PARTITION SUBALGEBRAS FOR MAXIMAL ALMOST DISJOINT FAMILIES

ALAN DOW AND JINYUAN ZHOU

Abstract. Partitioner Algebras are defined in [BW82] as a natural tool for studying the properties of maximal almost disjoint families of subsets of ω. We prove from PFA^{+} and ♢(S_{0}^{2}) that there exists a representable Boolean Algebra which contains a non representable subalgebra.

1. Introduction

For any maximal almost disjoint family M on ω, we use I_{M} to denote the ideal generated by the mad family M and the finite subsets of ω. A set A is a partitioner of M if, for all M ∈ M, either M \ A is finite or M ∩ A is finite. A partitioner A of M is called a non-trivial partitioner of M if A is not in I_{M}. The set of partitioners 𝔄 of M forms a subalgebra of P(ω). We some times use 𝔄_{M} to denote this subalgebra. The partitioner algebra P_{M} of M is the quotient algebra 𝔄/I_{M}. A Boolean Algebra is said to be representable if it is isomorphic to a partitioner algebra of some mad family on ω.

Baumgartner and Weese in [BW82] introduced the notions of partitioner algebra and representable algebra. In actual fact this study began with Mrówka’s ground breaking result that the partitioner algebra could be trivial (See [Mro77]). It has been an interesting and useful topic for more than twenty years to determine if a given Boolean Algebra is representable (eg. see [Teres80]) . Mrówka’s method was expanded in [BW82] and the authors gave the first fundamental theorem which says that, under CH, every Boolean Algebra with cardinality less or equal to 𝔠 is representable. It is also shown that in Cohen real extension any Boolean algebra containing a free Boolean Algebra with size 𝔠 is not representable. In the same paper, there was a number of open problems regarding the representability of a Boolean Algebra.

In answering question 7 of [BW82], Zbierski and Frankiewicz proved that it is consistent with MA (σ-linked) that 𝔠 is arbitrarily large and that each Boolean algebra of cardinality ≤ 𝔠 is indeed representable. Dow and Nyikos proved in [DN92] that the free algebra with ℵ_{1} many generators is representable. This answers question 5 of [BW82]. Dow and Frankiewicz in [DF91] proved that it is consistent that the class of representable algebras is not closed under the quotients and that there is an algebra of cardinality ℵ_{1} which is not representable.

The first problem in [BW82] is to decide if every subalgebra of a representable algebra is also representable. By Baumgartner and Weese’s result, the answer is yes under CH. In this paper, we prove that, with the presence of a supercompact cardinal, it is consistent there exists a nonrepresentable algebra which is a subalgebra of a representable algebra. We will actually prove that PFA^{+} and the existence of a ♢-sequence on {α ∈ ω_{2} : cf α = ω} imply the existence of such algebras. In section 3, we will contruct a partitioner algebra with a nonrepresentable subalgebra B ; the proof that B is not representable will occupy sections 4 through 6 .

Date: July 19, 1997.

1991 Mathematics Subject Classification. 03E35, 06E05; Secondary: 54B15, 54D80, 03E55.

Key words and phrases. Boolean Algebra, partioner, representable, PFA^{+}, forcing, mad family, coherence sequence .

1.1. Definitions and Notations. Our notation and definitions are standard. For any set S, we use |S| to denote the cardinality of S. For any cardinal number κ, [S]^{κ} is the set of subsets of S with cardinality κ. When we write f: A → B, we mean f is a function or a partial function from the set A to the set B. We will use Fin to denote the ideal of finite subsets of ω. For a family A of subsets of S, we use I_{A} to denote the minimal ideal on S containing A. For a family A of subsets of S and two subsets A and B of S, we write A ⊆_{A} B if A \ B ∈ I_{A}. Naturally, A =_{A} B means that A△B ∈ I_{A}. When S is ω and I is Fin, we use the standard notation, for example, A =^{∗} ∅ means that A is a finite set.

We call an almost disjoint family an ad family and a maximal almost disjoint family a mad family. A sequence {a_{α}: α < λ} of sets is increasing mod finite if for α < β < λ, a_{α} ⊆^{∗} a_{β}

If an infinite set a is almost contained in every member of a family B of sets then we say that a is a pseudo-intersection of B. A set a splits a set b if both b ∩ a and b \ a are infinite. So a is a partitioner of a mad family M iff a does not split any member of M.

Our standard references for forcing are [Kun80] and [She82]. We always assume a poset has a largest element 1. We use x̌ (and occasionally just x) to denote a canonical name for x in the ground model. Consequently, for a poset ℙ and a set X in the ground model, if τ is a nice ℙ-name for a subset of X, we will write τ in the following form, ⋃{{x} × π_{x}: x ∈ X} instead of the formally correct ⋃{{x̌} × π_{x}: x ∈ X}, where each π_{x} is an antichain of ℙ.

Definition 1.1. A sequence {(a_{α}, b_{α}) : α ∈ λ} is called a coherent sequence on ω provided that both {a_{α} : α ∈ λ} and {b_{α} : α ∈ λ} are increasing mod finite sequences on ω and for each α < β < λ,

(i) a_{α} ⊆ b_{α}, (ii) a_{α} =^{∗} a_{β} ∩ b_{α}. If A = {(a_{α}, b_{α}) : α ∈ λ} and B = {(c_{α}, d_{α}) : α ∈ ρ} are coherent sequences,

then A^{⌢}B is the sequence {(u_{α}, v_{α}) : α < λ + ρ} where (u_{α}, v_{α}) = (a_{α}, b_{α}) when α < λ and (u_{α}, v_{α}) = (c_{ξ}, d_{ξ}) when α = λ + ξ for some ξ < ρ. It is easy to see that A^{⌢}B will also be a coherent sequence if for all α < λ, a_{α} =^{∗} c_{0} ∩ b_{α} and b_{α} ⊆^{∗} d_{0}. In the case B is the singleton {(c_{0}, d_{0})}, we write A^{⌢}(c_{0}, d_{0}) instead of A^{⌢}{(c_{0}, d_{0})}. A set X is said to extend A, if for all α < λ, X ∩ b_{α} =^{∗} a_{α}.

If a ⊆ b are two sets, we adopt Shelah’s notation in [She82] using Ch_{b}^{a} to denote the function r from b to {0, 1} such that r^{−1}(1) = a.

Definition 1.2. Suppose A = {(a_{α}, b_{α}) : α ∈ λ} is a coherent sequence on ω. ℙ_{A} is the poset consisting of the partial functions p from ω to {0, 1} such that there is an α < λ such that p =^{∗} Ch_{b_{α}}^{a_{α}}, i.e the following holds,

1. dom p =^{∗} b_{α} and

2. p^{−1}(1) =^{∗} a_{α} For p and q in ℙ_{A}, p ≤ q if p ⊇ q.

Obviously, ℙ_{A} adds a subset X of ω which extends A.

For sets I and J and an infinite cardinal λ, the set Fn(I, J, λ) is the poset, ordered by reverse inclusion, of partial functions from I into J with cardinality less than λ .

When we write ∃^{∞} we mean “there are infinitely many” and by ∀^{∞} we mean “for all but only finitely many”. More generally, suppose λ is a cardinal number, then ∃^{λ} means “there exists λ many”; ∀^{λ} means “for all but fewer than λ many”. We give one more notation to finish this section. If f and g are any two functions, we use f \overline{∪} g to denote the function f ∪ g ↾(dom g \ dom f).

2. Preliminaries

Suppose λ is a cardinal. A poset ℙ is λ-directed closed if whenever D ⊆ P is directed and |D| < λ, then there is a p ∈ ℙ with p ≤ q for all q ∈ D. Obviously ℙ is λ-directed closed iff whenever γ < λ and {p_{ξ}: ξ < γ} is a decreasing sequence of elements of ℙ, then there is a p ∈ ℙ such that p ≤ p_{ξ} for all ξ < γ. In case λ is ω_{1}, ℙ is said to be countably closed.

Definition 2.1. [She82] A poset ℙ is called proper if, for any cardinal number λ and any stationary subset S of [λ]^{≤ω}, S remains stationary in the forcing extension V [G] for any ℙ-generic G over V.

We use S_{0}^{2} to denote the stationary set of all ordinals in ω_{2} with countable cofinality. For any ordinal α, lim α is the set of all limit ordinals below α.

The following theorem is well-known, but does not seem to be in the literature, so we only give an outline of its proof.

Theorem 2.2. If there is a supercompact cardinal, then there is a model V of ZFC such that

V ⊧ “PFA^{+} +♢(S_{0}^{2})” .

Proof. Suppose there exists a supercompact cardinal κ. By the theorem in [Lav78], we can assume κ is indestructible, i.e. for any forcing notion Q which is κ-directed closed, κ remains supercompact in V^{Q}. In the forcing extension by κ-directed closed forcing P = Fn(κ, 2, κ) , κ is still supercompact and there is a diamond sequence on {α ∈ κ : cf(α) = ω}. In V^{P}, let Q̇ be the usual forcing to get PFA. It is well-known that Q̇ forces PFA^{+}. Since, Q̇ is κ-cc, in V^{P ∗Q̇}, there is still a diamond sequence on {α ∈ κ : cf(α) = ω}. But κ = ω_{2} in the final model. □

It is well-known that Martin’s Maximum implies Chang’s Conjecture. In fact, PFA^{+} also implies Chang’s Conjecture (See Todorčević’s Theorem 3.25 and Theorem 3.27 in [Bek91] and the discussion after Corollary 3.6 on page 272 in [Vel92]). P. Koszmider proved in [Kos] that Chang’s Conjecture implies there is no strong chain of length ω_{2} in [ω_{1}]^{ω_{1}}(see the definition below). Therefore PFA^{+} implies there is no strong chain of length ω_{2}.

Definition 2.3. A sequence {X_{α} : α ∈ λ} in [ω_{1}]^{ω_{1}} is called a strong chain if for any α < β ∈ λ , X_{α} \ X_{β} is finite and X_{β} \ X_{α} is uncountable.

A weak Kurepa tree is a tree of height ω_{1} and cardinality ℵ_{1} with more than ℵ_{1} many ℵ_{1}-branches. The weak Kurepa Hypothesis (wKH) asserts that there is a weak Kurepa tree.

We will need to use the Open Coloring Axiom (OCA):

If X is a separable metric space and

[X]^{2} = K_{0} ∪ K_{1}

is a partition of [X]^{2} with K_{0} open, i.e. for any {x, y} ∈ K_{0}, there are open neighbourhoods U_{0} and U_{1} of x and y respectively, such that, for any x′ ∈ U_{0} and y′ ∈ U_{1}, {x′, y′} ∈ K_{0}.

Then either

1. there exists an uncountable 0-homogeneous subset A of X, i.e. [A]^{2} ⊆ K_{0}, or 2. X can be covered by countably many 1-homogeneous subsets, {A_{n}: n ∈ ω},

i.e. for each n, [A_{n}]^{2} ⊆ K_{1}. We now list several of the well known consequences PFA in the following Theo-

rem.

Theorem 2.4. Suppose PFA holds. Then the following are true:

1. OCA (Todorčević,[Tod89]),

2. wKH is false (Todorčević, [Tod81], see also [Bau84])

3. 2^{ℵ_{0}} = ω_{2} (Todorčević, [Bek91]).

Definition 2.5. A family {(a_{α}, b_{α}): α ∈ ω_{1}} is a gap if for each α < ω_{1}, β < ω_{1}, a_{α} ∩ b_{β} is finite and a_{α} ∩ b_{α} is empty. For a set a, we say a fills the gap if for all α, a_{α} ⊆^{∗} a and b_{α} ∩ a is finite. In this case we say that the gap can be filled.

Definition 2.6 ([Dow97, 2.6]). A family {(a_{α}, b_{α}) : α ∈ ω_{1}} is a Hausdorff-Luzin type family if for each α < ω_{1} and m ∈ ω,

{β < α : |a_{α} ∩ b_{β}| < m} is finite.

If, in addition, {(a_{α}, b_{α}) : α ∈ ω_{1}} is a gap, then we call it a Hausdorff-Luzin type gap.

It is easy to see that the property of being a Hausdorff-Luzin type family is absolute for models preserving ω_{1}. It is also easy to see that any Hausdorff-Luzin type gap can not be filled.

Definition 2.7 ([Dow97, 2.1]). Two ideals I and J can be countably separated if there are countably many sets X_{n} such that for each pair (I, J) ∈ I ×J, there is an n such that I is almost disjoint from X_{n} and J is almost contained in X_{n}. We will say that the ideals are strongly countably separated if the X_{n}’s have the additional property that X_{n} ∩ I is finite for each I ∈ I and each n ∈ ω (i.e. [X_{n}]^{ω} ∩ I = ∅ for each n).

If we say that a set of pairs {(I_{α}, J_{α}) : α ∈ ω_{1}} is (strongly) countably separated then we mean that the ideal generated by family {I_{α} : α ∈ ω_{1}} is (strongly) countably separated from the one generated by {J_{α} : α ∈ ω_{1}}. We recall the following result.

Theorem 2.8 ([Dow97]). Suppose I and J are ideals of countable sets with I a P -ideal. If I and J are countably separated, then they are strongly countably separated.

Moreover, if PFA holds. then either

1. I and J can be countably separated,

or 2. there are ℵ_{1}-generated subideals I′ ⊂ I and J′ ⊂ J that cannot be countably

separated.

3. Main Results

In this section, we will first construct a mad family and a subalgebra of its partitioner algebra. Then we list some basic facts about the mad family and the subalgebra. In the last subsection we will prove from Lemma 3.8 that the subalgebra is not representable and the proof of Lemma 3.8 will occupy the rest of the paper.

By Theorem 2.2, if there is a supercompact cardinal, then there is a model in which PFA^{+} and ♢(S_{0}^{2}) hold. Hence, for the rest of the paper, we will assume that PFA^{+} and ♢(S_{0}^{2}) hold in V. Therefore the following are true in our universe V:

1. OCA.

2. 𝔠 = ω_{2}.

3. there is no weak Kurepa tree.

4. There is no strong chain in [ω_{1}]^{ω_{1}} of length ω_{2}.

3.1. Constructions. The actual construction of 𝔅 is relatively short but we will spend considerable effort to prove that it works. We will first construct a mad family M and simultaneously an increasing mod finite sequence {a_{α}: α ∈ ω_{2}} of non trivial partitioners of M. The sequence {a_{α}: α ∈ ω_{2}} will generate a maximal proper ideal on ω. Using a different underlying set, it is easy to see that P_{M} × P_{M} is isomorphic to the partitioner algebra of the mad family M̃ defined as follows: M̃ = M ∪ M′, where M′ = {M × {0}: M ∈ M}. We will use PM̃ instead of P_{M} × P_{M}. 𝔅 will be generated by the union of the ideals generated by the two chains {a_{α} : α ∈ ω_{2}} and {a_{α} × {0}: α ∈ ω_{2}}. Formally we let 𝔅 be the set of equivalence classes in PM̃ of those partitioners x for which there is an α < ω_{2} such that either x or its complement is contained in a_{α} ∪ a_{α} × {0} . Obviously ω is a nontrivial partitioner of M̃ which is not in 𝔅.

We were initially motivated by the observation that 𝔅 itself cannot be the actual partitioner algebra of some mad family on ω∪ω×{0} because the gap structure from the a_{α}’s will ensure that ω itself will be an unwanted partitioner. This theme will be carried through the proof as our method will be to show that in any representation there will be a partitioner analogous to [ω]. Indeed if a potential representation does not have a partitioner analogous to [ω] then it is too wild and we will use Shelah’s no non-trivial automorphism techniques to kill it.

Let us first fix an enumeration, {C_{α} : α ∈ lim ω_{2}}, of [ω]^{ω} and choose a sufficiently large regular cardinal θ. For the elementary submodels 𝔐_{α} appearing below, we use T_{α} to denote the tree consisting of all countable coherent sequences in 𝔐_{α}. The ordering on T_{α} is end extension. In other words, for B and A in T_{α}, B ≺ A iff A = {(c_{ξ}, d_{ξ}) : ξ < β} and, for some γ ≤ β, B = {(c_{ξ}, d_{ξ}) : ξ < γ}. [T_{α}] is the set of all ℵ_{1}-branches of T_{α}.

We will define for each α < ω_{2}, 𝔐_{α}, M_{α} and a_{α} such that {𝔐_{α} : α < ω_{2}} is a ∈-chain of elementary submodels of H_{θ} each with cardinality ℵ_{1}, {M_{α} : α ∈ ω_{2}} is an increasing sequence of almost disjoint families on ω and {a_{α} : α ∈ ω_{2}} is an increasing, mod finite, sequence of infinite subsets of ω. Moreover, the following properties will hold.

1. 𝔐_{0} contains a ♢(S_{0}^{2}) sequence and {C_{α} : α ∈ lim ω_{2}} as elements, thus if α ∈ 𝔐_{α}, then C_{α} ∈ 𝔐_{α}.

2. [T_{α}] ⊆ 𝔐_{α+1}.

3. ∀α ∈ ω_{2}, a_{α} is co-infinite and M_{α} < a_{α}, i.e. ∀M ∈ M_{α}, M ⊆^{∗} a_{α}.

4. ∀α ∈ ω_{2}, {α, M_{α}, a_{α}} ∈ 𝔐_{α}, and if α < β, then M_{β} ∩ 𝔐_{α} = M_{α}.

5. ∀α ∈ lim ω_{2}, (i) M_{α} = ⋃_{β<α} M_{β}; (ii) M_{α} = M_{α+1} = M_{α+2} and M_{α+n} = M_{α+3}(n ≥ 3); (iii) for a ⊆ ω, if a ∈ 𝔐_{α+2}, then a is a partitioner of M_{α+3} \ M_{α}.

