A MODEL IN WHICH EVERY BOOLEAN ALGEBRA HAS MANY SUBALGEBRAS

JAMES CUMMINGS AND SAHARON SHELAH

Abstract. We show that it is consistent with ZFC (relative to large cardinals) that every infinite Boolean algebra B has an irredundant subset A such that 2^{|A|} = 2^{|B|}. This implies in particular that B has 2^{|B|} subalgebras. We also discuss some more general problems about subalgebras and free subsets of an algebra.

The result on the number of subalgebras in a Boolean algebra solves a question of Monk from [6]. The paper is intended to be accessible as far as possible to a general audience, in particular we have confined the more technical material to a “black box” at the end. The proof involves a variation on Foreman and Woodin’s model in which GCH fails everywhere.

1. Definitions and facts

In this section we give some basic definitions, and prove a couple of useful facts about algebras and free subsets. We refer the reader to [3] for the definitions of set-theoretic terms. Throughout this paper we are working in ZFC set theory.

Definition 1.1. Let M be a set. F is a finitary function on M if and only if F : M^{n} ⟶ M for some finite n.

Definition 1.2. M is an algebra if and only if M = (M, F) where M is a set, and F is a set of finitary functions on M which is closed under composition and contains the identity function.

In this case we say M is an algebra on M.

Definition 1.3. Let M = (M, F) be an algebra and let N ⊆ M. Then N is a subalgebra of M if and only if N is closed under all the functions in F. In this case there is a natural algebra structure on N given by N = (N, F ↾ N).

Definition 1.4. Let M = (M, F) be an algebra. Then

Sub(M) = { N : N is a subalgebra of M }.

Definition 1.5. Let M = (M, F) be an algebra. If A ⊆ M then

Cl_{M}(A) = { F (a⃗) : F ∈ F, a⃗ ∈ A }.

Cl_{M}(A) is the least subalgebra of M containing A.

1991 Mathematics Subject Classification. Primary 03E35, 03G05. Key words and phrases. Boolean algebras, free subsets, Radin forcing. The first author was supported by a Postdoctoral Fellowship at the Mathematics Institute,

Hebrew University of Jerusalem.

The second author was partially supported by the Basic Research Fund of the Israel Academy of Science. Paper number 530.

Definition 1.6. Let M = (M, F) be an algebra. If A ⊆ M then A is free if and only if

∀a ∈ A a ∉ Cl_{M}(A − {a}).

In the context of Boolean algebras, free subsets are more usually referred to as irredundant. We can use free sets to generate large numbers of subalgebras using the following well-known fact.

Fact 1.7. If A ⊆ M is free for M then |Sub(M)| ≥ 2^{|A|}.

Proof. For each B ⊆ A let N_{B} = Cl_{M}(B). It follows from the freeness of A that N_{B} ∩ A = B, so that the map B ⟼ N_{B} is an injection from PA into Sub(M).

The next fact (also well-known) gives us one way of generating irredundant subsets in a Boolean algebra.

Fact 1.8. If κ is a strong limit cardinal and B is a Boolean algebra with |B| ≥ κ then B has an irredundant subset of cardinality κ.

Proof. We can build by induction a sequence ⟨b_{α} : α < κ⟩ such that b_{β} is not above any element in the subalgebra generated by b⃗ ↾ β. The point is that a subalgebra of size less than κ cannot be dense, since κ is strong limit. It follows from the property we have arranged that we have enumerated an irredundant subset of cardinality κ.

We make the remark that the last fact works even if κ = ω. The next fact is a technical assertion which we will use when we discuss free subsets and subalgebras in the general setting.

Fact 1.9. Let μ be a singular strong limit cardinal. Let M = (M, F) be an algebra and suppose that μ ≤ |M| < 2^{μ} and |F| < μ. Then Sub(M) has cardinality at least 2^{μ}.

Proof. Let λ = cf(μ) + |F|, then 2^{λ} < μ since μ is singular and strong limit. Define

P = { A ⊆ M : |A| = cf(μ) }.

From the assumptions on M and μ,

2^{μ} = μ^{cf μ} ≤ |M|^{cf μ} = |P | ≤ 2^{μ. cf μ} = 2^{μ},

so that |P | = 2^{μ}.

Observe that if A ∈ P then |Cl_{M}(A)| ≤ λ. So if we define an equivalence relation on P by setting

A ≡ B ⟺ Cl_{M}(A) = Cl_{M}(B),

then the classes each have size at worst 2^{λ}. Hence there are 2^{μ} classes and so we can generate 2^{μ} subalgebras by closing representative elements from each class.

2. Pr(κ)

Definition 2.1. Let κ be a cardinal. P r(κ) is the following property of κ: for all algebras M = (M, F) with |M| = κ and |F| < κ there exists A ⊆ M free for M such that |A| = κ.

In some contexts we might wish to make a more complex definition of the form “P r(κ, μ, D, σ) iff for every algebra on κ with at most μ functions there is a free set in the σ-complete filter D” but P r(κ) is sufficient for the arguments here.

We collect some information about P r(κ).

Fact 2.2. If κ is Ramsey then P r(κ).

Proof. Let M = (M, F) be an algebra with |M| = κ, |F| < κ. We lose nothing by assuming M = κ. We can regard M as a structure for a first-order language L with |L| < κ. By a standard application of Ramseyness we can get A ⊆ κ of order type κ with A a set of order indiscernibles for M.

Now A must be free by an easy application of indiscernibility.

Fact 2.3. If P r(κ) holds and ℙ is a μ^{+}-c.c. forcing for some μ < κ then P r(κ) holds in V^{ℙ}.

