MORE FINE STRUCTURAL GLOBAL SQUARE SEQUENCES

MARTIN ZEMAN

Abstract. We extend the construction of a global square sequence in extender models from [8] to a construction of coherent non-threadable sequences and give a characterization of stationary reflection at inaccessibles similar to Jensen’s characterization in L.

AMS Subject Classification: 03E05, 03E45, 03E55

Keywords: Global quare sequence, fine structure, extender model, weakly compact cardinal, stationary reflection.

February 16, 2009

1 This note presents a fine structural construction of a so-called □(κ, A) sequence 2 for certain stationary subsets A of an inaccessible cardinal κ as well as a charac-3 terization of weakly compact cardinals in fine structural extender models in terms 4 of stationary reflection. These results extend analogous results of Jensen for the 5 constructible universe that originate in [3] and are described in more detail in [1]. 6 Although the characterization of weakly compact cardinals in an extender model 7 turns out to be exactly the same as in L, the proof requires a significant amount 8 of extra work. Also, the author believes that the proof presented in this paper is 9 more straightforward than that described in [3] and [1]. 10 The exposition in this paper is based on extender models with Jensen’s λ-11 indexing of extenders introduced in [4]; see [7] as a reference. The paper builds 12 on previous work on fine structural square sequences in extender models, in par-13 ticulsr on [5, 6] and [8]. We will frequently refer to [8]. However, no detailed 14 knowledge of arguments in [8] is necessary, as we will only need certain lemmata 15 from that paper which can be used as black boxes. In particular, all references 16 concerning protomice will be hidden in black boxes.

17 Theorem 0.1 (Main Theorem). Working in a fine-structural Jensen-style extender 18 model L[E], assume κ is an inaccessible cardinal that is not weakly compact and 19 A ⊆ κ is stationary. Then there is a stationary A′ ⊆ A and a sequence ⟨C_{τ} | τ < κ⟩ 20 satisfying the following conditions. 21 (a) C_{τ} is a closed unbounded subset of τ. 22 (b) Cτ̄ = C_{τ} ∩ τ̄ whenever τ̄ ∈ lim(C_{τ}). 23 (c) A′ ∩ lim(C_{τ}) = ∅.

24 A sequence satisfying (a) – (c) in the above theorem is called a □(κ, A′)-sequence. 25 Any such sequence is a □(κ)-sequence, that is, it cannot be threaded: If C ⊆ κ is 26 a closed unbounded set then C ∩ α ≠ C_{α} for some limit point α of C. From the 27 above theorem we obtain the following corollaries, the first of which is immediate.

Research partially supported by NSF grant DMS-0500799.

1 Corollary 0.2. Let L[E] be a Jensen-style extender model. The following di-2 chotomy is true in L[E] of any inaccessible cardinal κ. 3 • κ is weakly compact ⟹ every stationary subset of κ reflects at some κ̄ < κ. 4 • κ is not weakly compact ⟹ nonreflecting stationary subsets of κ are dense. 5 In particular, an inaccessible cardinal κ is weakly compact just in case that every 6 stationary subset of κ reflects at some κ̄ < κ.

7 The first clause in the above corollary is, of course, a ZFC consequence. It should 8 be noted that reflection points of stationary subsets whose existence is guaranteed 9 by weak compactness are regular. Not only the argument that is used to obtain 10 reflection points produces regular (in fact inaccessible) reflection points; the fact 11 that we have a global square sequence on singular cardinals in L[E] guarantees that 12 any reflection point of any stationary subset of an inaccessible cardinal κ must be 13 regular. The property that every stationary subset of κ reflects at some singular 14 ordinal κ̄ < κ or at an ordinal of fixed uncountable cofinality, if consistent with ZFC, 15 must have high consistency strength; however the exact result here is not known. 16 Even at small regular cardinals, the requirement that every stationary set reflects 17 at some ordinal of small cofinality implies the consistency of measurable cardinals 18 of high Mitchell order; see [2].

19 Corollary 0.3. Let V = L[E] be a Jensen-style extender model. Then for any 20 regular cardinal κ that is not weakly compact there is a Suslin κ-tree.

21 This follows from Jensen’s construction of higher Suslin trees in [3]. For successor 22 cardinals κ = μ^{+} where μ is not subcompact one uses ♢_{κ}(A) and □(κ, A) for a suit-23 able stationary A ⊆ κ; here the □(κ, A)-sequence is obtained from a □_{μ}-sequence 24 whose existence is guaranteed by [6]. If μ is subcompact then μ is inaccessible, 25 so GCH in L[E] makes it possible to construct a Suslin κ-tree “naively” by using 26 only a ♢_{κ}(S_{μ}^{κ})-sequence^{1} to seal off large antichains at limit stages of cofinality κ 27 in the construction, and adding all possible branches at limit stages of cofinality 28 smaller than κ. For inaccessible κ one constructs a Suslin κ-tree using ♢_{κ}(A) and 29 □(κ, A) as above; this time the existence of a □(κ, A)-sequence is guaranteed by 30 Theorem 0.1.

