THE TREE PROPERTY

JAMES CUMMINGS AND MATTHEW FOREMAN

Abstract. We construct a model in which there are no ℵ_{n}-Aronszajn trees for any finite n ≥ 2, starting from a model with infinitely many supercompact cardinals. We also construct a model in which there is no κ^{++}-Aronszajn tree for κ a strong limit cardinal of cofinality ω, starting from a model with a supercompact cardinal and a weakly compact cardinal above it.

1. Introduction

We will prove the following theorems.

Theorem 1. If “ZFC + there exist infinitely many supercompact cardinals” is consistent, then “ZFC + there are no ℵ_{n}-Aronszajn trees for 2 ≤ n < ω” is also consistent.

Theorem 2. If “ZFC + there exists a supercompact cardinal with a weakly compact cardinal above it” is consistent then “ZFC + there exists κ a strong limit cardinal of cofinality ω such that there are no κ^{++}-Aronszajn trees” is also consistent.

We start by recalling the definition of “κ-Aronszajn tree” and some related concepts.

Definition 1.1. Let κ be regular.

1. A κ-tree is a tree of height κ whose every level has size less than κ.

2. A κ-Aronszajn tree is a κ-tree with no cofinal branch.

3. κ has the tree property iff there is no κ-Aronszajn tree.

4. A λ^{+}-tree T is special iff there is a function F : T ⟶ λ such that x <_{T} y implies F (x) ≠ F (y).

Aronszajn trees arise naturally in infinitary combinatorics, and the tree property has been investigated by many set theorists. We list a few classical results.

• (König [6]) ℵ_{0} has the tree property.

• (Aronszajn, see [9]) ℵ_{1} does not have the tree property.

• (Specker [15]) If τ^{<τ} = τ then there is a special τ^{+}-tree. In particular CH implies that ℵ_{2} fails to have the tree property.

• If κ is strongly inaccessible, κ has the tree property if and only if κ is weakly compact.

• (Mitchell [14]) The theories “ZFC + there exists a weakly compact cardinal” and “ZFC + ℵ_{2} has the tree property” are equiconsistent.

1991 Mathematics Subject Classification. Primary 03E35, 03E55; Secondary 03E05.

Key words and phrases. Forcing, large cardinals, Aronszajn trees.

The first author was supported by grant number 517/94 of the Israel Science Foundation.

The second author was partially supported by DMS 9500484.

The construction from [14] is quite general. Let δ < κ with δ regular and κ weakly compact; there is a forcing which preserves cardinals up to δ^{+}, makes 2^{δ} = δ^{++} = κ, and preserves the tree property of κ in the generic extension. In the other direction, any cardinal with the tree property in V will be weakly compact in L.

It is natural to ask whether two small cardinals can simultaneously have the tree property. It turns out that starting from two weakly compact cardinals it is fairly easy to make a model where (for example) 2^{ℵ_{0}} = 2^{ℵ_{1}} = ℵ_{2}, 2^{ℵ_{2}} = 2^{ℵ_{3}} = ℵ_{4}, and both ℵ_{2} and ℵ_{4} have the tree property. However a naive approach will fail if one starts with two weak compacts and tries to make them into successive cardinals with the tree property.

This is more than just a technical problem; Magidor (see [1]) showed that getting a pair of successive cardinals to have the tree property requires at least a measurable cardinal. This result was improved by Foreman and Magidor to show that a strong cardinal is required. Abraham [1] showed that using large enough cardinals this situation is consistent; he started with δ < κ < λ which are respectively regular, supercompact and weakly compact and forced to make 2^{δ} = δ^{++} = κ, 2^{δ^{+}} = δ^{+++} = λ, preserving the tree property at κ and λ.

In the first part of this paper we will start with ω supercompact cardinals ⟨κ_{n} : n < ω⟩, and will collapse so that κ_{n} becomes ℵ_{n+2} and still enjoys the tree property. The idea is essentially to iterate Abraham’s forcing from [1], although there are several new technical problems that have to be coped with. The authors discovered the construction independently in slightly different versions; Cummings’ version is the one given here.

We will also discuss the tree property for κ^{++} where κ is singular strong limit. Since 2^{κ} = κ^{+} ⟹ (κ^{+})^{<κ^{+}} = κ^{+}, there will be a special κ^{++}-tree unless the Singular Cardinals Hypothesis fails at κ. Foreman showed that the tree property can hold in this situation, and in the second part of the paper we will give the proof. The method can be used to show that ℵ_{ω+2} can have the tree property.

The tree property at successors of singulars is a problem with a different flavour, since here there are connections with the existence of weak squares and much larger cardinals are used in the consistency proofs. It is proved in [11] that ℵ_{ω+1} can have the tree property. The consistency of ℵ_{ω+1} and ℵ_{ω+2} having the tree property simultaneously (or even of ℵ_{ω+1} having the tree property with 2^{ℵ_{ω}} > ℵ_{ω+1}) is open. We also do not know whether “the tree property holds for all ℵ_{n} with 2 ≤ n < ω and for ℵ_{ω+1}” is consistent.

We would like to thank Menachem Magidor for a very helpful discussion of the material of Section 7.

2. Preliminaries

In this section we collect some technical definitions and facts for later use. The impatient reader is advised to skip ahead to the next section and refer back if necessary.

2.1. Forcing conventions. We mostly follow the notation and forcing conventions of [7]. In particular a forcing ℙ is a preordering with a distinguished maximum element 1_{ℙ}, “p ≤ q” means that p is stronger than q, and “κ-closed” means that descending sequences of length less than κ have lower bounds. We will write “p ≤_{ℙ} q” when there is a possibility of confusion about which ordering is meant. For iterated forcing we mostly follow the conventions of [2].

Recall that a forcing is separative iff p ≤ q ⟺ p ⊩ q̂ ∈ Ġ. Not all of the forcings in this paper will have this property.

We will use V^{ℙ} to denote the class of ℙ-names, and will use V [ℙ] to denote a generic extension by some unspecified ℙ-generic filter. When we have a particular generic filter G in mind we will use V [G] to denote the extension by G.

We will need several refinements and variations of the standard chain condition and closure properties.

Definition 2.1. Let κ be regular, let ℙ be some forcing.

1. ℙ is κ-Knaster if and only if for all sequences ⟨p_{α} : α < κ⟩ from ℙ there is X unbounded in κ such that ⟨p_{α} : α ∈ X⟩ consists of pairwise compatible elements.

2. ℙ is κ-directed closed if and only if every directed subset of ℙ of size less than κ has a lower bound.

3. ℙ is canonically κ-directed closed if and only if every directed subset of ℙ of size less than κ has a greatest lower bound.

4. ℙ is < κ-distributive if and only if every family of fewer than κ many dense open sets has a non-empty intersection. This is equivalent to the property that no sequence of ordinals of length less than κ is added by ℙ.

The following easy fact will be useful later.

Lemma 2.2. Let ℙ be canonically κ-directed closed, let G be a ℙ-generic filter over V, and let A ⊆ G where V [G] ⊧ |A| < κ. Then A ∈ V, A is directed, and the greatest lower bound of A is a member of G.

2.2. Trees and forcing. It will be important in what follows to know that certain kinds of forcing cannot add branches to a tree.

Lemma 2.3 (Kunen/Tall [8]). Let κ be regular, let ℙ be κ-Knaster, and suppose that T is a tree of height κ with no cofinal branch. Then T has no cofinal branch in V [ℙ].

Proof. Suppose not, and let ḃ ∈ V^{ℙ} be a name for a cofinal branch of the tree T. Choose for each α a condition p_{α} such that p_{α} forces the element of b on level α to be a particular point x_{α}. Now using the Knaster property we may find X ⊆ κ of cardinality κ such that { p_{α} : α ∈ X } is pairwise compatible. But then { x_{α} : α ∈ X } is a cofinal linearly ordered subset of T in V, which can easily be extended to a cofinal branch. Contradiction.

It is crucial to the intended application of Lemma 2.3 that it applies to all trees of height κ, not just to κ-trees. By contrast, the next lemma is specific to κ-trees.

Lemma 2.4 (Silver). Let τ, κ be regular and suppose τ < κ ≤ 2^{τ}. Let ℙ be τ^{+}- closed, let T be a κ-tree. Then every branch of T in V [ℙ] is in fact a member of V.

Proof. We may assume that τ is minimal with 2^{τ} ≥ κ.

Let ḃ ∈ V^{ℙ} name a new branch. Now we may build by induction for each s ∈ ^{≤τ+1}2 conditions p_{s} and points x_{s} of T with the following properties.

1. If s properly extends t then p_{s} ≤ p_{t}, x_{s} >_{T} x_{t}.

2. p_{s} forces x_{s} is on the branch ḃ.

3. For each α the points { x_{s} : s ∈ ^{α}2 } are all on the same level of T, say level η_{α}.

4. For each s ∈ ^{<τ} 2 the points x_{s⌢0} and x_{s⌢1} are incompatible.

The minimal choice of τ ensures that for α < τ the set { x_{s} : s ∈ ^{α}2 } has size less than κ, so that we can choose η_{α+1}. The closure of ℙ guarantees that the construction works at limit stages.

This leads to a contradiction, because the level η_{τ} of T must have fewer than κ elements, yet we have constructed 2^{τ} many distinct ones.

2.3. Cohen forcing. We need a careful analysis of the properties of Cohen forcing, as this forcing will be the key building-block in the main construction.

Definition 2.5. Let κ be regular, X a set of ordinals. Then Add(κ, X) is the forcing whose conditions are functions p such that dom(p) ⊆ X, rge(p) ⊆ 2, | dom(p)| < κ. The ordering is by extension.

When we have defined ℙ = Add(κ, λ), and η < λ, we will often use “ℙ ↾ η” as a convenient shorthand for Add(κ, η). Notice that Add(κ, η) is a complete subordering of Add(κ, λ).

It is easy to see that Add(κ, λ) is canonically κ-directed closed, and a standard Δ-system argument (see [7]) shows that Add(κ, λ) is (2^{<κ})^{+}-Knaster. It turns out that if we want to add subsets to κ in a model where 2^{<κ} is big, via a forcing which has some reasonable chain condition, then it can be helpful to look at Cohen forcing defined in some inner model where 2^{<κ} is small. This is the motivation behind the following lemma, which is implicit in [1].

Lemma 2.6 (Abraham). Let τ < κ, and assume that V ⊧ “τ is regular” and V ⊧ “κ is inaccessible”. Let ℙ = Add(τ, η).

Let W ⊇ V be a model of ZFC such that

1. κ and τ are still cardinals in W.

2. If X ∈ W is a set of ordinals such that W ⊧ |X| < κ, then there is Y ⊇ X such that Y ∈ V and V ⊧ |Y | < κ.

Then ℙ has the κ-Knaster property in W.

Proof. [Sketch] Working in W, mimic the usual Δ-system argument. Use the inaccessibility of κ in V and the “covering” property to show that the argument goes through.

2.4. Projections. Another key idea in [1] is to use the concept of a projection from one forcing to another. We collect some more or less standard information about projections in a series of definitions and lemmas. The proofs are routine.

Definition 2.7. Let ℙ and ℚ be forcings. Then π : ℙ ⟶ ℚ is a projection if and only if

1. p_{1} ≤ p_{0} ⟹ π(p_{1}) ≤ π(p_{0}).

2. π(1_{ℙ}) = 1_{ℚ}.

3. ∀p ∈ ℙ ∀q ≤ π(p) ∃p_{1} ≤ p π(p_{1}) ≤ q.

Notice that π“ℙ is dense in ℚ, by setting p = 1_{ℙ} in clause 3.

Lemma 2.8. Let π : ℙ ⟶ ℚ be a projection. Then 1. If G is ℙ-generic over V, then H = { q : ∃p ∈ G π(p) ≤ q } is ℚ-generic

over V.