6. ∀α ∈ lim ω_{2}, there exists M ∈ M_{α+3} such that C_{α} ∩ M ≠^{∗} ∅ and either ω \ C_{α} ⊆^{∗} a_{α+1} or C_{α} ⊆^{∗} a_{α+1}.

7. For all α ∈ ω_{2}, ω \ a_{α} is infinite and there exist, in 𝔐_{α+1}, Cohen generic reals g_{0}^{α} and g_{1}^{α} over 𝔐_{α} such that (g_{0}^{α})^{−1}(0) ∪ (g_{1}^{α})^{−1}(1) ⊆ a_{α+1} .

It will be clear during the construction that most of the conditions will be trivially satisfied. For example, to get condition (1) and condition (2), we just put the necessary sets into 𝔐_{α}. Condition (6) ensures that {a_{α}: α < ω_{2}} will generate a maximal ideal and that ⋃_{α<ω_{2}} M_{α} is a mad family. The Cohen reals in condition (7), in some sense, guarantee that the mad family ⋃_{α<ω_{2}} M_{α} and its partitioner algebra are complicated enough to allow us to force that 𝔅 is not representable. Conditions (1) - (5) are mainly technical which will be used in the next three sections.

We will actually perform the induction along the non successor ordinals, i.e. for each limit α, we describe how to get {𝔐_{α+i} : i ∈ ω}, {M_{α+i} : i ∈ ω} and {a_{α+i} : i ∈ ω}. Suppose we are at now at some non-successor stage α. Case 1: α = 0 Take any elementary submodel 𝔐 of H_{θ} as 𝔐_{0} such that 𝔐 has cardinality ℵ_{1} and it contains a ♢(S_{0}^{2}) sequence and {C_{α}: α ∈ lim ω_{2}} as elements. Let M_{0} = ∅. Let a_{0} ∈ 𝔐_{0} be any infinite and co-infinite subset of ω. For α ∈ ω\{0}, proceed as in subcase 4 below. Case 2: α ∈ lim ω_{2} Let M_{α} = ⋃_{β<α} M_{β}. Since |M_{α}| ≤ ℵ_{1} and PFA holds, there exists a_{α} ⊆ ω such that ω \ a_{α} is infinite and

∀β < α, a_{β} ⊆^{∗} a_{α} .

Let 𝔐_{α} be an elementary submodel of H_{θ} such that |𝔐_{α}| ≤ ℵ_{1} and {𝔐_{β}, β < α}, {M_{β}, β < α} and {a_{β} : β < α} and {C_{α}, a_{α}} are elements of 𝔐_{α}. It is easy to see that conditions (1)-(7) are satisfied. Next we define, by induction on i, {𝔐_{α+i}, M_{α+i}, a_{α+i} : 0 < i ∈ ω} . Subcase 1 : i = 1 We have to find 𝔐_{α+1}, M_{α+1}, a_{α+1} and g_{0}^{α} and g_{1}^{α}. Let

T_{α} = {A ∈ 𝔐_{α} : A is a coherent sequence of countable length} .

Then T_{α} is a tree under the partial order ≺ defined above. Obviously T_{α} has cardinality not greater than ℵ_{1} and height ℵ_{1}. By Theorem 2, the weak Kurepa Hypothesis fails, thus T_{α} has at most ℵ_{1} many ℵ_{1}-branches, i.e. |[T_{α}]| ≤ ℵ_{1}. Since 𝔐_{α} has size ℵ_{1}, by MA_{ℵ_{1}}, there are countably many Cohen reals over 𝔐_{α}, say, {r_{n} : n ∈ ω} such that for any real a ⊆ ω in 𝔐_{α}, {r_{n}^{−1}(0): n ∈ ω} is independent on a.

Now let M_{α+1} = M_{α} and 𝔐_{α+1} be an elementary submodel of H_{θ} with cardinality ℵ_{1} and {[T_{α}], {r_{n} : n ∈ ω}, 𝔐_{α}} ⊆ 𝔐_{α+1}. Note that since ω_{1} is both a subset of 𝔐_{α+1} and an element of 𝔐_{α+1}, the sets [T_{α}], {r_{n} : n ∈ ω} and 𝔐_{α} are also subsets of 𝔐_{α+1}. Since a_{α} ∈ 𝔐_{α}, the set r_{0}^{−1}(0) ∪ r_{1}^{−1}(1) ∪ a_{α} is co-infinite. If ω \ C_{α} is almost contained in a_{α}, then we let a_{α+1} be r_{0}^{−1}(0) ∪ r_{1}^{−1}(1) ∪ a_{α}. Otherwise, a_{α} ∪ C_{α} is co-infinite, therefore, r_{0}^{−1}(0) ∪ r_{1}^{−1}(1) ∪ a_{α} ∪ C_{α} is also co-infinite. Let a_{α+1} = r_{0}^{−1}(0) ∪ r_{1}^{−1}(1) ∪ a_{α} ∪ C_{α}. Let g_{0}^{α} = r_{0} and g_{1}^{α} = r_{1}. Obviously, conditions (1)-(7) are satisfied. Subcase 2 : i = 2

We have to find 𝔐_{α+2}, M_{α+2} , a_{α+2}, g_{0}^{α+1} and g_{1}^{α+1}. Let M_{α+2} = M_{α}. Take any elementary submodel 𝔐 of H_{θ} containing a sequence of reals {r_{1,n} : n ∈ ω} which are mutually Cohen generic over 𝔐_{α+1}. Let 𝔐 also contain all the ℵ_{1} branches [T_{α+1}] of T_{α+1} and 𝔐_{α+1}. Let 𝔐_{α+2} be 𝔐. Let a_{α+2} = r_{1,0}^{−1}(0) ∪ r_{1,1}^{−1}(1) ∪ a_{α+1}. Let g_{0}^{α+1} = r_{1,0} and g_{1}^{α+1} = r_{1,1} Subcase 3: i = 3

We have to find 𝔐_{α+3}, M_{α+3}, a_{α+3}, g_{0}^{α+2}, and g_{1}^{α+2}.

Take an ultrafilter V on ω such that V ∈ 𝔐_{α+2} and {ω \ M : M ∈ M_{α}} ⊆ V and C_{α} ∈ V in the case that, for all M ∈ M_{α}, C_{α} ∩ M =^{∗} ∅. Again PFA implies there exists a pseudo-intersection M^{∗} of V ∩ 𝔐_{α+2}. Let

M_{α+3} = M_{α+2} ∪ {M^{∗}} .

Since M^{∗} is a pseudo-intersection of V ∩ M_{α+2}, M^{∗} is not in 𝔐_{α+2} and no subset of ω in 𝔐_{α+2} splits M^{∗}.

Let {r_{2,n} : n ∈ ω} be mutually Cohen reals over 𝔐_{α+2}. Let 𝔐_{α+3} be an elementary submodel of H_{θ} with cardinality ℵ_{1} such that {𝔐_{α+2}, M^{∗}, {r_{2,n} : n ∈ ω}, [T_{α+2}]} ⊆ 𝔐_{α+3}

By the choice of M^{∗}, it is easy to see that a_{α+2} ∪ M^{∗} is contained in some coinfinite subset of a in 𝔐_{α+2}. Therefore, a_{α+2} ∪ M^{∗} ∪ r_{2,0}^{−1}(0) ∪ r_{2,1}^{−1}(1) is co-infinite. Let a_{α+3} = a_{α+2} ∪ M^{∗} ∪ r_{2,0}^{−1}(0) ∪ r_{2,1}^{−1}(1) and g_{0}^{α+2} = r_{2,0} and g_{1}^{α+2} = r_{2,1}. Subcase 4: i > 3

In this case we let M_{α+i} = M_{α+3} and simply choose 𝔐_{α+i} ≺ H_{θ} such that {𝔐_{α+i−1}, [T_{α+i−1}], {r_{i−1,j} : j ∈ ω}} ⊆ 𝔐_{α+i}, where {r_{i−1,j} : j ∈ ω} are reals which are mutually Cohen over 𝔐_{α+i−1}. Let a_{α+i} = a_{α+i−1} ∪ r_{i−1,0}^{−1}(0) ∪ r_{i−1,1}^{−1}(1) and g_{0}^{α+i−1} = r_{i−1,0} and g_{1}^{α+i−1} = r_{i−1,1}.

Finally let M = ⋃_{α<ω_{2}} M_{α}.

3.2. Basic Facts and finally 𝔅. We first prove some basic facts about {𝔐_{α}, M_{α}, a_{α} : α ∈ ω_{2}}.

Fact 3.1. M is a mad family.

Proof. This is obvious by (6). □

Fact 3.2. {a_{α} : α ∈ ω_{2}} generates a maximal ideal on in P(ω).

Proof. By (6), for each subset a of ω, either a or ω \ a is contained in some a_{β}. □

Fact 3.3. 1. If a ∈ 𝔐_{α}, then a is a partitioner of M \ M_{α}. Moreover, if α is a limit and, for each β < α, a ∩ a_{β} is a partitioner of M_{β}, then, a is a partitioner of M.

2. each a_{α} is a non trivial partitioner of M.

Proof. Since M_{α} < a_{α}, (1) implies (2), so we prove (1). Let M ∈ M\M_{α} and set β to be the minimal ordinal such that M ∈ M_{β}. Then by (4), β must be a successor ordinal and β must be of the form ξ + 3 with ξ a limit. Since α < β = ξ + 3, a ∈ 𝔐_{ξ+2}. By (4), a is a partitioner of M_{ξ+3} \ M_{ξ} which contains M according to the choice of ξ. Therefore a does not split M. Suppose, in addition, α is a limit, and a ∩ a_{β} is a partitioner of M_{β} for each β < α. Let M ∈ M_{α}, then there is β < α such that M ∈ M_{β}. Since M ⊆^{∗} a_{β}, a does not split M. □

Recall that if M is an ad family on a countable subset S , we use I_{M} to denote the ideal on P(S) generated by M.

We need some additional notation to define 𝔅 which will actually be defined using a partitioner algebra on ω ∪ ω × {0} .

We use a′_{α} to denote a_{α}×{0}. Let M′ = {M ×{0} : M ∈ M} and M̃ = M∪M′. Obviously, M̃ is a mad family on ω ∪ ω × {0} and P_{M} × P_{M} is isomorphic to the partitioner algebra of M̃. Let 𝔄 denote the partitioner algebra of M̃.

Let 𝔅̃ = {a ⊆ ω ∪ ω × {0}: a is a partitioner of M̃ and ∃ β a ⊆^{∗} a_{β} ∪ a′_{β} or (ω ∪ ω × {0}) \ a ⊆^{∗} a_{β} ∪ a′_{β}}. It is obvious that 𝔅̃ forms a subalgebra of P(ω ∪ ω × {0}).

Finally let 𝔅 = 𝔅̃/IM̃. Obviously, 𝔅 is the subalgebra of the partitioner algebra 𝔄. The following obvious fact will play a significant role in determining that B is not representable.

Fact 3.4. There does not exist a ∈ 𝔅̃ such that, for all α ∈ ω_{2}, a_{α} \ a ∈ IM̃ and a′_{α} ∩ a ∈ IM̃ □

Since {a_{α} : α ∈ ω_{2}} generates a maximal ideal on ω, {ω \ a_{α} : α ∈ ω_{2}} is a base for a P_{ω_{2}}-ultrafilter on ω. Let us call this ultrafilter z.

Let X, X_{0} and X_{1} denote the Stone spaces of P_{M} × P_{M} , P_{M} and P_{M′} respectively. Then X is the topological union of X_{0} and X_{1}. Let z_{0} and z_{1} be the copies of the P_{ω_{2}} point z in X_{0} and X_{1} respectively. In X , we identify z_{0} and z_{1} to get a quotient space Y. Hence another way to describe 𝔅 is that it is the clopen algebra of the topological space Y.

We need more notation.

Definition 3.5. Let A be a coherent sequence. We call A a good coherent sequence if A is of the form: {(b_{α}, a_{μ(α)}) : α ∈ ρ} , where a_{μ(α)} are the non trivial partitioners of M in the above construction and b_{α} is also a nontrivial partitioner of M for all α ∈ ρ. A good coherent sequence A = {(b_{α}, a_{μ(α)}) : α < ρ} is called a reasonable sequence if for all α < ρ,

A_{α} = {(b_{ξ}, a_{μ(ξ)}) : ξ < α} ∈ 𝔐_{μ(α)} .

We will call a good coherent sequence with countable length a g-sequence and a reasonable coherent sequence with countable length an r-sequence.

3.3. Main Theorem. In this section we reduce the proof that B is not representable to a technical Lemma.

Theorem 3.6. The Boolean Algebra 𝔅 is not representable.

Definition 3.7. Let F be a function from [ω ∪ ω × {0}]^{ω} to [ω]^{ω} and let N be a mad family on ω. We say that F induces an isomorphism from 𝔅 to P_{N} if the map φ : [A] ↦ [F (A)] is an isomorphism from 𝔅 onto P_{N} . That is:

A ⟶^{F} F (A)
φ_{0}↓  ^{φ_{1}}↓
[A] ⟶^{φ} φ([A])

Where φ_{0} : 𝔅̃ → 𝔅 and φ_{1} : 𝔄_{N} → P_{N} are the natural maps.

It will be clear from the proof of Theorem 3.6 that the following lemma essentially says that there is a coherent sequence A extended by some set X while the image of A under F can not be extended modulo I_{N} by any subset of ω. The c_{α}’s and e_{α}’s are used to guarantee that no partitioner of N can exist as the image of X.

Lemma 3.8. If N is a mad family whose partitioner algebra is isomorphic to 𝔅, then there are a function F: [ω ∪ (ω × {0})]^{ω} → [ω]^{ω} and a reasonable coherent sequence A = {(b_{α}, a_{μ(α)}) : α ∈ ω_{1}} such that F induces an isomorphism from 𝔅 to P_{N} and A is extended by some set X ⊆ ω. Moreover, there exist a sequence C = {(c_{α+k}, e_{α+k}): α ∈ lim ω_{1}, 0 < k ≤ ω} and an increasing sequence Σ = {σ_{α} : α ∈ ω_{1}} in lim ω_{1} such that c_{ξ} and e_{ξ} are disjoint non trivial partitioners of N and the following hold

1. ∀α ∈ lim ω_{1}, k, l ∈ ℕ, c_{α+k} ⊆ c_{α+ω} and e_{α+k} ⊆ e_{α+ω} and k < l implies c_{α+k}, e_{α+k}, c_{α+l} and e_{α+l} are mutually disjoint and

c_{α+ω} =_{N} F (b_{α+ω})
e_{α+ω} =_{N} F (a_{μ(α+ω)}) \ F (b_{α+ω})
c_{α+k} =_{N} F (b_{α+k}) \ F (b_{α+k−1})
e_{α+k} =_{N} F (a_{μ(α+k)}) \ (F (a_{μ(α+k−1)}) ∪ F (b_{α+k})) .

2. For evey k ∈ N the family {(c_{σ_{α}+k}, e_{σ_{α}+k}): α ∈ ω_{1}} is a Hausdorff-Luzin type family.

Note that the role of the sequence C is simply to ensure that there is no suitable real which extends the image of A modulo I_{N}.

We introduce more notation. Let A = {(b_{α}, a_{μ(α)}) : α ∈ λ} be a good coherent sequence with λ a limit ordinal such that λ ≠ β + ω for any β. Suppose F induces an isomorphism from 𝔅 to some partitioner algebra P_{N}. Then the derived sequence C(A, F) = {(c_{α+k}, e_{α+k}): α ∈ lim λ and 0 < k ≤ ω} of A and F is defined by the following formulas,

d_{α+ω} = F (a_{μ(α+ω)})
c_{α+ω} = F (b_{α+ω}) ∩ d_{α+ω},
e_{α+ω} = d_{α+ω} \ c_{α+ω}
d_{α+k} = (F (a_{μ(α+k)}) \ ⋃_{i<k} F (a_{μ(α+i)})) ∩ d_{α+ω} ,
c_{α+k} = c_{α+ω} ∩ d_{α+k}
e_{α+k} = d_{α+k} \ c_{α+k}

It is obvious that the derived sequence satisfies (1) in Lemma 3.8.

We first complete the proof of Theorem 3.6 from Lemma 3.8.

Proof of Theorem 3.6: Suppose that 𝔅 is representable. We will derive a contradiction. Let N be a mad family which represents 𝔅. Let F, A, C, Σ be as in Lemma 3.8. Let μ = sup{μ(β): β < ω_{1}}. Observe that {A_{α}: α < ω_{1}} ∈ [T_{μ}]. so A ∈ 𝔐_{μ+1}. By elementarity we can assume that X ∈ 𝔐_{μ+1}. We can also assume that X ⊆ a_{μ}. Otherwise we use X ∩ a_{μ} to replace X. Let Y = F (X).

Since X ∈ 𝔐_{μ+1}, it follows from Fact 3.3 that X does not split any M ∈ M\M_{μ+1}. However for any M ∈ M_{μ+1}, there exists α < ω_{1} such that M ⊆^{∗} a_{μ(α)}.

Since X extends A, X ∩ a_{μ(α)} =^{∗} b_{α} and recall that b_{α} is a partitioner of M. Therefore X does not split M. We have proven that X is a partitioner of M. Furthermore, since X ⊆ a_{μ+1}, X ∈ 𝔅̃, i.e [X] ∈ 𝔅.

Therefore Y must be a non-trivial partitioner of N.

For α ∈ ω_{1}, since X extends A and F induces an isomorphism from 𝔅 to P_{N},

F (X) ∩ F (a_{μ(α)}) =_{N} F (b_{α}) .