Proof. Let Ṁ = (κ̂, ⟨Ḟ_{α} : α < λ⟩) name an algebra on κ with λ functions, λ < κ. For each a⃗ from κ and α < λ we may (by the chain condition of ℙ) enumerate the possible values of Ḟ_{α}(a⃗) as ⟨G_{α,β}(a⃗) : β < μ⟩.

Now define in V an algebra M^{∗} = (κ, ⟨G_{α,β} : α < λ, β < μ⟩) and use P r(κ) to get I ⊆ κ free for M^{∗}. Clearly it is forced that I be free for Ṁ in V^{ℙ}.

3. Building irredundant subsets

In this section we will define a combinatorial principle (∗) and show that if (∗) holds then we get the desired conclusion about Boolean algebras. We will also consider a limitation on the possibilities for generalising the result.

Definition 3.1 (The principle (∗)). The principle (∗) is the conjunction of the following two statements: S1. For all infinite cardinals κ, 2^{κ} is weakly inaccessible and

κ ≤ λ < 2^{κ} ⟹ 2^{λ} = 2^{κ}.

S2. For all infinite cardinals κ, P r(2^{κ}) holds.

Theorem 3.2. (∗) implies that every infinite Boolean algebra B has an irredundant subset A such that 2^{|A|} = 2^{|B|}.

Proof. Observe that a Boolean algebra can be regarded as an algebra with ℵ_{0} functions (just take the Boolean operations and close under composition).

Define a closed unbounded class of infinite cardinals by

C = { μ : ∃θ 2^{θ} = μ or μ is strong limit }

There is a unique μ ∈ C such that μ ≤ |B| < 2^{μ}. By the assumption S1, 2^{μ} = 2^{|B|}, so it will suffice to find a free subset of size μ. Since μ is infinite we can find B_{0} a subalgebra of B with |B_{0}| = μ, and it suffices to find a free subset of size μ in B_{0}. Case A: μ = 2^{θ}. In this case μ > ω, and S2 implies that P r(μ) holds, so that B_{0} has a free (irredundant) subset of cardinality μ.

Case B: μ is strong limit. In this case Fact 1.8 implies that there is an irredundant subset of cardinality μ.

At this point we could ask about the situation for more general algebras. Could every algebra with countably many functions have a large free subset, or a large number of subalgebras? There is a result of Shelah (see Chapter III of [9]) which sheds some light on this question.

Fact 3.3. If κ is inaccessible and not Mahlo (for example if κ is the first inaccessible cardinal) then κ ↛ [κ]_{κ}^{2}.

From this it follows immediately that if κ is such a cardinal, then we can define an algebra M on κ with countably many functions such that |Sub(M)| = κ.

In the next section we will prove that in the absence of inaccessibles some information about general algebras can be extracted from (∗). The model of (∗) which we eventually construct will in fact contain some quite large cardinals; it will be a set model of ZFC in which there is a proper class of cardinals α which are ℶ_{3}(α)-supercompact.

4. General algebras

In this section we prove that in the absence of inaccessible cardinals (∗) implies that certain algebras have a large number of subalgebras. By the remarks at the end of the last section, it will follow that the restriction that no inaccessibles should exist is essential.

Theorem 4.1. Suppose that (∗) holds and there is no inaccessible cardinal. Let κ and λ be infinite cardinals. If M = (M, F) is an algebra such that |M| = λ and |F| = κ, and 2^{κ} ≤ λ, then Sub(M) has cardinality 2^{λ}.

Proof. Let M, κ and λ be as above. Define a class of infinite cardinals C by

C = { μ : ∃θ 2^{θ} = μ or μ is singular strong limit. }

There are no inaccessible cardinals, so C is closed and unbounded in the class of ordinals.

Let μ be the unique element of C such that μ ≤ λ < 2^{μ}, where we know that μ exists because λ ≥ 2^{κ} and C is a closed unbounded class of ordinals. Since 2^{κ} ≤ λ < 2^{μ}, κ < μ.

Now let N be a subalgebra of M of cardinality μ; such a subalgebra can be obtained by taking any subset of size μ and closing it to get a subalgebra. S1 implies that 2^{λ} = 2^{μ}, so it suffices to show that the algebra N = (N, F ↾ N) has 2^{μ} subalgebras.

We distinguish two cases: Case A: μ is singular strong limit. In this case we may apply Fact 1.9 to conclude that N has 2^{μ} subalgebras. Case B: μ = 2^{θ}. It follows from S2 that P r(μ) is true, hence there is a free subset of size μ for N. But now by Fact 1.7 we may generate 2^{μ} subalgebras.

This concludes the proof.

For algebras with countably many functions the last result guarantees that there are many subalgebras as long as |M| ≥ 2^{ℵ_{0}}. In fact we can do slightly better here, using a result from [1].

Fact 4.2. If C is a closed unbounded subset of [ℵ_{2}]^{ℵ_{0}} then |C| ≥ 2^{ℵ_{0}}.

Corollary 4.3. If (∗) holds and there are no inaccessibles, then for every algebra M with |M| > ℵ_{1} and countably many functions |Sub(M)| = 2^{|M|}.

Proof. If |M| ≥ 2^{ℵ_{0}} we already have it. If ℵ_{2} ≤ |M| < 2^{ℵ_{0}} then apply the theorem we just quoted and the fact that 2^{|M|} = 2^{ℵ_{0}}.

5. How to make a model of (∗).

In this section we sketch the argument of [2] and indicate how we modify it to get a model in which (∗) holds. We begin with a brief review of our conventions about forcing. For us p ≤ q means that p is stronger than q, a forcing is λ-closed if every decreasing sequence of length less than λ has a lower bound, a forcing is λ-dense if it adds no < λ-sequence of ordinals, and Add(κ, λ) is the forcing for adding λ Cohen subsets of κ.