2. Let H be ℚ-generic over V. Let ℙ̄ = { p : π(p) ∈ H }, ordered as a suborder of ℙ. Then ℙ̄ is non-empty, and if G is ℙ̄-generic over V [H] then G is ℙgeneric over V. Moreover, π“G generates H.

3. Let G be ℙ-generic, and define H as in 1 and then ℙ̄ as in 2. Then G ⊆ ℙ̄, and G is ℙ̄-generic over V [H]. That is, we can factor forcing with ℙ as forcing with ℚ followed by forcing with ℙ̄ over V [ℚ].

Some projections have a a stronger property, which enables us to give a slightly different factor analysis.

Definition 2.9. Let ℙ and ℚ be forcings. Then π : ℙ ⟶ ℚ is a good projection if and only if

1. p_{1} ≤ p_{0} ⟹ π(p_{1}) ≤ π(p_{0}).

2. π(1_{ℙ}) = 1_{ℚ}.

3. For all p ∈ ℙ and q ≤ π(p) there is p_{1} ≤ p such that (a) π(p_{1}) = q.

(b) For all r ≤ p, if π(r) ≤ q then r ≤ p_{1}.

Notice that the p_{1} in part 3 of the definition is essentially unique, because if p′_{1} also has these properties then it is easy to see that p_{1} ≤ p′_{1} ≤ p_{1}. We let Ext(p, q) denote some extension of p with these properties.

Lemma 2.10. Let π : ℙ ⟶ ℚ be a good projection, and suppose that H is ℚgeneric. Let ℙ̄ = { p ∈ ℙ : π(p) ∈ H }, and define an ordering ≤^{∗} on ℙ̄ by

p ≤^{∗} q ⟺ ∃r ≤ π(p) (r ∈ H ∧ Ext(p, r) ≤ q).

Then forcing over V [H] with ℙ̄ ordered as a subset of ℙ is equivalent to forcing over V [H] with ℙ̄ ordered by ≤^{∗}.

In the case which we will be using later, ℙ is an iteration, ℚ = ℙ ↾ β is an initial segment of ℙ, and π : p ⟼ p ↾ β. In this case Ext(p, q) = q ⌢ (p ↾ dom(p) \ β).

Suppose that ℙ and ℚ are two forcings with the same underlying set but different orderings, and suppose that the identity function projects ℙ to ℚ. If G is ℙ-generic, G is directed as a subset of ℚ and generates a generic filter H on ℚ. If H is ℚgeneric then forcing with H considered as a suborder of ℙ produces a ℙ-generic G such that G ⊆ H, and G generates H.

It is important to notice that the constructions of this section can produce nonseparative posets.

2.5. Easton’s lemma. As we mentioned in Section 2.3 we will be interested in analysing the properties of forcings defined in inner models of the universe. The following lemmas will be used for this purpose.

Lemma 2.11 (Easton’s lemma). Let κ be regular. If ℙ has the κ-chain condition and ℚ is κ-closed, then

1. ⊩_{ℚ} ℙ has the κ-chain condition.

2. ⊩_{ℙ} ℚ is < κ-distributive.

3. If G is ℙ-generic over V and H is ℚ-generic over V, then G and H are mutually generic.

4. With the same assumptions as the last claim, if X ∈ V [G][H] is a set of ordinals and V [G][H] ⊧ |X| < κ, then there is Y ⊇ X in V such that V ⊧ |Y | < κ.

5. If ℝ is κ-closed, then ⊩_{ℙ×ℚ} ℝ is < κ-distributive..

Proof. [Sketch] We take each claim in turn.

1. If Ȧ is a ℚ-name for an antichain of size κ in ℙ, we can build a decreasing sequence of length κ in ℚ such that element α of the chain decides element α in the antichain. This gives us an antichain of size κ in V, contradiction.

2. If X is a sequence of ordinals of length less than κ in V [ℚ × ℙ] then the last claim implies that X has a name ṫ of size less than κ in V [ℚ]^{ℙ}. ℚ is κ-closed so ṫ ∈ V and X ∈ V [ℙ].

3. Every maximal antichain of ℙ in V [H] has size less than κ, so is in V.

4. X ∈ V [G] and ℙ is κ-c.c.

5. ℝ × ℚ is < κ-distributive in V [ℙ].

We make the remark that claims 4 and 5 will actually be true for any extension intermediate between V and V [ℙ × ℚ].

The next lemma is an easy result with the same flavour as Easton’s lemma.

Lemma 2.12. Let ℙ be κ-closed and let ℚ be < κ-distributive, then

1. ⊩_{ℚ} ℙ is κ-closed.

2. ⊩_{ℙ} ℚ is < κ-distributive.

The final lemma is rather ad hoc, and will be used to propagate some inductive claims in the main construction.

Lemma 2.13. Let τ be regular, and let 𝔸 = Add(τ, η) for some η. Let κ be inaccessible with τ < κ. Then

1. If ℚ is κ-c.c. and ℚ is a projection of ℙ × 𝕌, where ℙ is τ-c.c. and 𝕌 is τ-closed, then V [ℚ] ⊧ “𝔸 is κ-Knaster and < τ-distributive”.

2. Suppose that V [ℚ] ⊧ “ℚ^{∗} is a projection of Add(τ, ζ)_{V} × 𝕌^{∗}” and also that V [ℚ] ⊧ “𝕌^{∗} is κ-closed”. Then V [ℚ ∗ ℚ̇^{∗}] ⊧ “𝔸 is κ-Knaster”.

Proof. A series of routine applications of Easton’s Lemma and Lemma 2.6.

2.6. Elementary embeddings. We will assume familiarity with the theory of large cardinals and elementary embeddings, as developed for example in [5].

The following fact was proved by Laver in [10], and used by him there as a kind of prediction principle in a forcing iteration. Our application will be similar, we will use it to build some degree of “lookahead” into the main construction.

Lemma 2.14 (Laver). If κ is supercompact then there exists f : κ ⟶ V_{κ} with the following property; for all λ, for all x ∈ H_{λ^{+}}, there is j : V ⟶ M such that j(κ) > λ, ^{λ}M ⊆ M, and j(f)(κ) = x.

We will frequently be interested in lifting elementary embeddings onto some generic extension. The following lemma gives a necessary and sufficient condition for this to be possible.

Lemma 2.15 (Silver). Let k : M ⟶ N be an elementary embedding between inner models of ZFC. Let ℙ ∈ M be a forcing, and suppose that G is ℙ-generic over M, H is k(ℙ)-generic over N, and

k“G ⊆ H.

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

Proof. It is clear that if k^{+} exists, it must be given by the formula

k^{+}(ṫ^{G}) = k(ṫ)^{H}

for ṫ ∈ M^{ℙ}. It is fairly routine to use the Truth Lemma and elementarity to check that this definition works.

It is important to notice that in this result it need not be the case that M = V, that N ⊆ M, or that H ∈ M[G]. We also note a connection with the theory [12] of proper forcing; if ℙ ∈ N ≺ H_{θ} and π : N ⟶ N̄ is the transitive collapse map, then p ∈ ℙ is (N, ℙ)-generic precisely when it forces that G^{∗} =_{def} π“G_{ℙ} ∩ N is π(ℙ)-generic over N̄, and in this case we can lift π^{−1} to get an embedding from N̄[G^{∗}] to N[G].

Part 1. The tree property for ω successive cardinals

In this part of the paper we will prove Theorem 1.

3. The main forcing

3.1. Defining ℝ. Before we define the main forcing, a few words of motivation may be in order. The principal aim of the main forcing is to take cardinals τ < κ with τ regular and κ supercompact, and (while preserving all cardinals up to τ^{+}) to force that 2^{τ} = κ = τ^{++} and that κ retains the tree property. We aim to iterate the main forcing in a certain way, and stage n + 1 of the iteration will be defined using some forcing computed at stage n; this explains why an inner model is one of the parameters in the definition of the main forcing. The main forcing is also designed to make κ have the tree property in a very “indestructible” way, so the last parameter in the definition is a function from κ to V_{κ} whose rôle is to guess some information about subsequent forcing extensions.

Definition 3.1. Let V ⊆ W be two models of set theory, and suppose that we have τ, κ such that W ⊧ “τ = cf(τ) < κ and κ is inaccessible”. Let ℙ = Add(τ, κ)_{V} and suppose also that W ⊧ “ℙ is τ^{+}-c.c.and < τ-distributive”. Let F ∈ W be a function with F : κ ⟶ (V_{κ})_{W}. Define in W a forcing

ℝ = ℝ(τ, κ, V, W, F)

as follows. The definition is by induction; for each β ≤ κ we will define a forcing ℝ ↾ β, and we will finally set ℝ = ℝ ↾ κ. To start the induction, ℝ ↾ 0 is the trivial forcing.

(p, q, f) is a condition in ℝ ↾ β iff

1. p ∈ ℙ ↾ β = Add(τ, β)_{V}.

2. q is a partial function on β, |q| ≤ τ, dom(q) consists of successor ordinals, and if α ∈ dom(q) then q(α) ∈ W^{ℙ↾α} and ⊩_{ℙ↾α}^{W} q(α) ∈ Add(τ^{+}, 1)_{W [ℙ↾α]}.

3. f is a partial function on β, |f| ≤ τ, dom(f) consists of limit ordinals, and dom(f) is a subset of

{ α : ⊩_{ℝ↾α}^{W} F (α) is a canonically τ^{+}-directed-closed forcing }

4. If α ∈ dom(f) then f(α) ∈ W^{ℝ↾α} and ⊩_{ℝ↾α}^{W} f(α) ∈ F (α).

The conditions in ℝ ↾ β are ordered in the following way.

(p_{1}, q_{1}, f_{1}) ≤ (p, q, f)

if and only if 1. p_{1} ≤ p in ℙ ↾ β.

2. For all α ∈ dom(f), p_{1} ↾ α ⊩_{ℙ↾α}^{W} q_{1}(α) ≤ q(α).

3. For all α ∈ dom(q), (p_{1}, q_{1}, f_{1}) ↾ α ⊩_{ℝ↾α}^{W} f_{1}(α) ≤ f(α).

It is important to notice that if (p, q, f) ∈ ℝ and α < κ then the term q(α) depends only on ℙ ↾ α, while f(α) depends on ℝ ↾ α. It is also important that the definition is made in the model W, and the only appearance of V is as the model where the Cohen forcing ℙ is to be defined.

One should think of ℝ as aiming to add several objects to W.

1. A ℙ-generic object.

2. For each successor α, an Add(τ^{+}, 1)_{W [ℙ↾α]}-generic object over W [ℙ ↾ α].

3. For each appropriate limit α, an F (α)-generic object over W [ℝ ↾ α].

We need to analyse the properties of ℝ, which we will do in a series of lemmas.

3.2. Easy properties of ℝ.

Lemma 3.2. |ℝ| = κ, and ℝ has the κ-Knaster property.

Proof. An easy argument counting terms shows that |ℝ| = κ, the key point being that at each α there fewer than κ possibilities for q(α) or f(α). A standard Δsystem argument then shows that ℝ has the κ-Knaster property.

Lemma 3.3. ℝ can be projected to ℙ, ℝ ↾ α ∗ F (α), and ℙ ↾ α ∗ Add(τ^{+}, 1)_{W [ℙ↾α]}.

Proof. Define projection maps

π_{0} : (p, q, f) ⟼ p
π_{1} : (p, q, f) ⟼ ((p, q, f) ↾ α, f(α))
π_{2} : (p, q, f) ⟼ (p ↾ α, q(α))

It is routine to check that these are projections (this was in fact one motivation for the definition of ℝ).

It follows that if G is ℝ-generic then G induces a ℙ-generic object over V, g say. G also induces an Add(τ^{+}, 1)_{V [g↾α]}-generic object over V [g ↾ α] and a F (α)^{G↾α}- generic object over V [G ↾ α].

Lemma 3.4. ℝ adds at least κ subsets to τ.

Proof. ℝ projects to ℙ, ℙ adds at least κ subsets to τ.

