ORGANIC AND TIGHT J. CUMMINGS, M. FOREMAN, AND E. SCHIMMERLING

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1 ORGANIC AND TIGHT J. CUMMINGS, M. FOREMAN, AND E. SCHIMMERLING 1. Introduction This paper is motivated by the combinatorics of ℵ ω in L where there are canonical examples of scales, square sequences other structures of PCF theory. There, too, concepts such as tight sets, mutually stationary sequences, tightly stationary sequences have concrete reformulations in terms of fine structure. One reason for this analysis is that it leads to reasonable conjectures about singular cardinal combinatorics in ZFC alone. Another is that it also tells us what to expect in arbitrary extender models, which are models of the form L[E] where E is a coherent sequence of extenders. Moreover, somewhat surprisingly, there remain basic open problems about the combinatorics of ℵ ω in L. We use more or less stard terminology from PCF theory but only as it applies to ℵ ω. Thus, for any set X with X < ℵ ω, we have the characteristic function char X : n sup(x ℵ n ) we speak as if char X is always a function in the product ℵ n even though it actually belongs to n N ℵ n for some N. We say that X has eventual cofinality ℵ k iff cf(char X (n)) = ℵ k for all but finitely many n < ω. Recall that a sequence S n 1 < n < ω is mutually stationary iff S n is stationary in ℵ n for 1 < n < ω for every structure A in a countable language whose universe includes ℵ ω, there exists X A such that char X (n) S n for all but finitely many n < ω. The official definition of mutual stationarity says that there exists X whose characteristic function meets S n for all 1 < n < ω but our definition is equivalent in most cases. Here is an example, which is essentially the one analyzed in Section 7.2 Research partially supported by NSF DMS Grants (Cummings), (Foreman) (Schimmerling). 1

2 2 J. CUMMINGS, M. FOREMAN, AND E. SCHIMMERLING of Foreman Magidor [FM01]. Assume V = L. Define S k n+1 to be the set of α such that ℵ n < α < ℵ n+1, cf(α) > ω, J α = every set has cardinality ℵ n, if β α is least such that then J β+1 = α = ℵ n, ℵ n = ρ k+1 (J β ) < α ρ k (J β ). Then Sn+1 k is stationary for every pair k, n < ω, for every function e : ω ω, the sequence Sn e(k) 1 < n < ω is mutually stationary iff e is eventually constant. Given A {ℵ n n < ω}, a scale of length λ in A is a sequence f α α < λ of functions that is increasing cofinal in A relative to the order g < h iff g(n) < h(n) for all but finitely many n A. Shelah proved that there is a set A such that there exists a scale of length ℵ ω+1 on A. Moreover, if 2 ℵω = ℵ ω+1, then this holds with A = {ℵ n n < ω}. Most of the results in this paper have GCH built into their hypotheses. For this reason, we will refer to scales of length ℵ ω+1 on ℵ n simply as scales. In L, there is a canonical scale that we will define later in the paper. A scale is continuous iff for every limit β < ℵ ω+1 of uncountable cofinality, if there is an eub for f α α < β, then f β is such an eub. Here eub sts for exact upper bound, which means that f β is an upper bound for f α α < β in terms of <, for every g < f β, there exists α < β such that g < f α. Recall that a set X is tight iff X ℵ n is cofinal in (X ℵ n ). Suppose f α α < ℵ ω+1 is a continuous scale in ℵ n. It is a fact proved in [FM01] that, if X H(ℵ ω+1 ) X has eventual cofinality ℵ m for some m 1, then X is tight iff char X = f sup(x ℵω+1 ) sup(x ℵ ω+1 ) is a good ordinal. For the record, an ordinal γ < ℵ ω+1 is good iff there is an unbounded S γ such that for all but finitely many n < ω, f α (n) α S is a strictly increasing sequence of ordinals. Notice that whether an ordinal is good depends on the choice of continuous scale. Jensen s square principle ℵω says there exists a sequence C α α < ℵ ω+1 such that for every limit ordinal β < ℵ ω+1, C β is a club subset of β with type(c β ) ℵ ω,, for every α lim(c β ), C β α C α. Recall, also, that if ℵω holds, then modulo

