The Failure of Diamond on a Reflecting Stationary Set

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 364, Number 4, April 2012, Pages 1771–1795 S 0002-9947(2011)05355-9 Article electronically published on November 17, 2011

THE FAILURE OF DIAMOND ON A REFLECTING STATIONARY SET

MOTI GITIK AND ASSAF RINOT

Abstract. 1. It is shown that the failure of ♦S ,forasetS ⊆ℵω+1 that

reflects stationarily often, is consistent with GCH and APℵω ,relativetothe ∗ existence of a supercompact cardinal. By a theorem of Shelah, GCH and λ + entails ♦S for any S ⊆ λ that reflects stationarily often. 2. We establish the consistency of existence of a stationary subset of ω [ℵω+1] that cannot be thinned out to a stationary set on which the sup- function is injective. This answers a question of König, Larson and Yoshinobu in the negative. 3. We prove that the failure of a diamond-like principle introduced by Dˇzamonja and Shelah is equivalent to the failure of Shelah’s strong hypothesis.

0. Introduction 0.1. Background. Recall Jensen’s diamond principle [10]: for an inﬁnite cardinal + λ and a stationary set S ⊆ λ , ♦S asserts the existence of a sequence Aδ | δ ∈ S + such that the set {δ ∈ S | A ∩ δ = Aδ} is stationary for all A ⊆ λ . λ + It is easy to see that ♦λ+ implies 2 = λ , and hence it is natural to ask whether the converse holds. Jensen proved that for λ = ℵ0, the inverse implication fails (see [2]). However, for λ>ℵ0, a recent theorem of Shelah [19] indeed establishes the λ + inverse implication, and moreover, it is proved that 2 = λ entails ♦S for every λ+ { + | } stationary S which is a subset of E=cf( λ) := α<λ cf(α) =cf(λ) . The result of [19] is optimal: by a theorem of Shelah from [20], GCH is consistent

with the failure of ♦ λ+ for a regular uncountable cardinal λ. By another theorem Eλ of Shelah, from [17, §2], GCH is consistent with the failure of ♦S for a singular ⊆ λ+ cardinal λ and a stationary set S Ecf(λ). However, the latter happens to be a λ+ proper subset of Ecf(λ); more specifically, it is a non-reflecting stationary subset. This leads to the following question: Question. Suppose λ is a singular cardinal and 2λ = λ+. ♦ ⊆ λ+ 1 Must S hold for every S Ecf(λ) that reflects stationarily often? In [17, §3], Shelah answered the above question in the affirmative, provided ∗ that λ holds and that λ is a strong limit. Later, Zeman [21] applied ideas from [19] and eliminated the “strong limit” hypothesis. Then, in [15], the second author

Received by the editors May 20, 2009 and, in revised form, March 29, 2010. 2010 Mathematics Subject Classification. Primary 03E35; Secondary 03E04, 03E05. Key words and phrases. Diamond, sap, square, approachability, very good scale, reflection. This research was supported by the Israel Science Foundation (grant No. 234/08). The authors would like to thank the referee for his comments and corrections. 1We say that S reflects stationarily often iff there are stationarily many α<λ+ with cf(α) >ω such that S ∩ α is stationary.

c 2011 American Mathematical Society Reverts to public domain 28 years from publication 1771

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1772 MOTI GITIK AND ASSAF RINOT

introduced the Stationary Approachability Property at λ, abbreviated SAPλ,proved ∗ that SAPλ is strictly weaker than λ, and answered the above question positively inthepresenceofSAPλ. It was unknown whether the hypothesis SAPλ can be eliminated or even whether it is possible to replace it with the usual Approachability Property,APλ. In the present paper we answer the discussed question in the negative, and moreover, do so in the presence of APλ.LetReﬂλ denote the assertion that every λ+ stationary subset of Ecf(λ) reﬂects stationarily often; then the main result of this paper reads as follows. Theorem A. It is relatively consistent with the existence of a supercompact cardinal that all of the following hold simultaneously: (1) GCH;

(2) APℵω ;

(3) Reflℵω ; ℵω+1 (4) ♦S fails for some stationary S ⊆ Eω . Combining the preceding theorem with the results from [15], we now obtain a complete picture of the effect of weak square principles on diamond.2 Corollary (first three items are from [15]). For a singular cardinal λ: ∗ ⇒♦∗ (1) GCH +Reflλ + λ λ+ ; ⇒♦∗ (2) GCH +Reflλ +SAPλ λ+ ; + (3) GCH +Reflλ +SAPλ ⇒♦S for every stationary S ⊆ λ ; + (4) GCH +Reflλ +APλ ⇒♦S for every stationary S ⊆ λ . Once that the effect of weak square principles to diamond is well understood, it is natural to study which of the other combinatorial principles from [3, §4] is strong enough to impose an affirmative answer to Question 1. It turns out that even the strongest among these principles does not suffice. We prove: Theorem B. It is relatively consistent with the existence of a supercompact cardinal that there exists a singular cardinal λ for which all of the following hold simultaneously: (1) λ is a strong limit and 2λ = λ+; (2) there exists a very good scale for λ; ♦ ⊆ λ+ (3) S fails for some S Ecf(λ) that reflects stationarily often. To conclude the introduction, let us say a few words about the structure of the proof of Theorem A. We start with a supercompact κ and a singular cardinal λ above +ω ⊆ λ+ it, namely λ := κ . We add a generic stationary subset S Ecf(λ) and then kill ♦S by iteration, while preserving the stationarity of S and the supercompactness of κ.Sinceκ remains supercompact, Reflλ holds, so this already gives an example ♦ ⊆ λ+ of a model of GCH on which S fails for some stationary S Ecf(λ) that reflects stationarily often. Nevertheless, by cf(λ) <κ<λand a theorem of Shelah from [16], APλ fails in our model. For this, at the final stage of the proof, we move everything down to ℵω and use a method of Foreman and Magidor from [7] to

ensure APℵω .

2For the deﬁnition of ♦∗ , as well as Kunen’s theorem that ♦∗ ⇒♦ for every stationary λ+ λ+ S S ⊆ λ+,see[12].

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE FAILURE OF DIAMOND ON A REFLECTING STATIONARY SET 1773

The main problem that we address here is an iteration over the successor of a singular. More speciﬁcally, the main body of our proof is the argument that S remains stationary after the iteration for killing all diamond sequences over it.

0.2. Organization of this paper. In Section 1, we present a λ+-directed-closed λ++-c.c. notion of forcing for introducing a stationary subset of λ+ on which diamond fails. Then, in the presence of a supercompact cardinal, we appeal to this notion of forcing and construct three models in which diamond fails on a set that reﬂects stationarily often. In particular, Theorem A and Theorem B are proved in this section. In Section 2, we revisit a theorem by Dˇzamonja and Shelah from [4] in which, starting with a supercompact cardinal, they construct a model satisfying the failure of one of the consequences of diamond. Here, we establish that this particular consequence of diamond is quite weak. We do so by reducing the consistency strength of its failure to the level of existence of a measurable cardinal κ of Mitchell order κ++. In particular, the strength of its failure is lower than that of the failure of a weak square. In Section 3, we answer a question by K¨onig, Larson and Yoshinobu from [11], concerning stationary subsets of [λ+]ω. We do so by linking between the diamond principle and the behavior of the sup-function on generalized stationary sets. In Section 4, we state a couple of open problems.

0.3. Notation. Generally speaking, we follow the notation and presentation of [12] and [3]. Let us quickly review our less standard conventions. For cardinals κ<λ, λ { | } κ { ⊆ || | } λ <κ denote Eκ := α<λ cf(α)=κ and [λ] := X λ X = κ . E>κ and [λ] are defined analogously. We let Ord denote the class of ordinals. For sets of ordinals a, b,wewritea b or b a iff a is an initial segment of b,thatis,iffa ⊆ b and b ∩ sup(a)=a ∩ sup(a). Given a set of ordinals, a, we let cl(a):={sup(a ∩ α) | α ∈ Ord} denote its topological closure. For Z ⊆ Ord and distinct functions g, h ∈ Z 2, we let Δ(g, h):=min{α ∈ Z | g(α) = h(α)}.Forγ ∈ Ord and a function s : γ → 2, we denote s := {δ ∈ dom(s) | s(δ)=1}. For ordinals β ≤ γ, we designate the (possibly empty) open interval ((β,γ)) = {α ∈ γ | β ∈ α}. Our forcing conventions are as follows. We denote by p ≥ q the fact that p is stronger than q. For a ground model set, x, we denote its canonical name byx ˇ. For arbitrary sets of the generic extension, we designate names such asa ˙ and ∼a.If p1, q˙1, p2, q˙2 are conditions of a two-step iteration P∗Q and if p2 ≥ p1 P q˙1 =˙q2, then we slightly abuse notation by writing p2, q˙1 = p2, q˙2.

1. Negation of diamond 1.1. Forcing the failure of diamond. In this subsection, we present for a limit uncountable cardinal λ,aλ+-directed-closed, λ++-c.c., notion of forcing for introducing a stationary subset S ⊆ λ+ on which diamond fails. For simplicity, we shall be focusing on the case cf(λ)=ω. The general case is discussed in subsection 1.3. Definition 1.1. For a cardinal λ, we define the forcing notion S(λ+). S + → + ⊆ γ A condition s is in (λ )iffs : γ 2 is a function such that γ<λ and s Eω. A condition s is stronger than s, denoted s ≥ s,iffs ⊇ s. The following is obvious.