Lemma 3.5. ℝ collapses every cardinal between τ^{+} and κ to τ^{+}.

Proof. Let α be such a cardinal. ℝ projects to a forcing which makes 2^{τ} ≥ α and then adds a Cohen subset of τ^{+}. This forcing will collapse α to τ^{+}.

Lemma 3.6. Let η ≤ τ. If ℙ is η-closed in W then ℝ is η-closed in W.

Proof. Let μ < η and suppose that ⟨(p_{α}, q_{α}, f_{α}) : α < μ⟩ is a decreasing sequence in ℝ. By hypothesis we can find p ∈ ℙ such that p ≤ p_{α} for all α < μ. It is easy to see that for each β the condition p ↾ β forces the sequence ⟨q_{α}(β) : α < μ⟩ to be decreasing in Add(τ^{+}, 1)_{W [ℙ↾β]}; Add(τ^{+}, 1)_{W [ℙ↾β]} is τ^{+}-closed in W [ℙ ↾ β] so we may choose a ℙ ↾ β-term q(β) which is forced by p ↾ β to be a lower bound. The domain of q has cardinality at most τ.

Now we will define f(β) by induction on β in such a way that (p, q, f) ↾ β is a lower bound for ⟨(p_{α}, q_{α}, f_{α}) ↾ β : α < μ⟩. If we have this up to stage β then (p, q, f) ↾ β forces that ⟨f_{α}(β) : α < μ⟩ is decreasing in F (β); F (β) is τ^{+}-closed in W [ℝ ↾ β], so we may choose a ℝ ↾ β-term f(β) which is forced by (p, q, f) ↾ β to be a lower bound. The domain of f will also have cardinality at most τ, so the induction can proceed and at the end (p, q, f) is the lower bound that we require.

To sum up, we have seen that ℝ preserves cardinals greater than or equal to κ, collapses all cardinals strictly between τ^{+} and κ, and adds κ subsets to τ. To understand what is happening in W [ℝ] at τ^{+} and below, we will analyse ℝ as the projection of ℙ × 𝕌 for some τ^{+}-closed forcing U.

3.3. Factoring ℝ.

Definition 3.7. With the same data as for ℝ, 𝕌 = 𝕌(τ, κ, V, W, F) is the ordering with underlying set { (0, q, f) : (0, q, f) ∈ ℝ } and the partial ordering inherited from ℝ.

Lemma 3.8. 𝕌 has the κ-chain condition.

Proof. A standard Δ-system argument.

Lemma 3.9. In W, 𝕌 is a canonically τ^{+}-directed closed poset.

Proof. We write out explicitly the definition of the ordering on 𝕌. (0, q_{1}, f_{1}) ≤ (0, q_{0}, f_{0}) if and only if

1. dom(q_{0}) ⊆ dom(q_{1}), and ⊩_{ℙ↾α}^{W} q_{1}(α) ≤ q_{0}(α) for all α ∈ dom(q_{0}).

2. dom(f_{0}) ⊆ dom(f_{1}), and (0, q_{1}, f_{1}) ↾ α ⊩_{ℝ↾α}^{W} f_{1}(α) ≤ f_{0}(α) for all α ∈ dom(f_{0}).

Let { (0, q_{η}, f_{η}) : η < τ } be a directed set of conditions. Let us define A_{1} =_{def} ⋃_{η<τ} dom(q_{η}), and observe that |A_{1}| ≤ τ. We will define a function q with domain A_{1}. For α ∈ A_{1}, consider { q_{η}(α) : η < τ }. If η, ζ < τ then for some μ < τ we have that (0, q_{μ}, f_{μ}) is a common refinement of (0, q_{η}, f_{η}) and (0, q_{ζ}, f_{ζ}). In particular ⊩ q_{μ}(α) ≤ q_{η}(α), q_{ζ}(α). So we can look at { q_{η}(α) : η < τ } as a name in W^{ℙ↾α} for a directed set of size τ in Add(τ^{+}, 1)_{W [ℙ↾α]}, and find r(α) which is forced to be the greatest lower bound. In particular, ⊩ r(α) ≤ q_{η}(α) for all η < τ.

Now let A_{2} =_{def} ⋃_{η<τ} dom(f_{η}), and observe that |A_{2}| ≤ τ. We will define by induction on α a function g with domain A_{2} such that (0, r, g) ↾ α ⊩ g(α) ≤ f_{η}(α) for all α, η. Fix α. As we remarked already, if η, ζ < τ then for some μ < τ we have that (0, q_{μ}, f_{μ}) is a common refinement of (0, q_{η}, f_{η}) and (0, q_{ζ}, f_{ζ}), in particular (0, q_{μ}, f_{μ}) ↾ α ⊩ f_{μ}(α) ≤ f_{η}(α), f_{ζ}(α). By induction (0, r, g) ↾ α ≤ (0, q_{μ}, f_{μ}) ↾ α, so (0, r, g) ↾ α ⊩ f_{μ}(α) ≤ f_{η}(α), f_{ζ}(α). Now (0, r, g) ↾ α forces that { f_{η}(α) : η < τ } is directed. We define g(α) rather carefully, to be a name which denotes the greatest lower bound of { f_{η}(α) : η < τ } if that set is directed, and the trivial condition otherwise. In particular (0, r, g) ↾ α ⊩ g(α) ≤ f_{η}(α) for all η. At the end we have constructed a condition (0, r, g) which is a lower bound for the directed set { (0, q_{η}, f_{η}) : η < τ }.

It remains to be seen that (0, r, g) is a greatest lower bound. Let (0, s, h) be any condition such that for all η (0, s, h) ≤ (0, q_{η}, f_{η}). Clearly A_{1} ⊆ dom(s) and A_{2} ⊆ dom(h). For each α ∈ dom(s) we have ⊩ s(α) ≤ f_{η}(α) for all η, and since r(α) is forced to be a greatest lower bound ⊩ s(α) ≤ r(α). We attempt to show by induction that (0, s, h) ↾ α ⊩ h(α) ≤ g(α). If it is true below α then (0, s, h) ↾ α ≤ (0, r, g) ↾ α, so that (0, s, h) ↾ α forces that { f_{η}(α) : η < τ } is directed. It also follows from the hypothesis that (0, s, h) ↾ α ⊩ h(α) ≤ f_{η}(α) for all η, so that by our choice of g(α) (0, s, h) ↾ α ⊩ h(α) ≤ g(α). So the induction goes through, and at the end we have shown (0, s, h) ≤ (0, r, g). Hence (0, r, g) is the greatest lower bound.

Lemma 3.10. In ℝ, (p, q, f) is the greatest lower bound for (p, 0, 0) and (0, q, f).

Proof. (p, q, f) is clearly a lower bound. Suppose that (p_{1}, q_{1}, f_{1}) is also a lower bound. Then by definition p_{1} ≤ p, p_{1} ↾ α ⊩ q_{1}(α) ≤ q(α), and (p_{1}, q_{1}, f_{1}) ↾ α ⊩ f_{1}(α) ≤ f(α). That is to say (p_{1}, q_{1}, f_{1}) ≤ (p, q, f).

Lemma 3.11. With ℙ and 𝕌 as above 1. ℙ × 𝕌 is κ-c.c.

2. All τ-sequences of ordinals in W [ℙ × 𝕌] are in W [ℙ].

Proof. By Easton’s lemma, ℙ is τ^{+}-c.c. in W [𝕌]. Since 𝕌 is κ-c.c. in W, ℙ × 𝕌 is κ-c.c. Easton’s lemma also shows that all τ-sequences of ordinals from W [ℙ × 𝕌] are in W [ℙ].

Definition 3.12. π : ℙ × 𝕌 ⟶ ℝ is the function given by

π : (p, (0, q, f)) ⟼ (p, q, f).

Lemma 3.13. π is a projection.

Proof. It is clear that π preserves the identity and respects the ordering relation.

Let us have (p_{1}, q_{1}, f_{1}) ≤ (p_{0}, q_{0}, f_{0}) in ℝ. Observe that for all α we have p_{1} ↾ α ⊩ q_{1}(α) ≤ q_{0}(α). Define q̄(α) as a name with the following property; for G any ℙ ↾ α-generic object, q̄(α) interprets as q_{1}(α)^{G} if p_{1} ↾ α ∈ G, and interprets as q_{0}(α)^{G} if p_{1} ↾ α ∉ G. By construction, ⊩ q̄(α) ≤ q_{0}(α) and p_{1} ↾ α ⊩ q̄(α) = q_{1}(α).

Now we attempt to define by induction a term f̄(α) such that (p_{1}, q̄, f̄) ↾ α ⊩ f̄(α) = f_{1}(α) and (0, q̄, f̄) ↾ α ⊩ f̄(α) ≤ f_{0}(α). If we have done this for stages below α, then the conditions (p_{1}, q̄, f̄) ↾ α and (p_{1}, q_{1}, f_{1}) ↾ α are equivalent in ℝ ↾ α.

By hypothesis, (p_{1}, q_{1}, f_{1}) ↾ α ⊩ f_{1}(α) ≤ f_{0}(α). Define f̄(α) as follows: for any ℝ ↾ α-generic object G, the interpretation of f̄(α) is f_{1}(α)^{G} if (p_{1}, q̄, f̄) ↾ α ∈ G and f_{0}(α)^{G} otherwise. Now (p_{1}, q̄, f̄) ↾ α ⊩ f̄(α) = f_{1}(α), and ⊩ f̄(α) ≤ f_{0}(α), so we are done.

At the end of this construction we have shown that the conditions (p_{1}, q̄, f̄) and (p_{1}, q_{1}, f_{1}) are equivalent in ℝ and (0, q̄, f̄) ≤ (0, q_{0}, f_{0}), which is what is needed.

Recall that we also have projections ρ : ℝ ⟶ ℙ and σ : ℙ × 𝕌 ⟶ ℙ given by ρ : (p, q, f) ⟼ p and σ : (p, (0, q, f)) ⟼ p. These projections commute as in Figure 1. Consequently we may take it that W ⊆ W [ℝ] ⊆ W [ℙ × 𝕌].

Figure 1. Projections between ℙ × 𝕌, ℝ and ℙ.

Corollary 3.14. Let G be ℝ-generic over W, let g be the ℙ-generic object added by G. If X ∈ W [G] is a set of ordinals of size τ, then X ∈ W [g].

Proof. By Lemma 3.13 we can embed W [G] in a larger extension W [g][H], where H is 𝕌-generic over W [g]. As ℙ is τ^{+}-c.c. and 𝕌 is τ^{+}-closed, Easton’s lemma implies that X ∈ W [g].

Corollary 3.15. ℝ preserves τ^{+}, and forces that 2^{τ} = κ = τ^{++}.

Proof. If f ∈ W [G] and f : τ ⟶ τ_{W}^{+} then by Corolary 3.14 f ∈ W [g], and g is generic for τ^{+}-c.c. forcing so that f is bounded. Similarly Pτ ∩ W [G] = Pτ ∩ W [g], and standard arguments show that 2^{τ} = κ in W [g]. Finally, we saw in Lemma 3.5 that ℝ collapses everything in the interval [τ^{+}, κ) to τ^{+}.

Corollary 3.16. ℝ is < τ-distributive in W, so preserves cardinals less than or equal to τ.

Proof. If X ∈ W [G] is a set of ordinals of size less than τ, then X ∈ W [g] by Corollary 3.14. But ℙ is < τ-distributive in W, so X ∈ W.

Corollary 3.17. 𝕌 is κ-c.c. and τ^{+}-closed. In W [𝕌] we have τ^{++} = κ.

Proof. We already saw the closure and chain condition results. 𝕌 must collapse κ to τ^{++} because W [𝕌] ⊧ “ℙ is τ^{+}-c.c. and < τ-distributive”, and κ is collapsed to τ^{++} in W [𝕌 × ℙ].