3 ORGANIC AND TIGHT 3 the club filter over ℵ ω+1, if γ has uncountable cofinality, then γ is good. In fact, square principles very much weaker than ℵω suffice for this conclusion; see [FM97]. Thus, in contexts such as V = L, the equation displayed above characterizes tightness for sets with an uncountable eventual cofinality. This motivates the following definition. Definition 1. Let f α α < ℵ ω+1 be a continuous scale in ℵ n X be a set. Then X is organic to f α α < ℵ ω+1 iff there exists δ < ℵ ω+1 such that char X = f δ. We are not satisfied with our understing of the extent to which organicity depends on the choice of scale. Given continuous scales, f α α < ℵ ω+1 g α α < ℵ ω+1, there is a club C in ℵ ω+1 such that f α = g α for every α C. Can this be used, as in the case of tightness, to see that X is organic relative to f α α < ℵ ω+1 iff X is organic relative to g α α < ℵ ω+1 for many sets X? In Section 2, it is shown that the collection of sets that are inorganic relative to a fixed continuous scale is stationary (Theorem 3), as is the collection of organic sets that are not tight (Theorem 4). In Section 3, we work in L define a scale in a canonical way. Then, we give characterizations of tight sets organic sets in L in terms of fine structure (Theorem 7). Recall that a mutually stationary sequence S n 1 < n < ω is tightly stationary iff for every structure A in a countable language whose universe includes ℵ ω, there exists a tight X A such that char X (n) S n for all but finitely many n < ω. We introduce the following related concept. Definition 2. A mutually stationary sequence S n 1 < n < ω is organically stationary iff for every structure A in a countable language whose universe includes ℵ ω, there exists an organic X A such that for all but finitely many n < ω. char X (n) S n Cummings, Foreman Magidor [CFM] showed that a certain forcing notion adds a mutually stationary sequence that is not tightly stationary. In fact, their example is not organically stationary. We do not know whether there is a model in which every mutually stationary sequence is organically stationary, or every organically stationary sequence is tightly stationary. And, in spite of the progress in Section 3, we do not know the answers regarding L.

4 4 J. CUMMINGS, M. FOREMAN, AND E. SCHIMMERLING Since it is related to another purpose of this paper, we give an overview of the result from [CFM] mentioned in the previous paragraph. Donder, Jensen Stanley [DJS] introduced a combinatorial principle called Coherent Squares proved it holds in L. Coherent Squares involves ℵn -sequences C n α α < ℵ n+1 for n ω a continuous scale in ℵ n that are tied together in a particular way. Given Coherent Squares a sequence of limit ordinals γ n 1 < n < ω such that sup γ n = ℵ ω γ n < ℵ n, let T n+1 = {α ℵ n+1 Cof(ℵ 1 ) type(c n α) γ n }. In [CFM], it is shown that T n+1 1 < n < ω is not tightly stationary. An easy modification of their proof shows that it is not organically stationary. They also showed that if the Coherent Squares was obtained by forcing in a certain way, then T n+1 1 < n < ω is mutually stationary. Therefore, it is consistent that mutual organic stationarity are different. The authors of [CFM] asked whether the Coherent Squares in L has a similar effect; in Section 3 we show that it does not (Corollary 9). 2. About V A set X is said to be ℵ 1 -uniform iff ℵ 1 = X X cf(x ℵ n ) = ℵ 1 whenever 1 < n < ω. Theorem 3 is due to the first two authors. The various claims in its proof reiterate arguments from [CFM], which we include to make this paper more self-contained. Theorem 3. Assume GCH fix a continuous scale. Then the set is stationary. {X H(ℵ ω+1 ) X is ℵ 1 -uniform inorganic} Proof. The first claim is that if X H(ℵ ω+1 ) is ℵ 1 -uniform, then X ℵ n is ω-closed. To see this, let a X ℵ n with type(a) = ω. Let α = sup(a). Then α < sup(x ℵ n ). Let β = min(x α). Then α β. For contradiction, suppose that α < β. Then cf(β) > ℵ 1 as ℵ 1 X X is bounded in β. But now ℵ 1 = cf(x cf(β)) = cf(x β) = cf(x α) = ℵ 0. A corollary to the first claim is that, for every ℵ 1 -uniform X H(ℵ ω+1 ), there is a sequence C n 1 < n < ω such that type(c n ) = ω 1 C n is a club subset of X ℵ n whenever 1 < n < ω. Let A be an expansion of H(ℵ ω+1 ). The second claim is that the set {char X X A X is ℵ 1 -uniform}

