A global version of a theorem of Ben-David and Magidor ^{∗}^{†}

Arthur W. Apter ^{‡} Department of Mathematics Baruch College of CUNY New York NY 10010 USA awabb@cunyvm.cuny.edu http://math.baruch.cuny.edu/∼apter

James Cummings ^{§} Department of Mathematical Sciences Carnegie Mellon University Pittsburgh PA 15213 USA jcumming@andrew.cmu.edu http://www.math.cmu.edu/users/jcumming

October 4, 1999

Abstract We prove a consistency result about square principles and station-

ary reflection which generalises the result of Ben-David and Magidor [4].

∗1991 Mathematics Subject Classifications: primary 03E35; secondary 03E05, 03E55 †Keywords: Square, weak square, stationary reflection, Radin forcing, large cardinals ‡Supported by the Volkswagen-Stiftung (RiP-program at Oberwolfach). In addition,

this research was partially supported by PSC-CUNY Grants 665337 and 667379.

§Supported by the Volkswagen-Stiftung (RiP-program at Oberwolfach). In addition, this research was partially supported by NSF Grant DMS-9703945.

1 Introduction

In this paper we prove a consistency result about square principles and stationary reflection, which is a generalisation of a theorem from Ben-David and Magidor’s paper [4]. We begin by giving some pertinent facts and definitions.

Definition 1.1 Let κ be an uncountable regular cardinal, let S ⊆ κ be stationary, and let α < κ. Then S reflects at α iff cf(α) > ω and S ∩ α is stationary in α. S reflects iff there exists α < κ such that S reflects at α. S is non-reflecting iff S does not reflect. A sequence S⃗ = ⟨S_{i} : i < β⟩ of stationary subsets of κ reflects simultaneously iff there exists α < κ such that S_{i} reflects at α for every i < β; S⃗ reflects simultaneously to cofinality μ iff there exists α such that cf(α) = μ and S_{i} ∩ α is stationary for all i < β.

Large cardinals can be used to get instances of simultaneous reflection. In particular we will use the following fact (due to Solovay). Since we could not find a reference, we sketch the proof.

Fact 1.2 If κ is κ^{+} supercompact and S⃗ is a sequence of stationary subsets of { α < κ^{+} | cf(α) < κ } such that lh(S⃗) < κ, then there are unboundedly many μ < κ such that

1. μ is the successor of an inaccessible.

2. S⃗ reflects simultaneously to cofinality μ.

Proof: Let S⃗ = ⟨S_{α} : α < ν⟩ where ν < κ and each S_{α} is a stationary subset of { α < κ^{+} | cf(α) < κ }. Let ρ < κ. Let j : V ⟶ M be an elementary embedding such that crit(j) = κ, j(κ) > κ^{+} and ^{κ^{+}}M ⊆ M. Let λ = sup(j“κ^{+}), where it follows from the closure of M that cf_{M}(λ) = κ^{+} = κ_{M}^{+} and λ < j(κ^{+}).

j(S⃗) = ⟨j(S_{α}) : α < ν⟩ and j(ρ) = ρ < κ, and also κ is inaccessible in M. It will therefore suffice to show that

M ⊨ “j(S_{α}) ∩ λ is stationary for every α < ν”

and then appeal to the elementarity of j to see that

V ⊨ “there is an inaccessible ζ > ρ such that S⃗ reflects to cofinality ζ^{+}”

Clearly j“S_{α} ⊆ j(S_{α}) ∩ λ. If D ⊆ λ is club then it is easy to see that j^{−1}“D is < κ-club in κ^{+}, so that there is some β ∈ S_{α} with j(β) ∈ D. Thus j(S_{α}) ∩ λ is stationary (even in V!) and we are done.

♦

Definition 1.3 Let κ be an infinite cardinal, and let λ ≤ κ be a cardinal. Then a □_{κ,λ}-sequence is a sequence ⟨C_{α} : lim(α), α < κ^{+}⟩ such that

1. 1 ≤ |C_{α}| ≤ λ.

2. For every C ∈ C_{α}

(a) C is club in α.

(b) o.t.(C) ≤ κ.

(c) For all β ∈ lim(C), C ∩ β ∈ C_{β}.

We say that “□_{κ,λ} holds” iff there exists a □_{κ,λ}-sequence. A □_{κ,<λ}- sequence is defined as above with clause 1 replaced by “1 ≤ |C_{α}| < λ”.

The principle □_{κ,λ} was defined by Schimmerling [14] in his work on the core model for one Woodin cardinal. It is a common generalisation of two principles studied by Jensen [8], the principles □_{κ} = □_{κ,1} and □_{κ}^{∗} = □_{κ,κ}.

With these definitions in hand we are ready to state the main theorem of this paper.

Theorem 1 Let GCH hold and let κ be a κ^{+5}-supercompact cardinal. Then there exists a forcing poset ℙ such that in V^{ℙ}

1. κ is κ^{+5}-supercompact.

2. For every singular cardinal λ < κ

(a) There exists S ⊆ λ^{+} stationary such that if S⃗ is a sequence of stationary subsets of S and lh(S⃗) < cf(λ) then S⃗ reflects to cofinality μ for unboundedly many μ < λ.

(b) The combinatorial principle □_{λ,cf(λ)} holds.

Notice that truncating the generic extension by ℙ at κ will give a set model of the theory “ZFC + there exists a proper class of cardinals δ which are δ^{+4}-supercompact + every singular cardinal λ has the above properties”.

This theorem is (as we explain below) a generalisation of the following result of Ben-David and Magidor.

Fact 1.4 (Ben-David and Magidor [4]) If κ is κ^{+}-supercompact there is a generic extension in which

1. κ = ℵ_{ω}, κ^{+} = ℵ_{ω+1}.

2. □_{ℵ_{ω}} fails.

3. □_{ℵ_{ω}}^{∗} holds.

The model of [4] is built by using a modification of Magidor’s “supercompact Prikry forcing with interleaved forcing” from [10]. Apter and Henle [2] showed that a somewhat similar proof can be made to work using κ which is only κ^{+}-strongly compact. Using the ideas of [2] and methods of Gitik for iterating Prikry-type forcing, Apter was able to show

Fact 1.5 (Apter [1]) Con(ZFC + GCH + κ is an inaccessible limit of cardinals δ which are δ^{+} strongly compact) ⟹ Con(ZFC + κ is inaccessible + For every cardinal λ < κ, there exists a stationary S ⊆ λ^{+ω+1} such that if S′ ⊆ S is stationary, then S′ reflects at δ for unboundedly many δ < λ^{+ω+1} (so ¬□_{λ^{+ω}}) + For every cardinal λ < κ, □_{λ^{+ω},ω}).

In the model of [4] there is a uniform ultrafilter on ℵ_{ω+1} which is λindecomposable for ℵ_{0} < λ < ℵ_{ω}. This implies [4, Lemma 2.2] that every stationary subset of S = { α < ℵ_{ω+1} | cf(α) > ω } reflects, and from this stationary reflection principle it follows [4, Lemma 2.1] that □_{ℵ_{ω}} fails.

In the model of [4] the transfer principle (ℵ_{1}, ℵ_{0}) ⟶ (ℵ_{ω+1}, ℵ_{ω}) holds. It follows from this transfer principle that there is a special ℵ_{ω+1}-Aronszajn tree, and hence by work of Jensen [8] that □_{ℵ_{ω}}^{∗} holds.

Our model for Theorem 1 is built using Foreman and Woodin’s “supercompact Radin forcing with interleaved forcing” from their consistency proof [7] for “ZFC + GCH fails everywhere”. Their work builds on ideas from [10] and Radin’s paper [13].

It is worth remarking that the similarity between our model for Theorem 1 and the model of [4] is even more pronounced than has been shown so far. The methods of Section 4 of this paper can be used to show that in the model of [4] any finite sequence of stationary subsets of S reflects simultaneously, and it follows from Fact 1.8 that □_{ℵ_{ω},ω} holds in the model of [4].

The machinery that we use to get □_{λ,cf(λ)} to hold in the model of Theorem 1 is based on the following distinctive property of the forcing ℙ: every singular cardinal of V^{ℙ} below κ is inaccessible in V. This idea originates in the proof of Fact 1.8. Before stating Fact 1.8 we make a technical definition.

Definition 1.6 Let κ be regular and uncountable. X ⊆ κ is a > ω-club subset of κ if and only if X is unbounded in κ and X is closed under suprema of uncountable cofinality.

Remark 1.7 It is easy to see that the > ω-club subsets of κ generate a normal filter on κ. We shall refer to this filter as the “> ω-club filter”.

Fact 1.8 (Cummings and Schimmerling [6]) Let V ⊨ “κ is inaccessible”. Let W ⊇ V be a model such that

1. κ and κ^{+} are cardinals in W.

2. There is E ∈ W such that

(a) o.t.(E) = ω.

(b) For all D ∈ V, if V ⊨ “D is a > ω-club subset of κ” then E ∩D ≠ ∅.

Then W ⊨ □_{κ,ω}.

Remark 1.9 In particular if κ is measurable in V, and W is a generic extension by Prikry forcing at κ, then □_{κ,ω} holds in W.

In the context of Theorem 1 we are changing cofinalities to values other than ω, and will have to prepare the ground model in order to get an analogue to Fact 1.8. This preparation uses ideas of Baumgartner.

We now quote some facts whose proofs can be found in [5]. Taken together they indicate that the result of Theorem 1 is close to being optimal.

Fact 1.10 (Cummings, Foreman and Magidor [5]) Let λ be singular. If □_{λ,μ} holds for μ < λ and S ⊆ λ^{+} is stationary then there exists T⃗ = ⟨T_{i} : i < cf(λ)⟩ such that each T_{i} is a stationary subset of S and T⃗ does not reflect simultaneously to cofinality μ for any μ with cf(λ) < μ < λ.

Fact 1.11 (Schimmerling) Let λ be singular. If □_{λ,μ} holds for μ < cf(λ) and S ⊆ λ^{+} is stationary, then there exists T ⊆ λ stationary and δ < λ such that T does not reflect at any point of cofinality greater than δ.