Therefore, c_{α+ω} ⊆_{N} Y and e_{α+ω} ∩ Y =_{N} ∅.

For each k ∈ ℕ, if c_{α+k} \ Y is infinite, then there exists N_{k} ∈ N such that N_{k} ⊆^{∗} c_{α+k} \ Y ⊆ c_{α+ω} \ Y. Since c_{α+k} (k ∈ ω) are mutually disjoint, the N_{k}’s are distinct. Since c_{α+ω} ⊆_{N} Y, there are at most finitely many such N_{k}. Therefore, there are at most finitely many k such that c_{α+k} ⊈^{∗} Y. Similarly there are only finitely many k such that e_{α+k} ∩ Y ≠^{∗} ∅. Therefore we can find an uncountable subset Γ of ω_{1} and k ∈ ℕ and finite sets H, K such that for all α ∈ Γ,

c_{σ_{α}+k} \ Y = H and e_{σ_{α}+k} ∩ Y = K .

Let m > max(H ∪ K) , and α ∈ Γ such that α ∩ Γ is infinite. Then, for all β ∈ α ∩ Γ, c_{σ_{α}+k} ∩ e_{σ_{β}+k} ⊆ m. Therefore {β < α: c_{σ_{α}+k} ∩ e_{σ_{β}+k} ⊆ m} is infinite. But according to (2) of Lemma 3.8, {(c_{σ_{α}+k}, e_{σ_{α}+k}): α ∈ ω_{1}} is a Hausdorff-Luzin type family which is the contradiction we seek. □

Our strategy to prove Lemma 3.8 is first to get such a coherent sequence in some proper forcing extension. Then we use PFA to ensure that such a coherent sequence exists in the ground model. Then, since we are using PFA to produce such a coherent sequence in the ground model, we are only allowed to use ω_{1} many dense subsets to determine the fact that no suitable subsets of ω extends the image of A modulo I_{N}. However, a Hausdorff-Luzin type family can be determined via ω_{1} many dense subsets. The next three sections are actually to construct such a proper forcing and such a coherent sequence in the forcing extension. In other words, we will use the next three sections to prove the following Theorem 3.9 . We omit the standard dense set argument which shows how Lemma 3.8 follows from Theorem 3.9. It turns out that it is very difficult to get such a proper forcing notion.

Theorem 3.9. If N is a mad family whose partitioner algebra is isomorphic to 𝔅, then there exist a function F that induces an isomorphism from 𝔅 to P_{N} and a proper forcing notion 𝔻 such that, in V^{𝔻}, there exist a good coherent sequence A = {(b_{α}, a_{μ(α)}) : α ∈ ω_{1}} and a sequence Σ = {σ_{α} : α ∈ ω_{1}} in lim ω_{1} such that the following are true.

(i) For all α ∈ ω_{1}, A_{α} is an r-sequence and A can be extended by some subset X of ω.

(ii) F, A, C(A, F) and Σ satisfy the conclusions of Lemma 3.8. Where C(A, F) = {(c_{α+k}, e_{α+k}): α ∈ lim ω_{1} and 0 < k ≤ ω} is the derived sequence of A and F

4. Separating Mad Families

In this section we will prove Theorem 4.1, which will be the foundation for the whole proof of Theorem 3.9.

We are motivated by the picture that either the {N_{i,j}}’s or J_{n}’s listed therein are like the columns in ω × ω and the F (a_{α})’s are like the points below a graph of a function. By the construction of 𝔅, if F induces an isomorphism from 𝔅 to some partitioner algebra, then {(F (a_{α}), F (a′_{α})): α < ω_{2}} will be very similar to an unfilled gap. However, by PFA, each (ω_{2}, ω_{2}^{∗})-gap can be filled. It is essentially this situation which led us to believe that 𝔅 should not be representable.

Theorem 4.1. If N is a mad family on ω such that 𝔅 is isomorphic to the partitioner algebra P_{N} of N, then there exists a map F from [ω ∪ (ω × {0})]^{ω} to [ω]^{ω} such that F induces an isomorphism from 𝔅 to P_{N} and there exist Ñ = {N_{i,j} : i, j ∈ ω} ⊆ N and a partition of ω , J̃ = {J_{n} : n ∈ ω} such that either (i)-(ii) hold or (iii)-(v) hold where (i)-(v) are as follows:

For α ∈ ω_{2}, let X_{α} = {N ∈ N : N ⊆^{∗} F (a_{α})}.

(i) X_{α} ∩ {N_{i,j} : j ∈ ω} is finite for all α ∈ ω_{2} , i ∈ ω; (ii) If I ∈ [N]^{ω} and there are infinitely many i ∈ ω such that I ∩ {N_{i,j} : j ∈ ω} ≠

∅}, then there exists an α such that I ∩ X_{α} ≠^{∗} ∅; (iii) for all α ∈ ω_{2}, for all m ∈ ω, F (a_{α}) ∩ J_{m} =^{∗} ∅; (iv) if a ⊆ ω and there are infinitely many m ∈ ω such that a ∩ J_{m} ≠ ∅, then there

exists an α such that a ∩ F (a_{α}) ≠^{∗} ∅;

(v) For any N′ ∈ [⋃_{α∈ω_{2}} X_{α}]^{ω}, there is a β such that N′ ∩ X_{β} is infinite.

4.1. Aligning the mad family N. Suppose N is a mad family on ω which represents 𝔅 and that φ : 𝔅 → P_{N} is an isomorphism induced by F : [ω ∪(ω ×{0})]^{ω} → [ω]^{ω}. Obviously, if F induces φ and if F′ : [ω ∪ (ω × {0})]^{ω} → [ω]^{ω} is such that for any non trivial partitioner A ∈ 𝔅̃, F (A) is a partitioner of N and F (A) =_{N} F′(A) , then F′ also induces φ.

We will prove that either there exists F_{1} and Ñ satisfying the properties (i) and (ii) in Theorem 4.1 or there exist F_{2} and J̃ satisfying the properties (iii)-(v).

For the rest of this chapter we assume that N is a mad family representing 𝔅 and we fix an isomorphism φ from 𝔅 to P_{N}. If F : [ω ∪ (ω × {0})]^{ω} → [ω]^{ω} induces φ, we set X_{α}^{F} = {N ∈ N : N ⊆^{∗} F (a_{α})} and Z_{α}^{F} = {N ∈ N : N ⊆^{∗} F (a′_{α})}. It follows that {X_{α}^{F} : α < ω_{2}} and {Z_{α}^{F} : α < ω_{2}} are increasing, mod finite, subsets of N.

For any subset A of ω, we will also use N (A) to denote the set {N ∈ N : N ⊆^{∗} A}. The next lemma produces a set P which looks very much like a partitioner of N which should not exist (i.e. this P corresponds to the illegal partitioner [ω] of 𝔅 in PM̃). Recall we have PFA^{+} and ♢(S_{0}^{2}) available.

Lemma 4.2. There is a function F : [ω ∪ (ω × {0})]^{ω} → [ω]^{ω} that induces φ and for which the following hold.

1. There exists {Y_{n} : n ∈ ω} ⊆ P(N) and {Y′_{n} : n ∈ ω} such that Y′_{n} is a countable subset of Y_{n} and,

(a) for any countable subset K of N, if K is almost disjoint with X_{α}^{F} for all α in ω_{2}, then K ⊆^{∗} Y_{n} for some n;

(b) For all α ∈ ω_{2}, X_{α}^{F} ∩ Y_{n} = X_{α}^{F} ∩ Y′_{n} is finite.

2. for all λ ∈ ω_{2}, there exists μ_{λ} such that

⋃_{ξ<λ} X_{ξ}^{F} \ X_{μ_{λ}}^{F} is countable.

3. there exists P ⊆ ω, such that

(∀α ∈ ω_{2})F (a_{α}) ⊆ P and F (a′_{α}) ∩ P = ∅.

Proof. Fix any function f which induces the isomorphism φ and such that, for each α ∈ ω_{2}, f(a_{α}) ∩ f(a′_{α}) is empty. Let I = {I ∈ [N]^{ω} : (∃α)I ⊆^{∗} X_{α}^{f}} and let J = {I ∈ [N]^{ω} : (∃α)I ⊆^{∗} Z_{α}^{f}} . It follows that I and J are P-ideals and we now verify that I and J are ω_{1}-separated, i.e. given sequences {I_{α} : α < ω_{1}} and {J_{α} : α < ω_{1}} from I and J respectively, there is an H such that for all α ∈ ω_{1}, I_{α} ⊆^{∗} H and H ∩J_{α} is finite. First pick ξ < ω_{2} such that , for all α < ω_{1}, I_{α} ⊆^{∗} X_{ξ}^{f} and J_{α} ⊆^{∗} Z_{ξ}^{f}. Since X_{ξ}^{f} ∩ Z_{ξ}^{f} is finite, simply set H = X_{ξ}^{f}. By Theorem 2.8, I and J are strongly countably separated. Therefore there are {X(n) : n ∈ ω} such that for any I ∈ I and J ∈ J, there exists n such that I ⊆^{∗} X(n) and X(n) ∩ J is finite. Actually, since I and J are both P-ideals, there is a single n_{0} such that X(n_{0}) separates I and J. To see this, suppose that it is not true. Then, for any n in ω, there exists I_{n} ∈ I and J_{n} ∈ J such that either I_{n} ⊈^{∗} X(n) or X(n) ∩ J_{n} is infinite. Since they are P-ideals, fix I ∈ I and J ∈ J so that, for all n ∈ ω , I_{n} ⊆^{∗} I and J_{n} ⊆^{∗} J. But now there is m such that X(m) separates I and J which implies that X(m) separates I_{m} and J_{m} which contradicts the choice of I_{m} and J_{m}.

Let X be X(n_{0}). Obviously, for all α ∈ ω_{2} , X_{α}^{f} ⊆^{∗} X and X ∩ Z_{α}^{f} is finite. We may therefore assume X_{α}^{f} ⊆ X and X ∩ Z_{α}^{f} is empty for all α < ω_{2} because we can replace f by f′ where f′(a_{α}) = f(a_{α})\⋃(X_{α}^{f} \X) and f′(a′_{α}) = f(a′_{α})\⋃(Z_{α}^{f} ∩X).

Let K = {I ∈ [X]^{ω} : (∀α ∈ ω_{2}) I ∩ X_{α}^{f} =^{∗} ∅}. Again I and K are ω_{1}-separated. By Theorem 2.8, I and K can be strongly countably separated, say, by {Y_{n} : n ∈ ω}. Hence, for each α ∈ ω_{2}, X_{α}^{f} ∩ Y_{n} is finite and each K ∈ K, K is almost contained in some Y_{n}. Claim 1 There are Y′_{n} ∈ [Y_{n}]^{ω} such that, for all α ∈ ω_{2} , X_{α}^{f} ∩ ⋃_{n∈ω} Y_{n} \ Y′_{n} is finite.

Proof of the Claim 1: If this were not the case, we can construct an ω_{1}-sequence {α_{ξ} : ξ < ω_{1}} in ω_{2} such that, for all η < ω_{1} ,

(X_{α_{η}}^{f} \ ⋃_{ξ<η} X_{α_{ξ}}^{f}) ∩ ⋃_{n∈ω} Y_{n} is infinite.

The construction is straightforward. Take any α ∈ ω_{2} as α_{0}. At stage η, let Ỹ_{n} = Y_{n} ∩ ⋃_{ξ<η} X_{α_{ξ}}. If the claim fails, there must be an α such that X_{α}^{f} ∩ ⋃_{n∈ω} Y_{n} \ Ỹ_{n} is infinite, i.e., (X_{α}^{f} \ ⋃_{ξ<η} X_{α_{ξ}}^{f}) ∩ ⋃_{n∈ω} Y_{n} is infinite. Take such α as α_{η} .

Let λ be sup{α_{ξ} : ξ ∈ ω_{1}}. Then , for all η < ω_{1} ,

Γ_{η} = (X_{λ}^{f} ∩ X_{α_{η}}^{f} \ ⋃_{ξ<η} X_{α_{ξ}}^{f}) ∩ ⋃_{n∈ω} Y_{n} is infinite.

But the Γ_{η}’s are disjoint. Hence X_{λ}^{f} ∩ ⋃_{n∈ω} Y_{n} is uncountable. This contradicts the choice of Y_{n}’s, which finishes the proof of Claim 1.

By again modifying f, we can assume that, for all α, X_{α}^{f} ∩ ⋃_{n∈ω}(Y_{n} \ Y′_{n}) = ∅, i.e., for all α , X_{α}^{f} ∩ Y_{n} ⊆ Y′_{n}. Claim 2. For all λ ∈ ω_{2}, there exists μ_{λ} ≥ λ such that ⋃_{α<λ} X_{α}^{f} \ X_{μ_{λ}}^{f} and ⋃_{α<λ} Z_{α}^{f} \ Z_{μ_{λ}}^{f} are both countable. Proof of Claim 2: We only give the proof for the X_{α}^{f}’s.

Fix a λ ∈ ω_{2} and let S_{λ} be ⋃_{α<λ} X_{α}^{f} \ X_{λ}^{f}. Then |S_{λ}| ≤ ℵ_{1} and we just have to consider the case when |S_{λ}| = ℵ_{1}. Set S_{λ} = {N_{β} : β < ω_{1}}. Since Chang’s Conjecture holds and by Lemma 2, there are no strong chains of length ω_{2} on S_{λ}. Therefore there is some μ_{0} such that, for all μ > μ_{0}, (X_{μ}^{f} \ X_{μ_{0}}^{f}) ∩ S_{λ} is countable.

Subclaim There is a countable subset S of S_{λ} and μ_{λ} > μ_{0} such that

for all μ > μ_{λ}, (S_{λ} \ S) ∩ (X_{μ}^{f} \ X_{μ_{λ}}^{f}) is finite.

Proof of Subclaim: Suppose otherwise. We can then easily define a sequence {α_{ξ}^{0}, α_{ξ}^{1}, μ_{ξ}^{0}, μ_{ξ}^{1} : ξ ∈ ω_{1}} such that

1. for ξ < η < ω_{1}, α_{ξ}^{0} < α_{ξ}^{1} < α_{η}^{0}; μ_{0} < μ_{ξ}^{0} < μ_{ξ}^{1} < μ_{η}^{0};

2. (X_{μ_{ξ}^{1}}^{f} \ X_{μ_{ξ}^{0}}^{f}) ∩ {N_{β} : β ∈ (α_{ξ}^{0}, α_{ξ}^{1})} is infinite.

Let μ_{ω_{1}} be sup{μ_{ξ}^{1} : ξ < ω_{1}}. Then, for all ξ < ω_{1}, (X_{μ_{ω_{1}}}^{f} \ X_{μ_{0}}^{f}) ∩ {N_{β} : β ∈ (α_{ξ}^{0}, α_{ξ}^{1})} is infinite. Hence, (X_{μ_{ω_{1}}}^{f} \ X_{μ_{0}}^{f}) ∩ S_{λ} is uncountable contradicting the choice of μ_{0}. This finishes the proof of Subclaim.

Let A be S_{λ} \ (S ∪ X_{μ_{λ}}^{f}). We prove that A is countable. We have for any μ < ω_{2}, A ∩ X_{μ}^{f} is finite. For any I ∈ [A]^{ω}, there is an n such that I ⊆^{∗} Y_{n}. But A ∩ Y_{n} ⊆ Y′_{n}, therefore, I ⊆^{∗} Y′_{n}. Thus A ⊆^{∗} ⋃_{n∈ω} Y′_{n} and A is countable. Therefore, ⋃_{α<λ} X_{α}^{f} \ X_{μ_{λ}}^{f} =^{∗} S_{λ} \ X_{μ_{λ}}^{f} ⊆ (A ∪ S) is countable. This finishes the proof of the Claim 2.

Observe that (1) and (2) of the lemma holds for (our revised) f and {Y_{n}, Y′_{n} : n ∈ ω}. Next we modify f to some F such that (1)-(3) of the lemma holds for F and {Y_{n}, Y′_{n} : n ∈ ω}

Let T = {(a, b) ∈ P(ω) × P(ω) : a ∩ b = ∅ and ∃α a ⊆^{∗} f(a_{α}), b ⊆ f(a′_{α})} Let K_{0} = {{(a, b), (c, d)} ∈ [T]^{2} : a ∩ d ≠ ∅ or c ∩ b ≠ ∅}. Let K_{1} = [X]^{2} \ K_{0}. K_{0} is open with respect to the topology on T inherited from P(ω). Claim 3 There is no 0-homogeneous set with cardinality ℵ_{1}

Proof of Claim 3 Suppose there is an ℵ_{1} sized 0-homogeneous set {(c_{α}, d_{α}) : α < ω_{1}}. For each α ∈ ω_{1} , let ξ_{α} < ω_{2} be such that c_{α} ⊆^{∗} f(a_{ξ_{α}}) and d_{α} ⊆^{∗} f(a′_{ξ_{α}}). Let λ = sup{ξ_{α} : α < ω_{1}}. By Claim 2, there is a μ_{λ} such that ⋃_{α∈ω_{1}} X_{ξ_{α}}^{f} \ X_{μ_{λ}}^{f} and ⋃_{α∈ω_{1}} Z_{ξ_{α}}^{f} \ Z_{μ_{λ}}^{f} are countable. Therefore there exist an uncountable subset Γ of ω_{1} and finite subsets N_{1} and N_{2} of N, such that for each α ∈ Γ, X_{ξ_{α}}^{f} \ X_{μ_{λ}}^{f} = N_{1} and Z_{ξ_{α}}^{f} \Z_{μ_{λ}}^{f} = N_{2}. Thus for each α ∈ Γ , f(a_{ξ_{α}}) ⊆^{∗} f(a_{μ_{λ}})∪⋃ N_{1} and f(a′_{ξ_{α}}) ⊆^{∗} f(a′_{μ_{λ}}) ∪ ⋃ N_{2} . Of course, for all α ∈ Γ, ⋃ N_{1} ⊆^{∗} f(a_{ξ_{α}}) and ⋃ N_{2} ⊆^{∗} f(a′_{ξ_{α}}). Let e_{1} = (f(a_{μ_{λ}}) ∪ ⋃ N_{1}) \ ⋃ N_{2} and e_{2} = (f(a′_{μ_{λ}}) ∪ ⋃ N_{2}) \ ⋃ N_{1}. Obviously e_{1} ∩ e_{2} is finite and, for all α ∈ Γ, c_{α} ⊆^{∗} e_{1} and d_{α} ⊆^{∗} e_{2}.