5 ORGANIC AND TIGHT 5 is closed in (ℵ n Cof(ℵ 1 )) with the product of the discrete topologies on the coordinate spaces. Say lim char X i = α n 1 < n < ω. i This means that lim sup(x i ℵ n ) = α n i whenever 1 < n < ω. In turn, this means that whenever 1 < n < ω, there exists k(n) < ω such that for all i > k(n), sup(x i ℵ n ) = α n. By the corollary to the first claim, we have Cn i such that type(cn) i = ω 1 Cn i is a club subset of X i ℵ n for i < ω 1 < n < ω. Let D n = Cn. i k(n)<i<ω Then type(d n ) = ω 1 D n is a club subset of X i ℵ n whenever 1 < n < ω k(n) < i < ω. In particular, sup(d n ) = α n for 1 < n < ω. Let E = 1<n<ω X = Hull A (E). It is straightforward to verify that char X = α n 1 < n < ω. By the second claim, there is a tree T such that D n [T ] = {char X X A X is ℵ 1 -uniform}. The third claim is that T has a stationary branching subtree. To see this, play the following game. I A 0 ℵ 2 Cof(ℵ 1 ) A 1 ℵ 3 Cof(ℵ 1 ) II α 0 A 0 α 1 A 1 where every A n must be nonstationary A n T or else player I loses. Also, α n n < ω [T ] or else II loses. If both players survive, then II wins I loses. If player II has a winning strategy, τ, then the set of sequences α i i < n coming from plays according to τ is a stationary branching subtree of T. Since the game is determined, the other possibility is that player I has a winning strategy, σ. Let X A be ℵ 1 -uniform with σ X. Put Clearly, α n = sup(x ℵ n+2 ). α n n < ω [T ].

6 6 J. CUMMINGS, M. FOREMAN, AND E. SCHIMMERLING We will derive a contradiction by showing that player II can play these ordinals against σ survive. The verification is by induction. Let A 0 = σ( ). Then A 0 X. Since A 0 is nonstationary X H(θ), there exists C X such that C is club in ℵ 2 A 0 C =. Then α 0 C, so α 0 A 0. Let A 1 = σ( α 0 ). Then A 1 σ( β ). β ℵ 2 Cof(ℵ 1 ) The set on the right is an element of X is a nonstationary subset of ℵ 3, so it does not have α 1 as an element. Hence α 1 A 1. Continuing in this way, we complete the proof of the third claim. The final claim is that there is an X A such that X is ℵ 1 -uniform inorganic. To see this, note that if X Y are organic sets, then char X char Y or vice-versa. On the other h, if S is a stationary branching subtree of T, then S has branches char X char Y that are -incomparable. Theorem 4. Assume GCH fix a continuous scale. Then the set is stationary. {X H(ℵ ω+1 ) X is ℵ 1 -uniform, organic but not tight} Proof. We modify the proof of Zapletal s result (found in [FM01]) that the set of non-tight X H(ℵ ω+1 ) with eventual uncountable cofinality is stationary. Let A be a structure whose underlying set is H(ℵ ω+1 ). Let M A be transitive with M = ℵ ω. Let γ = sup(or M). Let X α α < ω 1 be a continuous chain of countable elementary submodels of M δ α α < ω 1 be an increasing sequence of ordinals such that δ 0 γ, char Xα < f δα. ran(f δα ) X α+1. Let δ = sup α<ω1 δ α X = α<ω 1 X α. Then char X is an eub for f η η < δ hence so is f δ. Therefore char X = f δ. 3. About L In this section, we relate the notions we have been discussing to the fine structure of L. These results are due to the third author. Here is some of the notation that we will use. If M is a structure n ω, then ρ n (M) p n (M) are the n-th projectum the n-th stard parameter of M respectively. If κ is an ordinal but not a cardinal of L, then β(κ) is the least β κ such that ρ ω (J β ) < κ, n(κ) is the