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1774 MOTI GITIK AND ASSAF RINOT

Lemma 1.2. For every infinite cardinal λ: (1) S(λ+) has the (2λ)+-c.c.; (2) S(λ+) is λ+-directed-closed; (3) every increasing sequence of conditions in S(λ+) of length <λ+ has a least upper bound. For future needs, it is useful to introduce the following set-name: S˙(λ+):={δ,ˇ s|s ∈ S(λ+)&δ ∈ s}. Clearly, if G is S(λ+)-generic, then S˙(λ+)isanamefortheset {s | s ∈ G}. We now consider a natural forcing notion for Killing a given Diamond sequence. Definition 1.3. For a set S ⊆ Ord and a sequence of sets A with dom(A )=S,we define the forcing notion KD(S, A ). A condition p is in KD(S, A )iffp =(x, c), where c is a closed set of ordinals, x ⊆ max(c) < sup(S), and x ∩ δ = A (δ) for all δ ∈ c ∩ S. A condition (x, c)isstrongerthan(x,c), denoted (x, c) ≥ (x,c), iff x x and c c.

Notice that indeed if A = Aδ | δ ∈ S is a ♦S sequence in V ,thenA will cease to be so in V KD(S,A). For a strong limit singular cardinal λ with 2λ = λ+ and a subset S ⊆ λ+,let KAD(S) denote the forcing notion for Killing All Diamond sequences over S.That is, KAD(S)isa(≤ λ)-support iteration ++ ++ (Pα | α ≤ λ , R˙α | α<λ ) P ++ P R˙ such that 0 is a trivial forcing, and− for→ all α<λ , α forces that α is a name KD ˇ P for the forcing (S,A∼α), whereas A∼α is a α-name for a sequence chosen by a book-keeping function in such a way that all potential diamond sequences are han- dled at some stage. The existence of such a function follows from the cardinal arithmetic hypothesis and the λ++-c.c. of KD(S, A ). Definition 1.4. Let Q(λ+):=S(λ+) ∗ KAD(S˙(λ+)). We also define Q(λ+). A condition s, k is in Q(λ+) iff all of the following hold: (1) s, k∈Q(λ+); (2) s decides the support of k to be, say, supp(k); ∈ S + ∗P k k 3 (3) for all α supp(k), k(α)isan (λ ) α-canonical name for a pair (xα,cα); ∈ ∈ ∩ k S + ∗ P (4) for all α supp(k)andδ s cα, s, k (λ ) α decides Aα(δ); k k ∈ (5) max(cα)=sup(xα)=sup(dom(s)) for all α supp(k). Lemma 1.5. Every increasing sequence of conditions in Q(λ+) of length <λ+ has a least upper bound (within Q(λ+)). + Proof. Suppose sβ,kβ|β<θ is an increasing sequence of conditions in Q (λ ), + with θ<λ . Puts ¯ := β<θ sβ and γ := sup(dom(¯s)). Now, define s, k by letting: • s :=s ¯ ∪{(γ,0)} if γ ∈ dom(¯s); otherwise, let s :=s ¯; • s decides the support of k to be β<θ supp(kβ);

3 k k In particular, xα and cα are elements of the ground model.

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE FAILURE OF DIAMOND ON A REFLECTING STATIONARY SET 1775

• for all α ∈ supp(k): k { kβ | ∈ } xα := xα β<θ& α supp(kβ) , k { kβ | ∈ } cα := cl( cα β<θ& α supp(kβ) ). To establish that s, k∈Q(λ+), let us show that s, k∈Q(λ+), namely, that s k ∈ KAD(S˙(λ+)). So, suppose that t ∈ S(λ+), t ≥ s, α ∈ supp(k), and ∈ ∩ k k δ t cα.Sincemax(cα)=γ,wemayfocusont γ + 1. In other words, we may ∈ ∩ k assume that t = s.Byδ s cα and the deﬁnition of s, let us pick some β<θ such that δ ∈ sβ and α ∈ supp(kβ). kβ kβ Next, by property (5), we have max(cα )=sup(dom(sβ)) ≥ δ, and hence δ ∈ cα . + kβ By property (4), sβ,kβ S(λ ) ∗ Pα decides A α(δ), and hence xα ∩ δ = A α(δ). kβ kβ kβ Finally, by property (5), we have sup(xα )=max(cα ) ≥ δ, and hence xα ∩ δ = k ∩ k ∩ xα δ,soxα δ = Aα(δ). So, s, k is a legitimate element of Q(λ+), and it is clearly an upper bound of the above sequence. Finally, notice that if s, k is not the least upper bound, then s¯∪{(γ,1)},k is a legitimate condition as well; hence, in this case, the least upper bound is just s,¯ k.

Lemma 1.6. Q(λ+) is λ+-directed closed. Proof. Virtually the same proof as the preceding.

Lemma 1.7. Assume λ is a strong limit singular cardinal and 2λ = λ+.Then: (1) |Q(λ+)| = λ++; (2) Q(λ+) has the λ++-c.c.; (3) Q(λ+) is dense in Q(λ+). Proof. (1) is obvious, and (2) follows from a standard Δ-system argument. ≤ ++ Q S + ∗ P Q (3) To simplify the notation, for all β λ ,let β := (λ ) β and β := + Q (λ ) Qβ. Note that the proof of Lemma 1.5 shows that every increasing Q + sequence of conditions in β of length <λ has a least upper bound. We now ≤ ++ Q Q prove by induction on β λ that β is dense in β. Q S + ∗ P Q Induction base: For β =0,wehave β = (λ ) 0 = β. Induction step: Suppose the claim holds for α and that s, k is a given element Q Q of β,forβ = α + 1. We would like to ﬁnd a condition in β which is stronger than s, k. + + Since S(λ )isλ -closed and Pβ is a (≤ λ)-support iteration, we may assume that s already decides the support of k. To avoid trivialities, we may also assume that α ∈ supp(k). Q + Q Q Since α is a λ -closed, dense subset of α and k(α)isa α-name for a pair + of bounded subsets of λ , let us pick a condition s0,k0≥s, k and a pair (x, c) Q ∈ Q Q such that s0,k0 α α, k0(α)isthe α-canonical name for (x, c), and

s0,k0 Qα k(α)=k0(α). + Next, for all δ<λ ,letDδ denote the collection of all conditions s ,k ∈Qβ such that all of the following holds •s ,k ≥s0,k0; • Q ∈ Q s ,k α α;

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1776 MOTI GITIK AND ASSAF RINOT

•s ,k Qα decides A α(δ); • k (α)=k0(α). Q + Q Q Since α is a λ -closed, dense subset of α, we get that each Dδ is dense in β ∈ above s0,k0 , so let us pick a condition s1,k1 δ∈c Dδ. Let ∪ ∪ ∪ { k1 | ∈ ∩ } γ := sup x c dom(s1) max(ci ) i supp(k1) α .

Now, we take the condition s∗,k∗ as follows: • s∗ : γ + ω +1→ 2 is the function satisfying for all δ ≤ γ + ω, s(δ),δ∈ dom(s ), s∗(δ)= 1 0, otherwise;

∗ ∗ • s decides the support of k to be supp(k1); ∗ ∗ • for all i ∈ supp(k ), k (i) is the canonical name for the pair (xi,ci), where k1 ∪ k1 ∪{ } (xi ((γ,γ + ω)),ci γ + ω ),i<α, (xi,ci):= (x ∪ ((γ,γ + ω)),c∪{γ + ω}),i= α.

∗ ∗ ∗ ∗≥ ≥ ∗ k k Then s ,k s1,k1 s, k ,sup(dom(s )) = max(ci )=sup(xi )=γ + ω ∈ ∗ ∗ ∗ Q ∈ Q ∩ k∗ ∩ ≥ ∗ for all i supp(k ), s ,k α α,andt cα = s1 c for every t s ,so the fact that s1,k1 is in δ∈c Dδ entails clause (4) of Deﬁnition 1.4. Namely, we ∗ ∗∈Q have found s ,k β, which is stronger than s, k . Limit step: Suppose β is a limit ordinal and s, k∈Qβ. Clearly, we may assume that s decides the support of k. To avoid trivialities, we may also assume that sup(supp(k)) = β. In particular, cf(β) <λ. Let βα | α

•s0,k0 = s, k; • sα decides the support of kα; •s ,k Q ∈ Q ; α α βα βα ++ ++ • kα (λ \ βα)=k (λ \ βα). The successor stage simply utilizes the induction hypothesis, so let us show how to handle the limit stage of the recursion. Suppose α ≤ cf(β) is a limit ordinal and sε,kε|ε<α is deﬁned. Q | ≤ Fix ε<α. By the induction hypothesis, sη,kη βε ε η<α is an increasing sequence of conditions in Q of length <λ+,soletq denote its least βε ε upper bound. Then, for all ε<α,letsε,kε denote the least upper bound of the Q | ≤ weakly increasing sequence qη βε ε η<α. ς ς Q Q ≤ ≤ Notice that s ,k βε = s ,k βε whenever ς ε <α. We now deﬁne sα,kα and then argue that it is indeed a legitimate condition. Thus, let sα,kα be the condition satisfying: • 0 sα = s ; • ε ∪ sα decides the support of kα to be ε<α supp(k ) supp(k);

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE FAILURE OF DIAMOND ON A REFLECTING STATIONARY SET 1777

• for all i ∈ supp(kα) ∩ βα: ε ε kα(i)=k (i), where ε := min{ε | i ∈ supp(k )};

