS4) asserting that ^{∗}W β,q̄ ; N̄ ∈ N̄ for β = max(q̄). This β is obviously ≥ λ^{∗}(N̄), which is sufficient to guarantee that EN̄ ; top | λ^{∗}(N̄) ∈ N̄. Also, σ̃ is not necessarily an embedding between two premice, since it only preserves Σ_{0} formulae in the language of coherent structures if N^{∗} is of type C. To take care of this, notice that if Ñ = N||γ̃ where γ̃ is the index of E_{top}^{N} | λ̃ and λ̃ = sup(σ̃′′λ(F̄)) then σ̃ : N̄ → Ñ is as needed. This completes the verification in the remaining case where λ(E_{top}^{N^{∗}}) = μ.

We are now ready to apply the Condensation Lemma to N̄ and σ̃. If follows that one of the clauses (b), (c) or (d) of the Condensation Lemma must apply. To see that (a) cannot be true notice that either ωϱ_{N}^{n+1} = α in which case N̄ and N have different (n + 1)-st projecta or else α is a successor cardinal in N (here we use the fact that α is cardinal-correct in the collapsing structure for τ) in which case we have some κ such that κ^{+N̄} = ᾱ < α = κ^{+N}. In either case N̄ ≠ N.

CARDINAL TRANSFER PROPERTIES IN EXTENDER MODELS 23

Let q̄^{∗} = pN̄ ∩ λ(F̄) if λ(E_{top}^{N^{∗}}) > μ and q̄^{∗} = dᾱ ; N̄ otherwise. That q̄ is as in the statement of the lemma, i.e. q̄ = q̄^{∗} − ᾱ, follows from the above discussion and some easy applications of (FS1) – (FS4). If (a) or (d) of the Condensation Lemma applies then N̄ is as in (A) in the statement of the current lemma. That M̄ = N̄(μ, q̄^{∗}), follows from the way we defined N̄. If (c) of the Condensation Lemma applies then N̄ is as in (B) in the statement of the current lemma. To see this notice that N̄ = Ult^{∗}(Q, E_{δ}^{N}) where Q is the collapsing level for ᾱ in N, δ ≤ ht(Q) and E_{δ}^{N} has a single generator κ = cr(E_{δ}^{N}). This κ is a cardinal predecessor of ᾱ in Q, and so in N̄. By general properties of fine ultrapowers, no projectum of N̄ can be in the interval (κ, π_{Q}(κ)] where π_{Q} is the associated ultrapower map. We know that ωϱn+1 ; N̄ ≤ ᾱ, so necessarily ωϱn+1 ; N̄ ≤ κ. Now κ is a cardinal in N and N̄ ∈ N, so we actually have ωϱn+1 ; N̄ = κ. As E_{δ}^{Q} is an extender from the Q-sequence, π_{Q}(p_{N^{∗}}) = pN̄. From properties of fine ultrapowers we then see that N̄ = h̃n+1 ; N̄(κ ∪ {pN̄, κ}), so N̄ is sound above κ + 1. Notice that min(pN̄) ≥ π_{Q}(κ) > ᾱ. If λ(E_{top}^{N^{∗}}) > μ this together with the equality q̄ = (pN̄ ∩λ(F̄)−ᾱ proved above tells us that q̄ = pN̄ ∩ λ(F̄), so the soundness of N̄ above κ + 1 gives the first part of (B). If λ(E_{top}^{N^{∗}}) = μ then κ̄ =^{def} cr(EN̄ ; top) = cr(EN̄^{∗} ; top) < μ. (Recall N̄^{∗} = N^{∗}(M̄) = N^{∗}.) As Q is sound, d_{Q} exists and Q is Dodd solid by (DP1). Working in the language for coherent structures, general properties of fine ultrapowers yield N̄ = h∗ ; N̄(κ ∪ {π_{Q}(d_{Q}), κ}). Using (DP2), we infer π_{Q}(d_{Q}) = dᾱ ; N̄ − α = q̄ where the latter equality was established above. The conclusion then follows immediately.

⊓ ⊔(Lemma 1.15)

2. The Morass Construction

We now give a proof of Theorem 1.2, that is, we describe a construction of the Gap-1 Morass introduced in Definition 1.1. The construction uses techniques that resemble – to some extent – to the construction of a □_{κ}-sequence in [15], in that we split the class of all ordinals of interest into two disjoint subclasses, say 𝒯^{0} and 𝒯^{1}, and define collapsing structures on each of these classes separately. However, in contrast with that construction, it is not possible to define morass embeddings on each of these classes separately, as the construction will force us to consider morass embeddings σ_{τ̄,τ} where τ̄ ∈ 𝒯^{0} and τ ∈ 𝒯^{1}. Apart from this there are other

24 ERNEST SCHIMMERLING AND MARTIN ZEMAN

differences. For instance, in order to verify certain morass axioms for σ_{τ̄,τ} where both τ̄ and τ are in 𝒯^{0}, we need some information on all morass embeddings σ_{τ̄′,τ′} where τ′ <_{0} τ, and so we also have to consider ordinals τ′ ∈ 𝒯^{1}.

The section is divided into two subsections. In the first subsection we define the morass. The morass axioms are verified in the second subsection.

2.1. Defining the Morass. We begin with the definition of S(0) and S(1). Recall that we work in a fixed extender model W = L[E]. In order to avoid discussing certain trivial issues, it will be convenient to restrict ourselves to ordinals with sufficient closure properties. Definition 2.1. An acceptable structure J_{α}^{A} is good just in case that J_{ζ}^{E} is a ZFC^{−}-model for arbitrarily large ζ < α.

We then set:

S(0) = the class of all cardinal-correct ordinals α such that J_{α}^{E} is good.

To define S_{α} for α ∈ S(0), we proceed by cases.

(a) α is regular. Then S_{α} = S(0) ∩ (α, α^{+}).

(b) α is a singular cardinal. We put τ into S_{α} just in case that τ ∈ S(0), τ > α and α is regular in J_{τ}^{E}.

(c) α is not a cardinal. We distinguish two cases.

(c1) E_{α} = ∅. We let τ ∈ S_{α} just in case that J_{τ}^{E} is good, α is the largest

cardinal in J_{τ}^{E} and is regular in J_{τ}^{E}. (c2) E_{α} ≠ ∅. Let W′_{α} = Ult(W, E_{α}) and E′ = E^{W′_{α}}. We let τ ∈ S_{α} just in

case that J_{τ}^{E′} is good and α is the largest cardinal in J_{τ}^{E′}.

If (c2) in the above definition applies then λ(E_{α}) is a limit cardinal in W, as α is cardinal-correct. Also, α fails to be a cardinal in J_{α+1}^{E} in this case. Notice that α is a successor cardinal in J_{τ}^{E′}, so there is no need to demand the regularity of α in J_{τ}^{E′} in (c2).

That S(0) and all S_{α} satisfy all requirements in the definition of a morass stated above the morass axioms is obvious, unless E_{α} ≠ ∅. In this case, max(S_{α}) = α^{+W′_{α}} = π_{α}(ν) where ν = cr(E_{α})^{++W} and π_{α} is the ultrapower map associated with Ult(W, E_{α}). Since λ is a cardinal in W, the ultrapower map associated with Ult(J_{α}^{E}, E_{α}) agrees with π_{α} on all elements of J_{λ}^{E}, and since λ is a limit cardinal,

CARDINAL TRANSFER PROPERTIES IN EXTENDER MODELS 25

ν < λ. If α′ ∈ S(0) is larger than α then ⟨J_{α}^{E}, E_{α}⟩ ∈ J_{α′}^{E}. By the goodness of α′, the ultrapower Ult(J_{α}^{E}, E_{α}) together with its associated ultrapower map are in J_{α′}^{E} and so π_{α} ↾ λ ∈ J_{α′}^{E}. It follows that max(S_{α}) = π_{α}(ν) ∈ J_{α′}^{E}. Following the definition of a morass, we set

S(1) = ⋃{S_{α}; α ∈ S(0)}.

The next task is to define the morass ordering ≺ and morass embeddings σ_{τ̄,τ}. Both are induced by hulls over canonically chosen collapsing structures for τ. We chose such structures for all τ ∈ S(1) − S^{+} where

τ ∈ S^{+} ⟷ τ = max(S_{α}) for some α ∈ S(0) with E_{α} ≠ ∅.

We are thus ignoring the elements of S^{+} in our construction. This might look suspicious at first glance, but notice that the morass axiom (M3) allows the option that each τ ∈ S^{+} is minimal with respect to the morass ordering ≺. Furthermore, it will be clear from the construction that situations where τ ≺ τ′ and τ ∈ S^{+} never occur. Before turning to the definition of our collapsing structures we set:

N_{τ} = the collapsing structure for τ (see Definition 1.5)
E^{τ} = E^{N_{τ}}
n_{τ} = n(τ) (see remarks below Definition 1.8)
p_{τ} = p_{N_{τ}}
d_{τ}^{β} = d_{N_{τ}}^{β} whenever applicable
h̃_{τ} = h̃_{N_{τ}}^{n_{τ} +1}.

Notice that these definitions do make sense for any τ that is a local successor of α in W or in Ult(W, E_{α}) (more precisely: for τ such that α is the largest cardinal in J_{τ}^{E} or J_{τ}^{E′}) and we will occasionally use this fact. Notice also that if τ is not the largest element of S_{α}, then

(5)  ωϱ_{N_{τ}}^{n_{τ} +1} = ωϱ_{N_{τ}}^{ω} = α(τ).

Indeed, the ultimate projectum of N_{τ} cannot drop below α(τ), as otherwise α(τ) would be collapsed in any J_{τ′}^{E} where τ < τ′ ∈ S_{α(τ)}. Thus, in this case we

26 ERNEST SCHIMMERLING AND MARTIN ZEMAN

have p_{τ} = p_{N_{τ}}. If τ is the largest element of S_{α}, then the only way the ultimate projectum can drop below α is the following:

(6)  ωϱ_{N_{τ}}^{ω} = κ where κ^{+N_{τ}} = α(τ).

This is the case because α is cardinal-correct in W: As κ is a cardinal in W, any bounded subset x ∈ W of κ must be an element of J_{κ}^{E} ⊆ N_{τ}. However, it does not follow that ωϱ_{N_{τ}}^{n_{τ} +1} = κ. It is straightforward to find examples of ordinals τ such that cf(α(τ)) > ω and there are unboundedly many ᾱ < α(τ) that are of the form ωϱ_{Nτ̄}^{nτ̄+1} = α(τ̄) for some τ̄ and at the same time ωϱ_{Nτ̄}^{nτ̄+2} < ᾱ (in fact, such τ̄ satisfy τ̄ ≺ τ for ≺ introduced below). Finally recall that if ωϱ_{N_{τ}}^{ω} < α(τ) then p_{τ} ∩ α(τ) has at most one element.

We now turn to the collapsing structures. As already mentioned above, we split S(1) into two disjoint classes 𝒯^{0} and 𝒯^{1}, and define the collapsing structures on each of these classes separately.

• 𝒯^{0} = S(1) − 𝒯^{1}.

• 𝒯^{1} is the class of all τ ∈ S(1) satisfying: (a) α(τ) is a successor cardinal in N_{τ}, say α(τ) = μ^{+N_{τ}}.

(b) There is a q such that (μ, q) is a divisor for N_{τ} strong above α(τ). If τ ∈ 𝒯^{1} and (μ, q) is as in (b) then (μ, q) is a divisor for N_{τ} strong above α(τ) with largest possible μ, so (μ, q) = (μ(N_{τ}), q(N_{τ})); see Definition 1.11. We let

(μ_{τ} , q_{τ}) = (μ(N_{τ}), q(N_{τ}))

Definition 2.2. Given a τ ∈ S(1), we define the canonical structure M_{τ} as follows.

M_{τ} = \{ \begin{matrix}N_{τ} & if τ ∈ 𝒯^{0} \\ N_{τ} (μ_{τ} , q_{τ}) & if τ ∈ 𝒯^{1} \end{matrix}

Obviously, M_{τ} is a collapsing structure for τ. For τ ∈ 𝒯^{1} the structure M_{τ} agrees with N_{τ} up to λ(E_{top}^{M_{τ}})^{+N_{τ}}, that is, far beyond α(τ). For τ ∈ 𝒯^{1} we further set:

λ_{τ} = λ(N_{τ} (μ_{τ} , q_{τ}))
ϑ_{τ} = ϑ(M_{τ})
r_{τ} = r(N_{τ}) = p_{τ} − q_{τ}
h_{τ} = h_{M_{τ}} .

CARDINAL TRANSFER PROPERTIES IN EXTENDER MODELS 27

By the above definition, the only divisors we are considering are those with μ_{τ} being the cardinal predecessor of α(τ). This makes the choice of the canonical divisor straightforward when comparing the current situation with that in the construction of a square sequence where we have to allow more possibilities for μ_{τ}.

The discussion above Definition 1.11 yields the characterization of (μ_{τ} , q_{τ}) as the divisor (μ, q) for N_{τ} strong above α(τ) with largest possible μ and shortest possible q. This characterization be considered an alternative definition of (μ_{τ} , q_{τ}), and indeed the entire definition of a Gap-1 Morass can be dubbed with the choice of the canonical divisor based on this characterization. Notice that if we replace the requirement that (μ_{τ} , q_{τ}) is strong above α with the requirement that q_{τ} is shortest possible, the definition still does make sense, but most likely does not define a Gap-1 Morass. We will namely run into difficulties when trying to verify that the new definition really yields a morass. The difficulties occur in the verification of axioms (M4) and (M5) in the crucial Case 2. In those arguments, the only known way to verify that the interpolation argument yields a protomouse together with its canonical divisor is to use the fact that we start from a protomouse associated with a divisor which is strong above the corresponding α(τ). And we find it plausible that there are levels N_{τ} that admit divisors of the form (μ, q) where μ is the cardinal predecessor of α(τ) and q is shortest possible but (μ, q) is not strong above α(τ). Whether it is possible to eliminate any reference to strong divisors in the construction of a Gap-1 Morass is unclear, though we find it unlikely.

The next task is to define the morass ordering ≺ and the morass embeddings σ_{τ̄,τ}. Given τ ∈ S(1), we state a list of conditions that determine the ≺-predecessors of τ. We will treat the two cases τ ∈ 𝒯^{0} and τ ∈ 𝒯^{1} separately. We stress again that the definition of ≺ allows the possibility that τ ∈ 𝒯^{1} and τ̄ ∈ 𝒯^{0} for τ̄ ≺ τ.

Definition 2.3. Given τ ∈ 𝒯^{0} and τ̄ ∈ S(1), we let τ̄ ≺ τ just in case the following hold. Let (ᾱ, α) = (α(τ̄), α(τ)).

(T0A) τ̄ ∈ 𝒯^{0}. (T0B) nτ̄ = n_{τ}. (T0C) There is a Σ_{1}^{(n_{τ})}-preserving embedding σ : Nτ̄ → N_{τ} satisfying:

(T0C1) ᾱ = cr(σ) and σ(ᾱ) = α. If τ ∈ N_{τ} then τ̄ ∈ Nτ̄ and σ(τ̄) = τ.

(T0C2) σ(pτ̄ − ᾱ) = p_{τ} − α.

28 ERNEST SCHIMMERLING AND MARTIN ZEMAN

(T0C3) σ is Q-preserving.

Recall that a map σ : N̄ → N is Q-preserving just in case that σ preserves all Q-formulae, that is, formulae of the form (∀u)(∃v ⊇ u)ψ(v) where ψ(v) is a Σ_{1}-formula that has no free occurence of u. (T0C3) is superfluous except in the case where n_{τ} = 0 and τ is the height of N_{τ}, in which case we need (T0C3) to verify morass axiom (M0). This is the only reason for including (T0C3) in Definition 2.3. Actually, a much weaker property than Q-preservation is sufficient for the verification, so we work with stronger (T0C3) only for convenience.

By Definition 2.2 the structures M_{τ} and N_{τ} are equal for τ ∈ 𝒯^{0}, so we can write M_{τ} instead of N_{τ} in Definition 2.3. Given τ̄ and τ as in Definition 2.3, we have σ : h̃τ̄(ξ, pτ̄ − ᾱ) ↦ h̃_{τ} (ξ, p_{τ} − α) whenever ξ < ᾱ. It follows that the map σ is uniquely determined and its range is precisely h̃_{τ} (ᾱ ∪ {p_{τ} − α}).

The requirement that σ(τ̄) = τ in (T0C1) is superfluous and we included it only for convenience. Indeed, if τ ∈ N_{τ} then τ̄ ∈ Nτ̄ as required in (T0C1) and σ(τ̄) is a cardinal in N_{τ}, as σ is Σ_{1}-preserving. So actually σ(τ̄) = τ, as α is the largest cardinal in J_{σ(τ̄)}^{E^{τ}}. In most cases the requirement τ ∈ N_{τ} ⟹ τ̄ ∈ Nτ̄ is also superfluous, as one can prove that if τ ∈ N_{τ} then τ ∈ rng(σ). However, if n_{τ} = 0, N_{τ} is passive and ht(N_{τ}) is a successor ordinal we do not see how to prove that τ ∈ rng(σ) in general. Actually, we believe this is false. It is possible that neither of the requirements on τ̄ and τ is necessary, since we always have σ′′τ̄ ⊆ τ, no matter whether we impose any requirements on τ̄ and τ or not. Even then, the construction becomes much cleaner if we include them.

The morass map σ_{τ̄,τ} will be just the obvious restriction of σ, i.e.

σ_{τ̄,τ} = σ ↾ τ̄.

It will be convenient to disregard the distinction between the two maps and write σ_{τ̄,τ} to denote both. Clearly, there is no danger of confusion.

Because σ_{τ̄,τ} is entirely determined by the hull h̃_{τ} (ᾱ ∪ {p_{τ} − α}), we conclude that

(7)  ≺ ↾ (S(1) × 𝒯^{0}) = ≺ ↾ (𝒯^{0} × 𝒯^{0})

is a tree ordering. Indeed, if τ^{∗} ≺ τ then τ ∈ 𝒯^{0}. Letting α^{∗} = α(τ^{∗}), we have rng(σ_{τ^{∗},τ}) ⊂ rng(σ_{τ̄,τ}) or vice versa, depending on whether α^{∗} ≤ ᾱ or ᾱ ≤ α^{∗}. It

CARDINAL TRANSFER PROPERTIES IN EXTENDER MODELS 29

is clear that σ_{τ̄,τ}^{−1} ∘ σ_{τ^{∗},τ} witnesses that τ^{∗} ≺ τ̄ in the former case and σ_{τ^{∗},τ}^{−1} ∘ σ_{τ̄,τ} witnesses that τ̄ ≺ τ^{∗} in the latter case.

We now complete the definition of the morass ordering. For τ ∈ 𝒯^{1}, we defined μ_{τ} , q_{τ} , λ_{τ} and h_{τ} above. If τ ∈ 𝒯^{0} and N_{τ} is a pluripotent premouse with cr(E_{top}^{N_{τ}}) being the cardinal predecessor of α(τ) in N_{τ}, we let

μ_{τ} = cr(E_{top}^{N_{τ}})
ϑ_{τ} = ϑ(N_{τ})  (= μ_{τ}^{+N_{τ}})
q_{τ} = d_{N_{τ}}
λ_{τ} = λ(E_{top}^{N_{τ}})
h_{τ} = h_{N_{τ}}^{∗} .

The morass ordering ≺ ↾ S(1) × 𝒯^{1} is defined as follows. Definition 2.4. Given a τ ∈ 𝒯^{1} and a τ̄ ∈ S(1), we let τ̄ ≺ τ just in case (T1A), (T1B) and (T1C) below hold with (ᾱ, α) = (α(τ̄), α(τ)). (T1A) τ̄ satisfies either (T1A1) or (T1A2) below.

(T1A1) τ̄ ∈ 𝒯^{1};

(T1A2) τ̄ ∈ 𝒯^{0} and Mτ̄ = Nτ̄ is a pluripotent premouse such that cr(E_{top}^{M_{τ}}) is

the cardinal predecessor of α(τ) in M_{τ}.

(T1B) There is an embedding σ : Mτ̄ → M_{τ} that is Σ_{1}-preserving with respect to the language for coherent structures and such that both (T1B1) and (T1B2) below hold.

(T1B1) ᾱ = cr(σ) and σ(ᾱ) = α. (T1B2) σ(qτ̄ − ᾱ) = q_{τ} − α.

(T1C) If τ̄ ∈ 𝒯^{0} then there is a τ̃ ∈ 𝒯^{0} with α(τ̃) = ϑ_{τ} such that (T1A) and (T1B)

hold with (τ̄, ᾱ) replaced by (τ̃, α(τ̃)). To motivate the above definition we remark that the following situation is pos-

sible: τ̄, τ ∈ 𝒯^{0} and τ̄ ≺ τ but if we let τ′ = sup(σ′′_{τ̄,τ} τ̄), then τ ∈ 𝒯^{1}. In this sense, the elements of 𝒯^{1} can be viewed as the missing limits. When this situation arises, M_{τ} = N_{τ} is pluripotent and α(τ) is the cardinal successor of cr(E_{top}^{N_{τ}}) in N_{τ}. The same is true with τ replaced by τ̄. On the other hand, M_{τ′} is a protomouse that Σ_{0}-embeds into N_{τ}.

30 ERNEST SCHIMMERLING AND MARTIN ZEMAN

We can now make remarks similar to those for τ ∈ 𝒯^{0}. As before we can see that σ_{τ̄,τ} : hτ̄(ξ, qτ̄) ↦ h_{τ} (ξ, q_{τ}), so σ_{τ̄,τ} is uniquely determined and that it is the uncollapsing map associated with its range h_{τ} (ᾱ∪{q_{τ} −α}). That τ ∈ M_{τ} is trivial, since λ_{τ} > τ. From this we immediately have that τ ∈ rng(σ_{τ̄,τ}). Notice also that μτ̄ = μ_{τ}. The reason for including clause (T1C) will be explained below.