Foreman and Woodin begin the construction in [2] with a model V in which κ is supercompact, and in which for each finite n they have arranged that ℶ_{n}(κ) is weakly inaccessible and ℶ_{n}(κ)^{<ℶ_{n}(κ)} = ℶ_{n}(κ). They force with a rather complex forcing ℙ, and pass to a submodel of V^{ℙ} which is of the form V^{ℙ^{π}} for ℙ^{π} a projection of ℙ.

ℙ here is a kind of hybrid of Magidor’s forcing from [5] to violate the Singular Cardinals Hypothesis at ℵ_{ω} and Radin’s forcing from [8]. Just as in [5] the forcing ℙ does too much damage to V and the desired model is an inner model of V^{ℙ}, but in the context of [2] it is necessary to be more explicit about the forcing for which the inner model is a generic extension.

The following are the key properties of ℙ^{π}.

P1. κ is still inaccessible (and in fact is ℶ_{3}(κ)-supercompact) in V^{ℙ^{π}}. P2. ℙ^{π} adds (among other things) a generic club of order type κ in κ. In what

follows we will assume that a generic G for ℙ^{π} is given, and enumerate this club in increasing order as ⟨κ_{α} : α < κ⟩.

P3. In V each cardinal κ_{α} reflects to some extent the properties of κ. In particular in V each κ_{α} is measurable, and for each n we have that ℶ_{n}(κ_{α}) is weakly inaccessible and ℶ_{n}(κ_{α})^{<ℶ_{n}(κ_{α})} = ℶ_{n}(κ_{α}).

P4. For each α, if p ∈ ℙ^{π} is a condition which determines κ_{α} and κ_{α+1} then ℙ^{π} ↾ p (the suborder of conditions which refine p) factors as

ℙ_{α} × Add(ℶ_{4}(κ_{α}), κ_{α+1}) × ℚ_{α},

where

(a) ℙ_{α} is κ_{α}^{+}-c.c. if α is limit and ℶ_{4}(κ_{β})^{+}-c.c. if α = β + 1.

(b) All bounded subsets of ℶ_{4}(κ_{α+1}) which occur in V^{ℙ^{π}↾p} have already ap-

peared in the extension by ℙ_{α} × Add(ℶ_{4}(κ_{α}), κ_{α+1}).

Given all this it is routine to check that if we truncate V^{ℙ^{π}} at κ then we obtain a set model of ZFC in which GCH fails everywhere, and moreover exponentiation follows the pattern of clause 1 in (∗).

The reader should think of ℙ^{π} as shooting a club of cardinals through κ, and doing a certain amount of work between each successive pair of cardinals. Things have been arranged so that GCH will fail for a long region past each cardinal κ_{α} on the club, so that what needs to be done is to blow up the powerset of a well-chosen point in that region to have size κ_{α+1}.

For our purposes we need to change the construction of [2] slightly, by changing the forcing which is done between the points of the generic club to blow up powersets. The reason for the change is that we are trying to get P r(λ) to hold whenever λ is of the form 2^{η}, and in general it will not be possible to arrange that P r(λ) is preserved by forcing with Add(λ, ρ).

The solution to this dilemma is to replace the Cohen forcing defined in V by a Cohen forcing defined in a well chosen inner model. We need to choose this inner model to be small enough that Cohen forcing from that model preserves P r(λ), yet large enough that it retains some degree of closure sufficient to make the arguments of [2] go through. For technical reasons we will be adding subsets to ℶ_{5}(κ_{α}), rather than ℶ_{4}(κ_{α}) as in the case of [2].

To be more precise we will want to replace clauses P3 and P4 above by P3^{∗}. For each n < 6 it is the case in V that κ_{α} is measurable, ℶ_{n}(κ_{α}) is weakly

inaccessible, and ℶ_{n}(κ_{α})^{<ℶ_{n}(κ_{α})} = ℶ_{n}(κ_{α}). Also P r(ℶ_{n}(κ_{α})) holds for each such n.

For all possible (κ_{α}, κ_{α+1}) there is a notion of forcing Add^{∗}(ℶ_{5}(κ_{α}), κ_{α+1}) (which we will denote by ℝ_{α} in what follows) such that

(a) ℝ_{α} is ℶ_{4}(κ_{α})-closed, ℶ_{4}(κ_{α})^{+}-c.c. and ℶ_{5}(κ_{α})-dense.

(b) ℝ_{α} adds κ_{α+1} subsets to ℶ_{5}(κ_{α}).

(c) In V^{ℝ_{α}} the property P r(ℶ_{n}(κ_{α})) still holds ^{1} for n < 6. P4^{∗}. If p determines κ_{α} and κ_{α+1} then ℙ^{π} ↾ p factors as

ℙ_{α} × ℝ_{α} × ℚ_{α},

where

(a) ℙ_{α} is κ_{α}^{+}-c.c. if α is limit, and ℶ_{5}(κ_{β})^{+}-c.c. if α = β + 1.

(b) ℝ_{α} = Add^{∗}(ℶ_{5}(κ_{α}), κ_{α+1}) is as above.

(c) All bounded subsets of ℶ_{5}(κ_{α+1}) in V^{ℙ^{π}↾p} are already in the extension

by ℙ_{α} × ℝ_{α}. We’ll show that given a model in which P1, P2, P3^{∗} and P4^{∗} hold we can

construct a model of (∗).