31 1. The construction

32 We will work in a fixed model L[E] where E is a Jensen-style extender sequence, 33 that is, an extender sequence with λ-indexing of extenders. The predicate E is thus 34 also fixed. Throughout the construction we will use the Condensation Lemma for 35 premice; this is Lemma 2.2 in [8] or Lemma 9 in [7]. We will often make use of the 36 following simple consequence of the Condensation Lemma.

37 Proposition 1.1. Assume that σ : M̄ → M be a Σ_{0}-preserving embedding where 38 M is an L[E]-level and M̄ = ⟨J Ē ; ᾱ, Ē_{ωα}⟩ is an acceptable J-structure. Let τ = cr(σ) 39 and δ = τ^{+M̄}; here we allow the option that δ = ht(M̄) if τ is the largest cardinal 40 in M̄. If τ is a limit cardinal in M̄ then Ē ↾ δ = E ↾ δ.

41 From now on assume that κ is an inaccessible cardinal that is not weakly com-42 pact. As it is typical with constructions of □-like principles, we begin with identi-43 fying canonical structures assigned to ordinals τ < κ. As κ is not weakly compact, 1 there is a κ-tree on κ without cofinal branch; we fix the <_{E}-least one. Thus

1S_{μ}^{κ} = {ξ < κ | cf(ξ) = μ}

(1)  T = the <_{E}-least κ-tree on κ without cofinal branch.

2 Obviously, T ∈ J_{κ^{+}}^{E} and (1) defines T inside J_{κ^{+}}^{E}. We will write T ↾ τ to denote the 3 restriction of T to τ, that is, T ↾ τ is the tree on τ with tree ordering <_{T} ∩(τ × τ).

4 Lemma 1.2. There is a closed unbounded set of cardinals 𝒞 ⊆ κ and a map 5 τ ↦ δ_{τ} < τ^{+} such that for every τ ∈ 𝒞 we have 6 • τ is the largest cardinal in J_{δ_{τ}}^{E} and is inaccessible in J_{δ_{τ}}^{E}. 7 • T ↾ τ is a τ-tree in J_{τ}^{E} with no cofinal branch in J_{δ_{τ}}^{E}. 8 • T ↾ τ is an initial segment of T, that is, for all ξ ∈ T ↾ τ and all ζ ∈ T we 9 have ζ <_{T} ξ ⇒ ζ ∈ T ↾ τ.

10 Proof. Let ⟨X_{ξ} | ξ < κ⟩ be a continuous chain of elementary substuctures of J_{κ^{+}}^{E} 11 such that each X_{ξ} is of size τ_{ξ} = X_{ξ} ∩ κ ∈ κ. Clearly T ∈ X_{ξ} for all ξ < κ, as T 12 is definable in J_{κ^{+}}^{E}. By Proposition 1.1, each X_{ξ} collapses to some J_{δ(ξ)}^{E}. The fact 13 that X_{ξ} ≺ J_{κ^{+}}^{E} guarantees that τ_{ξ} is the largest cardinal in J_{δ(ξ)}^{E}, is inaccessible in 14 J_{δ(ξ)}^{E} and the tree T collapses to T ↾ τ_{ξ} ∈ J_{δ(ξ)}^{E} that has no cofinal branch in J_{δ(ξ)}^{E}. 15 So we can let 𝒞 = {τ_{ξ} | ξ < κ} and δ_{τ_{ξ}} = δ(ξ). 16 To see that T ↾ τ_{ξ} is an initial segment of T it suffices to show that the α-th 17 level of T ↾ τ_{ξ} agrees with the α-th level of T for all α < τ_{ξ}. Fix such an α. By 18 elementarity, there is a bijection f_{α} : θ_{α} → T_{α} in X_{ξ} where T_{α} is the α-th level of T 19 and θ_{α} is its size. Since T is a κ-tree, θ_{α} < κ so θ_{α} < τ_{ξ}. Then T_{α} = rng(f_{α}) ⊆ X_{ξ}, 20 as θ_{α} ⊆ X_{ξ}. ⊓ ⊔

21 Let τ ∈ 𝒞. Since T ↾ τ is an initial segment of T and T has height κ, the tree 22 T ↾ τ has a cofinal branch in L[E]. For τ ∈ 𝒞 we let

δ′_{τ} = the maximal δ such that T ↾ τ has no cofinal branch in J_{δ}^{E}.

23 By the above proposition, δ′_{τ} ≥ δ_{τ}. We would like to pick L[E] || δ′_{τ} as our canonical 24 structure, but the fact that τ may be collapsed inside L[E] || δ′_{τ} or even definably 25 collapsed over L[E] || δ′_{τ} does not allow to make this choice for each τ ∈ 𝒞. If a 26 cofinal branch of T ↾ τ is introduced later or at the same time when τ is singularized, 27 τ will be treated the same way as in the construction of a global square sequence. 28 This motivates our choice of the canonical structure. We define 29 • 𝒞^{0} = the set of all τ ∈ 𝒞 such that τ is singular in J_{δ′_{τ} +1}^{E}. 30 • 𝒞^{1} = 𝒞 − 𝒞^{0}. 31 and 32 • N_{τ} = the singularizing level of L[E] for τ if τ ∈ 𝒞^{0}. 33 • N_{τ} = L[E] || δ′_{τ} = ⟨J_{δ′_{τ}}^{E} , E_{ωδ′_{τ}} ⟩ if τ ∈ 𝒞^{1}.