The verification of the fact that ≺ ↾ (S(1) × 𝒯^{1}) is a tree ordering requires a little argument. First of all, we have to argue that ≺ is transitive. More precisely, we want to see that τ^{∗} ≺ τ̄ and τ̄ ≺ τ imply τ^{∗} ≺ τ. The point here is to show that σ′ = σ_{τ̄,τ} ∘ σ_{τ^{∗},τ̄} satisfies (T1B) with τ^{∗} in place of τ̄, and thereby witnesses that τ^{∗} ≺ τ. This is obvious, unless τ ∈ 𝒯^{1} and τ̄ ∈ 𝒯^{0} with Nτ̄ of type B. In this case the map σ_{τ^{∗},τ̄} is a map between two premice, and the conditions (T0A) – (T0C) hold with (τ^{∗}, τ̄) in place of (τ̄, τ). What we need to see is that (T1A) – (T1C) are satisfied with (τ^{∗}, τ̄) in place of (τ̄, τ). The language for premice is richer than that for coherent structures, so σ_{τ^{∗},τ̄} is automatically Σ_{1}-preserving in this weaker sense. Comparing the definitions of ≺ for 𝒯^{0} and 𝒯^{1}, it is clear that only (T1B2) needs a verification. By (DP1), it suffices to show that eᾱ ; Nτ̄ ∈ rng(σ_{τ^{∗},τ̄}) where ᾱ = α(τ̄), as the facts that σ_{τ^{∗},τ̄} is Σ_{1}-preserving with respect to the language for coherent structures and σ_{τ^{∗},τ̄}(γ_{N_{τ^{∗}}}) = γ_{Nτ̄} guarantee that σ_{τ^{∗},τ̄}^{−1}(eᾱ ; Nτ̄) is equal to e_{N_{τ^{∗}}}^{α^{∗}}. Here of course, α^{∗} = α(N_{τ^{∗}}). Now σ_{τ^{∗},τ̄} satisfies (T0C) with (τ^{∗}, τ̄) in place of (τ̄, τ), so we conclude that γ_{Nτ̄} ∈ hτ̄(ᾱ ∪ {p_{Nτ̄} ∪ e}) for some e ∈ rng(σ_{τ^{∗},τ̄}). Let ē be the <^{∗}-least such e; we claim that ē = eᾱ ; Nτ̄. Otherwise eᾱ ; Nτ̄ <^{∗} ē, so rng(σ_{τ^{∗},τ̄}) would contain some e′ <^{∗} ē such that γ_{Nτ̄} ∈ hτ̄(ᾱ ∪ {p_{Nτ̄} ∪ e′}), a contradiction with the minimality of ē.

We also have to check that {τ̄ ∈ S(1); τ̄ ≺ τ} is linearly ordered by ≺. That is, given τ^{∗} ≺ τ and τ̄ ≺ τ, we want to see that τ^{∗} and τ̄ are compatible. Say, τ^{∗} < τ̄. If τ̄ ∈ 𝒯^{1} or Nτ̄ is not of type B, the conclusion follows by the same argument as in the case τ ∈ 𝒯^{0}. Now assume that τ̄ ∈ 𝒯^{0} and Nτ̄ is of type B. Letting σ^{∗} = σ_{τ̄,τ}^{−1} ∘ σ_{τ^{∗},τ}, we want to verify that σ^{∗} witnesses τ^{∗} ≺ τ̄. This time σ^{∗} is Σ_{1}- preserving with respect to the language for coherent structures. We show that N_{τ^{∗}} is a type B premouse and σ^{∗}(γ_{τ^{∗}}) = γτ̄; from this it automatically follows that σ^{∗} is Σ_{1}-preserving with respect to the language for premice. This de facto verifies τ^{∗} ≺ τ̄, as (T0C2) is an immediate consequence of (DP1) and of the Σ_{1}-elementarity of σ^{∗}.

CARDINAL TRANSFER PROPERTIES IN EXTENDER MODELS 31

So let us turn to the verification that N_{τ^{∗}} is a type B premouse and σ^{∗}(γ_{τ^{∗}}) = γτ̄. As Nτ̄ is a type B premouse, dα(τ̄) ; τ̄ ≠ ∅ and its largest element β̄ is at least λ_{Nτ̄}^{∗}. ^{1} It follows that d_{τ^{∗}}^{α(τ^{∗})} ≠ ∅ and σ^{∗}(β^{∗}) = β̄ ≥ λ_{Nτ̄}^{∗} where β^{∗} = max(d_{τ^{∗}}^{α(τ^{∗})}). Letting λ^{∗} = {ξ ∈ On | σ^{∗}(ξ) < λ_{Nτ̄}^{∗}}, obviously λ^{∗} ≤ β^{∗} < λ(E_{top}^{N_{τ^{∗}}}), and it is easy to verify that λ^{∗} is a cutpoint of E_{top}^{N_{τ^{∗}}}: For f : μ_{τ^{∗}} → μ_{τ^{∗}} in N_{τ^{∗}} and every α < λ^{∗} we have σ^{∗}(E_{top}^{N_{τ^{∗}}} (f)(α)) = E_{top}^{Nτ̄}(f)(σ^{∗}(α)) < λ_{Nτ̄}^{∗}, so E_{top}^{N_{τ^{∗}}} (f)(α) < λ^{∗}. It follows that N_{τ^{∗}} is a type B premouse, since it has a cutpoint and it cannot be a type C premouse for the trivial reason that n_{τ^{∗}} = 0. From the definition of λ^{∗} we get σ^{∗}(λ^{∗}) ≥ λ_{Nτ̄}^{∗}, so the equality must hold as σ^{∗}(λ^{∗}) is a cutpoint of Nτ̄. Thus λ^{∗} is the largest cutpoint of N_{τ^{∗}} ^{2}, and the Σ_{1}-elementarity of σ^{∗} with respect to the language of coherent structures then yields the equality σ^{∗}(γ_{τ^{∗}}) = γτ̄.

The above discussion verifies that ≺ is a tree ordering on S(1). We can now focus on the verification of the morass axioms. Before we turn to this, let us make the following remark. By the definition of ≺, any ≺-predecessor of an element of 𝒯^{0} must again be in 𝒯^{0}. On the other hand, we allow that a ≺-predecessor of an element of 𝒯^{1} can be in 𝒯^{0}. If this happens, say τ̄ ∈ 𝒯^{0} is a ≺-predecessor of some τ ∈ 𝒯^{1}, then the set of all ≺-predecessors of τ splits into two ≺-intervals [τ̂, τ̃] and (τ̃, τ) where τ̃ is given by (T1C), the former interval is contained in 𝒯^{0} and the latter in 𝒯^{1}. We included requirement (T1C) in Definition 2.4 because in situations where τ ∈ 𝒯^{1}, τ̄ ∈ 𝒯^{0} and σ : Mτ̄ → M_{τ} is as in (T1B) we do not have any Condensation Lemma that would allow us to conclude that Mτ̄ is a level of L[E]. We will give more details on this issue below the verification of morass axiom (M3). It is not clear to us whether (T1C) can be omitted.

2.2. Morass Axioms. In this section we will systematically go over the morass axioms and verify that the definitions from the section 2.1 actually yield a realization of Gap-1 Morass in W.

Axiom (M0). Recall first that it is E_{α} alone that decides whether the membership to S_{α} is determined in W or in Ult(W, E_{α}). If E_{α} = ∅ then for each τ, the membership “τ ∈ S_{α}” is determined by examining the structure J_{τ}^{E} (and not J_{τ}^{E′},

1See section 1.2.

2The property of being a cutpoint of N is Π_{1}(N), see[21].

32 ERNEST SCHIMMERLING AND MARTIN ZEMAN

see (c2) in the definition of S_{α}). Similarly, if E_{α} ≠ ∅ then for each τ, this membership is determined by examining J_{τ}^{E′}. That σ_{τ̄,τ} has sufficient preservation degree to guarantee (M0) follows immediately unless τ ∈ 𝒯^{0}, ht(N_{τ}) = τ and n_{τ} = 0. In this case we use (T0C3): Since σ_{τ̄,τ} : Nτ̄ → N_{τ} is Q-preserving, τ̄ is a <_{0}-successor whenever τ is. ⊓ ⊔(M0)

Axiom (M1). The first part of (M1) is obvious. The second part of (M1), that is, the equality σ_{τ^{∗},τ} = σ_{τ̄,τ} ∘ σ_{τ^{∗},τ̄} was verified in the discussions following Definition 2.3 and Definition 2.4 where we proved that ≺ is a tree ordering. ⊓ ⊔(M1)

Axiom (M2). Clauses (a) and (b) of (M2) are clear. Clause (c) of (M2) requires a little discussion. Since τ̄′ < τ̄, the structure N_{}_{τ̄′} is an initial segment of Nτ̄. Since τ̄ is the cardinal successor of α(τ̄) = α(τ̄′) in Nτ̄, the structure N_{}_{τ̄′} is a proper initial segment of J Eτ̄ ; τ̄. It follows that N_{τ′} = σ_{τ̄,τ} (N_{}_{τ̄′}) is a proper initial segment of J_{τ}^{E^{τ}}. Thus, any first-order statement over N_{}_{τ̄′} is Σ_{0}-expressible over J Eτ̄ ; τ̄ using N_{}_{τ̄′} as an additional parameter. Such a statement is preserved under σ_{τ̄,τ}. Consequently, n_{}_{τ̄′} = n_{τ′} and σ_{τ̄,τ} (p_{}_{τ̄′}) = p_{τ′}. For pluripotent N_{}_{τ̄′} we also have σ_{τ̄,τ} (d_{}_{τ̄′}) = d_{τ′}. If α is a limit cardinal then so is ᾱ and both τ̄ and τ are in 𝒯^{0}. Otherwise, letting μ be the cardinal predecessor of α, it follows from the above discussion that the following statements are equivalent:

• there is a q̄ such that (μ, q̄) is a divisor for N_{}_{τ̄′} strong above ᾱ;

• there is a q such that (μ, q) is a divisor for N_{τ′} strong above α.

This means that τ̄′ ∈ 𝒯^{1} just in case that τ′ ∈ 𝒯^{1}, and for τ̄′ ∈ 𝒯^{1} we have σ_{τ̄,τ} (M_{}_{τ̄′}) = M_{τ′} and σ_{τ̄,τ} (q_{}_{τ̄′}) = q_{τ′}. All this together, the uniqueness of the morass maps discussed below Definition 2.3 and Definition 2.4, and the fact that the map σ_{τ̄,τ} ↾ N_{}_{τ̄′} : N_{}_{τ̄′} → Nτ̄, resp. σ_{τ̄,τ} ↾ M_{}_{τ̄′} : M_{}_{τ̄′} → Mτ̄ is fully elementary imply that σ_{τ̄′,τ′} = σ_{τ̄,τ} ↾ N_{}_{τ̄′}, resp. σ_{τ̄′,τ′} = σ_{τ̄,τ} ↾ M_{}_{τ̄′}. For τ̄′ ∈ 𝒯^{0}, the Qcondition follows from the full elementarity of σ_{τ̄′,τ′}. For τ̄′ ∈ 𝒯^{1}, clause (T1C) is vacuously true. ⊓ ⊔(M2)

Axiom (M3). We split our discussion into two cases, depending on whether τ is in 𝒯^{0} or 𝒯^{1}.