• for all i ∈ supp(kα) \ βα:

kα(i)=k(i). Q ε ε Evidently, for all ε<α,wehave sα,kα βε = s ,k . Since cf(βα) <λ, P P | Q the forcing βα is the inverse limit of ε ε<βα , and hence sα,kα βα ,the Q restriction of our proposed condition, is in βα . By arguments which, by now, are standard, we moreover have s ,k Q ∈ Q . α α βα βα Q ≥ Q ++ \ ++ \ Finally, as sα,kα βα s, k βα , kα (λ βα)=k (λ βα)and Q ++ \ ∈ KAD ˙ + ++ \ s, k βα k (λ βα) (S(λ )) (λ βα), we obtain Q ++ \ ∈ KAD ˙ + ++ \ sα,kα βα kα (λ βα) (S(λ )) (λ βα), + and hence sα,kα is a legitimate condition of Q(λ ). Thus, the recursion may ∈Q indeed be carried out, and we end up with a condition scf(β),kcf(β) β which is stronger than s, k, as requested. The next theorem is the core of our proof. We encourage the reader to notice the ⊆ λ+ 4 role of the fact that S concentrates on the critical coﬁnality, i.e., that S Ecf(λ). λ + Theorem 1.8. Suppose λ>cf(λ)=ω is a strong limit and 2 = λ . If G ∗ H is Q(λ+)-generic, then letting S := G we have V [G][H] |= S is stationary. Proof. Fix a name E˙ and a condition s∗,k∗∈Q(λ+) forcing that E˙ is a club subset of λ+. Clearly, we may assume that 0 ∈ supp(k∗)andthat| supp(k∗)| = λ. Thus, our goal is to ﬁnd an extension of s∗,k∗ that forces that E meets S.Fix a large enough regular cardinal χ and an elementary submodel M ≺H(χ),<χ) satisfying: •|M| = λ; • ∩ + ∈ λ+ λ

4 λ + Recall that by [19], 2 = λ entails ♦S for every S which does not concentrate on the critical coﬁnality.

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1778 MOTI GITIK AND ASSAF RINOT

Definition 1.8.2 (Branching extensions). For a condition s, k∈Q(λ+), a set Z ∈ [λ++]≤λ, a function g : Z → 2, and an ordinal γ ≥ sup(dom(s)), we shall g g ≤ define s, k γ .Let s, k γ = s ,k be the χ-least extension of s, k such that: •s,k∈Q(λ+); • s : γ + ω +1→ 2 is the function satisfying for all δ ≤ γ + ω: s(δ),δ∈ dom(s), s(δ):= 0, otherwise; • supp(k) = supp(k) ∪ Z; • for all i ∈ supp(k): ⎧ ⎨⎪(((γ + g(i),γ+ ω)), {γ + ω}) ,i∈ Z \ supp(k), (xk ,ck )= xk ∪ ((γ + g(i),γ+ ω)),ck ∪{γ + ω} ,i∈ Z ∩ supp(k), i i ⎩⎪ i i k ∪ k ∪{ } (xi ((γ,γ + ω)),ci γ + ω ), else. To see that the definition is good, just notice that for all t ≥ s and i ∈ supp(k), ∩ k ∅ ∈ k ∩ k ∩ ∈ ∩ k ∩ k if t ci = ,theni supp(k), and xi δ = xi δ for all δ t ci = s ci . { } ⊆ g Evidently, if g, γ, s, k M,then s, k γ is in M as well.

Deﬁnition 1.8.3 (Mixing two conditions). Given q0 = s0,k0,q1 = s1,k1 in + ++ + + Q (λ ), and β<λ such that s0,k0 S(λ ) ∗ Pβ ≤s1,k1 S(λ ) ∗ Pβ ,let mix(q1,β,q0)bethe<χ-least condition s ,k such that: •s,k∈Q(λ+); • if γ<λ+ is the minimal ordinal such that there exists a function s : γ + ω +1→ 2 with s ⊇ s1 and s (γ + ω)=0,thens is a witness to that; • supp(k ) = (supp(k1) ∩ β) ∪ supp(k0); • for all i ∈ supp(k): k k k (x 1 ∪ ((max(c 1 ),γ+ ω)),c 1 ∪{γ + ω}),i<β, (xk ,ck )= i i i i i k0 ∪ k0 k0 ∪{ } (xi ((max(ci ),γ+ ω)),ci γ + ω ), else.

It is not hard to see that the definition is good and that mix(q1,β,q0) ∈ M whenever {q1,β,q0}⊆M. Notice that mix(q1,β,q0) depends only on q0 and q1 + S(λ ) ∗ Pβ and that this notion also makes sense in case that β =0. Z ++ ≤λ + Claim 1.8.4. Suppose g1,g0 ∈ 2foragivensetZ ∈ [λ ] and that q ∈ Q (λ ). ≤ + g1 g0 g0 g1 α g0 If β Δ(g0,g1)andα, γ < λ , then mix(qγ ,β,qγ )=qγ ,andmix((qγ ) ,β,qγ ) g0 is a well-defined extension of qγ . Proof. This follows immediately from the above definitions, and we encourage the reader to digest these definitions by verifying this claim.

+ + Put τ := M ∩λ and pick a countable set of limit ordinals {τn | n<ω}⊆M ∩λ ∅ with supn<ω τn = τ.DenoteZ−1 := . We now recursively deﬁne a sequence of elements of M, (γn,Yn,Zn, Fn) | n<ω, in such a way that for all n<ω: + (1) γn <λ ; { | }⊆ ++ λ { | }⊆ ++ <λ (2) Yn n<ω [λ ] and Zn n<ω [λ ] are increasing chains, and n<ω Yn = n<ω Zn; Zn + (3) for every g ∈ 2, we deﬁne a condition qg ∈ Q (λ ) in such a way that ≤ F { | ∈ Zn } m

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE FAILURE OF DIAMOND ON A REFLECTING STATIONARY SET 1779

(4) for every g ∈ Zn 2, there exists some r ∈ Q (λ+) such that \ ∗ ∗ ≥ g (Zn Zn−1) ≥ ≥ qg rγn r s ,k ; (5) for every g, h ∈ Zn 2andβ<λ++,ifg β = h β,then + + qg S(λ ) ∗ Pβ = qh S(λ ) ∗ Pβ;

(6) if s, k∈Fn,thens, k decides a value for min(E \ τn), and dom(s) >τn. { k∗ | ∈ ∗ } ∗ We commence with letting γ0 := sup max(ci ) i supp(k ) , Y0 := supp(k ), ∈ Z0 { | and Z0 := ψY0 “λ0. Next, we would like to deﬁne qg for all g 2. Let gj λ0 Z0 j<2 } be the <χ-least injective enumeration of 2. We shall deﬁne an upper λ0 triangular matrix of conditions, {qjl | j ≤ l<2 }, in such a way that for all j ≤ j ≤ l ≤ l < 2λ0 , the following holds: gj ∗ ∗ ≤ ≤ ≤ (a) rγ0 qjj qjl qjl ,wherer := s ,k ; τ + (b) p 0 ≤ qjj for some condition p ∈ Q (λ ); + + (c) qjl S(λ ) ∗ Pβ = qjl S(λ ) ∗ Pβ whenever gj β = gj β. Z0 Once we have that, for each g ∈ 2 we pick the unique j such that gj = g and let λ qg be the least upper bound of the increasing sequence qjl | j ≤ l<2 0 .Then item (a) takes care of requirement (4), item (b) establishes requirement (6), and item (c) yields requirement (5).

Thus, the jth row of the matrix is responsible for the condition qgj . The actual definition of the matrix, however, is obtained along the columns. Namely, we define λ λ {qjl | j ≤ l<2 0 } by induction on l<2 0 . g0 τ0 Induction base: Let q00 := (rγ0 ) . λ0 Successor step: Suppose l<2 and {qjl | j ≤ l} has already been defined. We would like to define {qjl | j ≤ l },forl := l +1. Put α := sup{Δ(gj ,gl )+1| j

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1780 MOTI GITIK AND ASSAF RINOT

λ Limit step: Suppose l < 2 0 is a limit ordinal and {qjl | j ≤ l

Zn λn Clearly, our main task is deﬁning qg for all g ∈ +1 2. Let {gj | j<2 +1 } be the Zn+1 <χ-least bijective enumeration of 2. We shall now deﬁne an upper triangular λn λn matrix {qjl | j ≤ l<2 +1 } in such a way that for all j ≤ j ≤ l ≤ l < 2 +1 ,the following holds:

gj (Zn+1\Zn) ≤ ≤ ≤ (a) (qgj Zn )γn+1 qjj qjl qjl ; τn + (b) p +1 ≤ qjj for some condition p ∈ Q (λ ); + + (c) qjl S(λ ) ∗ Pβ = qjl S(λ ) ∗ Pβ whenever gj β = gj β. As in the base case, once the matrix is defined, for each g ∈ Zn+1 2, we pick the unique j such that gj = g, and let qg be the least upper bound of the increasing λn sequence qjl | j ≤ l<2 +1 . \ g0 (Zn+1 Zn) τn+1 Induction base: Let q00 := ((qg0Zn )γn+1 ) . λn Successor step: Suppose l<2 +1 ,and{qjl | j ≤ l} has already been defined. We would like to define {qjl | j ≤ l },forl := l +1. Put α := sup{Δ(gj ,gl )+1| j