Our forcing notation is fairly standard. We write p ≤ q when p is a stronger condition than q. Add(κ, λ) is the poset to add λ Cohen subsets of κ, Coll(κ, < λ) is the Levy collapse to make every ordinal in [κ, λ) have cardinality κ. A poset is κ-closed when every descending sequence of length less than κ has a lower bound, and is κ-directed closed when every directed subset of size less than κ has a lower bound. RO(ℙ) is the complete Boolean algebra corresponding to the poset ℙ.

The second author would like to thank Matt Foreman for telling him Fact 2.7, and showing him a proof that if κ is κ^{+}-supercompact then any finite sequence of stationary subsets of κ^{+} reflects after doing Prikry forcing at κ.

2 Building squares

In this section we discuss the machinery we use to get □_{λ,cf(λ)}-sequences in the model of Theorem 1. We will define a combinatorial principle □_{κ}^{B} (due to Baumgartner) and prove the following lemma, which is a generalisation of Fact 1.8. The point here is that when the cofinality of κ is changed to be uncountable we need some amount of square in the ground model to see that the desired form of square holds in the extension.

Lemma 2.1 Let V ⊨ “κ is inaccessible and □_{κ}^{B}”. Let W ⊇ V be a model such that

1. κ and κ^{+} are cardinals in W and λ is regular in W.

2. There is E ∈ W such that

(a) E is closed and unbounded in κ.

(b) o.t.(E) = λ.

(c) For all D ∈ V, if V ⊨ “D is a > ω-club subset of κ” then E∩D ≠ ∅.

Then W ⊨ □_{κ,λ}.

Remark 2.2 If κ is measurable and A ⊆ κ is a > ω-club subset of κ then A ∈ U for every normal measure U on κ. It follows that if C is a generic club of order type λ added by any of the standard forcings for changing cofinality (Prikry forcing [12] [where the generic set C, having order type ω, is trivially considered as being club in κ], Magidor forcing [11] or Radin forcing [13]) then the technical condition 2 in the statement of Lemma 2.1 holds; in fact C will be eventually contained in every > ω-club subset of κ from the ground model. This will also hold for the modified Radin forcing ℙ_{a} which we define in the proof of Theorem 1.

It is also easy to see that condition 2 holds if W ⊨ cf(κ) = λ ≥ ℵ_{2}. We do not know whether it holds always when W ⊨ cf(κ) = λ = ℵ_{1} or W ⊨ cf(κ) = λ = ω.

Before giving the proof of this lemma we discuss some ideas that are used in the proof, define □_{κ}^{B} and show that this principle is consistent with κ being a large cardinal.

2.1 Good matrices

We will use the idea of a good matrix, which comes from the proof of Fact 1.8. In the interest of making this paper self-contained we have reproduced some results from [6].

Definition 2.3 Let κ be inaccessible and let S = { α < κ^{+} | cf(α) < κ }. A good matrix is an array of sets

⟨C_{αi} : α ∈ S, i ∈ X_{α}⟩

such that

1. C_{αi} is club in α.

2. X_{α} contains a > ω-club subset of κ.

3. o.t.(C_{αi}) < κ.

4. If i ∈ X_{α} and β ∈ lim C_{αi} then i ∈ X_{β} and C_{αi} ∩ β = C_{βi}.

5. If i, j ∈ X_{α} and i < j then C_{αi} ⊆ C_{αj}.

6. If α, β ∈ S with β < α then β ∈ lim C_{αi} for some i ∈ X_{α} (and thus for all large i ∈ X_{α} by the preceding clause).

Fact 2.4 (Cummings and Schimmerling [6]) Let the cardinal κ be inaccessible. Then there exists a good matrix for κ.

Remark 2.5 When building a good matrix the hardest stages are those of the form β + ω where cf(β) = κ, which are treated in Case 5 of the inductive definition below. To prepare for these stages, we need to make sure in Case 4 that if ω < cf(α) < κ then X_{α} is as large as possible.

Accordingly when ω < cf(α) < κ we choose X_{α} as the “maximally fat” set of indices i on which it is possible to define C_{αi}. X_{α} will be in the > ω-club filter on κ but is not necessarily an actual > ω-club subset of κ.

Proof:[Fact 2.4] We construct a good matrix by induction on α ∈ S.

Case 1: α = ω. We set X_{ω} = κ and C_{ωi} = ω for all i.

Case 2: α = β + ω for limit β with cf(β) < κ (that is to say β ∈ S). We set X_{α} = X_{β} and C_{αi} = C_{βi} ∪ [β, α) for all i ∈ X_{α}.

Clearly C_{αi} is club in α. X_{α} = X_{β} and so X_{α} is in the > ω-club filter, and o.t.(C_{αi}) = o.t.(C_{βi}) + ω < κ. If i ∈ X_{α} and γ ∈ lim C_{αi} then either γ ∈ lim C_{βi} or γ = β. In the former case we have by induction that i ∈ X_{γ} and C_{γi} = C_{βi} ∩ γ, in the latter that i ∈ X_{β} = X_{γ} and C_{γi} = C_{βi}: in either case C_{αi} ∩ γ = C_{γi}.

If i, j ∈ X_{α} with i < j then by induction C_{βi} ⊆ C_{βj}, so that C_{αi} ⊆ C_{αj}. Finally if γ ∈ S ∩ α then either γ ∈ S ∩ β or γ = β: if γ ∈ S ∩ β then by induction γ ∈ lim C_{βi} for some i and then γ ∈ lim C_{αi} for the same i, while if γ = β then γ ∈ lim C_{αi} for every i ∈ X_{α}.

Case 3: cf(α) = ω and α is a limit of limit ordinals. We choose ⟨α_{m} : m < ω⟩ an increasing sequence of ordinals in S which is cofinal in α. We set

X_{α} = { i < κ | ∀m < ω i ∈ X_{α_{m}} and ∀m < n < ω α_{m} ∈ lim C_{α_{n}i} }.

X_{α} is in the > ω-club filter because it is a final segment of ⋂_{j} X_{α_{j}}.

We observe that if i ∈ X_{α} then C_{α_{m}i} = C_{α_{n}i} ∩ α_{j} for all m < n < ω. We now set C_{αi} = ⋃_{m} C_{α_{m}i} for all i ∈ X_{α}.

C_{αi} is club in α because every initial segment is an initial segment of C_{α_{m}i} for some m. A similar argument shows that o.t.(C_{αi}) < κ. If β ∈ lim C_{αi} then β ∈ lim C_{α_{m}i} for some m, and by induction i ∈ X_{β} and C_{βi} = C_{α_{m}i}∩β = C_{αi} ∩ β.

If i, j ∈ X_{α} with i < j then by induction C_{α_{m}i} ⊆ C_{α_{m}j} for all m < ω, so that C_{αi} ⊆ C_{αj}. Finally if β ∈ S ∩ α then β ∈ S ∩ α_{m} for some m, and so by induction β ∈ lim C_{α_{m}i} for all large i ∈ X_{α_{m}}; it follows that β ∈ lim C_{αi} for any large enough i ∈ X_{α}.

Case 4: ω < cf(α) < κ. Let cf(α) = ρ say. As in Case 3 we fix ⟨α_{m} : m < ρ⟩ an increasing and continuous sequence of members of S which is cofinal in α. We define

Y_{α} = { i < κ | ∀m < ρ i ∈ X_{α_{m}} and ∀m < n < ρ α_{m} ∈ lim C_{α_{n}i} }.

Note that the precise nature of Y_{α} depends on the sequence ⟨α_{m} : m < ρ⟩ used in its definition. Exactly as in Case 3 Y_{α} is in the > ω-club filter, and if i ∈ Y_{α} then C_{α_{m}i} = C_{α_{n}i} ∩ α_{m} for all m < n < ρ.

We now let

X_{α} = { i < κ | ∃E club in α ∀γ ∈ lim(E) (i ∈ X_{γ} and E ∩ γ = C_{γi}) }.

If i ∈ Y_{α} and we let E = ⋃_{m} C_{α_{m}i} then it is easy to check that E witnesses i ∈ X_{α}, so that Y_{α} ⊆ X_{α}. We observe that

• X_{α} is independent of the choice of the sequence ⟨α_{m} : m < ρ⟩.

• By its definition and clause 4 in the definition of a good matrix, X_{α} is the “maximally fat” set of indices i for which we can hope to define C_{αi}.

Suppose that i ∈ X_{α} and E, E′ are both clubs in α witnessing this. Then E ∩ E′ is club in α and

E = ⋃_{γ∈lim(E∩E′)} C_{γi} = E′.

For each i ∈ X_{α}, we now define C_{αi} to be the unique E which is club in α and is such that ∀γ ∈ lim(E) E ∩ γ = C_{γi}. Notice that if i ∈ Y_{α} then automatically C_{αi} = ⋃_{m} C_{α_{m}i}.

Since every initial segment of C_{αi} is an initial segment of C_{γi} for some γ < α, o.t.(C_{αi}) < κ. If β ∈ lim C_{αi} then β ∈ lim C_{γi} for some γ ∈ lim C_{αi}, and we have by induction that i ∈ X_{β} and C_{βi} = C_{γi} ∩ β = C_{αi} ∩ β.

Let i, j ∈ X_{α} with i < j. Let C_{αi} = E and C_{αj} = F. Then

E = ⋃_{γ∈lim(E∩F)} C_{γi} ⊆ ⋃_{γ∈lim(E∩F)} C_{γj} = F,

that is to say that C_{αi} ⊆ C_{αj}. Finally we may argue as in Case 3 that S ∩ α ⊆ ⋃_{i∈Y_{α}} lim C_{αi}, which suffices since Y_{α} ⊆ X_{α}.

Case 5: α = β + ω where cf(β) = κ. We fix ⟨β_{i} : i < κ⟩ an increasing and continuous sequence of members of S which is cofinal in β. Let

Z = { i < κ | ∀j < i i ∈ X_{βj} and ∀j < k < i β_{j} ∈ lim C_{β_{k}i} }.