34 Notice that even if τ ∈ 𝒞^{0} we have ht(N_{τ}) ≥ δ_{τ}, so T ↾ τ ∈ N_{τ} for all τ ∈ 𝒞. 35 We first define the sets C_{τ} witnessing Theorem 0.1 for τ ∈ lim(𝒞). We will treat 36 the cases τ ∈ 𝒞^{i}, i = 0, 1 separately and show that the two constructions do not 37 interfere. We begin with 𝒞^{0}, as here we can use the global square sequence of [8]. 38 Let ⟨C′_{τ} | τ ∈ 𝒮 ∩ κ⟩ denote the global square sequence from [8]^{2} where 𝒮 is the 39 class of all singular cardinals. So each C′_{τ} is a closed subset of τ that is unbounded 1 whenever τ has uncountable cofinality, the sequence of sets C′_{τ} is fully coherent and 2 otp(C′_{τ}) < τ for each τ ∈ 𝒮. The class 𝒮 is divided into two disjoint classes 𝒮^{0} and 3 𝒮^{1} and the sets C′_{τ} satisfy the inclusions C′_{τ} ⊆ 𝒮^{i} whenever τ ∈ 𝒮^{i} for i = 0, 1. We 4 first make the following observation.

2This is the sequence which is denoted by ⟨C_{τ} | τ ∈ 𝒮⟩ in [8]; here we write C′_{τ} instead of C_{τ}, as C_{τ} will be the final sequence produced in this paper.

5 Lemma 1.3. If τ ∈ lim(𝒞) ∩ 𝒞^{0} and cf(τ) > ω then there is some γ < τ such that 6 C′_{τ} ∩ 𝒞 − γ ⊆ 𝒞^{0}.

7 Proof. Obviously, C′_{τ} ∩ 𝒞 is unbounded in τ. As τ ∈ 𝒞^{0}, the canonical structure 8 N_{τ} is the singularizing L[E]-level for τ. Let τ̄ ∈ C′_{τ} and N̄ be the singularizing 9 L[E]-level for τ̄. By the construction in [8], there is a Σ_{0}-preserving map σ_{τ̄,τ} 10 such that σ_{τ̄,τ} : N̄ → N_{τ} if τ ∈ 𝒮^{0} and σ_{τ̄,τ} : M̄ → M_{τ} if τ ∈ 𝒮^{1}; here M̄ and 11 M_{τ} are the canonical protomice assigned to τ̄ and τ. In our situation we have 12 T ↾ τ ∈ N_{τ}, as τ ∈ 𝒞. First assume τ ∈ 𝒮^{0}. If τ̄ ∈ C′_{τ} ∩ 𝒞 is large enough that 13 T ↾ τ ∈ rng(σ_{τ̄,τ}) then T ↾ τ̄ ∈ N̄, σ_{τ̄,τ} (T ↾ τ̄) = T ↾ τ and T ↾ τ̄ has no cofinal 14 branch in Nτ̄. These conclusions are consequences of the Σ_{0}-elementarity of the 15 map; the former two follow by an argument similar to that in proof of Lemma 1.2 16 and the latter one follows from the fact that the nonexistence of a cofinal branch can 17 be expressed as a Π_{1}-statement, so it is preserved backward under σ_{τ̄,τ}. Hence that 18 T ↾ τ̄ has no cofinal branch in the singularizing structure for τ̄, and consequently 19 Nτ̄ = N̄. Now assume τ ∈ 𝒮^{1}. The conclusion then follows from the fact that M_{τ} 20 and N_{τ} compute the cardinal successor of τ the same way and they agree below this 21 common successor, and the same is true of the structures M̄ and N̄ and cardinal 22 τ̄ to which they are assigned. The same argument as above can be then used with 23 the map σ_{τ̄,τ} which is now a map between two protomice. As before we conclude 24 that N̄ = Nτ̄. It follows that τ̄ ∈ 𝒞^{0} and the same conclusion can be made for any 25 τ′ such that τ̄ ≤ τ′ < τ, so it suffices to let γ = τ̄. ⊓ ⊔

For τ ∈ lim(𝒞) ∩ 𝒞^{0} we let
γ_{τ} = the least γ ≤ τ such that C′_{τ} ∩ 𝒞 − γ ⊆ 𝒞^{0}

26 and define C_{τ}^{∗} as follows. 27 • If C′_{τ} ∩ 𝒞 − γ_{τ} is unbounded in τ we let C_{τ}^{∗} = C′_{τ} ∩ 𝒞 − γ_{τ}; 28 • otherwise C_{τ}^{∗} is the <_{E}-least sequence of order type ω converging to τ. 29 Lemma 1.3 together with the properties of the sets C′_{τ} guarantee that this definition 30 makes sense, each C_{τ}^{∗} is a closed unbounded subset of τ and C∗ ; τ̄ = C_{τ}^{∗} ∩ τ̄ whenever 31 τ̄ ∈ lim(C_{τ}^{∗}). Thus, for τ ∈ lim(𝒞) ∩ 𝒞^{0}, (a) and (b) in Theorem 0.1 hold with C_{τ}^{∗} 32 in place of C_{τ}. 33 We next define sets C_{τ}^{∗} for τ ∈ lim(𝒞)∩𝒞^{1}. The definition of the sets C_{τ}^{∗} is based 34 on the following observation, which is a direct consequence of the fact that τ ∈ 𝒞^{1}.

