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Abstract

We prove that some problems in the theory of compactifications of the integers are not decided by the usual axioms of set theory.

We prove in Chapter 2 through Chapter 5 that it is consistent that there be a partitioner algebra on ω which has a subalgebra which is not representable by any partitioner algebra. The present construction makes use of a supercompact cardinal.

In Chapter 6, we prove that the Continuum Hypothesis implies that, for every ultrafilter p on ω, the ultrafilter space ω ∪ {p} can be embedded into some pseudoradial zero-dimensional Hausdorff space.

In Chapter 7, it is established that in any forcing extension by adding enough Cohen reals the space ω ∪ {p} can not be embedded into any regular pseudoradial space, for any P-point p.

In Chapter 8, we prove that in some forcing extension as in Chapter 7 there is a special ultrafilter p, such that ω ∪ {p} can be embedded into some pseudoradial and zero-dimensional Hausdorff space.

In Chapter 9, we construct, in ZFC, an ultrafilter p on ω with a special almost disjoint refinement and a strong Noetherian base.

In Chapter 10, we dicsuss some possible implications of our results and our new techniques and give a list of open problems.

Acknoledgments

I give my sincere thanks to my supervisor, Professor Alan Dow, for his advice, guidance and encouragement.

I would also like to thank Professor Hao-Xuan Zhou who helped me to study Set-Theoretic Topology when I was in Sichuan University, China from 1987 to 1988.

I would like to thank Professor Juris Steprāns for teaching me descriptive set theory and Professor Steve Watson for teaching me the interesting subjects in general topology.

I wish to thank the external examinal, Porfessor Peter Klass Hart, for his numerous suggestions in producing the much improved final version of this thesis.

I would like to thank members of the Toronto Set Theory Seminar for creating a stimulating and friendly research environment.

Finally, my sincere appreciation goes to my wife, Wei Wei, for her support and understanding during my study at York University.

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Contents

Abstract iv Acknowledgments v Chapter 1. Introduction and Preliminaries 1 1.1. Definitions and Notations 10 1.2. Preliminaries 13 Chapter 2. A Non Representable Boolean Algebra 21 2.1. Definitions for Representable Algebras 21 2.2. Constructions 22 2.3. Basic Facts and 𝔅 26 2.4. 𝔅 is not representable 28 Chapter 3. Separating Mad Families 40 3.1. Rearranging the mad family N 41 3.2. Proof of Theorem 3.0.1 46 Chapter 4. Extending Coherent Sequences 49

CONTENTS vii

4.1. An Extension Lemma 49 4.2. Proof of the main lemma 53 Chapter 5. Collapsing ℵ_{2} and a Good Coherent Sequence 62 5.1. Collapsing ℵ_{2} 62 5.2. Forcing a Good Coherent Sequence 70 5.3. Hausdorff-Luzin type gaps 83 Chapter 6. Pseudoradialness under CH 89 6.1. Definition 89 6.2. Under CH 90 Chapter 7. Pseudoradialness in the Cohen model 93 7.1. Main Theorem 93 7.2. Some facts 94 7.3. Proof 97 Chapter 8. Pseudoradialness for a Special Ultrafilter 99 8.1. Reflecting Self-Indexed Stationary Set 99 8.2. A Special Space ω ∪ {p} 101 Chapter 9. Another Ultrafilter On ω 109 Chapter 10. Conclusions 114 Bibliography 118 Index 121

CHAPTER 1 Introduction and Preliminaries

This thesis is an investigation into some problems arising from the general study of topological compactifications of the integers. This study occupies a central place in the field because it impacts so broadly on the class of compact spaces with countable convergence properties. Topology itself has two separate roots: geometry and analysis. One may regard topology’s role in analysis as the language of convergence. In this context the need to fully understand countable convergence in compact spaces is of paramount importance.

Let us briefly explain the notions of limit points and convergence of sequences which are what ever Calculus is based on. When a point is infinitesimally close to a set , the point is described as a limit point of the set. Topology has made precise the notion of infinitesimal closeness (but has had to abandon as insufficient the naive notion of distance in the conventional sense). Putting a topological structure on a set is actually to define which points are limit points of which sets. A simple converging sequence is modeled on the intuition that the elements in the sequence are continually getting closer to the limit. However with no adequate notion of “closer” available, let us recognize that this simply means that every infinite subset of the set still has the point as a limit. This simplicity is very helpful and should perhaps be contrasted with the extreme case we study in Chapter 6 in which one

2 1. INTRODUCTION AND PRELIMINARIES

has that if the set is partitioned into two, precisely one of them still has the point as a limit.