We claim that Z is in the > ω-club filter. To see this first observe that if D = { i < κ | ∀j < i i ∈ X_{β_{j}} } then D is a diagonal intersection of sets in the > ω-club filter, and therefore is in the > ω-club filter. Define F : [κ]^{2} ⟶ κ by setting F (j, k) equal to the least i ∈ X_{β_{k}} with β_{j} ∈ lim C_{β_{k}i}, and let C be the club set of i < κ such that F “[i]^{2} ⊆ i. If i ∈ D ∩ C then

1. Since i ∈ D, ∀j < i i ∈ X_{β_{j}}.

2. If j, k < i then since i ∈ C we have F (j, k) < i, and by definition F (j, k) ∈ X_{β_{k}} and β_{j} ∈ lim C_{β_{k}F (j,k)}. Since i ∈ D we also have i ∈ X_{β_{k}}, and so by the properties of a good matrix C_{β_{k}F (j,k)} ⊆ C_{β_{k}i} and so β_{j} ∈ lim C_{β_{k}i}.

It follows that D ∩ C ⊆ Z, and D ∩ C is easily seen to be in the > ω-club filter.

We let X_{α} = { ζ ∈ D ∩ C | cf(ζ) > ω }. Let i ∈ X_{α} and consider the construction at level β_{i}; since cf(i) > ω and the sequence ⟨β_{n} : n < κ⟩ is continuous, cf(β_{i}) = cf(i) > ω and the relevant clause of the definition is Case 4.

If we let E = ⋃_{j<i} C_{β_{j}i} then the fact that i ∈ Z and the coherence properties of the good matrix imply that ∀γ ∈ lim(E) E ∩ γ = C_{γi}, so that by the definition of X_{β_{i}} and C_{β_{i}i} from Case 4 i ∈ X_{β_{i}} and C_{β_{i}i} = ⋃_{j<i} C_{β_{j}i}. Note that if i ∈ Z, the sequence B = ⟨β_{j} : j < i⟩ is very likely different from the sequence A = ⟨α_{m} : m < cf(i)⟩ used in the definition of Y_{β_{i}}. Even though A and B agree on a club of order type cf(i), this isn’t necessarily enough to allow us to infer that i ∈ Y_{α} for every α ∈ A, something that would be critical in allowing us to infer that i ∈ Y_{β_{i}}. The “maximal fatness” of X_{β_{i}}, however, ensures this isn’t a problem and that i ∈ X_{β_{i}}.

We define

C_{αi} = C_{β_{i}i} ∪ {β_{i}} ∪ [β, α).

Clearly C_{αi} is club in α, and o.t.(C_{αi}) = o.t.(C_{β_{i}i}) + ω < κ. If γ ∈ lim C_{αi} then either γ ∈ lim C_{β_{i}i} or γ = β_{i}, and in either case it is easy to see that i ∈ X_{γ} and C_{γi} = C_{β_{i}i} ∩ γ = C_{αi} ∩ γ.

Let i, j ∈ X_{α} with i < j. By induction

C_{β_{i}i} = ⋃_{k<i} C_{β_{k}i} ⊆ ⋃_{k<i} C_{β_{k}j} ⊆ ⋃_{k<j} C_{β_{k}j} = C_{β_{j}j}.

Since C_{β_{j}j} is club in β_{j} and C_{β_{i}i} is cofinal in β_{i}, it follows that β_{i} ∈ C_{β_{j}j}. Therefore by definition C_{αi} ⊆ C_{αj}.

Finally let γ ∈ S ∩α, and observe that since β ∉ S we have S ∩α = S ∩β. Find i such that γ < β_{i}, and then j ∈ X_{α} such that i < j and γ ∈ lim C_{β_{i}j}. Since C_{β_{j}j} = ⋃_{k<j} C_{β_{k}j}, γ ∈ lim C_{β_{j}j}.

This concludes the proof.

♦

2.2 The principle □_{κ}^{B}

The following version of □_{κ} was studied by Baumgartner. Its main interest for us is that unlike the original □_{κ} principle it is consistent for κ supercompact.

Definition 2.6 Let κ be regular. A □_{κ}^{B}-sequence is a sequence ⟨C_{α} : α ∈ T ⟩ where

1. T is a set of limit ordinals less than κ^{+}.

2. { α < κ^{+} | cf(α) = κ } ⊆ T.

3. For all α ∈ T, C_{α} is a club subset of α with o.t.(C_{α}) ≤ κ.

4. If α ∈ T and β ∈ lim(C_{α}) then β ∈ T and C_{β} = C_{α} ∩ β.

As usual we say “□_{κ}^{B} holds” if there is a □_{κ}^{B}-sequence.

Fact 2.7 (Baumgartner [3]) Let κ be regular. Then there exists a forcing poset ℙ = ℙ(κ) such that

1. ℙ is κ-directed closed.

2. ℙ is strategically closed for the game of length κ + 1.

3. ⊩_{ℙ} “□_{κ}^{B} holds”.

Proof: We force with the set of initial segments of successor length of such a sequence. More formally let ℙ be the set of sequences ⟨C_{α} : α ∈ s⟩ where

1. s is a bounded set of limit ordinals less than κ^{+}, with a maximal element γ.

2. { α < γ | cf(α) = κ } ⊆ s.

3. For all α ∈ s, C_{α} is a club subset of α with o.t.(C_{α}) ≤ κ.

4. If α ∈ s and β ∈ lim(C_{α}) then β ∈ s and C_{β} = C_{α} ∩ β.

ℙ is ordered as follows: if p = ⟨C_{α} : α ∈ s⟩ and q = ⟨D_{α} : α ∈ t⟩ then p ≤ q iff t = s ∩ (max(t) + 1) and C_{α} = D_{α} for all α ∈ t.

Now we take the claims made in the theorem one by one. We adopt the convention “a sequence is a function is a set of ordered pairs”, so that ⟨C_{α} : α ∈ s⟩ = { (α, C_{α}) | α ∈ s }.

1. Since the partial ordering on ℙ is treelike it suffices to show that ℙ is κ-closed. Let ⟨p_{i} : i < ζ⟩ be a decreasing ζ-sequence of conditions for some ζ < κ, and let s_{i} = dom(p_{i}) and γ_{i} = max(s_{i}). Let β^{∗} = sup_{i} γ_{i}, β = β^{∗}+ω, c = [β^{∗}, β). Define p = ⋃_{i} p_{i}∪{(β, c)}. It is routine to check that p is a condition, the key point being that since cf(β^{∗}) = cf(ζ) < κ we are under no obligation to put β^{∗} into the domain of p.

2. We describe a winning strategy for player II in the game of length κ+1 played on ℙ. Let the move made at stage i be p_{i} = ⟨C_{α} : α ∈ s_{i}⟩, with max(s_{i}) = γ_{i}. Player II will play to guarantee that if i, j are even with i < j then γ_{i} ∈ lim(C_{γ_{j}}) and C_{γ_{i}} = C_{γ_{j}} ∩ i. Suppose we have reached an even stage α and it is the turn of Player II.

Case I. α is a successor ordinal, say α = β + 2. We set γ_{α} = γ_{β+1} + ω, C_{γ_{α}} = C_{γ_{β}} ∪ {γ_{β}} ∪ [γ_{β+1}, γ_{α}), s_{α} = s_{β+1} ∪ {γ_{α}}, and finally p_{α} = p_{β+1} ∪ {(γ_{α}, C_{γ_{α}})}. Case II. α is a limit ordinal with α ≤ κ. We set γ_{α} = sup_{j<α} γ_{j}, C_{γ_{α}} = { γ_{j} | j < α }, and finally p_{α} = ⋃_{γ<α} p_{γ} ⋃{(γ_{α}, C_{γ_{α}})}.

It is routine to check that this is a successful strategy for player II. The key points are that II always plays a set of order type less than or equal to κ, and that if α is limit and β ∈ lim(C_{γ_{α}}) then β = γ_{δ} for limit δ < α and so

C_{γ_{δ}} = { γ_{j} | j < δ } = C_{γ_{α}} ∩ γ_{δ}.

3. It is easy to see that for all p ∈ ℙ and β < κ^{+} there exists q ≤ p such that max(dom(q)) > β. The proof is a routine induction on β, using the same idea as in the proof of strategic closure to get through stages of cofinality κ. By strategic closure, ℙ adds no κ-sequence of ordinals so preserves all cardinals and cofinalities up to κ^{+}. It follows that if G is ℙ-generic then ⋃ G is a □_{κ}^{B}-sequence in V [G].

♦

We will need to know later that □_{κ}^{B} is consistent with κ being a large cardinal.

We will use the following basic facts, which originated in Silver’s work on Reverse Easton forcing.

Fact 2.8 Let M and N be models of ZFC, let k : M ⟶ N be an elementary embedding. Let ℙ ∈ M be a forcing poset and let G and H be such that

1. G is ℙ-generic over M.

2. H is k(ℙ)-generic over N.

3. ∀p ∈ G k(p) ∈ H.

Then there is a unique elementary embedding k^{+} : M[G] ⟶ N[H] such that k^{+} extends k and k^{+}(G) = H.

Fact 2.9 Let M and N be models of ZFC with M ⊆ N. Let λ be an Ncardinal. Let ℙ ∈ M be a forcing poset such that

1. N ⊨ “ℙ is λ-closed”.

2. { A ∈ M | A is a maximal antichain in ℙ } has cardinality λ in N.

Then for every p ∈ ℙ there exists H ∈ N such that p ∈ H and H is ℙ-generic over M.

Theorem 2 Let κ be κ^{+n}-supercompact for some n with 0 < n < ω, and let GCH hold. Let ℙ_{κ+1} be the Reverse Easton iteration of length κ + 1 where at every inaccessible α ≤ κ we force with ℙ(α)_{V^{ℙα}} (where ℙ(α) is the forcing to add □_{α}^{B} defined in Fact 2.7).