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE FAILURE OF DIAMOND ON A REFLECTING STATIONARY SET 1781

\ gjβ (Zn+1 Zn) In addition, by property (a), we have (q )γ ≤ q .Denoter := gjβ Zn n+1 jβ l l \ gl (Zn+1 Zn) + + S ∗ P ≤ S ∗ P (qgl Zn )γn+1 . It follows that rl (λ ) β qjβ l (λ ) β, and hence mix(qjβ l,β,rl ) is a well-deﬁned extension of rl . S + ∗ P By property (c), we get that β<γ<αimplies qjβ l (λ ) β = qjγ l S + ∗ P ≤ (λ ) β and mix(qjβ l,β,rl )=mix(qjγ l,β,rl ) mix(qjγ l,γ,rl ). For all β<α,letpβ := mix(qjβ l,β,rl ). Then, we have just established that g (Zn+1\Zn) | ≥ l pβ β<α is an increasing sequence of conditions, with p0 (qgl Zn )γn+1 , + + and pβ S(λ ) ∗ Pβ = pγ S(λ ) ∗ Pβ for all β<γ<α.Bycf(α) <λ,letpα be the least upper bound of the sequence, pβ | β<α. Then, for all j

g (Zn+1\Zn) l Denote rl := (qgl Zn )γn+1 .Byitem(a) and the above equation, we get S + ∗ P ≤ S + ∗ P that rl (λ ) β qjβ (λ ) β, and hence mix(qjβ ,β,rl ) is a well-deﬁned | extension of rl .By(c), if we write pβ := mix(qjβ ,β,rl ), then pβ β<α \ gl (Zn+1 Zn) is an increasing sequence of conditions, with p ≥ (q )γ ,andp 0 gl Zn n+1 β + + S(λ ) ∗ Pβ = pγ S(λ ) ∗ Pβ for all β<γ<α. Let pα be the least upper bound of the sequence, pβ | β<α. Then, for all j< τn+1 l ,wehave(j)asabove,soletql l := (pα) and qjl := mix(ql l , Δ(gj ,gl ),qj) for all j

Claim 1.8.5. {γn,Zn,Yn, Fn}⊆M for all n<ω. Proof. Easy. Put Z := n<ω Zn. We now state and prove several claims that should gradually clarify the role of the above construction. Claim 1.8.6. For every g ∈ Z 2, if s, k is the least upper bound of the increasing | sequence qgZn n<ω , then supp(k)=Z.

Proof. Fix n<ωand sn,kn∈Fn. By property (4) and Deﬁnition 1.8.2, we have Zn \ Zn−1 ⊆ supp(kn). By deﬁnition of Yn+1, we also have supp(kn) ⊆ Yn+1.

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1782 MOTI GITIK AND ASSAF RINOT

| It follows that if s, k is the least upper bound of an increasing sequence sn,kn ∈ F n<ω n<ω n,then Z = (Zn \ Zn−1) ⊆ supp(k) ⊆ Yn+1 = Z. n<ω n<ω

Deﬁnition 1.8.7. For every β<λ++, h : Z ∩ β → 2, and i ∈ Z ∩ β, we deﬁne a h h pair (xi ,ci ). Pick a function g : Z → 2 that extends h.Foreachn<ω,denotesn,kn :=

qgZn , and then let

h { kn | ∈ } xi := xi n<ω& i supp(kn) , h { kn | ∈ } ci := cl( ci n<ω& i supp(kn) ). Notice that by property (5) of the construction, the above deﬁnition does not depend on the choice of the function g. Claim 1.8.8. For all β<λ++, h : Z ∩ β → 2, and i ∈ Z ∩ β,wehave

h h sup(xi )=max(ci )=τ. → Proof. Pick g : Z 2 extending h.Fixn<ω, and denote sn,kn = qgZn .By + property (6) of the construction, we have dom(sn) >τn.Bysn,kn∈Q (λ )∩M, we have

+ ∩ kn kn ≥ τ = λ M>sup(xi )=max(ci )=sup(dom(s)) τn. The conclusion now follows.

Claim 1.8.9. There existss ¯ : τ → 2 such that for all g ∈ Z 2, if s, k is the least | ∈{ ∪{ }} upper bound of the increasing sequence qgZn n<ω ,thens s,¯ s¯ (τ,0) . n Proof. By property (5) of the construction, there exists a sequence s | n<ω ∈F n n such that for every n<ωand s, k n,wehaves = s.Put¯s := n<ω s . + If s ,k ∈Fn,thens ∈ M, and hence dom(s ) ⊆ τ = M ∩ λ .Inparticular, dom(¯s) ⊆ τ. Z ∈ | Finally, for every function g 2, the increasing sequence qg Zn n<ω is F an element of n<ω n, so by the proof of Lemma 1.5, if s, k is the least upper bound of the above sequence, then either s =¯s or s =¯s ∪{(τ,0)}.

Claim 1.8.10. For every g ∈ Z 2, if q is an upper bound of the increasing sequence | ∈ ˙ qgZn n<ω ,thenq τˇ E.

Proof. Fix g and q as above. For all n<ω, by property (6) and Fn ⊆ M, ≤ + ∩ ∈ ˙ there exists αn with τn αn <λ M = τ such that qgZn αˇn E.Since sup{τn | n<ω} = τ,wegetthatsup{αn | n<ω} = τ, and hence q forces that τ is an accumulation point of the club E.

Thus, our main task is to argue the existence of some g ∈ Z 2 for which there exists | an upper bound of the increasing sequence qgZn n<ω which is compatible with s¯ ∪{(τ,1)}, ∅. Evidently, such a condition will force that S meets E.Akeyfact for ensuring the existence of such a function is the next claim.

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE FAILURE OF DIAMOND ON A REFLECTING STATIONARY SET 1783

Claim 1.8.11. Suppose A ⊆ τ. ∈ h For every i Z,thereexistssomem<2 such that xi = A for every function h : Z ∩ β → 2 with i<β<λ++ and h(i)=m.

Proof. Suppose i ∈ Z.Letn<ωbe the unique ordinal such that i ∈ Zn \ Zn−1. + Since ((γn +0,γn + ω)) =(( γn +1,γn + ω)) and γn + ω<τ= M ∩ λ , let us pick some m<2 such that ((γn + m, γn + ω)) = A ∩ ((γn,γn + ω)). To see that m works, suppose that h is as in the hypothesis and that g ∈ Z 2 extends h.Inparticular,g(i)=m.Put sn,kn := qgZn . Then by property (4) \ ≥ g (Zn Zn−1) ≥ of the construction, sn,kn rγn r for some condition r,andsoby Deﬁnition 1.8.2, this means that i ∈ supp(k)and

kn ∩ ∩ xi ((γn,γn + ω)) = ((γn + g(i),γn + ω)) = ((γn + m, γn + ω)) = A ((γ,γ + ω)).

h kn ∩ kn ∩ kn kn In particular xi sup(xi ) = A sup(xi ). As xi xi , we conclude that h xi = A. Let D denote the set of all s, k such that all items of Definition 1.4, except for (5), are satisfied, and where instead we require that for all α ∈ supp(k): k k (5a) either sup(xα)=max(cα)=sup(dom(s)) or ∈ ∗ k k k∗ k∗ (5b) α supp(k )and(xα,cα)=(xα ,cα ). Then D ⊇ Q(λ+) is a dense set and, by a proof similar to the one of Lemma 1.5, every increasing sequence of length <λ+ of elements of D has a minimal upper bound within D. For each α<λ++,letD denote the dense open subset of S(λ+) ∗ P that −→ α α → { ∈ | S + ∗ P ∈ } decides Aα(τ), and let Dα := p D p (λ ) α Dα denote its cylindric extension within D. Put θ := otp(Z) and let {εα | α<θ} be the increasing enumeration of Z.We now define by induction an increasing sequence of conditions pα = sα,kα|α<θ and a chain of functions {hα : Z ∩ (εα +1)→ 2 | α<θ} in such a way that for all α<θ: ∪{ } ∗≤ ∈ → (a) s¯ (τ,1) ,k sα,kα Dε ; α ∗ (b) Z ∩ (εα +1)⊆ supp(kα) ⊆ supp(k ) ∪ (εα +1); hα hα kα kα (c) x x and c c for all i ∈ Z ∩ (εα +1); i i ∗i ∗ i kα kα k k ∈ \ (d) (xi ,ci )=(xi ,ci ) for all i supp(kα) (εα +1). Induction base: Since 0 ∈ supp(k∗) ⊆ Z,wehaveε = 0. Pick s , ∅ ∈ D with −→ 0 0 0 ∪{ } 5 ∅ τ s¯ (τ,1) s0. In particular, s0, decides A0(τ)tobe,say,A0 .SinceD0 is a dense open set, we may moreover assume that dom(s0) is the successor of a limit ordinal γ and that s0(γ)=0. h { }→ τ 0 By Claim 1.8.11, pick a function h0 : ε0 2 such that A0 = xε , and let ∗ 0 p0 := s0,k0 be the condition in D such that supp(k0) = supp(k )andforall i ∈ supp(k0): h0 h0 (x ∪ ((τ,γ)),c ∪{γ}),i= ε , (xk0 ,ck0 ):= i i 0 i i k∗ k∗ (xi ,ci ), else.

5 + Recall that D0 ⊆ S(λ ) ∗ P0 and that P0 is the trivial forcing {∅}, {(∅, ∅)} .