Simple countable converging sequences were the motivation to study topological space in order to offer calculus a firm base which does not depend on the geometric intuition when various pathological phenomena in convergence of trigonometric series were discovered early last century. However, the notion of countable convergence sequence is not general enough to encapsulate the conceptual ideas of convergence both in the countable and the uncountable settings. This was soon discovered. It is a constant challenge to understand to what degree the notion of converging sequence is sufficient. The question is even more relevant today as the new development in the study of the compact spaces with countable convergence properties tightness has influential applications in other branches of mathematics, particularly, Functional Analysis.

Several important classes of spaces arising in Functional Analysis do not have simple a convergence structure as one might expect. For example, when one starts with even the unit interval and considers the topology of pointwise convergence on the space of continuous real-valued functions, the space, while quite nice in many respects, has no globally adequate notion of distance, is not first countable (i.e. countable local basis at each point). It is not Fréchet (i.e. if x is a limit of A then there is a sequence from A converging to x) and it is not even sequential (i.e. if A is not closed, there should be a sequence from A converging to a point outside of A). This space does have the property called countable tightness (i.e. if x is a limit of A, then it is a limit of a countable subset of A).

However compactness dramatically improves the situation (Grothendieck): a countably compact subset is compact and Fréchet. This is true of all Eberlein compact spaces and many of its generalizations. Each of them is known to have countable tightness but not much more beyond countable tightness can be said

1. INTRODUCTION AND PRELIMINARIES 3

universally.

While we know in certain situations, compact spaces with countable tightness do have a large supply of converging sequence, we do not really understand this class of spaces quite well. Working from below we have metric spaces, Fréchet spaces and sequential spaces whose topology is determined by the countable converging sequences. Working from above, we try to extrapolate those properties which guarantee a large supply of the convergence sequence, such as sequentially compact, in order to analyze how large a class of such spaces is to gain more insight into the properties of function spaces which distinguish them from the wider class of spaces with countable tightness.

These are some of the reasons that the class of compact spaces with countable tightness or countable convergence properties are both fundamental and interesting in the study of topology. The even more basic concept is lurking in the background and this is what is called an almost disjoint family of countable sets.

A family of infinite subset of a countable set is an almost disjoint family if each two of its members have finite intersection. If an almost disjoint family is not a sub family of any other almost disjoint family, then it is called a maximal almost disjoint family. To give a simple but illustrative example, let us note that any family of converging sequences of rational numbers which are convergent to different points is actually an almost disjoint family. Thus we see the countable set of rational numbers supports rather large almost disjoint families. One can see the similarities in the following brief explanation about the important roles of the almost disjoint families in the study of convergence. The first step in computing the set of limit points for a set in an abstract space in our classes is best viewed just in this way: consider the maximal almost disjoint family of converging subsequences. The structure of this almost disjoint family determines which points are the limit points of the set.

4 1. INTRODUCTION AND PRELIMINARIES

We now recall some of the more explicit appearances of almost disjoint families in the topological constructions and their significant role in set theory. Mrówka constructed what is now called the Ψ-space associated with a maximal almost disjoint family to give the first example of a pseudocompact space which is not countably compact. Later on Terasawa and Mrówka [Ter81, Mro77] constructed various maximal almost disjoint families and, via the Ψ-spaces, solved a number of problems in dimension theory and the theory of compactifications of ω. P. Simon’s elegant result on the product of Fréchet spaces is another such example. P. Simon constructed an example of two compact Fréchet spaces whose product is not Fréchet. However, the vital part of the constructions need a subtle analysis of general maximal almost disjoint families. Both examples actually demonstrated intimate intrinsic connections between the properties of an almost disjoint family and those of a topological space. Another place where the almost disjoint families play an equally important role is the study of βω (the Stone-Céch compactification of the countable discrete set of integers) or equivalently the combinatorial structure on the Boolean algebra of infinite subsets of ω. Since all countable discrete spaces are viewed as the same spaces, we will use the set of natural numbers ω as our standard countable discrete space. We know that any fundamental result about βω will enjoy a substantial long-term degree of significance because of the diverse and numerous ways in which βω arises in mathematics.