Then in V^{ℙ_{κ+1}}

1. V -cardinals greater than or equal to κ are preserved.

2. GCH holds.

3. κ is κ^{+n}-supercompact.

Proof: ℙ_{κ} is a κ-c.c. forcing of size κ, so forcing with ℙ_{κ} preserves cardinals greater than or equal to κ and GCH holds at and above κ in V^{ℙ_{κ}}.

We saw in the proof of Fact 2.7 that ℙ(κ) adds no κ-sequences of ordinals, so it preserves all cardinals and cofinalities up to κ^{+}. 2^{κ} = κ^{+} in V^{ℙ_{κ}} so ℙ(κ) has cardinality κ^{+}, hence it is κ^{++}-c.c. and preserves all cardinals and cofinalities above κ^{+}. Standard arguments show that GCH still holds at and above κ in V^{ℙ_{κ+1}}, and a routine induction using the above analysis shows that GCH holds below κ in V^{ℙ_{κ+1}}.

To show the preservation of supercompactness, suppose that G is ℙ_{κ}- generic over V and g is ℙ(κ)-generic over V [G]. Fix j : V ⟶ M which is the ultrapower map defined from some supercompactness measure on P_{κ}(κ^{+n}). Notice that κ^{+n+1} = κ_{M}^{+n+1} < j(κ). In particular if we let η be the least M-inaccessible greater than κ then η > κ^{+n+1}. |P_{κ}κ^{+n}| = κ^{+n}, and it follows using GCH that j(κ^{++}) < κ^{+n+2}.

j(ℙ_{κ+1}) is an iterated forcing in M, and the resemblance between V and M implies that ℙ_{κ+1} is an initial segment of j(ℙ_{κ+1}). By an easy chain condition argument V [G][g] ⊨ ^{κ^{+n}}M[G][g] ⊆ M[G][g].

Let ℝ be the forcing j(ℙ_{κ})/G ∗ g. Then in M[G][g] the forcing ℝ is η-closed and has j(κ) maximal antichains. |j(κ)| = κ^{+n+1}, so working in V [G][g] we may build a descending κ^{+n+1}-chain of conditions in ℝ to decide each antichain in M[G][g]. Thus we may build H ∈ V [G][g] which is ℝgeneric over M[G][g].

M[G][g][H] ⊆ V [G][g], and V [G][g] ⊨ ^{κ^{+n}}M[G][g][H] ⊆ M[G][g][H]. Since j“G ⊆ G ∗ g ∗ H, working in V [G][g] we may lift j to get a new embedding (which we also denote by j, without risk of confusion) j : V [G] ⟶ M[G][g][H].

Now we use Silver’s “master condition” argument. Consider p =_{def} ⋃ j“g. p ∈ M[G][g][H] and p is a sequence ⟨D_{β} : β ∈ A⟩, where A is a bounded subset of j(κ^{+}) because

M[G][g][H] ⊨ cf(sup(j“κ^{+})) = κ^{+} < j(κ^{+}).

The only way in which p falls short of being a condition in ℙ(j(κ)) is that A does not have a largest element. Since sup(A) = sup(j“κ^{+}) and M ⊨ cf(sup(j“κ^{+})) = κ^{+} < j(κ), we are not obliged to define a club at sup(A). Accordingly we let γ = sup(A)+ω, let D_{γ} = { sup(A) + n | n < ω } and A^{∗} = A∪{γ}. Now p^{∗} =_{def} ⟨D_{β} : β ∈ A^{∗}⟩ is a condition and ∀q ∈ g p^{∗} ≤ j(q).

In M[G][g][H], the forcing ℙ(j(κ)) is κ^{+n+1}-closed and has j(κ^{++}) antichains. Since |j(κ^{++})| = κ^{+n+1} and

V [G][g] ⊨ ^{κ^{+n}}M[G][g][H] ⊆ M[G][g][H],

we may build a chain of conditions in ℙ(j(κ)) to decide each antichain in M[G][g][H], and so build h ∈ V [G][g] which is ℙ(j(κ))-generic over the model M[G][g][H]. Taking the first element of that chain to be p^{∗} we may assume that p^{∗} ∈ h and hence that j“g ⊆ h.

So we may lift j again to get j : V [G][g] ⟶ M[G][g][H][h]. This map is defined in V [G][g] and witnesses that κ is κ^{+n}-supercompact in V [G][g].

♦

2.3 Proof of Lemma 2.1

We are now ready to prove Lemma 2.1. Proof:[Lemma 2.1] As before, let

S = { α < κ^{+} | cf(α) < κ }.

Let

⟨C_{αi} : α ∈ S, i ∈ X_{α}⟩

be a good matrix and let ⟨D_{α} : α ∈ T ⟩ witness the truth of □_{κ}^{B}. We will define a □_{κ,λ}-sequence ⟨E_{α} : α < κ^{+}⟩ in W. Let E ∈ W be a club in κ which has order type λ and meets every > ω-club from V, and let E = ⟨κ_{i} : i < λ⟩ with κ_{i} increasing.

Case 1: V ⊨ cf(α) < κ, α ∉ T.

Let E_{α} = { C_{ακ_{j}} | κ_{j} ∈ X_{α} }. Case 2: V ⊨ cf(α) < κ, α ∈ T.

Let E_{α} = { C_{ακ_{j}} | κ_{j} ∈ X_{α} } ∪ {D_{α}}. Case 3: V ⊨ cf(α) = κ (which implies that α ∈ T).

Let E_{α} = {D_{α}}.

We verify that this is a □_{κ,λ}-sequence. Clearly each E_{α} is a family of closed unbounded subsets of α which have order type at most κ, and the cardinality of E_{α} is at most λ. Each E_{α} is non-empty by our assumption on the club E.

Let C ∈ E_{α} and let β ∈ lim C. There are two possibilities.

1. V ⊨ cf(α) < κ and C = C_{ακ_{j}} for some j < λ with κ_{j} ∈ X_{α}. By the coherence properties of a good matrix, we have κ_{j} ∈ X_{β} and C ∩ β = C_{βκ_{j}}. The definition of E_{β} now tells us that C ∩ β ∈ E_{β}.

2. C = D_{α} for some α ∈ T. By the coherence properties of a □_{κ}^{B}-sequence, β ∈ T and C ∩ β = D_{β}. Again the definition of E_{β} tells us that C ∩ β ∈ E_{β}.

♦

Remark 2.10 If λ = ω (that is to say we are in the case of Fact 1.8) then we do not need the assumption that □_{κ}^{B} holds in V. In this case we just set

E_{α} = { C_{ακ_{j}} | κ_{j} ∈ X_{α} }

for all α with V ⊨ cf(α) < κ, and set E_{α} = {X_{α}} for some X_{α} cofinal in κ with order type ω when V ⊨ cf(α) = κ.

3 Radin forcing

We assume in this section that GCH holds and that κ is a κ^{+5}-supercompact cardinal. We describe a partial ordering ℙ which will have the properties that

1. κ is κ^{+5}-supercompact in V^{ℙ}.

2. If μ < κ is any limit cardinal in V^{ℙ} then V ⊨ “μ is μ^{+}-supercompact”.

The forcing ℙ will be obtained by a minor modification in the construction of Foreman and Woodin’s model where GCH fails everywhere [7].

In [7] Foreman and Woodin are working with a cardinal κ which is ℶ_{ω}(κ)- supercompact and is such that ℶ_{n}(κ) is weakly inaccessible for each n. They show (see Section 7 of their paper) that it is possible to construct some forcing ℙ such that

1. ℙ preserves cardinals.

2. κ is ℶ_{3}(κ)-supercompact in V^{ℙ}.

3. The forcing ℙ adds a club set C ⊆ κ such that if C is enumerated in increasing order as ⟨κ_{i} : i < κ⟩ then

(a) Each κ_{i} is a large cardinal in V.

(b) The forcing adds an Add(ℶ_{4}(κ_{i}), κ_{i+1})-generic object to V for each i < κ.

Truncating V^{ℙ} at κ then gives a model in which GCH fails everywhere. We will build ℙ in a very similar way so that

1. κ is κ^{+5}-supercompact in V^{ℙ}.

2. The forcing ℙ adds a club set C ⊆ κ such that if C is enumerated in increasing order as ⟨κ_{i} : i < κ⟩ then

(a) κ_{0} = ℵ_{0}, κ_{i} is a large cardinal in V for i > 0.

(b) The forcing adds a Coll(κ_{i}^{+6}, < κ_{i+1})-generic object to V for each i < κ.

(c) Cardinals not lying in the intervals (κ_{i}^{+6}, κ_{i+1}) are preserved.

We will assume that the reader has a copy of [7] to hand. Rather than reproducing the excellent exposition in that paper with minor changes, we will mostly just give references to that paper and leave the reader to fill in the details. We attempt to use notation which parallels that of [7]. In particular we define

P_{γ}δ = { X ⊆ δ | o.t.(X) < γ, X ∩ γ ∈ γ }.

Given x, y ∈ P_{γ}δ we define a relation “x ≺ y” by

x ≺ y ⟺ (x ⊆ y ∧ o.t.(x) < y ∩ γ).

This definition really depends on the values of γ and δ but these should always be clear from the context.

We begin by setting σ = κ^{+5}, ρ = κ^{+4}, λ^{∗} = κ^{+3}. For the rest of this paper we fix j such that

1. crit(j) = κ, j(κ) > σ, ^{σ}M ⊆ M.

2. j arises as the ultrapower map associated with some normal fine ultrafilter U on the set P_{κ}σ.

We will define a sequence M⃗ = ⟨M_{α} : α < λ^{∗}⟩ such that M_{0} = j“λ^{∗} and for α > 0 M_{α} is a measure on

Z = P_{κ}λ^{∗} × V_{κ} × V_{κ}.