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1784 MOTI GITIK AND ASSAF RINOT

Let us show that p0 is indeed a well-defined condition. Assume towards a contradic- ∈ ≥ ∈ ∩ k0 tion that this is not the case. Hence, we can fix i supp(k0), t s0 and δ t ci −→ ∗ k0 ∩ ∈ ∩ k ∈ ∗ such that Ai(δ)=xi δ. Clearly, if i>ε0,thenδ t ci implies that δ s , ∗ ∗ contradicting the fact that s ,k is a legitimate condition. So i = ε0 =0. h0 Since δ ∈ c, we get from Claim 1.8.8 that either δ ∈{τ,γ} or δ ∈ s¯.Itis 0 −→ k0 ∩ impossible that δ = τ, because the choice of h0 ensured that x0 δ = A0(δ). It is also impossible that δ = γ, because t ≥ s0 and s0(γ)=0.So,δ ∈ s¯. → n n Pick a function g : Z 2 extending h0, and let s ,k := qgZn for all n< h h n 0 kn 0 kn ∪{ } ω.Then¯s = n<ω s , x0 = n<ω x0 and c0 = n<ω c0 τ . Pick n< ∈ n n∈Q + kn kn ω such that δ sn.Bys ,k (λ ), we have sup(x0 )=max(x0 )= −→ h n ≥ 0 ∩ kn ∩ ∈ n ∩ kn sup(dom(s )) δ,andthenA0(δ)=x0 δ = x0 δ while δ s c0 , contradicting the basic fact that sn,kn is a legitimate condition. → Thus, p0 is a well-defined element of Dε0 and satisfies requirements (a)–(d). Successor step: Suppose that for some α<θ, pα has already been defined. Pick → + p = s ,k ≥pα in D with dom(s )=γ+1 and s (γ) = 0 for some limit γ<λ . εα+1 −−−→ S + ∗ P τ Then p (λ ) εα+1 decides Aεα+1 (τ)tobe,say,Aεα+1 . By Claim 1.8.11, pick a h +1 ∩ → τ α function hα+1 : Z (εα+1 +1) 2 extending hα such that Aεα+1 = xεα+1 . Finally, let pα+1 = sα+1,kα+1 be the condition in D such that: • sα+1 = s ; ∗ • supp(kα+1) = (supp(k ) ∩ εα+1) ∪{εα+1}∪supp(k ); • for all i ∈ supp(kα+1): ⎧ ⎪ k k ⎨⎪(xi ,ci ),i<εα+1, k k hα+1 hα+1 α+1 α+1 (xi ,ci ):= (x ∪ ((τ,γ)),c ∪{γ}),i= ε , ⎪ i i α+1 ⎩ k∗ k∗ (xi ,ci ), else.

Assume towards a contradiction that pα+1 is not−→ a well-defined condition. Fix ∈ ∈ ∩ kα+1 kα+1 ∩ i supp(kα+1)andδ sα+1 ci such that Ai(δ)=xi δ.Ifi = εα+1, then we get a contradiction to the fact that s∗,k∗ and s,k are legitimate conditions. If i = εα+1, then Claim 1.8.8 and the choice of the function hα+1 ensures that δ ∈ s ∩ τ.Butthen,δ ∈ s¯, and so by the exact same argument as α+1 −−−→ kα+1 ∩ the induction base, we have Aεα+1 (δ) = xεα+1 δ. Clearly, pα+1 satisfies requirements (a)–(d). Limit step: Suppose that pβ | β<α and {hβ : Z ∩ (εβ +1)→ 2 | β<α} has already been defined for some limit ordinal α<θ. + Since α<θ<λ ,lets, k be the least upper bound of pβ | β<α.Note k k k∗ k∗ ∈ \ ≥ → that (xi ,ci )=(xi ,ci ) for all i supp(k) εα.Fix s ,k s, k in Dεα with dom(s)=γ +1ands(γ) = 0 for some limit γ<λ+.Thens,k S(λ+) ∗ P −−→ εα τ ∩ → decides Aεα (τ)tobe,say,Aεα .Puth := β<α hβ and pick hα : Z (εα +1) 2 h τ α extending h such that Aεα = xεα . Finally, let pα = sα,kα be the condition in D such that: • sα = s ; ∗ • supp(kα) = (supp(k ) ∩ εα) ∪{εα}∪supp(k );

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE FAILURE OF DIAMOND ON A REFLECTING STATIONARY SET 1785

• for all i ∈ supp(kα): ⎧ k k ⎨⎪(xi ,ci ),i<εα, h h (xkα ,ckα ):= α ∪ α ∪{ } i i ⎪(xi ((τ,γ)),ci γ ),i= εα, ⎩ ∗ ∗ k k (xi ,ci ), else.

Then pα is a well-defined condition that satisfies all the requirements. This completes the construction. Put g := α<θ hα, and let p be an upper bound of the increasing sequence, pα | α<θ. By Claim 1.8.6, Definition 1.8.7, Claim 1.8.9 and clauses (a) through (c) of the induction, p is an upper bound of | the increasing sequence qgZn n<ω, and so by Claim 1.8.10 and clause (a) of the induction, p τˇ ∈ E˙ ∩ S.˙

1.2. Applications. Utilizing the poset from the previous subsection and the existence of a supercompact cardinal, we now consider three models in which diamond fails on a set that reﬂects stationarily often. Theorem 1.9. It is relatively consistent with the existence of a supercompact cardinal that all of the following hold simultaneously: (1) GCH;

(2) APℵω ;

(3) Reflℵω ; ℵω+1 (4) ♦S fails for some stationary S ⊆ Eω . Proof. We take as our ground model the model from [7, §5]. That is, GCH holds, +ω+1 κ is a supercompact cardinal, there exists Cα | α<κ which is a very weak | ∈ κ+ω+1 square sequence, and there exists Dα α E≥κ which is a partial square sequence. We shall not define these concepts here; instead, we just mention two important facts. The first is that the properties of these sequences are indestructible under cofinality-preserving forcing which are ω-distributive. The second is that in the generic extension by the Lévy collapse, Col(ω1,< κ), these two sequences are

combined to witness APℵω . Let P denote the iteration of length κ + 1 with backward Easton support, where for every inaccessible α ≤ κ,weforcewithQ(α+ω+1) from Definition 1.4, and for accessible α<κ, we use trivial forcing. Let G be P-generic over V . Then by Lemmas 1.6, 1.7, and a well-known argument of Silver (see [1, §11]), κ remains supercompact in V [G]. Also, the very weak square sequence and the partial square sequence remain as such. ⊆ κ+ω+1 By Theorem 1.8, there exists in V [G] a stationary subset S Eω such that ♦S fails. Finally, let H be Col(ω1,<κ)-generic over V [G]. Work in V [G][H]. Then ℵ ℵ +ω ℵ +ω+1 2 = κ, ω = κ , ω+1 = κ ,andGCH +APℵω holds. Since Col(ω1,< κ) satisfies the κ-c.c., S remains stationary, and ♦S still fails (for if {A˙δ | δ ∈ S} is a name for a ♦S -sequence in V [G][H], then {Aδ := {A ⊆ δ |∃p ∈ Col(ω1,< κ)(p ˇ ˙ }| ∈ } ♦− A = Aδ) δ S would be a S -sequence in V [G]; see [12]). Finally, since κ is κ+ω+1-supercompact in V [G], an argument of Shelah yields § that Reflℵω holds (see [1, 10]).

The next theorem shows that it is possible to have the failure of ♦S for a set S which reﬂects in an even stronger sense.

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1786 MOTI GITIK AND ASSAF RINOT

Theorem 1.10. It is relatively consistent with the existence of a supercompact cardinal that there exists a singular cardinal λ for which all of following hold simultaneously: (1) λ is a strong limit of countable coﬁnality and 2λ = λ+; ⊆ λ+ (2) there exists a stationary S Ecf(λ) such that: { ∈ λ+ | ∩ } (a) α E>ω S α contains a club in α is stationary; (b) ♦S fails.

Proof. Start with a model of MM.Putλ := ω.Sinceλ is a singular strong limit, we get from [5] that 2λ = λ+, and hence Q(λ+) is well deﬁned, so let us work in + the generic extension, V Q(λ ). + Since Q(λ )isℵ2-directed closed, we get from Larson’s theorem [13] that MM is preserved, and by the additional “good properties” of Q(λ+), the cardinal structure λ + ⊆ λ+ is preserved as well. Then λ = ω,2 = λ , and there exists a stationary S Ecf(λ) such that ♦S fails. Finally, clause (a) is an immediate consequence of the fact that MM implies Friedman’s problem (see [5]).

Analysis of the models from Theorems 1.9 and 1.10 yields that these models ∗ satisfy a certain strong form of reflection, namely, Refl ([λ+]ω), and hence there exists no very good scale (or even a better scale)forλ in these models. We now consider a third model, establishing that a very good scale has no effect on the validity of diamond for reflecting stationary sets. Theorem 1.11. It is relatively consistent with the existence of a supercompact cardinal that there exists a singular cardinal κ for which all of following hold simultaneously: (1) κ is a strong limit of countable cofinality and 2κ = κ+; (2) there exists a very good scale for κ; ♦ ⊆ κ+ (3) S fails for some S Ecf(κ) that reflects stationarily often. Proof. Start with a model of GCH in which there exists a supercompact cardinal, κ.Letλ := κ+ω.