By shrinking Γ, if necessary, we can assume that there exist n and disjoint subsets L_{1}, L_{2} of n such that, for each α ∈ Γ, c_{α} \ n ⊆ e_{1}, d_{α} \ n ⊆ e_{2}, c_{α} ∩ n = L_{1} and d_{α} ∩ n = L_{2}.

It is easy to see that for any α, β in Γ, c_{α} ∩ d_{β} = c_{β} ∩ d_{α} = ∅. This contradicts the assumption that {(c_{α}, d_{α}), (c_{β}, d_{β})} ∈ K_{0} and finishes the proof of Claim 3.

By OCA, T can be written as ⋃_{n∈ω} T (n) , where each T (n) is a 1-homogeneous set. Since, for any α ∈ ω_{2}, f(a_{α}) ∩ f(a′_{α}) is empty, (f(a_{α}), f(a′_{α})) ∈ T. There exists an n_{0} such that there are cofinally many α in ω_{2}, such that (f(a_{α}), f(a′_{α})) ∈ T (n_{0}). Let P = ⋃{f(a_{α}) : (f(a_{α}), f(a′_{α})) ∈ T (n_{0})}. Then it is easy to see that, for all α ∈ ω_{2}, f(a_{α}) ⊆_{N} P and f(a′_{α}) ∩ P =_{N} ∅. So, for all α ∈ ω_{2}, there are finite subsets N_{α}^{1}, N_{α}^{2} such that f(a_{α})\⋃ N_{α}^{1} ⊆^{∗} P and (f(a′_{α})\⋃ N_{α}^{2})∩P is finite. Now define F so that F (a_{α}) = P ∩ f(a_{α}) \ ⋃ N_{α}^{1} and F (a′_{α}) = f(a′_{α}) \ (⋃ N_{α}^{2} ∩ P) and for any other A ∈ dom f, F (A) = f(A). Obviously F and P satisfy the conclusions in (3) of the lemma. Also, since, for all α ∈ ω_{2} , X_{α}^{F} ⊆ X_{α}^{f} =^{∗} X_{α}^{F}, it follows that (1) and (2) of the lemma still hold for F and {Y_{n}, Y′_{n} : n ∈ ω}. □

Remark: For any F′ such that for all A, F′(A) ⊆ F (A) and F (A) \ F′(A) ∈ I_{N}, F′ still satisfies Lemma 4.2.

We finish this section with a proof of Theorem 4.1.

Proof of Theorem 4.1:

Let F, P, {Y_{n}, Y′_{n} : n ∈ ω} satisfies the Lemma 4.2 and for each α < ω_{2}, let X_{α} = {N ∈ N : N ⊆^{∗} F (a_{α})} We proceed by cases. Case 1 : There are infinitely many n such that Y_{n} \ ⋃_{k<n} Y_{k} is infinite.

Let {n_{i} : i ∈ ω} = {n : Y_{n} \ ⋃_{k<n} Y_{k} is infinite }. For each i, fix an infinite countable set N_{i} such that

Y′_{n_{i}} \ ⋃_{k<n_{i}} Y_{k} ⊆ N_{i} ⊆ Y_{n_{i}} \ ⋃_{k<n_{i}} Y_{k}

Let N_{i} = {N_{i,j} : j ∈ ω}. Obviously, (i) and (ii) hold. Case 2: There are only finitely many n such that Y_{n} \ ⋃_{k<n} Y_{k} is infinite.

In this case, there is an m ∈ ω, such that, for all n > m , Y_{n} ⊆^{∗} ⋃_{k≤m} Y_{k}. Therefore, for all countable subsets I of N, if I ∩ X_{α} is finite for all α ∈ ω_{2} , then I ⊆^{∗} ⋃_{k≤m} Y_{k}. Since each X_{α} ∩ ⋃_{k≤m} Y_{k} is finite, we can assume that, for all α ∈ ω_{2}, X_{α}∩⋃_{k≤m} Y_{k} is empty (i.e. by replacing F (a_{α}) with F (a_{α})\⋃(X_{α}∩⋃_{k≤m} Y_{k})).

Let X = ⋃_{α∈ω_{2}} X_{α} and Y = ⋃_{k≤m} Y_{k}.

Observe that for any infinite countable subset I of X, I is not mod finite contained in Y and so there is an α such that I ∩ X_{α} is infinite. We record this as claim 1. Claim 1: For all I ∈ [X]^{ω}, there exists α such that I ∩ X_{α} is infinite.

Now we are ready to find {J_{n} : n ∈ ω}. Let T = {(a, b) ∈ [ω]^{ω} × [ω]^{ω} : (∃α)a ⊆^{∗} F (a_{α}) and (∀α)b ∩ F (a_{α}) =^{∗} ∅ and a ∩ b = ∅}. Let K_{0} = {{(a, b), (c, d)} ∈ [T]^{2} : a ∩ d ≠ ∅ or c ∩ b ≠ ∅}. K_{0} is an open coloring for the natural separable metric topology on T. With a similar argument as in the proof of Claim 3 in Lemma 4.2, we can easily get the following Claim 2: There is no uncountable 0-homogeneous set in T

By OCA we let T = ⋃_{n∈ω} T (n) where each T (n) is 1-homogeneous. For all n, let S(n) = ⋃{b : (∃a)(a, b) ∈ T (n)} . Let E be the set of all n such that there are cofinally many α in ω_{2} such that there exists b with (F (a_{α}), b) ∈ T (n). For any b such that b is almost disjoint with each F (a_{α}), there exist k and n such that there are cofinally many α in ω_{2} such that (F (a_{α}), b \ k) ∈ T (n). Hence n ∈ E and b ⊆^{∗} S(n). It is easy to see that for any n ∈ E, and any α ∈ ω_{2}, there exists β > α such that (F (a_{β}), b) ∈ T (n) for some b ⊆ ω, which implies F (a_{β}) ∩ S(n) is empty. Since F (a_{α}) ⊆_{N} F (a_{β}), F (a_{α}) ∩ S(n) is in I_{N}. Claim 3: For all n ∈ E, there is a ∈ I_{N} such that, for all α < ω_{2} , F (a_{α})∩S(n) ⊆^{∗} a.

Proof of Claim 3 Suppose the claim fails for some n ∈ E. Then we can find an infinite set {N_{i} : i ∈ ω} ⊆ N such that, for all i in ω, N_{i} ∩ S(n) is infinite and N_{i} is contained in some F (a_{α_{i}}). Clearly {N_{i} : i ∈ ω} ⊆ X. Therefore, by Claim 1, there is an α such that {N_{i} : i ∈ ω} ∩ X_{α} is infinite. Without loss of generality, we assume that N_{i} ∈ X_{α} for all i, i.e. for all i ∈ ω, N_{i} ⊆^{∗} F (a_{α}). Since each N_{i} intersects S(n) on an infinite set, F (a_{α}) ∩ S(n) ∉ I_{N}. This contradicts the fact that n ∈ E.

According to Claim 3, for each n ∈ E, there exists a finite subset N′_{n} of X, such that, for all α ∈ ω_{2}, F (a_{α})∩S(n) ⊆^{∗} ⋃ N′_{n}. Now for n ∈ E, let S^{0}(n) = S(n)\⋃ N′_{n}. Suppose b ⊆ ω is such that, for all α ∈ ω_{2}, b ∩ F (a_{α}) is finite. Let n ∈ E be such that b ⊆^{∗} S(n). Since N′_{n} (n ∈ E) are subsets of X, b ∩ ⋃ N′_{n} is finite and so b ⊆^{∗} S^{0}(n) .

We enumerate E as {n_{k} : k < ρ}, where ρ ≤ ω. For k < ρ, let J_{k} = S^{0}(n_{k}) \ ⋃_{i<k} S^{0}(n_{i}). Obviously for all α ∈ ω_{2} and k < ρ, F (a_{α}) ∩ S^{0}(n_{k}) is finite, and for any b such that, for all α ∈ ω_{2}, b ∩ F (a_{α}) is finite, there is k < ρ, such that b ⊆^{∗} ⋃_{i≤k} J_{i}.

The next Claim will finish the proof of the Theorem. Claim 4: There are infinitely many k ∈ ρ such that J_{k} is infinite.

Proof of Claim 4. Assume there is k_{0} ∈ ω such that, for all k ∈ ρ ,

S^{0}(n_{k}) ⊆^{∗} ⋃_{i≤k_{0}} S^{0}(n_{i}).

Let Q = ⋃_{i≤k_{0}} S^{0}(n_{i}). For any α ∈ ω_{2}, F (a_{α}) ∩ Q is finite. By (3) of Lemma 4.2, F (a′_{α})∩F (a_{β}) is empty for all α and β in ω_{2} . Therefore F (a′_{α}) ⊆^{∗} Q for all α ∈ ω_{2}. If Q is a partitioner of N, then there is a partitioner A of M̃ such that A ∈ 𝔅′ and F (A) =_{N} Q. Note that since F (A) =_{N} Q, it follows that for all α ∈ ω_{2}, a_{α} ∩ A ∈ IM̃ and a′_{α} \ A ∈ IM̃. Therefore, by Fact 3.4, we have that Q is not a partitioner of N. On the other hand, for any fix N ∈ N, if N ∈ X, then there is some α such that N ⊆^{∗} F (a_{α}), so N ∩ Q ⊆^{∗} F (a_{α}) ∩ Q is finite. If N ∉ X, then N is disjoint with F (a_{α}) for all α ∈ ω_{2}. In this case, there is k such that N ⊆^{∗} S(n_{k}). Since N ∉ X and N′_{n_{k}} ⊆ X, N ∉ N′_{n_{k}}. Therefore N ⊆^{∗} S^{0}(n_{k}) ⊆^{∗} Q. Thus Q is a partitioner of N; this contradiction completes the proof. □

Henceforth we assume F, Ñ and J̃ satisfy the conclusion of Theorem 4.1.

Since Ñ and J̃ are countable, we will assume, for convenience, that they are members of 𝔐_{λ} for all λ.

5. Extending Coherent Sequences

5.1. An Extension Lemma. Recall that (See page 2) for a coherent sequence A = {(c_{α}, d_{α}) : α ∈ β} on ω, ℙ_{A} is the poset consisting of the partial functions p from ω to {0, 1} such that for any p ∈ ℙ_{A}, there is an α < β such that p =^{∗} Ch_{d_{α}}^{c_{α}}. For p and q in ℙ_{A}, p < q if p ⊇ q.

In this section, we prove a preparatory lemma which will allow us to construct a good coherence sequence in the next section.

Recall that our strategy is to find a good coherent sequence A_{ω_{1}} such that there is a partitioner X of M such that [X] is in 𝔅 and, for all α ∈ ω_{1}, X ∩ a_{μ(α)} =^{∗} b_{α} while there is no Y ⊆ ω such that, for all α ∈ ω_{1}, Y ∩ F (a_{μ(α)}) =_{N} F (b_{α}). This, of course, will contradict the fact that F induces an isomorphism from 𝔅 to the partitioner algebra P_{N}. The preparatory lemma will ensure that certain good coherent sequences can be extended while keeping certain commitments.

In order to ensure that, in V^{ℙ_{A}}, no partitioner of N will extend, mod I_{N}, the image of A_{ω_{1}} under F, we have to consider all possible partitioners of N. That is, we will need to prove that for all possible names τ of partitioners of N,

⊩_{ℙ_{Aω_{1}}} “ τ ∩ F (a_{μ(α+1)}) \ F (a_{μ(α)}) ≠_{N} F (b_{α+1}) \ F (b_{α})” .

In constructing the coherent sequences A_{ω_{1}}, we will handle one name, τ, at a time. Lemma 5.1 will guarantee that, under certain situations, we can extend A_{α} to A_{α+1} and take care of τ = τ_{α} from a suitable list.

Lemma 5.1. Suppose μ is a limit in ω_{2}. Let A = {(b_{ξ}, a_{μ(ξ)}) : ξ ≤ β} be a good coherent sequence such that μ = μ(β) and A ∈ 𝔐_{μ}. Let τ ∈ 𝔐_{μ} be a nice ℙ_{A}-name for a subset of ω. Then there exist p ∈ ℙ_{A} and a ℙ_{A}-generic, g, over 𝔐_{μ} and such that g ∈ 𝔐_{μ+1}, p ⊆ g and for any μ′ ≥ μ + 2, there are λ′ < λ < ω_{2} with μ′ ≤ λ′ and {R_{n} : n ∈ ω} ⊆ P(ω) such that the following hold.

1. A′ = A^{⌢}(g^{−1}(1) ∩ a_{λ}, a_{λ}) is a good coherent sequence,

2. ∀A ⊆ ω, if ∃^{∞}n A ∩ R_{n} ≠^{∗} ∅, then A ∉ I_{N},

3. p ∪ g ↾a_{λ} ⊩_{ℙ_{A′}} “ ∀^{∞}n R_{n} ∩ (τ△F (g^{−1}(1))) ∩ (F (a_{λ′}) \ F (a_{μ′})) ∉ I_{N} ”,

4. {R_{n} : n ∈ ω} ∪ {F (g^{−1}(1)), F (a_{λ′}), F (a_{μ′})} ⊆ 𝔐_{λ}.

Note if g ∈ 𝔐_{μ+1} is ℙ_{A}-generic over 𝔐_{μ}, then g^{−1}(1) is a partitioner of M. Therefore if a is a partitioner of M and a_{μ(β)} ⊆^{∗} a, then A^{⌢}(g^{−1}(1) ∩ a, a) is a good coherent sequence . We will denote A^{⌢}(g^{−1}(1) ∩ a, a) by A(g, a). Since (1) is guaranteed by the fact that g ∈ 𝔐_{μ+1} and that g is ℙ_{A}-generic over 𝔐_{μ}, we will not verify condition (1) in the proofs. Also, by condition (2), condition (3) is actually saying that

p ∪ g ↾a_{λ} ⊩_{ℙ_{A_{α+1}}} “ τ ∩ F (a_{μ(α+1)}) \ F (a_{μ(α)}) ≠_{N} F (b_{α+1}) \ F (b_{α})” .

It will be clear when applying this lemma in the next section that other conditions are simply there to help preserve the above forcing statement when p ∪ g ↾ a_{λ} is considered as an element in ℙ_{A_{ω_{1}}} and the forcing notion is ℙ_{A_{ω_{1}}} in the above forcing statement.

We break the proof into three Lemmas. We will skip the proof of Lemma 5.2 from Lemma 5.3 since it is an straightforward pressing-down argument. We will first prove Lemma 5.1 from Lemma 5.2. We give, in the next section, the longest proof, that of Lemma 5.3, which, using the structures from Theorem 4.1, basically tells us why F cannot induce an isomorphism from B to the partitioner algebra of N.

Lemma 5.2. Let A, μ, and τ be as in the Lemma 5.1. Then there exist g ∈ 𝔐_{μ+1} and p ∈ ℙ_{A} such that g is ℙ_{A}-generic over 𝔐_{μ}, p ⊆ g and there are cofinally many λ’s in ω_{2} satisfying the following property:

(†) ∀ρ, γ such that μ + 2 ≤ ρ < λ ≤ γ

p ∪ g ↾a_{γ} ⊩_{ℙ_{A′}} “ (τ△F (g^{−1}(1))) ∩ (F (a_{λ}) \ F (a_{ρ})) ∉ I_{N} ” ,

where A′ = A(g, a_{γ})

Lemma 5.3. Let A, μ, τ be as in Lemma 5.1. There exist a ℙ_{A}-generic g over 𝔐_{μ} and p ∈ ℙ_{A} such that g ∈ 𝔐_{μ+1} and p ⊆ g. Moreover, for all ρ ≥ μ, q ∈ ℙ_{A}, and m ∈ ω, there is λ_{0} = λ_{0}(ρ, q) ∈ ω_{2} such that for all λ > λ_{0}, γ ≥ λ, m ∈ ω, and N′ ∈ [N]^{<ω}, either p ∪ q ∪ g ↾a_{γ} is not a condition of ℙ_{A′} or

p ∪ q ∪ g ↾a_{γ} ⊮_{ℙ_{A′}} “ (τ△F (g^{−1}(1))) ∩ (F (a_{λ}) \ F (a_{ρ})) ⊆ m ∪ ⋃ N′” ,

where A′ = A(g, a_{γ})

Now we prove Lemma 5.1

Proof of Lemma 5.1 from Lemma 5.2: Let p, g be as the Lemma 5.2. Let μ′ = μ + 2. Take any increasing sequence {λ_{n} : n ∈ ω} such that λ_{0} > μ′ and each λ_{n} satisfies (†) in Lemma 5.2. Let λ′ = sup{λ_{n} : n ∈ ω} and fix λ > λ′ large enough so that 𝔐_{λ} contains {F (a_{λ_{i}}) : i ∈ ω} ∪ {F (a_{λ′}), F (a_{μ}), F (a_{μ′}), F (g^{−1}(1))}.