The notion of almost disjoint family dates back to the work of Cantor at the turn of last century ([Lus47]) and was investigated in more details by great mathematicians like Erdös, Sierpiński and Tarski early this century. The theory of the almost disjoint families on a countable set has deep consequences in Set-Theory and Model Theory (§12 of [CN74]). However, the properties of almost disjoint families are very problematic to determine and sensitive to the axioms of set theory. For example, the minimal size of a maximal almost disjoint family varies with mo-

1. INTRODUCTION AND PRELIMINARIES 5

dels but Martin’s Axiom decides that each maximal almost disjoint family must have cardinality 𝔠.

Our first result focuses primarily on the relatively recent attempts to classify almost disjoint families. Baumgartner and Weese [BW82] had the interesting idea to investigate the properties of maximal almost disjoint families (henceforth mad families) by virtue of their algebra of partitioners (see page 21). In actual fact this study began with Mrowka’s ground breaking result that the partitioner algebra could be trivial. This was also cleverly modified by Terasawa (see [Mro77, Ter81]). However Baumgartner and Weese initiated the general study of these partitioner algebras. In terms of partitioner algebras, Terasawa and Mrówka basically constructed various mad families whose partitioner algebras have required properties. In [BW82], Boolean algebras isomorphic to some partitioner algebras are called representable. There have been many interesting results since that time aimed at determining which Boolean algebras are representable. (see [DN92], [DF91], [Fra85], [FZ90] and [FZ88].) In Baumgartner and Weese’s paper [BW82], there were a number of open problems regarding representability. Several have been answered but not till now, the first question which is to determine if each subalgebra of a representable algebra is still representable. By the results in [BW82], the answer is trivially yes under CH. However, the question of whether the answer could be negative in some model of set theory remained open for more than a decade. In this thesis our contribution towards the study of representable algebras is to answer this question. We prove that from the existence of a supercompact cardinal a model of Martin’s Axiom can be constructed in which the answer is negative. We actually prove that in any model of PFA^{+} and ♢(S_{0}^{2}) there exists a partitioner algebra containing a subalgebra which is not representable (see 2.4.1).

Later chapters mainly discuss the possibility of embedding subspaces of the

6 1. INTRODUCTION AND PRELIMINARIES

Stone-Céch compactification βω of the form ω∪{p} into some pseudoradial spaces. Before we explain why this is an interesting topic, let us first remark that the above mentioned spaces ω ∪ {p} have no converging sequence while the pseudoradial spaces are expected to have sufficiently many converging sequences to decide their structure.

A radial space is such a space where one can get the all limit point of a subset by adding all the limit points of well-ordered sequences. In a pseudoradial spaces, one has to iteratively taking such limit points. Radial and pseudoradial spaces were first introduced by Herrlich [Her67] in 1967 as generalizations of Fréchet spaces and sequential spaces. The same classes of topological spaces were then studied by a number of topologists such as Mrówka, Rajagopalan and Arhangel′skiĭ (see, for example, [DMR67], [SMS73], [Arh78b], [Arh78a] and [AIG86]). However, pseudoradial spaces has recently received new attention, particularly after Shapirovskiĭ’s remarkable theorem on the equivalence of sequential compactness and pseudoradialness in compact spaces (see [Sha90]). Shapirovskiĭ’s result stimulated several interesting results regarding sequential compactness and pseudoradialness (see [DJSS96], [JS93]). By studying Fréchet and sequential spaces, topology offers a powerful tool in study of the convergent sequences. The development of the study of radial spaces and pseudoradial spaces show that the transfinite sequence not only serves as a natural more general concept than the countable convergent sequences but also plays an important role which is especially useful in inductive constructions.

Many properties known to be true for Fréchet or sequential spaces are not clear for radial or pseudoradial spaces. In this thesis our contributions are towards the fundamental problem of deciding how pseudoradial spaces differ from other topological spaces. In doing so, we solve several open problems.

In [AIG86], the authors posed a problem of finding an internal characterization

1. INTRODUCTION AND PRELIMINARIES 7

of those spaces which are subspaces of pseudoradial spaces. It is know that each topological space, X, is a quotient space of the topological sum of its so-called prime spaces {X_{p}: p ∈ X}, where X_{p} is obtained from X by making each point except p isolated while p retains its neighbourhoods. Since quotient maps preserve sub-pseudoradialness, it naturally becomes the first step to examine the spaces with only one non isolated point. Further more, by a theorem in [Zho94], it is often sufficient to examine all countable subspace of X. Therefore, the countable spaces with only one non isolated point becomes very important. Among them, the subspaces ω ∪ {p} of βω, again become most interesting in this area, where p are ultrafilters on ω. Our work again involves the questions and techniques in study of the Stone-Céch compactification βω including manipulation of almost disjoint families on the countable set ω.

