TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 357, Number 12, Pages 4813–4837 S 0002-9947(05)04007-9 Article electronically published on June 29, 2005
COFINALITY OF THE NONSTATIONARY IDEAL
PIERRE MATET, ANDRZEJ ROS LANOWSKI, AND SAHARON SHELAH
Abstract. We show that the reduced cofinality of the nonstationary ideal NSκ on a regular uncountable cardinal κ may be less than its cofinality, where the reduced cofinality of NSκ is the least cardinality of any family F of nonsta- tionary subsets of κ such that every nonstationary subset of κ can be covered by less than κ many members of F. For this we investigate connections of κ the various cofinalities of NSκ with other cardinal characteristics of κ and we also give a property of forcing notions (called manageability) which is pre- served in <κ–support iterations and which implies that the forcing notion preserves non-meagerness of subsets of κκ (and does not collapse cardinals nor changes cofinalities).
0. Introduction
Let κ be a regular uncountable cardinal. For C ⊆ κ and γ ≤ κ,wesaythatγ is a limit point of C if (C ∩ γ)=γ>0. C is closed unbounded if C is a cofinal subset of κ containing all its limit points less than κ.AsetA ⊆ κ is nonstationary if A is disjoint from some closed unbounded subset C of κ. The nonstationary subsets of κ form an ideal on κ denoted by NSκ.Thecofinality of this ideal, cof(NSκ), is the least cardinality of a family F of nonstationary subsets of κ such that every nonstationary subset of κ is contained in a member of F.Thereduced cofinality of NSκ, cof(NSκ), is the least cardinality of a family F⊆NSκ such that every nonstationary subset of κ can be covered by less than κ many members of F.This paper addresses the question of whether cof(NSκ)=cof(NSκ). Note that + κ κ ≤ cof(NSκ) ≤ cof(NSκ) ≤ 2 ,
so under GCH we have cof(NSκ)=cof(NSκ). Let κ2 be endowed with the κ–box product topology, 2 itself considered discrete. ⊆ κ We say that a set W 2isκ–meager if there is a sequence Uα : α<κ of dense κ open subsets of 2 such that W ∩ Uα = ∅.Thecovering number for the category α<κ κ of the space 2, denoted cov(Mκ,κ), is the least cardinality of any collection X of
Received by the editors March 3, 2003. 2000 Mathematics Subject Classification. Primary 03E05, 03E35, 03E55. Key words and phrases. Nonstationary ideal, cofinality. The second author thanks the University Committee on Research of the University of Nebraska at Omaha for partial support. He also thanks his wife, Ma lgorzata Jankowiak–Ros lanowska, for supporting him when he was preparing the final version of this paper. The research of the third author was partially supported by the Israel Science Foundation. Publication 799.
c 2005 American Mathematical Society 4813
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 4814 PIERRE MATET, ANDRZEJ ROS LANOWSKI, AND SAHARON SHELAH κ–meager subsets of κ2 such that X = κ2. It is not hard to verify that <κ cov(Mκ,κ) ≤ cof(NSκ) ≤ cof(NSκ) .
It follows that if cof(NSκ) < cov(Mκ,κ) and the Singular Cardinals Hypothesis + holds true, then cf(cof(NSκ)) <κand cof(NSκ)=(cof(NSκ)) .Weprove: Theorem 0.1. Assume GCH. Then there is a κ–complete, κ+–cc forcing notion P such that +ω +(ω+1) P “ cof(NSκ)=κ and cof(NSκ)=κ ”.
What about the consistency of “cof(NSκ) is regular and cof(NSκ) < cof(NSκ)”? We establish: Theorem 0.2. It is consistent, relative to the existence of a cardinal ν such that ++ N ℵ ℵ o(ν)=ν , that cof( Sω1 )= ω+1 and cov(Mℵ1,ℵ1 )= ω+2. The structure of the paper is as follows. In Section 1, for each infinite cardinal <µ µ ≤ κ we introduce the <µ–cofinality cof (NSκ)andthe<µ–dominating number <µ dκ , and we show that these two numbers are equal. Section 2 is concerned with <µ cl,<µ a variant of dκ denoted by dκ (where cl stands for “club”). We establish that <µ cl,<µ dκ = dκ if µ>ω. N N κ Sκ is the smallest normal ideal on κ. Section 3 deals with Sκ,λ, the small- P <µ N κ est κ–normal ideal on κ(λ). We compute cof ( Sκ,λ) and give examples of <µ N κ N κ situations when cof ( Sκ,λ) < cof( Sκ,λ). In the following section we present some basic facts regarding the ideal of κ– meager subsets of κ2 and its covering number. The final three sections of the paper present the consistency results mentioned in Theorems 0.1, 0.2 above. First, in Section 5 we introduce manageability,aproperty of <κ–complete κ+–cc forcing notions which implies preservation of non-meagerness of subsets of κκ and which can be iterated. Next, in Section 6, we define one-step forcing and verify that it has all required properties. The final section gives the applications obtained by iterating this forcing notion. Notation 0.3. Our notation is rather standard and compatible with that of classical textbooks (like Jech [6]). In forcing we keep the older (Cohen’s) convention that a stronger condition is the larger one. Some of our conventions are listed below. (1) For a forcing notion P,ΓP stands for the canonical P–name for the generic filter in P. With this one exception, all P–names for objects in the extension via P will be denoted with a dot above (e.g.τ ˙, X˙ ). The weakest element of P will be denoted by ∅P (and we will always assume that there is one, and that there is no other condition equivalent to it). In iterations, if ∗ Q¯ = Pζ , Q˙ ζ : ζ<ζ and p ∈ lim(Q¯ ), then we keep the convention that p(α)=∅˙ for α ∈ ζ∗ \ Dom(p). Q˙ α (2) Ordinal numbers will be denoted by α, β, γ, δ, ε, ζ, ξ and also by i, j (with possible sub- and superscripts). Infinite cardinal numbers will be called θ, ι, µ, ν, τ (with possible sub- and superscripts); κ is our fixed regular uncountable cardinal, λ will denote a fixed cardinal >κ(in Section 3). (3) By χ we will denote a sufficiently large regular cardinal and by H(χ)the family of all sets hereditarily of size less than χ. Moreover, we fix a well ∗ H ordering <χ of (χ).
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use COFINALITY OF THE NONSTATIONARY IDEAL 4815
(4) A bar above a letter denotes that the object considered is a sequence; usually X¯ will be Xi : i<ζ,whereζ denotes the length of X¯.Foraset A and a cardinal µ, the set of all sequences of members of A of length µ (length <µ, respectively), will be denoted by µA (<µA, respectively).