The definition of M⃗ will be recursive, and will go as follows: for 0 < α < λ^{∗} we will define some function g_{α} such that dom(g_{α}) = A_{α} ∈ M_{α} and g_{α} : A_{α} → V_{κ}, and will then set

M_{α} = { X ⊆ Z | (M(0), ⟨M_{β} : 0 < β < α⟩, ⟨g_{β} : 0 < β < α⟩) ∈ j(X) }.

It is routine to check that (M(0), ⟨M_{β} : 0 < β < α⟩, ⟨g_{β} : 0 < β < α⟩) ∈ j(Z), so M_{α} is a κ-complete measure on Z for all α < λ^{∗}. The argument of Lemma 3.1 from [7] shows that Ult(V, M_{α}) is closed under λ^{∗}-sequences.

We will need to know that M_{α} concentrates on triples whose second entries are constructed in a manner similar to that in which M⃗ is constructed; we prove this by a reflection argument. We define j_{0} : V ⟶ M_{0}^{∗} to be the ultrapower map arising from the ultrafilter

U_{0} = { x ∈ P_{κ}ρ | j“ρ ∈ j(x) }.

Defining F : P_{κ}σ ⟶ P_{κ}ρ by F (x) = x ∩ ρ it is clear that X ∈ U_{0} ⟺ F^{−1}[X] ∈ U. It follows that F induces a map from M_{0}^{∗} = Ult(V, U_{0}) to M = Ult(V, U) given by k : [f]_{U_{0}} ⟼ [f ∘ F]_{U}, and it is routine to check that k is an elementary embedding and that k ∘ j_{0} = j.

Since ρ + 1 ⊆ rge(k), crit(k) > ρ. Since ρ_{M_{0}^{∗}}^{+} = ρ_{M}^{+} = σ but ρ_{M_{0}^{∗}}^{++} < ρ_{M}^{++} it follows that crit(k) = ρ_{M_{0}^{∗}}^{++}. We also know that H_{σ}^{M_{0}^{∗}} = H_{σ}^{M} = H_{σ}, and can argue in a standard way that k ↾ H_{σ} = id. If we now define a sequence N⃗ in the same way as M⃗ is defined, save that j is replaced by j_{0}, then a routine inductive proof shows that N⃗ = M⃗; the key point is that for every α ≤ λ^{∗} the sequence M⃗ ↾ α lies in M_{0}^{∗} and is fixed by k.

By GCH we see that U_{0} ∈ M. Let i_{0} : M ⟶ N = Ult(M, U_{0}) be the ultrapower map. i_{0} ↾ H_{σ} = j_{0} ↾ H_{σ} and so if N⃗_{0} is defined like M⃗ using i_{0} in place of j then N⃗_{0} = N⃗ = M⃗. It follows that M_{α} will concentrate on triples whose second entries are sequences of measures constructed in the same manner as M⃗.

We defer for the moment the question of exactly how the function g_{α} is defined; we will return to this point after some general discussion of the construction of ℙ. The first step in the construction of ℙ will be to build some forcing ℝ which adds to the universe sequences ⟨x_{i} : i < κ⟩ and ⟨G_{i} : i < κ⟩ such that 1. x_{i} ∈ P_{κ}λ^{∗}, i < j ⟹ x_{i} ≺ x_{j}.

2. The sequence x⃗ is continuous in the sense that if α < κ is a limit ordinal then x_{α} = ⋃_{i<α} x_{i}.

3. If we let κ_{i} = x_{i} ∩ κ then G_{i} is Coll(κ_{i}^{+6}, < κ_{i+1})-generic over V.

4. λ^{∗} = ⋃_{i<κ} x_{i}.

The forcing ℝ does too much damage to the cardinal structure of V, in particular it collapses κ_{i}^{+} for many i; we will accordingly define ℙ as a projection of ℝ which adds ⟨(κ_{i}, G_{i}) : i < κ⟩ but preserves all those cardinals not explicitly collapsed by the G_{i}.

The forcing ℝ is best considered as a generalised version of Prikry forcing. A condition will contain some information about the g’s and x’s along with some constraint on the g’s and x’s which can be added.

By its definition, M_{α} (α > 0) will concentrate on the set of triples (u, v, w) such that

1. u ∈ P_{κ}λ^{∗}.

2. Defining κ_{u} = u ∩ κ and λ_{u} = o.t.(u),

(a) dom(v) = dom(w) = [1, β) for some β < λ_{u}.

(b) v(α) is a measure on Z_{u} = P_{κ_{u}}λ_{u} × V_{κ_{u}} × V_{κ_{u}}.

(c) w(α) is a function such that dom(w(α)) ∈ v(α) and rge(w(α)) ⊆ V_{κ_{u}}.

The building blocks of the forcing ℝ will be triples of this general form. A condition will determine a finite sequence of triples ⟨(u_{i}, v_{i}, w_{i}) : i < n⟩ such that u_{0} ≺ u_{1} ≺ … ≺ u_{n−1}. If dom(v_{i}) = [1, β_{i}) then the condition will associate with the triple (u_{i}, v_{i}, w_{i}) a sequence ⟨A_{i}(j) : 1 ≤ j < β_{i}⟩ such that A_{i}(j) ∈ v_{i}(j) for j ∈ [1, β_{i}). The idea is that the sets A_{i}(j) will constrain the triples (u, v, w) that can be interpolated between (u_{i−1}, v_{i−1}, w_{i−1}) and (u_{i}, v_{i}, w_{i}) when the condition is extended.

Unfortunately the measure v_{i}(j) concentrates on the set Z_{u_{i}}, while a candidate for interpolation is a triple (u, v, w) as above where u ∈ P_{κ}λ^{∗} and u ≺ u_{i}. While v and w are of the right form (because λ_{u} < κ_{u_{i}}), u is not literally a member of P_{κ_{u_{i}}}λ_{u_{i}}. We define some functions that will be used to deal with this problem.

If u, u′ ∈ P_{κ}λ^{∗} and u ≺ u′ then (since u ⊆ u′) there is a natural map i_{uu′} : λ_{u} → λ_{u′} induced by the inclusion map from u to u′. Notice that i_{uu′} ↾ κ_{u} = id. Abusing notation, we also denote by i_{uu′} the map from Z_{u} to Z_{u′} given by

i_{uu′}(a, b, c) = (i_{uu′}(a), b, c).

In the situation of the last paragraph, the appropriate question to ask will be whether i_{uu_{i}}(u, v, w) ∈ A_{i}(j) where j = i_{uu_{i}}(lh(v)).

Similarly if u ∈ P_{κ}λ^{∗} the inclusion map induces a natural map i_{u} : λ_{u} → λ^{∗} with i_{u} ↾ κ_{u} = id. As in the preceding paragraph we abuse notation slightly and define i_{u} : Z_{u} ⟶ Z by i_{u}(a, b, c) = (i_{u}(a), b, c).

It is clear from the definitions that if u ≺ v then i_{u} = i_{v} ∘i_{uv}, and similarly if u ≺ v ≺ w then i_{uw} = i_{vw} ∘ i_{uv}.

Following [7] we will henceforth identify the triple (u, v, w) with the pair (u ⌢ v, w), and will also regard u ⌢ v as lying in P_{κ}λ^{∗}. Following the notation of [7] we usually write such a pair as (u⃗, w⃗), where with the conventions we are now using

1. u⃗ is a sequence whose domain is some ordinal β; we write lh(u⃗) = β and sometimes with a mild abuse of notation lh((u⃗, w⃗)) = β. dom(w⃗) = [1, β).

2. u_{0} is an element of P_{κ}λ^{∗}. We will define κu⃗ = u_{0} ∩ κ, λu⃗ = o.t.(u_{0}). We also let κ_{(u⃗,w⃗)} = κu⃗ and λ_{(u⃗,w⃗)} = λu⃗.

3. For 0 < α < β, u_{α} is a measure on P_{κu⃗}λu⃗ × V_{κu⃗}, and w_{α} is a function with domain lying in u(α) and with range a subset of V_{κu⃗}.

Definition 3.1 (u⃗, w⃗) is a good pair if conditions 1, 2, and 3 above hold. If (u⃗, w⃗) is a good pair and u_{0} ≺ x then i_{u_{0}x}(u⃗, w⃗) = (u^{∗}, w⃗) where u^{∗} is the sequence obtained from u⃗ by replacing u_{0} with i_{u_{0}x}(u_{0}).

The sequence of lemmas and definitions which follows runs exactly parallel to the corresponding discussion in [7]. We leave the proofs to the reader and just make a few motivating remarks. We will state some results about the sequence M⃗ before defining the functions g_{α}, but this is not problematic because these results do not depend on the exact definition of the g’s.

Lemma 3.2 Fix a well-ordering ⊲ of H_{λ^{+6}}. There exists a sequence of mutually disjoint sets ⟨A_{α} : 0 < α < λ^{∗}⟩ such that if (u⃗, h⃗) ∈ A_{β} then

1. (u⃗, h⃗) is a good pair.

2. lh(u⃗) < λu⃗, iu⃗(lh(u⃗)) = β.

3. The structure ⟨H_{sup(u_{0})^{+5}}, ∈, (u⃗, h⃗), ⊲⟩ is elementarily equivalent to the structure ⟨H_{sup(j“λ^{∗})^{+5}}^{M}, ∈, (M⃗ ↾ β, g⃗ ↾ β), ⊲⟩.

The point of this lemma is that M_{α} concentrates on sequences which resemble (M⃗ ↾ α, g⃗ ↾ α). In particular if (u⃗, h⃗) is in A_{α} then since U_{0} ∈ H_{sup(j“λ^{∗})^{+5}}^{M} it will follow that there is some elementary embedding i such that

1. i witnesses that κu⃗ is κ+4 ; u⃗-supercompact.

2. u⃗ and h⃗ are constructed from i in the same way that M⃗ and g⃗ are constructed from j.

As in [7] we may define a class U_{∞} of pairs (u⃗, h⃗) such that if (u⃗, h⃗) ∈ U_{∞} then

1. (u⃗, h⃗) has the properties from clauses 1, 2 and 3 of Lemma 3.2.