In [AIG86], the authors asked whether, for each ultrafilter p on ω, the space ω ∪ {p} is a subspace of some pseudoradial space (sub pseudoradial space), where ω ∪ {p} takes the subspace topology in βω, i.e. p is the only non-isolated point and a neighbourhood of p is A ∪ {p} for A ∈ p. It is proved in [Zho94] that, under the assumption 𝔭 = 𝔠 , ω ∪ {p} is a subspace of a pseudoradial Hausdorff space for p ∈ βω \ ω. As a consequence, under 𝔭 = 𝔠, each space with countable tightness is a subspace of some pseudoradial space. However the pseudoradial space constructed in [Zho94] is not regular. J. Vaughan [Vau90] asked if it is possible to make the pseudoradial space in [Zho94] regular. In [Nyi92] P. Nyikos asked whether each topological space can be embedded into a pseudoradial space (Problem 6.22 ). These are clearly very natural questions for which one needs to know the answer in order to understand the subspaces of pseudoradial spaces. In this thesis we will settle these problems in the class of regular spaces and, for brevity, we will not specifically mention that the spaces under discussion are assumed to be regular.

8 1. INTRODUCTION AND PRELIMINARIES

It is proved that there is a forcing extension in which for some ultrafilter p, the space ω ∪ {p} can be embedded into some regular pseudoradial space while the same is not true for all ultrafilters, specifically not for P-point ultrafilters. It is also proved that, under CH, the space ω ∪ {p} can be embedded into a zero-dimensional Hausdorff pseudoradial space for every ultrafilter p on ω.

Therefore, in the class of regular spaces, for ultrafilters p on ω, the sub pseudoradialness of ω ∪ {p} is totally independent of ZFC and there are plenty of spaces which are not sub pseudoradial.

As a by-product, we produce, in ZFC, an ultrafilter on ω with a strong Noetherian base (see page 109 for definition). By our results, it appears very likely that 𝔠 ≤ ω_{2} implies that ω ∪ {p} is sub pseudoradial for this particular ultrafilter p.

All our results and proofs occurs in Chapter 2 through Chapter 9. In Chapter 10, we will give a more specific summary of our work as well as a list of open problems. We will also try to explore possible implications of our results and our new techniques.

This chapter also contains the notations and preliminaries. The section 1.2 can be skipped. The reader can come back when some theorems or notations are used in reading the later chapters.

In Chapter 2 through chapter 5, we deal with representable algebras.

In Chapter 2, we will construct a partitioner algebra and find a subalgebra 𝔅 which is not representable. We will prove that 𝔅 is not representable based on a theorem whose proof will take the following three chapters.

In Chapter 3, we prove a structural theorem which reflects our motivations in constructing the non representable algebra 𝔅 in the way we did.

In Chapter 4, we will prove a key lemma used in the construction of a coherent sequence (see page 11) in Chapter 5. The lemma will show that we can extend a

1. INTRODUCTION AND PRELIMINARIES 9

coherent sequence with certain requirements.

In Chapter 5, we give the proof of the theorem cited in Chapter 2 to prove that 𝔅 is not representable. We construct a special coherent sequence in some forcing extension by a proper forcing.

In Chapter 6 through Chapter 8, we prove some results about embedding the spaces ω ∪ {p} into some pseudoradial spaces.

In Chapter 6, we give the relevant definitions and prove that under CH, for every ultrafilter p on ω, the space ω∪{p} with the usual topology can be embedded into some pseudoradial zero-dimensional Hausdorff space.

In Chapter 7, we prove that in any forcing extension by adding ℵ_{2} many Cohen reals to a model of CH, for all P-point p, the space ω ∪ {p} can not be embedded into any regular pseudoradial spaces.