Corollary 3.18. If X ∈ W [G] is a set of ordinals of size τ, then X is covered by a set of size τ in W.

Proof. X ∈ W [g] and ℙ is τ^{+}-c.c. in W, so the claim follows by a standard chain condition argument.

3.4. The forcing 𝕊. W [ℙ × 𝕌] is a generic extension of W [ℝ], and we need some information about this extension.

Definition 3.19. Let G be ℝ-generic over W. Then define in W [G] a forcing 𝕊 = 𝕊(τ, κ, V, W, F, G) whose conditions are { (p, (0, q, f)) : (p, q, f) ∈ G }, ordered as a suborder of ℙ × 𝕌.

𝕊 is a version of the forcing to expand G to an ℙ×𝕌-generic object which projects to G. We need to analyse the properties of 𝕊. For a careful analysis of this sort of forcing, in a slightly different setting, we refer the reader to Foreman’s paper [3].

Lemma 3.20. Let G be generic for ℝ and let 𝕊 =_{def} 𝕊(τ, κ, V, W, F, G). Then W [G] ⊧ “𝕊 is < τ^{+}-distributive, τ-closed and κ-c.c.”. In particular forcing with 𝕊 over W [G] does not collapse cardinals.

Proof. The κ-chain condition and < τ^{+}-distributivity follow at once from our earlier remarks about ℙ × 𝕌. It remains to show τ-closure.

In W [G], let ⟨(p_{ζ}, (0, q_{ζ}, f_{ζ})) : ζ < μ⟩ be a decreasing μ-sequence of conditions from 𝕊 for some μ < τ. By Corollary 3.16, this sequence is a member of W. Let p = ⋃_{ζ} p_{ζ}, then (by Lemma 2.2 or elementary facts about Cohen forcing) p ∈ g, where g is the ℙ-generic added by ℝ.

Since the sequence ⟨(0, q_{ζ}, f_{ζ}) : ζ < μ⟩ is decreasing in 𝕌, we may perform the construction of Lemma 3.9 to get (0, q̄, f̄) which is a greatest lower bound in 𝕌 for this sequence. We now claim that (p, q̄, f̄) ∈ G.

We already know that p ∈ g. Fix a successor α < κ, then by the definition of q̄(α) we see that q̄^{g↾α} is the greatest lower bound for the sequence ⟨q_{ζ}(α)^{g↾α} : ζ < μ⟩ in the forcing Add(τ^{+}, 1)_{W [g↾α]}. If we let G_{α}^{0} be the Add(τ^{+}, 1)_{W [g↾α]}-generic filter added by G then we know that q_{ζ}(α)^{g↾α} ∈ G_{α}^{0} for all ζ, and so by another application of Lemma 2.2 q̄(α)^{g↾α} ∈ G_{α}^{0}.

For each relevant limit α < κ let G_{α}^{1} denote the F (α)^{G↾α}-generic filter added by G. We will prove by induction on α that f̄(α)^{G↾α} ∈ G_{α}^{1}. Suppose that we have done this up to stage α, so that in particular (0, q̄, f̄) ↾ α ∈ G ↾ α. Since (0, q̄, f̄) is a lower bound for ⟨(0, q_{ζ}, f_{ζ}) : ζ < μ⟩ the condition (0, q̄, f̄) ↾ α forces that ⟨f_{ζ}(α) : ζ < μ⟩ is decreasing, so that ⟨f_{ζ}(α)^{G↾α}⟩ is a decreasing sequence of members of G_{α}^{1}. What is more q̄(α)^{G↾α} is the greatest lower bound of this sequence, so that applying Lemma 3.9 one more time q̄(α)^{G↾α} ∈ G_{α}^{1} and we are done.

3.5. ℝ^{∗} and 𝕌^{∗}. To conclude the analysis of ℝ, we need to look at the forcing obtained when we factor ℝ over one of its initial segments. Fix β < κ and consider the projection π_{β} : ℝ ⟶ ℝ ↾ β given by restriction. It is easy to see that this is a good projection, in particular if G_{β} is ℝ ↾ β-generic then we may consider the forcing to prolong G_{β} to an ℝ-generic object as given by the following definition.

Definition 3.21. Given β < κ and G_{β} generic over W for ℝ, define

ℝ^{∗} = ℝ^{∗}(τ, κ, V, W, F, G_{β}) = { r ∈ ℝ : r ↾ β ∈ G_{β} },

with the ordering on ℝ^{∗} given by r_{1} ≤ r_{0} ⟺ ∃s ∈ G_{β} Ext(r_{1}, s) ≤ r_{0}. We observe that here Ext(r_{1}, s) is just the extension of r_{1} in which r_{1} ↾ β is replaced by s.

Definition 3.22. Given β and G_{β} as above, define 𝕌^{∗} = 𝕌^{∗}(τ, κ, V, W, F, G_{β}) = { (0, q, f) : (0, q, f) ∈ ℝ^{∗} }, ordered as a suborder of ℝ^{∗}.

Definition 3.23. With β and G_{β} as above, let ℙ^{∗} be { p ∈ ℙ : (p, 0, 0) ∈ ℝ^{∗} }, ordered as a suborder of ℙ.

It is easy to see that ℙ^{∗} is essentially ℙ ↾ κ − β = Add(τ, κ − β)_{V}.

Lemma 3.24. If we define π on domain ℙ^{∗} × 𝕌^{∗} by π : (p, (0, q, f)) ⟼ (p, q, f), then π is a projection from ℙ^{∗} × 𝕌^{∗} to ℝ^{∗}.

Proof. It follows from lemma 3.10 that (p, q, f) ∈ ℝ^{∗}. The rest is also fairly routine.

Lemma 3.25. 𝕌^{∗} is τ^{+}-closed in W [G_{β}].

Proof. The proof is similar to that of Lemma 2.18 in [1]. Let τ̇ name a descending τ-sequence in 𝕌^{∗}. Let g_{β} be the ℙ ↾ β-generic object added by G_{β}. We may assume that τ̇ ∈ W^{ℙ↾β}, because all τ-sequences in W [G_{β}] come from W [g_{β}]. τ̇_{η} will denote the canonical term for entry η in the sequence named by τ̇. We adopt from [1] the convention that

L(0, q, f) = q
R(0, q, f) = f

Let G_{γ} be ℝ ↾ γ-generic, let g_{γ} be the associated ℙ ↾ γ-generic object. Then τ̇_{η}^{g_{γ}↾β} is a condition in 𝕌^{∗}, so that [R(τ̇_{η}^{g_{γ}↾β})]^{G_{γ}} ∈ F (γ)^{G_{γ}} . Similarly, [L(τ̇_{η}^{g_{γ}↾β})]^{g_{γ}} ∈ Add(τ^{+}, 1)_{W [g_{γ}]}.

We will define in W a condition (0, q^{∗}, f^{∗}) ∈ ℝ such that

• dom(q^{∗}) ⊆ κ − β, dom(f^{∗}) ⊆ κ − β.

• dom(q^{∗}) is the set of γ ≥ β such that for some η < τ, γ is a potential member of the domain of L(τ̇_{η}).

• dom(f^{∗}) is the set of γ ≥ β such that for some η < τ, γ is a potential member of the domain of R(τ̇_{η}).

• For all γ ≥ β, if G_{γ} is ℝ ↾ γ-generic and (0, q^{∗}, f^{∗}) ↾ γ ∈ G_{γ} , then ∀η < τ τ̇_{η}^{g_{γ}↾β} ↾ γ ∈ G_{γ}.

We observe that by τ^{+}-c.c. for ℙ in W, the domains are not too big. We will start by setting (0, q^{∗}, f^{∗}) ↾ β = (0, 0, 0). Suppose we have defined (0, q^{∗}, f^{∗}) ↾ γ successfully. We will now define f^{∗}(γ). Let G_{γ} be ℝ ↾ γ-generic, and assume that (0, q^{∗}, f^{∗}) ↾ γ ∈ G_{γ}. By our induction hypothesis, ∀η < τ τ̇_{η}^{g_{γ}↾β} ↾ γ ∈ G_{γ}. We will work in W [G_{γ}].

Claim . Define a τ-sequence of conditions in F (γ)^{G_{γ}} by p(η) = [R(τ̇_{η}^{g_{γ}↾β})]^{G_{γ}} . Then this is a decreasing sequence.

Proof. Let ζ < η < τ, and suppose that

τ̇_{ζ}^{g_{γ}↾β} = (0, q, f)
τ̇_{η}^{g_{γ}↾β} = (0, q̄, f̄)

Notice that p(ζ) = f(γ)^{G_{γ}}, p(η) = f̄(γ)^{G_{γ}}. What is more (0, q, f) ↾ γ and (0, q̄, f̄) ↾ γ are in G_{γ}. We may choose s ∈ G_{γ} ↾ β such that s ⊩ τ̇_{ζ} = (0, q, f), s ⊩ τ̇_{η} = (0, q̄, f̄), and Ext(s, (0, q̄, f̄)) ≤ (0, q, f) in ℝ. Now Ext(s, (0, q̄, f̄)) ↾ γ ∈ G_{γ}, so that (by the definition of extension in ℝ) p(η) = f̄(γ)^{G_{γ}} ≤ f(γ)^{G_{γ}} = p(ζ).

This proves the claim.

Now we choose f^{∗}(γ) to be a name, forced by (0, q^{∗}, f^{∗}) ↾ γ to be a lower bound for that sequence p⃗. We observe for the record that if we assume (0, q^{∗}, f^{∗}) ↾ γ ∈ G_{γ}, then f^{∗}(γ)^{G_{γ}} ≤ [R(τ̇_{η}^{g_{γ}↾β})]^{G_{γ}} . The choice of q^{∗}(γ) is similar. Let G_{γ}, g_{γ} be as usual, where we assume that (0, q^{∗}, f^{∗}) ↾ γ ∈ G_{γ}. Working in W [g_{γ}] define a sequence q in Add(τ^{+}, 1) by q(η) = [L(τ̇_{η}^{g_{γ}↾β})]^{g_{γ}} . Working much as before we can show that q is decreasing. Now choose q^{∗}(γ) to be a ℙ ↾ γ-name for a lower bound.

We check that the induction hypothesis goes through. Suppose that (0, q^{∗}, f^{∗}) ↾ γ + 1 ∈ G_{γ+1}, and let η < τ. Suppose that τ̇_{η}^{g_{γ+1}↾β} = t_{η} = (0, q_{η}, f_{η}). Then t_{η} ↾ γ ∈ G_{γ}, and by construction we know that q^{∗}(γ)^{g_{γ}} ≤ q_{η}(γ)^{g_{γ}} and f^{∗}(γ)^{G_{γ}} ≤ q_{η}(γ)^{G_{γ}}. So t_{η} ↾ γ + 1 ∈ G_{γ+1}.

Limits do not present a problem, so that the construction of (0, q^{∗}, f^{∗}) can proceed. We finish by showing that we have constructed a lower bound.

Claim . Let G_{β} be ℝ ↾ β-generic. Then (0, q^{∗}, f^{∗}) is a lower bound in 𝕌^{∗} for the sequence τ̇^{g_{β}}.

Proof. Let ζ < τ, and suppose τ̇_{ζ}^{g_{β}} = (0, q, f). Choose r ∈ G_{β} such that r ≤ (0, q, f) ↾ β and r ⊩ τ̇_{ζ} = (0, q, f). Now by construction

r ↾ γ ⊩ q^{∗}(γ) ≤ q(γ)

Ext((0, q^{∗}, f^{∗}) ↾ γ, r) ⊩ f^{∗}(γ) ≤ f(γ)

for each γ, so that Ext((0, q^{∗}, f^{∗}), r) ≤ (0, q, f).

With the last claim, the proof is done.

4. The final model

4.1. Building the model. Let ⟨κ_{n} : n < ω⟩ be an increasing sequence of supercompact cardinals. We will build a model by iterating the forcing described in the last section, in such a way that κ_{n} becomes ℵ_{n+2} in the final model.