CARDINAL TRANSFER PROPERTIES IN EXTENDER MODELS 33

Case 1: τ ∈ 𝒯^{0}. Suppose ᾱ < α is a limit point of C(τ). Our aim is to show that ᾱ ∈ C(τ). Because ≺ is a tree ordering, we have a linear diagram

⟨N_{τ^{∗}}, σ_{τ^{∗},τ′}; τ^{∗}, τ′ ≺ τ & α(τ^{∗}), α(τ′) ∈ C(τ) ∩ ᾱ & τ^{∗} < τ′⟩.

Consider the direct limit ⟨N̄, σ̄_{τ^{∗}}; τ^{∗} ≺ τ & α(τ^{∗}) ∈ C(τ) ∩ ᾱ⟩ of this diagram. Let σ̄ : N̄ → N_{τ} be the unique embedding satisfying σ̄ ∘ σ̄_{τ^{∗}} = σ_{τ^{∗},τ} for all τ^{∗}. A straightforward argument that involves an inductive computation of projecta in fine structural direct limits ^{3} yields:

• N̄ is well-founded, and we will consider it to be transitive;

• all direct limit maps σ̄_{τ^{∗}} : N_{τ^{∗}} → N̄ are Σ_{1}^{(n)}-preserving where n = n_{τ};

• cr(σ̄_{τ^{∗}}) = α(τ^{∗}) and σ̄_{τ^{∗}}(α(τ^{∗})) = ᾱ whenever τ^{∗} ≺ τ and α(τ^{∗}) ∈ C(τ)∩ᾱ. From these clauses we easily infer

(8)  cr(σ̄) = ᾱ,
(9)  σ̄(ᾱ) = α

and

(10)  σ̄ is Σ_{1}^{(n)}-preserving.

We should stress that all computations are done in the language for premice. Our aim is to show that N̄ is the collapsing structure for τ̄ = ᾱ^{+N̄} in the sense of Definition 1.5. The main tool here will be the Condensation Lemma (Lemma 1.4). In the following we verify that the triple N̄, N_{τ} and σ satisfies the assumptions of the Condensation Lemma.

We begin by noticing:

(11)  N̄ is a premouse of the same type as N_{τ}.

This follows easily if n_{τ} > 0; see [21], Chapter 9. If n_{τ} = 0 then N_{τ} must be a premouse of type A or B and the same applies to all premice N_{τ^{∗}}; namely, they all must be of the same type as N_{τ}. By construction, N̄ must be a type A or B potential premouse. Since premousehood in both these cases is a Π_{2}-property in the language for premice [21], it is downwards preserved under σ̄.

3This argument goes through abstractly for any acceptable structures, see e.g. preliminaries in [15] or Section 5 in [9])

34 ERNEST SCHIMMERLING AND MARTIN ZEMAN

Let p̄ = σ̄_{τ^{∗}}(p_{τ^{∗}} − α(τ^{∗})); this value obviously does not depend on τ^{∗}. Also, σ̄(p̄) = p_{τ} − α. This, the soundness of N_{τ} and (10) together yield

(12)  h̃n+1 ; N̄(ᾱ ∪ {p̄}) = N̄

which in turn gives

(13)  ωϱn+1 ; N̄ ≤ ᾱ < ωϱn ; N̄.^{4}

Clauses (8), (10) and (13) enable us to apply the first part of the Condensation Lemma to the embedding σ̄ : N̄ → N_{τ} and conclude:

(14)  N̄ is solid.

Let β̄ ∈ p̄ and τ^{∗} ≺ τ and β^{∗} ∈ p_{τ^{∗}} be such that σ̄_{τ^{∗}}(β^{∗}) = β̄. Since the map σ̄_{τ^{∗}} is Σ_{1}^{(n)}-preserving and N_{τ^{∗}} is solid, σ̄_{τ^{∗}}(W_{N_{τ^{∗}}}^{β^{∗},p_{τ^{∗}}}) ∈ N̄ is a generalized witness for β̄ with respect to p̄ and N̄. This together with the solidity of N̄ and (12) then guarantees ([21], Lemma 1.12.5):

(15)  p̄ = pN̄ − ᾱ

It follows that N̄ is sound above ᾱ. As an immediate consequence we see that N̄ is a collapsing structure for τ̄ = ᾱ^{+N̄}.

The soundness of N̄ above ᾱ allows us to apply the second part of the Condensation Lemma. Our immediate goal is proving that N̄ = Nτ̄. Notice that it suffices to rule out possibility (c) in the Condensation Lemma, as this will tell us that exactly one of the following holds:

• Eτ ; ᾱ is not an extender and N̄ is the initial level of N_{τ} collapsing τ̄.

• Eτ ; ᾱ is an extender and N̄ is the initial level of Ult(N_{τ} , Eᾱ) collapsing τ̄. Since N_{τ} agrees with W below α (so, in particular, Eτ ; ᾱ = Eᾱ), we conclude that N̄ is the initial segment of W or Ult(W, Eᾱ) collapsing τ̄, which means N̄ = Nτ̄.

Assume for a contradiction that clause (c) in the Condensation Lemma holds. Then N̄ is of the form Ult^{∗}(Q, E_{δ}^{N_{τ}}) where Q is sound and μ =^{def} cr(E_{δ}^{N_{τ}}) is the cardinal predecessor of ᾱ in both Q and N̄. Moreover, E_{δ}^{N_{τ}} has a single generator, namely μ. It follows that N̄ is sound above μ + 1, and no projectum of N̄ can lie in the interval (μ, π(μ)] where π is the associated ultrapower map. Since ᾱ < π(μ) and

4The sharp inequality on the right follows from the fact that α(τ^{∗}) < ωϱ_{N_{τ}^{∗}}^{n} for all τ^{∗}.

CARDINAL TRANSFER PROPERTIES IN EXTENDER MODELS 35

(13) holds, we must have ωϱn+1 ; N̄ ≤ μ. So h̃n+1 ; N̄(μ∪{pN̄, μ}) = N̄. Since N̄ is the fine ultrapower of Q by an internal measure, pN̄ = π(p_{Q}). So min(pN̄ − μ) ≥ π(μ). ^{5} Applying (15), we obtain N̄ = h̃n+1 ; N̄(μ ∪ {p̄, μ}). But μ ∪ {p̄, μ} ⊆ rng(σ̄_{τ^{∗}}) for each τ^{∗}, and the fact that σ̄_{τ^{∗}} is Σ_{1}^{(n)}-preserving yields rng(σ̄_{τ^{∗}}) = N̄, so σ̄_{τ^{∗}} = id. This contradicts e.g. (8), and thereby rules out clause (c) in the Condensation Lemma.

Summarizing the above discussion, we have N̄ = Nτ̄ and, by (15), also p̄ = pτ̄−ᾱ. It is also clear that if τ ∈ N_{τ} then τ^{∗} ∈ N_{τ^{∗}} for all τ^{∗} and τ̄ = σ_{τ^{∗}}(τ^{∗}) ∈ Nτ̄. This, the fact that σ̄(p̄) = p_{τ} − α, (8) – (10) and (13) verify (T0B) and (T0C) except the Q-condition, which we verify now. Fix a Σ_{1}^{(n)}-formula φ(u, v) and some q̄ ∈ Nτ̄. Pick some τ^{∗} such that α(τ^{∗}) ∈ C(τ) ∩ ᾱ and q̄ ∈ rng(σ̄_{τ^{∗}}). Let σ̄_{τ^{∗}}(q^{∗}) = q̄ and σ̄(q̄) = q. If Nτ̄ ⊧ φ(ξ, q̄) for cofinally many ξ < τ̄ then N_{τ^{∗}} ⊧ φ(ξ, q^{∗}) for cofinally many ξ < τ^{∗} since σ̄_{τ^{∗}} is Σ_{1}^{(n)}-preserving. But then the Q-condition for σ_{τ^{∗},τ} guarantees that N_{τ} ⊧ φ(ξ, q) for cofinally many ξ < τ.

It remains to verify that τ̄ ∈ 𝒯^{0}. Clearly, ᾱ ∈ S(0). Furthermore, the structure J Eτ̄ ; τ̄ is good, as follows from the preservation properties of σ̄. The ordinal ᾱ is the largest cardinal in J Eτ̄ ; τ̄ and is regular there. So we conclude that τ̄ ∈ S(1) and ᾱ = α(τ̄). In the following we rule out the possibilitiy ᾱ ∈ 𝒯^{1}. If τ̄ were in 𝒯^{1} then ᾱ would be a successor cardinal in Nτ̄, say ᾱ = μ^{+Nτ̄} and (μ, q̄) would be a divisor for Nτ̄ strong above ᾱ where q̄ is some bottom part of pτ̄ if Nτ̄ fails to be pluripotent and q̄ = dτ̄ if Nτ̄ is pluripotent. ^{6} Let τ^{∗} ∈ C(τ) ∩ ᾱ. The pair (μ, q̄− ᾱ) would be in the range of σ̄_{τ^{∗}}, say (μ, q̄) = σ̄_{τ^{∗}}(μ, q^{∗}) where q^{∗} is a bottom part of p_{τ^{∗}} − α(τ^{∗}) or q^{∗} = d_{τ} − α(τ^{∗}). ^{7} By (b1) or (b2) of Lemma 1.12 and the fact that σ̄_{τ^{∗}} is Σ_{1}^{(n)}-preserving, the pair (μ, q^{∗}) would be a divisor for N_{τ^{∗}} strong above α(τ^{∗}). Contradiction, as τ^{∗} ∈ 𝒯^{0}. This completes the verification that C(τ) is closed.

In the following we show that C(τ) is unbounded in α, granting that τ is not the largest element of S_{α}. Let τ′ ∈ S_{α} be larger than τ. Then α is regular in

5In fact, min(pN̄) ≥ π(μ). This follows from the fact that μ is a cardinal in W, as ᾱ is cardinalcorrect in W. So actually ωϱ_{Q}^{n+1} = ω_{Q}^{ω} = μ = ωϱn+1 ; N̄ = ωϱω ; N̄ and p̄ = pN̄ = π(p_{Q}). However, these facts have no relevance in showing that (c) in the Condensation Lemma does not occur.

6The reader may consult (2) in section 1.3.

7Recall that d_{N}^{β} = d_{N} − β; see the discussion below Definition 1.3.