(2)  τ ∩ h̃_{N_{τ}}^{n+1}(α ∪ {p_{N_{τ}} }) is bounded in τ whenever α < τ and n ∈ ω.

35 The sets C_{τ}^{∗} are defined as follows.

36 • C_{τ}^{∗} is the set of all τ̄ ∈ τ ∩ 𝒞^{1} satisfying: Nτ̄ is a premouse of the same type 37 as N_{τ} and there is a Σ^{∗}-preserving embedding σ_{τ̄,τ} such that: 38 (i) τ̄ = cr(τ) and σ_{τ̄,τ} (τ̄) = τ. 39 (ii) σ_{τ̄,τ} (p_{Nτ̄}) = p_{N_{τ}}. 40 (iii) σ_{τ̄,τ} (T ↾ τ̄) = T ↾ τ.

1 Clause (iii) in the above definition is superfluous, as it is easy to see that (ii) holds 2 even if we drop it. We include it as a part of the definition in order to simplify 3 the matters. Clearly, the map σ_{τ̄,τ} is the unique Σ^{∗}-preserving map σ : Nτ̄ → N_{τ} 4 satisfying (i) and (ii).

5 Lemma 1.4. If τ ∈ lim(𝒞) ∩ 𝒞^{1} and cf(τ) > ω then C_{τ}^{∗} is unbounded in τ.

6 Proof. Given some τ^{∗} < τ we find τ̄ ∈ C_{τ}^{∗} such that τ^{∗} ≤ τ̄. As is typical for 7 constructions of square sequences, we will look for the right kind of hulls. Let 8 • n ∈ ω be such that ωϱ_{N_{τ}}^{n+1} ≤ τ < ωϱ_{N_{τ}}^{n};

9 Such an n exists, as there is a cofinal branch through T ↾ τ in J_{δ′_{τ} +1}^{E} − J_{δ′_{τ}}^{E} and such 10 a branch, being a subset of τ, is Σ^{∗}-definable over J_{δ′_{τ} +1}^{E} || δ′_{τ} = N_{τ}. Let x ∈ [τ]^{<ω} 11 be such that T ↾ τ ∈ h̃_{N_{τ}}^{n+1}(x ∪ {p_{N_{τ}} }), W_{N_{τ}}^{ν,p_{τ}} ∈ h̃_{N_{τ}}^{n+1}(x ∪ {p_{N_{τ}} }) whenever ν ∈ p_{N_{τ}}, 12 and some cofinal branch through T ↾ τ is Σ_{1}^{(n)}(N_{τ})-definable from x and p_{N_{τ}}. Such 13 an x exists, as these tasks require only finite amount of information. Define a 14 sequence ⟨τ_{k}, X_{k} | k ∈ ω⟩ of ordinals below τ and hulls as follows.

τ_{0} = max(x ∪ {τ^{∗}}) + 1
X_{k} = ⋃_{ℓ∈ω} h̃_{N_{τ}}^{ℓ+1}(τ_{k} ∪ {p_{N_{τ}} })
τ′_{k+1} = sup(τ ∩ X_{k})
τ_{k+1} = min(𝒞 − τ′_{k+1}) + 1

15 By (2), each τ ∩h̃_{N_{τ}}^{ℓ+1}(τ_{κ}∪{p_{N_{τ}} }) is smaller than τ, granting that τ_{k} < τ. Since τ has 16 uncountable cofinality, also τ_{k+1} < τ, which enables us to run induction on k and 17 then conclude that also τ̄ = sup({τ_{k} | k ∈ ω}) is below τ. Letting X = ⋃_{k∈ω} X_{k} 18 we have τ ∩ X = τ̄. In the following we show that τ̄ ∈ C_{τ}^{∗}. 19 Notice first that since the ordinals τ_{k} are strictly increasing and each interval 20 (τ_{k}, τ_{k+1}) has nonempty intersection with 𝒞, the supremum τ̄ is a limit point of 𝒞, 21 so τ̄ ∈ 𝒞. We next observe:

(3)  X = h̃_{N_{τ}}^{n+1}(τ̄ ∪ {p_{N_{τ}} }).

22 By construction, each z ∈ X is of the form h̃_{N_{τ}}^{ℓ+1}(i, ⟨z̄, p_{N_{τ}} ⟩) for some ℓ ≥ n, i ∈ ω 23 and z̄ ∈ [τ̄]^{<ω}. Assume ℓ > n. The function h̃_{N_{τ}}^{ℓ+1}(u, ⟨v, p_{N_{τ}} ⟩) can be expressed as a 24 composition h̃_{N_{τ}}^{n+1}((u)_{0}^{2}, ⟨h((u)_{1}^{2}, v), p_{N_{τ}} ⟩) where h : ω × J_{τ}^{E} → J_{τ}^{E} is a partial good 25 Σ_{1}^{(ℓ+1)}-function; see [7], Section 1.8 for details. If z̄ ∈ [τ_{k}]^{<ω} then h((i)_{1}^{2}, z̄) ∈ J_{τ′_{k}}^{E}, 26 and since there is a uniformly Σ_{1}-definable surjection of τ′_{k} onto J_{τ′_{k}}^{E} we can replace 27 the above value of h with some finite z′ ∈ [τ′_{k}]^{<ω}. So there is some j < ω such that 28 z = h̃_{N_{τ}}^{n+1}(j, ⟨z′, p_{N_{τ}} ⟩). This proves (3). 29 Let N̄ be the transitive collapse of X and σ : N̄ → N_{τ} be the inverse to the 30 Mostowski collapsing isomorphism. Then τ̄ = cr(σ) and σ(τ̄) = τ. Moreover, it 31 follows from (3) and the construction of X that