For each n ∈ ω, let R_{n} = F (a_{λ_{n+1}})\⋃_{k≤n} F (a_{λ_{k}}). Since each R_{n} is a non-trivial partitioner of N and {R_{n} : n ∈ ω} is pairwise disjoint, {R_{n} : n ∈ ω} satisfies (2). Obviously {R_{n} : n ∈ ω} is a member of 𝔐_{λ}. Therefore (4) is also true. Next we prove that (3) holds.

For each n in ω, it is easy to see that R_{n} ∩F (a_{λ′})\F (a_{μ′}) =_{N} F (a_{λ_{n+1}})\F (a_{λ_{n}}). Since λ_{n+1} satisfies (†), we can set λ = γ and ρ = λ_{n} and A′ = A(g, a_{λ}) and conclude that

p ∪ g ↾a_{λ} ⊩_{ℙ_{A′}} “ (τ△F (g^{−1}(1))) ∩ (F (a_{λ_{n+1}}) \ F (a_{λ_{n}})) ∉ I_{N} ” .

Therefore

p ∪ g ↾a_{λ} ⊩_{ℙ_{A′}} “ R_{n} ∩ (τ△F (g^{−1}(1))) ∩ (F (a_{λ′}) \ F (a_{μ′})) ∉ I_{N} ” .
□

The following is an easy corollary of the above proof.

Corollary 5.4. If {R_{n} : n ∈ ω} satisfies (1) and (2) of Lemma 5.1, then for any λ ≥ λ′′ ≥ λ′,

p ∪ g ↾a_{λ} ⊩_{ℙ_{A′}} “ ∀^{∞}n R_{n} ∩ (τ△F (g^{−1}(1))) ∩ (F (a_{λ′′}) \ F (a_{μ′})) ∉ I_{N} ” ,

where A = A(g, a_{λ}). □

5.2. Proof of the main lemma. We need some preliminaries to prove Lemma 5.3.

Lemma 5.5. Suppose A = {(c_{α}, d_{α}) : α ≤ β} is a coherent sequence with length β + 1. Then for any g : ω → {0, 1}, g is a ℙ_{A}-generic if and only if g ↾d_{β} =^{∗} Ch_{d_{β}}^{c_{β}} and g ↾ω \ d_{β} is a Cohen real on ω \ d_{β}.

Proof. Trivial. □

Suppose A = {(c_{α}, d_{α}) : α ≤ β} is a coherent sequence and g : ω → ω is such that

g ↾d_{β} =^{∗} Ch_{d_{β}}^{c_{β}} .

Let D be a family of predense subset of ℙ_{A}. We say g is D-generic if for each E ∈ D, there is e ∈ E such that e ⊆ g. Hence if g is ℙ_{A}-generic over some model 𝔐 which contains A as an element, then g is a D_{𝔐}-generic, where D_{𝔐} is the set of all predense subset of ℙ_{A} in 𝔐.

Let E be a subset of ℙ_{A}. We use E(n) to denote the set {r ∈ ℙ_{A} : n ⊆ dom r and ∃ q ∈ E, r ≤ q}. Obviously, we have

Fact 5.6. E is a predense subset of ℙ_{A} if and only if E(n) is a dense and open subset of ℙ_{A}. □

For any bijection π : {0, 1}^{n} → {0, 1}^{n}, we use E(π) to denote the following set

\{r ∈ ℙ_{A}: n ⊆ dom r and π(r ↾n) \overline{∪} r ∈ E(n)\} .

So E(π) is obtained from E(n) by changing the roots of the elements of E(n) according to π. The following fact is obvious.

Fact 5.7. E(n) is dense in ℙ_{A} if and only if E(π) is dense in ℙ_{A} . □

We call a family D of predense subsets of ℙ_{A} closed if for any E ∈ D, n ∈ ω, E(n) ∈ D and for any permutation π on {0, 1}^{n}, E(π) ∈ D. For any family of predense subsets D of ℙ_{A}, we use cl D to denote the smallest closed set containing D. It is easy to see that cl D is definable from D. In particular, if D is in some elementary submodel 𝔐 of some H_{λ}, then cl D is also in 𝔐. Obviously, for all α such that A ∈ 𝔐_{α}, if D is the set of all predense subset of ℙ_{A} in 𝔐_{α}, then D must be closed. Following lemma is the main reason to introduce the closeness here.

Lemma 5.8. Suppose A = {(c_{α}, d_{α}) : α < β} is a coherent sequence and D is a family of predense subsets of ℙ_{A}. If g is a D-generic and D is closed, then

1. any finite modification of g is still a D-generic,

2. for any E ∈ D, E is still predense in ℙ_{A(g,d)} for any d ⊆ ω such that for all α < β, d_{α} ⊆^{∗} d.

Proof. Obvious. □ The following corollary collects these results together.

Corollary 5.9. Let 𝔐 be an elementary submodel of H_{θ} and suppose A = {(c_{α}, d_{α}) : α < β} ∈ 𝔐 is a coherent sequence. If g is ℙ_{A}-generic over 𝔐, then any finite modification of g is also ℙ_{A}-generic over 𝔐 and for any predense subset E of ℙ_{A} in 𝔐, E remains predense in ℙ_{A′} whenever A′ = A(g, d) and d almost contains each d_{α}. □

We will also need the following easy lemma. For any subset A of a poset P, we will use A^{⊥} to denote the set of all elements of P that are incompatible with each element in A.

Lemma 5.10. Suppose A = {(c_{α}, d_{α}) : α < β} is a coherent sequence. Let τ be a nice ℙ_{A}-name for a subset of some set S. Let g be D generic for some closed family D of predense subsets of ℙ_{A}. For s ∈ S, let A_{s} = {p ∈ ℙ_{A} : (š, p) ∈ τ} and B_{s} = A_{s}^{⊥}. If, for all s ∈ S, A_{s} ∪ B_{s} is in D, then for any p ∈ ℙ_{A}, and any d ⊆ ω such that d almost contains d_{α} for all α < β,

p ⊩_{ℙ_{A}} “ s ∈ τ” iff p ⊩_{ℙ_{A(g,d)}} “ s ∈ τ”

and

p ⊩_{ℙ_{A}} “ s ∉ τ” iff p ⊩_{ℙ_{A(g,d)}} “ s ∉ τ”

Proof. Trivial. □

Now we are ready to prove Lemma 5.3.

Proof of Lemma 5.3: Let μ, A and τ be as given in the Lemma 5.1. Recall that we have fixed Ñ, J̃ and F satisfying the conclusions of Theorem 4.1 and that X_{α} = {N ∈ N : N ⊆^{∗} F (a_{α})} (α ∈ ω_{2}) .

We remind the reader that we must show that there exist a ℙ_{A}-generic g over 𝔐_{μ} and p ∈ ℙ_{A} such that g ∈ 𝔐_{μ+1} and p ⊆ g. Moreover, for all ρ ≥ μ and q ∈ ℙ_{A}, there is λ_{0} = λ_{0}(ρ, q) ∈ ω_{2} such that for all λ > λ_{0}, γ ≥ λ, m ∈ ω and N′ ∈ [N]^{<ω}, either p ∪ q ∪ g ↾a_{γ} is not a condition of ℙ_{A′} or

p ∪ q ∪ g ↾a_{γ} ⊮_{ℙ_{A′}} “ (τ△F (g^{−1}(1))) ∩ (F (a_{λ}) \ F (a_{ρ})) ⊆ m ∪ ⋃ N′” .

where A′ = A(g, a_{γ}).

Our plan is to find infinitely many N_{i} ∈ N (i ∈ ω) and λ_{0} such that each N_{i} meet τ△F (g^{−1}(1)) on an infinite set and all but finitely N_{i} are contained in F (a_{λ}) \ F (a_{ρ}) for all λ ≥ λ_{0}. The existence of such λ_{0} will be guaranteed by the choice of N_{i} together with (i) or (v) of Theorem 4.1. The choice of g will depend on whether τ is “small” or “big”. If τ is small, we choose g such that g^{−1}(0) is a subset of a_{μ+1}, then F (g^{−1}(1)) will be big enough to contain mod finite all N_{i} which is disjoint with τ. If τ is “big”, we choose g such that g^{−1}(1) is a subset of a_{μ+1}, then F (g^{−1}(1)) will be small enough to avoid each N_{i} which meet τ on an infinite set. We will proceed by different cases in Theorem 4.1 combined with the “small-big” possibilities. Case 1: {N_{i,j} : i, j ∈ ω} satisfies (i) and (ii) of the Theorem 4.1. Subcase 1. There is a p ∈ ℙ_{A} such that

p ⊩_{ℙ_{A}} “ ∃^{∞}i∃^{∞}j τ ∩ N_{i,j} =^{∗} ∅” .

This is the case when τ is small. Let g_{0} = g_{0}^{μ} ∈ 𝔐_{μ+1} be the Cohen real over 𝔐_{μ} defined in the construction of section 3 and then g_{0}^{−1}(0) ⊆^{∗} a_{μ+1}. Let g = p\overline{∪}Ch_{a_{μ(β)}}^{b_{β}} \overline{∪}g_{0}. Then according to Lemma 5.5, g is ℙ_{A}-generic over 𝔐_{μ}. Clearly g^{−1}(0) ⊆^{∗} a_{μ+1}. Consider any ρ ≥ μ and q ∈ ℙ_{A}. If p and q are incompatible, there is nothing to prove. Therefore we assume that p and q are compatible. Let g̃ = q \overline{∪} g. Then g̃ is also ℙ_{A} generic over 𝔐_{μ} and g̃ ∈ 𝔐_{μ+1}. By elementarity, (p ⊩_{ℙ_{A}} “ ∃^{∞}i∃^{∞}j τ ∩N_{i,j} =^{∗} ∅”)^{𝔐_{μ}} . Since p ⊂ g̃ ∈ 𝔐_{μ+1} and (p ⊩_{ℙ_{A}} “ ∃^{∞}i∃^{∞}j τ ∩ N_{i,j} =^{∗} ∅”)^{𝔐_{μ}}, there are {i_{k} : k ∈ ω} ⊆ ω and infinite subsets d_{k} of ω, for each k ∈ ω, such that, for all j ∈ d_{k}, τg̃∩N_{i_{k},j} is finite, where τg̃ = val(τ, g̃). Since g^{−1}(1) is in 𝔐_{μ+1}, by Fact 3.3, g^{−1}(1) is a partitioner of M. Consequently F (g^{−1}(1)) and F (g^{−1}(0)) are partitioners of N. Recall that for any subset A of ω we use N (A) to denote the set {N ∈ N : N ⊆^{∗} A}.

Since X_{ρ} ∩ {N_{i,j}: j ∈ ω} and X_{μ+2} ∩ {N_{i,j}: j ∈ ω} are finite, there is an h such that, for all i ∈ ω,

N (F (a_{ρ}) ∪ F (a_{μ+2}) ∪ F (g^{−1}(0)) ∪ ω \ F (g^{−1}(1))) ∩ {N_{i,j} : j ∈ ω}

⊆ {N_{i,j} : j < h(i)}

For all k ∈ ω, let j_{k} ∈ d_{k}\h(i_{k}), then there exists λ_{0} such that {N_{i_{k},j_{k}} : k ∈ ω}∩X_{λ_{0}} is infinite. For all λ > λ_{0}, X_{λ_{0}} ⊆^{∗} X_{λ}, hence {N_{i_{k},j_{k}} : k ∈ ω} ∩ X_{λ} is infinite.

For all λ, γ ∈ ω_{2}, m ∈ ω and N′ ∈ [N]^{<ω} such that γ ≥ λ > λ_{0}, there is k ∈ ω such that N = N_{i_{k},j_{k}} ∈ X_{λ} \ N′. Then N ⊆^{∗} F (a_{λ}) \ F (a_{ρ}) and N ∩ τg̃ is finite. Since N ⊆^{∗} F (g^{−1}(1)), there exists l in the set

N ∩ (F (g^{−1}(1)) \ τg̃) ∩ (F (a_{λ}) \ F (a_{ρ})) \ (m ∪ ⋃ N′) .

Since l ∉ τg̃, there exists r ⊆ g̃ = q \overline{∪} g such that r ⊩_{ℙ_{A}} “ l ∉ τ”. Let A′ = ℙ_{A(g,a_{γ})}. Since g is ℙ_{A}-generic over 𝔐_{μ} and τ, ℙ_{A} are in 𝔐_{μ}, by Lemma 5.10,

r ⊩_{ℙ_{A′}} “ l ∉ τ” .

Therefore,

r ⊩_{ℙ_{A′}} “ l ∈ (τ△F (g^{−1}(1))) ∩ (F (a_{λ}) \ F (a_{ρ})) \ (m ∪ ⋃ N′)” .

Since, r ⊆ q \overline{∪} g and q and p are compatible, either p ∪ q ∪ g ↾a_{γ} is not a condition or r is compatible with it. Thus either p ∪ q ∪ g ↾a_{γ} is not a condition or

p ∪ q ∪ g ↾a_{γ} ⊮_{ℙ_{A′}} “ l ∉ (τ△F (g^{−1}(1))) ∩ (F (a_{λ}) \ F (a_{ρ})) \ (m ∪ ⋃ N′)” .

Hence,

p ∪ q ∪ g ↾a_{γ} ⊮_{ℙ_{A′}} “ (τ△F (g^{−1}(1))) ∩ (F (a_{λ}) \ F (a_{ρ})) ⊆ (m ∪ ⋃ N′)” .

Subcase 2 : ∃p ∈ ℙ_{A} ∃k (i > k → p ⊩ “ ∀^{∞}j τ ∩ N_{i,j} infinite ”) .

Let g ∈ 𝔐_{μ+1} be a ℙ_{A}-generic over 𝔐_{μ} such that p ⊆ g and g^{−1}(1) ⊆^{∗} a_{μ+1}. As in subcase 1, g is made using g_{1}^{μ}. Let ρ ≥ μ, q ∈ ℙ_{A}. Again we assume that p and q are compatible. Let g̃ be q \overline{∪}g. Then g̃ is also ℙ_{A}-generic over 𝔐_{μ} containing p. As in the previous subcase, we can find a function f_{0} ∈ 𝔐_{μ+1} ∩^{ω}ω such that

∀i > k ∀j ≥ f_{0}(i) τg̃ ∩ N_{i,j} is infinite.

Take an h, greater than f_{0}, such that for all i ∈ ω,

{N_{i,j} : j ∈ ω} ∩ N (A) ⊆ {N_{i,j} : j < h^{∗}(i)},

where A = F (g^{−1}(1)) ∪ (ω \ F (g^{−1}(0))) ∪ F (a_{ρ}) ∪ F (a_{μ+1}) ∪ F (a_{μ+2}). Again since Ñ satisfies (ii) of Theorem 4.1, there exists λ_{0} such that X_{λ_{0}} ∩ {N_{i,h^{∗}(i)} : i ∈ ω} is infinite and, for all λ > λ_{0}, X_{λ} ∩ {N_{i,h^{∗}(i)} : i ∈ ω} is also infinite. The rest of the proof is similar to the subcase 1. Case 2 {J_{n} : n ∈ ω} satisfies (iii) to (v) of Theorem 4.1.

Observe that in this case, for any N ∈ N, either there are only finitely many i such that N ∩ J_{i} is not empty or N is almost disjoint with all J_{n} (n ∈ ω). To see this, let N ∈ N. If there exists λ such that N ⊆^{∗} F (a_{λ}), then by property (iii), N is almost disjoint with each J_{n} (n ∈ ω). Otherwise, since each F (a_{λ}) is a partitioner, N ∩ F (a_{λ}) is finite for all λ < ω_{2}. By property (iv), there are at most finitely many i ∈ ω such that N ∩ J_{i} is not empty.

We will only give the proof for the case when there exists p ∈ ℙ_{A}, p ⊩ “ ∀^{∞}nJ_{n} ⊆^{∗} τ” because the proof for the other case is almost the same.

As in Case 1, we can take g ∈ 𝔐_{μ+1}, a ℙ_{A}-generic real over 𝔐_{μ} such that g^{−1}(1) ⊆^{∗} a_{μ+1}. Again g is made using g_{1}^{μ}. As in the previous case, F (g^{−1}(1)) is a partioner of N. Let ρ ≥ μ, q ∈ ℙ_{A}. Let h ∈ 𝔐_{μ+2} such that ^{ω}ω ∩ 𝔐_{μ+1} <^{∗} h . Let A = F (g^{−1}(1)) ∪ (ω \ F (g^{−1}(0))), then A ⊆_{N} F (a_{μ+1}), i.e. A \ F (a_{μ+1}) ∈ I_{N}. Hence there are at most finitely many i such that A \ F (a_{μ+2}) ∩ J_{i} is not finite. Take f_{0} ∈ ^{ω}ω and k ∈ ω, such that, for all n > k,

J_{n} ∩ (F (g^{−1}(1)) ∪ (ω \ F (g^{−1}(0))) ∪ F (a_{ρ})) ⊆ f_{0}(n)^{.}

Let h^{∗} ∈ ^{ω}ω such that h^{∗} > {f_{0}, h}.

Define {A_{i}, β_{i}, N_{i} : i ∈ ω} as follows.