<µ 1. cof (NSκ)
Definition 1.1. (1) For a set A and a cardinal µ, Pµ(A)={a ⊆ A : |a| <µ}. (2) Given two infinite cardinals µ ≤ τ, u(µ, τ) is the least cardinality of a collection A ⊆Pµ(τ) such that Pµ(τ)= P(a). a∈A Definition 1.2. Let S be an infinite set and J an ideal on S (containing all singletons). (1) cof(J ) is the least cardinality of any X⊆Jsuch that for every A ∈J, there is B ∈X with A ⊆ B. (2) add(J ) is the least cardinality of any X⊆Jsuch that X ∈J/ . (3) For an infinite cardinal µ ≤ add(J ), cof<µ(J ) is the least cardinality of a X⊆J ∈J Y∈P X family such that for every A ,thereis µ( ) such that A ⊆ Y. J (4) We let cof(J )=cof With these preliminaries out of the way, we can concentrate on ideals on κ.If there is a family of size κ+ω of pairwise almost disjoint cofinal subsets of κ,then there is a κ–complete ideal J on κ such that cof(J ) < cof(J ) (see Matet and Pawlikowski [9]). Proposition 1.6. Suppose J is a normal ideal on κ and κ is a limit cardinal. Then cof(J )=cof<µ(J ) for some infinite cardinal µ<κ. Proof. Assume that the conclusion fails. Fix X⊆Jsuch that |X | = cof(J )and J = {P( X):X ∈Pκ(X )}. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 4816 PIERRE MATET, ANDRZEJ ROS LANOWSKI, AND SAHARON SHELAH Set Y = {A ∪ β : A ∈X & β ∈ κ}.Notethat|Y| = cof( J ). For each infinite cardinal µ<κwe may select a set Bµ ∈J so that Bµ Y for any Y ∈Pµ(Y). ∈ Now let B be the set of all α<κsuch that α Bµ for some infinite cardinal µ<α . ∈J J ∈P X ⊆ Since B (by normality of ), there must be X κ( ) such that B X. Let τ be any infinite cardinal such that |X| <τ<κ.ThenBτ ⊆ (A ∪ (τ + 1)), A∈X which is a contradiction. Arguing as in Proposition 1.6, we get: Proposition 1.7. Suppose J is a κ–complete ideal on κ and ν is an uncountable limit cardinal <κ. Then there is an infinite cardinal µ<ν such that cof<ν (J )= cof<µ(J ). Moreover, the least such µ is either ω, or a successor cardinal. <µ The remainder of this section is concerned with cof (NSκ). Let us recall the definition of the bounding number bκ: κ Definition 1.8. The bounding number bκ is the least cardinality of any F⊆ κ with the property that for every g ∈ κκ,thereisf ∈F such that |{α<κ: g(α) ≤ f(α)}| = κ. The following is proved in Matet and Pawlikowski [9]. Proposition 1.9. (i) cof(NSκ) ≥ bκ. (ii) If cof(NSκ)=bκ,thencof(NSκ)=cof(NSκ). Proposition 1.10. Let µ be an infinite cardinal ≤ κ.Then <µ <µ either cf(cof (NSκ)) <µ, or cf(cof (NSκ)) ≥ bκ. <µ Proof. Suppose to the contrary that µ ≤ cf(cof (NSκ)) = τ License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use COFINALITY OF THE NONSTATIONARY IDEAL 4817 <ω N Note that dκ = dκ. Landver [7] established that cof( Sκ)=dκ. His result can be generalized as follows. ≤ <µ N <µ Theorem 1.12. Let µ be an infinite cardinal κ.Thencof ( Sκ)=dκ . <µ N <µ ≤ Proof. Set τ =cof ( Sκ). First we will argue that dκ τ. Select a family C of size τ of closed unbounded subsets of κso that for every closed unbounded ∈P C \ {∅} ⊆ ∈P C \ {∅} subset D of κ,thereisX µ( ) with X D.ForU ω( ) define κ fU ∈ κ by fU (α)=min U \ (α +1) .NotethatfV (α) ≤ fU (α) whenever κ V ∈P(U) \ {∅}.Nowgiveng ∈ κ,letD be the set of all limit ordinals δ<κsuch that g(α) <δfor every α<δ.PickX ∈Pµ(C) \ {∅} so that X ⊆ D. Define ∈ κ h κ by h(α)=sup fU (α):U ∈Pω(X) \ {∅} . We are going to show that g (∀α<κ)(g(α) < sup{f(α):f ∈ Fg}). For f ∈ F,letCf be the set of all limit ordinals α<κsuch that f(β) <αfor every β<α. Easily NS = {A ⊆ κ :(∃g ∈ F)(A ∩ C = ∅)} κ g (see, e.g., [9]) and (as Cf ⊆ Cg for every g ∈ F) it follows that τ ≤|F|. f∈Fg It follows from Propositions 1.6 and 1.7 and Theorem 1.12 that to determine the <µ value of cof (NSκ) for every infinite cardinal µ ≤ κ, it suffices to compute dκ and τ dκ for every infinite cardinal τ<κ. cl,<µ 2. dκ <µ F⊆κ It is straightforward to check that dκ is the least cardinality of a family κ such that κ (∀g ∈ κ)(∃F ∈Pµ(F))(| α ∈ κ : g(α) ≥ sup{f(α):f ∈ F } | <κ). In this section we discuss the variant that arises if we replace “has cardinality <κ” by “is nonstationary”. cl F⊆κ Definition 2.1. (1) dκ is the least cardinality of a family κ with the property that for every g ∈ κκ,thereisf ∈F such that {α ∈ κ : g(α) ≥ f(α)}∈NSκ. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 4818 PIERRE MATET, ANDRZEJ ROS LANOWSKI, AND SAHARON SHELAH ≤ cl,<µ (2) For an infinite cardinal µ κ, dκ is the least cardinality of a family κ κ F⊆ κ with the property that for every g ∈ κ,thereisF ∈Pµ(F)such that α ∈ κ : g(α) ≥ sup{f(α):f ∈ F } ∈NSκ. cl,<ω cl cl ≥ Note that dκ = dκ . It is simple to check that cf(dκ ) bκ. Theorem 2.2. For every uncountable cardinal µ ≤ κ, cl,<µ <µ dκ = dκ . Theorem 2.2 easily follows from the next two lemmas. ≤ cl,<µ cl,<τ Lemma 2.3. Let µ be an uncountable limit cardinal κ.Thendκ = dκ for some infinite cardinal τ<µ. Proof. The proof is similar to that of Proposition 1.7. Suppose that the conclusion fails. Fix a family F⊆κκ such that |F| = dcl,<µ and κ κ (∀g ∈ κ)(∃F ∈Pµ(F))( α ∈ κ : g(α) ≥ sup{f(α):f ∈ F } ∈NSκ). κ For each infinite cardinal τ<µwe may select gτ ∈ κ so that for every F ∈Pτ (F) we have α ∈ κ : gτ (α) ≥ sup{f(α):f ∈ F } ∈N/ Sκ. κ Define g ∈ κ so that g(α) ≥ gτ (α) for every infinite cardinal τ<µsuch that τ<α.NowpickF ∈Pµ(F) such that α ∈ κ : g(α) ≥ sup{f(α):f ∈ F } ∈NSκ. Let τ be any infinite cardinal with |F | <τ<µ. Obviously, F ∈P (F)and τ α ∈ κ : gτ (α) ≥ sup{f(α):f ∈ F } ∈NSκ, a contradiction. To establish the following lemma, we adapt the proof of Theorem 5 in Cummings and Shelah [3]. ≤ <µ cl,<µ Lemma 2.4. Let µ be a regular uncountable cardinal κ.Thendκ = dκ . Proof. Select a family F⊆κκ such that (a) every member of F is increasing, |F| cl,<µ (b) = dκ ,and κ (c) for each g ∈ κ,thereisF ∈Pµ(F) such that α ∈ κ : g(α) ≥ sup{f(α):f ∈ F } ∈NSκ. We claim that the family F ∗ def= f ∈ κκ : ∃α, β < κ ∃g ∈F fβ ≡ α & f[β,κ)=g[β,κ) κ is <µ–dominating. So let g ∈ κ. Stipulate that g−1 = g. By induction on n ∈ ω κ choose a closed unbounded subset Cn of κ, gn,hn ∈ κ and Fn ∈Pµ(F)sothat (i) Cn+1 ⊆ Cn, (ii) gn−1(α) < sup{f(α):f ∈ Fn} for all α ∈ Cn, (iii) hn(β)=min(Cn \ (β + 1)), (iv) gn(β)=sup Rng(gn−1(hn(β)+1)) . License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use COFINALITY OF THE NONSTATIONARY IDEAL 4819 ≤ ≤ ≤ ∈ Note that, by (iii) and (iv), g(β) g0(β) g1(β) ... for all β κ.Set F = Fn and ζ =sup{min(Cn):n ∈ ω}. We are going to show that g(γ) < n∈ω sup{f(γ):f ∈ F } whenever ζ<γ<κ. To this end suppose that ζ<γ<κ.By (i), there are m ∈ ω and ξ ∈ κ such that ξ =sup(γ ∩ Cn) whenever m ≤ n<ω. By (iii), hm(ξ) ≥ γ and so (by (iv)) g(γ) ≤ gm−1(γ) ≤ gm(ξ). Since γ>ζwe also have γ ∩ Cm+1 = ∅. Hence ξ ∈ Cm+1 and consequently, by (ii), g(γ) ≤ gm(ξ) < sup{f(ξ): f ∈Fm+1}≤sup{f(γ): f ∈Fm+1}≤sup{f(γ): f ∈ F }. ω ≤ cl ≤ Theorem 2.2 implies that dκ dκ dκ. We mention that it was shown in cl Cummings and Shelah [3] that dκ = dκ if κ> ω. <µ N κ 3. cof ( Sκ,λ) Throughout this section λ denotes a fixed cardinal >κ. Our object of study will N κ P N be the ideal Sκ,λ,a κ(λ) version of Sκ. Definition 3.1. For a regular uncountable cardinal ν and a cardinal τ ≥ ν, Jν,τ is the set of all A ⊆Pν (τ) such that for some a ∈Pν (τ)wehave{b ∈ A : a ⊆ b} = ∅. It is straightforward to check that Jν,τ is a ν–complete ideal on Pν (τ). + Definition 3.2. (1) An ideal J of Pκ(λ)isκ–normal if given A ∈J and f : A −→ κ such that f(a) ∈ a ∩ κ for all a ∈ A,thereisB ∈J+ ∩P(A) such that f is constant on B. P J N κ (2) The smallest κ–normal ideal on κ(λ) containing κ,λ is denoted by Sκ,λ. κ (3) For f ∈ (Pκ(λ)) we let def Cf = {a ∈Pκ(λ):a ∩ κ = ∅ and f(α) ⊆ a}. α∈a∩κ The following lemma is due to Abe. Lemma 3.3 (Abe [1]). Let A ⊆Pκ(λ).Then ∈N κ ∃ ∈ κ P ∩ ∅ A Sκ,λ if and only if ( f ( κ(λ)))(A Cf = ). <µ N κ Our purpose in this section is to compute the value of cof ( Sκ,λ). We will <µ need an analogue of dκ defined in Definition 3.4(1) below. Definition 3.4. Let µ ≤ κ be an infinite cardinal. κ,<µ X P (1) dκ,λ is the least cardinality of a family of functions from κ to κ(λ) with the property that κ ∀g ∈ (Pκ(λ)) ∃X ∈Pµ(X ) ∀α ∈ κ g(α) ⊆ f(α) . f∈X + + (2) cov(λ, κ ,κ ,µ) is the least cardinality of a family X⊆Pκ+ (λ) such that ∀B ∈Pκ+ (λ) ∃X ∈Pµ(X ) B ⊆ X . Theorem 3.5. Let µ be an infinite cardinal ≤ κ.Then <µ N κ κ,<µ { <µ + + } cof ( Sκ,λ)=dκ,λ =max dκ , cov(λ, κ ,κ ,µ) . Theorem 3.5 is an immediate consequence of Lemmas 3.6–3.9 below. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 4820 PIERRE MATET, ANDRZEJ ROS LANOWSKI, AND SAHARON SHELAH Lemma 3.6. Let µ be an infinite cardinal ≤ κ.Then + + ≤ <µ N κ cov(λ, κ ,κ ,µ) cof ( Sκ,λ). κ Proof. By3.3wemaypickafamilyX⊆ (Pκ(λ)) with the property that |X | = <µ N κ −→ P ∈P X cof ( Sκ,λ) and for every function g : κ κ(λ)thereisX µ( ) such that Cf ⊆ Cg.Forf ∈X,letBf = κ ∪ f(α) ∈Pκ+ (λ). f∈X α<κ ∈P −→ P Suppose now that B κ+ (λ). Pick a function g : κ κ(λ) such that B ⊆ g(α). There is X ∈Pµ(X ) such that Cf ⊆ Cg. We are going to show α<κ f∈X that B ⊆ Bf . To this end suppose α<κand let us argue that g(α) ⊆ Bf . f∈X f∈X For n<ωlet an ∈Pκ( Bf ) be defined by f∈X a0 = {α} and an+1 = an ∪ f(β), f∈X β∈a ∩κ n and let a = an.Thenα ∈ a ∈ Cf ⊆ Cg and consequently g(α) ⊆ a ⊆ n<ω f∈X Bf . f∈X ≤ <µ ≤ <µ N κ Lemma 3.7. Let µ be an infinite cardinal κ.Thendκ cof ( Sκ,λ). <µ N ≤ <µ N κ Proof. By Theorem 1.12, it suffices to establish that cof ( Sκ) cof ( Sκ,λ). X⊆κ P |X | <µ N κ Let a family ( κ(λ)) be such that =cof ( Sκ,λ)and ∀ ∈N κ ∃ ∈P X \ {∅} ∩ ∅ B Sκ,λ X µ( ) B Cf = . f∈X For f ∈X,letZf be the set of all limit ordinals α<κsuch that (∀β<α)(f(β) ∩ κ ⊆ α). Plainly, Zf is a closed unbounded subset of κ. Now given a closed unbounded subset T of κ,setBT = {a ∈Pκ(λ):a ∩ κ/∈ T }. A simple argument (see, ∈N κ ∈P X \ {∅} e.g., [10]) shows that BT Sκ,λ. Hence there is XT µ( ) such that BT ∩ Cf = ∅. We will show that Zf ⊆ T .Thusletα ∈ Zf . Setting ∈ ∈ ∈ f XT f XT f XT a = α ∪ f(β), it is easy to see that a ∩ κ = α and a ∈ Cf . It follows f∈XT β<α f∈XT that α = a ∩ κ ∈ T . ≤ <µ N κ ≤ κ,<µ Lemma 3.8. Let µ be an infinite cardinal κ.Thencof ( Sκ,λ) dκ,λ . Proof. The inequality easily follows from the following observation. κ Suppose h : κ −→ P κ(λ)andX ∈Pµ (Pκ(λ)) are such that ∀α<κ h(α) ⊆ f(α) . f∈X Then Cf ⊆ Ch. f∈X Lemma 3.9. Let µ be an infinite cardinal ≤ κ.Then κ,<µ ≤ { <µ + + } dκ,λ max dκ , cov(λ, κ ,κ ,µ) . License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use COFINALITY OF THE NONSTATIONARY IDEAL 4821 Proof. Fix F⊆κκ so that |F| = d<µ and κ κ ∀h ∈ κ ∃F ∈Pµ(F) ∀α<κ h(α) ≤ sup{f(α):f ∈ F } . + + Also, fix X⊆Pκ+ (λ) such that |X | =cov(λ, κ ,κ ,µ)and Pκ+ (λ)= {P( X):X ∈Pµ(X )}. onto For each a ∈X, select a mapping ϕa : κ −→ a.Nowforf ∈F and a ∈X define gf,a : κ −→ P κ(λ)by gf,a(α)={ϕa(δ):δ Next pick F ∈Pµ(F) such that (∀α<κ)(h(α) ≤ sup{f(α):f ∈ F }). Then ∀α<κ g(α) ⊆ gf,a(α) . f∈F a∈X Another formula worth noting is: <µ N κ { <µ N <µ J } cof ( Sκ,λ)=max cof ( Sκ), cof ( κ+,λ) . This identity follows from Theorems 1.12 and 3.5 and the next proposition. Proposition 3.10. Let µ be an infinite cardinal ≤ κ.Thencov(λ, κ+,κ+,µ)= <µ cof (Jκ+,λ). Proof. The result easily follows from the following observation. Suppose that X⊆Pκ+ (λ)andX ∈Pµ(X ) \ {∅}.Then {c ∈Pκ+ (λ):a ⊆ c} = {c ∈Pκ+ (λ): X ⊆ c}, a∈X and therefore for each b ∈Pκ+ (λ), b ⊆ X if and only if {c ∈Pκ+ (λ):a ⊆ c}⊆{c ∈Pκ+ (λ):b ⊆ c}. a∈X <µ N κ N κ We next consider special cases when cof ( Sκ,λ) < cof( Sκ,λ). Lemma 3.11. Let µ be an infinite cardinal ≤ κ.Thencov(λ, κ+,κ+,µ) ≥ λ. Proof. It is shown in Matet, P´ean and Shelah [11] that cof(Jκ+,λ) ≥ λ.Now + + observe that (by Proposition 3.10) cov(λ, κ ,κ ,µ) ≥ cof(Jκ+,λ). Lemma 3.12. Suppose λ is singular and µ is a cardinal such that cf(λ) <µ≤ κ. Then cov(λ, κ+,κ+,µ) ≤ sup{u(κ+,ν):κ<ν<λ}. Proof. Let λξ : ξ License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 4822 PIERRE MATET, ANDRZEJ ROS LANOWSKI, AND SAHARON SHELAH Proposition 3.13. Let µ be an uncountable cardinal ≤ κ.Then <µ N κ { <µ +ω} cof Sκ,κ+ω =max dκ ,κ . Proof. By Lemmas 1.4, 3.11 and 3.12 we have cov(κ+ω,κ+,κ+,µ)=κ+ω, so the result follows from Theorem 3.5. <ω1 ≤ +ω Thus, if dκ κ ,then <ω1 N κ N κ cof ( Sκ,κ+ω ) < cof( Sκ,κ+ω ). Lemma 3.14. Assume the Singular Cardinals Hypothesis. If λ ≥ 2κ,then λ+ if cf(λ) ≤ κ, u(κ+,λ)= λ otherwise. Proof. Plainly, λ ≤ u(κ+,λ) ≤ λκ. It follows immediately that u(κ+,λ)=λ if cf(λ) >κ. For the other case, use the well–known fact (see, e.g., [10]) that cf(u(κ+,λ)) ≥ κ+. Proposition 3.15. Assume the Singular Cardinals Hypothesis. If λ ≥ 2κ and ℵ0 ≤ µ ≤ κ,then λ+ if µ ≤ cf(λ) ≤ κ, cof<µ(NSκ )= κ,λ λ otherwise. + + ≥ ≥ ≥ <µ Proof. By Lemma 3.11, cov(λ, κ ,κ ,µ) λ dκ dκ , so by Theorem 3.5, <µ N κ + + cof ( Sκ,λ)=cov(λ, κ ,κ ,µ). Case: cf(λ) >κ. By Lemmas 3.11 and 3.14 we have λ ≤ cov(λ, κ+,κ+,µ) ≤ u(κ+,λ) ≤ λ, and hence cov(λ, κ+,κ+,µ)=λ. Case: µ ≤ cf(λ) ≤ κ. By Lemma 3.14 we know that <µ cov(λ, κ+,κ+,µ) ≤ u(κ+,λ) ≤ λ+ and λ+ ≤ u(κ+,λ) ≤ cov(λ, κ+,κ+,µ) . Since λ<µ = λ, it follows that cov(λ, κ+,κ+,µ)=λ+. Case: cf(λ) <µ. By Lemmas 3.11, 3.12 and 3.14 we have λ ≤ cov(λ, κ+,κ+,µ) ≤ sup{u(κ+,ν):κ<ν<λ}≤λ, and consequently cov(λ, κ+,κ+,µ)=λ. Thus, if the Singular Cardinals Hypothesis holds, µ ≤ κ and λ ≥ 2κ,then <µ N κ N κ cof ( Sκ,λ) < cof( Sκ,λ) if and only if cf(λ) <µ. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use COFINALITY OF THE NONSTATIONARY IDEAL 4823 4. cov(Mκ,κ) Let us recall some basic facts and definitions related to the combinatorics of the κ κ–meager ideal Mκ,κ on 2. Definition 4.1. (1) The Baire number n(X) of a topological space X (also called the Novak number of X) is the least number of nowhere dense subsets of X needed to cover X. (2) For a topological space X and a cardinal µ,the(<µ)–complete ideal of sub- sets of X generated by nowhere dense subsets of X is denoted by M<µ(X); M<µ+ (X) will be also denoted by Mµ(X). The ideal Mµ(X)isthe ideal of µ–meager subsets of X. (3) The space κκ (respectively κ2) is endowed with the topology obtained by <κ <κ taking as basic open sets ∅ and Os for s ∈ κ (respectively s ∈ 2), κ κ where Os = {f ∈ κ : s ⊆ f} (respectively Os = {f ∈ 2:s ⊆ f}). κ κ κ (4) The ideals of κ–meager subsets of κ, 2 are denoted by Mκ,κ and Mκ,κ, respectively. Remark 4.2. (1) Clearly, for a topological space X, n(X) is the least number of open dense subsets of X with empty intersection. If µ ≤ n(X), then M<µ(X) is a proper ideal (i.e., X/∈ M<µ(X)). (2) Following the tradition of the set theory of the reals, we may consider the covering number cov(M (X)) of the ideal M (X): <µ <µ cov(M<µ(X)) = min{|A| : A⊆M<µ(X)& A = X}. By the definition, n(X)=cov(M<ℵ0 (X)). But also for every µ Lemma 4.3. Suppose that X is a topological space, µ License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 4824 PIERRE MATET, ANDRZEJ ROS LANOWSKI, AND SAHARON SHELAH Now, let π : <κκ −→ <κ2 be such that <κ • π()=, π(s α)=F (π(s),α)fors ∈ κ,and <κ • if sζ : ζ<ξ⊆ κ is –increasing, ξ<κ, s = sζ , ζ<ξ then π(s)= π(sζ). ζ<ξ Then π induces a one-to-one mapping π∗ : κκ −→ κ2:η → π(ηζ). The range ζ<κ of π∗ is Rng(π∗)={ρ ∈ κ2:(∀α<κ)(∃β<κ)(α<β& ρ(β)=0)}. Plainly, Rng(π∗) is the intersection of κ many open dense subsets of κ2. Moreover, π∗ is a homeomorphism from κκ onto Rng(π∗). Therefore, using Lemma 4.3, we get n(κκ)=n(Rng(π∗)) = n(κ2). The rest should be clear (remember Remark 4.2(2,3)). Proposition 4.5. cov(Mκ,κ) ≤ dκ. κ Definition 4.6. Cµ,κ is the forcing notion for adding µ Cohen functions in κ with <κ–support. Thus a condition in Cµ,κ is a function q such that Dom(q) ⊆ µ × κ, Rng(q) ⊆ κ and |q| <κ. The order of Cµ,κ is the inclusion. <κ ≥ Proposition 4.7. Assume 2 = κ<µ.Then Cµ,κ “ cov(Mκ,κ) µ ”. 5. Manageable forcing notions In this section we introduce a property of forcing notions which is crucial for the consistency results presented later: (θ, µ, κ)–manageability. This property has three ingredients: an iterable variant of κ+–cc (see Definition 5.1), κ–completeness and a special property implying preservation of non-meagerness of subsets of κκ (see Proposition 5.9). Since later we will work with <κ–support iterations, we also prove a suitable preservation theorem (see Theorem 5.11). Throughout the section we will assume that our fixed (uncountable) regular cardinal κ satisfies 2<κ = κ (so also κ<κ = κ). Definition 5.1 (See Shelah [15, Definition 1.1] and [16, Definition 7]). Let P be a forcing notion and ε<κa limit ordinal. cc P (1) We define a game ε,κ( ) of two players, Player I and Player II. A play lasts ε steps, and at each stage α<εof the playq ¯α, p¯α,ϕα are chosen so that: 0 + 0 + + • q¯ = ∅P : i<κ , ϕ : κ −→ κ : i → 0. • If α>0, then Player I picksq ¯α,ϕα such that α α +⊆P (i)q ¯ = qi : i<κ satisfies ∀ ∀ + β ≤ α ( β<α)( i<κ )(pi qi ), (ii) ϕα : κ+ −→ κ+ is regressive, i.e., (∀i<κ+)(ϕα(i) < 1+i). • α α +⊆P Player II answers, choosing a sequencep ¯ = pi : i<κ such ∀ + α ≤ α that ( i<κ )(qi pi ). License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use COFINALITY OF THE NONSTATIONARY IDEAL 4825 If at some stage of the game Player I does not have any legal move, then he loses. If the game lasted ε steps, Player I wins a play q¯α, p¯α,ϕα : α<ε if there is a club C of κ+ such that for each distinct members i, j of C satisfying cf(i)=cf(j)=κ and (∀α<ε)(ϕα(i)=ϕα(j)), the set { α }∪{ α } pi : α<ε pj : α<ε has an upper bound in P. P ∗ ε (2) The forcing notion satisfies condition ( )κ if Player I has a winning strat- cc P egy in the game ε,κ( ). ∗ ε + Remark 5.2. Condition ( )κ is a strong version of κ –cc (easily, if ε<κis limit, ε P ∗ ε P + κ = κ,and satisfies ( )κ,then satisfies κ –cc). This condition