2. Each measure in u⃗ concentrates on a subset of U_{∞}. Henceforth we assume that all pairs (u⃗, h⃗) which we consider are drawn from U_{∞}.

At this point we are finally ready to be more precise about the definition of the functions g_{α} involved in the definition of M⃗. At the same time we will define a map π, which will eventually be used to define the forcing ℙ as a projection of the forcing ℝ.

g_{α} will be chosen so that dom(g_{α}) ∈ M_{α} and g_{α}(u⃗, h⃗) ∈ Coll(κ+6 ; u⃗, < κ) for all (u⃗, h⃗) ∈ dom(g_{α}).

We define a map π whose domain is U_{∞}. π(u⃗, f⃗) will be defined by induction on κu⃗ to be (v⃗, G⃗) where

1. lh(v⃗) = lh(u⃗), dom(G⃗) = dom(f⃗).

2. v_{0} = κu⃗.

3. For 0 < α < lh(v⃗)

(a) v_{α} is the measure on V_{v_{0}} defined by

v_{α} = { X ⊆ V_{v_{0}} | π^{−1}[X] ∈ u(α) }.

(b) G_{α} is a certain filter on the Boolean algebra given by the ultraproduct Q(v⃗, α) = ∏_{a} RO(Coll(κ_{a}^{+6}, < κ))/v_{α}. G_{α} is generated by elements of the form [b(u⃗, f⃗, A⃗, α)]_{v_{α}}, where we define

b(u⃗, f⃗, A⃗, α)(c) = ⋁ { f_{α}(a) | a ∈ A_{α}, π(a) = c }

for each c ∈ V_{v_{0}} and each sequence A⃗ such that lh(A⃗) = lh(v⃗) and A_{α} ∈ v_{α} for all α.

We define U_{∞}^{π} = { π(a) | a ∈ U_{∞} }.

Exactly as in [7] we may now choose the sequence of functions g_{α} in such a way that π(M⃗, g⃗) = (w⃗, F⃗) with each of the F_{α} an ultrafilter on the appropriate Boolean algebra. To be more precise we use the closure of Ult(V, M_{α}) to choose the g_{α} in such a way that for every b ∈ Q(w⃗, α) there is A⃗ such that [b(M⃗, g⃗, A⃗, α)]_{w_{α}} either refines or is incompatible with b.

A key point in the construction that follows is this: if k⃗ is a sequence of functions with B_{α} = dom(k_{α}) ∈ w_{α}, B_{α} ⊆ dom(g_{α}) and k_{α}(a) ≤ g_{α}(a) for all a then [b(M⃗, k⃗, b⃗, α)]_{w_{α}} ∈ F_{α} for all α. This will be crucial in the proof that ℙ collapses only those cardinals which it ought to.

We are now ready to define the forcing ℝ. In fact we will define for each a ∈ U_{∞} a forcing ℝ_{a}. Our final ℝ will be ℝ_{a} where a is a certain initial segment of (M⃗, g⃗).

Following [7] we say that (u⃗, f⃗, A⃗, k⃗, s) is a suitable quintuple iff EITHER

1. (u⃗, f⃗) ∈ U_{∞}.

2. A⃗ is a sequence such that A_{α} ∈ u_{α} for 0 < α < lh(u⃗).

3. k⃗ is a sequence such that for 0 < α < lh(u⃗)

(a) dom(k_{α}) = A_{α}.

(b) For all a ∈ A_{α}, k_{α}(a) ∈ Coll(κ_{a}^{+6}, < κu⃗) and k_{α}(a) ≤ f_{α}(a).

4. s ∈ Coll(κ+6 ; u⃗ < κ).

OR

1. u⃗ = ⟨ℵ_{0}⟩, where by convention we set κu⃗ = ℵ_{0}.

2. f⃗ = A⃗ = k⃗ = 0.

3. s ∈ Coll(ℵ_{6}, < κ).

When u_{0} ≺ x we let i_{u_{0}x}(u⃗, f⃗, A⃗, k⃗, s) = (u^{∗}, f⃗, A⃗, k⃗, s), where u^{∗} is the result of replacing u_{0} by i_{u_{0}x} in u. Definition 3.3 Let a = (u⃗, f⃗) ∈ U_{∞}. A condition in ℝ_{a} is a sequence

⟨t_{1}, …, t_{n}, (u⃗, f⃗, A⃗, k⃗)⟩

such that 1. Each t_{i} is a suitable quintuple t_{i} = (u⃗_{i}, f⃗_{i}, A⃗_{i}, k⃗_{i}, s_{i}).

2. t_{1} = (⟨ℵ_{0}⟩, 0, 0, 0, s_{1}).

3. (u⃗_{i})_{0} ≺ (u⃗_{i+1})_{0} for 1 ≤ i < n and (u⃗_{i})_{0} ∈ P_{κu⃗}λu⃗ for 1 ≤ i ≤ n.

4. A⃗ is a sequence of measure one sets for u⃗. dom(k_{δ}) = A_{δ} and k_{δ}(a) ∈ Coll(κ_{a}^{+6}, < κu⃗) for all a ∈ A_{δ}.

5. s_{i} ∈ Coll(κ+6 ; u⃗_{i}, < κu⃗_{i+1}) for 1 ≤ i < n and s_{n} ∈ Coll(κ+6 ; u⃗_{n}, < κu⃗).

Intuitively this condition is intended to give a certain amount of information about a pair of sequences x⃗ and G⃗ where 1. x⃗ is a continuous and ≺-increasing chain of sets from P_{κu⃗}λu⃗ with union

λu⃗.

2. G_{i} is Coll(κ_{x_{i}}^{+6}, < κ_{x_{i+1}})-generic. The ordering on ℝ_{a} will be defined accordingly in Definition 3.5. Definition 3.4 Let q^{0} = (u^{0}, f⃗^{0}, A⃗^{0}, k⃗^{0}, s^{0}) and q^{1} = (u^{1}, f⃗^{1}, A⃗^{1}, k⃗^{1}, s^{1}) be suitable quintuples.

1. q^{1} shrinks q^{0} iff (a) (u^{1}, f⃗^{1}) = (u^{0}, f⃗^{0}).

(b) A_{α}^{1} ⊆ A_{α}^{0} for all α.

(c) k_{α}^{1}(a) ≤ k_{α}^{0}(a) for all a ∈ A_{α}^{1}.

(d) s^{1} ≤ s^{0}.

2. q^{1} is addable to q^{0} iff (a) (u^{1})_{0} ≺ (u^{0})_{0} (so we may define a map i = i_{(u^{1})_{0}(u^{0})_{0}}).

(b) i(lh(u^{1})) < lh(u⃗_{0}).

(c) i(u^{1}, f⃗^{1}) ∈ A_{i(lh(u^{1}))}^{0}.

(d) i“A_{δ}^{1} ⊆ A_{i(δ)}^{0} for 0 < δ < lh(u^{1}).

(e) k_{δ}^{1}(a) ≤ k_{i(δ)}^{0}(i(a)) for all a ∈ A_{δ}^{1}, 0 < δ < lh(u^{1}).

(f) s^{1} ≤ k_{i(lh(u^{1}))}^{0}(i(u^{1}, f⃗^{1})).

Notice that the definition of addability did not involve s^{0}, so that with a mild abuse of language we may say that “(u^{1}, f⃗^{1}, A⃗^{1}, k⃗^{1}, s^{1}) is addable to (u^{0}, f⃗^{0}, A⃗^{0}, k⃗^{0})”.

Definition 3.5 Let a = (u⃗, f⃗) ∈ U_{∞} and let

p^{0} = ⟨t_{1}^{0}, …, t_{n}^{0}, (u⃗, f⃗, A⃗^{0}, k⃗^{0})⟩
p^{1} = ⟨t_{1}^{1}, …, t_{m}^{1}, (u⃗, f⃗, A⃗^{1}, k⃗^{1})⟩

be two conditions in ℝ_{a}, where t_{e}^{d} = (u_{e}^{d}, f⃗_{e}^{d}, A⃗_{e}^{d}, k⃗_{e}^{d}, s_{e}^{d}).

Then p^{1} ≤ p^{0} iff

1. n ≤ m.

2. A_{α}^{1} ⊆ A_{α}^{0} and a ∈ A_{α}^{1} ⟹ k_{α}^{1}(a) ≤ k_{α}^{0}(a) for all 0 < α < lh(u⃗).

3. For every i with 1 ≤ i ≤ n there is j with 1 ≤ j ≤ m such that t_{j}^{1} shrinks t_{i}^{0}.

4. For each j with 1 ≤ j ≤ m one of the following statements holds:

(a) t_{j}^{1} is addable to t_{1}^{0}.

(b) There exists i such that 1 ≤ i ≤ n and t_{j}^{1} shrinks t_{i}^{0}.

(c) There exists i such that 1 ≤ i < n, (u_{i}^{0})_{0} ≺ (u_{j}^{1})_{0} ≺ (u_{i+1}^{0})_{0} and t_{j}^{1} is addable to t_{i+1}^{0}.

(d) (u_{n}^{0})_{0} ≺ (u_{j}^{1})_{0} and i_{(u_{j}^{1})_{0}}(t_{j}^{1}) is addable to (u⃗, f⃗, A⃗^{0}, k⃗^{0}).

We are now able to define the projected forcing ℙ. A suitable quintuple for ℙ is (v⃗, G⃗, B⃗, b⃗, s) where

EITHER

1. (v⃗, G⃗) ∈ U_{∞}^{π}.

2. B⃗ is a sequence of measure one sets for v⃗.

3. b_{γ} is a function with domain B_{γ} such that a ∈ B_{γ} ⟹ b_{γ}(a) ∈ RO(Coll(κ_{a}^{+6}, < κv⃗)).

4. [b_{γ}]_{v_{γ}} ∈ G_{γ}.

5. s ∈ Coll(κ+6 ; v⃗, < κ).

OR

1. v⃗ = ⟨ℵ_{0}⟩, where by convention κv⃗ = ℵ_{0}.