Let A_{0} = {min(J_{n} \ h^{∗}(n)) : n ∈ ω}. Since N is a mad family, there exists N_{0} such that N_{0} ∩ A_{0} is infinite. Obviously, there are infinitely many n such that N_{0} ∩ A_{0} ∩ J_{n} is not empty. According to (iv) of Theorem 4.1, there is a β_{0} such that N_{0} ∩ A_{0} ∩ F (a_{β_{0}}) is infinite. Hence, N_{0} ⊆^{∗} F (a_{β_{0}}), i.e. N_{0} ∈ X_{β_{0}}. At stage i, suppose we have defined {A_{j}, β_{j}, N_{j}: j < i} such that N_{j} ⊆^{∗} F (a_{β_{j}}) and A_{j} \ F (g^{−1}(1)) ∪ F (a_{ρ}) is infinite. Now let A_{i} = {min(J_{n} \ (h^{∗}(n) ∪ ⋃_{j<i} F (a_{β_{j}}))): n ∈ ω}. Let N_{i} ∈ N such that N_{i} ∩ A_{i} is infinite. Similarly, we can find a β_{i} such that N_{i} ⊆^{∗} F (a_{β_{i}}). Obviously for each i ∈ ω and l ∈ ω, N_{i} ∩ (⋃_{n>l} J_{n} \ h^{∗}(n)) is infinite and N_{i} \ (F (g^{−1}(1)) ∪ F (a_{ρ})) is infinite, thus N_{i} ∩ (F (g^{−1}(1)) ∪ F (a_{ρ})) is finite.

By (v) of the Theorem 4.1, there is a λ_{0} such that {N_{i} : i ∈ ω} ∩ X_{λ_{0}} is infinite. Consequently, for all λ > λ_{0}, {N_{i} : i ∈ ω} ∩ X_{λ} is infinite. For any λ > λ_{0}, γ ≥ λ and N′ ∈ [N]^{<ω} and m ∈ ω, take i such that N = N_{i} ∈ X_{λ} \ N′. Let g̃ = q \overline{∪} g. If q is not compatible with p, we are done. So assume q is compatible with p. In this case, p ⊆ g̃. Since p ⊩ “ ∀^{∞}n J_{n} ⊆^{∗} τ” there exist k and f ∈ 𝔐_{μ+1} such that, for all n > k, J_{n} \ τg̃ ⊆ f(n). Therefore, since h^{∗} > f, there is a k, such that

⋃_{n>k} J_{n} \ h^{∗}(n) ⊆ τg̃ .

It follows that N ∩ τg̃ is infinite. By the choice of N_{i}, N = N_{i} ⊆^{∗} (F (a_{λ}) \ F (a_{ρ})) \ (F (g^{−1}(1)) ∪ ⋃ N′). Hence a = (τg̃ \ F (g^{−1}(1))) ∩ (F (a_{λ}) \ F (a_{ρ})) \ (m ∪ ⋃ N′) is infinite. Take l ∈ a and choose r ⊆ g̃ such that r ⊩_{ℙ_{A}} “ l ∈ τ”. Hence, as did in the first case, we have

r ⊩_{ℙ_{A′}} “ l ∈ (τ△F (g^{−1}(1))) ∩ (F (a_{λ}) \ F (a_{ρ})) \ m ∪ ⋃ N′” ,

where A′ = A(g, a_{γ}). If q ∪ p ∪ g ↾ a_{γ} is a function, then r is compatible with it, thus

q ∪ p ∪ g ↾a_{γ} ⊮_{ℙ_{A′}} “ (τ△F (g^{−1}(1))) ∩ (F (a_{λ}) \ F (a_{ρ})) ⊆ m ∪ ⋃ N′” .
□

6. Collapsing ℵ_{2} and Good Coherent Sequence

In this section, we will find a proper forcing notion 𝔻 which forces that there is a coherent sequence with the properties stated in the Theorem 3.9. At the end of this section we will summarize the results and complete the proof of Theorem 3.9.

6.1. Collapsing ℵ_{2}. Let ℙ = Fn(ω_{1}, ω_{2}, ℵ_{1}) with the reverse inclusion as its partial order. This poset ℙ is often called the countable condition collapse of ω_{2} to ω_{1}.

We will prove several results to get ready to prove Theorem 3.9. The following theorem is the culmination of this Chapter.

Theorem 6.1. If G is a ℙ-generic filter, then in V [G], there is a good coherent sequence A = {(b_{α}, a_{μ(α)}) : α < ω_{1}} such that

(a) ℙ_{A} is ccc,

(b) ℙ_{A} forces that for all Y ⊆ ω,

{α : Y ∩ F (a_{μ(α+1)}) \ F (a_{μ(α)}) ≠_{N} F (b_{α+1}) \ F (b_{α})} is uncountable

(c) {(b_{ξ}, a_{μ(ξ)}) : ξ < α} ∈ 𝔐_{μ(α)}.

In this subsection, we prove some lemmas and then prove Theorem 6.1.

Since 𝔠 = ω_{2}, we can enumerate ^{ω}2 as {d_{α} : α < ω_{2}} such that each function d ∈ ^{ω}2 appears in the enumeration only once. Let ℚ be the set of countable partial functions f from {(α, β) : β < α < ω_{1}} to {0, 1} such that dom f = ⋃_{ξ∈A}{ξ}×ξ for some countable subset A of ω_{1}. ℚ is a poset with the reverse inclusion as its order. It is well-known that 𝔠 = ω_{2} implies that ℚ is forcing equivalent to ℙ. However, for the purpose of the following lemma, we define an isomorphism e from ℙ to ℚ as follows.

For each α < ω_{1}, let π_{α} : α → ω be a bijection. Define for f ∈ ℙ, e(f) so that dom e(f) is ⋃_{α∈dom(f)}{α} × α and for α ∈ dom f, β < α,

e(f)(α, β) = d_{f(α)}(π_{α}(β)) .

It is easy to see that e is an isomorphism. Obviously e is order preserving. To see it is a bijection, we define a function c : ℚ → ℙ such that e ∘ c = id_{ℚ} and c ∘ e = id_{ℙ}. For any g ∈ ℚ, let A be the countable subset of ω_{1} such that dom g = ⋃_{α∈A}{α}×α. For all α ∈ A, let h^{α} : ω → 2 be such that

h^{α}(n) = g(α, π_{α}^{−1}(n))

and define c(g) ∈ ℙ such that dom c(g) = A and c(g)(α) is the unique ordinal η such that h^{α} = d_{η}. Obviously, for all α ∈ A, and β < α, e(c(g))(α, β) = d_{c(g)(α)}(π_{α}(β)) = h^{α}(π_{α}(β)) = g(α, π_{α}^{−1}π_{α}(β)) = g(α, β). Therefore e(c(g)) = g. It is also easy to see that for any f ∈ ℙ, c(e(f)) = f.

For any filter G on ℙ, g = ⋃ G : ω_{1} → ω_{2}^{V} is a partial function. We can define e(g) = ⋃{e(f) : f ∈ G}. Obviously G is ℙ-generic iff e[G] is ℚ-generic.

When G is a ℙ-generic we also call g a ℙ-generic and e(g) a ℚ-generic.

We fix a ℙ-generic g : ω_{1} → ω_{2}^{V}. Let s_{α} = {β < α : e(g)(α, β) = 1}, i.e. s_{α} = g_{α}^{−1}(1), where g_{α} : α → {0, 1} is defined by g_{α}(β) = e(g)(α, β). Let δ_{α} = sup{g(ξ) : ξ < α}.

Let ġ be the ℙ-name for the ℙ-generic g. Let ṡ_{α} and δ̇_{α} be the names for s_{α} and δ_{α} respectively. The proof of the following Lemma is again standard; we will omit it.

Lemma 6.2. If G is ℙ-generic, then in V [G], {s_{α} : α ∈ ω_{1} and cf α = ω} is a ♢-sequence on {α ∈ ω_{1} : cf α = ω} such that for any X ⊆ ω_{1} and f : X → ω_{2}^{V} there are stationarily many α such that X ∩ α = s_{α} and f ↾s_{α} ∈ 𝔐_{δ_{α}+1}. □

The following Corollary is what will actually be used the the proof of Theorem 6.1.

Corollary 6.3. If G is ℙ-generic, then in V [G], there is a ♢-sequence {(S_{α}, T_{α}) ∈ P(ω_{1}) × P(ω_{1}) : α ∈ ω_{1} and cf α = ω} such that for any X, Y ⊂ ω_{1} and f_{0} : X → Z, f_{1} : Y → Z, where Z is in 𝔐_{0} with cardinality ω_{2} in V, then there are stationarily many α such that X ∩ α = S_{α}, Y ∩ α = T_{α}, f_{0} ↾ S_{α} ∈ 𝔐_{δ_{α}+1} and f_{1} ↾T_{α} ∈ 𝔐_{δ_{α}+1}. □

6.2. Forcing a Good Coherent Sequence. We list some notations and properties in the next lemma to make the proof of Theorem 6.1 shorter.

Lemma 6.4. Suppose A_{α} ≺ A_{β} (α < β) are coherent sequences, R = {R_{n} : n ∈ ω} ⊆ P(ω) and τ is a nice ℙ_{A_{α}}-name. Then the following are true:

1. Each of the following are predense subsets of ℙ_{A_{α}},

(i) D_{n}(A_{α}, R, τ) = D_{n}^{0}(A_{α}, R, τ) ∪ (D_{n}^{0}(A_{α}, R, τ))^{⊥}, where D_{n}^{0}(A_{α}, R, τ) = {q ∈ ℙ_{A_{α}} : ∃i > n, q ⊩_{ℙ_{Aα}} “ τ ∩ R_{i} ≠^{∗} ∅”}.

(ii) E_{i,n}(A_{α}, R, τ) for each i, n, where,

E_{i,n}(A_{α}, R, τ) = {q ∈ ℙ_{A_{α}} : q ⊩_{ℙ_{Aα}} “ τ ∩ R_{i} ⊆ n”

or ∃k ∈ R_{i} \ n, q ⊩_{ℙ_{Aα}} “ k ∈ τ”} .

(iii) H_{n}^{τ} = A_{n}^{τ} ∪ (A_{n}^{τ})^{⊥}, where A_{n}^{τ} = {p ∈ ℙ_{A_{α}}: (n, p) ∈ τ}.

2. If p ∈ ℙ_{A_{α}} and

p ⊩_{ℙ_{Aα}} “ ∃^{∞}i R_{i} ∩ τ ≠^{∗} ∅” ,

then, for all n in ω, D_{n}^{0}(A_{α}, R, τ) is dense below p.

3. If D_{n}(A_{α}, R, τ), E_{i,n}(A_{α}, R, τ) and H_{n}^{τ} are predense in ℙ_{A_{β}}, and

p ⊩_{ℙ_{Aα}} “ ∃^{∞}i R_{i} ∩ τ ≠^{∗} ∅” ,

then

p ⊩_{ℙ_{A_{β}}} “ ∃^{∞}i R_{i} ∩ τ ≠^{∗} ∅” .

4. If H_{n}^{τ} are predense in ℙ_{A_{β}}, then for any subsets A, B of ω,

1 ⊩_{ℙ_{A_{β}}} “ τ′ = (τ ∩ A)△B” ,

where τ′ is the ℙ_{A_{α}}-name defined as follows,

τ′ = {⟨n, p⟩ ∈ τ, n ∈ A \ B} ∪ {⟨n, 1⟩ : n ∈ B \ A}

∪ {{n} × (A_{n}^{τ})^{⊥} : n ∈ A ∩ B}.

Proof. (1) and (2) are trivial. (4) is also straightforward. We prove (3) by contradiction. Take p′ ∈ ℙ_{A_{β}} and n ∈ ω such that

∀i > n p′ ⊩_{ℙ_{A_{β}}} “ R_{i} ∩ τ =^{∗} ∅” .

Since D_{n}(A_{α}, R, τ) is predense in ℙ_{A_{β}}, there exists q ∈ D_{n}(A_{α}, R, τ) such that p′ and q are compatible. Since q is compatible with p and p ⊩_{ℙ_{Aα}} “ ∃^{∞}iR_{i} ∩τ ≠^{∗} ∅”, q must be in D_{n}^{0}(A_{α}, R, τ). Therefore, there is i > n such that q ⊩_{ℙ_{Aα}} “ τ ∩R_{i} ≠^{∗} ∅”. By the choice of p′, we can find r ∈ ℙ_{A_{β}} and m ∈ ω such that r ≤ q ∪ p′ and r ⊩_{ℙ_{A_{β}}} “ τ ∩ R_{i} ⊆ m”. Since E_{i,m} is predense in ℙ_{A_{β}}, there is some s ∈ E_{i,m} which is compatible with r. Since s is compatible with q and q ⊩_{ℙ_{Aα}} “ τ ∩ R_{i} ≠^{∗} ∅”, s ⊮_{ℙ_{Aα}} “ τ ∩ R_{i} ⊆ m”. Therefore there is a k ∈ R_{i} \ m such that s ⊩_{ℙ_{Aα}} “ k ∈ τ”. It follows that A_{k}^{τ} is predense below s in ℙ_{A_{α}}. We claim that A_{k}^{τ} is also predense below s in ℙ_{A_{β}}. Indeed, since H_{k}^{τ} is predense in ℙ_{A_{β}}, it is sufficient to see that any element in (A_{k}^{τ})^{⊥} is incompatible with s. But this is obvious from the definition of (A_{k}^{τ})^{⊥}. Therefore, we have s ⊩_{ℙ_{A_{β}}} “ k ∈ τ”. However, from the choice of r , s ∪ r ⊩_{ℙ_{A_{β}}} “ τ ∩ R_{i} ⊆ m”. Contradiction. □

In the sequel, we will use A_{α} to denote {(b_{ξ}, a_{μ(ξ)}) : ξ < α} ∈ 𝔐_{μ(α)}. Let D denote the set of all the countable partial functions from ω to {0, 1}. Recall that |D| is ω_{2}. Since ℙ is ω_{1}-closed, the definition of D is absolute between V and V^{ℙ}. Since ℙ collapses ω_{2}^{V} to ω_{1}, we can enumerate, in V^{ℙ}, D as {f_{α} : α < ω_{1}}.

Proof of Theorem 6.1: Let G be a ℙ-generic over V and let g be ⋃ G. Recall we use δ_{α} to denote sup{g(β) : β < α}. Fix a ♢-sequence satisfying Corollary 6.3. Since ℙ collapses ω_{2}^{V} to ω_{1}, we can enumerate all the τ of the form

⋃_{n∈ω}{n} × A_{n}

with A_{n} ∈ [D]^{≤ω} as {τ_{α} : α < ω_{1}} such that for each τ, {α ∈ lim ω_{1} : τ = τ_{α}} is uncountable.

Note that since ℙ is ω_{1}-closed, there is no new countable coherent sequence in V^{ℙ} and for any countable coherent sequence A′, there is no new countable nice ℙ_{A′}-name.

In V [G], we will define, by induction on α, A_{α}, D_{α}, R^{α} = {R_{n}^{α} : n ∈ ω}, ρ′_{α}, ρ′′_{α}, p_{α}, τ′_{α} and g_{α} such that A_{α} = {(b_{β}, a_{μ(β)}) : β < α} is a good coherent sequence. For each infinite countable ordinal α, D_{α} is a family of predense subsets of ℙ_{A_{α}} which is closed. Moreover, for each infinite ordinal α and β the following are true.

1. if α < β, then A_{α} is an initial segment of A_{β}.

2. A_{α+1} ∈ 𝔐_{μ(α)}, and b_{α} = g_{α}^{−1}(1) ∩ a_{μ(α)}; if α = β + 1, then g_{α} ∈ 𝔐_{μ(β)+1}.

3. if α ∈ lim ω_{1} and τ_{α} ∈ 𝔐_{μ(α)} is a ℙ_{A_{α+1}}-name, then μ(α) + 2 < ρ′′_{α} < ρ′_{α} < μ(α + 1) and the following are true, (i) F (a_{μ(α)}), F (a_{ρ′_{α}}), F (a_{ρ′′_{α}}), F (g_{α+1}^{−1}(1)∩a_{ρ′_{α}}), F (g_{α+1}^{−1}(1)∩a_{ρ′′_{α}}), F (b_{α}), p_{α}, τ′_{α} and {R_{n}^{α} : n ∈ ω} are elements of 𝔐_{μ(α+1)} and τ′_{α} is a ℙ_{A_{α+2}} name for a subset of ω defined as in Lemma 6.4 with A = F (a_{ρ′_{α}}) \ F (a_{ρ′′_{α}}) and B = F (g_{α+1}^{−1}(1) ∩ a_{ρ′_{α}}) \ F (g_{α+1}^{−1}(1) ∩ a_{ρ′′_{α}}). Thus 1 forces that

τ′_{α} = (τ_{α} ∩ (F (a_{ρ′_{α}}) \ F (a_{ρ′′_{α}})))△(F (g_{α+1}^{−1}(1) ∩ a_{ρ′_{α}}) \ F (g_{α+1}^{−1}(1) ∩ a_{ρ′′_{α}})) .

Moreover, p_{α} ∈ ℙ_{A_{α+1}} is such that p_{α} ∪ Ch_{a_{μ(α+1)}}^{b_{α+1}} ∈ ℙ_{A_{α+2}} and

p_{α} ∪ Ch_{a_{μ(α+1)}}^{b_{α+1}} ⊩_{ℙ_{A_{α+2}}} “ ∃^{∞}i, R_{i}^{α} ∩ τ′_{α} ≠^{∗} ∅” .

(ii) For all A ⊆ ω, if ∃^{∞}i, A ∩ R_{i}^{α} ≠^{∗} ∅, then A ∉ I_{N}.

4. (i) ω < α ≤ β → D_{α} ⊆ D_{β}; if β is a limit, then D_{β} = ⋃_{α<β} D_{α}.

(ii) D_{α+1} ⊆ 𝔐_{μ(α)}.