was used in a number of papers, e.g., to obtain a series of consistency results on partition relations; see Shelah and Stanley [17], [18], Shelah [14], [15], [16]. Its primary use comes from the fact that it is preserved in <κ–support iterations. Proposition 5.3 (See Shelah [15, Iteration Lemma 1.3] and [16, Theorem 35]). <κ Let ε<κbe a limit ordinal, κ = κ . Suppose that Q¯ = Pξ, Q˙ ξ : ξ<γ is a <κ–support iteration such that for each ξ<γ, Q˙ ∗ ε Pξ “ ξ satisfies ( )κ ”. P ∗ ε Then γ satisfies ( )κ. Definition 5.4. A forcing notion P is <θ–complete if every ≤P–increasing chain of length less than θ has an upper bound in P.Itis<θ–lub–complete if every ≤P–increasing chain of length less than θ has a least upper bound in P. Definition 5.5. Let θ and µ be cardinals such that θ<κand µ<κ = µ.LetP be a <θ+–lub–complete forcing notion. ≺ H ∈ ∗ P P ∈ ⊆ (1) A model N ( (χ), ,<χ)is(,κ,µ)–relevant if ,µ N, µ N, |N| = µ and <κN ⊆ N. (2) For a (P,κ,µ)–relevant model N we define a game m(N,θ,P)oftwoplay- ers, He and She, as follows. A play lasts θ moves, and in the ith move conditions pi,qi ∈ P are chosen so that: • qi ∈ N ∩ P, qi ≤ pi, • (∀j License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 4826 PIERRE MATET, ANDRZEJ ROS LANOWSKI, AND SAHARON SHELAH Definition 5.7. Let N be a (P,κ,µ)–relevant model, and let q ∈ N ∩ P, p ∈ P be such that q ≤ p. We say that a pair (q∗,p∗)isan N–cover for (q, p), if • q ≤ q∗ ∈ N ∩ P, p ≤ p∗ ∈ P, q∗ ≤ p∗,and • every condition q ∈ N ∩ P stronger than q∗ is compatible with p∗. Lemma 5.8. Suppose that P is a <θ+–lub–complete forcing notion, N is a (P,κ,µ)– relevant model, and She has a winning strategy in the game m(N,θ,P).Then: (1) For all conditions q ∈ N ∩P and p ∈ P such that q ≤ p, there is an N–cover (q∗,p∗) for (q, p). (2) N ∩ P <◦ P. m Proof. (1) Consider a play qi,pi : i<θ of (N,θ,P) in which He starts with ∗ q0 = q, p0 = p, and then he always plays the <χ–first legal moves, and She uses ∗ ∗ her winning strategy. Let q ∈ N ∩ P, p ∈ P be least upper bounds of qi : i<θ, ∗ ∗ pi : i<θ, respectively. Plainly, as She won the play, the pair (q ,p )isan N–cover for (q, p). (2) Suppose that A⊆N ∩ P is a maximal antichain in N ∩ P, but p ∈ P is ∗ ∗ incompatible with all members of A.Let(q ,p )beanN–cover for (∅P,p). The condition q∗ is compatible with some q ∈A,soletq+ ∈ N ∩P be such that q+ ≥ q∗, q+ ≥ q ∈A. By the choice of (q∗,p∗) we know that the conditions q+ and p∗ are compatible, and hence q and p are compatible, a contradiction. The rest follows from the elementarity of N. Proposition 5.9. Assume θ<κ≤ µ = µ<κ <τ. Suppose that a set Y ⊆ κκ cannot be covered by the union of less than τ nowhere dense subsets of κκ,andP is a weakly (θ, µ, κ)–manageable forcing notion not collapsing cardinals. Then P “ Y is not the union of <τ nowhere dense subsets of κκ ”. Proof. Let P be weakly (θ, µ, κ)–manageable with a witness x ∈H(χ). Suppose toward contradiction that a condition q ∈ P is such that q “ Y is the union of <τ nowhere dense subsets of κκ ”. Passing to a stronger condition if needed, we may assume that for some ι<τ and P–names A˙ ξ (for ξ<ι)wehave: <κ <κ • q “ A˙ ξ ⊆ κ &(∀s ∈ κ)(∃t ∈ A˙ ξ)(s ⊆ t)”,and • q “(∀y ∈ Y )(∃ξ<ι)(∀t ∈ A˙ ξ)(t y)”. P ≺ H ∈ ∗ ˙ For each ζ<ιpick a ( ,κ,µ)–relevant model Nζ ( (χ), ,<χ) such that q, Aξ : ξ<ι,x,ζ ∈ Nζ .Then| Nζ | = ι · µ<τ,sowemaypickay ∈ Y such that ζ<ι κ y ∈Ofor all open dense subsets O of κ from Nζ . By our assumptions, there ζ<ι are ξ<ιand p ≥ q such that p “(∀t ∈ A˙ ξ)(t y)”. ∗ ∗ Let (q ,p )beanNξ–cover for (q, p) (there is one by Lemma 5.8(1)). Put <κ ∗ A = {s ∈ κ :(∃q ≥ q )(q s ∈ A˙ ξ)}. κ Clearly, A ∈ Nξ, A ⊆ Nξ,andO = Os ∈ Nξ is an open dense subset of κ. s∈A ∗ Hence s ⊆ y for some s ∈ A.Letq ∈ Nξ ∩ P be a condition stronger than q and License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use COFINALITY OF THE NONSTATIONARY IDEAL 4827 ∗ such that q s ∈ A˙ ξ. The condition q is compatible with p ,andsowithp.Take a condition q+ stronger than both q and p.Then + + q “ s ∈ A˙ ξ & s ⊆ y ”andq “(∀t ∈ A˙ ξ)(t y)”, a contradiction. <κ Corollary 5.10. Suppose that θ<κ≤ µ = µ and cov(Mκ,κ) >µ.LetP be a (θ, µ, κ)–manageable forcing notion. Then V P “ cov(Mκ,κ) ≤ cov(Mκ,κ) ”. Proof. Remembering Proposition 4.4, apply Proposition 5.9 to τ =cov(Mκ,κ)= κ κ cov(Mκ,κ)andY = κ to get V P “(κκ) is not the union of <τ nowhere dense sets ”. ≤ κ But this clearly implies P “ τ cov(Mκ,κ)=cov(Mκ,κ)”. <κ Theorem 5.11. Assume that θ<κ≤ µ = µ .LetQ¯ = Pξ, Q˙ ξ : ξ<γ be <κ–support iteration such that for each ξ<γ, Q˙ Pξ “ ξ is (θ, µ, κ)–manageable ”. Then Pγ is (θ, µ, κ)–manageable. Proof. Let θ, κ, µ and Q¯ be as in the assumptions of the theorem. First note that the limits of <κ–support iterations of <κ–complete forcing no- ∗ θ + <κ tions satisfying the condition ( )κ are <κ–complete κ –cc (as κ = κ; remember Proposition 5.3). Therefore no such iteration collapses cardinals nor changes cofi- nalities nor adds sequences of ordinals of length <κ. Hence the assumed properties P of θ, κ and µ hold in all intermediate extensions V ξ and our assumption on Q˙ ξ’s is meaningful. P + ∗ θ Plainly, γ is <κ–complete, <θ –lub–complete and satisfies condition ( )κ.We have to show that Pγ is weakly (θ, µ, κ)–manageable. For ξ<γletx ˙ ξ be a Pξ–name for a witness for Q˙ ξ being weakly manageable ≺ H ∈ ∗ P and letx ¯ = x˙ ξ : ξ<γ. Suppose that N ( (χ), ,<χ)isa( γ ,κ,µ)–relevant model such that (¯x, Q¯ ) ∈ N. + Since each Pξ is <κ–complete and satisfies the κ –cc (and κ +1⊆ N), we know that if ξ ∈ N ∩ γ and Gξ ⊆ Pξ is generic over V,theninV[Gξ]wehave ∗ ∩ ≺ H ∈ V[Gξ] <κ ⊆ N[Gξ] V = N and N[Gξ] ( (χ), ,<χ) and N[Gξ] N[Gξ]. Q˙ Gξ ∈ Q˙ Gξ Since clearly ξ N[Gξ], we conclude that N[Gξ]is( ξ ,κ,µ)–relevant, and Gξ ∈ m Q˙ Gξ x˙ ξ N[Gξ]. Therefore, She has a winning strategy in the game (N[Gξ],θ, ξ ). Let st˙ ξ be a Pξ–name