7 ORGANIC AND TIGHT 7 least n < ω such that ρ n+1 (J β(κ) ) < κ, M κ is the n-th mastercode structure for J β(κ). We will write L α instead of J α in contexts where it is clear that they are equal, which happens exactly when α = ω α. The following fact is useful. Theorem 5. Assume that V = L. Let X L ℵω with X < ℵ ω. Then X has an eventual cofinality. Proof. Say X = ℵ l. Let π : L κω X. Say π(κ n ) = ℵ n for every n < ω. Then cf(κ n ) = cf(sup(π κ n )) = cf(sup(x ℵ n )) = cf(char X (n)) for every n < ω. We mainly care about n > l since crit(π) = κ l+1. Recall that OR M κ = ρ n (J β(κ) ) ρ 1 (M κ ) = ρ n+1 (J β(κ) ). In our case, ρ 1 (M κω ) is either 1 or an uncountable cardinal of L κω. Hence, there is l m < ω such that Soundness tells us that ω ρ 1 (M κω ) = κ m. M κω = Hull Mκω 1 (κ m p 1 (M κω )). Thus, all cardinals of M κω between κ m+1 κ ω are collapsed definably over M κω. Hence, M κω = M κn whenever m < n < ω. By a stard fine structural calculation, cf(κ n+1 ) = cf(or M κω ) whenever m n < ω. To see this, define a function by setting h : OR M κω κ n+1 h(σ) = sup(κ n+1 Hull Mκω σ 1 (κ m p 1 (M κω )). Then, h is non-decreasing, i.e., cofinal, i.e., So we are done. σ τ = h(σ) h(τ), sup(ran(h)) = κ n+1. When working in L, it is often useful to consider the set Λ n+1 of local successors of ℵ n. What we mean is that α Λ n+1 iff ℵ n < α < ℵ n+1 ℵ n is the largest cardinal of L α. Since Λ n+1 is club in ℵ n+1, we often slur over the difference between Λ n+1 ℵ n+1. Consider an arbitrary α Λ ω+1. Then ρ 1 (M α ) = ℵ ω M α = Hull Mα 1 (ℵ ω p 1 (M α )).

8 8 J. CUMMINGS, M. FOREMAN, AND E. SCHIMMERLING Define f α (n + 1) = sup(ℵ n+1 Hull Mα 1 (ℵ n p 1 (M α ))). We will refer to f α α Λ ω+1 as the canonical scale in L, which is justified by the naturalness of its definition the following result. Theorem 6. Assume V = L. Then f α α Λ ω+1 is a continuous scale in Λ n+1. Proof. First we show that the sequence is < -increasing. Consider ordinals α < β in Λ ω+1. Pick m < ω so that If m n < ω, then α Hull M β 1 (ℵ m p 1 (M β )). f α (n + 1) Hull M β 1 (ℵ n p 1 (M β )) so f α (n + 1) < f β (n + 1). Next we show that the sequence is < -cofinal. Given g Λ n+1, pick α Λ ω+1 with g J α. Let m < ω be large enough that g Hull M β 1 (ℵ m p 1 (M β )). Easily we see that g(n + 1) < f α (n + 1) whenever m n < ω. Finally, we show that the scale is continuous. Let β be a limit point of Λ ω+1 of uncountable cofinality. We show that f β is an eub for Assume that g < f β. Let f α α β Λ ω+1. σ = sup(or Hull M β 1 (ran(g) p 1 (M β )). Then σ < OR M β because the ordinal height of M β has the same uncountable cofinality as β. The proof is like the stard calculation at the end of the proof of Theorem 5. Let M be the Mostowski collapse of Hull M β σ 1 (ℵ ω p 1 (M β )) α = ℵ M ω+1. Then M = M α, α β Λ ω+1 g < f α. The following theorem characterizes organic tight relative to the canonical scale in terms of fine structure.