(iii) If α,τ′_{α}, τ_{α}, ρ′_{α}, ρ′′_{α} and R^{α} and g_{α+1} are as in (3), then for all n and i in ω, D_{n}(A_{α+2}, R^{α+1}, τ′_{α}), E_{i,n}(A_{α+2}, R^{α+1}, τ′_{α}) and H_{n}^{τ′_{α}} are in D_{α+2}. (iv) If α is a limit and S_{α}, T_{α} ∈ 𝔐_{μ+1}, where μ = sup{μ(ξ) : ξ < α} and A_{α}, {f_{ξ} : ξ ∈ T_{α}} ∈ 𝔐_{μ+1} and {f_{ξ} : ξ ∈ T_{α}} is predense in ℙ_{A_{α}}, then

{f_{ξ} : ξ ∈ T_{α}} ∈ D_{α+1} .

(v) For α > ω, g_{α} is a D_{α}-generic.

5. μ(α) > δ_{α}.

The basic idea involved in (3) and (4) is to keep various predense subsets of ℙ_{A_{α}} still predense in ℙ_{A_{β}} when α < β. It is not difficult to see that (4) will guarantee that the final poset ℙ_{A} is ccc. To achieve (b), we handle each Y at stage α + 2 when τ_{α} is a name for Y. In (3) we pose a condition which ensures that ℙ_{A_{α+2}} will force that Y ∩ F (a_{μ(α+1)}) \ F (a_{μ(α)}) is not equal to F (b_{α+1}) \ F (b_{α})} mod I_{N}. We then preserve this condition by preserving certain predense subsets of ℙ_{A_{α+2}}.

First of all, we show that if A = {(b_{α}, a_{μ(α)}) : α < ω_{1}} satisfies (1) to (5), then A satisfies (a) to (c). To check that ℙ_{A} is ccc, suppose K is a maximal antichain in ℙ_{A}, and K = {f_{α} : α ∈ Γ}. Let Σ = {α : f_{α} ∈ ℙ_{A}}. Recall D = {f_{α}: α < ω_{1}} is an enumeration of all countable partial functions from ω to {0, 1} (see the discussion after Lemma 6.4). Let f : ω_{1} → D be defined by f(α) = f_{α}. Then by Corollary 6.3, E = {α : Σ∩α = S_{α}, Γ∩α = T_{α}, f ↾S_{α} ∈ 𝔐_{δ_{α}+1} and f ↾T_{α} ∈ 𝔐_{δ_{α}+1}} is stationary in ω_{1}. Let C_{1} be the set of α for which ℙ_{A_{α}} = {f_{ξ} : ξ ∈ Σ ∩ α} and for which K ∩ ℙ_{A_{α}} is a maximal antichain in ℙ_{A}. C_{2} = {α ∈ ω_{1} : δ_{α} = sup{μ(β) : β < α}}. Obviously C_{1} and C_{2} are cubs on ω_{1}. Take α ∈ E ∩ C_{1} ∩ C_{2}, then Σ ∩ α = S_{α} and Γ ∩ α = T_{α}. Moreover ℙ_{A_{α}} = {f_{ξ} : ξ ∈ Σ ∩ α} = ran f ↾ S_{α} ∈ 𝔐_{δ_{α}+1}. Similarly, K ∩ ℙ_{A_{α}} = ran f ↾T_{α} ∈ 𝔐_{δ_{α}+1}. By (i) and (iv) of (4), K ∩ ℙ_{A_{α}} is predense in ℙ_{A_{β}} for all β > α. Therefore K ∩ ℙ_{A_{α}} is predense in ℙ_{A}. This means K = K ∩ ℙ_{A_{α}} and so is countable. Condition (c) is immediate by (2).

Now we show, by contradiction, that (b) is true. Since ℙ_{A} is homogeneous, we have 1 forces that

∃x ⊆ ω{α : x ∩ F (a_{μ(α+1)}) \ F (a_{μ(α)}) ≠_{N} F (b_{α+1}) \ F (b_{α})} is countable .

Since ℙ_{A} is ccc, there exist a nice ℙ_{A}-name τ for a subset of ω and η ∈ ω_{1} such that, for all α > η,

1 ⊩ “ τ ∩ F (a_{μ(α+1)}) \ F (a_{μ(α)}) =_{N} F (b_{α+1}) \ F (b_{α})” .

Since ℙ is ccc, τ is a countable name. Therefore τ is in V. Therefore there are uncountably many limit α such that τ = τ_{α}. By (5), {μ(α) : α < ω_{1}} is cofinal in ω_{2}^{V}. Therefore, there is a limit α such that τ = τ_{α} ∈ 𝔐_{μ(α)} and α > η.

We claim that p_{α} ∪ Ch_{a_{μ(α+1)}}^{b_{α+1}} ∈ ℙ_{A_{α+2}} and

p_{α} ∪ Ch_{a_{μ(α+1)}}^{b_{α+1}} ⊩_{ℙ_{A}} “ τ ∩ (F (a_{μ(α+1)}) \ F (a_{μ(α)}) ≠_{N} F (b_{α+1}) \ F (b_{α}))”

This will contradict that α > η.

We have to prove that

p_{α} ∪ Ch_{a_{μ(α+1)}}^{b_{α+1}} ⊩_{ℙ_{A}} “ (τ ∩ (F (a_{μ(α+1)}) \ F (a_{μ(α)})))△(F (b_{α+1}) \ F (b_{α})) ∉ I_{N} ” .

By (4) of Lemma 6.4 and (i) and (iii) of (4) in the above construction, we have

1 ⊩_{ℙ_{A}} “ τ′_{α} = (τ_{α} ∩ (F (a_{ρ′_{α}}) \ F (a_{ρ′′_{α}})))△(F (g_{α+1}^{−1}(1) ∩ a_{ρ′_{α}}) \ F (g_{α+1}^{−1}(1) ∩ a_{ρ′′_{α}}))” .

Since A is a good coherent sequence, it is easy to see that 1 forces that τ′_{α} is contained, mod I_{N}, in (τ_{α} ∩(F (a_{μ(α+1)})\F (a_{μ(α)})))△(F (b_{α+1})\F (b_{α})) . Therefore it is sufficient to prove

p_{α} ∪ Ch_{a_{μ(α+1)}}^{b_{α+1}} ⊩_{ℙ_{A}} “ τ′_{α} ∉ I_{N} ” .

It is obvious that property (ii) of (3) is absolute. Therefore, it suffices to prove that

p ∪ Ch_{a_{μ(α+1)}}^{b_{α+1}} ⊩_{ℙ_{A}} “ ∃^{∞}i τ′_{α} ∩ R_{i} ≠^{∗} ∅” .

Since

p ∪ Ch_{a_{μ(α+1)}}^{b_{α+1}} ⊩_{ℙ_{A_{α+2}}} “ ∃^{∞}i τ′_{α} ∩ R_{i} ≠^{∗} ∅” ,

by Lemma 6.4, it is sufficient to see that D_{n}(A_{α+2}, R^{α}, τ′_{α}), E_{i,n}(A_{α+2}, R^{α}, τ′_{α}) and H_{n}^{τ′} are predense in ℙ_{A} = ℙ_{A_{ω_{1}}} . However by (i) and (iii) of (4), for all β > α + 2,

D_{n}(A_{α+2}, R^{α}, τ′_{α}), E_{i,n}(A_{α+2}, R^{α}, τ′_{α}) and H_{n}^{τ′_{α}} are predense in ℙ_{A_{β}}, therefore they are predense in ℙ_{A_{ω_{1}}} .

Next we construct the sequences satisfying (1) to (5). We will proceed by induction. Notice that for R^{α}, ρ′_{α}, ρ′′_{α} and τ′_{α} we only have to define them when α is of the form β + 1 for β a limit ordinal. For simplicity, we will not mention them in any other case in the following construction. For g_{α}, we will define it at stage α + 1. We will construct p_{α} at stage α + 2.

We let A_{0} be ∅. At stage α we proceed according to the following cases. Case 1 α is a limit.

We let A_{α} = ⋃_{ξ<α} A_{ξ} = {(b_{ξ}, a_{μ(ξ)}) : ξ < α}. Let μ be the least ordinal bigger than sup{μ(ξ) : ξ < α} such that A_{α} ∈ 𝔐_{μ}. Let D_{α} = ⋃_{β<α} D_{β}. Obviously, D_{α} is closed. For all ξ < α and D ∈ D_{ξ}, we prove that D is predense in ℙ_{A_{α}}. Take any p ∈ ℙ_{A_{α}}, we can find η < α such that ξ < η and p ∈ ℙ_{A_{η}}. But by the induction hypothesis, D is predense in ℙ_{A_{η}}, therefore there is q ∈ D such that p and q are compatible. Case 2 α = β + 1 and A_{β} ∈ 𝔐_{μ+1}, where β is a limit and μ = sup{μ(ξ): ξ ∈ β}.

By the construction of {𝔐_{α} : α ∈ ω_{2}} in Subsection 3.1, there is a g in 𝔐_{μ+2} which is Fn(ω, 2)-generic over 𝔐_{μ+1}. Since ℙ_{A_{β}} and Fn(ω, 2) are both countable and atomless, they have isomorphic dense subsets. Therefore, there is a ℙ_{A_{β}}-generic g_{β} over 𝔐_{μ+1} such that g_{β} ∈ 𝔐_{μ+2}. Let μ(β) = μ + 2 and b_{β} = g_{β}^{−1}(1) ∩ a_{μ(β)}. By Fact 3.3, g_{β}^{−1}(1) is a partitioner of M. Thus b_{β} is a partitioner of M. Since g_{β} is generic over 𝔐_{μ+1}, by Lemma 5.8, every set D in 𝔐_{μ+1} which is predense in ℙ_{A_{β}} is still predense in ℙ_{A_{β+1}}. However, by the induction hypothesis, D_{β} ⊆ 𝔐_{μ+1}. Therefore, every member D of D_{β} is still predense in ℙ_{A_{β+1}}. Now let D_{β+1} be all the predense subsets of ℙ_{A_{β+1}} which are in 𝔐_{μ(β)} = 𝔐_{μ+2}. Obviously, D_{β+1} is closed and D_{β} ⊆ D_{β+1}. It is also clear that {f_{ξ} : ξ ∈ T_{β}} ∈ D_{β+1} provided {f_{ξ} : ξ ∈ T_{β}} is predense in ℙ_{A_{β}} and it is in 𝔐_{μ+1}. Case 3 α = β + 1, β is a limit and A_{β} ∉ 𝔐_{μ+1}, where μ = sup{μ(ξ) : ξ < β} .

Note ξ < β implies that A_{ξ} is in 𝔐_{μ}. Let g in 𝔐_{μ+1} be a Cohen generic real over 𝔐_{μ}. Let λ be big enough such that {A_{β}, D_{β}} ⊆ 𝔐_{λ} and λ > δ_{β}. Choose a function f ∈ ^{ω}ω ∩ 𝔐_{λ+1} which dominates 𝔐_{λ}. Also, in 𝔐_{λ}, choose {β_{n} : n ∈ ω} cofinal in β. We define h ∈ ^{ω}ω such that h(n + 1) = f(h(n) + 1). Since f ∈ 𝔐_{λ+1}, h ∈ 𝔐_{λ+1}. Observe that for each increasing f′, if f′ <^{∗} f then, for almost all n, h(n) < f′(h(n)) < f′(h(n) + 1) < h(n + 1). Let x_{n} = a_{μ(β_{n})} \ ⋃_{j<n} a_{μ(β_{j})} and x = ⋃_{n∈ω} x_{n} \ h(n). Let g_{β} ∈ ^{ω}ω be such that g_{β} agrees with g on ω \ x and, for each n, agrees with g_{β_{n}} on x_{n} \ h(n). Clearly g_{β} is in 𝔐_{λ+1}. Let μ(β) be λ + 1 and b_{β} be a_{λ+1} ∩ g_{β}^{−1}(1). Let D_{β+1} = D_{β}. We check that A_{β+1} is a good coherent sequence and g_{β} is a D_{β}-generic. It will then follow that E ∈ D_{β}, E will still be predense in P_{A_{β+1}}. Let us use g_{n} to denote the restriction of the function g_{β_{n}} on x_{n}.

First we prove that A_{β+1} is a good coherent sequence. It suffices to prove that g_{β}^{−1}(1) is a partitioner of M. Let N be in M. We need to show that g_{β}^{−1}(1) does not split N. Let ξ be such that N first appears in M_{ξ+1}. Since g_{β} ∈ M_{λ+1}, by Fact 3.3, g_{β}^{−1}(1) does not split any member of M which is not in 𝔐_{λ+1}. Hence we can assume that ξ < λ. We may assume that N is not contained, mod finite, in a_{β} for any β < μ since then N is contained in a_{μ(β_{n})} for some n and g_{β} almost agrees with g_{n} on a_{μ(β_{n})}. Therefore, N is almost disjoint with each x_{n}, in which case ξ > μ. Consequently, ξ ≥ μ + 2. Now f and h are dominating over 𝔐_{λ}. Thus N is almost disjoint from x. On the other hand, since g^{−1}(1) is in 𝔐_{μ+1} and N is not in 𝔐_{μ+1}, g^{−1}(1) does not split N. Finally, noting that the restrictions of g and g_{β} on ω \ x are the same, we deduce that g_{β}^{−1}(1) does not split N.

Next we prove that g_{β} is a D_{β}-generic.

Given E in D_{β}, we must prove that there exists an e in E, such that e ⊆ g_{β}. Let ξ ∈ β such that E ∈ D_{ξ}. Take an m ∈ ω such that β_{m} > ξ. Then, for all e ∈ E, dom(e) is almost contained in a_{μ(β_{m})}. Define, in 𝔐_{λ}, a strictly increasing function F_{E} from ω to ω such that

1. F_{E}(0) = m,

2. If F_{E}(n) = l, then for all s ∈ F n(l, 2) and k such that m ≤ k ≤ l, there is an e ∈ E such that

e ⊆ s \overline{∪} ⋃_{j≤k} g_{j} \overline{∪} g and dom(e) \ ⋃_{j≤m} x_{j} ⊆ F_{E}(n + 1) .

We first show that if we have F_{E} in 𝔐_{λ} with above property, then we can find an e such that e ⊆ g_{β}. Indeed, since F_{E} <^{∗} f, for almost all n, there is a k in ω such that h(n) ≤ F_{E}(k) < F_{E}(k + 1) < h(n + 1). Choose such n > m and k. Let s = g ↾⋃_{j≤n} h(j) ∩ x_{j}, then s ∈ Fn(F_{E}(k), 2) and

g_{β} ↾ ⋃_{j≤n} x_{j} \overline{∪} g = s \overline{∪} ⋃_{j≤n} g_{j} \overline{∪} g .

Therefore there is an e ∈ E such that

e ⊆ g_{β} ↾ ⋃_{j≤n} x_{j} \overline{∪} g

and

dom(e) \ ⋃_{j≤m} x_{j} ⊆ F_{E}(k + 1) < h(n + 1) .

Since dom(e) \ ⋃_{j≤m} x_{j} ⊆ h(n + 1) and g_{β} ↾ (h(n + 1) \ ⋃_{j≤n} x_{j}) = g ↾ (h(n + 1) \ ⋃_{j≤n} x_{j}), it is clear that e ⊆ g_{β}.

The construction of F_{E} is straightforward. Suppose we have defined F_{E}(n) = l. For each s ∈ Fn(l, 2) and k such that m ≤ k ≤ l, since s \overline{∪} ⋃_{j≤k} g_{j} \overline{∪} g is a P_{A_{β_{k}+1}} generic over 𝔐_{μ(β_{j})}, we can find an e(s, k) ∈ E such that e(s, k) ⊆ s \overline{∪} ⋃_{j≤k} g_{j} \overline{∪} g. Recall that dom(e(s, k)) ⊆^{∗} a_{μ_{βm}} =^{∗} ⋃_{j≤m} x_{j}. Let F_{E}(n + 1) be bigger than l and large enough such that for all s ∈ Fn(l, 2) and k with m ≤ k ≤ l, F_{E}(n+1) contains dom(e(s, k)) \ ⋃_{j≤m} x_{j}. Case 4 α = β + 2 and β is a limit ordinal

We have to define μ(β + 1), g_{β+1}, p_{β}, ρ′_{β}, ρ′′_{β} and R^{β} satisfying the required properties. Let τ be a nice ℙ_{A_{β+1}}-name for a subset of ω such that if τ_{β} is a nice ℙ_{A_{β+1}}-name for a subset of ω and τ_{β} ∈ 𝔐_{μ(β)}, then τ is τ_{β}.

Let μ be μ(β). By Lemma 5.1, there are p ∈ ℙ_{A_{β+1}} and g in 𝔐_{μ+1} such that g is a ℙ_{A_{β+1}}-generic over 𝔐_{μ} and there are μ′, λ′ and λ such that μ + 2 < μ′ < λ′ < λ and δ_{β+1} < λ and A_{β+1} ∈ 𝔐_{λ}. There are also {R_{n} : n ∈ ω} ⊆ P(ω) such that

(i) For all A ⊆ ω, if ∃^{∞}n A ∩ R_{n} ≠^{∗} ∅, then A ∉ I_{N}; (ii) p ∪ g ↾a_{λ} ⊩_{ℙ_{A′}} “ ∀^{∞}n R_{n} ∩ ((τ△F (g^{−1}(1))) ∩ (F (a_{λ′}) \ F (a_{μ′}))) ∉ I_{N} ”, where

A′ = A_{β+1}(g, a_{λ}). (iii) {R_{n} : n ∈ ω} ∪ {F (g^{−1}(1)), F (a_{μ′}), F (a_{λ′})} ⊆ 𝔐_{λ}.