(4)  X = h̃_{N_{τ}}^{ℓ+1}(τ̄ ∪ {p_{N_{τ}} }) whenever ℓ ≥ n,

32 so the map σ is Σ_{1}^{(ℓ)}-preserving for all such ℓ hence Σ^{∗}-preserving. As x, T ↾ τ ∈ X, 33 we have x, T ↾ τ̄ ∈ N̄ and σ(x, T ↾ τ̄) = (x, T ↾ τ). For x this is immediate, for T ↾ τ 34 this follows from the fact that T ↾ τ ∈ rng(σ) by an argument similar to that in the 1 proof of Lemma 1.2. Since p_{τ} ∈ X we have some p̄ ∈ N̄ such that σ(p̄) = p_{τ}. From 2 (3) we obtain N̄ = h̃n+1 ; N̄(τ̄ ∪ {p̄}) which implies that ωϱn+1 ; N̄ ≤ τ̄ and p̄ ∈ Rn+1 ; N̄.^{3} 3 By construction τ̄ is a limit cardinal in L[E], so actually ωϱω ; N̄ = ωϱn+1 ; N̄ = τ. 4 The structure N̄ is a premouse of the same type as N; this follows from the Σ^{∗}- 5 elementarity of σ. 6 An application of the Condensation Lemma to the map σ : N̄ → N_{τ} then 7 yields that N̄ is a solid premouse. The choice of the set x at the beginning of 8 the construction guarantees that for each ν ∈ p_{τ} the standard witness W_{N_{τ}}^{ν,p_{τ}} is an 9 element of X, so its preimage Q_{ν} under σ is a generalized witness for ν̄ = σ^{−1}(ν) ∈ p̄ 10 with respect to M and p̄. ^{4} So p̄ = pN̄ and N̄ is sound above τ̄ by [7], Lemma 1.12.5. 11 One more application of the Condensation Lemma to the map σ : N̄ → N then 12 gives us the following options: (a) N̄ = core(N), (b) N̄ is a proper initial segment 13 of N, (c) N̄ is an ultrapower of an initial segment N′ of N with critical point 14 equal to the cardinal predecessor of τ̄ in N′ and (d) N̄ is a proper initial segment 15 of Ult(N, Eτ̄). Here option (a) is impossible as N̄ and N have different ultimate 16 projecta and options (c) and (d) are impossible as τ̄ is a limit cardinal in N̄. Thus, 17 N̄ is a proper initial segment of N_{τ} and thereby an initial segment of L[E]. 18 So far we have obtained an initial segment N̄ of L[E] and a Σ^{∗}-preserving map 19 σ : N̄ → N_{τ} such that τ^{∗} < τ̄ = cr(σ) is inaccessible in N̄ and σ(τ̄, pN̄, T ↾ τ̄) = 20 (τ, p_{τ} , T ↾ τ). Obviously, T ↾ τ̄ has no cofinal branch in N̄, as T ↾ τ has no cofinal 21 branch in N_{τ} and σ is sufficiently elementary. In order to verify that τ̄ ∈ C_{τ}^{∗} 22 we have to verify that τ̄ ∈ 𝒞^{1} which amounts to showing that N̄ = Nτ̄. This is 23 equivalent to saying that τ̄ is regular in J_{β+1}^{E} and T ↾ τ̄ has a cofinal branch in 24 J_{β+1}^{E} where β = ht(Nτ̄). The former follows immediately from the construction of 25 X, as the Σ^{∗}-elementarity of σ implies that h̃ℓ+1 ; N̄(τ_{k} ∪ {pN̄}) is bounded in τ̄ for all 26 k, ℓ ∈ ω. As any function f : τ̄ → τ̄ that is an element of J_{β+1}^{E} is definable over N̄ 27 and therefore can be expressed in the form h̃ℓ+1 ; N̄(− ∪ {q ∪ pN̄}) for some ℓ ∈ ω and 28 q ∈ [τ̄]^{<ω}, such function cannot singularize τ̄. 29 To see that J_{β+1}^{E} contains a cofinal branch through T ↾ τ̄ it suffices to show that 30 such a branch is Σ^{∗}-definable over N̄. Let b ∈ J_{δ′_{τ} +1}^{E} be a cofinal branch through

31 T ↾ τ. Similarly as with f above, it follows that b is Σ_{1}^{(ℓ)}(N_{τ})-definable from p_{N_{τ}} 32 and some q ∈ [τ]^{<ω} for some ℓ ∈ ω. Let φ(u, v) be a Σ_{1}^{(ℓ)}-formula that defines b, 33 that is, for each ξ < τ we have

ξ ∈ b ⟺ N_{τ} ⊧ φ(ξ, q ∪ p_{N_{τ}}).