In Chapter 8, we first introduce the notion of a reflecting self-indexed stationary set and then show that the existence of such a stationary set together with the assumption 𝔠 = ω_{2} will imply that for the ultrafilter p which will be discussed in Chapter 9, the space ω ∪ {p} is a subspace of some pseudoradial and zerodimensional Hausdorff space. We then deduce that there is a forcing extension in which, for all P-points p, the space ω ∪ {p} can not be embedded into any regular pseudoradial spaces while for our special ultrafilter p, ω ∪ {p} can be embedded into some pseudoradial and zero-dimensional Hausdorff space.

In Chapter 9, we construct, in ZFC, an ultrafilter p with a special almost disjoint refinement (see page 101) and a strong Noetherian base.

We give some basic definitions and notations and the preliminaries in the next two sections.

10 1. INTRODUCTION AND PRELIMINARIES

1.1. Definitions and Notations

Our notations and definitions are standard. Most of the notations and definitions for set theory can be found in [Kun80]. Notations and definitions for general topology can usually be found in R. Engelking’s book General Topology [Eng77]. Other good references are [KV84] and [She82]. The terminologies in model theory can be found in [CK73]

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.

For a set S and a family I of subsets of S, I is an ideal on S if, (1) for any I in I, any subset of I is in I, (2) for any I and J in I, I ∪ J is also in I. For a family A of subsets of S, we use I_{A} to denote the minimal ideal on S containing A. For an ideal I on S and two subsets A and B of S, we write A ⊆_{I} B if A \ B ∈ I. Naturally, A =_{I} B means that A△B ∈ I.

We will use Fin to denote the ideal of finite subsets of ω. When S is ω and I is Fin, we have the following standard notations.

If A and B are subsets of ω, then A ⊆^{∗} B means that A \ B is finite. We write A =^{∗} B when A△B is finite. Hence A ≠^{∗} ∅ means that A is an infinite set. If A ∩ B =^{∗} ∅, i.e. A ∩ B is finite, then we say that A and B are almost disjoint. An almost disjoint family M on ω is a subset of P(ω) such that the members of M are pairwise almost disjoint. We also call an almost disjoint family an ad family and a maximal almost disjoint family a mad family. We will call a sequence {a_{α}: α < λ} in P(ω) increasing (decreasing) mod finite if for α < β < λ, a_{α} ⊆^{∗} a_{β} (a_{α}^{∗} ⊇ a_{β}). We will also use the natural generalizations to other countable sets.

For a subset A of ω, A is co-infinite if ω \ A is infinite and A is co-finite if ω \ A

1.1. DEFINITIONS AND NOTATIONS 11

is finite. If A and B are subsets of P(ω) we write A < B or B > A if for all a ∈ A and for all b ∈ B , a ⊆^{∗} b. If A is a singleton containing only the element a, then we write a < B instead of {a} < B. Similarly we will write A < b or b > A, etc. In case a < B, we call a a pseudo-intersection of B.

If a and b are subsets of ω, we say a splits b if both b ∩ a and b \ a are infinite. If a does not split any member of M, we call a a partitioner of M.

We will sometimes call a function from ω to ω a real. We use ℕ to denote all positive integers, i.e. ℕ = ω \ {0}.

We call a partially ordered set a poset and sometimes a forcing notion. If P is a poset and E_{1}, E_{2} ⊆ P, we write E_{1} ⊥ E_{2} when, for all p ∈ E_{1} and q ∈ E_{2}, q and p are incompatible (p and q have no common extension). In case E_{1} is singleton say, {p}, we write p ⊥ E_{2} instead of {p} ⊥ E_{2}. We use E^{⊥} to denote the set {q ∈ P: q ⊥ E}. Thus E_{1} ⊥ E_{2} iff E_{1} ⊆ (E_{2})^{⊥}.

Our standard references for forcing are [Kun80] and [She82]. We always assume a poset has a largest element 1. We use x̌ to denote a canonical name for x in the ground model. To make notations less complicated, we will sometimes use x in place of x̌ in the discussions. 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 one ⋃{{x̌}×π_{x}: x ∈ X}, where each π_{x} is an antichain of ℙ. This abuse of terminology is quite standard.

Definition 1.1.1. 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,

• a_{α} ⊆ b_{α},

• a_{α} =^{∗} a_{β} ∩ b_{α}.

A set X is said to uniformize A, if for all α < λ, X ∩ b_{α} =^{∗} a_{α}.

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

Definition 1.1.2. 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 uniformizes A.

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})}.