Definition 4.1. Fix ⟨F_{n} : n < ω⟩ such that F_{n} : κ_{n} ⟶ V_{κ_{n}} has the “diamond” property described in Lemma 2.14. We will define a forcing iteration ℝ_{ω} of length ω.

1. The first stage is ℚ_{0} =_{def} ℝ(ℵ_{0}, κ_{0}, V, V, F_{0}), and ℝ_{1} =_{def} ℚ_{0}.

2. Define a name Ḟ_{1} in V^{ℚ_{0}} for a function from κ_{1} to (V_{κ_{1}})_{V [ℚ_{0}]}, by setting Ḟ_{1}^{G_{0}}(α) = F_{1}(α)^{G_{0}} if F_{1}(α) is a ℚ_{0}-name and Ḟ_{1}^{G_{0}}(α) = 0 otherwise. Then define ℚ̇_{1} to be the canonical name for ℝ(ℵ_{1}^{V} , κ_{1}, V, V [ℚ_{0}], F_{1}^{∗}) where F_{1}^{∗} is the realisation of Ḟ_{1} in V [ℚ_{0}]. and ℝ_{2} =_{def} ℚ_{0} ∗ ℚ̇_{1}.

3. Suppose that we have defined the iteration up to stage n, where n ≥ 2. Let ℝ_{n} =_{def} ℚ_{0} ∗ … ∗ ℚ̇_{n−1}. As at stage one, define a ℝ_{n}-name Ḟ_{n} by Ḟ_{n}^{G_{n}}(α) = F_{n}(α)^{G_{n}} if F_{n}(α) is a ℝ_{n}-name and Ḟ_{n}^{G_{n}}(α) = 0 otherwise. Then define ℚ̇_{n} to name ℝ(κ_{n−2}, κ_{n}, V [ℝ_{n−1}], V [ℝ_{n}], F_{n}^{∗}) where F_{n}^{∗} is the interpretation of F_{n} in V [ℝ_{n}].

4. ℝ_{ω} is the inverse limit of ⟨ℝ_{n} : n < ω⟩.

It is not yet clear that this definition is legitimate, because we can only define ℝ(τ, κ, V, W, F) when we know that certain things are true in W; namely τ must be regular, κ must be inaccessible, and Add(τ, κ)_{V} must be τ^{+}-c.c. and < τdistributive. We will now do an inductive analysis of ℙ_{n} and ℚ̇_{n}, which will show among other things that the definition is a valid one. The case n = 0 needs special treatment; this case is essentially Abraham’s analysis of the main forcing from [1].

Lemma 4.2. Let ℙ_{0} =_{def} Add(ℵ_{0}, κ_{0}) and 𝕌_{0} =_{def} 𝕌(ℵ_{0}, κ_{0}, V, V, F_{0}).

1. |ℚ_{0}| = κ_{0}, and ℚ_{0} is κ_{0}-Knaster. In particular (a) All V -cardinals greater than or equal to κ_{0} are preserved in V [ℚ_{0}].

(b) All V -inaccessibles greater than κ_{0} remain inaccessible in V [ℚ_{0}].

(c) All sets of size less than κ_{0} in V [ℚ_{0}] are covered by sets of size less than

κ_{0} in V.

2. ℚ_{0} is a projection of ℙ_{0} × 𝕌_{0}, and V [ℙ_{0}] ⊆ V [ℚ_{0}] ⊆ V [ℙ_{0} × 𝕌_{0}].

3. All ω-sequences of ordinals from V [ℚ_{0}] are in V [ℙ_{0}].

4. ℚ_{0} preserves ℵ_{1}, and 2^{ℵ_{0}} = κ_{0} = ℵ_{2} in V [ℚ_{0}].

5. Add(ℵ_{0}, η)_{V} has the ℵ_{1}-Knaster property in V [ℚ_{0}].

6. For all η, V [ℚ_{0}] ⊧ “Add(ℵ_{1}, η)_{V} is < ℵ_{1}-distributive and κ_{0}-Knaster”.

Proof. We take the claims in turn.

1. This follows from Lemma 3.2 and standard facts about forcing.

2. This is exactly the content of Lemma 3.13.

3. This follows from 3.14.

4. Immediate by Corollary 3.15.

5. Add(ℵ_{0}, η)_{V} = Add(ℵ_{0}, η)_{V [ℚ_{0}]}.

6. Use clause 1 from Lemma 2.13, with τ = ℵ_{1}, κ = κ_{0}, ℚ = ℚ_{0}, ℙ = ℙ_{0}, and 𝕌 = 𝕌_{0}.

The definition of ℚ_{1} is now seen to be legitimate, because in V [ℚ_{0}] we know that ℵ_{1}^{V} is regular, κ_{1} is inaccessible, and Add(ℵ_{1}, κ_{1})_{V} is < ℵ_{1}-distributive and ℵ_{2}-Knaster. We now turn to the general case. In the statement of the following lemma, when we refer to “ℵ_{i}” we mean ℵ_{i} in the sense of V [ℝ_{n}]; a key point is that the values of the cardinals ℵ_{i} for i ≤ n + 1 are already fixed in V [ℝ_{n}], namely ℵ_{0} and ℵ_{1} are as in the ground model and ℵ_{i} = κ_{i−2} for 2 ≤ i ≤ n + 1.

Lemma 4.3. Let n ≥ 1. Let ℝ_{n} = ℚ_{0} ∗ … ℚ_{n−1}, ℝ_{n−1} = ℚ_{0} ∗ … ℚ_{n−2}, ℙ_{n} = Add(ℵ_{n}, κ_{n})_{V [ℝ_{n−1}]} and 𝕌̇_{n} = 𝕌(ℵ_{n}, κ_{n}, V [ℝ_{n−1}], V [ℝ_{n}], F_{n}^{∗}).

1. In V [ℝ_{n}], cardinal arithmetic follows the pattern 2^{ℵ_{i}} = ℵ_{i+2} = κ_{i} for i < n. κ_{i} is inaccessible for i ≥ n.

2. V [ℝ_{n}] ⊧ “ℚ_{n} is < ℵ_{n}-distributive, κ_{n}-Knaster and ℵ_{n−1}-closed”, and also V [ℝ_{n}] ⊧ “|ℚ_{n}| = κ_{n}”. In particular

(a) All V -cardinals greater than or equal to κ_{n} are preserved in V [ℝ_{n} ∗ ℚ̇_{n}].

(b) All V -inaccessibles greater than κ_{n} remain inaccessible in V [ℝ_{n} ∗ ℚ̇_{n}].

(c) All sets of ordinals of size less than κ_{n} in V [ℝ_{n} ∗ ℚ̇_{n}] are covered by sets

of size less than κ_{n} in V [ℝ_{n}].

3. All ℵ_{n−1}-sequences of ordinals from V [ℝ_{n} ∗ ℚ̇_{n}] are in V [ℝ_{n−1} ∗ ℙ̇_{n−1}].

4. All cardinals up to ℵ_{n} are preserved in V [ℝ_{n} ∗ ℚ̇_{n}].

5. V [ℝ_{n}] ⊧ “ℚ_{n} is a projection of ℙ_{n} × 𝕌_{n}”, and V [ℝ_{n} ∗ ℙ̇_{n}] ⊆ V [ℝ_{n} ∗ ℚ̇_{n}] ⊆ V [ℝ_{n} ∗ (ℙ̇_{n} × 𝕌̇_{n})].

6. All ℵ_{n}-sequences of ordinals from V [ℝ_{n} ∗ ℚ̇_{n}] are in V [ℝ_{n} ∗ ℙ̇_{n}].

7. ℵ_{n+1} (which is κ_{n−1}) is preserved in V [ℝ_{n} ∗ ℚ̇_{n}]. Cardinal arithmetic in V [ℝ_{n} ∗ ℚ̇_{n}] follows the pattern 2^{ℵ_{i}} = ℵ_{i+2} = κ_{i} for i ≤ n.

8. Add(ℵ_{n}, η)_{V [ℝ_{n−1}]} is ℵ_{n+1}-Knaster in V [ℝ_{n} ∗ ℚ̇_{n}], for any ordinal η.

9. V [ℝ_{n} ∗ ℚ̇_{n}] ⊧ “Add(ℵ_{n+1}, η)_{V [ℝ_{n}]} is < ℵ_{n+1}-distributive and κ_{n}-Knaster”, for any ordinal η.

Proof. We prove the lemma by induction on n ≥ 1, using Lemma 4.2 to get some information in the case n = 1. Notice that by induction the forcing ℙ_{n} has the right distributivity and chain condition in V [ℝ_{n}], and that ℵ_{n} and κ_{n} are respectively regular and inaccessible in V [ℝ_{n}]; it is therefore legitimate to define ℚ̇_{n}.

1. This is immediate by induction.

2. By Lemma 3.2, |ℚ_{n}| = κ_{n} and ℚ_{n} is κ_{n}-Knaster in V [ℝ_{n}]. By Lemma 3.16, ℚ_{n} is < ℵ_{n}-distributive in V [ℝ_{n}]. For the closure, observe that ℙ_{n} is ℵ_{n}-closed in V [ℝ_{n−1}] and that (by induction) ℚ_{n−1} is < ℵ_{n−1}-distributive in V [ℝ_{n−1}], so that ℙ_{n} is ℵ_{n−1}-closed in V [ℝ_{n}]. By Lemma 3.6 ℚ_{n} is ℵ_{n−1}-closed in V [ℝ_{n}].

3. Since ℚ_{n} is < ℵ_{n}-distributive, every ℵ_{n−1}-sequence of ordinals from V [ℝ_{n}∗ℚ̇_{n}] is in V [ℝ_{n}] = V [ℝ_{n−1} ∗ ℚ̇_{n−1}]. By induction, every ℵ_{n−1}-sequence of ordinals from V [ℝ_{n−1} ∗ ℚ̇_{n−1}] is in V [ℝ_{n−1} ∗ ℙ̇_{n−1}].

4. This follows immediately from the last claim.

5. Apply Lemma 3.13 in V [ℝ_{n}].

6. Apply Corollary 3.14 in V [ℝ_{n}].

7. By Corollary 3.15, ℵ_{n+1} is preserved in V [ℝ_{n} ∗ ℚ̇_{n}]. Since ℚ_{n} is < ℵ_{n}- distributive in V [ℝ_{n}], it follows that all cardinals up to ℵ_{n} are preserved and that we still have 2^{ℵ_{i}} = κ_{i} = ℵ_{i+2} for i < n in V [ℝ_{n} ∗ ℚ̇_{n}]. By Corollary 3.15 again, 2^{ℵ_{n}} = κ_{n} = ℵ_{n+2} in V [ℝ_{n} ∗ ℚ̇_{n}].

8. Apply clause 2 of Lemma 2.13 in V [ℝ_{n−1}] with τ = ℵ_{n}, ℚ = ℚ_{n−1}, ℚ^{∗} = ℚ_{n}.

9. Apply clause 1 of Lemma 2.13 in V [ℝ_{n}] with τ = ℵ_{n+1}, ℚ = ℚ_{n}.

The fact that the closure of ℚ_{n} in V [ℝ_{n}] increases with n enables us to see that the inverse limit ℝ_{ω} is well-behaved. We make this precise in the following lemma.

Lemma 4.4. Let G_{ω} be ℝ_{ω}-generic. Let X ∈ V [G_{ω}] be a κ_{n}-sequence of ordinals. Then X ∈ V [G_{0}][…][G_{n}][G_{n+1}][g_{n+2}], where G_{0} ∗ … G_{n+1} ∗ g_{n+2} is the initial segment of G_{ω} generic for ℚ_{0} ∗ … ℚ_{n+1} ∗ ℙ_{n+2}.