36 ERNEST SCHIMMERLING AND MARTIN ZEMAN

J_{τ′}^{E^{τ′}}. ^{8} Choose an ordinal α^{∗} < α. In J_{τ′}^{E^{τ′}}, we can carry out the standard elementary chain construction to get a fully elementary substructure X ≺ N_{τ} such that (α^{∗} + 1) ∪ {α, p_{N_{τ}} } ⊂ X and ᾱ = X ∩ α ∈ α. Letting N̄ be the transitive collapse of X, the inverse to the associated collapsing map σ̄ : N̄ → N_{τ} is fully elementary. This is more than sufficient to transfer all information about the fine structure of N_{τ} to N̄, as this information is first order expressible over N_{τ} in the parameters α and p_{N_{τ}}. Letting p̄ ∈ N̄ be the σ̄-preimage of p_{τ}, we conclude that ωϱn+1 ; N̄ = ᾱ < ωϱn ; N̄, all clauses (8) – (12) are satisfied with current ᾱ, σ̄, N̄ and p̄, p̄ = pN̄, and N̄ is solid and sound. ^{9} Here (8) and (9) follow for current ᾱ and σ̄ from the fact that X ∩ α = ᾱ and the rest is a direct consequence of the full elementarity of σ̄. We apply the Condensation Lemma to σ̄ : N̄ → N_{τ} and conclude that N̄ = Nτ̄ where τ̄ = σ̄^{−1}(τ) if τ ∈ N̄ ^{10} and τ̄ = ht(N̄) otherwise. This time, the failure of (c) in the Condensation lemma fails follows immediately from the fact that N̄ is sound.

It remains to verify that τ̄ ∈ 𝒯^{0}, since the rest goes through as before. But this is an immediate consequence of the full elementarity of σ̄: If τ̄ were in 𝒯^{1} then (μ, q̄) would be a strong divisor for Nτ̄ where μ is a cardinal predecessor of ᾱ in Nτ̄. Using the full elementarity of σ̄ and (b1) or (b2) of Lemma 1.12 we conclude that (μ, σ̄(q̄)) is a strong divisor for N_{τ}, a contradiction with the fact that τ ∈ 𝒯^{0}. This shows that ᾱ = α(τ̄) ∈ C(τ) and ᾱ > α^{∗}, and thereby completes the proof of the unboundedness of C(τ) in α(τ).

Once we know that C(τ) is unbounded in α, it follows that every ζ < τ is in the range of some σ_{τ^{∗},τ}; it suffices to fix some ξ < α such that ζ = h̃_{τ}^{n+1}(ξ, p_{τ}) and choose τ^{∗} so that α(τ^{∗}) > ξ. As a direct consequence we obtain

τ = ⋃{σ′′_{τ^{∗},τ} τ^{∗}; τ^{∗} ≺ τ},

which verifies the last clause of (M3) and thus completes Case 1 in the verification of (M3). ⊓ ⊔((M3), Case 1)

8Recall that E^{τ} = E^{τ′} ↾ τ, see (c2) in the definition of S_{α}.

9The present notation is consistent with that used above.

10Since σ̄ is fully elementary, τ ∈ rng(σ̄) in this case.

CARDINAL TRANSFER PROPERTIES IN EXTENDER MODELS 37

Case 2: τ ∈ 𝒯^{1}. Towards the closedness of C(τ), we again consider a limit point ᾱ of C(τ) and show that it is in C(τ). This time M_{τ} is a protomouse, μ_{τ} is the cardinal predecessor of α = α(τ) and ϑ_{τ} < α. If there is some τ^{∗} ≺ τ with τ^{∗} ∈ 𝒯^{0} then, by (T1C), we also have a τ̃ ≺ τ with τ̃ ∈ 𝒯^{0} and α̃ = α(τ̃) = ϑ_{τ}. By the discussion at the end of the previous subsection, the set {τ^{∗}; τ^{∗} ≺ τ} splits into two intervals, namely [τ̂, τ̃] and (τ̃, τ). Accordingly, C(τ) splits into two intervals, namely C(τ̃) and C(τ) ∩ (α̃, α). But we already know that C(τ̃) is closed, as this falls under Case 1. Thus, no matter whether there are τ^{∗} ≺ τ that are in 𝒯^{0} or not, it suffices to consider the case where ᾱ > ϑ_{τ}, i.e. where ᾱ = α(τ̄) for some τ̄ ≺ τ such that τ̄ ∈ 𝒯^{1}.

We again have a linear diagram ^{11}

⟨M_{τ^{∗}}, σ_{τ^{∗},τ′}; τ^{∗}, τ′ ≺ τ & α(τ^{∗}), α(τ′) ∈ C(τ) ∩ (α̃, ᾱ) & τ^{∗} < τ′⟩.

We form the direct limit of the above diagram with the direct limit structure M̄ and the direct limit maps σ̄_{τ^{∗}} : M_{τ^{∗}} → M̄. We also define the map σ̄ : M̄ → M_{τ} in the canonical way. The following list summarizes relevant information about these objects. Most of the clauses below are obtained in the same manner as in Case 1, but some require an additional explanation. In the following we assume that τ^{∗} is as above, i.e. τ^{∗} ≺ τ and α(τ^{∗}) ∈ C(τ) ∩ ᾱ.

(i) M̄ is well-founded and we will consider it to be transitive.

(ii) All maps σ̄_{τ^{∗}} : M_{τ^{∗}} → M̄ as well as σ̄ : M̄ → M_{τ} are Σ_{1}-preserving and

cofinal, and σ_{τ^{∗},τ} = σ̄ ∘ σ̄_{τ^{∗}}. (iii) σ̄_{τ^{∗}}(α(τ^{∗})) = ᾱ. (iv) cr(σ̄) = ᾱ and σ̄(ᾱ) = α.

(v) M̄ is a coherent structure and ϑ(M̄) = ϑ_{τ} = ϑ_{τ^{∗}}.

(vi) hτ̄(α ∪ {q̄}) = M̄ for q̄ = σ̄_{τ^{∗}}(q_{τ^{∗}} − α(τ^{∗})). ^{12} (vii) σ̄(q̄) = q_{τ} − α_{τ}. (viii) For every β ∈ q̄ there is a generalized witness Qβ,q̄ ; M̄ ∈ M̄ for β with respect

to q̄ and M̄.

11We consider α̃ = 0 if {τ^{∗}; τ^{∗} ≺ τ} ∩ 𝒯^{0} = ∅.

12Of course σ̄_{τ^{∗}}(q_{τ^{∗}} − α(τ^{∗})) is independent of τ^{∗}.

38 ERNEST SCHIMMERLING AND MARTIN ZEMAN

The maps σ̄_{τ^{∗}} and σ̄ in (ii) are Σ_{1}-preserving, as all maps σ_{τ^{∗},τ̄} are Σ_{1}-preserving; the cofinality of σ̄ is a consequence of our choice of ᾱ to be larger than ϑ_{τ}. This choice of ᾱ also guarantees (v). The preservation properties of the maps σ̄_{τ^{∗}} give the rest. Regarding (vi), we additionally apply the fact that h_{τ^{∗}}(α(τ^{∗}) ∪ {q_{τ^{∗}}}) = M_{τ^{∗}} for all τ^{∗} as above. To see the last clause, we apply the fact that for each β^{∗} ∈ q_{τ^{∗}} there is a generalized witness Q_{M_{τ^{∗}}}^{β,q_{τ^{∗}}} ∈ M_{τ^{∗}} with respect to q_{τ^{∗}} and M_{τ^{∗}}. The rest is standard.

Our aim is to prove that M̄ = Mτ̄ where τ̄ = ᾱ^{+M̄} = σ̄_{τ^{∗}}(τ^{∗}). Here we apply our substitute for the Condensation Lemma, namely Lemma 1.15, to the embedding σ̄ : M̄ → M_{τ}. Assumptions (a) – (c) of the lemma were verified above. We will argue that conclusion (A) of the lemma must apply, and thereby obtain that M̄ is of the form N̄(μ, q̄^{∗}) where N̄ is the collapsing structure for τ̄ and q̄^{∗} is such that q̄ = q̄^{∗} − ᾱ. Thus, N̄ = Nτ̄ and M̄ = Nτ̄(μ, q̄^{∗}); the lemma also tells us that ᾱ = α(τ̄) and q̄^{∗} is a bottom part of pτ̄ − ᾱ or q̄ = dᾱ ; τ̄.

To see that conclusion (B) in Lemma 1.15 fails, argue by contradiction. Assuming that (B) holds, we have ᾱ ∈ h̃n+1 ; N̄(μ ∪ {r̄, q̄, μ}) where N̄ = N(M̄), n = n_{τ} and r̄ = pN̄ − λ(EM̄ ; top). By Proposition 1.13, we can find some function f ∈ J_{ϑ}^{E} and ξ < μ such that ᾱ = EM̄ ; top(f)(q̄, ξ) where E = EM̄ and ϑ = ϑ(M̄) = ϑ_{τ}. Let τ^{∗} ≺ τ, τ^{∗} < τ̄ be large enough such that τ^{∗} ∈ 𝒯^{1}. Then ϑ_{τ^{∗}} = ϑ. Since σ̄_{τ^{∗}} is Σ_{1}-preserving and f, ξ, q̄ ∈ rng(σ̄_{τ^{∗}}), we conclude that ᾱ ∈ rng(σ̄_{τ^{∗}}). This is a contradiction, as ᾱ = cr(σ̄_{τ^{∗}}).

Recall that q̄^{∗} = pτ̄ ∩ max(q̄ + 1) if N^{∗} is passive or active with λ(E_{top}^{N^{∗}}) > μ and q̄^{∗} = dN̄ otherwise. To conclude that M̄ = Mτ̄, we have to verify that (μ, q̄^{∗}) is a divisor for Nτ̄ strong above ᾱ. Here we apply Proposition 1.13; see also the discussion concerning (4). Let x ⊆ λ(EM̄ ; top) be in the hull h̃τ̄(μ ∪ {pτ̄ − ᾱ}). By Proposition 1.13 we can find a function f ∈ J_{ϑ}^{Eτ̄} and an ordinal η < μ such that x = EM̄ ; top(f)(q̄, η) where ϑ = ϑ(M̄) = ϑ_{τ}. Since σ̄ is Σ_{1}-preserving and the identity on ᾱ, σ̄(x) = E_{top}^{M_{τ}} (f)(q_{τ} − α(τ), η). Again by Proposition 1.13, now used in the reversed direction, we conclude that σ̄(x) ∈ h̃_{τ} (μ ∪ {p_{τ} − α(τ)}). As (μ, q_{τ}) is a divisor for N_{τ} strong above α(τ) and σ̄ is the identity on μ, we apply (S2) and conclude x ∩ μ = σ(x) ∩ μ ∈ J_{ϑ}^{E^{τ}} = J_{ϑ}^{Eτ̄}. If N^{∗} is active and E_{top}^{N^{∗}} = μ we proceed similarly but work with dτ̄ and d_{τ} instead of pτ̄ and p_{τ}. This proves that the divisor

CARDINAL TRANSFER PROPERTIES IN EXTENDER MODELS 39

(μ, q̄^{∗}) for Nτ̄ is strong above ᾱ, and thereby completes the proof that M̄ = Mτ̄. It follows that τ̄ ∈ 𝒯^{1}; that τ̄ ≺ τ is witnessed by the map σ̄. So ᾱ = α(τ̄) ∈ C(τ), which verifies the closedness of C(τ).