Definition 1.1.3. For sets I and J and an infinite cardinal λ, Fn(I, J, λ) = {p : |p| < λ ∧ p is a function ∧ dom p ⊆ I}. Order Fn(I, J, λ) by: p ≤ q ↔ q ⊆ p. Fn(I, J) is Fn(I, J, ω).

To simplify the notations, we will sometimes use logical symbols ∀, ∃, → , ↔, ∨, ∧ as well as ¬ as logical connectives in the standard way. Also, 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 less 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).

1.2. Preliminaries

In this section, we state some axioms and theorems in set theory which will be needed. We will also introduce more notions and definitions related to these theorems.

Definition 1.2.1. [Kun80] A poset ℙ has the countable chain condition (ccc) iff every antichain in ℙ is countable. More generally ℙ satisfies the λ-cc if every antichain in ℙ has cardinality less than λ. Therefore, ω_{1}-cc is simply ccc.

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 1.2.2. Suppose S is a set. C ⊆ [S]^{≤ω} is called a cub in [S]^{≤ω}, if (1) for any countable subset B of S, there is an A ∈ C such that B ⊆ A, and (2) for any countable increasing sequence {A_{n}: n ∈ ω} in C,

⋃_{n∈ω} A_{n} is in C .

E ⊆ [S]^{≤ω} is called stationary, if E meets every cub on [S]^{≤ω}.

For a function f from [A]^{<ω} to A, we use C(f) to denote the set {B ∈ [A]^{≤ω}: ∀F ∈ [B]^{<ω}, f(F) ∈ B}. Obviously, C(f) is a cub on [A]^{ω}. It is also not difficult to prove that C contains a cub on A iff there is a function f from [A]^{<ω} to A such that C(f) ⊆ C.

14 1. INTRODUCTION AND PRELIMINARIES

Proposition 1.2.3. Suppose ℙ is a ccc poset and A is any set. If Ċ is a ℙname for a cub on [A]^{<ω} and G is a ℙ-generic over V, then there is a cub C_{0} on [A]^{<ω} such that C_{0} is from V and C_{0} ⊆ val(Ċ, G).

Proof: Let C = val(Ċ, G). Let f be a function from [A]^{<ω} to A such that C(f) ⊆ C. Since ℙ is ccc, by Lemma 6.5.5 of [Kun80], there is a function F from [A]^{<ω} to [A]^{ω} such that for any B ∈ [A]^{<ω}, f(B) ⊆ F (B) and F is in the ground model. Let C_{0} be the set of countable subsets B of A such that for each E ∈ [B]^{<w}, F (E) ⊆ B. Clearly C_{0} is a cub and is a subset of C. □

Definition 1.2.4. [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.

By the above proposition, every ccc poset is proper. It is also easy to prove that every countably closed poset is proper.

Definition 1.2.5. PFA is the following statement:

Suppose P is a proper poset, {D_{α}: α ∈ ω_{1}} are dense subsets of P.

Then there exists a filter G ⊆ P, such that

∀α < ω_{1} G ∩ D_{α} ≠ ∅ .

PFA^{+} is the following statement:

Suppose P is a proper poset, {D_{α}: α ∈ ω_{1}} are dense subsets of P and {τ_{α}: α ∈ ω_{1}} are P -names for a subset of ω_{1} such that ∀α < ω_{1} ⊩_{P} “ τ_{α} is stationary subset of ω_{1}”. Then there exists a filter G ⊆ P, such that

(1) ∀α < ω_{1} G ∩ D_{α} ≠ ∅,

(2) ∀α < ω_{1} S_{α}(G) = {ξ < ω_{1} : ∃p ∈ G p ⊩_{P} “ ξ ∈ τ_{α}”} is statio-

nary in ω_{1}.

1.2. PRELIMINARIES 15

For a cardinal κ, and a stationary set E on κ, ♢(E) is the following statement: There exists a sequence {S_{α}: α ∈ E} such that for any subset X of κ, there are stationary many α ∈ E such that X ∩ α = S_{α}.

We call such sequence a ♢(E) − sequence. Therefore, ♢ is ♢(ω_{1}). We use S_{0}^{2} to denote the stationary set of all ordinals in ω_{2} with countable cofinality. Thus a ♢(S_{0}^{2})-sequence is of the form

{S_{α}: α ∈ ω_{2} and cf α = ω}

and such that for any subset X of ω_{2}, for stationary many α ∈ S_{0}^{2}, X ∩ α = S_{α}. The following theorem is well-known, so we only give an outline of its proof.