As argued in the previous cases, g^{−1}(1) is a partitioner of M. Now let μ(β +1) = λ, g_{β+1} = g, b_{β+1} = g^{−1}(1) ∩ a_{λ}, ρ′_{β} = λ′ and ρ′′_{β} = μ′. Let p_{α} = p and for each n, R_{n}^{β} = R_{n}. If τ_{β} ∈ 𝔐_{μ(β)} is a nice ℙ_{A_{β}}-name, then τ′_{α} is defined as in Lemma 6.4. It is easy to see that

1 ⊩_{ℙ_{A′}} “ τ′_{β} =_{N} (τ△F (g^{−1}(1))) ∩ (F (a_{λ′}) \ F (a_{μ′}))” .

Otherwise, let τ′_{β} = ∅. Since τ′_{β} is definable from the sets in 𝔐_{μ(β+1)}, τ′_{β} is also in 𝔐_{μ(β+1)}. For the same reason, all the following dense subsets of ℙ_{A_{β+2}} are in

𝔐_{μ(β+1)}: D_{n}(A_{β+2}, R^{β}, τ′_{β}), E_{i,n}(A_{β+2}, R^{β}, τ′_{β}) and H_{n}^{τ′_{β}}.

Since D_{β+1} ⊆ 𝔐_{μ(β)}, g_{β+1} is a D_{β+1}-generic. Thus each member of D_{β+1} is still predense in ℙ_{A_{β+2}}. Let D_{β+2} be the set of all of predense subsets of ℙ_{A_{β+2}} in 𝔐_{μ(β+1)}. Case 5 α = β + n + 1, where β is a limit ordinal and n ≥ 2.

Since A_{β+n} ∈ 𝔐_{μ(β+n−1)}, there is a g in 𝔐_{μ(β+n−1)+1} which is ℙ_{A_{β+n}}-generic over 𝔐_{μ(β+n−1)}. Take any limit λ such that λ is bigger than both δ_{β+n} and μ(β + n − 1) and 𝔐_{λ} contains A_{β+n} and D_{β+n}. Let μ(β + n) be λ, g_{β+n} be g, and let D_{β+n+1} be the set of all predense subsets of ℙ_{A_{β+n+2}} which are in 𝔐_{λ}. It is easily checked that all the conditions from (1) to (5) are satisfied. □

By Theorem 6.1, if we take the poset ℙ ∗ ℙ_{A} and the good coherent sequence A, we are almost done provided A is in V. We try to use PFA, via a dense-subset argument, to show that such A can exist in V. However, the property (b) can not be decided by ℵ_{1} many dense subsets of ℙ ∗ ℙ_{A}. There are actually ℵ_{2} many reals in the forcing extension. We fix this in the rest of the section.

In the next section, we work on an improved good coherent sequence which will be given by the next theorem. We will basically force, by a ccc forcing notion ℚ, countably many Hausdorff-Luzin type gaps simultaneously which will ensure (b) is true. However, any Hausdorff-Luzin type gap can be decided by ℵ_{1} many dense subsets. We start with the next theorem.

Theorem 6.5. Let G be a ℙ generic filter over V. Suppose A = {(b_{α}, a_{μ(α)}) : α ∈ ω_{1}} satisfies the conclusion of Theorem 6.1. Then in V [G], there is a subsequence {δ(α) : α < ω_{1}} of ω_{1} such that

(i) ∀α, Ã_{α} = {(b_{δ(ξ)}, a_{μ(δ(ξ))}) : ξ < α} is in 𝔐_{μ(δ(α))}, (ii) ℙÃ_{ω_{1}} forces that for all Y ⊆ ω there exists β in ω_{1} such that either ∀ξ > η >

β,

Y ∩ [F (a_{μ(δ(ξ))}) \ (F (a_{μ(δ(η))}) ∪ F (b_{μ(δ(ξ))}))] ∉ I_{N} ;

or ∀ξ > η > β,

(F (b_{μ(δ(ξ))}) \ F (b_{μ(δ(η))})) \ Y ∉ I_{N} .

We denote the least such β by β_{Y}. (iii) ℙÃ_{ω_{1}} is ccc.

Proof. Let H be a ℙ_{A}-generic filter over V [G]. Since CH holds in V [G] and ℙ_{A} is a ccc poset with size ω_{1}, CH still holds in V [G][H]. In V [G][H], let {Y_{α} : α ∈ ω_{1}} enumerate P(ω). For α ∈ ω_{1}, let

Γ_{α} ={ξ ∈ ω_{1} : Y_{α} ∩ (F (a_{μ(ξ+1)}) \ F (a_{μ(ξ)})) ≠_{N} F (b_{ξ+1}) \ F (b_{ξ})},
Γ_{α}^{0} ={ξ ∈ Γ_{α} : Y_{α} ∩ [F (a_{μ(ξ+1)}) \ (F (a_{μ(ξ)}) ∪ F (b_{ξ+1}))] ∉ I_{N} },
Γ_{α}^{1} ={ξ ∈ Γ_{α} : (F (b_{ξ+1}) \ F (b_{ξ})) \ Y_{α} ∉ I_{N} }.

Clearly Γ_{α} = Γ_{α}^{0} ∪ Γ_{α}^{1}. Therefore, for each α, there is i(α) ∈ {0, 1} such that Γ_{α}^{i(α)} is uncountable. Let C_{α} be the set of all the limit points of Γ_{α}^{i(α)}. Then C_{α} is a cub in ω_{1} and there is an element of Γ_{α}^{i(α)} between each two elements of C_{α}. It is easy to see that if i(α) = 0, then for any η < ξ ∈ C_{α},

Y_{α} ∩ [F (a_{μ(ξ)}) \ (F (a_{μ(η)}) ∪ F (b_{ξ}))] ∉ I_{N} .

Similarly, if i(α) = 1, then for all η < ξ ∈ C_{α},

(F (b_{ξ}) \ F (b_{η})) \ Y_{α} ∉ I_{N} .

Set C = △_{δ<ω_{1}}C_{δ} = {ξ : ∀δ < ξ, ξ ∈ C_{δ}}. For all α < ω_{1}, if η < ξ ∈ C and α < η

then

i(α) = 0 → Y_{α} ∩ F (a_{μ(ξ)}) \ (F (a_{μ(η)}) ∪ F (b_{ξ})) ∉ I_{N} ,

and

i(α) = 1 → (F (b_{ξ}) \ F (b_{η})) \ Y_{α} ∉ I_{N} .

Since C is a cub in V [G][H] and ℙ_{A} is ccc, C contains a cub C^{0} in V [G]. Furthermore, since ℙ is a countably closed forcing, any countable subset of A is actually in V. By the construction of A, {μ(α) : α < ω_{1}} is cofinal in ω_{2}^{V}. Therefore we can find a subsequence C^{1} = {δ(α) : α < ω_{1}} of C^{0} such that for all α ∈ ω_{1},

{(b_{δ(ξ)}, a_{μ(δ(ξ))}) : ξ ∈ α} ∈ 𝔐_{μ(δ(α))} .

Obviously ℙ_{A} = ℙÃ and ℙÃ is ccc and condition (ii) holds. □

From this point on we will only use the sequence Ã in Theorem 6.5, but for convenience we simply assume A = Ã.

Therefore we have the following

Theorem 6.6. In V^{ℙ}, there is a good coherent sequence A = {(b_{α}, a_{μ(α)}) : α ∈ ω_{1}} such that

(i) ∀α, A_{α} is an r-sequence. (ii) ℙ_{A} is ccc, (iii) ℙ_{A} forces that, for all Y ⊆ ω, there exists β such that either

∀η, ξ > β, (η < ξ → Y ∩ F (a_{μ(ξ)}) \ (F (a_{μ(η)}) ∪ F (b_{ξ})) ∉ I_{N}),

or

∀η, ξ > β, (η < ξ → (F (b_{ξ}) \ F (b_{η})) \ Y ∉ I_{N}).
□

For the good coherent sequence A, ℙ_{A} adds an X ⊆ ω such that X ∩a_{μ(α)} =^{∗} b_{α} for all α ∈ ω_{1}. We will show that there is a ccc forcing ℚ in V^{ℙ∗ℙ_{A}} such that ℚ adds some Hausdorff-Luzin type gap which will imply that there is no Y ⊆ ω such that F (X) = Y. We intend to apply PFA as follows. Since ℙ ∗ ℙ_{A} ∗ ℚ is proper and we only need ℵ_{1} many dense subsets to determine the above A, X and these Hausdorff-Luzin gaps as well as the related properties, we can get them in V.

In the next subsection we will modify the usual techniques in order to add what might be called countably many simultaneous Hausdorff-Luzin type gaps.

6.3. Hausdorff-Luzin type gaps.

Definition 6.7. Suppose I = {c_{α} : α ∈ ω_{1}} and J = {e_{α} : α ∈ ω_{1}} are two sequences of infinite subsets of ω. We say that I is near to J if for any subset Y of ω, either {α : c_{α} ⊆^{∗} Y } is countable or {α : Y ∩ e_{α} is finite } is countable. We say I and J are tight or {(c_{α}, e_{α}) : α ∈ ω_{1}} is tight if, for any pair Σ and Γ of uncountable subsets of ω_{1} and n ∈ ω, there exist uncountable subsets Σ′ and Γ′ of Σ and Γ respectively such that

| ⋂_{α∈Σ′} c_{α} ∩ ⋂_{α∈Γ′} e_{α}| ≥ n .

Fact 6.8. If {c_{α} : α ∈ ω_{1}} is near to {e_{α} : α ∈ ω_{1}}, then {c_{α} : α ∈ ω_{1}} and {e_{α} : α ∈ ω_{1}} are tight .

Proof. Suppose Σ and Γ are uncountable subsets of ω_{1} and n ∈ ω. We will define uncountable sets {Σ_{i}, Γ_{i} : i ≤ n} and integers {m_{i} : i < n} such that

1. Σ_{n} ⊆ ⋯ ⊆ Σ_{0} ⊆ Σ, Γ_{n} ⊆ ⋯ ⊆ Γ_{0} ⊆ Γ and m_{0} < ⋯ < m_{n−1}.

2. m_{i} ∈ ⋂_{α∈Σ_{i+1}} c_{α} ∩ ⋂_{α∈Γ_{i+1}} e_{α} Obviously, if we have the above sequence and we let, Σ′ = Σ_{n} and Γ′ = Γ_{n} then

{m_{0}, …, m_{n−1}} ⊆ ⋂_{α∈Σ′} c_{α} ∩ ⋂_{α∈Γ′} e_{α} .

Let Σ_{0} = Σ and Γ_{0} = Γ. At stage i > 0, let Y be the set of n ∈ ω which is in c_{ξ} for uncountably many ξ in Σ_{i}. Then the set of ξ ∈ Σ_{i} for which a_{ξ} ⊈^{∗} Y is countable. Since {c_{α} : α ∈ ω_{1}} is near to {e_{α} : α ∈ ω_{1}}, for all but countably many ξ in Γ_{i}, Y ∩ e_{ξ} is infinite. Let Γ′_{i} = {ξ ∈ Γ_{i} : Y ∩ e_{ξ} ≠^{∗} ∅}. Γ′_{i} is uncountable. For each ξ ∈ Γ′_{i}, take n_{ξ} ∈ Y ∩ e_{ξ} \ max{m_{j} : j < i}. There is a n^{∗} such that n^{∗} = n_{ξ} for uncountably many ξ. Let m_{i} = n^{∗} and Σ_{i+1} = {ξ ∈ Σ_{i} : n ∈ c_{ξ}} and Γ_{i+1} = {ξ ∈ Γ′_{i} : n^{∗} = n_{ξ}}. □

Fact 6.9. If {(c_{α}, e_{α}) : α ∈ ω_{1}} is tight, then there is a ccc forcing ℚ with size ω_{1} such that 1 forces

∃I ∈ [ω_{1}]^{ω_{1}}, ∀k ∈ ω, {(c_{α+k} e_{α+k}) : α ∈ I} is a Hausdorff-Luzin type family.

Proof. Let ℝ = {(H, n) : H ∈ [ω_{1}]^{<ω}, n ∈ ω}. Define (H, n) < (K, m) if the following holds

1. K ⊆ H and m ≤ n;

2. if β < α, α ∈ K, and β ∈ H \ K then for all k < m, |c_{α+k} ∩ e_{β+k}| > m. It is easy to see that (ℝ, <) is a poset. Claim 1 (ℝ, <) is ccc.

Suppose ℝ has an uncountable antichain A. By the △-system Lemma, we can assume A has the form {(H_{α}, n) : α ∈ ω_{1}} for some n and that {H_{α} : α ∈ ω_{1}} forms a △-system with root H. We can also assume H_{α} = H ∪ {ξ_{1}^{α}, …, ξ_{m}^{α}} for all α and that, for α < β, H < {ξ_{1}^{α}, …, ξ_{m}^{α}} < {ξ_{1}^{β}, …, ξ_{m}^{β}}. By applying the fact that {(c_{α}, e_{α}) : α ∈ ω} is tight finitely many times, we can find uncountable subsets Γ_{1} and Γ_{2} of ω_{1} such that for each k < n and for all α ∈ Γ_{1} and β ∈ Γ_{2} and i, j ∈ {1, …, m},

|c_{ξ_{i}^{α}+k} ∩ e_{ξ_{j}^{β}+k}| > n and |e_{ξ_{i}^{α}+k} ∩ c_{ξ_{j}^{β}+k}| > n .

Now it is easy to see that for any α ∈ Γ_{1} and β ∈ Γ_{2}, (H_{α} ∪ H_{β}, n) is stronger than both (H_{α}, n) and (H_{β}, n).

Next we prove that forcing with a subset of ℝ will add many Hausdorff-Luzin gaps as described. Let ℝ_{α} = {(H, n) ∈ ℝ, H ∩ α = ∅}.

Since ℝ is ccc, the proof of the following claim is straightforward. Claim 2 There exits α such that for all β > α, D_{β}^{α} = {(H, n) ∈ ℝ_{α} : H ⊈ β} is dense in ℝ_{α}.

Fix an α^{∗} which satisfies Claim 2. Let ℚ be ℝ_{α^{∗}}. Obviously ℚ is ccc and ℚ adds a subset I as required. We are done. □

Working in V^{ℙ}, let {(c_{α+k}, e_{α+k}) : α ∈ lim ω_{1} and 1 < k ≤ ω} be the derived sequence of A and F (See the paragraph following Lemma 3.8 for the definition). We first prove the following lemma.

Lemma 6.10. In V^{ℙ∗ℙ̇_{A}}, {c_{α} : α is a successor } is near to {e_{α} : α is a successor }.

Proof. Let Y ⊆ ω. Suppose there are uncountably many successor ordinals α such that c_{α} ⊆ Y. Recall that β_{Y} is the minimal ordinal β such that either

∀η, ξ > β (η < ξ → Y ∩ F (a_{μ(ξ)}) \ (F (a_{μ(η)}) ∪ F (b_{ξ})) ∉ I_{N}),

or

∀η, ξ > β (η < ξ → (F (b_{ξ}) \ F (b_{η})) \ Y ∉ I_{N}).

We will show that for any successor α > β_{Y}, e_{α} ∩ Y is infinite. Take α > β_{Y} such that c_{α} ⊆ Y. Since c_{α} =_{N} F (b_{α}) \ F (b_{α−1}), it is not true that

∀η, ξ > β_{Y} (η < ξ → (F (b_{ξ}) \ F (b_{η})) \ Y ∉ I_{N}).

Therefore

∀η, ξ > β_{Y} , (η < ξ → Y ∩ F (a_{μ(ξ)}) \ (F (a_{μ(η)}) ∪ F (b_{ξ})) ∉ I_{N}),

and so for all successor α > β_{Y},

Y ∩ F (a_{μ(α)}) \ (F (a_{μ(α−1)}) ∪ F (b_{α})) ∉ I_{N} .

But

e_{α} =_{N} F (a_{μ(α)}) \ (F (a_{μ(α−1)}) ∪ F (b_{α})) ,

hence Y ∩ e_{α} is infinite since it is not in I_{N}. □ By Fact 6.9, we have

Theorem 6.11. In V^{ℙ∗ℙ̇_{A}}, there is a ccc forcing ℚ with size ω_{1} such that

1 ⊩_{ℚ} “∃I ∈ [ω_{1}]^{ω_{1}} such that ∀k ∈ ℕ,

{(c_{α+k}, e_{α+k}) : α ∈ I} is a Hausdorff-Luzin type family”

□

By Theorem 6.6 and Theorem 6.11, we now are ready to prove Theorem 3.9. Proof of Theorem 3.9: Let Q̇ be the ℙ ∗ ℙ_{A}-name for the poset in The-

orem 6.11 and let 𝔻 be the the poset ℙ ∗ ℙ_{A} ∗ Q̇. Obviously in V^{ℙ}, ℙ_{A} adds an X which extends A. Let {(c_{α+k}, e_{α+k}): α ∈ lim ω_{1} and k ∈ ℕ or k = ω} be the derived sequence obtained before Lemma 6.10 and Σ be the sequence I ∈ V^{𝔻} obtained by Theorem 6.11. We enumerate I by {σ_{α} : α ∈ ω_{1}}. It is easily checked that these sequences and sets satisfy the statement of the theorem. □

This completes the proof that PFA^{+} +♢(S_{0}^{2}) implies there is a subalgebra of a representable algebra which is not representable. We should remark that we have no information as to whether the hypotheses can be weakened. The most interesting questions are to ask if PFA^{+} can be weakened to PFA and, still more important, if the consistency can be established relative to ZFC without any large cardinal assumption.

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Department of Mathematics, York University, 4700 Keele Street, North York, Ontario, Canada M3J 1P3

   E-mail address: Alan Dow:  dowa@mathstat.yorku.ca
   E-mail address: Jinyuan Zhou: ysma9202@mathstat.yorku.ca