Finally we prove that C(τ) is unbounded in α. The structure of the argument is the same as in Case 1. Given α^{∗} < α, we pick some τ′ > τ in S_{α} and use the elementary chain construction inside J_{τ′}^{E^{τ′}} for some to construct a coherent structure M̄, and ordinal ᾱ < α, a finite set of ordinals q̄ and a fully elementary embedding σ̄ : M̄ → M_{τ} such that:

(i) ᾱ = cr(σ̄) > α^{∗} and σ̄(ᾱ) = α.

(ii) ϑ_{τ} = ϑ(M̄) < ᾱ.

(iii) ωϱ1 ; M̄ = ᾱ.

(iv) σ̄(q̄) = q_{τ}.

(v) M̄ = hM̄(ᾱ ∪ {q̄}).

(vi) For every β ∈ q̄ there is a generalized witness Qβ,q̄ ; M̄ ∈ M̄ The first clause here is again guaranteed by construction, as α ∈ rng(σ̄) and is regular in J_{τ′}^{E^{τ′}}; the rest follows from the full elementarity of σ̄ and the fact that we put α and q_{τ} into the range of σ̄. Exactly as in the above proof of the closedness, using Lemma 1.15 and Lemma 1.12 we then show that M̄ = Mτ̄ where τ̄ = ᾱ^{+M̄}. To see that conclusion (B) in Lemma 1.15 does not apply in this case is easier than in the proof of closedness, as it is an immediate consequence of (iii) above. It follows that τ ∈ 𝒯^{1}, α^{∗} < α(τ̄) and τ̄ ≺ τ. This completes the proof that C(τ) is unbounded in α in Case 2.

The verification of the rest of (M3) is easy and practically the same as in Case 1. ⊓ ⊔(M3)

Remark. Case 2 in the verification of morass axiom (M3) is the only place in the entire construction where we make use of requirement (T1C) from Definition 2.4, and the situation arising in that argument is the very reason why we introduce (T1C) in the definition of morass ordering and morass embeddings. As already indicated at the end of the last section, the issue here is that we do not see how to verify that M̄ is a level of M_{τ} in the situation where τ ∈ 𝒯^{1}, n_{τ} = 0, M̄ is a pluripotent premouse, M_{τ} is a protomouse such that cr(EM̄ ; top) = cr(E_{top}^{M_{τ}}) is the cardinal predecessor of both ᾱ and α(τ), and there is a Σ_{1}-preserving map

40 ERNEST SCHIMMERLING AND MARTIN ZEMAN

σ̄ : M̄ → M_{τ} that also preserves all relevant parameters. This happens because of the lack of any kind of condensation lemma that could be applied in such a situation. The way we approach the issue is demanding the existence of the largest τ̃ ∈ 𝒯^{0} that is a ≺-predecessor of τ, as this allows us to apply the standard condensation lemma, namely Lemma 1.4. The largest τ̃ that is a ≺-predecessor of τ is precisely the ordinal τ̃ satisfying ϑτ̃ = ϑ_{τ} as demanded in (T1C). Of course, the question arises whether (T1C) is not too strong, that is, whether τ̃ as above exists. We will see in verifications of morass axioms (M4) and (M5) that such τ̃ exists in all relevant situations.

Axiom (M4). Assume τ̄ ≺ τ and let τ′ < τ be the supremum of σ′′_{τ̄,τ} τ̄. Our aim is to show that τ̄ ≺ τ′. ^{13} We will again consider two cases.

Case 1: τ ∈ 𝒯^{0} and it is not the case that N_{τ} is both pluripotent and cr(E_{top}^{M}) is the cardinal predecessor of α in N_{τ} (equivalently, in W).

Obviously, the same applies to Nτ̄. Let n = n_{τ}. The Interpolation Lemma ([21], Section 3.6) applied to σ_{τ̄,τ} : Nτ̄ → N_{τ} yields a transitive structure N′ together with maps σ̄ and σ′ satisfying:

• σ̄ : Nτ̄ → N′ is Σ_{0}^{(n)}-preserving and cofinal;

• σ′ : N′ → N_{τ} is Σ_{0}^{(n)}-preserving;
• cr(σ̄) = ᾱ and σ̄(ᾱ) = α;
• cr(σ′) = τ′ and σ′(τ′) = τ.

We would like to use the Condensation Lemma to argue that N′ = N_{τ′}. For this, we need to verify first that N′ is a premouse of the same type as N_{τ} and Nτ̄ (note that both these structures are premice of the same type). This follows from preservation properties of the maps σ̄ and σ′. Here we verify merely that N′ is a potential premouse, which reduces to proving that E_{top}^{N′} is a total extender on N′ whenever N′ is active. If n > 0 this follows from the fact that σ̄ is Σ_{2}-preserving. If n = 0 and cr(E_{top}^{N_{τ}}) ≥ α we have cr(E_{top}^{Nτ̄}) ≥ ᾱ, and the conclusion follows from the fact that σ̄, being a coarse (pseudo)ultrapower ([21], Section 3.6) embedding, maps τ̄ cofinally into τ′. This observation is sort of trivial, but we decided to include it, as this is the place which makes Case 1 different from Case 2, and the issue

13We are, of course, omitting the trivial case τ′ = τ.

CARDINAL TRANSFER PROPERTIES IN EXTENDER MODELS 41

that arises in Case 2 at this place is the reason for introducing protomice in the entire construction. In the remaining case we have μ =^{def} cr(E_{top}^{Nτ̄}) < κ where κ is the cardinal predecessor of ᾱ = cr(σ̄) in Nτ̄. It follows that both structures Nτ̄ and N′ contain the same subsets of μ and σ̄ is the identity on this power set. ^{14} This verifies that N′ is a potential premouse. That N′ is of the same type as the two other premice and that it satisfies the initial segment conditions, follows from the lemmata in [21], Section 9.1.

Now, using the soundness and solidity of Nτ̄ and the preservation properties of the embedding σ̄, we conclude:

(i) N′ = h̃_{N′}^{n+1}(α ∪ {σ̄(pτ̄)}); (ii) ωϱ_{N′}^{n+1} ≤ α < ωϱ_{N′}^{n}; (iii) N′ is solid; (iv) σ̄(pτ̄) − α = p_{N′} − α;

(v) N′ is sound above α.

The verification of these facts is standard. Clause (i) follows from the soundness of Nτ̄ and the preservation degree of σ̄. Clause (ii) follows from (i), clause (iii) follows from (ii) combined with the first part of the Condensation Lemma ^{15} , (iv) follows from (i), the solidity of Nτ̄ and the preservation degree of σ̄ and (v) follows from (i) and (iv). The arguments are the same as in Case 1 in the verification of (M3).

Now we can apply the second part of the Condensation Lemma to conclude that N′ = N_{τ′}. This amounts ruling out clauses (c) and (d) in the Condensation Lemma. To see that (c) fails, notice that (c) would imply that N′ is not sound above α which contradicts (v) above. To see that (d) fails, notice that (d) would imply that N′ is a proper initial segment of Ult(N_{τ} , E_{τ′}^{τ}), so its ultimate projectum must be at least τ′. This contradicts (ii) above. It follows that N′ is the initial level of N_{τ} collapsing τ′, so N′ = N_{τ′}. Obviously, N_{τ′} is a proper initial segment of N_{τ}, as we are assuming that τ′ < τ.

14Actually, all of this is vacuous: The argument at the beginning of Case 2 shows that μ < ᾱ is never the case if τ′ < τ, but this information does not simplify the proof in the present case.

15Recall that this is Lemma 1.4

42 ERNEST SCHIMMERLING AND MARTIN ZEMAN

Summarizing the results, we have just verified (T0B) and (T0C). ^{16} To see that τ′ ≺ τ, it suffices to prove that τ′ ∈ 𝒯^{0}. But this follows by exactly the same argument as in the verification of (M3), Case 1.

The verification that τ′ ≺ τ is now complete. The equality σ_{τ̄,τ′} ↾ τ̄ = σ_{τ̄,τ} ↾ τ̄ is an immediate consequence of the fact that σ′ ↾ τ′ = id. ⊓ ⊔((M4), Case 1)

Case 2: Either τ ∈ 𝒯^{0} and N_{τ} is pluripotent and cr(E_{top}^{N_{τ}}) is the cardinal predecessor of α in N_{τ} or else τ ∈ 𝒯^{1}.

Either of these two options yields:

(16)  τ̄ ∈ 𝒯^{0}, i.e. Nτ̄ is a pluripotent premouse.

(17) μτ̄ = μ_{τ} =^{def} μ is the cardinal predecessor of ᾱ in Nτ̄ (and of α in N_{τ}).

In the former case where τ ∈ 𝒯^{0} this is just obvious. In the latter case where τ ∈ 𝒯^{1}, it is clear that μ is the cardinal predecessor of ᾱ in Mτ̄. What we have to verify is that Mτ̄ = Nτ̄ is a pluripotent premouse. That is, we have to show that τ̄ ∈ 𝒯^{0}. Assume for a contradiction that τ̄ ∈ 𝒯^{1}. Then ϑ(Mτ̄) = ϑ(M_{τ}), since ϑ(Mτ̄) < ᾱ. It follows that the domains of E_{top}^{Mτ̄} and E_{top}^{M_{τ}} agree, so σ_{τ̄,τ}, being the identity on these domains, maps Mτ̄ cofinally into M_{τ}. Let H_{τ} (x, y, w, z) be a Σ_{0}(M_{τ})-relation such that for every parameter p ∈ M_{τ} we have

y = h_{τ} (x, p) ⟷ (∃z)H_{τ} (x, y, p, z).

For ζ < ht(M_{τ}) define a partial function g_{τ}^{ζ} : J_{ζ}^{E^{τ}} → J_{ζ}^{E^{τ}} and a sequence of ordinals ⟨τ(ζ); ζ < ht(M_{τ})⟩ by

y = g_{τ}^{ζ}(x) ⟷ (∃z ∈ J_{ζ}^{E^{τ}})H_{τ} (x, y, q_{τ} , z)

and

τ(ζ) = sup(τ ∩ (g_{τ}^{ζ})′′α).

Obviously, each g_{τ}^{ζ} is an element of M_{τ}, so τ(ζ) < τ, as τ = α^{+M_{τ}} is a regular cardinal in M_{τ}. Furthermore, the sequence ⟨τ(ζ)⟩_{ζ} is monotonic and converges to τ; the latter follows from the fact that h_{τ} (α ∪ {q_{τ} }) = M_{τ}. Let gζ ; τ̄ and τ̄(ζ) be defined in the same way over Mτ̄. Then σ_{τ̄,τ} (gζ ; τ̄, τ̄(ζ)) = (g_{τ}^{σ_{τ̄,τ} (ζ)}, τ(σ_{τ̄,τ} (ζ))). Thus, if τ^{∗} < τ, we can find some ζ such that τ(ζ) ≥ τ^{∗} and, by the cofinality of σ_{τ̄,τ}, some

16We omitted the trivial verification of the Q-condition.

