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

Size: px

Start display at page:

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

Evelyn Sherman
5 years ago
Views:

1 ORGANIC AND TIGHT J. CUMMINGS, M. FOREMAN, AND E. SCHIMMERLING 1. Introduction This report 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 mutual stationarity tightness have concrete reformulations in terms of fine structure. Several kinds of new questions come out of this analysis. First, surprisingly, there remain basic open problems regarding what is true in L. Second, it leads to reasonable conjectures about the combinatorics of ℵ ω in ZFC alone without V = L. Third, it gives us expectations about arbitrary singular cardinals in extender models L[E]. 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 this is slightly incorrect. 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 countable structure A 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 this is equivalent. The following is one of several examples analyzed by Foreman Magidor [FM2001]. Assume V = L. Define S k n+1 to be the set of α between ℵ n ℵ n+1 such that, if β α is least such that J α = every set has cardinality ℵ n J β+1 = α = ℵ n, 1

2 2 J. CUMMINGS, M. FOREMAN, AND E. SCHIMMERLING then 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. A scale is a sequence f α α < ℵ ω+1 that is increasing unbounded in ℵ n relative to the order g < h iff g(n) < h(n) for all but finitely many n < ω. The 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. That is, 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 ). Consider an arbitrary continuous scale f α α < ℵ ω+1 in ℵ n. By [FM2001], 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. This can be stated without reference to goodness if ℵ ω holds since then the set of good ordinals contains a club in ℵ ω+1. We introduce the following variant. 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. In some contexts, the definition above is not sensitive to the choice of scale. (If f α α < ℵ ω+1 g α α < ℵ ω+1 are continuous scales, then there is a club C in ℵ ω+1 such that f α = g α for every α C.) There are times, too, when the scale is clear from context. For example, when we work in L, we always mean the canonical scale that we will define. In such settings, we suppress mention of f α α < ℵ ω+1. The first two authors showed that the set of inorganic sets with eventual uncountable cofinality is stationary; see Section 2. One purpose of this report is to give practical characterizations of tightness organicity in L in terms of fine structure. This is done in Section 3. Recall that a mutually stationary sequence S n 1 < n < ω is tightly stationary iff for every countable structure A 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.

3 ORGANIC AND TIGHT 3 Definition 2. A mutually stationary sequence S n 1 < n < ω is organically stationary iff for every countable structure A 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 [CFM2006] showed a certain forcing notion adds a mutually stationary sequence that is not tightly stationary. Since this is related to other purposes of this report, we give an overview. Donder, Jensen Stanley [DJS] introduced a combinatorial principle called Coherent Squares proved it holds in L. Coherent Squares involves n -sequences Cα n 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 [CFM2006], 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 added by forcing, then T n+1 1 < n < ω is mutually stationary. Therefore, it is consistent that is mutual organic stationarity are different. In spite of what seems like progress, we have not answered the basic question of how mutual, organic tight stationarity are related in L. 2. About V A set X is said to be ℵ 1 -uniform iff ℵ 1 = X X cf(sup(x ℵ n )) = ℵ 1 whenever 1 < n < ω. The following result is due to the first second authors. The same technique was used for another purpose in [CFM2006]. Theorem 3. Assume GCH. Then the set is nonstationary. {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

4 4 J. CUMMINGS, M. FOREMAN, AND E. SCHIMMERLING 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} 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,

5 ORGANIC AND TIGHT 5 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 ]. 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 organic. 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. 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 [FM2001]) that 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.

6 6 J. CUMMINGS, M. FOREMAN, AND E. SCHIMMERLING 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 Here is some of the fine structure notation that we will use in our discussion of L. 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 least n < ω such that ρ n+1 (J β(κ) ) < κ, M κ is the n-th mastercode structure for J β(κ). The following fact is useful. Theorem 5. Assume that V = L. Let X L ℵω X has an eventual cofinality. with X < ℵ ω. Then Proof. 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, we have l m < ω such that ρ 1 (M κω ) = κ m. Then M κω = M κn whenever m < n < ω. By a stard fine structural calculation, cf(κ n ) = cf(or M κω ) whenever m < n < ω, so we are done. 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 Define M α = Hull Mα 1 (ℵ ω p 1 (M α )). 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 following. Theorem 6. Assume V = L. Then f α α Λ ω+1 is a continuous scale in Λ n+1.