for such a strategy. We may assume that the strategy st˙ ξ is such that (∗)ifi<θis even and q = p = ∅˙ , i i Q˙ ξ then st˙ instructs Her to play q = p = ∅˙ . ξ i+1 i+1 Q˙ ξ m We define a strategy st for Her in the game (N,θ,Pγ )asfollows.Atanoddstage i<θof the game, the strategy st first instructs Her to choose (side) conditions − − ∈ P ∈ ∩ P ∈ P qi ,pi γ and only then pick conditions qi N γ and pi γ which are to be played. These conditions will be chosen so that if qj,pj : j License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 4828 PIERRE MATET, ANDRZEJ ROS LANOWSKI, AND SAHARON SHELAH m P − − of (N,θ, γ) in which She uses st, and qj ,pj are the side conditions picked by Her(foroddj ≤ i), then − ∩ − (α)i Dom(qi)=Dom(qi )=Dom(pi−1) N,Dom(pi )=Dom(pi−1), ≤ − ≤ ≤ − ≤ − ≤ ≤ (β)i pi−1 pi pi, qi−1 qi pi , qi−1 qi pi, ∗ ∗ − − ≤ (γ)i letting (qj ,pj )be(qj ,pj )ifj i is odd and (qj,pj)ifj (Remember our assumption (∗)onst˙ ξ and our convention regarding ∅P stated in − − P Q˙ Notation 0.3(1).) Let qi (ξ)andpi (ξ)be ξ–names for members of ξ such that − − ∈ ≤ − qi ξ Pξ “ qi (ξ) N[ΓPξ ]&qi−1(ξ) qi (ξ)”, and − − − ˙ pi ξ Pξ “(qi (ξ),pi (ξ)) is what stξ tells Her to play ∗ ∗ as the answer to qj (ξ),pj (ξ):j pθξ P “ pθ(ξ) is a least upper bound of pi(ξ):i<θ ”. ξ We may also assume that Dom(qθ)= Dom(qi)=Dom(pθ) ∩ N. i<θ License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use COFINALITY OF THE NONSTATIONARY IDEAL 4829 Let q ∈ N ∩ Pγ be a condition stronger than qθ (and thus stronger than all qi for i<θ). We define a condition p ∈ Pγ as follows. First, we declare that Dom(p)= Dom(q)∪Dom(pθ), and p(ξ)=q(ξ)forξ ∈ Dom(q)\Dom(pθ), and p(ξ)=pθ(ξ)for ξ ∈ Dom(pθ) \ Dom(q)=Dom(pθ) \ N. Now suppose that ξ ∈ Dom(q) ∩ Dom(pθ) and we have already defined pξ so that qξ ≤ pξ and pθξ ≤ pξ. Then, by our choices, ∗ p ξ Pξ “ the sequence qj(ξ),pj (ξ):j<θ is a legal play of m Q˙ ˙ (N[ΓPξ ],θ, ξ) in which She uses the strategy stξ,and ∈ q(ξ) N[ΓPξ ] is stronger than all qj(ξ)forj<θ”. ∗ − (Above, pj are as in the definition of st: either pj or pj , depending on the parity of j.) Consequently, pξ “ q(ξ)andpθ(ξ) are compatible ”, so we may pick a Pξ–name p(ξ) for a condition in Q˙ ξ such that pξ “ q(ξ) ≤ p(ξ)andpθ(ξ) ≤ p(ξ)”. This completes the choice of p ∈ Pγ . Plainly, p is an upper bound of q and pθ showing that they are compatible. Remark 5.12. Note that if P is weakly (θ, µ, κ)–manageable, then it satisfies the µ+–cc. Hence we may use a slight modification of the proof of Theorem 5.11 to show (by induction on γ)that <κ if θ<κ= κ , Q¯ = Pξ, Q˙ ξ : ξ<γ is a (<κ)–support iteration of <κ–complete weakly (θ, κ, κ)–manageable forcing notions, then Pγ is weakly (θ, κ, κ)–manageable and κ–complete (and thus also κ+–cc). 6. The one-step forcing In this section we introduce a forcing notion Q for adding a small family of functions in κκ which τ–dominates κκ ∩ V. Iterating this type of forcing notions τ we will get models with dκ small. Our forcing is (of course) manageable for suitable parameters, and thus it preserves non-meagerness of subsets of κ. Throughout this section we assume the following. Context 6.1. (i) τ =cf(τ) < cf(κ)=κ =2<κ, ≤ (ii)µ ¯ = µα : α<τ is an increasing sequence of regular cardinals, κ µ0, κ κ (iii) | µα| =2 and π : µα −→ κ is a bijection. α<τ α<τ We will write πη for π(η). Also, for a set u ⊆ µα we let α<τ T (u) def= {ηα : α<τ & η ∈ u}. Definition 6.2. (1) We define a forcing notion Q = Q(π,µ,¯ κ) as follows. ¯ p p ¯p p A condition in Qis a tuple p =(i, u, f,g)=(i ,u , f ,g ) such that (a) i<κ, u ∈Pκ µα , α<τ ¯ (b) f = fσ : σ ∈ T (u) and fσ : i −→ κ for σ ∈ T (u), License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 4830 PIERRE MATET, ANDRZEJ ROS LANOWSKI, AND SAHARON SHELAH (c) g : u −→ i +1,andifη ∈ u, g(η) ≤ j πη(j) < sup{fηα(j):α<τ}. The order of Q is such that for p, q ∈ Q we have p ≤ q if and only if p ≤ q p ⊆ q p ⊆ q p ⊆ q ∈ p i i , u u , g g and fσ fσ for σ T (u ). p (2) For a set U ⊆ µα we let QU = {p ∈ Q : u ⊆ U}, and for a condition α<τ q ∈ Q we put qU = iq,uq ∩ U, f¯qT (uq ∩ U),gq(uq ∩ U) . κ Proposition 6.3. (1) Q is a <κ–lub–complete forcing notion of size 2 . (2) Let U ⊆ µα be of size ≤ κ.Then|QU|≤κ. α<τ Proof. (1) Plainly, (Q, ≤) is a partial order of size 2κ. To prove the completeness ∗ ∗ suppose that pξ : ξ<ξ is an ≤–increasing sequence of members of Q and ξ <κ. Put iq =supipξ ,uq = upξ ,gq = gpξ ∗ ξ<ξ ∗ ∗ ξ<ξ ξ<ξ p ∗ q { ξ ∈ pξ } ∈ q q q ¯q q ∈ and fσ = fσ : ξ<ξ & σ T (u ) for σ T (u ). Clearly, q =(i ,u , f ,g ) ∗ Q is the least upper bound of pξ : ξ<ξ . (2) Should be clear. Q ∗ ε Proposition 6.4. The forcing notion satisfies the condition ( )κ (see Definition 5.1(2)) for any limit ordinal ε<κ. Proof. Let ε<κbe a limit ordinal. To give the winning strategy for Player I in cc Q the game ε,κ( ) we need two technical observations. Claim 6.4.1. If p, q ∈ Q are such that ip = iq and gp(up ∩ uq)=gq(up ∩ uq) and p q ∈ p ∩ q fσ = fσ for σ T (u ) T (u ), then the conditions p, q have a least upper bound. Proof of the Claim. Let ir = ip = iq, ur = up ∪ uq, gr = gp ∪ gq and p ∈ p r fσ if σ T (u ), fσ = q ∈ q fσ if σ T (u ). Then r =(ir,ur, f¯r,gr) ∈ Q is the least upper bound of p, q. + Claim 6.4.2. Suppose q¯ = qj : j<κ ⊆Q. Then there is a regressive function + + ϕq¯ : κ −→ κ such that + if j + Proof of the Claim. Take a sequence ηξ : ξ<κ ⊆ µα such that for each j< α<τ + q + κ of cofinality κ and an α License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use COFINALITY OF THE NONSTATIONARY IDEAL 4831 + and F (U)= F (Uj). Note that |F (Uj)|≤κ and |F (U)|≤κ .Choosea j<κ+ + function ψ1 : κ −→ F (U) such that + ∀j<κ cf(j)=κ ⇒ Rng(ψ1j)=F (Uj) . Finally, let c : κ+ × κ+ −→ κ+ be a bijection such that ∀j<κ+ cf(j)=κ ⇒ Rng(c(j × j)) = j . + + + Now let ϕq¯ : κ −→ κ be a regressive function such that for j<κ of cofinality κ we have + + ¯qj