CARDINAL TRANSFER PROPERTIES IN EXTENDER MODELS 43

ζ̄ with σ_{τ̄,τ} (ζ̄) ≥ ζ. It follows that τ > σ_{τ̄,τ} (τ̄(ζ̄)) = τ(σ_{τ̄,τ} (ζ̄)) ≥ τ(ζ) ≥ τ^{∗}, which tells us that σ_{τ̄,τ} maps τ̄ cofinally into τ. We have thus reached contradiction with our assumption that τ′ < τ. ⊓ ⊔(16),(17)

Following the same strategy as in Case 1, we apply the Interpolation lemma to the map σ_{τ̄,τ} : Mτ̄ → M_{τ}. We obtain a transitive structure M′ and embeddings σ̄ and σ′ satisfying

• σ̄ : Mτ̄ → M′ is Σ_{0}-preserving and cofinal;

• σ′ : M′ → M_{τ} is Σ_{0}-preserving;
• cr(σ̄) = ᾱ and σ̄(ᾱ) = α;
• cr(σ′) = τ′ and σ′(τ′) = τ.

Since Mτ̄ is a premouse, the first two clauses immediately imply that M′ is a coherent structure and

(18)  ϑ(M′) = ᾱ < α = μ^{+M′}.

To see the equality on the left, recall that both Mτ̄ and M′ are amenable structures, so by the cofinality of σ̄ we have

E_{top}^{M′} = ⋃{σ̄(E_{top}^{Mτ̄} ∩ J_{π̄(ξ)}^{Eτ̄}); ξ < ᾱ)}

where π̄ is the ultrapower embedding associated with Ult(J Eτ̄ ; τ̄, E_{top}^{Mτ̄}). The maps σ̄(E_{top}^{Mτ̄} ∩ J_{π̄(ξ)}^{Eτ̄}) and E_{top}^{Mτ̄} ∩ J_{π̄(ξ)}^{Eτ̄} have the same domains for each ξ < ᾱ, which implies that also E_{top}^{Mτ̄} and E_{top}^{M′} have the same domain, i.e. ϑ(M′) = ᾱ. ⊓ ⊔(18)

From the soundness and solidity of Mτ̄ above ᾱ and (DP1) we obtain that hτ̄(ᾱ∪{qτ̄}) = Mτ̄ and for each β ∈ qτ̄−ᾱ there is a generalized witness Q_{Mτ̄}^{β,qτ̄} ∈ Mτ̄ with respect to qτ̄ and β. Standard methods then yield:

• M′ = h_{M′}(α ∪ {q′}) where q′ = σ̄(qτ̄ − α).

• For each β ∈ q′ there is a generalized witness Q_{M′}^{β,q′} ∈ M′ with respect to

q′ and M′. This verifies the assumptions of Lemma 1.14. It follows that N′ = N(M′) is the collapsing level for τ′ in N and (μ, q′) is a divisor for N′. The former easily implies that N′ = N_{τ′}. The latter follows from the fact that α is actually the ultimate projectum of N′, so q′ is a bottom part of p_{τ′} or q′ = d_{τ′}. To see that

44 ERNEST SCHIMMERLING AND MARTIN ZEMAN

M′ = M_{τ′} we verify that (μ, q′) is strong for N_{τ′}. ^{17} First consider the case where λ(E_{top}^{N^{∗}}) > μ where N^{∗} = (N_{τ′})^{∗}(μ, q′), i.e. N^{∗} is the collapsing level for ᾱ in N_{τ′}. This is equivalent to saying that EN_{τ′} ; ᾱ = ∅. Now use criterion (S2) from the discussion of strong divisors in section 1.3. If a ⊂ μ is an element of the transitive collapse of h̃_{τ′}(μ ∪ {p_{τ′}}) then, using the notation described in (4), a is of the form E_{top}^{M′}(f)(q′, ξ) ∩ μ for some ξ < μ and a suitable f. Since ξ, f and a are essentially subsets of μ, they are not moved by σ̄, so

σ̄(a) = a = E_{top}^{M′}(f)(q′, ξ) ∩ μ = σ̄ (E_{top}^{Mτ̄}(f)(q̄ − ᾱ, ξ) ∩ μ) .

This tells us that a = E_{top}^{Mτ̄}(f)(q̄− ᾱ, ξ) ∩ μ, so a ∈ Mτ̄. But ᾱ is a cardinal in Mτ̄, so in fact a ∈ J Eτ̄ ; ᾱ = J E^{τ′} ; ᾱ ⊂ N^{∗}. By (S2), (μ, q′) is strong for N′. If EN_{τ′} ; ᾱ ≠ ∅, the argument is the same, but uses the version of (S2) for d_{N_{τ′}}.

We have thus proved that τ̄ ≺ τ′. The above discussion verifies (T1A) and (T1B); (T1C) is trivially satisfied, as we can let τ̃ = τ̄. So σ̄ = σ_{τ̄,τ′}. ⊓ ⊔(M4)

Remark. It is this very last part of Case 2 in the verification of morass axiom (M4) where we make use of the fact that the divisors used in our construction are strong. There are only two places in the entire construction of Gap-1 Morass where the requirement on strongness is used in a substantial way; this is one of them. The other one is in the verification of morass axiom (M5) where an interpolation is used. In the case of (M0) – (M3) it is possible to avoid the use of strong divisors and use divisors of the form (μ, q) with shortest possible q instead. However, we do not see how to show that the divisor (μ, q′) for N′ that arises in the interpolation argument has shortest possible q′ among all divisors of the form (μ, q) for N′ unless we work with strong divisors. This is caused by the fact that the two interpolation maps fail to have sufficient preservation degree that would allow us to use Lemma 1.12 to transfer the information on strongness of the respective divisors from M_{τ} or Mτ̄ to M_{τ′}.

Notice also that the situation in Case 2 above is a reason for introducing protomice into our construction. If both M_{τ} and Mτ̄ are pluripotent premice such that cr(E_{top}^{M_{τ}}) = cr(E_{top}^{Mτ̄}) is the cardinal predecessor of α(τ) (and so the cardinal predecessor of α(τ̄)) then the structure M_{τ′} that arises in the interpolation argument

17“Strong” is in this case equivalent to “strong above α”.

CARDINAL TRANSFER PROPERTIES IN EXTENDER MODELS 45

is necessarily a protomouse even though the inputs in the argument are premice. We thus arrive at the situation where where τ̄ ∈ 𝒯^{0} and τ′ ∈ 𝒯^{1}. It should be noted that the situation in Case 2 above and the analogous situation in Case 2 in the verification of (M5) constitute the only reason for introducing protomice in our construction. That is, in all other arguments throughout the construction protomice never arise in situations where inputs are premice.

Finally note that the interpolation argument gives rise to situation that is required in (T1C), that is, we have τ̄ ≺ τ′ where τ′ ∈ 𝒯^{1}, τ̄ ∈ 𝒯^{0} and ϑτ̄ = ϑ_{τ′}. Recall that we included (T1C) in the definition of the morass ordering and morass embeddings since without (T1C) we don’t see how to verify axiom (M3) in the case where τ ∈ 𝒯^{1} and τ̄ ∈ 𝒯^{0} is such that α(τ̄) is a limit point of C(τ). Recall also that the issue here is that we do not have any condensation lemma allowing us to conclude that if σ̄ : M̄ → M_{τ} is a sufficiently preserving map and M̄ is a premouse then M̄ is a level of M_{τ}. Ideally, we wish we could arrange that all ≺-predecessors of elements of 𝒯^{1} are again elements of 𝒯^{1}, but we have seen above this is not possible. So we have to deal with two issues simultaneously: We have to allow ≺-predecessors of τ′ ∈ 𝒯^{1} that are in 𝒯^{0}, and at that same time we must make sure that if some τ^{∗} ∈ 𝒯^{0} is a ≺-predecessor of a τ′ ∈ 𝒯^{1} then we also have a ≺-predecessor τ̄ of τ′ that is in 𝒯^{0} and satisfies the requirement ϑτ̄ = ϑ_{τ′}. The argument in Case 2 above as well as that in Case 2 in the verification of (M5) guarantee this.

Axiom (M5). Assume that τ̄, τ and α̂ satisfy the hypothesis of (M5). We will find τ̂ as in the conclusion of (M5). At some point, our proof will break up into two cases: τ ∈ 𝒯^{0} and τ ∈ 𝒯^{1}. By our earlier calculations, in the first case, we will have that τ̄, τ̂ ∈ 𝒯^{0}. As we pointed out earlier, in the second case both τ̄ ∈ 𝒯^{0} and τ̄ ∈ 𝒯^{1} are possible. In both of these subcases, it turns out that τ̂ ∈ 𝒯^{1}. The verification uses the hypothesis of (M5) that the range of σ_{τ̄,τ} is unbounded in τ.

To simplify the notation, we will write σ for σ_{τ̄,τ} and τ′ for σ(τ̄′) whenever τ̄′ <_{0} τ̄. We will use an interpolation argument to construct Nτ̂ and the map σ_{τ̂,τ}. In order to satisfy the assumptions of the interpolation lemma, we need to find τ̂, a structure J Ê ; τ̂ and Σ_{0}-preserving maps

σ_{0}^{0} : J Eτ̄ ; τ̄ → J Ê ; τ̂ and σ_{1}^{0} : J Ê ; τ̂ → J_{τ}^{E^{τ}}

46 ERNEST SCHIMMERLING AND MARTIN ZEMAN

such that σ_{0}^{0} is cofinal and σ_{1}^{0} ∘ σ_{0}^{0} = σ. Notice that these requirements together with the cofinality of σ imply that σ_{1}^{0} is cofinal as well.

Notice that for each τ̄′ <_{0} τ̄ the ordinal τ̂′ as in (M5) is unique. From the argument in the verification of morass axiom (M2) we obtain that if τ̄′, τ̄′′ <_{0} τ̄ with, say τ̄′ < τ̄′′ then E^{τ̂′} = E^{τ̂′′} ↾ τ̂′ and

(19)  σ_{τ̄′,τ̂′} ↾ J_{τ̄′}^{Eτ̄} = σ_{τ̄′′,τ̂′′} ↾ J_{τ̄′}^{Eτ̄}
(20)  σ_{τ̂′,τ′} ↾ J_{τ̂′}^{E^{τ̂′}} = σ_{τ̂′′,τ′′} ↾ J_{τ̂′}^{E^{τ̂′}}

where τ̂′ ≺ τ′, τ̂′′ ≺ τ′′ and α(τ̂′) = α̂ = α(τ̂′′), as guaranteed by the hypothesis of (M5). This allows us to set

J Ê ; τ̂ = ⋃{J_{τ̂′}^{E^{τ̂′}}; τ̂′ ≺ τ′ for some τ̄′ <_{0} τ̄}
σ_{0}^{0} = ⋃{σ_{τ̄′,τ̂′} ↾ J_{τ̄′}^{E^{τ̄′}}; τ̄′ ≺ τ′ for some τ̄′ <_{0} τ̄}
σ_{1}^{0} = ⋃{σ_{τ̂′,τ′} ↾ J_{τ̂′}^{E^{τ̂′}}; τ̂′ ≺ τ′ for some τ̄′ <_{0} τ̄}

For τ̄′ <_{0} τ̄ we have τ̄′ ≺ τ̂′ ≺ τ′, and since the maps along the branches of the morass ordering ≺ commute,

σ ↾ τ̄′ = σ_{τ̄′,τ′} ↾ τ̄′ = (σ_{τ̂′,τ′} ↾ τ̂′) ∘ (σ_{τ̄′,τ̂′} ↾ τ̄′) = (σ_{0}^{0} ↾ τ̄′) ∘ (σ_{1}^{0} ↾ τ̂′).