7 ORGANIC AND TIGHT 7 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 < -unbounded. 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 must show that f β is an eub for f α α β Λ ω+1. Assume that g < f β. Let 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. Let M be the Mostowski collapse of Hull M β σ 1 (ℵ ω p 1 (M β )). Then M = M α for some α β Λ ω+1 g < f α. The following is our characterization of organic tight. Theorem 7. Assume V = L. Let X L ℵω+1 such that X < ℵ ω X has eventual uncountable cofinality. Say, for every n ω, Let Then π : L κω+1 X π(κ n ) = ℵ n. Q = ult(m κω, π, ℵ ω ). X is organic Q is wellfounded X is tight M κω+1 = M κω.

8 8 J. CUMMINGS, M. FOREMAN, AND E. SCHIMMERLING 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 be the ultrapower map. Also set define maps ψ n ψ n,n M κω Q 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 Recall that M κω is sound, that is, M κω ρ 1 (M κω ) = κ m. = 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)).

9 ORGANIC AND TIGHT 9 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 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 M κω = M κω+1. We remark that clause (2) of the theorem does not allow us to assume that Q is wellfounded for this characterization of tight; we will return to this point. 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 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 {f i i < ω} Hull Mκω σ 1 (κ m p 1 (M κω )).

10 10 J. CUMMINGS, M. FOREMAN, AND E. SCHIMMERLING 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 )) 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

11 ORGANIC AND TIGHT 11 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 ) = sup(ψ nκ n+1 ) = ψ n (κ n+1 ) = ℵ Qn n+1 for n < ω. Because M κω is sound, Q n is too whenever m n < ω, i.e., Q n = Hull Qn 1 (ℵ n p 1 (Q n )). Thus, Q n = M charx (n+1) whenever m n < ω. It also follows that ψ n,n is the inverse of the Mostowski collapse of 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 for n < ω. Since f δ (n + 1) = ℵ M f δ (n+1) n+1 Q n = M fδ (n+1) char X (n + 1) = f δ (n + 1) for all sufficiently large n < ω, the result follows. 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?

12 12 J. CUMMINGS, M. FOREMAN, AND E. SCHIMMERLING 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 X has eventual uncountable cofinality. Then is eventually constant for n < ω. type(c charx (n+1)) By canonical we mean the one defined by Jensen. answers a question from [CFM2006]. The following 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. For Γ ℵ n+1, we say that C α α lim(γ) is 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 describe certain features of how this sequence is defined. 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 so we cannot explain why this works.

13 ORGANIC AND TIGHT 13 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, Then M α = ℵ l exists. 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, for n + 1 k l, Define functions F κk let κ n+1 = α M κk = M α. with domain θ by F κk : η sup(κ k Hull Mα 1 (η d p 1 (M α ) s)) C κk = ran(f κk ). 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 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 ).

14 14 J. CUMMINGS, M. FOREMAN, AND E. SCHIMMERLING such that X < ℵ ω X has eventual uncount- Now let X L ℵω able 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, because Q n = Hull Qn 1 (ℵ n p 1 (Q n )) 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 hence Therefore, F charx (n+1) = ψ n F κn+1, 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.

15 ORGANIC AND TIGHT 15 References [CFM2001] J. Cummings, M. Foreman M. Magidor, Squares, scales stationary reflection, J. Math. Logic 1 (2001) [CFM2003] J. Cummings, M. Foreman M. Magidor, The non-compactness of square, J. Symbolic Logic 68 (2003) [CFM2004] J. Cummings, M. Foreman M. Magidor, Canonical structure in the universe of set theory: part one, Ann. Pure Appl. Logic 129 (2004) [CFM2006] J. Cummings, M. Foreman M. Magidor, Canonical structure in the universe of set theory: part two, Ann. Pure Appl. Logic 142 (2006) [DJS] H.-D. Donder, R.B. Jensen L.J. Stanley, Condensation-coherent global square systems, in Recursion theory (Ithaca, N.Y., 1982), , Proc. Sympos. Pure Math., 42, Amer. Math. Soc., Providence, RI, 1985 [FM2001] 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) [S] R.-D. Schindler, Mutual stationarity in the core model, in Logic Colloquium 01, , Lect. Notes Log., 20, Assoc. Symbol. Logic, Urbana, IL, 2005

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

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