ϕq¯(j)=c(min{α<κ : ψ0(α)=qjUj}, min{α<κ : ψ1(α)=f T (Uj)}). Easily, ϕq¯ is as required. Now we may complete the proof of Proposition 6.4. Consider the following cc Q strategy st for Player I in the game ε,κ( ). Suppose that the players arrived at stage α>0 of the play and they have already constructed a sequence q¯β, p¯β,ϕβ : + β β<α. Then, for each j<κ , the sequence pj : β<α is increasing, so Player α α I can take its least upper bound qj . This determinesq ¯ played by Player I; the α function ϕ played at this stage is the ϕq¯α given by Claim 6.4.2. One easily verifies that the strategy st is a winning one (remember Claim 6.4.1). Theorem 6.5. Suppose θ and ι are cardinals such that θ =cf(θ) <κ≤ ι = ι<κ. Then the forcing notion Q is (θ, ι, κ)–manageable. Proof. For each σ ∈ µβ fix a sequence ησ ∈ µα such that σ ⊆ ησ.Let α<τ β<α α<τ η¯ = ησ : σ ∈ µβ. α<τ β<α Suppose that N is a (Q,κ,ι)–relevant model such that (¯η, µ,¯ π) ∈ N. ∈ Q + − Foraconditionp we define conditions clN (p)=q and clN (p)=r by • ir = iq = ip, r p p q p p • u =(u ∩ N) ∪{ησ : σ ∈ T (u ) ∩ N}, u = u ∪{ησ : σ ∈ T (u ) ∩ N}, • r p ∈ p ∩ q p ∈ p q r p fσ = fσ for σ T (u ) N, fσ = fσ for σ T (u ), and fσ(j)=fσ(j)=i for σ ∈ T (uq) \ T (up), j q q r q r q p r g (η)ifη ∈ u , i = i ,u= u ∪ u ,g(η)= and: gp(η)ifη ∈ up \ uq , License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 4832 PIERRE MATET, ANDRZEJ ROS LANOWSKI, AND SAHARON SHELAH ∈ q r q ∈ p \ q if σ T (u ), then fσ = fσ ,andifσ T (u ) T (u ), then p ⊆ r r { ⊆ ∈ p} p ≤ q fσ fσ and fσ(j)=sup πη(j):σ η u +1 fori j ∗ ∗ Claim 6.5.2. Suppose that a sequence pζ : ζ<ζ⊆Q is increasing, ζ <κis + ∗ ∗ a limit ordinal, and clN (pζ )=pζ+1 for all even ζ<ζ.Letp be the least upper ∗ − ∗ − ∗ bound of pζ : ζ<ζ .ThenclN (p ) is the least upper bound of clN (pζ ):ζ<ζ . − ≤ − ∗ Proof of the Claim. It follows from Claim 6.5.1(3) that clN (pζ) clN (p )(forζ< ∗ − ∗ ζ ). To show that clN (p ) is actually the least upper bound it is enough to note that − − cl (p∗) p∗ p ∗ cl (p ) ∗ i N = i =sup{i ζ : ζ<ζ } =sup{i N ζ : ζ<ζ }, and − p∗ p ∗ cl (p ) ∗ u ∩ N = {u ζ+1 ∩ N : ζ<ζ & ζ even} = {u N ζ : ζ<ζ & ζ even}, − p∗ p ∗ cl (p ) ∗ {ησ : σ ∈ T (u ) ∩ N} = {ησ : σ ∈ T (u ζ ) ∩ N & ζ<ζ }⊆ {u N ζ : ζ<ζ }, − − cl (p∗) p∗ cl (p ) ∗ so u N = u ∩ N = {u N ζ : ζ<ζ }. Now we may describe a strategy st for Her in the game m(N,θ,Q). Suppose that i<θis even and (qi,pi) is His move at this stage of the play (so qi ∈ N ∩ P, ≤ ∈ P − + qi pi ). Then st instructs Her to play qi+1 =clN (pi)andpi+1 =clN (pi). It follows from Claim 6.5.1(1) that (qi+1,pi+1) is a legal move. It follows from Claims 6.5.2 and 6.5.1(2) that the strategy st is a winning one. Thus we have shown that Q is weakly (θ, ι, κ)–manageable (remember Proposi- tion 6.3(1)). The rest follows from Propositions 6.3 and 6.4. ˙ Definition 6.6. We define Q–names fσ (for σ ∈ µβ)and˙g by α<τ β<α ˙ { p ∈ ∈ p } Q “ fσ = fσ : p ΓQ & σ T (u ) ”, p Q “˙g = {g : p ∈ ΓQ} ”. Proposition 6.7. (1) Q “ g˙ : µα −→ κ ”. α<τ ˙ (2) For each σ ∈ µβ we have Q “ fσ : κ −→ κ ”. α<τ β<α (3) For each η ∈ µα, α<τ ˙ Q “ ∀j<κ g˙(η) ≤ j ⇒ πη(j) < sup{fηα(j):α<τ} ”. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use COFINALITY OF THE NONSTATIONARY IDEAL 4833 Proof. For η ∈ µα and i<κlet α<τ p i p Iη = {p ∈ Q : η ∈ u } and I = {p ∈ Q : i 7. The models Theorem 7.1. Assume that (a) κ =2<κ, (b) µ is a cardinal such that cf(µ) <κ<µ<µcf(µ) =2κ, (c) there is an increasing sequence µ¯ = µα : α The forcing notion P is built as the limit of a <κ–support iteration Q¯ = Pξ, Q˙ ξ : + + ξ<κ . The names Q˙ ξ are defined by induction on ξ<κ so that κ (α) Pξ has a dense subset of size 2 , and for all cardinals θ, ι satisfying cf(θ)=θ<κ≤ ι = ι<κ, (β) the forcing notion Pξ is (θ, ι, κ)–manageable and Q˙ (γ) Pξ “ ξ is (θ, ι, κ)–manageable ”. So suppose that Pξ is already defined (and clauses (α), (β) hold). Then Pξ is <κ– complete κ+–cc, and hence the properties of κ, µ andµ ¯ stated in (a)–(c) hold in License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 4834 PIERRE MATET, ANDRZEJ ROS LANOWSKI, AND SAHARON SHELAH P V ξ .TakeaP –nameπ ˙ such that ξ ξ −→ κ Pξ “˙πξ : µα κ is a bijection ”, α and let Q˙ ξ be a Pξ–name for the forcing notion Q(˙πξ, µ,¯ κ). Then clause (γ)holds (remember Theorem 6.5). It follows from Proposition 6.3(1) that the demand (α) is preserved at successor stages and it is preserved at limits ≤ κ+ by the support we use. By Theorem 5.11, P ≤ + P P the clause (β) holds for each ξ (ξ κ ). So our = κ+ satisfies (i)+(ii). + ˙ξ ∈ ξ P For ξ<κ let fσ (for σ µβ)and˙g be ξ+1–names for functions α Therefore we have a P0–name P˙ for a forcing notion satisfying 7.1(i)–(iv). (Note +(ω+1) P∗ that P0 “cov(Mκ,κ)=κ ”, so the assumption of 7.1(iv) holds.) Let = P0 ∗ P˙ . Note that Theorem 0.1 follows from Corollary 7.2, Theorem 1.12, and the fact that cov(Mκ,κ) ≤ cof(NSκ). Theorem 7.3. Assume that (a)–(c) of Theorem 7.1 hold and (d) ν is a regular cardinal such that µ<ν<2κ. Then there is a forcing notion Pν satisfying (i)+(ii) of Theorem 7.1 and + τ (iii) P “ d = ν for every cardinal τ satisfying cf(µ) ≤ τ<κ”, ν ν κ − V ≤ (iv) Pν “ cov(Mκ,κ) cov(Mκ,κ) ”. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use COFINALITY OF THE NONSTATIONARY IDEAL 4835 Proof. The forcing notion Pν is the limit of <κ–support iteration Q¯ = Pξ, Q˙ ξ : ξ< ν,whereQ˙ ξ are defined as in the proof of Theorem 7.1 (so the only difference is the P cf(µ) ≤ length of the iteration). As there, ν satisfies (i)+(ii) and Pν “ dκ ν ”. Since − cov(Mκ,κ) >κand Pν is (ℵ0,κ,κ)–manageable, we get (iv) (by Corollary 5.10). + F˙ P To show that (iii)ν holds, suppose that is a ν –name for a family of functions κ in κ of size <ν.ThenF˙ is essentially a Pξ–name for some ξ<ν.SincePξ+κ P P ξ adds a subset of κ which is Cohen over V ξ ,(κκ)V is not a dominating family in P (κκ)V ν , and hence for any τ<κ, F˙ Pν “ is not τ–dominating ” (remember that Pν is <κ–complete). Now, Theorem 0.2 follows from Theorem 1.12 and the following corollary. Corollary 7.4. It is consistent, relative to the existence of a cardinal ν such that ++ ¯ ℵ ℵ o(ν)=ν , that dℵ1 = ω+1 and cov(Mℵ1,ℵ1 )= ω+2. Proof. Gitik and Woodin (see Gitik [4] also Gitik and Merimovich [5]) constructed a ℵn ℵ0 ℵω ++ model of “ 2 < ℵω for every n<ω,2 = ℵ1 and 2 = ℵω+2 ”fromo(ν)=ν . Add ℵω+2 Cohen subsets of ℵ1 (with countable support) to that model and then apply Theorem 7.3 (with κ = ℵ1, µ = ℵω and ν = ℵω+1). Theorem 7.5. Assume that (a) κ =cf(κ)=2<κ, n<ω, (b) µ0,µ1,...,µn are cardinals such that µ0 >µ1 >...