Lemma 5.1. Let G be generic for a modified version of ℙ^{π} obeying P1, P2, P3^{∗} and P4^{∗}. If we define V_{1} = V [G], then V_{1} is a model of ZFC in which (∗) holds.

Proof. S1 is proved just as in [2].

It follows from our assumptions that in V_{1} we have the following situation:

• μ is strong limit if and only if μ = κ_{λ} for some limit λ.

• Cardinal arithmetic follows the pattern that for n < 5

ℶ_{n}(κ_{α}) ≤ θ < ℶ_{n+1}(κ_{α}) ⟹ 2^{θ} = ℶ_{n+1}(κ_{α}),

while

ℶ_{5}(κ_{α}) ≤ θ < κ_{α+1} ⟹ 2^{θ} = κ_{α+1}.

We need to show that for all infinite λ we have P r(2^{λ}). The proof divides into two cases.

Case 1: 2^{λ} is of the form ℶ_{m}(κ_{η}) where 1 ≤ m ≤ 5 and η is a limit ordinal. By the factorisation properties in P4^{∗} above it will suffice to show that P r(2^{λ}) holds in the extension by ℙ_{η} × ℝ_{η}.

1The hard work comes in the case n = 5, density guarantees that the property survives for n < 5.

Certainly P r(2^{λ}) holds in the extension by ℝ_{η}, because P3^{∗} says just that. Also we have that κ_{η}^{+} < ℶ_{1}(κ_{η}) ≤ 2^{λ}, and ℙ_{η} is κ_{η}^{+}-c.c. so that applying Fact 2.3 we get that P r(2^{λ}) holds in the extension by ℙ_{η} × ℝ_{η}.

Case 2: 2^{λ} is of the form ℶ_{m}(κ_{β+1}) where 0 ≤ m ≤ 5. Again it suffices to show that

P r(2^{λ}) holds in the extension by ℙ_{β+1} × ℝ_{β+1}.

P r(2^{λ}) holds in V^{ℝ_{β+1}}, ℶ_{5}(κ_{β})^{+} < κ_{β+1} ≤ 2^{λ} and ℙ_{β+1} is ℶ_{5}(κ_{β})^{+}-c.c. so that as in the last case we can apply Fact 2.3 to get that P r(2^{λ}) in the extension by ℙ_{β+1} × ℝ_{β+1}.

This concludes the proof.

6. The preparation forcing

Most of the hard work in this paper comes in preparing the model over which we intend (ultimately) to do our version of the construction from [2].

We’ll make use of Laver’s “indestructibility” theorem from [4] as a labour-saving device. At a certain point below we will sketch a proof, since we need a little more information than is contained in the statement.

Fact 6.1 (Laver). Let κ be supercompact, let η < κ. Then there is a κ-c.c. and η-directed closed forcing ℙ, of cardinality κ, such that in the extension by ℙ the supercompactness of κ is indestructible under κ-directed closed forcing.

Now we will describe how to prepare the model over which we will do Radin forcing. Let us start with six supercompact cardinals enumerated in increasing order as ⟨κ_{i} : i < 6⟩ in some initial model V_{0}.

6.1. Step One. We’ll force to make each of the κ_{i} indestructibly supercompact under κ_{i}-directed closed forcing. To do this let 𝕊_{0} ∈ V_{0} be Laver’s forcing to make κ_{0} indestructible under κ_{0}-directed closed forcing, and force with 𝕊_{0}. In V_{0}^{𝕊_{0}} the cardinal κ_{0} is supercompact by construction, and the rest of the κ_{i} are still supercompact because |S_{0}| = κ_{0} and supercompactness survives small forcing.

Now let 𝕊_{1} ∈ V_{0}^{𝕊_{0}} be (as guaranteed by Fact 6.1) a forcing which makes κ_{1} indestructible and is κ_{0}-directed closed. In V_{0}^{𝕊_{0}∗𝕊_{1}} we claim that all the κ_{i} are supercompact and that both κ_{0} and κ_{1} are indestructible. The point for the indestructibility of κ_{0} is that if ℚ ∈ V_{0}^{𝕊_{0}∗𝕊_{1}} is κ_{0}-directed closed then 𝕊_{1} ∗ ℚ is κ_{0}-directed closed in V_{0}^{𝕊_{0}}, so that κ_{0} has been immunised against its ill effects and is supercompact in V_{0}^{𝕊_{0}∗𝕊_{1}∗ℚ}.

Repeating in the obvious way, we make each of the κ_{i} indestructible. Let the resulting model be called V.

6.2. Step Two. Now we will force over V with a rather complicated κ_{0}-directed closed forcing. Broadly speaking we will make κ_{0} have the properties that are demanded of κ_{α} in P3^{∗} from the last section.

Of course κ_{0} will still be supercompact after this, and we will use a reflection argument to show that we have many points in κ_{0} which are good candidates to become points on the generic club.

To be more precise we are aiming to make a model W in which the following list of properties holds: K1. κ_{0} is supercompact. K2. For each i < 5, 2^{κ_{i}} = κ_{i+1} and κ_{i+1} is a weakly inaccessible cardinal with

κ_{i+1}^{<κ_{i+1}} = κ_{i+1}.