Proof. It is easy to see that ℝ_{ω}/ℝ_{m} is κ_{m−3}-closed, so that X ∈ V [G_{0}][…][G_{n+3}]. Since ℚ_{n+3} is < κ_{n+1}-distributive in V [G_{0}][…][G_{n+2}], X ∈ V [G_{0}][…][G_{n+2}]. Finally all κ_{n}-sequences of ordinals in V [G_{0}][…][G_{n+2}] are in V [G_{0}][…][G_{n+1}][g_{n+2}], so we are done.

In the interests of brevity, we will denote V [G_{0}][…][G_{n}] by V_{n}. Notice that g_{n+2} is generic over V_{n+1} = V_{n}[G_{n+1}] for ℙ_{n+2} ∈ V_{n}, so that we can consider G_{n+1} and g_{n+2} as mutually generic over V_{n}.

Corollary 4.5. 2^{ℵ_{n}} = κ_{n} = ℵ_{n+2} in V [G_{ω}].

Proof. Immediate from Lemma 4.4 and Lemma 4.3.

4.2. Why the model works. We are now ready to begin the argument that all of the cardinals κ_{n} have the tree property in V [G_{ω}]. By Lemma 4.4, if T ∈ V [G_{ω}] is a κ_{n}-tree then T ∈ V_{n+1}[g_{n+2}]. We will show that there are no κ_{n}-Aronszajn trees in V_{n+1}[g_{n+2}].

We begin our work in V. Let λ be some cardinal greater than sup { κ_{n} : n ∈ ω }. Using the “diamond” property of F_{n}, choose j : V ⟶ M such that

1. crit(j) = κ_{n}, j(κ_{n}) > λ, ^{λ}M ⊆ M.

2. j(F_{n})(κ_{n}) is the canonical ℝ_{n}-name for the canonical ℚ_{n}-name for

Add(κ_{n}, κ_{n+2})_{V [ℝ_{n}∗ℚ̇_{n}]} × 𝕌(ℵ_{n+1}, κ_{n+1}, V [ℝ_{n}], V [ℝ_{n} ∗ ℚ̇_{n}], F_{n+1}^{∗}).

Notice that j(F_{n})(κ_{n}) is a name for a κ_{n}-directed closed forcing in V [ℝ_{n} ∗ ℚ̇_{n}].

Our aim is now to force over V_{n+1}[g_{n+2}] to get a new model V_{n+1}[g_{n+2}][X], in such a way that in V_{n+1}[g_{n+2}][X] we may define an extension of j to a (generic) elementary embedding j : V_{n+1}[g_{n+2}] ⟶ N ⊆ V_{n+1}[g_{n+2}][X]. We will break up the construction into a series of stages. To keep the notation simple, we will denote all the extensions of the original j : V ⟶ M by “j” also.

4.2.1. Stage one. Since j(ℝ_{n}) = ℝ_{n}, it is easy to lift the embedding j onto the extension by ℝ_{n}. We will then have an embedding j : V_{n−1} ⟶ V_{n−1} extending j : V ⟶ M.

4.2.2. Stage two. We force over V_{n+1}[g_{n+2}] with

𝕊 =_{def} 𝕊(ℵ_{n+1}, κ_{n+1}, V_{n−1}, V_{n}, F_{n+1}^{∗}, G_{n+1}).

Since G_{n+1} and g_{n+2} are mutually generic, we can look at the resulting extension as being obtained from V_{n+1} by forcing first with 𝕊 and then with ℙ_{n+2}.

𝕊 (by design) will expand V_{n+1} to a new model V_{n}[g_{n+1}×u_{n+1}], where g_{n+1}×u_{n+1} is generic for ℙ_{n+1} × 𝕌_{n+1}. So forcing with 𝕊 over V_{n+1}[g_{n+2}] gives us a model V_{n}[g_{n+1} × u_{n+1} × g_{n+2}], with g_{n+1} × u_{n+1} × g_{n+2} generic for ℙ_{n+1} × 𝕌_{n+1} × ℙ_{n+2} over V_{n}.

4.2.3. Stage three. It is at this point that the careful choice of j pays off. If we consider the construction of j(ℚ_{n}) in M[G_{0}][…][G_{n−1}] then it is clear that

• j(ℚ_{n}) ↾ κ_{n} = ℚ_{n}.

• At stage κ_{n}, the forcing at the third coordinate will be

Add(κ_{n}, κ_{n+2})_{V_{n}} × 𝕌(ℵ_{n+1}, κ_{n+1}, V_{n−1}, V_{n}, F_{n+1}^{∗}),

• If we let M_{i} =_{def} M[G_{0}][…][G_{i}] then the agreement between V and M implies that this forcing equals

Add(κ_{n}, κ_{n+2})_{M_{n}} × 𝕌(ℵ_{n+1}, κ_{n+1}, M_{n−1}, M_{n}, F_{n+1}^{∗}),

This means that we can look at the model M_{n}[g_{n+2}×u_{n+1}] as a generic extension of M_{n−1} by j(ℚ_{n}) ↾ κ_{n} + 1. We will now force over V_{n}[g_{n+1} × u_{n+1} × g_{n+2}] to get H_{n} a j(ℚ_{n})-generic object, with H_{n} ↾ κ_{n} + 1 = G_{n} ∗ (g_{n+2} × u_{n+1}). Notice that g_{n+1} is generic over M_{n−1}[H_{n}] for ℙ_{n+1}.

Since ℚ_{n} is κ_{n}-c.c it is easy to see that j ↾ ℚ_{n} is a complete embedding from ℚ_{n} into j(ℚ_{n}), so we may lift j : V_{n−1} ⟶ V_{n−1} to get j : V_{n} ⟶ M_{n−1}[H_{n}].

4.2.4. Stage four. Recall that ℙ_{n+1} = Add(ℵ_{n+1}, κ_{n+1})_{V_{n−1}}. ℙ_{n+1} is ℵ_{n+2}-Knaster (that is κ_{n}-Knaster) in V_{n}, so that j ↾ ℙ_{n} is a complete embedding of ℙ_{n} into j(ℙ_{n}). We can be more precise, ℙ_{n} is isomorphic via j ↾ ℙ_{n} to Add(ℵ_{n+1}, j“κ_{n+1})_{M_{n−1}}, which is equal to Add(ℵ_{n+1}, j“κ_{n+1})_{V_{n−1}} by the agreement between V and M.

We force over V_{n}[H_{n}][g_{n+1}] with Add(ℵ_{n+1}, j(κ_{n+1}) − j“κ_{n+1})_{V_{n−1}} to get a generic object h_{n+1} for j(ℙ_{n}) such that j“g_{n+1} ⊆ h_{n}. We can now lift j : V_{n} ⟶ M_{n−1}[H_{n}] to get j : V_{n}[g_{n+1}] ⟶ M_{n−1}[H_{n}][h_{n+1}].

4.2.5. Stage five. By the construction and the closure of M, we know that j“u_{n+1} ∈ M_{n−1}[H_{n}]. H_{n} collapses κ_{n+1} to have cardinality ℵ_{n+1}, and j(𝕌_{n}) is j(κ_{n})-directed closed where j(κ_{n}) = (ℵ_{n+2})_{M_{n−1}[H_{n}]}. Therefore we may find t such that t ≤ j(q) for all q ∈ u_{n+1}.

We now force over V_{n}[H_{n}][h_{n}] with j(U_{n}), below the condition t, to get a generic object x_{n+1} such that j“u_{n+1} ⊆ x_{n+1}. h_{n+1} and x_{n+1} are mutually generic over M_{n−1}[H_{n}] by Easton’s lemma, and h_{n+1} × x_{n+1} generates a filter H_{n+1} generic for j(ℝ_{n+1}) over M_{n−1}[H_{n}].

We claim that j“G_{n+1} ⊆ H_{n+1}. To see this recall that G_{n+1} is generated by g_{n+1} × u_{n+1}, so that if (p, q, f) ∈ G_{n+1} we may find p̄ ∈ g_{n+1} and (0, q̄, f̄) ∈ u_{n+1} such that (p̄, q̄, f̄) ≤ (p, q, f) in ℝ_{n+1}. Now j(p̄) ∈ h_{n+1}, (0, j(q̄), j(f̄)) ∈ x_{n+1}, so that (j(p̄), 0, 0) and (0, j(q̄), j(f̄)) are both in H_{n+1}. Their greatest lower bound is j((p̄, q̄, f̄)), so this condition must be in H_{n+1}; moreover j((p̄, q̄, f̄)) ≤ j((p, q, f)) in j(ℝ_{n+1}), so that j((p, q, f)) ∈ H_{n+1}.

We may now lift j : V_{n}[g_{n+1}] ⟶ M_{n−1}[H_{n}][h_{n+1}], to get an embedding j : V_{n+1} ⟶ M_{n−1}[H_{n}][H_{n+1}].

4.2.6. Stage six. j“g_{n+2} ∈ M_{n−1}[H_{n}] by construction and the closure of M, so if we let s =_{def} ⋃ j“g_{n+2} then s is a condition in j(ℙ_{n+2}) = Add(j(κ_{n}), j(κ_{n+2}))_{M_{n−1}[H_{n}]}. So we can force over V_{n}[H_{n}][h_{n}][x_{n+1}] with j(ℙ_{n+2}) below s, and get a generic object h_{n+2} such that j“g_{n+2} ⊆ h_{n+2}.

We finish the construction by lifting j : V_{n+1} ⟶ M_{n−1}[H_{n}][H_{n+1}], to get j : V_{n+1}[g_{n+2}] ⟶ M_{n−1}[H_{n}][H_{n+1}][h_{n+2}].

Now that we have lifted j, we can return to the proof that κ_{n} has the tree property in V_{n+1}[g_{n+2}]. Suppose that T is a κ_{n}-Aronszajn tree in V_{n+1}[g_{n+2}]. If we apply j to T we get a tree j(T) of height j(κ_{n}) > κ_{n}, which has an initial segment j(T) ↾ κ_{n} that is isomorphic to T. It follows that T ∈ M_{n−1}[H_{n}][H_{n+1}][h_{n+2}], and T has a branch b in that model.

Now we will locate the tree T and its branch more precisely. Because of the resemblance between V and M, T ∈ M_{n+1}[g_{n+2}]. κ_{n+1} is collapsed to ℵ_{n+1} in M_{n−1}[H_{n}][H_{n+1}][h_{n+2}], so applying some familiar arguments b ∈ M_{n−1}[H_{n}][h_{n+1}].

Notice that M_{n+1}[g_{n+2}] = M_{n−1}[G_{n}][G_{n+1}][g_{n+2}] and

M_{n−1}[H_{n}][h_{n+1}] = M_{n−1}[G_{n}][g_{n+2} × u_{n+1}][H_{n}^{∗}][g_{n+1} × h_{n+1}^{∗}],

decomposing H_{n} as G_{n}∗(g_{n+2}×u_{n+1})∗H_{n}^{∗} and h_{n+1} as g_{n+1}×h_{n+1}^{∗}. We will analyse the forcing that takes us from M_{n+1}[g_{n+2}] to M_{n−1}[H_{n}][h_{n+1}], and show that this forcing cannot add a cofinal branch to a κ_{n}-Aronszajn tree. This contradiction will finish the proof that κ_{n} has the tree property in V_{n+1}[g_{n+2}].

In the analysis that follows, we will use repeatedly and without comment the resemblance between V and M.