34 We first observe that q can be taken from [τ̄]^{<ω}. This is the case, as the statement 35 “the set of all ξ < τ satisfying N_{τ} ⊧ φ(ξ, q ∪ p_{N_{τ}}) determines a cofinal branch 36 through T ↾ τ” can be expressed in a Π_{1}^{(ℓ+2)}-manner, namely as the conjunction of

\begin{matrix}(∀ξ^{ℓ+1}, ζ^{ℓ+1}) \\ [(φ(ξ^{ℓ+1}, q ∪ p_{N_{τ}}) & φ(ζ^{ℓ+1}, q ∪ p_{N_{τ}})) ⟶ (ξ^{ℓ+1} <_{T} ζ^{ℓ+1} ∨ ζ^{ℓ+1} <_{T} ξ^{ℓ+1})] \end{matrix}

37 and

(∀ξ^{ℓ+2})(∃ζ^{ℓ+1})(ζ^{ℓ+1} > ξ^{ℓ+2} & φ(ζ^{ℓ+2}, q ∪ p_{N_{τ}})).

3See [7], Section 1.5 4See ([7], Section 1.12 or ([8])

1 The former expresses that b determines a branch through T ↾ τ and the latter 2 expresses that the branch is cofinal. This conjunction is a statement about q and 3 p_{N_{τ}}; denote it by ψ(q, p_{N_{τ}}). As q witnesses that N_{τ} ⊧ (∃z^{ℓ+3})ψ(z, p_{N_{τ}}) and X is 4 closed under good Σ^{∗}-functions, there also must be a witness q̄ ∈ X. Then q̄ ∈ [τ̄]^{<ω} 5 and N̄ ⊧ ψ(q̄, pN̄). It follows that {ξ < τ̄ | N̄ ⊧ φ(ξ, q̄∪ pN̄)} determines a cofinal 6 branch through T ↾ τ̄. Such branch is Σ_{1}^{(ℓ)}-definable over N̄ in parameters. This 7 completes the proof of the fact that τ̄ ∈ 𝒞^{1} and thereby the proof of the lemma. ⊓ ⊔

8 Lemma 1.5. If τ ∈ lim(𝒞) ∩ 𝒞^{1} then C_{τ}^{∗} is closed.

9 Proof. Let τ̄ be a limit point of C_{τ}^{∗}. We show that τ̄ ∈ C_{τ}^{∗}. As in the previous 10 lemma, let 11 • n be such that ωϱ_{N_{τ}}^{n+1} ≤ τ < ωϱ_{N_{τ}}^{n}.

12 We first observe that if τ^{∗} ∈ C_{τ}^{∗} then ωϱ_{N_{τ^{∗}}}^{n+1} ≤ τ̄ < ωϱ_{N_{τ^{∗}}}^{n}. The inequality on

13 the right follows from the fact that N_{τ} satisfies the Σ_{1}^{(n)}-statement (∃ξ^{n})(τ < ξ^{n}) 14 and this statement is preserved under σ. The inequality on the left follows from 15 the fact that h̃_{N_{τ}}^{n+1}(τ ∪ {p_{N_{τ}} }) = N_{τ} and σ preserves Σ_{1}^{(n)}-statements. Consider 16 the diagram ⟨N_{τ^{∗}}, σ_{τ^{∗},τ′} | τ^{∗} ≤ τ′ ∈ τ̄ ∩ C_{τ}^{∗}⟩; let ⟨N̄, σ_{τ^{∗}} | τ^{∗} ∈ τ̄ ∈ C_{τ}^{∗}⟩ be its 17 direct limit with the direct limit maps σ_{τ^{∗}} : N_{τ^{∗}} → N̄ and let σ̄ : N̄ → N_{τ} be 18 the canonical embedding of the direct limit N̄ into N_{τ} satisfying σ̄ ∘ σ_{τ^{∗}} = σ_{τ^{∗},τ}. 19 Standard considerations yield that N̄ can be viewed as a premouse of the same type 20 as N_{τ} and all N_{τ^{∗}}, all maps σ_{τ^{∗}} and σ̄ are Σ^{∗}-preserving and ωϱn+1 ; N̄ ≤ τ̄ < ωϱn ; N̄. If 21 p̄ is the common value of σ_{τ^{∗}}(p_{N_{τ^{∗}}}) then obviously σ̄(p̄) = p_{N_{τ}}. By the properties 22 of n recorded above, h̃_{N_{τ}}^{n+1}(τ ∪ {p_{N_{τ}} }) = N_{τ} so h̃n+1 ; N̄(τ̄ ∪ {p̄}) = N̄, as follows from 23 preservation properties of σ̄. An application of the Condensation Lemma to the map 24 σ̄ : N̄ → N_{τ} yields that N̄ is solid. Since each N_{τ^{∗}}, being a proper initial segment 25 of L[E], is sound, for each ν ∈ p_{N_{τ^{∗}}} the standard witness W_{N_{τ^{∗}}}^{ν,p_{N_{τ^{∗}}}} is an element of 26 N_{τ^{∗}} and its image under σ_{τ^{∗}} is a generalized witness for σ_{τ^{∗}}(ν) with respect to N̄ 27 and p̄, as σ_{τ^{∗}} is sufficiently elementary. This way we conclude that for each element 28 of p̄ there is in N̄ a generalized witness with respect to N̄ and p̄, and exactly as 29 in the proof of Lemma 1.4 then conclude that N̄ is sound and p̄ = pN̄. One more 30 application of the Condensation Lemma then yields, exactly as in Lemma 1.4 that 31 N̄ is an initial segment of N_{τ}. Obviously τ̄ = cr(σ), σ(τ̄, pN̄) = τ, p_{N_{τ}}, the cardinal 32 τ̄ is inaccessible in N̄ and N̄, being a limit point of C_{τ}^{∗}, is a limit point of 𝒞 hence 33 τ̄ ∈ 𝒞. It remains to prove that τ̄ ∈ 𝒞^{1}. As σ is Σ^{∗}-preserving, this follows exactly 34 as in the proof of Lemma 1.4.^{5} ⊓ ⊔