9 ORGANIC AND TIGHT 9 Theorem 7. Assume V = L. Let X L ℵω+1 such that X < ℵ ω. Suppose that the eventual cofinality of X is uncountable. Say π : L κω+1 X, for every n ω, π(κ n ) = ℵ n. Let Q = ult(m κω, π, ℵ ω ). Then, relative to the canonical scale, X is organic Q is wellfounded X is tight M κω+1 = M κω. Recall that if X has eventual uncountable cofinality X is tight, then X is organic. Above, Q is the ultrapower of M κω by the extender of length ℵ ω derived from π. It is the ultrapower formed by using coordinates a [ℵ ω ] <ω functions f : [κ n ] a M κω with n < ω f M κω. This makes sense precisely because M κω has the same bounded subsets of κ ω as dom(π) = L κω+1. Recall that there is a bounded subset of κ ω that is not an element of M κω but is Σ 1 definable over M κω. Thus, in terms of the natural ordering on mastercode structures for levels of L, if M κω M, then the extender of length ℵ ω derived from π cannot be applied to M. In particular, note that M κω+1 M κω. Proof. Let ψ : M κω Q be the ultrapower map. Also set define maps ψ n ψ n,n M κω Q n = ult(m κω, π, ℵ n ) ψ n according to the diagram ψ Q n Q n ψn,n ψ n Q Take m < ω such that ρ 1 (M κω ) = κ m.

10 10 J. CUMMINGS, M. FOREMAN, AND E. SCHIMMERLING Recall that M κω is sound, that is, M κω = Hull Mκω 1 (κ m p 1 (M κω )). First we assume that Q is wellfounded show that X is organic. This part of the proof does not use the hypothesis that X has eventual uncountable cofinality. Identify Q with its Mostowski collapse. Then ψ is a cofinal Σ 1 -elementary embedding from M κω to Q The soundness of M κω ψ κ ω+1 = π κ ω+1. translates into that of Q, namely, Q = Hull Q 1 (ℵ ω p 1 (Q)). Let δ be the cardinal successor of ℵ ω in Q. Then Q = M δ. Observe that if λ is the cardinal successor of κ ω in M κω, then δ = sup(ψ λ). Note too that κ ω+1 λ γ δ where For n < ω, we have that γ = sup(π κ ω+1 ) = sup(x ℵ ω+1 ). ψ n κ n+1 = π κ n+1 ℵ Qn n+1 = ψ n (κ n+1 ) = sup(ψ nκ n+1 ) = sup(π κ n+1 ) = sup(x ℵ n+1 ) = char X (n + 1). If m n < ω, then Q n is the Mostowski collapse of Thus Hull Q 1 (ℵ n p 1 (Q)). f δ (n + 1) = sup(ℵ n+1 Hull M δ 1 (ℵ n p 1 (M δ ))) = ℵ Qn n+1. We have seen that for m n < ω, char X (n + 1) = ℵ Qn n+1 = f δ (n + 1). In particular, X is organic. Using the hypothesis that X has eventual uncountable cofinality, we have also seen that if Q is wellfounded, then X is tight δ = γ λ = κ ω+1 M κω = M κω+1. Now drop the assumption that Q is wellfounded. We claim that Q n is wellfounded for every n < ω. Suppose otherwise. Then we have a sequence of coordinates a i i < ω from [ℵ n ] <ω a sequence of functions f i i < ω from M κω such that dom(f i ) = [κ n ] a i

11 ORGANIC AND TIGHT 11 ψ(f i+1 )(a i+1 ) ψ(f i )(a i ) for i < ω. By a stard calculation that we have used before, cf(x ℵ n+1 ) = cf(κ n+1 ) = cf(or M κω ) whenever m n < ω. Since X has eventual uncountable cofinality, cf(or M κω ) > ω. Therefore, there exists σ < OR M κω large enough so that Mκω σ {f i i < ω} Hull1 (κ m p 1 (M κω )). Let Mκω σ G : Hull1 (κ n p 1 (M κω )) M be the Mostowski collapse. Then M L κn+1 = dom(π). We have a commutative diagram M ψ π(m) ult(m, π, ℵ n ) that implies ult(m, π, ℵ n ) is wellfounded. On the other h, dom(g(f i )) = [κ n ] a i ψ(g(f i+1 ))(a i+1 ) ψ(g(f i ))(a i ) for i < ω, which implies that ult(m, π, ℵ n ) is illfounded. This completes the proof of the claim. If M κω = M κω+1, then Q = ult(m κω+1, π, ℵ ω ) the argument of the previous paragraph can be applied with n = ω to see that Q is wellfounded. Therefore, if M κω = M κω+1, then X is tight. Finally, assume that X is organic. Say char X = f δ. We will conclude that Q is wellfounded by showing Q = M δ. Note that Q is the direct limit of the structures Q n under the maps ψ n,n for n < n < ω. Let ψ : Q n M δ be the inverse of the Mostowski collapse of Hull M δ 1 (ℵ n p 1 (M δ ))