We begin by considering what happens when we force over M_{n+1}[g_{n+2}] with 𝕊, to get the model M_{n}[g_{n+1} × u_{n+1} × g_{n+2}] (see Stage Two above). By Lemma 3.20, M_{n+1} ⊧ “𝕊 is ℵ_{n+1}-closed, < ℵ_{n+2}-distributive and ℵ_{n+3}-c.c.”. Since g_{n+2} is generic over M_{n+1} for < ℵ_{n+2}-distributive forcing, 𝕊 is still ℵ_{n+1}-closed in M_{n+1}[g_{n+2}]. It follows that forcing with 𝕊 over M_{n+1}[g_{n+2}] preserves cardinals up to ℵ_{n+1}; since 2^{ℵ_{n}} = ℵ_{n+2} = κ_{n} in M_{n+1}[g_{n+2}], it follows from Lemma 2.4 that T has no cofinal branch in M_{n}[g_{n+1} × u_{n+1} × g_{n+2}].

g_{n+2} × u_{n+1} is generic over M_{n} for ℵ_{n+2}-directed closed forcing and g_{n+1} is generic over M_{n} for ℵ_{n+2}-Knaster forcing, so by Easton’s lemma it follows that all ℵ_{n+1}-sequences of ordinals from M_{n}[g_{n+1} × u_{n+1} × g_{n+2}] are in M_{n+1}[g_{n+2}]. In particular κ_{n} is still a cardinal (namely ℵ_{n+2}) in M_{n}[g_{n+1} × u_{n+1} × g_{n+2}], and so T is still a κ_{n}-Aronszajn tree in that model.

Now we will force over M_{n}[g_{n+1} × u_{n+1} × g_{n+2}] to add h_{n+1}^{∗}. h_{n+1}^{∗} is generic for ℙ_{n+1}^{∗} =_{def} Add(ℵ_{n+1}, j(κ_{n+1}) − j“κ_{n+1}), and we know by Lemma 2.13 that M_{n} ⊧ “ℙ_{n+1}^{∗} is < ℵ_{n+1}-distributive and κ_{n}-Knaster”.

We claim that ℙ_{n+1}^{∗} is still κ_{n}-Knaster in M_{n}[g_{n+1} × u_{n+1} × g_{n+2}]. We will use Lemma 2.6 to see this. ℵ_{n+1} is still regular in M_{n}[g_{n+1} × u_{n+1} × g_{n+2}], so to finish the proof of the claim suppose that X ∈ M_{n}[g_{n+1} ×u_{n+1} ×g_{n+2}] is a set of ordinals with |X| < κ_{n}. Then |X| ≤ ℵ_{n+1} in M_{n}[g_{n+1} × u_{n+1} × g_{n+2}], and so as we proved already X ∈ M_{n}[g_{n+1}]. Since g_{n+1} is generic over M_{n} for ℵ_{n+2}-c.c. (that is κ_{n}-c.c.) forcing and G_{n} is generic over M_{n−1} for κ_{n}-c.c. forcing, there is Y ∈ M_{n−1} such that |X| < κ_{n} in M_{n−1} and X ⊆ Y.

We also claim that ℙ_{n+1}^{∗} is still < ℵ_{n+1}-distributive in M_{n}[g_{n+1}×u_{n+1}×g_{n+2}]. To see this suppose that Y is an ℵ_{n}-sequence of ordinals in M_{n}[h_{n+1} × u_{n+1} × g_{n+2}]. By Easton’s lemma u_{n+1} × g_{n+2} is generic for < ℵ_{n+2}-distributive forcing over M_{n}[h_{n+1}], so Y ∈ M_{n}[h_{n+1}]. Since j(ℙ_{n}) is < ℵ_{n+1}-distributive in M_{n}, Y ∈ M_{n} and we are done.

It follows from these claims that κ_{n} is still a cardinal (in fact is still ℵ_{n+2}) in M_{n}[h_{n+1} × u_{n+1} × g_{n+2}]. Since ℙ_{n+1}^{∗} is κ_{n}-Knaster in M_{n}[g_{n+1} × u_{n+1} × g_{n+2}], Lemma 2.3 implies that T is still a κ_{n}-Aronszajn tree in M_{n}[h_{n+1} × u_{n+1} × g_{n+2}].

Finally we will force over M_{n}[h_{n+1} ×u_{n+1} ×g_{n+2}] with ℝ_{n}^{∗} = j(ℙ_{n})/G_{n} ∗(u_{n+1} × g_{n+2}), to get the model M_{n−1}[H_{n}][h_{n+1}]. We know from Lemmas 3.24 and 3.25 that M_{n}[u_{n+1} × g_{n+2}] ⊧ “ℝ_{n}^{∗} is a projection of ℙ_{n}^{∗} × 𝕌_{n}^{∗}”, where ℙ_{n}^{∗} = Add(ℵ_{n}, j(κ_{n}) − κ_{n})_{M_{n−2}} and M_{n}[u_{n+1} × g_{n+2}] ⊧ “𝕌_{n}^{∗} is ℵ_{n+1}-closed”. We will finish the argument by showing that T can have no branch in any extension of M_{n}[h_{n+1} × u_{n+1} × g_{n+2}] by ℙ_{n}^{∗} × 𝕌_{n}^{∗}.

We have already shown that every ℵ_{n}-sequence of ordinals in M_{n}[h_{n+1} × u_{n+1} × g_{n+2}] is in M_{n}, so 𝕌_{n}^{∗} is still ℵ_{n+1}-closed in M_{n}[h_{n+1} × u_{n+1} × g_{n+2}]. Since 2^{ℵ_{n}} = ℵ_{n+2} = κ_{n} in M_{n}[h_{n+1} × u_{n+1} × g_{n+2}], another application of Lemma 2.4 shows that T has no cofinal branch in M_{n}[h_{n+1} × u_{n+1} × g_{n+2}][𝕌_{n}^{∗}]. Forcing with 𝕌_{n}^{∗} preserves cardinals up to ℵ_{n+1} and collapses ℵ_{n+2} (that is κ_{n}) to be an ordinal of cofinality ℵ_{n+1}, so that in the extension by 𝕌_{n}^{∗} we may find a tree T_{0} of height ℵ_{n+1} and a cofinal map from T_{0} to T.

We now claim that ℙ_{n}^{∗} is ℵ_{n+1}-Knaster in M_{n}[h_{n+1} × u_{n+1} × g_{n+2}][𝕌_{n}^{∗}]. To see this we apply Lemma 2.6. ℵ_{n} is still regular in M_{n}[h_{n+1} × u_{n+1} × g_{n+2}][𝕌_{n}^{∗}], so let X ∈ M_{n}[h_{n+1} × u_{n+1} × g_{n+2}][𝕌_{n}^{∗}] be a set of ordinals of size less than ℵ_{n+1} (that is κ_{n−1}). Then |X| ≤ ℵ_{n} in M_{n}[h_{n+1} × u_{n+1} × g_{n+2}][𝕌_{n}^{∗}], and so by the closure of 𝕌_{n}^{∗} we have X ∈ M_{n}[h_{n+1} × u_{n+1} × g_{n+2}]. We saw already that this implies X ∈ M_{n}, and actually we can go further and see that X ∈ M_{n−1}[g_{n}]; this is a κ_{n−1}-c.c. extension of M_{n−2}, so that X is covered by a set in M_{n−2} of size less than κ_{n−1}.

By Lemma 2.3, forcing with ℙ_{n}^{∗} over M_{n}[h_{n+1} ×u_{n+1} ×g_{n+2}][𝕌_{n}^{∗}] adds no cofinal branch to T_{0}, and therefore adds no cofinal branch to T. This concludes the proof of Theorem 1.

Part 2. The tree property at the double successor of a singular

In this part of the paper we will give Foreman’s proof that it is consistent to have a double successor of a singular cardinal with the tree property. It is possible to modify the construction along the lines of [3] or [4] to get a model in which ℵ_{ω+2} has the tree property.

5. Preliminaries

Starting with κ < λ where κ is supercompact and λ is weakly compact, we will build a generic extension in which

1. κ is a singular cardinal of cofinality ω.

2. κ^{+} is preserved.

3. 2^{κ} = κ^{++} = λ.

4. λ has the tree property.

We start by doing Laver’s construction from [10] to make κ indestructibly supercompact under any κ-directed-closed forcing. Laver’s poset has cardinality κ, so will preserve the weak compactness of λ. We will denote by V the resulting model in which κ is indestructibly supercompact and λ is weakly compact.

Now let ℙ =_{def} Add(κ, λ), so that by construction κ is supercompact in V [ℙ]. Let U̇ be a ℙ-name for a normal ultrafilter on κ in the model V [ℙ].

Lemma 5.1. There is A ⊆ λ a set of Mahlo cardinals such that 1. If α ∈ A and G is Add(κ, λ)-generic over V then U̇^{G} ∩ V [G ↾ α] ∈ V [G ↾ α].

2. A is in the weakly compact filter on λ.

Proof. Given β < λ, we can use the inaccessibility of λ and the κ^{+}-c.c. for ℙ to find F (β) < λ such that for every canonical ℙ ↾ β-name Ȧ for a subset of κ we have [Ȧ ∈ U̇]_{RO(ℙ)} ∈ RO(ℙ ↾ F (β)). A standard argument shows that the set of Mahlo limit points of F is in the weakly compact filter, so we define A to be this set.

For each α ∈ A, let U_{α} =_{def} U̇^{G} ∩ V [G ↾ α]. Clearly, U_{α} is a normal measure on κ in the model V [G ↾ α].

Working in V [ℙ] we define ℚ to be the Prikry forcing constructed from the normal measure U. We also define ℚ_{α} to be the Prikry forcing constructed from the measure U_{α} in V [ℙ ↾ α].

Lemma 5.2. Let x be a cofinal ω-sequence in κ, which is ℚ-generic over V [G]. Then for each α ∈ A, x is ℚ_{α}-generic over V [G_{α}].

Proof. This is immediate from Mathias’ theorem [13] that an ω-sequence is Prikry generic for a measure V if and only if every set of V -measure one contains a final segment of the ω-sequence.

Let π_{α} denote the natural projection map from P ∗ ℚ̇ to RO(ℙ ↾ α ∗ ℚ_{α}).

Lemma 5.3. ℙ ∗ ℚ has the κ^{+}-Knaster property.

Proof. Let ⟨(p_{α}, q_{α}) : α < κ^{+}⟩ be a κ^{+} sequence of conditions. Refining if necessary, we may as well assume that p_{α} decides the lower part of q_{α} for each α. Since there are only κ possible lower parts, we may find a subsequence of length κ^{+} on which the lower part of q_{α} is constant, say with value s.

Now Add(κ, λ) has the κ^{+}-Knaster property, so we may thin out even further to find A unbounded in κ^{+} such that if α, β ∈ A then p_{α} is compatible with p_{β} and the lower parts of q_{α} and q_{β} are equal. This implies that (p_{α}, q_{α}) is compatible with (p_{β}, q_{β}), so we are done.

Notice that the same proof applies to ℙ ↾ α ∗ ℚ_{α}.

6. The main forcing

Let G be ℙ-generic over V and let x be ℚ-generic over V [G]. We have seen that for each α ∈ A we may view the submodel V [G ↾ α][x] of V [G][x] as a generic extension of V by ℙ ↾ α ∗ ℚ_{α}. For each α, we define ℝ_{α} = Add(κ^{+}, 1)_{V [ℙ↾α∗ℚ_{α}]}.

Now we are ready to define the main forcing ℝ.

Definition 6.1. Conditions in ℝ are triples of the form (p, q, r) where 1. (p, q) ∈ ℙ ∗ ℚ̇.

2. r is a partial function with dom(r) ⊆ A, | dom(r)| ≤ κ.

3. For each α ∈ dom(r), r(α) is a ℙ ↾ α ∗ ℚ_{α}-name for a condition in ℝ_{α}.

Given conditions (p_{0}, q_{0}, r_{0}) and (p_{1}, q_{1}, r_{1}), (p_{1}, q_{1}, r_{1}) ≤ (p_{0}, q_{0}, r_{0}) iff

1. (p_{1}, q_{1}) ≤ (p_{0}, q_{0}) in ℙ ∗ ℚ.

2. dom(r_{0}) ⊆ dom(r_{1}), and π_{α}((p_{1}, q_{1})) ⊩ r_{1}(α) ≤ r_{0}(α) for every α ∈ dom(r_{0}).