For any μ > κ_{5} there is a forcing ℚ_{μ} ⊆ V_{μ} such that (if we define ℚ_{η} = ℚ_{μ}∩V_{η} for η with κ_{5} < η < μ)

K3. |ℚ_{η}| = η^{<κ_{5}}, ℚ_{η} is κ_{4}-directed closed, κ_{5}-dense and κ_{4}^{+}-c.c. K4. In W^{ℚ_{η}}, 2^{κ_{5}} ≥ η and P r(κ_{i}) holds for i < 6. K5. If η < ζ ≤ μ then ℚ_{η} is a complete suborder of ℚ_{ζ}. For each i < 5 let ℙ_{i} be the Cohen conditions (as computed by V) for adding

κ_{i+1} subsets of κ_{i}. We will abbreviate this by ℙ_{i} = Add(κ_{i}, κ_{i+1})_{V}. Let ℙ_{i,j} = ∏_{i≤n≤j} ℙ_{n}.

Let V_{1} = V [ℙ̇_{0,3}]. In V_{1} we know that 2^{κ_{i}} = κ_{i+1} for i < 4, and that κ_{5} is still supercompact since |ℙ_{0,3}| = κ_{4}. In V_{1} define a forcing as in Fact 6.1 for making κ_{5} indestructible. More precisely choose ℝ ∈ V_{1} such that

• In V_{1}, ℝ is κ_{5}-c.c. and κ_{4}^{+50}-directed closed with cardinality κ_{5}.

• In V_{1}[ℝ̇] the supercompactness of κ_{5} is indestructible under κ_{5}-directed closed forcing.

Let V_{2} = V_{1}[ℝ̇]. Define ℚ_{μ} = Add(κ_{5}, μ)_{V_{2}}, and observe that for every η ℚ_{η} = Add(κ_{5}, η)_{V_{2}}. This is really the key point in the construction, to choose ℚ_{μ} in exactly the right inner model (see our remarks in the last section just before the definition of P3^{∗}).

Finally let W = V_{2}[ℙ̇_{4}], where we stress that ℙ_{4} is Cohen forcing as computed in V.

We’ll prove that we have the required list of facts in W. K1. κ_{0} is supercompact.

Proof. Clearly ℙ_{0,3} ∗ ℝ is κ_{0}-directed closed in V. ℙ_{4} is κ_{4}-directed closed in V so is still κ_{0}-directed closed in V_{2}, hence ℙ_{0,3} ∗ ℝ ∗ ℙ_{4} is κ_{0}-directed closed in V. By the indestructibility of κ_{0} in V, κ_{0} is supercompact in W.

K2. For each i < 5, 2^{κ_{i}} = κ_{i+1} and κ_{i+1} is a weakly inaccessible cardinal with κ_{i+1}^{<κ_{i+1}} = κ_{i+1}. Proof. In V_{1} we have this for i < 4. ℙ_{4} is κ_{4}-closed and κ_{4}^{+}-c.c. in V, ℙ_{0,3} is κ_{3}^{+}-c.c. even in V [ℙ̇_{4}] so by the usual arguments with Easton’s lemma ℙ_{4} is κ_{4}-dense and κ_{4}^{+}-c.c. in V_{1}. ℝ is κ_{4}^{+50}-closed in V_{1} so that ℙ_{4} is κ_{4}-dense and κ_{4}^{+}-c.c. in V_{2}. Moreover, by the closure of ℝ we still have the claim for i < 4 in V_{2}. κ_{5} is inaccessible (indeed supercompact) in V_{2} so that after forcing with ℙ_{4} we have the claim for i < 5. Recall that ℚ_{η} = Add(κ_{5}, η)_{V_{2}}.

K3. |ℚ_{η}| = η^{<κ_{5}}, ℚ_{η} is κ_{4}-directed closed, κ_{5}-dense and κ_{4}^{+}-c.c.

Proof. The cardinality statement is clear. ℚ_{η} is κ_{5}-directed closed in V_{2}, so that (by what we proved about the properties of ℙ_{4} in V_{2} during the proof of K2) ℚ_{η} is κ_{4}-directed closed and κ_{5}-dense in W. Since ℚ_{η} is κ_{5}-closed in V_{2}, ℙ_{4} is κ_{4}^{+}-c.c. in V_{2}[ℚ̇_{η}]. ℚ_{η} is κ_{5}^{+}-c.c. in V_{2} so that ℚ_{η} × ℙ_{4} is κ_{5}^{+}-c.c. in V_{2}, so that finally ℚ_{η} is κ_{5}^{+}-c.c. in W.

K4. In W^{ℚ_{η}}, 2^{κ_{5}} ≥ η and P r(κ_{i}) holds for i < 6.

Proof. Easily 2^{κ_{5}} ≥ η. We break up the rest of the proof into a series of claims.

Claim . For i < 4, P r(κ_{i}) holds in W [ℚ̇_{η}].

Proof. κ_{i} is supercompact in V [ℙ̇_{i,3}], because ℙ_{i,3} is κ_{i}-directed closed. ℙ_{0,i−1} has κ_{i−1}^{+}-c.c. in that model so that P r(κ_{i}) holds in V_{1}. Also we know that W [ℚ̇_{η}] = V_{1}[ℝ̇][ℙ̇_{4}][ℚ̇_{η}] is an extension of V_{1} by κ_{4}-dense forcing, so P r(κ_{i}) is still true in W [ℚ̇_{η}].

Claim . P r(κ_{4}) holds in W [ℚ̇_{η}].

Proof. W = V_{2}[ℙ̇_{4}] = V [ℙ̇_{0,4}][ℝ̇]. Arguing as in the previous claim, P r(κ_{4}) holds in V [ℙ_{0,4}]. ℙ_{4} is κ_{4}^{+}-c.c. in V_{1} so ℝ is κ_{4}^{+}-dense in V [ℙ̇_{0,4}],hence P r(κ_{4}) holds in W. Finally ℚ_{η} is κ_{5}-dense in W, so P r(κ_{4}) holds in W [ℚ̇_{η}].