Theorem 1.2.6. 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}.

Let P = Fn(κ, 2, κ). Then it is easy to show that in V^{P} there is a ♢-sequence on {ξ: cf κ = ω}. Since P is κ-directed closed, κ is still supercompact. In V^{P}, let Q be the usual proper forcing notion to get a model of PFA in which 𝔠 = κ = ω_{2}. (for example, see [Dev83], Section 5). It is easy to verify that forcing with Q actually results in a model of PFA^{+}. Hence, in V^{P ∗Q̇}, PFA^{+} is true. Since Q is also κ-cc with cardinality κ, there is again a ♢-sequence on {ξ ∈ κ: cf ξ = ω} in V^{P ∗Q̇}. □

We will need a result of Koszmider which is a consequence of a principle called Chang’s Conjecture. Chang’s Conjecture:

16 1. INTRODUCTION AND PRELIMINARIES

For any structure of the form A = ⟨ω_{2}, ω_{1}, (f_{i})_{i<ω}⟩, there is an elementary submodel B of A such that the universe B of B has cardinality ω_{1} and A ∩ ω_{1} is countable.

The reader can refer to [CK73] for more details. We recall results in [Bek91] about the relationship between PFA^{+} and Chang’s Conjecture.

Definition 1.2.7. An ω_{1} ϵ-chain of submodels of H_{θ} is a sequence ⟨N_{ζ}: ζ < ω_{1}⟩ such that

(1) ⟨N_{θ}: ζ < η⟩ ∈ N_{η+1}

(2) N_{ζ} = ⋃_{ξ<ζ} N_{ξ} for ζ limit,

(3) N_{ζ} is a countable submodel of H_{θ}

Definition 1.2.8. We say S ⊆ [H_{θ}]^{ω} reflects if there is an ω_{1} ϵ-chain ⟨N_{ζ}: ζ < ω_{1}⟩ such that {ζ: N_{ζ} ∈ S} is stationary.

The following theorem is a combination of Theorem 3.25 and Theorem 3.27 in [Bek91].

Theorem 1.2.9 (S. Todorcěvić). If for all cardinals θ > ω_{1}, each stationary set in [H_{θ}]^{ω} reflects, then Chang’s Conjecture holds. □

It is essentially folklore that PFA^{+} implies that every stationary set in [H_{θ}]^{ω} reflects for any cardinal θ > ω_{1}. Therefore we have,

Proposition 1.2.10 (S. Todorcěvić). PFA^{+} implies Chang’s Conjecture. □

For our application of Chang’s Conjecture we need the following notion.

Definition 1.2.11. A sequence {X_{α} : α ∈ λ} in [ω_{1}]^{ω_{1}} is called a strong chain if for any α and β in λ such that, for α < β, X_{α} \ X_{β} is finite and X_{β} \ X_{α} is uncountable.

1.2. PRELIMINARIES 17

Here is the theorem by P. Koszmider.

Lemma 1.2.12. [Kos] Under the Chang’s Conjecture, there is no strong chain of length ω_{2} in [ω_{1}]^{ω_{1}}. □

We still need more consequences of PFA.

Theorem 1.2.13 (Theorem 3.16, [Bek91]). PFA implies 2^{ℵ_{0}} = ℵ_{2}. □

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.

Theorem 1.2.14 (Theorem 7.10, [Bau84]). PFA implies that every tree of height ω_{1} and cardinality ℵ_{1} is special and therefore wKH is false. □

If G is a Fn(ω, 2)-generic over V, then we call g = ⋃ G or g^{−1}(0) a Cohen real over V. If A is a ground model subset of ω, then G∩Fn(A, 2) is a Fn(A, 2)-generic. Therefore we also call g ↾A or g^{−1}(1) ∩ A a Cohen real on A.

If 𝔐 is a model of a suitable fragment of ZFC (eg. ZFC^{−} = ZFC - Power Set Axiom), we call a real r ∈ ^{ω}2 a Cohen generic real over 𝔐, if {p ∈ Fn(ω, 2): p ⊆ r} a generic filter on Fn(ω, 2) over 𝔐.

We say that a family A of subsets of ω is independent on a subset B of ω if for any two disjoint finite subsets F and H of A,

B ∩ (⋂ F \ ⋃ H) is infinte.

We will need the following fact. The proof is a standard application of MA so we will omit it.