We observe that the definition is very similar to that of the main forcing from Mitchell’s paper [14]. The only difference is that ℙ ↾ α is replaced by the more complex forcing ℙ ↾ α ∗ ℚ_{α}. Lemmas 6.2, 6.3 and 6.4 are proved as for the main forcing from [14] (see [1] or the first part of this paper).

Lemma 6.2. Let 𝕌 be the partial ordering consisting of elements of ℝ of the form (0, 0, r) with the induced partial ordering. Let ρ : (ℙ ∗ ℚ) × 𝕌 ⟶ ℝ be given by ρ : ((p, q), (0, 0, r)) ⟼ (p, q, r). Then

1. 𝕌 is κ^{+}-closed.

2. ρ is a projection map which commutes with the natural projections from ℝ and ℙ ∗ ℚ × 𝕌 to ℙ ∗ ℚ (so that in a natural way V [ℙ ∗ ℚ] ⊆ V [ℝ] ⊆ V [ℙ ∗ ℚ × 𝕌]).

3. V [ℝ] and V [ℙ ∗ ℚ] have the same κ-sequences.

Lemma 6.3. ℝ is λ-Knaster. It preserves all cardinals except for those in the interval (κ^{+}, λ), which it collapses to κ^{+}. In the model V [ℝ], 2^{κ} = λ = κ^{++}.

Given β ∈ A, we will define ℝ ↾ β in the following way: conditions are triples of the form (p, q, r) where

1. (p, q) ∈ ℙ ↾ β ∗ ℚ̇_{β}.

2. r is a partial function with dom(r) ⊆ A ∩ β, | dom(r)| ≤ κ.

3. For each α ∈ dom(r), r(α) is a ℙ ↾ α ∗ ℚ_{α}-name for a condition in ℝ_{α}. The ordering is defined just as for ℝ. It is easy to see that if β is in A then ℝ ↾ β preserves all cardinals except those in (κ^{+}, β) and forces 2^{κ} = κ^{++} = β. Moreover V [ℝ ↾ β] is a submodel of V [ℝ], and if X ∈ V [ℝ] is a bounded subset of λ then X ∈ V [ℝ ↾ β] for all sufficiently large β.

Lemma 6.4. Let β be in A. Then ℝ/ℝ ↾ β is (in V [ℝ ↾ β]) a projection of (ℙ ∗ ℚ/ℙ ↾ β ∗ ℚ_{β}) × 𝕌^{∗} for some κ^{+}-closed forcing 𝕌^{∗}.

To finish the proof it will suffice to prove that (ℙ ∗ ℚ/ℙ ↾ β ∗ ℚ_{β}) is κ^{+}-Knaster in V [ℝ ↾ β ∗ 𝕌^{∗}]:

Lemma 6.5. If (ℙ ∗ ℚ/ℙ ↾ β ∗ ℚ_{β}) is κ^{+}-Knaster in V [ℝ ↾ β ∗ 𝕌^{∗}] for every β ∈ A, then there is no κ^{++}-Aronszajn tree in V [ℝ].

Proof. The proof follows exactly the same lines as the proof of Theorem 1. If T is a κ^{++}-Aronszajn tree in V [ℝ] then, by the reflection properties of λ, there is β ∈ A such that T ↾ β is an Aronszajn tree in V [ℝ ↾ β].

Clearly T ↾ β has a branch in V [ℝ] (fix a point on level β of T and look at the points below). Since ℝ/ℝ ↾ β is a projection of (ℙ ∗ ℚ/ℙ ↾ β ∗ ℚ_{β}) × 𝕌^{∗}, T ↾ β will have a branch in V [ℝ ↾ β][ℙ ∗ ℚ/ℙ ↾ β ∗ ℚ_{β} × 𝕌^{∗}].

Since 𝕌^{∗} is κ^{+}-closed, T will have no branch in V [ℝ ↾ β][𝕌^{∗}]. 𝕌^{∗} collapses β to be some ordinal of cardinality and cofinality κ^{+}, so that there is a tree T_{0} of height κ^{+} which embeds cofinally into T ↾ β. Now supposedly forcing with ℙ∗ℚ/ℙ ↾ β∗ℚ_{β} will add a branch to T and (therefore a branch to T_{0}), but this is impossible because κ^{+}-Knaster forcing cannot add a branch to a branchless tree of height κ^{+}.

7. Analysing ℙ ∗ ℚ/ℙ ↾ β ∗ ℚ_{β}

We saw in the last section that to finish the proof of Theorem 2 we just need to prove the following result.

Lemma 7.1. (ℙ ∗ ℚ/ℙ ↾ β ∗ ℚ_{β}) is κ^{+}-Knaster in V [ℝ ↾ β ∗ 𝕌^{∗}] for every β ∈ A.

Proof. We start with a claim in the same spirit as Lemma 2.6.

Claim . Let V ⊆ W be two inner models of set theory. Suppose that 1. κ is a limit cardinal in W.

2. κ_{V}^{+} = κ_{W}^{+} .

3. If X ∈ W is a set of ordinals, W ⊧ |X| ≤ κ, then ∃Y ∈ V such that X ⊆ Y and V ⊧ |Y | ≤ κ.

4. V ⊧ κ^{<κ} = κ.

Let ⟨x_{α} : α < κ^{+}⟩ ∈ W be a κ^{+}-sequence of sets of ordinal such that x_{α} ∈ V, |x_{α}| < κ. Then there exists X ⊆ κ^{+} unbounded such that ⟨x_{α} : α ∈ X⟩ forms a Δ-system.

Proof. We work in W. Let A = ⋃_{α<κ^{+}} x_{α}, and let F ∈ W be a bijection F : A ≃ κ^{+}. Let y_{α} = F “x_{α}. Thinning out we may assume that |y_{α}|_{W} = μ for some fixed μ < κ.

Let T_{0} be the stationary set { α < κ^{+} : cf(α) = μ^{+} }. For each α ∈ T_{0} we have sup(y_{α} ∩ α) < α, so by Fodor’s lemma there is T_{1} ⊆ T_{0} stationary and a fixed β < κ^{+} such that ∀α ∈ T_{1} sup(y_{α} ∩ α) = β.

Now F^{−1}“β is a set of size κ lying in W, so by our assumptions there is Z ⊇ F^{−1}“β such that |Z| = κ and Z ∈ V. Now since V ⊧ κ^{<κ} = κ there are only κ possibilities for x_{α} ∩ Z, so we may find T_{2} ⊆ T_{1} stationary and a fixed v such that ∀α ∈ T_{2} x_{α} ∩ Z = v. Then ∀α ∈ T_{2} y_{α} ∩ α = y_{α} ∩ β = F “x_{α} ∩ Z = F “v.

It is now easy to finish, building inductively an unbounded set T_{3} ⊆ T_{2} such that ⟨y_{α} : α ∈ T_{3}⟩ (and hence ⟨x_{α} : α ∈ T_{3}⟩) is a Δ-system. This finishes the proof of the claim.

We observe that the assumptions of the claim hold if we set W = V [ℝ ↾ β][𝕌^{∗}]. Our next task is to analyse the forcing ℙ ∗ ℚ/ℙ ↾ β ∗ ℚ_{β}. It is important to

notice that this forcing is not just Add(κ, λ − β)_{V}, although it does add a generic object for that forcing. The point is that given g ∗ x which is ℙ ↾ β ∗ ℚ_{β}-generic, the forcing ℙ ∗ ℚ/ℙ ↾ β ∗ ℚ_{β} will add an Add(κ, λ − β)-generic object H with the property that x is actually ℚ-generic over V [g × H].

Claim . Let p^{∗} = (p, (x, Ẋ)) ∈ ℙ ↾ β ∗ ℚ_{β}, and let q^{∗} = (q, (s, Ȧ)) ∈ ℙ ∗ ℚ. Then p^{∗} forces that q^{∗} is not a condition in ℙ ∗ ℚ/ℙ ↾ β ∗ ℚ_{β} if and only of one of the following conditions holds:

1. q ↾ β is incompatible with p.

2. q ↾ β is compatible with p, s ⊈ x and x ⊈ s.

3. q ↾ β is compatible with p, x extends s and q ∪ p forces that x − s is not a subset of Ȧ.

4. q ↾ β is compatible with p, s extends x and (q ↾ β) ∪ p forces that s − x is not a subset of Ẋ.

Proof. To say that p^{∗} ⊩ q^{∗} ∉ ℙ∗ℚ/ℙ ↾ β∗ℚ_{β} is to say that there is no generic object G ∗ x ∋ q^{∗} such that G ↾ β ∗ x ∋ p^{∗}. It is easy to check that each of the conditions above rules out the existence of such a G. On the other hand if they all fail then we can force below an appropriate condition r to manufacture an appropriate G; r is (p ∪ q, (t, Ȧ ∩ Ẋ)) where t is the longer of x and s.

To finish the proof of Lemma 7.1, suppose for a contradiction that we have a condition ((p, (x, Ẋ), f), u) ∈ ℝ ↾ β ∗ 𝕌^{∗} which forces that ⟨ṙ_{α} : α < λ^{+}⟩ is a counterexample to the κ^{+}-Knaster property for ℙ ∗ ℙ/ℙ ↾ β ∗ ℚ_{β}.

Working in V [ℝ ↾ β ∗ 𝕌^{∗}] for the moment, suppose that r_{α} is a name for a condition (q_{α}, (s_{α}, Ȧ_{α})). We may refine and thin out to assume that there is a fixed s such that q_{α} ⊩ s_{α} = s for all α. Refining ((p, (x, Ẋ), f), u) if necessary we may assume that it decides the value of s, and what is more that x extends s and p ∪ q_{α} ⊩ x − s ⊆ Ẋ for every α.

So now ((p, (x, Ẋ), f), u) forces ⟨r_{α}^{∗} : α < κ^{+}⟩ is a counterexample to the Knaster property, where r_{α}^{∗} names a condition of the form (q_{α}, (s, Ȧ_{α})) and s ⊆ x. Applying Lemma in V [ℝ ↾ β ∗ 𝕌^{∗}] we may assume that ((p, (x, Ẋ), f), u) forces the sequence ⟨q_{α} : α < κ^{+}⟩ to consist of mutually compatible conditions in ℙ.

We claim that ((p, (x, Ẋ), f), u) forces ⟨r_{α}^{∗} : α < κ^{+}⟩ to be mutually compatible. Otherwise we can find α and β and a refinement ((q, (y, Ẏ), g), v) of ((p, (x, Ẋ), f), u) which decides r_{α}^{∗} and r_{β}^{∗} and forces them incompatible. Let us say that the condition ((q, (y, Ẏ), g), v) forces that r_{α}^{∗} = (q_{α}, (s, Ȧ_{α})) and r_{β}^{∗} = (q_{β}, (s, Ȧ_{β})).

It follows that (q, (y, Ẏ)) must force that (q_{α}, (s, Ȧ_{α})) and (q_{β}, (s, Ȧ_{β})) are conditions in ℙ ∗ ℚ/ℙ ↾ β ∗ ℚ_{β}, and that (q_{α} ∪ q_{β}, (s, Ȧ_{α} ∩ A_{β})) is not a condition. But now q ∪ q_{α} ∪ q_{β} is a condition in ℙ because q, q_{α} and q_{β} are pairwise compatible, so that it must be the case that q ∪ q_{α} ∪ q_{β} ⊩ y − s ⊈ Ẏ. This is absurd because (q ∪ q_{α} ∪ q_{β}, (y, Ẏ)) refines (q, (y, Ẏ)) so should force that (q_{α}, (s, Ȧ_{α})) ∈ ℙ ∗ ℚ/ℙ ↾ β ∗ ℚ_{β}.

This concludes the proof of Lemma 7.1, and with it the proof of Theorem 2.

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Mathematics Institute, Hebrew University, Givat Ram, 91904 Jerusalem, ISRAEL

  E-mail address: cummings@math.huji.ac.il

Mathematics Department, University of California, Irvine, Irvine CA 92717, USA

  E-mail address: mforeman@math.uci.edu