2. G⃗ = B⃗ = b⃗ = 0.

3. s ∈ Coll(ℵ_{6}, < κ). In the natural way we may define a map π which takes suitable quintuples

for ℝ to suitable quintuples for ℙ. We let π(u⃗, f⃗, A⃗, k⃗, s) = (v⃗, G⃗, B⃗, b⃗, s) where

1. π(u⃗, f⃗) = (v⃗, G⃗).

2. B_{γ} = π“A_{γ}.

3. b_{γ} = b(u⃗, k⃗, A⃗, γ). For the special case of a quintuple of form q = (⟨ℵ_{0}⟩, 0, 0, 0, s) we set π(q) = q.

Now we define ℙ_{c} for every c ∈ U_{∞}^{π}. Definition 3.6 Let c = (v⃗, G⃗) ∈ U_{∞}^{π}. A condition in ℙ_{c} is a sequence

⟨t_{1}, …, t_{n}, (v⃗, G⃗, B⃗, b⃗)⟩

such that 1. Each t_{i} is a suitable quintuple for ℙ, say t_{i} = (v⃗_{i}, G⃗_{i}, B⃗_{i}, b⃗_{i}, s_{i}).

2. t_{1} = (⟨ℵ_{0}⟩, 0, 0, 0, s_{1}).

3. κv⃗_{i} < κv⃗_{i+1} for 1 ≤ i < n and κv⃗_{n} < κv⃗.

4. B⃗ is a sequence of measure one sets for v⃗. b_{γ} is a function such that dom(b_{γ}) = B_{γ}, b_{γ}(a) ∈ RO(Coll(κ_{a}^{+6}, < κv⃗)) for all a ∈ B_{γ}, and [b_{γ}]_{v_{γ}} ∈ G_{γ}.

5. s_{i} ∈ Coll(κ+6 ; v⃗_{i}, < κv⃗_{i+1}) for 1 ≤ i < n and s_{n} ∈ Coll(κ+6 ; v⃗_{n}, < κv⃗).

If p = ⟨t_{1}, …, t_{n}, (v⃗, G⃗, B⃗, b⃗)⟩ ∈ ℙ_{c} then we will divide p into a “lower part” and an “upper part”, to be more precise we let lp(p) = ⟨t_{1}, …, t_{n}⟩ and up(p) = (v⃗, G⃗, B⃗, b⃗). Definition 3.7 Let q^{0} = (v^{0}, G⃗^{0}, B⃗^{0}, b⃗^{0}, s^{0}) and q^{1} = (v^{1}, G⃗^{1}, B⃗^{1}, b⃗^{1}, s^{1}) be suitable quintuples for ℙ.

1. q^{1} shrinks q^{0} iff

(a) (v^{1}, G⃗^{1}) = (v^{0}, G⃗^{0}).

(b) B_{α}^{1} ⊆ B_{α}^{0} for all α.

(c) b_{α}^{1}(a) ≤ b_{α}^{0}(a) for all a ∈ B_{α}^{1}.

(d) s^{1} ≤ s^{0}.

2. q^{1} is addable to q^{0} iff

(a) There is γ < lh(v^{0}) such that (v^{1}, G⃗^{1}) ∈ B_{γ} and s^{1} ≤ b_{γ}^{0}(v^{1}, G⃗^{1}).

(b) There is an increasing e : lh(v^{1}) ⟶ lh(v^{0}) such that B_{α}^{1} ⊆ B_{e(α)}^{0}

and b_{α}^{1}(a) ≤ b_{e(α)}^{0}(a) for all 0 < α < lh(v^{1}) and all a ∈ B_{α}^{1}.

Definition 3.8 Let c = (v⃗, G⃗) ∈ U_{∞}^{π} and let

p^{0} = ⟨t_{1}^{0}, …, t_{n}^{0}, (v⃗, G⃗, B⃗^{0}, b⃗^{0})⟩
p^{1} = ⟨t_{1}^{1}, …, t_{m}^{1}, (v⃗, G⃗, B⃗^{1}, b⃗^{1})⟩

be two conditions in ℙ_{c}, where t_{e}^{d} = (v_{e}^{d}, G⃗_{e}^{d}, B⃗_{e}^{d}, b⃗_{e}^{d}, s_{e}^{d}).

Then p^{1} ≤ p^{0} iff

1. n ≤ m.

2. B_{α}^{1} ⊆ B_{α}^{0} and a ∈ B_{α}^{1} ⟹ b_{α}^{1}(a) ≤ b_{α}^{0}(a) for all 0 < α < lh(v⃗).

3. For every i with 1 ≤ i ≤ n there is j with 1 ≤ j ≤ m such that t_{j}^{1}

shrinks t_{i}^{0}.

4. For each j with 1 ≤ j ≤ m one of the following statements holds:

(a) t_{j}^{1} is addable to t_{1}^{0}.

(b) There exists i such that 1 ≤ i ≤ n and t_{j}^{1} shrinks t_{i}^{0}.

(c) There exists i such that 1 ≤ i < n, (v_{i}^{0})_{0} < (v_{j}^{1})_{0} < (v_{i+1}^{0})_{0} and t_{j}^{1} is addable to t_{i+1}^{0}.

(d) (v_{n}^{0})_{0} < (v_{j}^{1})_{0} and t_{j}^{1} is addable to (v⃗, G⃗, B⃗^{0}, b⃗^{0}).

At this point we are finally ready to define the forcings ℙ and ℝ. To do this we observe that by GCH there are only 2^{2^{κ}} = κ^{++} many measures on V_{κ}, so that if (w⃗, F⃗) = π(M⃗, g⃗) then the sequence w⃗ must contain a repetition. Accordingly we define λ_{1} < λ^{∗} to be minimal such that for some λ_{0} < λ_{1} we have w_{λ_{0}} = w_{λ_{1}}. We then set ℝ = ℝ_{(M⃗↾λ_{1},g⃗↾λ_{1})} and ℙ = ℙ_{(w⃗↾λ_{1},F⃗↾λ_{1})}.

The following properties are then proved by easy adaptations of proofs in [7].

Lemma 3.9 ℙ_{a} has the κ_{a}^{+}-chain condition.

Lemma 3.10 ℙ_{a} adds to the universe a sequence a⃗ of members of U_{∞}^{π} such that if κ_{i} = κ_{a_{i}} then κ_{0} = ℵ_{0} and κ⃗ is continuous, increasing and cofinal in κ_{a}. lh(a_{i}) > 1 exactly when i is a limit ordinal. ℙ_{a} also adds a sequence G⃗ such that G_{i} is Coll(κ_{i}^{+6}, < κ_{i+1})-generic for all i < lh(κ⃗) = lh(G⃗).

Lemma 3.11 The generic club added by ℙ_{a} is eventually contained in every set X ⊆ κ_{a} such that { v | κ_{v} ∈ X } is measure one for all the measures appearing in a. In particular the generic club is eventually contained in every > ω-club subset of κ_{a}.

Lemma 3.12 Let a = (v⃗, G⃗) ∈ U_{∞}^{π}, let

p = ⟨t_{1}, …, t_{n}, (v⃗, G⃗, B⃗, b⃗)⟩ ∈ ℙ_{a}

with t_{i} = (v⃗_{i}, G⃗_{i}, B⃗_{i}, b⃗_{i}, s_{i}).

1. If lh(v_{i}) > 1 then

ℙ_{a}/p ≃ ℙ_{(v⃗_{i},G⃗_{i})}/q × ℙ_{a}/r

where

q = ⟨t_{1}, …, t_{i−1}, (v⃗_{i}, G⃗_{i}, B⃗_{i}, b⃗_{i})⟩

and

r = ⟨(⟨κv⃗_{i}⟩, ⟨⟩, ⟨⟩, ⟨⟩, s_{i}), t_{i+1}, …, t_{n}, (v⃗, G⃗, B⃗, b⃗)⟩.

2. If also lh(v⃗_{i+1}) = 1 then ℙ_{a}/p ≃ ℙ_{(v_{i},G_{i})}/q×Coll(κ+6 ; v⃗_{i}, < κv⃗_{i+1})/s_{i}×ℙ_{a}/r^{∗}

where

r^{∗} = ⟨t_{i+1}, …, t_{n}, (v⃗, G⃗, B⃗, b⃗)⟩.

Lemma 3.13 Let F be ℙ_{a}-generic with p ∈ F. Let a⃗ be the sequence from U_{∞}^{π} added by F, and let G⃗ be the corresponding sequence of generic collapses.

If lh(a_{i}) > 1 then V [F] = V [F_{1} × F_{2}] where

1. F_{1} is ℙ_{a_{i}}-generic.

2. F_{2} is ℙ_{a}-generic.

3. F_{1} adds a⃗ ↾ i and G⃗ ↾ i.

4. F_{2} adds a^{∗}, G⃗^{∗} where a_{j}^{∗} = a_{i+j} and G_{j}^{∗} = G_{i+j}.

Abusing notation, we will denote F_{1} by “F ↾ i”.

If p, q are conditions in ℙ_{a} we say that p is a direct extension of q and write p ≤^{∗} q when p and q are sequences of the same length. To put it another way, p is obtained from q by merely shrinking the quintuples which are present in p and the relevant parts of the final quadruple. We write p ≤^{∗} q when p is a direct extension of q.

Lemma 3.14 With the same hypotheses and notation as Lemma 3.12, if b ∈ RO(ℙ_{a}) is a boolean value then there exist a maximal antichain A ⊆ ℙ_{(v⃗_{i},G⃗_{i})}/q and a condition r^{∗} ≤^{∗} r such that (s, r^{∗}) decides b for all s ∈ A.

Lemma 3.15 Let F be ℙ_{a}-generic and let κ⃗, G⃗ be as in Lemma 3.10. Let λ < κ_{a} be a cardinal and let i be the largest limit ordinal such that κ_{i} ≤ λ. Let n < ω be minimal such that λ < κ_{i+n}^{+6}.