35 Lemma 1.6. If τ ∈ lim(𝒞) ∩ 𝒞^{1} and τ̄ ∈ lim(C_{τ}^{∗}) then τ̄ ∈ lim(𝒞) ∩ 𝒞^{1} and

36 C∗ ; τ̄ = C_{τ}^{∗} ∩ τ̄.

37 Proof. Since τ ∈ lim(𝒞) ∩ 𝒞^{1}, the condition τ̄ ∈ lim(C_{τ}^{∗}) implies τ̄ ∈ lim(𝒞) ∩ 𝒞^{1}, 38 so C_{τ}^{∗} and C∗ ; τ̄ are defined in the same way. If τ^{∗} ∈ Cτ̄ then τ^{∗} ∈ 𝒞^{1} and we have 39 the map σ_{τ^{∗},τ̄} : N_{τ^{∗}} → Nτ̄ witnessing the memebership of τ^{∗} to C∗ ; τ̄. But then 40 σ_{τ̄,τ} ∘ σ_{τ^{∗},τ̄} : N_{τ^{∗}} → N_{τ} witnesses the membership of τ^{∗} to C_{τ}. Conversely, if 1 τ^{∗} ∈ C_{τ}^{∗} ∩ τ̄ then τ^{∗} ∈ 𝒞^{1} and there is a map σ_{τ^{∗},τ} witnessing the membership of 2 τ^{∗} to C_{τ}. Since both σ_{τ^{∗},τ} and σ_{τ̄,τ} are Σ^{∗}-preserving and τ^{∗} < τ̄ we have

5Alternatively, one can consider a definition of a cofinal branch of T ↾ τ^{∗} over N_{τ^{∗}} from parameters p_{N_{τ∗}} and q ∈ [τ^{∗}]^{<ω} for some/any τ^{∗} ∈ C_{τ}^{∗} ∩ τ̄ and show that the same definition over N̄ defines a cofinal branch through T ↾ τ̄ from pN̄ and q. This works, as σ_{τ^{∗}} is Σ^{∗}-preserving.

rng(σ_{τ^{∗},τ}) = ⋃_{ℓ∈ω} h̃_{N_{τ}}^{ℓ+1}(τ^{∗} ∪ {p_{N_{τ}} }) ⊆ ⋃_{ℓ∈ω} h̃_{N_{τ}}^{ℓ+1}(τ̄ ∪ {p_{N_{τ}} }) = rng(σ_{τ̄,τ}),

3 so (σ_{τ̄,τ})^{−1} ∘ σ_{τ^{∗},τ} : N_{τ^{∗}} → Nτ̄ witnesses the membership of τ^{∗} to C∗ ; τ̄. ⊓ ⊔

4 So far we have constructred sets C_{τ}^{∗} for τ ∈ lim(𝒞) such that (a) and (b) in 5 Theorem 0.1 hold with C_{τ}^{∗} in place of C_{τ}. Given stationary set A ⊆ κ, we find 6 stationary A′ ⊆ A and refine C_{τ}^{∗} into C_{τ} that will satisfy all conclusions of the 7 theorem. We let 8 • A′ = the set of all τ ∈ 𝒞 for which there are an L[E]-level P = J_{β}^{E} and a 9 parameter a ∈ P such that: 10 (a) P ⊧ ZFC^{−}, τ is the largest cardinal in P, is inaccessible in P and 11 T ↾ τ has no cofinal branch in P. 12 (b) For every X ≺ P satisfying X ∩ τ ∈ τ and p ∈ X we have X ∩ τ ∉ A. 13 The same proof as that of [8], Theorem 3.21 shows that the set A′ is stationary 14 in κ. Notice that the only difference between A′ in this paper and A′ in [8] is the 15 additional requirement in (a) above that T ↾ τ has no cofinal branch in P and the 16 restriction of the set A′ to elements of the closed unbounded set 𝒞.

17 Lemma 1.7. Let τ ∈ lim(𝒞). If τ̄ ∈ lim(C_{τ}^{∗}) ∩ A′ then there is some τ^{∗} ∈ C_{τ}^{∗} ∩ τ̄ 18 such that A is disjoint with C_{τ}^{∗} ∩ (τ^{∗}, τ̄).