Claim . P r(κ_{5}) holds in W [ℚ̇_{η}].

Proof. W [ℚ̇_{η}] = V_{2}[ℙ̇_{4}][ℚ̇_{η}] = V_{2}[ℚ̇_{η}][ℙ̇_{4}]. As ℚ_{η} is κ_{5}-directed closed in V_{2}, κ_{5} is supercompact in V_{2}[ℚ_{η}]. ℙ_{4} is κ_{4}^{+}-c.c. in V_{2}[ℚ̇_{η}], so that P r(κ_{4}) is still true in W [ℚ̇_{η}].

This finishes the proof of K4. K5. If η < ζ ≤ μ then ℚ_{η} is a complete suborder of ℚ_{ζ}.

Proof. This is immediate by the uniform definition of Cohen forcing. We finish this section by proving that in W we can find a highly supercompact

embedding j : W ⟶ M with critical point κ_{0} such that in M the cardinal κ_{0} has some properties resembling K1–K5 above. The point is that there will be many α < κ_{0} which have these properties in W, and eventually we’ll ensure that every candidate to be on the generic club added by the Radin forcing has those properties.

For the rest of this section we will work in the model W unless otherwise specified.

Definition 6.2. A pair of cardinals (α, κ) is sweet iff (setting α_{n} = ℶ_{n}(α)) 1. α is measurable.

2. For each i < 6, α_{i} is weakly inaccessible and α_{i}^{<α_{i}} = α_{i}.

There is a forcing ℚ_{κ}^{α} ⊆ V_{κ} such that (setting ℚ_{η}^{α} = ℚ_{κ}^{α} ∩ V_{η})

3. |ℚ_{η}^{α}| = η^{<α_{5}}, ℚ_{η}^{α} is α_{4}-directed closed, α_{5}-dense and α_{4}^{+}-c.c.

4. In the generic extension by ℚ_{η}^{α}, 2^{α_{5}} ≥ η and P r(α_{i}) holds for i < 6.

5. If η < ζ ≤ κ then ℚ_{η}^{α} is a complete suborder of ℚ_{ζ}^{α}.

Theorem 6.3. Let λ = ℶ_{50}(κ_{5}). In W there is an embedding j : W ⟶ M such that crit(j) = κ_{0}, ^{λ}M ⊆ M, and in the model M the pair (κ_{0}, j(κ_{0})) is sweet.

Proof. Recall that we started this section with a model V_{0}, and forced with some 𝕊_{0} to make κ_{0} indestructible. Let 𝕊 be the forcing that we do over V_{0}^{𝕊_{0}} to get to the model V_{2}, and recall that we force over V_{2} with ℙ_{4} to get W. 𝕊 ∗ ℙ_{4} is κ_{0}-directed closed in V_{0}^{𝕊_{0}}.

Fix a cardinal μ much larger than λ, such that μ^{<κ} = μ. As in [4], fix in V_{0} an embedding j_{0} : V_{0} ⟶ N such that

1. crit(j_{0}) = κ_{0}, μ < j_{0}(κ_{0}).

2. V_{0} ⊧ ^{μ}N ⊆ N.

3. j_{0}(𝕊_{0}) = 𝕊_{0} ∗ 𝕊 ∗ ℙ_{4} ∗ ℝ, where ℝ is μ^{+}-closed in N^{𝕊_{0}∗𝕊∗ℙ_{4}}. Let G_{0} be 𝕊_{0}-generic over V_{0}, let g_{1} ∗ g_{2} be 𝕊 ∗ ℙ_{4}-generic over V_{0}[G_{0}]. Let W = V_{0}[G_{0}][g_{1}][g_{2}] and let N^{+} = N[G_{0}][g_{1}][g_{2}].

Now N^{+} is closed under μ-sequences in W so that the factor forcing ℝ is μ^{+}- closed in W. Let H be ℝ-generic over W, then it is easy to see that j_{0} lifts to

j_{1} : V_{0}[G_{0}] ⟶ N^{+}[H].

Figure 1. Factorisation of j_{2}.

Now we know that j_{1} ↾ 𝕊 ∗ ℙ_{4} ∈ N[G_{0}], and g_{1} ∗ g_{2} ∈ N^{+}[H], so that j[g_{1} ∗ g_{2}] ∈ N^{+}[H]. By the directed closure of j(𝕊 ∗ ℙ_{4}) in N^{+}[H] we can find a “master condition” p (a lower bound in j(𝕊 ∗ ℙ_{4}) for j[g_{1} ∗ g_{2}]) and then force over W [H] with j(𝕊 ∗ ℙ_{4}) ↾ p to get a generic X and an embedding

j_{2} : W ⟶ N^{∗} = N^{+}[H][X].

Of course this embedding is not in W but a certain approximation is. Observe that ℝ ∗ j(𝕊 ∗ ℙ_{4}) is μ^{+}-closed in W. Now factor j_{2} through the ultrapower by

U = { A ⊆ P_{κ}μ : j[μ] ∈ j_{2}(A) },

to get a commutative triangle as in Figure 1.

By the closure U ∈ W, so j is an internal ultrapower map. It witnesses the μ-supercompactness of κ_{0} in the model W, because (as can easily be checked) U is a fine normal measure on P_{κ}μ.

It remains to check that (κ_{0}, j(κ_{0})) is sweet in M. We’ll actually check that (κ_{0}, j_{2}(κ_{0})) is sweet in N^{∗}, this suffices because crit(k) > μ > κ_{0}.