As an immediate consequence we obtain

(21)  σ_{1}^{0} ∘ σ_{1}^{0} = σ ↾ τ̄

and all three maps are cofinal.

We are now ready to apply the Interpolation Lemma. We again consider two separate cases.

Case 1: τ ∈ 𝒯^{0}.

In this case τ̄ ∈ 𝒯^{0} by (T0A). From the Interpolation Lemma we obtain a coherent structure N̂, and maps σ_{0}^{1} and σ_{1}^{1} where each σ_{i}^{1} is an extension of σ_{i}^{0}, and:

• σ_{0}^{1} : Nτ̄ → N̂ is Σ_{0}^{(n)}-preserving and cofinal at the level n;
• σ_{1}^{1} : N̂ → N_{τ} is Σ_{0}^{(n)}-preserving and cofinal at the level n;
• σ_{1}^{1} ∘ σ_{0}^{1} = σ

CARDINAL TRANSFER PROPERTIES IN EXTENDER MODELS 47

where n = n_{τ} = nτ̄. It follows that both maps σ_{0}^{1} and σ_{1}^{1} are Σ_{1}^{(n)}-preserving. The cofinality of σ_{0}^{1} is a consequence of the fact that the maps σ and σ_{0}^{1} are cofinal and σ = σ_{1}^{1} ∘ σ_{0}^{1}. The cofinality of both σ and σ_{1}^{1} follows by a standard argument from the fact that these maps are Σ_{0}^{(n)}-preserving, map τ̄ cofinally into τ, resp. τ̂ and τ̄ ⊆ h̃τ̄(α(τ̄) ∪ (pτ̄ − α(τ̄))). [For σ, the argument runs as follows: Since h̃τ̄ is a good Σ_{0}^{(n)}(Nτ̄) function, it has a functionally absolute Σ_{1}^{(n)} definition, say y = h̃τ̄(x, pτ̄ − α(τ̄)) iff (∃z^{n})H̄(z^{n}, x, y, pτ̄ − α(τ̄)) where H̄ is Σ_{0}^{(n)}(Nτ̄). If ωϱ = sup(σ′′ωϱ_{Nτ̄}^{n}) < ωϱ_{N_{τ}}^{n} and H is Σ_{0}^{(n)}(N_{τ}) by the same definition as H̄, we could define a function h mapping α(τ) cofinally into τ by letting y = h(x) iff (∃z^{n} ∈ J_{ϱ}^{E^{τ}})H(z^{n}, x, y, p_{τ} −α(τ)). This function would be an element of N_{τ}, which is impossible, as it would collapse τ inside N_{τ}.]

N_{τ} is a premouse and σ_{1}^{1} is Σ_{1}-preserving if N_{τ} is of type A or B and Σ_{1}^{(1)}- preserving if N_{τ} is of type C. Since premousehood is a Π_{2}-property in the former case and Π_{1}^{1}-property in the latter case, N̂ must be a premouse of the same type as N_{τ}. It may be instructive to see directly that a situation where protomice result from an interpolation argument analogous to that in the verification of (M4) does not arise here. It cannot happen that N_{τ} and Nτ̄ are pluripotent and their top extenders have a common critical point μ that is the common cardinal predecessor of α(τ) and α(τ̄), as the fact that σ maps τ̄ cofinally into τ would imply that τ is collapsed inside ⟨J_{ζ}^{E^{τ}} , E_{top}^{N_{τ}} ∩ J_{ζ}^{E^{τ}} ⟩ ∈ N_{τ} where ζ is the image of ϑτ̄ under the ultrapower map by E_{top}^{N_{τ}}. ^{18} Using the soundness and solidity of Nτ̄, standard arguments yield:

(i) N̂ = h̃n+1 ; N̂(α̂ ∪ {p̂}) where p̂ = σ_{0}^{1}(pτ̄ − ᾱ).

(ii) For every β ∈ p̂ there is a generalized witness Qβ,p̂ ; N̂ with respect to p̂ and N̂. (iii) ωϱn+1 ; N̂ ≤ α̂, p̂ = pN̂ − α and N̂ is sound above α̂ and solid.

Regarding (iii), that ωϱn+1 ; N̂ ≤ α̂ is immediate by (i). From this combined with the first part of the Condensation Lemma we obtain the solidity of N̂. Once we know that N̂ is solid, (i) and (ii) yield p̂ = pN̂ − α, and thereby also the soundness of N̂ above α̂ ([21], Lemma 1.12.5).

Now we apply the Condensation Lemma to the triple σ_{1}^{1} : N̂ → N_{τ} and conclude that N̂ = Nτ̂. Here we proceed exactly as in the discussion of the closedness of

18This can be seen by an argument of the same kind as in the previous paragraph.

48 ERNEST SCHIMMERLING AND MARTIN ZEMAN

C(τ) in (M3), Case 1. It suffices to rule out possibility (c) in the Condensation Lemma. As in that argument we show that (c) would imply h̃n+1 ; N̂(μ ∪ {p̂, μ}) = N̂ where μ is the cardinal predecessor of α̂ in N̂. Then μ would be the cardinal predecessor of α(τ̄) in Nτ̄, so μ < α(τ̄) = cr(σ_{0}^{1}). Since σ_{0}^{1} is Σ_{1}^{(n)}-preserving and μ ∪ {p̂, μ} ⊂ rng(σ_{0}^{1}), we would obtain rng(σ_{0}^{1}) = N̂. But σ_{0}^{1}, being an extension of σ_{0}^{0}, has critical point ᾱ, a contradiction.

Once we know that N̂ = Nτ̂, the verification of the fact that τ̂ ∈ 𝒯^{0}, as well as that of the Q-condition, is literally the same as in (M3), Case 1. The key point

here is that σ_{1}^{1}(μ, α̂, pτ̂ − α̂) = (μ, α, p_{τ} − α) where μ < α̂ and that σ_{1}^{1}(dα̂ ; τ̂) = d_{τ}^{α}

whenever applicable. ⊓ ⊔((M5), Case 1)

Remark. We would like to point out that in the above argument, we are unable to prove that ωϱ_{Nτ̂}^{ω} = α(τ̂). Thus, it may happen that this projectum is the cardinal predecessor of α(τ̂), even in the case where ωϱ_{Nτ̄}^{ω} = α(τ̄). It may namely happen that the single ordinal in p_{Nτ̂} ∩α(τ̂) is not in the range of σ_{0}^{1}. The situation here can equivalently be described as τ̄ ≺ τ̂ where τ̂ is the largest element in S_{α(τ̂)}. Verification of (M5) is the only place in the entire construction of a Gap-1 Morass where it may happen that we introduce ordinals that are of the form max(S_{β}) for some β, even though the inputs are not of this form. Verifications of all other axioms can be arranged so that starting from β < max(S_{β}) we produce an ordinal β′ that is smaller than max(S_{β′}).

Case 2: τ ∈ 𝒯^{1}.

In this case Mτ̄ is either a protomouse or else a pluripotent premouse such that cr(E_{top}^{Mτ̄}) = μ = cr(E_{top}^{M_{τ}}) is the common cardinal predecessor of α(τ) and α(τ̄) and ϑ_{τ} = ϑτ̄ = α(τ̄). ^{19} From the interpolation argument we obtain a coherent structure M̂, and maps σ_{0}^{1} and σ_{1}^{1} where σ_{i}^{1} is an extension of σ_{i}^{0} for i = 0, 1, and:

• σ_{0}^{1} : Mτ̄ → M̂ is Σ_{0}-preserving and cofinal;

• σ_{1}^{1} : M̂ → M_{τ} is Σ_{0}-preserving and cofinal;

• σ_{1}^{1} ∘ σ_{0}^{1} = σ.

19That ϑ_{τ} = ϑτ̄ follows by an argument similar to that at the end of the paragraph immediately above (i) in Case 1.

CARDINAL TRANSFER PROPERTIES IN EXTENDER MODELS 49

It follows that both maps σ_{0}^{1} and σ_{1}^{1} are Σ_{1}-preserving. The cofinality of these maps can be verified as in Case 1. [In fact, there is another way of seeing this: Just recall that ϑτ̄ = ϑ_{τ} and, by construction, this value is equal to ϑ(M̂). The rest follows from the fact that we deal with coherent structures and both maps σ_{0}^{1} and σ_{1}^{1} agree with the identity map up to ϑ.]

Our aim is to prove that M̂ = Mτ̂. The main tool we are going to use for this purpose is Lemma 1.15. In the following we verify the assumptions of the lemma. First notice the important fact that M̂ is a coherent structure whose top extender is not weakly amenable since ϑ(M̂) = ϑτ̄ ≤ α(τ̄) < α̂ = μ^{+M̂}. This can be seen similarly as in the verification of Case 2 in (M4) or directly using the Σ_{1}-elementarity of σ_{1}^{1}. Using an argument analogous to that in the verification of (M4), Case 2 we obtain:

(i) M̂ = hN̂(α̂ ∪ {q̂}) where q̂ = σ_{0}^{1}(qτ̄ − ᾱ).

(ii) For every β ∈ q̂ there is a generalized witness Qβ,q̂ ; M̂ with respect to q̂ and M̂. Both clauses follow from the corresponding properties of Mτ̄. This verifies assumptions (a) and (b) in Lemma 1.15; assumption (c) is obviously satisfied as σ_{1}^{1}∘σ_{0}^{1} = σ. We can thus apply Lemma 1.15 to the triple σ_{1}^{1} : M̂ → M_{τ}. To get the desired conclusion, we verify that conclusion (B) in the lemma does not apply. Here we proceed as in Case 2 in the verification of (M3), applying Proposition 1.13: Letting ϑ = ϑ(M̂) = ϑτ̄ = ϑ_{τ} and E = EM̂, we find a function f ∈ J_{ϑ}^{E} and an ordinal ξ < μ such that α̂ = EM̂ ; top(f)(q̂, ξ). We then get a contradiction by observing that α̂ ∈ rng(σ_{0}^{1}), whereas α̂ = cr(σ_{0}^{1}). Notice that unlike in the case of (M3), here we also have to consider the situation wehre τ̄ ∈ 𝒯^{0}. Due to our notation, both cases τ̄ ∈ 𝒯^{0} and τ̄ ∈ 𝒯^{1} are treated the same way.

Finally we have to verify that the divisor (μ, pM̂ ∩ λ(EM̂ ; top)) for Nτ̂ is strong above α̂ = α(τ̂), but this can be done exactly as in the verification of (M4), Case 2.

⊓ ⊔(M5)

This completes the construction of a Gap-1 Morass in L[E].

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Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213-3890, USA

E-mail address: eschimme@andrew.cmu.edu

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

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