19 Proof. For τ ∈ 𝒞^{0} this was proved in [8], Lemma 3.22. For τ ∈ 𝒞^{1} the same 20 argument goes through. If there is a pair (P, a) ∈ Nτ̄ witnessing the membership 21 of τ̄ to A′ the argument can be literally repeated: Given τ′ ∈ C_{τ}^{∗} ∩ τ̄ large enough 22 that (P, a) is in the range of σ_{τ′,τ̄}, let P′ ∈ N_{τ′} be such that σ_{τ′,τ̄}(P′) = P; then 23 X = σ_{τ′,τ̄}[P′] ≺ P and a ∈ X, so τ′ = X ∩ τ̄ ∉ A. In the remaining case we 24 conclude that P = J_{δ′ ; τ̄}^{E} where recall that δ′ ; τ̄ = ht(Nτ̄). This is the case, as T ↾ τ̄ has 25 a cofinal branch in J_{δ′_{τ̄+1}}^{E}. As P ⊧ ZFC^{−} and Nτ̄ projects to τ̄, necessarily E_{top}^{Nτ̄} is an

26 extender with λ(E_{top}^{Nτ̄}) = τ̄. Since σ_{τ′,τ̄} is Σ^{∗}-preserving, E_{top}^{N_{τ′}} is an extender with 27 λ(E_{top}^{N_{τ′}}) = τ′ and the two extenders have the same critical point μ < τ′. Moreover, 28 since both τ′ and τ̄ are limit cardinals, both N_{τ′} and Nτ̄ compute the cardinal 29 successor of μ the same way as L[E]; denote this common successor by ϑ. As both 30 these premice are coherent structures, J_{δ′_{τ′}}^{E} = Ult(J_{ϑ}^{E}, E_{top}^{N_{τ′}}), J_{δ′ ; τ̄}^{E} = Ult(J_{ϑ}^{E}, E_{top}^{Nτ̄}) 31 and it follows immediately that σ_{τ′,τ̄} : π′(f)(α) ↦ π̄(f)(σ_{τ′,τ̄}(α)) and therefore is 32 fully elementary. Hence if a ∈ rng(σ_{τ′,τ̄}) then X = rng(σ_{τ′,τ̄}) ≺ P. It follows that 33 τ′ = X ∩ τ̄ ∉ A. ⊓ ⊔

34 For τ ∈ lim(𝒞) we can now define sets C_{τ} as in [8]. We first let

δ_{τ} = the least δ ≤ τ such that A ∩ C_{τ}^{∗} − δ = ∅.

35 We then let

C_{τ} = C_{τ}^{∗} − ⋃{(δτ̄, τ̄) | τ̄ ∈ lim(C_{τ}^{∗}) ∩ A′}.

36 Then the sets C_{τ} are obviously closed. If A′ ∩ lim(C_{τ}^{∗}) is bounded in τ then C_{τ} is 37 clearly unbounded; otherwise C_{τ} is unbounded because it follows from its definition 38 that A′ ∩ lim(C_{τ}^{∗}) ⊆ C_{τ}. The coherency of sets C_{τ} follows from the coherency of 1 sets C_{τ}^{∗} and the uniformity of the definition of C_{τ}. Finally lim(C_{τ}) ∩ A′ = ∅, as 2 every element of A′ is a successor point of C_{τ}. 3 It remains to define the sets C_{τ} for τ ∉ lim(𝒞). Notice that A′ ⊆ 𝒞, which 4 simplifies the matters. The complement of lim(𝒞) can be written as the union of 5 disjoint open intervals that are bounded in κ. We assume that these intervals are 6 maximal. Let (α, β) be such an interval. Then α, β ∈ lim(𝒞) by maximality. The 7 set C_{β} is defined above, and it has no limit points in the interval (α, β). For each 8 τ ∈ (α, β) we can thus let C_{τ} = τ − (α + 1). Obviously, this definition does not 9 collide with the definition in the case where τ ∈ lim(𝒞) and satisfies (a) – (c) in 10 Theorem 0.1. This completes the entire construction.

11 References

12 [1] Devlin, K. J., Constructibility, Springer, 1984 13 [2] Cox, S., Covering Theorems for the core model, with an application to stationary set 14 reflection, to appear 15 [3] Jensen, R. B., The fine structure of the constructible hierarchy, Annals of Mathematical 16 Logic 4 (1972), 229-308 17 [4] Jensen, R. B., A new fine structure for higher core models, handwritten notes, Berlin, 1997 18 [5] Schimmerling, E. and Zeman, M., Square in core models, Square in core models, Bulletin 19 of Symbolic Logic 7(3), (2001), 305 – 314 20 [6] Schimmerling, E. and Zeman, M., Characterization of □_{κ} in core models, Journal of Math-21 ematical Logic 4(1)1, (2004), 1 – 72 22 [7] Zeman, M., Inner Models and Large Cardinals, de Gruyter Series in Logic and its Appli-23 cations 5, de Gruyter (2002) 24 [8] Zeman, M., Global square sequences in extender models, submitted, preprint available at 25 http://math.uci.edu/∼mzeman/RESEARCH

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

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