Clauses 1 and 2 are immediate by the closure of M inside the model W. Now to witness sweetness let us set

ℚ_{j_{2}(κ_{0})}^{κ_{0}} = Add(κ_{5}, j_{2}(κ_{0}))_{V_{0}[G_{0}][g_{1}]} = Add(κ_{5}, j_{2}(κ_{0}))_{N[G_{0}][g_{1}]}.

Clause 3 is true in N^{+} by the same arguments that we used for W above. N^{∗} is an extension of N^{+} by highly closed forcing so that Clause 3 is still true in N^{∗}. Clause 5 is easy so we are left with Clause 4.

Let h be ℚ_{η}^{κ_{0}}-generic over N^{∗} for ℚ_{η}^{κ_{0}}. Then h is generic over N^{+}. By the chain condition of ℚ_{η}^{κ_{0}} and the closure of N^{+} in W, W and N^{+} see the same set of antichains for ℚ_{η}^{κ_{0}}. Hence h is ℚ_{η}^{κ_{0}}-generic over W, and we showed that P r(κ_{i}) holds in W [h]. But then by closure again P r(κ_{i}) holds in N^{+}[h], by Easton’s lemma H ∗ X is generic over N^{+}[h] for highly dense forcing so that finally P r(κ_{i}) holds in N^{∗}[h].

7. The final model

In this section we will indicate how to modify the construction of [2], so as to force over the model W obtained in the last section and produce a model of (∗). The modification that we are making is so minor that it seems pointless to reproduce the long and complicated definitions of ℙ and ℙ^{π} from [2]; in this section we just give a sketch of what is going on in [2], an indication of how the construction there is to be modified, and then some hints to the diligent reader as to how the proofs in [2] can be changed to work for our modified forcing.

As we mentioned in Section 5, the construction of [2] involves building a forcing ℙ and then defining a projection ℙ^{π}; the idea is that the analysis of ℙ provides information which shows that ℙ^{π} does not too much damage to the cardinal structure of the ground model. We start by giving a sketchy account of ℙ^{π} as constructed in [2].

Recall the the aim of ℙ^{π} is to add a sequence ⟨(κ_{α}, F_{α}) : α < κ⟩ such that:

1. κ⃗ enumerates a closed unbounded subset of κ.

2. F_{α} is Add(ℶ_{4}(κ_{α}), κ_{α+1})-generic over V.

3. In the generic extension, cardinals are preserved and κ is still inaccessible.

A condition will prescribe finitely many of the κ_{α}, and for each κ_{α} that it prescribes it will give some information about F_{α}; it will also place some constraint on the possibilities for adding in new values of κ_{α} and for giving information about the corresponding F_{α}. The idea goes back to Prikry forcing, where a condition prescribes an initial segment of the generic ω-sequence and puts constraint on the possibilities for adding further points.

To be more specific ℙ^{π} is built from a pair (w⃗, F⃗) where w(0) = κ and w(α) is a measure on V_{κ} for α > 0. w(α) will concentrate on pairs (u⃗, h⃗) that resemble (w⃗ ↾ α, F⃗ ↾ α); in particular if we let κu⃗ =_{def} u(0), κu⃗ will be a cardinal that resembles κ. F(α) will be an ultrafilter on a Boolean algebra ℚ(w⃗, α), which consists of functions f such that dom(f) ∈ w(α) and f(u⃗, k⃗) ∈ RO(Add(ℶ_{4}(κu⃗), κ)), with the operations defined pointwise.

Conditions in ℙ^{π} will have the form

⟨(u⃗_{0}, k⃗_{0}, A⃗_{0}, f⃗_{0}, s_{0}), …, (u⃗_{n−1}, k⃗_{n−1}, A⃗_{n−1}, f⃗_{n−1}, s_{n−1}), (w⃗, F⃗, B⃗, g⃗)⟩

where 1. (u⃗_{i}, k⃗_{i}) ∈ V_{κ}, κu⃗_{0} < … κu⃗_{n−1} < κ.

2. dom(f_{i}(α)) = A_{i}(α) ∈ u_{i}(α), dom(g(α)) = B(α) ∈ w(α).

3. f_{i}(α) ∈ k_{i}(α), g(α) ∈ F(α).

4. s_{i} ∈ Add(ℶ_{4}(κ_{u_{i}}), u_{i+1}) for i < n − 1, s_{n−1} ∈ Add(ℶ_{4}(κ_{u_{n−1}}), κ).

The aim of this condition is to force that the cardinals κ_{u_{i}} are on the generic club, and that the conditions s_{i} are in the corresponding generics. It is also intended to force that if we add in a new cardinal α and Add condition s between κ_{u_{i}} and κ_{u_{i+1}} then there is a pair (v⃗, h⃗) in some A_{i+1}(β) such that α = κv⃗ and s ≤ k_{i+1}(β)(v⃗, h⃗). This motivates the ordering (which we do not spell out in detail here); a condition is refined either by refining the “constraint parts” or by adding in new quintuples that obey the current constraints and which impose constraints compatible with the current ones.

The key fact about ℙ^{π} is that given a condition p, any question about the generic extension can be decided by refining the constraint parts of p (this is modelled on Prikry’s well-known lemma about Prikry forcing). It will follow from this that bounded subsets of κ are derived from initial segments of the generic as in P4 b) from Section 5, which in turn will imply that ℙ^{π} has the desired effect of causing the GCH to fail everywhere below κ. It remains to be seen that forcing with ℙ^{π} preserves the large cardinal character of κ; this is done via a master condition argument which succeeds because w⃗ is quite long (it has a so-called repeat point) and the forcing Add(ℶ_{4}(λ), μ) is ℶ_{4}(λ)-directed-closed.