Let G^{∗} = F ↾ i × ∏_{j<n} G_{i+j}. Then G^{∗} is ℙ_{a_{i}} × ∏_{j<n} Coll(κ_{i+j}^{+6}, < κ_{i+j+1})- generic and (Pλ)_{V [F]} = (Pλ)_{V [G^{∗}]}. In particular, if λ < κ and does not lie in an interval (κ_{i}^{+6}, κ_{i+1}) then λ is preserved.

Lemma 3.16 Let G be ℙ-generic. Then κ is κ^{+5}-supercompact in V [G].

One more remark is in order before we begin to exploit the forcing ℙ. If A ⊆ κ and κ ∈ j(A) then every measure w_{α} will concentrate on the set of a with κ_{a} ∈ A. Accordingly by forcing below an appropriate condition we may ensure that the generic sequence κ⃗ added by ℙ consists of points from A.

4 Proof of the main theorem

We are now in a position to prove Theorem 1. Proof:[Theorem 1] We begin with a model (V_{0} say) in which GCH holds and κ is κ^{+5}-supercompact. We do a Reverse Easton iteration of the sort described in Theorem 2 and obtain a model V in which

1. GCH holds.

2. κ is κ^{+5}-supercompact.

3. □_{κ}^{B} holds.

Let

A = { α < κ | α is α^{+}-supercompact and □_{α}^{B} holds }.

It is easy to see that κ ∈ j(A).

Working in V, we now use j to build a forcing notion ℙ as in the preceding section. Let G_{ℙ} be ℙ-generic over V and let a⃗, κ⃗, G⃗ be as in Lemma 3.10. Forcing below a suitable condition we may assume that κ_{i} ∈ A for all i > 0.

We know by the work of the last section that 1. κ is κ^{+5}-supercompact in V [G_{ℙ}].

2. If a cardinal λ is in (κ_{i}^{+6}, κ_{i+1}) for some i then λ is collapsed to have cardinality κ_{i}^{+6} in V [G_{ℙ}]. Otherwise λ is preserved.

The following claim will establish Theorem 1.

Claim 4.1 Let λ < κ be a singular cardinal of V [G_{ℙ}]. Then in V [G_{ℙ}]

1. λ = κ_{i} for some limit i < κ.

2. The combinatorial principle □_{λ,cf(λ)} holds.

3. If S = { α < λ^{+} | cf(α) < λ }_{V} then S is stationary. Moreover if S⃗ is a sequence of stationary subsets of S and lh(S⃗) < cf(λ) then S⃗ reflects to cofinality μ for unboundedly many μ < λ.

Proof:

We take the various claims in turn.

1. Since λ is singular, λ is a limit cardinal. We have collapsed all but finitely many cardinals in each interval [κ_{j}, κ_{j+1}) so the only possibility is that λ = κ_{i} for some limit i. Notice that we must have lh(a_{i}) > 1 because sequences of length 1 correspond to successor points on the sequence κ⃗.

2. Since λ = κ_{i}, λ ∈ A and so V ⊨ □_{λ}^{B}. λ_{V}^{+} = λ_{V [G_{ℙ}]}^{+} by Lemma 3.15, so V [G_{ℙ}] ⊨ □_{λ,cf(λ)} by Lemma 3.11 and Lemma 2.1.

3. We know λ = κ_{i} where i is limit and lh(a_{i}) > 1. By Lemma 3.15 we know (Pλ^{+})_{V [G_{ℙ}]} = (Pλ^{+})_{V [G_{ℙ}↾λ]}. Since ℙ_{a_{i}} is λ^{+}-c.c. the set S is still stationary in V [G_{ℙ} ↾ i], and so is stationary in V [G_{ℙ}]. It suffices to prove the desired reflection statement in V [G_{ℙ} ↾ λ]. We will do this using the fact that λ ∈ A, so that λ is λ^{+}-supercompact in V. Let G_{j}^{∗} = G_{ℙ} ↾ j for j ≤ lh(a⃗). Let μ = cf_{V [G_{ℙ}]}(λ) = cf_{V [G_{i}^{∗}]}(λ). Until further notice we work in V [G_{i}^{∗}]. Fix a sequence ⟨τ_{j} : j < μ⟩ which is increasing and cofinal in i. Then ⟨κ_{τ_{j}} : j < μ⟩ is increasing and cofinal in κ_{i} = λ. If the reflection claim fails we can find p ∈ G_{i}^{∗}, ⟨Ṡ_{k} : k < ν⟩ for some ν < μ, and σ < λ such that p ⊩ Ṡ_{k} ⊆ Ŝ and

p ⊩_{ℙ_{a_{i}}}^{V} “⟨Ṡ_{k} : k < ν⟩ reflects simultaneously at no cofinality > σ”.

Let S_{k} be the realisation of the term Ṡ_{k} by the generic G_{i}^{∗}. Then S_{k} = ⋃_{j<μ} S_{k}^{j} where

S_{k}^{j} = { α | ∃q ∈ G_{i}^{∗} lp(q) ∈ G_{τ_{j}}^{∗}, q ⊩ α ∈ Ṡ_{k} }.

Notice that S_{k}^{j} increases with j. For each k < ν there exists j < μ such that S_{k}^{j} is stationary. Increasing j further we may also arrange that S_{k}^{j} ∩ X_{j} is stationary where

X_{j} = { α < κ^{+} | cf(α) < κ_{τ_{j}} }_{V} .

Since ν < μ = cf(μ) we may find j so large that (a) S_{k}^{j} ∩ X_{j} is stationary for all k.

(b) lp(p) ∈ G_{τ_{j}}^{∗}.

We will now work in V [G_{τ_{j}}^{∗}] until further notice. Let

T_{k} = { α | ∃q ∈ ℙ_{a_{i}} lp(q) ∈ G_{τ_{j}}^{∗}, q ⊩ α ∈ Ṡ_{k} ∩ X_{j} }.

T_{k} ∈ V [G_{τ_{j}}^{∗}] ⊆ V [G_{i}^{∗}], S_{k}^{j} ∩ X_{j} ⊆ T_{k} and S_{k}^{j} ∩ X_{j} is stationary in V [G_{i}^{∗}], so clearly T_{k} is stationary in V [G_{τ_{j}}^{∗}]. Now λ is still λ^{+}-supercompact in V [G_{τ_{j}}^{∗}] because |ℙ_{a_{τ_{j}}} | < λ. T_{k} = ⋃_{p∈G_{τ_{j}}^{∗}} { α | p ⊩ α ∈ Ṫ_{k} }, and so there is p_{k} ∈ G_{τ_{j}}^{∗} such that Y_{k} = { α | p_{k} ⊩ α ∈ Ṫ_{k} } is stationary. That is to say, for every k we may find Y_{k} ⊆ T_{k} such that Y_{k} is stationary and Y_{k} ∈ V. ⟨Y_{k} : k < ν⟩ is not necessarily in V because p_{k} depends on k and G_{τ_{j}}^{∗}, but since ℙ_{a_{τ_{j}}} is λ-c.c. we may find a family X ∈ V of stationary subsets of X_{j} such that |X| < λ and { Y_{k} | k < ν } ⊆ X. We now work in V. Appealing to Fact 1.2 we may find γ such that all the Y_{k} reflect at γ, and cf(γ) is the successor of an inaccessible with max{σ, κ_{τ_{j}}^{+6}} < cf(γ) < λ. Choose D ⊆ γ club such that o.t.(D) = cf(γ). Using the “strong factorisation” property from Lemma 3.14, we work in V [G_{τ_{j}}^{∗}] and choose for each k < ν and each β ∈ Y_{k} ∩ D an upper part q(k, β) such that

∃r ∈ G_{τ_{j}}^{∗} r ⌢ q(k, β) ⊩ β ∈ Ṡ_{k}.

Since max{|ℙ_{a_{τ_{j}}} |, ν, cf(γ)} < λ we may find a set Y of upper parts for ℙ_{a_{i}} such that Y ∈ V, |Y | < λ and { q(k, β) | k < ν, β ∈ Y_{k} ∩ D } ⊆ Y. The filters which appear in a_{i} are all λ-complete so we may find a single upper part q such that q shrinks all the q(k, β) as well as the upper part of p. Shrinking further if necessary we can guarantee that the condition ⟨q⟩ forces that the generic club added by ℙ_{a_{i}} has minimal entry greater than cf(γ). We force below ⟨q⟩ to get an H which is ℙ_{a_{i}}-generic over V [G_{τ_{j}}^{∗}]. Let μ^{∗} be the least entry on the generic club added by H. We force h which is Coll(κ_{τ_{j}}^{+6}, < μ^{∗})_{V} -generic over V [G_{τ_{j}}^{∗}][H]. By the second part of Lemma 3.12, G_{τ_{j}}^{∗} × h × H can be rearranged as a generic object G^{†} for ℙ_{a_{i}}. What is more we have arranged that p ∈ G^{†} and that Y_{k} ∩ D ⊆ Ṡ_{k}^{G^{†}} for all k. We will reach a contradiction by showing that Y_{k} ∩ D is stationary in V [G^{†}]. Since o.t.(D) = cf_{V} (γ) we may collapse Y_{k} ∩ D to get Y′_{k} a stationary subset of cf_{V} (γ). Since μ^{∗} > cf_{V} (γ) it will suffice to show that Y′_{k} is stationary in V [G_{τ_{j}}^{∗} × h]. Y′_{k} consists of ordinals of cofinality less than κ_{τ_{j}}. cf(γ) is the successor of an inaccessible (θ say), so □_{θ}^{∗} holds. So Coll(κ_{τ_{j}}^{+6}, < μ^{∗}) will preserve (see [9]) the stationarity of Y′_{k}. |ℙ_{τ_{j}}| = κ_{τ_{j}}^{+} so Y′_{k} (and thus Y_{k} ∩ D) will be stationary in V [G_{τ_{j}}^{∗} × h] as required. We now have a contradiction, so the simultaneous reflection property holds.

♦

This concludes the proof of Theorem 1.

♦

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