>µn >κ and cf(µ0) < cf(µ1) <... cf(µ ) κ (c) (µ) =2 for =0,...,n, (d) for =0,...,n, there is an increasing sequence µ¯ = µα : α ∀ ≤ cf(µ) { } ( α κ (e) cov(Mκ,κ)=2 . Then there is a forcing notion P such that (i) P has a dense subset of size 2κ, (ii) P is (θ, ι, κ)–manageable for all cardinals θ, ι satisfying cf(θ)=θ<κ≤ ι = ι<κ, cf(µ) κ (iii) P “ dκ = µ for =0,...,n and cov(Mκ,κ)=2 ”. + + Proof. Let A0,...,An be a partition of κ into sets of size κ . The forcing notion + P is the limit of a <κ–support iteration Q¯ = Pξ, Q˙ ξ : ξ<κ defined as in the proof of Theorem 7.1, but • ∈ P κ if ξ A,then˙πξ is a ξ–name for a bijection from µα onto κ,and α<τ Q˙ ξ is Q(˙πξ, µ¯ ,κ). We argue that P has the required properties similarly as in Theorem 7.1. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 4836 PIERRE MATET, ANDRZEJ ROS LANOWSKI, AND SAHARON SHELAH Remark 7.6. Of course the assumption (e) in Theorem 7.5 is not very important: we may start with adding 2κ many Cohen subsets of κ. Corollary 7.7. It is consistent, relative to the existence of a strong cardinal, that ℵ ℵ d 1 = ℵ , d 0 = ℵ and cov(Mℵ ℵ )=ℵ . ℵ2 ω1 ℵ2 ω1+ω 2, 2 ω1+ω+1 Proof. As was pointed out to us by Moti Gitik, by work of Magidor [8], Merimovich [12], and Segal [13], it is consistent relative to a strong cardinal that ℵ ℵ 1 ℵ ℵ 0 ℵ (a) 2 = 2, ω1 = ω1 ,and (b) for every α<ω1, ℵ ℵ ℵ 1 ℵ α+ω+1 ℵ α+ω+1 = α+ω+1 and 2 = α+ω+2, and ℵ ℵ ℵ 1 ℵ 0 ℵ (c) ω1 = ω1+ω = ω1+ω+1. ℵ ℵ ℵ After adding ω1+ω+1 Cohen subsets of 2 (with < 2 support) to a model of (a)– (c) we will get a model in which the assumptions of Theorem 7.5 are satisfied for ℵ ℵ ℵ κ = 2, n =2,µ0 = ω1+ω, µ1 = ω1 . References [1] Yoshihiro Abe. A hierarchy of filters smaller than CFκλ. Archive for Mathematical Logic, 36:385–397, 1997. MR1477763 (98m:03098) [2] Bohuslav Balcar and Petr Simon. Disjoint refinement. In Handbook of Boolean algebras,vol- ume 2, pages 333–388. North-Holland, Amsterdam, 1989. MR0991597 [3] James Cummings and Saharon Shelah. Cardinal invariants above the continuum. Annals of Pure and Applied Logic, 75:251–268, 1995. math.LO/9509228. MR1355135 (96k:03117) [4] Moti Gitik. The negation of the singular cardinal hypothesis from o(κ)=κ++. Annals of Pure and Applied Logic, 43:209–234, 1989. MR1007865 (90h:03037) ℵ ℵ [5] Moti Gitik and Carmi Merimovich. Possible values for 2 n and 2 ω . Annals of Pure and Applied Logic, 90:193–241, 1997. MR1489309 (2000a:03084) [6] Thomas Jech. Set theory. Academic Press, New York, 1978. MR0506523 (80a:03062) [7] Avner Landver. Singular Baire numbers and related topics. Ph.D. thesis, University of Wis- consin, Madison, 1990. [8] Menachem Magidor. On the singular cardinals problem I. Israel J. Math., 28:1–31, 1977. MR0491183 (58:10449a) [9] Pierre Matet and Janusz Pawlikowski. Q–pointness, P –pointness and feebleness of ideals. Journal of Symbolic Logic, 68:235–261, 2003. MR1959318 (2004c:03057) [10] Pierre Matet, C´edric P´ean, and Saharon Shelah. Cofinality of normal ideals on Pκ(λ), I. Archive for Mathematical Logic. math.LO/0404318. [11] Pierre Matet, C´edric P´ean, and Saharon Shelah. Cofinality of normal ideals on Pκ(λ), II. Israel Journal of Mathematics,toappear. [12] Carmi Merimovich. Extender-based Radin forcing. Transactions of the American Mathemat- ical Society, 355:1729–1772, 2003. MR1953523 (2004d:03108) [13] Miri Segal. Master Thesis. The Hebrew University of Jerusalem, 1996. Menachem Magidor, adviser. [14] Saharon Shelah. A weak generalization of MA to higher cardinals. Israel Journal of Mathe- matics, 30:297–306, 1978. MR0505492 (58:21606) [15] Saharon Shelah. Strong Partition Relations Below the Power Set: Consistency, Was Sierpi´nski Right, II? In Proceedings of the Conference on Set Theory and its Applications in honor of A.Hajnal and V.T.Sos, Budapest, 1/91,volume60ofColloquia Mathematica Societatis Janos Bolyai. Sets, Graphs, and Numbers, pages 637–638. 1991. math.LO/9201244. MR1218224 (95b:03052) [16] Saharon Shelah. Was Sierpi´nski right? IV. Journal of Symbolic Logic, 65:1031–1054, 2000. math.LO/9712282. MR1791363 (2001k:03092) License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use COFINALITY OF THE NONSTATIONARY IDEAL 4837 [17] Saharon Shelah and Lee Stanley. Generalized Martin’s axiom and Souslin’s hypothesis for higher cardinals. Israel Journal of Mathematics, 43:225–236, 1982. Corrections in [Sh:154a]. [18] Saharon Shelah and Lee Stanley. Corrigendum to: “Generalized Martin’s axiom and Souslin’s hypothesis for higher cardinals” [Israel Journal of Mathematics 43 (1982), no. 3, 225–236; MR 84h:03120]. Israel Journal of Mathematics, 53:304–314, 1986. Departement de Mathematiques,´ Universite´ de Caen – CNRS, BP 5186, 14032 Caen Cedex, France E-mail address: [email protected] Department of Mathematics, University of Nebraska at Omaha, Omaha, Nebraska 68182-0243 E-mail address: [email protected] URL: http://www.unomaha.edu/logic Institute of Mathematics, The Hebrew University of Jerusalem, 91904 Jerusalem, Israel – and – Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08854 E-mail address: [email protected] URL: http://shelah.logic.at License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use