12 12 J. CUMMINGS, M. FOREMAN, AND E. SCHIMMERLING ψ n,n : Q n Q n be the inverse of the Mostowski collapse of Hull e Q n 1 (ℵ n p 1 ( Q n )). Then M δ is the direct limit of the structures Q n under the maps ψ n,n for n < n < ω. It is enough to show that Q n = Q n ψ n,n = ψ n,n for all sufficiently large n < n < ω. Now Q n is wellfounded, ψ n is continuous at κ n+1, so ψ n κ n+1 = π κ n+1 char X (n + 1) = sup(x ℵ n+1 ) = sup(π κ n+1 ) for n < ω. Because M κω Thus, = sup(ψ nκ n+1 ) = ψ n (κ n+1 ) = ℵ Qn n+1 is sound, Q n is too whenever m n < ω, i.e., Q n = Hull Qn 1 (ℵ n p 1 (Q n )). Q n = M charx (n+1) whenever m n < ω. It also follows that ψ n,n Mostowski collapse of is the inverse of the Hull Q n 1 (ℵ n p 1 (Q n )) whenever m n < n < ω. In other words, ψ n,n is determined by Q n the same way that ψ n,n is determined by Q n for large enough n < n < ω. From the definition of f δ it is clear that f δ (n + 1) = ℵ M f δ (n+1) n+1 Q n = M fδ (n+1) for n < ω. Since char X (n + 1) = f δ (n + 1) for all sufficiently large n < ω, the result follows.

13 ORGANIC AND TIGHT 13 There is are ZFC questions that comes out of the previous theorem. Given X H(ℵ ω ), let π : M X be the Mostowski collapse π(κ n ) = ℵ n for n < ω. Call X firm iff for every transitive model N of enough set theory, if H(κ ω ) N = H(κ ω ) M, then ult(n, π, ℵ ω ) is wellfounded. In L, firm implies organic with the right interpretation of enough set theory they are equivalent for X with eventual uncountable cofinality. What happens if V L? Theorem 8. Assume V = L. Let C α α < ℵ n+1 be the canonical ℵn -sequence for n < ω. Suppose that X L ℵω X < ℵ ω. Assume that the eventual cofinality of X is uncountable. Then is eventually constant for n < ω. type(c charx (n+1)) By canonical we mean the one defined by Jensen. The theorem gives us the following answer to a question from [CFM] related to Coherent Square their example of a mutually stationary sequence that is not tightly stationary. Corollary 9. Assume V = L. Let C α α < ℵ n+1 be the canonical ℵn -sequence for n < ω. Let γ n 1 < n < ω be a sequence such that sup γ n = ℵ ω γ n ℵ n. Put T n+1 = {α < ℵ n+1 cf(α) > ω type(c α ) γ n }. Then T n+1 1 < n < ω is not mutually stationary. Sketch of proof of Theorem 8. For Γ ℵ n+1, define C α α lim(γ) to be a ℵn (Γ)-sequence iff every β lim(γ), (1) C β is club in β Γ, (2) type(c β ) ℵ n (3) if α lim(c β ), then C α = C β α. Then ℵn (Γ) is equivalent to ℵn for every club Γ in ℵ n via the Mostowski collapse Γ ℵ n+1. In L, we usually work with the canonical ℵn (Λ n+1 )-sequence instead. In order to sketch the proof of the theorem, we must isolate certain features of how this sequence is defined. The definition of the sequence can be found in the original source [J] although the notation is different. It is also found in [DJS].