Step 1. Let P1 denote the iteration of length κ + 1 with backward Easton support, where for every inaccessible α ≤ κ,weforcewithQ(α+ω+1), and for accessible α<κ, we use trivial forcing. Let V1 denote the generic extension by P1. Then, ⊆ λ+ in V1, GCH holds, κ is supercompact, and there exists a stationary S Ecf(λ) on which ♦S fails. + Step 2. Work in V1. Fix a normal ultraﬁlter U over Pκ(λ ) and the corresponding embedding j : V1 → M Ult(V1, U). Let P2 denote the iteration of length κ +1 with backward Easton support, where for every inaccessible α ≤ κ,weforcewith Add(α, α+ω+1), adding α+ω+1-Cohen functions from α to α, and for accessible α<κ, we use trivial forcing.

Let G be P2-generic over V1,andworkinV2 := V1[G]. Then, by standard arguments (see [1, §11]), we may ﬁnd a set G∗ such that the embedding j lifts to ∗ ∗ + an embedding j : V1[G] → M[G ]. Let Fα : κ → κ | α<λ denote the generic ∗ functions introduced by the κth-stage of P2.Asin[8],wemaychooseG in such a + κ λ + way that Fj∗(α)(κ)=α for all α<λ .Thus,2 =2 = λ .

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE FAILURE OF DIAMOND ON A REFLECTING STATIONARY SET 1787

+ As P2 has the κ -c.c., S remains stationary, and ♦S still fails. (By the chain P ♦ ♦− ♦− ♦ condition of 2,if S holds in V2,then S holds in V1. However, S entails S , { ∈ λ+ | ∩ } and the latter fails in V1.) Let T := α E<κ cf(α) >ω,S α is stationary . λ+ + Since S is a stationary subset of E<κ and κ is λ -supercompact (as witnessed by j∗), the set T is stationary. Note that the cardinal structure in V2 is the same as in V . ∗ +ω+1 ∗ + ∗ ∗ Step 3. Work in V2.LetU := {X ⊆Pκ(κ ) | j “λ ∈ j (X)}.ThenU is a normal ultrafilter extending U. For every n<ω,letUn denote the projection of ∗ +n ∗ U to Pκ(κ ). Next, let Q, ≤, ≤ denote the variation of supercompact Prikry forcing from [8, Definition 2.9]. That is, instead of working with a single measure, we work with the sequence of measures Un | n<ω.By[8,§2], we have: (a) Q, ≤ satisfies the λ+-c.c.; (b) Q, ≤ does not add new bounded subsets to κ; (c) in the generic extension by Q, ≤,(κ+n)V changes its cofinality to ω,for every n<ω.

Let V3 denote the generic extension by Q, ≤.WorkinV3.Thenκ is a strong limit cardinal of countable cofinality and 2κ =(λ+)V2 = κ+.SinceQ, ≤ has the + + λ -c.c., the sets S and T remain stationary subset of κ ,and♦S still fails. Claim 1.11.1. S reflects stationarily often. ∗ + Proof. Recall that we work in V3.PutT :={α<κ | cf(α) >ω,S∩α is stationary}. Since T is stationary, to show that S reflects stationarily often, it suffices to es- ⊆ ∗ ∈ κ+ ∩ κ+ tablish that T T . For this, it suffices to prove that if α E>ω E<κ and C is a club subset of α then there exists a club C ⊆ C lying in V2. But this is V2 obvious: fix, in V2, a continuous function π :cf (α) → α whose image is cofinal in α.PutC := C ∩ Im(π). Then C is a club subset of C and π−1[C] is a club subset + + V2 ∈ κ ∩ κ V2 V3 of cf (α). By property (c) and α E>ω E<κ,wehavecf (α)=cf (α), so −1 property (b) entails that π [C ]isinV2, and hence C is in V2, as requested. Claim 1.11.2. There exists a very good scale for κ.

Proof. Let Pn | n<ω denote the supercompact Prikry sequence introduced by Q ≤ ∩ , over V2.Foreachn<ω, consider the inaccessible cardinal κn = Pn κ.For each α<κ+ =(λ+)V2 , deﬁne a function t ∈ κ+ω+1 as follows: α n<ω n +ω+1 Fα(κn),Fα(κn) <κn tα(n):= (n<ω). 0, otherwise + Then, by Proposition 2.21 of [8], tα | α<κ is a very good scale.

1.3. Uncountable cofinality. It is worth mentioning that via a straightforward modification to the proofs of subsection 1.1, it is possible to handle singular cardinals of uncountable cofinality as well. More specifically, we have: Theorem 1.12. Suppose λ is a strong limit singular cardinal, and 2λ = λ+ Then there exists a notion of forcing Q(λ+), satisfying: (1) Q(λ+) is λ+-directed closed; (2) Q(λ+) has the λ++-c.c.;

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1788 MOTI GITIK AND ASSAF RINOT

(3) |Q(λ+)| = λ++; Q(λ+) ♦ ⊆ λ+ (4) in V , S fails for some stationary S Ecf(λ). Remark. In particular, it is possible to obtain the failure of diamond on a stationary subset of λ+ in the presence of a supercompact cardinal in the interval (cf(λ),λ). Outline of the proof of Theorem 1.12. Q(λ+) is a dense subset of the forcing no- λ+ tion for injecting a Cohen subset of Ecf(λ) followed by the iteration for killing all potential diamond sequences over it. To argue that the generic Cohen set remains stationary, one takes an elementary submodel M ≺H(χ),<χ) with ∩ + ∈ λ+

2. Negation of guessing In [4], Dˇzamonja and Shelah considered a particular consequence of diamond and established the consistency of its negation. To state their result, we need the following two deﬁnitions.

+ Definition 2.1. For a function f : λ → cf(λ), let κf denote the minimal cardinal- + cf(λ) + ity of a family P⊆[λ ] with the property that whenever Z ⊆ λ is such that | ∩ −1{ }| + ∈P ∩ βcardinality λ+ induces a non-trivial λ+ function, that is, a function f ∈ cf(λ) with κf > 0. + Definition 2.2. For a singular cardinal λ,wesaythatλ -guessing holds iff κf ≤ + λ+ for all f ∈ λ cf(λ). We refer the reader to [4] for background and motivation for this definition, but + let us just mention the obvious fact that ♦λ+ implies λ -guessing. In [4, §2], the consistency of the negation of λ+-guessing has been established: Theorem 2.3 (Dˇzamonja-Shelah). It is relatively consistent with the existence of a supercompact cardinal that there exist a strong limit singular cardinal λ and a + λ + function f : λ → cf(λ) such that κf =2 >λ . Here, we reduce the large cardinal hypothesis significantly by establishing that any model with a strong limit singular cardinal λ with 2λ >λ+ will do. Moreover, in such a model, any non-trivial function is a counterexample. Theorem 2.4. If λ is a strong limit singular cardinal, then λ+ λ {κf | f ∈ cf(λ)} = {0, 2 }.

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE FAILURE OF DIAMOND ON A REFLECTING STATIONARY SET 1789

Proof. Since λ is a strong limit, the next lemma tells us that any non-trivial func- cf(λ) λ tion, f, satisﬁes κf = λ =2 .

Lemma 2.5. Suppose λ is an inﬁnite cardinal, and 2cf(λ) + λ

+ Proof. Suppose f : λ → cf(λ) is a function with κf > 0. Fix β 0, we have |Aβ| = λ ≥ λ ,soletus

−1 Proof. For every β

cf(λ) cf(λ) + cf(λ) Let {gi | i<λ } be an injective enumeration of λ.Byλ ≤ κf ≤ λ , we avoid trivialities and assume that λcf(λ) >λ+. Thus, it suﬃces to establish the following. ∈ + cf(λ) { cf(λ) | ∩ } Claim 2.5.2. For all a [λ ] , Ia := i<λ sup(f[a Zgi ]) = cf(λ) has cardinality ≤ λ+.

cf(λ) + Proof. Assume towards a contradiction that a ∈ [λ] is such that |Ia| >λ . cf(λ) + By |Ia|≥(2 ) and the Erd¨os-Rado theorem, let us pick a set I ⊆ Ia with | | I > cf(λ) and an ordinal γ cf(λ) and an ordinal β>γ ∩ ∈ ∈ ∈ ∩ such that β = min(f[a Zgi ]) for all i I . Finally, for all i I , pick δi a Zgi such that f(δi)=β.Since|a| =cf(λ) < |I |, we may ﬁnd distinct i0,i1 ∈ I for

which δi0 = δi1 . Finally, by δ ∈ Z , δ ∈ Z and δ = δ ,wehave i0 gi0 i1 gi1 i0 i1 g β = g f(δ )=ψ (δ )=ψ (δ )=g f(δ )=g β, i0 i0 i0 δi0 i0 δi1 i1 i1 i1 i1 which contradicts the existence of some γ<βwith gi0 (γ) = gi1 (γ).

By combining the arguments of the above proof with the ones from [14], it is cf(λ) + possible to obtain a lower bound on κf even without assuming 2 ≤ λ .Namely, cf(λ)=ω,thenκf ≥ pp(λ) for every non-trivial f. The corollary follows:

Corollary 2.6. The following are equivalent: (1) Shelah’s strong hypothesis; (2) λ+-guessing holds for all singular cardinal λ.

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1790 MOTI GITIK AND ASSAF RINOT

3. The sup-function on stationary subsets of [λ+]ω In this section, we shall supply a negative answer to the following question.