We can now describe how we modify the construction of [2]. To bring our notation more into line with [2], let κ = κ_{0}, let V = W, let j : V ⟶ M be the embedding constructed at the end of the last section. For each α < κ such that (α, κ) is sweet let us fix ⟨ℚ_{η}^{α} : η ≤ κ⟩ witnessing this. Our modification of [2] is simply to restrict attention to those pairs (u⃗, k⃗) such that (κu⃗, κ) is sweet (there are sufficiently many because (κ, j(κ)) is sweet in M) and then to replace the forcing Add(ℶ_{4}(κu⃗), κ) by the forcing ℚ_{κ}^{κu⃗}. It turns out that the proofs in [2] go through essentially unaltered, because they only make appeal to rather general properties (chain condition, distributivity, directed closure) of Cohen forcing.

There is one slightly subtle point, which is that is we will need a certain uniformity in the dependence of ℚ_{η}^{α} on η, as expressed by the equation ℚ_{η}^{α} = ℚ_{ζ}^{α} ∩ V_{η}.

The real point of this is that when we consider a condition in j(ℙ^{π}) which puts onto the generic club a point α < κ followed by the point κ, the forcing which is happening between α and κ is j(ℚ)_{κ}^{α}; it will be crucial that j(ℚ)_{κ}^{α} = j(ℚ_{κ}^{α}) ∩ V_{κ} = ℚ_{κ}^{α}, that is the same forcing that would be used in ℙ^{π} between α and κ.

We conclude this section with a short discussion of the necessary changes in the proofs. We assume that the reader has a copy of [2] to hand; all references below are to theorem and section numbers from that paper.

Section Three: We will use j : V ⟶ M to construct the master sequence (M⃗, g⃗). Recall that we chose j to witness ℶ_{50}(κ_{5})-supercompactness of κ_{0}, it can be checked that this is enough to make all the arguments below go through.

When we build (M⃗, g⃗) we will choose each g_{α} so that dom(g_{α}) contains only (u⃗, h⃗) with (κu⃗, κ) sweet, and then let g_{α}(u⃗, h⃗) ∈ ℚ⃗_{κ}^{κu⃗}. As in [2], if j_{α} : V ⟶ N_{α} is the ultrapower by M_{α} then W ⊧ ^{κ_{3}}N_{α} ⊆ N_{α}. If we let F be the function given by

F : (u⃗, h⃗) ⟼ ℚ_{κ}^{κu⃗}

then [F]_{M_{α}} is κ_{4}-closed in N_{α}, hence is κ_{3}^{+}-closed in V.

This closure will suffice to make appropriate versions of 3.3, 3.4 and 3.5 go through.

Section Four: 4.1 is just as in [2]. A version of 4.2 goes through because ℚ_{κ}^{κ_{a}} ⊆ V_{κ}.

Section Five: In the definition of “suitable quintuple” we demand that k_{δ}(b) ∈ ℚ_{κ_{u}}^{κ_{b}} and s ∈ ℚ_{κ}^{κ_{u}}.

The definition of “addability” is unchanged. Clauses d) and e) of that definition still make sense because (e.g.) ℚ_{κ_{v}}^{κ_{a}} is a complete suborder of ℚ_{κ_{w}}^{κ_{a}}.

Versions of 5.1 and 5.2 go through without change.

In the definition of “condition in ℙ” we demand that k_{δ}(b) ∈ ℚ_{κ_{w}}^{κ_{b}}, s ∈ ℚ_{κ_{u_{i+1}}}^{κ_{u_{i}}}. The ordering on ℙ is as before.

The definitions of “canonical representatives” and of “upper and lower parts” are unchanged, as is 5.3.

We can do a “local” version of our argument for 3.3 above to show 5.4 goes through. Similarly 5.5, 5.6 and 5.8 go through.

Section Six: We need to make the obvious changes in the definitions of “suitable quintuple” and of the projected forcing ℙ^{π}. The proofs all go through because the forcing ℚ_{η}^{α} has the right chain condition and closure.

Section Seven: The master condition argument of this section goes through because the forcing notion j(α ⟼ ℚ_{κ}^{α})(κ) is κ_{4}-directed closed.

8. Conclusion

Putting together the results of the preceding sections we get the following result.

Theorem 8.1 (Main Theorem). If Con(ZFC + GCH + six supercompact cardinals) then Con(ZFC + every infinite Boolean algebra B has an irredundant subset A such that 2^{|A|} = 2^{|B|}.

Shelah has shown that the conclusion of this consistency result implies the failure of weak square, and therefore needs a substantial large cardinal hypothesis.

References

[1] J. Baumgartner and A. Taylor, Saturation properties of ideals in generic extensions I, Transactions of the American Mathematical Society 270 (1982), 557–574.

[2] M. Foreman and H. Woodin, GCH can fail everywhere, Annals of Mathematics 133 (1991), 1–35.

[3] T. Jech, Set Theory, Academic Press, 1978.

[4] R. Laver, Making the supercompactness of κ indestructible under κ-directed-closed forcing, Israel Journal of Mathematics 29 (1978), 385–388.

[5] M. Magidor, On the singular cardinals problem I, Israel Journal of Mathematics 28 (1977), 1-31.

[6] J. D. Monk, Cardinal Functions on Boolean Algebras, Birkhaüser, 1990.

[7] Handbook of Boolean Algebras, ed. J. D. Monk and Robert Bonnet, North-Holland, 1989.

[8] L. B. Radin, Adding closed cofinal sequences to large cardinals, Annals of Mathematical Logic 22 (1982), 243–261.

[9] S. Shelah, Cardinal Arithmetic, Oxford University Press (to appear).

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

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

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

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