14 14 J. CUMMINGS, M. FOREMAN, AND E. SCHIMMERLING Fix α lim(λ n+1 ). The definition of C α determined by M α. Recall that α equals either ℵ Mα n+1 or OR M α, M α is sound, so M α = Hull Mα 1 (ℵ n p 1 (M α )). If α has uncountable cofinality, then there is a limit ordinal θ ℵ n, a sequence of ordinals d = d i i < j with ℵ n > d 0 > > d j 1 θ a sequence of ordinals s = s i i < j with s 0 < < s j 1 < OR M α such that C α is the image of θ under the non-decreasing function η sup(α Hull Mα 1 (η d p 1 (M α ) s)). If α has countable cofinality, then there are two possibilities: either the definition of C α has the same form as above or it is an arbitrary set of type ω unbounded in α. We will not explain the choice of d, s θ here so we cannot explain why this works. We are especially interested in α Λ n+1 whose associated M α has a slightly rich cardinal structure. For l ω, define Λ n l to be the set of α between ℵ n ℵ n+1 such that for every k < l, M α = ℵ l exists. Then Λ n+1 = Λ n n+1 Λ n n+2 Λ n ω. Now we look again at the canonical ℵn (Λ n+1 )-sequence for a fixed α Λ n l where l n + 1. For k < l, set κ k = ℵ Mα k. We also set κ l = ℵ Mα l or κ l = OR M α depending on whether M α has an l-th infinite cardinal. Thus κ n+1 = α, for n + 1 k l, M κk = M α. Define functions F κk with domain θ by let F κk : η sup(κ k Hull Mα 1 (η d p 1 (M α ) s)) C κk = ran(f κk ).

15 ORGANIC AND TIGHT 15 Notice that the definition of C κk is consistent with that of C α described earlier in that C κn+1 = C α κ k Λ n+1 for every k > n + 1. Fine structure calculations based on the definitions of s, d θ that we have omitted show that if n + 1 k l, then C κk is a club subset of κ k Λ n l. They also show that if n + 1 k < k l η < η < θ, then hence F κk (η) < F κk (η ) F κk (η) < F κk (η ), type(c κk ) = type(c κk ). And they show coherence: if κ lim(c κk ), then C κ = C κk κ. The case k = n+1 just repeats the coherence clause (3) for ℵn+1 (Λ n+1 ). Now let X L ℵω such that X < ℵ ω X has eventual uncountable cofinality. Let π : L κω X π(κ n ) = ℵ n for n < ω. Fix m < ω such that κ m = ρ 1 (M κω ). We will use the fact from the previous paragraph that for m n < ω, type(c κn+1 ) = type(c κm+1 ) Let Q n = ult(m κω, π, ℵ n ) ψ n : M κω Q n be the ultrapower map. By ideas earlier in this section: the hypothesis of eventual uncountable cofinality implies that Q n is wellfounded; since M κω is sound, Q n = Hull Qn 1 (ℵ n p 1 (Q n )) because ψ n κ n+1 = π κ n+1, Q n = M charx (n+1) for m n < ω. Fine structure calculations using the definition of s, d θ show that F charx (n+1) = ψ n F κn+1,

16 16 J. CUMMINGS, M. FOREMAN, AND E. SCHIMMERLING hence Therefore, C charx (n+1) = ψ nc κn+1. type(c charx (n+1)) = type(c κn+1 ) = type(c κm+1 ) for m n < ω the theorem follows. The authors thank Philip Welch for pointing out that Corollary 9 also follows from a result in [KW]. References [CFM] J. Cummings, M. Foreman M. Magidor, Canonical structure in the universe of set theory: part two, Ann. Pure Appl. Logic 142 (2006) [DJS] H. Donder, R. Jensen L. Stanley, Condensation-coherent global square systems, in Recursion theory (Ithaca, N.Y., 1982), , Proc. Sympos. Pure Math., 42, Amer. Math. Soc., Providence, 1985 [FM97] M. Foreman M. Magidor, A very weak square principle, J. Symb. Logic 62 (1997) [FM01] M. Foreman M. Magidor, Mutually stationary sequences of sets the non-saturation of the non-stationary ideal on P κ (λ), Acta Math. 186 (2001) [J] R. Jensen, The fine structure of the constructible hierarchy, Ann. Math. Logic 4 (1972) [KW] P. Koepke P. Welch, On the strength of mutual stationarity, in Set theory (Centre de recerca Matematica, Barcelona, ), , Trends Math., Birkhäuser, Basel, 2006