Question (König-Larson-Yoshinobu, [11]). Let λ>ω1 be a successor cardinal. Is ℵ it possible to prove in ZFC that every stationary B⊆[λ] 0 can be thinned out to a stationary A⊆Bon which the sup-function is 1-1? We refer the reader to [11] for motivation and background concerning this question. Recall that a set A⊆P(λ)issaidtobestationary (in the generalized sense) iff for every function f :[λ]<ω → λ,thereexistssomeA ∈Asuch that f“[A]<ω ⊆ A. Now, it is obvious that if cf([λ]ω, ⊆) >λ,thenB := [λ]ω is a counterexample to the above question. In particular, any model on which the singular cardinals hypothesis fails gives a negative answer. Thus, in this section, we shall focus on answering the above question in the context of GCH. Definition 3.1. Given a set X⊆P(λ), denote S(X ):={sup(X) | X ∈X}. ⊆ λ+ Definition 3.2. For an infinite cardinal λ and a stationary set S E<λ,consider the following three principles: + <λ (a) (1)S asserts that there exists a stationary X⊆[λ ] such that: • the sup-function on X is 1-to-1; • S(X ) ⊆ S. + <λ (b) (λ)S asserts that there exists a stationary X⊆[λ ] such that: • the sup-function on X is (≤ λ)-to-1; • S(X ) ⊆ S. ♣− A | ∈ (c) S asserts that there exists a sequence δ δ S such that: <λ • for all δ ∈ S, Aδ ⊆ [δ] and |Aδ|≤λ; • if Z is a cofinal subset of λ+, then the following set is stationary:

{δ ∈ S |∃A ∈Aδ(sup(A ∩ Z)=δ)} . ♣− The principle S has been considered in [15] and was found to be the GCH-free version of ♦S . From this, we easily get the following. ⊆ λ+ Lemma 3.3. Suppose λ is an inﬁnite cardinal and S E<λ is stationary. Then (1) ⇒ (2) ⇒ (3) ⇒ (4),andif2λ = λ+, then moreover, (4) ⇒ (1),where:

(1) ♦S; (2) (1)S; (3) (λ)S; ♣− (4) S . <ω Proof. (1) ⇒ (2) By ♦S , pick a collection {fδ :[δ] → δ | δ ∈ S} such that for + <ω + <ω every f :[λ ] → λ there exists some δ ∈ S with f [δ] = fδ.Foreachδ ∈ S, pick a cofinal Yδ ⊆ δ of minimal order-type and find Xδ ⊇ Yδ with |Xδ| = |Yδ| such <ω that fδ“[Xδ] ⊆ Xδ. It is now easy to see that X := {Xδ | δ ∈ S} is as requested. (2) ⇒ (3) is obvious. (3) ⇒ (4) Let X exemplify (λ)S.Foreachδ ∈ S,letAδ := {X ∈X |sup(X)= } A | ∈ ♣− ⊆ + δ .Toseethat δ δ S witness S , we fix a cofinal subset Z λ and + a club C ⊆ λ , and argue that there exists δ ∈ C ∩ S and A ∈Aδ such that sup(Z ∩ A)=δ. Define a function f :[λ+]<ω → λ+ as follows: min(Z \ sup(σ)), |σ| is odd, f(σ):= min(C \ sup(σ)), |σ| is even.

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE FAILURE OF DIAMOND ON A REFLECTING STATIONARY SET 1791

Since X is stationary, we may pick some X ∈X such that f“[X]<ω ⊆ X.Put 7 4 δ := sup(X). Then X ∈Aδ, f“[X] is a cofinal subset of Z ∩ δ,andf“[X] is a cofinal subset of C ∩ δ.Inparticular,sup(X ∩ Z)=sup(X ∩ C)=δ,soδ ∈ C and we are done. λ + ♦ ⇔ λ + ♣− Finally, if 2 = λ , then a theorem from [15] stating that S 2 = λ + S yields (4) ⇒ (1). Remark. The preceding lemma improves an unpublished result by Matsubara and Sakai, who established the implication (3) ⇒ (1) under stronger cardinal arithmetic assumptions. Since, under mild cardinal arithmetic hypothesis, the above principles coincide, it is interesting to study whether these principles can be separated. The reader who is only interested in the promised solution to the above-mentioned question may now skip to Theorem 3.6 below. Proposition 3.4. If the singular cardinals hypothesis fails, then for some singular ♣− λ+ cardinal λ, S holds, while (λ)S fails, for S := E=cf( λ) ♣− Proof. By [15], λ+ holds for every infinite cardinal λ, so let us focus on the E=cf( λ) second component. Suppose that the singular cardinals hypothesis fails. By a theorem of Shelah [18], in this case, there exists a singular cardinal λ such that cf(λ)=ω and cov(λ, λ, cf(λ)+, 2) >λ+. This means that for every X⊆[λ]<λ of size λ+,thereexistssomex ∈ [λ]ω such that x ⊆ X for all X ∈X. Now assume + <λ towards a contradiction that X witnesses (λ) λ+ .Inparticular,X⊆[λ ] and E=cf( λ) |X | = λ+. Pick a function x : ω → λ with Im(x) ⊆ X for all X ∈X. Define f :[λ+]<ω → λ+ by f(σ):=x(|σ|) for all σ ∈ [λ+]<ω.SinceX is stationary, there exists some X ∈Xsuch that f“[X]<ω ⊆ X. In particular, Im(x) ⊆ X, contradicting the choice of x.

Proposition 3.5. (1)S ⇒♦S for any uncountable cardinal λ and any stationary ⊆ λ+ S Eω . ⊆ λ+ Proof. Suppose λ is an uncountable cardinal, S Eω and (1)S holds, as witnessed by a stationary set X . Use Cohen forcing to blow up the continuum above λ+;then ♦S fails. Finally, since Cohen forcing is proper, X remains stationary, so it still witnesses (1)S. Answering the above-mentioned question in the negative, we now prove: Theorem 3.6. It is relatively consistent with ZFC that the GCH holds and there ω exists a stationary subset B⊆[ℵω+1] that cannot be thinned out to a stationary A⊆Bon which the sup-function is injective.

ℵω+1 Proof. Start with a model of GCH in which ♦S fails for some stationary S ⊆ Eω (by appealing to the forcing from [17, §2] or by forcing with Q(ℵω+1) from section 1).

ω Claim 3.6.1. B := {X ∈ [ℵω+1] | sup(X) ∈ S} is stationary. <ω Proof. Suppose f :[ℵω+1] →ℵω+1 is a given function. Since {δ ∈ℵω+1 | f“[δ]<ω ⊆ δ} is a club and S is stationary, we may ﬁx some δ ∈ S such that

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1792 MOTI GITIK AND ASSAF RINOT

f“[δ]<ω ⊆ δ. Pick Y ∈ [δ]ω with sup(Y )=δ. Then there exists X ⊇ Y such that |X| = |Y | and f“[X]<ω ⊆ X. Such an X is in B, so we are done.

Now, suppose A⊆Bis stationary on which the sup-function is injective. Then (1)S(A) holds, and as S(A) ⊆ S,also(1)S holds, contradicting Lemma 3.3 and the fact that ♦S fails.

Let us emphasize that the above theorem does not require large cardinals. As- suming large cardinals, one can obtain a stronger counterexample:

Theorem 3.7. It is relatively consistent with the existence of a supercompact car- ω dinal that the GCH holds and there exists a stationary subset B⊆[ℵω+1] which is large in the following two senses: ω (1) if X ∈ [ℵω+1] and sup(X) ∈ S(B),thenX ∈B; ω1 (2) for every stationary A⊆B,thereexistssomeX ∈ [ℵω+1] with ω1 ⊆ X ω and cf(otp(X)) = ω1 such that A∩[X] is stationary. Still, the sup-function is not injective on any stationary subset of B.

Proof. Consider the model from Theorem 1.9: GCH holds, and there exists a sta- ℵ ω+1 ω tionary subset S ⊆ Eω such that ♦S fails. Put B := {X ∈ [ℵω+1] | sup(X) ∈ S}. Then (1) is obvious, and by Lemma 3.3, the sup-function is not injective on any stationary subset of B. Finally, clause (2) follows from the general fact from [5] that if κ is a supercom- ∗ pact cardinal, then V Col(ω1,<κ) |=Reﬂ ([κ+ω+1]ω).

∗ Note that SAPℵ (and hence ) necessarily fails in a model consisting of such ω ℵω B, hence, the high consistency strength.

Discussion. To obtain a ﬁner understanding of Theorem 3.6, we now discuss a more direct argument which allows us to point our ﬁnger at the role of the injectivity of the sup-function.

We start with a model of GCH and let λ := ℵω. We consider the forcing notion X(λ+)asanalternativetoS(λ+) from Section 1. A condition X is in X(λ+)iffX : [α]ω → 2forsomeα<λ+. A condition X is stronger than X iff X ⊇X. To study the injectivity of the sup-function, we also consider X1(λ+), where X∈X1(λ+)iff X∈X(λ+)andforallτ<λ+,theset{X ∈ [λ+]ω |X(X)=1& sup(X)=τ} contains at most a single element. Let S˙(λ+):={sup(ˇX), X | X ∈X ∈X(λ+)&X (X)=1} be the canonical λ+ X + X1 + name for the generic subset of Eω introduced by (λ )andby (λ ). Now, instead of forcing with Q(λ+)=S(λ+)∗KAD(S˙(λ+)), we shall force with P(λ+):= X(λ+) ∗ KAD(S˙(λ+)). For the sake of our comparison, we also define P1(λ+):= X1(λ+) ∗ KAD(S˙(λ+)). The same arguments as in Section 1 shows that P(λ+)andP1(λ+) satisfy the λ++-c.c. and that they have a dense subset in which every increasing sequence of conditions of length <λ+ has an upper bound. We now sketch the changes to be made to the proof of Theorem 1.8 to show that if G∗H is P(λ+)-generic and S := {X ∈ [λ+]ω |∃X∈G(X ∈ dom(X )&X (X)=1)}, then V [G][H] |= S is stationary.

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE FAILURE OF DIAMOND ON A REFLECTING STATIONARY SET 1793

Instead of fixing a name for a club E ⊆ λ+, we fix a name for a function e :[λ+]<ω → λ+. Instead of deciding the value for min(E \ α), we decide e [α]<ω, utilizing the fact that P(λ+) does not add bounded subsets of λ+.Then the analogue of Claim 1.8.10 is that for all g ∈ Z 2, if q is an upper bound of the | increasing sequence qgZn n<ω ,thenq forces that τ is a closure point of e and, moreover, q decides e [τ]<ω to be, say, eg :[τ]<ω → τ. While in Section 1, we didn’t care about E ∩ τ and only focused on the fact that τ is a (closure) point of E. Here we really need to know e [τ]<ω. Notice, however, g0 g1 that if g0 = g1, then it is possible that e = e . This is a subtle point, and we shall get back to it at the end of the discussion. From here, we continue smoothly until we reach to the construction of the sequence of conditions pα = Xα,kα|α<θ and the chain of functions {hα : Z ∩ (εα +1)→ 2 | α<θ}. At the induction base, instead of choosing s0, ∅ ∈ D0 ω withs ¯ ∪{(τ,1)} s0, we first pick an arbitrary cofinal subset X ∈ [τ] and then X ∅ ∈ X ⊇ X∪{¯ } choose 0, D0such that 0 (X, 1) . Once the construction is com- X pleted, we let g := α<θ hα and let ,k be an upper bound for the increasing sequence, pα | α<θ. X | <ω Then ,k is an upper bound of qgZn n<ω , and hence decides e [τ] to be eg :[τ]<ω → τ. Pick a cofinal X ⊆ [τ]ω which is closed under eg.Then X ∪{(X, 1)}\{(X, 0)},k is a legitimate condition because sup(X)=τ is + already forced by X0,k0 to be an element of S(λ ), and it moreover forces that there exists some X ∈X with eg“[X]<ω ⊆ X, as requested. ω So, in V [G][H], S is a stationary subset of [ℵω+1] , GCH holds, and ♦S(S) fails. 1 + + Now, what would have gone wrong had we forced with P (λ ) instead of P(λ )? X¯ ω → ∈ Z X We know that there exists : η<τ [η] 2 such that if g 2and g,kg is a | X ω X ∩ ω minimal upper bound of qgZn n<ω ,thendom( g)=[τ] and g η<τ [η] = X¯. Clearly, there is no way of ensuring that for some g there already exists an X ∈ dom(X¯) which is closed under eg and X¯(X) = 1. However, this is a density argument, so we may consider extensions of X¯. Now, for all g,sinceτ is a closure point of eg, there indeed exists a cofinal subset ω g Xg ∈ [τ] which is closed under e , and it is tempting to just take a function ω X :[τ] → 2 extending X¯ andsuchthatX (Xg) = 1, and to consider X ,kg.So, here is the problem — how do we know that the latter is a legitimate condition?−→ ∈ kg ∈ kg ∩ As τ ci for all i Z, we need, in particular, to establish that xi τ = A i(τ) for all i ∈ Z. In the above construction, we did so by “throwing in” a countable + cofinal subset of τ to X0, thus forcing that τ is in S(λ ). This time, we are allowed to throw in only a single cofinal subset of τ to X0, so we need to throw in a cofinal subset of τ which is closed under eg for all g ∈ Z 2, at once. But, this turns out to be impossible.

4. Open problems Let λ denote a singular cardinal. Probably the most interesting open question in this area is the following question of Shelah:

λ + Question 1. Does 2 = λ imply ♦ λ+ ?DoesGCH imply ♦ λ+ ? Ecf(λ) Ecf(λ)

λ+ By [21] and the fact that Ecf(λ) reﬂects stationarily often, a negative answer to ∗ the above question witnesses the failure of λ, so large cardinals are necessary.

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1794 MOTI GITIK AND ASSAF RINOT

⊆ λ+ Question 2. Suppose S Ecf(λ) reflects stationarily often; must NSλ+ S be non-saturated? ∗ By [15], a negative answer to the above question witnesses the failure of λ λ+ (actually, of SAPλ). Note that by [9], NSλ+ Ecf(λ) is indeed non-saturated. Also note that if one does not require reflection, then by results of Woodin and Foreman § (see [6, 8]), NSℵω+1 S can consistently be saturated for some stationary (non- ℵω+1 reflecting) S ⊆ Eω .

References

[1] James Cummings. Iterated forcing and elementary embeddings. In Matthew Foreman and Akihiro Kanamori, editors, Handbook of set theory, volume II, pages 775–883. Springer- Verlag, 2010. [2] Keith J. Devlin and H˙avard Johnsbr˙aten. The Souslin problem. Lecture Notes in Mathemat- ics, Vol. 405. Springer-Verlag, Berlin, 1974. [3] Todd Eisworth. Successors of Singular Cardinals. Handbook of set theory, pages 1229–1350. Springer, Dordrecht, 2010. MR2768694 [4] Mirna Dˇzamonja and Saharon Shelah. On versions of ♣ on cardinals larger than ℵ1. Math. Japon., 51(1):53–61, 2000. MR1739051 (2001m:03089) [5] M. Foreman, M. Magidor, and S. Shelah. Martin’s maximum, saturated ideals, and nonregular ultrafilters. I. Ann. of Math. (2), 127(1):1–47, 1988. MR924672 (89f:03043) [6] Matthew Foreman. Ideals and generic elementary embeddings. In Matthew Foreman and Akihiro Kanamori, editors, Handbook of set theory, volume II, pages 885–1147. Springer- Verlag, 2010. [7] Matthew Foreman and Menachem Magidor. A very weak square principle. J. Symbolic Logic, 62(1):175–196, 1997. MR1450520 (98i:03062) [8] Moti Gitik and Assaf Sharon. On SCH and the approachability property. Proc. Amer. Math. Soc., 136(1):311–320 (electronic), 2008. MR2350418 (2008m:03102) [9] Moti Gitik and Saharon Shelah. Less saturated ideals. Proc. Amer. Math. Soc., 125(5):1523– 1530, 1997. MR1363421 (97g:03055) [10] R. Björn Jensen. The fine structure of the constructible hierarchy. Ann. Math. Logic, 4:229– 308; erratum, ibid. 4:443, 1972. With a section by Jack Silver. MR0309729 (46:8834) [11] Bernhard König, Paul Larson, and Yasuo Yoshinobu. Guessing clubs in the generalized club filter. Fund. Math., 195(2):177–191, 2007. MR2320769 (2008f:03064) [12] Kenneth Kunen. Set theory, volume 102 of Studies in Logic and the Foundations of Mathe- matics. North-Holland Publishing Co., Amsterdam, 1980. An introduction to independence proofs. MR597342 (82f:03001) [13] Paul Larson. Separating stationary reflection principles. J. Symbolic Logic, 65(1):247–258, 2000. MR1782117 (2001k:03094) [14] Assaf Rinot. A topological reflection principle equivalent to Shelah’s strong hypothesis. Proc. Amer. Math. Soc., 136(12):4413–4416, 2008. MR2431057 (2009h:03058) [15] Assaf Rinot. A relative of the approachability ideal, diamond and non-saturation. J. Symbolic Logic, 75(3):1035–1065. 2010. MR2723781 [16] Saharon Shelah. On successors of singular cardinals. In Logic Colloquium ’78 (Mons, 1978), volume 97 of Stud. Logic Foundations Math., pages 357–380. North-Holland, Amsterdam, 1979. MR567680 (82d:03079) [17] Saharon Shelah. Diamonds, uniformization. J. Symbolic Logic, 49(4):1022–1033, 1984. MR771774 (86g:03083) [18] Saharon Shelah. Cardinal arithmetic,volume29ofOxford Logic Guides. The Clarendon Press Oxford University Press, New York, 1994. Oxford Science Publications. MR1318912 (96e:03001) [19] Saharon Shelah. Diamonds. Proc. Amer. Math. Soc., 138(6):2151–2161, 2010. MR2596054 [20] Charles I. Steinhorn and James H. King. The uniformization property for ℵ2. Israel J. Math., 36(3-4):248–256, 1980. MR597452 (82a:03051)

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE FAILURE OF DIAMOND ON A REFLECTING STATIONARY SET 1795

[21] Martin Zeman. Diamond, GCH and weak square. Proc. Amer. Math. Soc., 138(5):1853–1859, 2010. MR2587470

School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel E-mail address: [email protected] URL: http://www.math.tau.ac.il/~gitik School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel URL: http://www.math.tau.ac.il/~rinot Current address: The Fields Institute for Research in Mathematical Sciences, 222 College Street, Toronto, Ontario, Canada M5T 3J1 E-mail address: [email protected] URL: http://www.assafrinot.com

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use