Proposition 1.2.15 (MA). Suppose 𝔐 is a model of ZFC^{−} with size ℵ_{1}. Then there are countably many Cohen generics over 𝔐, {r_{n}: n ∈ ω}, such that for any a ⊆ ω in 𝔐, {r_{n}^{−1}(1): n ∈ ω} is independent on a. □

18 1. INTRODUCTION AND PRELIMINARIES

We will denote this by saying that {r_{n}: n ∈ ω} are mutually Cohen over 𝔐. We will need to use the Open Coloring Axiom (OCA). The OCA is the following

statement 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}.

Theorem 1.2.16 (S. Todorcěvić, [Tod89]). PFA implies OCA.

We will need the following lemma to be able to apply OCA.

Lemma 1.2.17. Suppose T ⊆ P(ω)×P(ω) and K_{0} = {{(a, b), (c, d)} ∈ [T]^{2} : a∩ d ≠ ∅ or c ∩ b ≠ ∅}. If T inherits the usual topology from P(ω), then K_{0} is an open set in [T]^{2} with respect to this separable metric topology on T.

Proof: Let σ be the topology on P(ω) generated by

{[p] : p ∈ Fn(ω, 2)}

where [p] = {A ⊆ ω : A ∩ dom p = p^{−1}(0)}. It is obvious that (P(ω), σ) is homomorphic to the Cantor set. Hence (P(ω) × P(ω), σ × σ) is separable metric space. Let τ be the subspace topology of T in (P(ω)×P(ω), σ ×σ). Let x = (a, b) and y = (c, d) and assume {x, y} ∈ K_{0}. We have to prove that there are open

1.2. PRELIMINARIES 19

sets U and V containing x and y respectively such that, for any x′ ∈ U and y′ ∈ V, {x′, y′} ∈ K_{0}. Without loss of generality, we assume that a ∩ d is not empty. Choose m ∈ a ∩ d. Let p = {⟨m, 1⟩}. Since x ≠ y, there are disjoint open sets U′ and V′ such that x ∈ U′ and y ∈ V′. Let U = U′ ∩ ([p] × P(ω)) and V = V′ ∩ (P(ω) × [p]). Clearly {{x′, y′}: x′ ∈ U and y′ ∈ V } ∩ [T]^{2} ⊆ K_{0}. □

An ideal I is a P-ideal if for any countable sequence {I_{n} : n ∈ ω} in I, there exists I in I, such that, for all n ∈ ω, I_{n} is almost contained in I.

Definition 1.2.18. 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 1.2.19 (Definition 2.6, [Dow97]). 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 properties of being a Hausdorff-Luzin type family is absolute for models preserving ω_{1}. It is also easy to see that any Hausdorf-Luzin type gap can not be filled.

Definition 1.2.20 (Def 2.1, [Dow97]). 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).

20 1. INTRODUCTION AND PRELIMINARIES

If we say that a set of pairs {(I_{α}, J_{α}) : α ∈ ω_{1}} are (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 quote two results about separating ideals to finish this chapter. Lemma 1.2.21 (Lemma 2.2, [Dow97]). If I and J are countably separated

and I is a P -ideal, then they are strongly countably separated.

Theorem 1.2.22 (Theorem 2.3, [Dow97]). Suppose PFA holds. If I and J are ideals of countable sets with I a P -ideal then either

(1) I and J can be countably separated,

or

(2) there are ℵ_{1}-generated subideals I′ ⊂ I and J′ ⊂ J which cannot be countably separated.

CHAPTER 2 A Non Representable Boolean Algebra

In [BW82], the first open problem is to decide if each subalgebra of a representable algebra is also representable. Baumgartner and Weese proved in [BW82] that, under CH, any Boolean Algebra with cardinality ≤ 𝔠 is representable. Therefore, under CH, the answer to their first question is positive. By assuming the existence of a supercompact cardinal, we will show that it is consistent that the answer to the above question is negative. In this chapter, we will construct a partitioner algebra with a non representable subalgebra. Then by applying a technical theorem we will prove that subalgebra is not representable. However, the proof of the technical theorem will take the next three chapters.

2.1. Definitions for Representable Algebras

Let M be a mad family on ω. For a ⊆ ω, if for each M ∈ M, either M ⊆^{∗} a or M ∩ a =^{∗} ∅, then a is called a partitioner of M. A partitioner a is a non trivial partitioner of M, if a is not in I_{M}, the ideal generated by the finite subsets of ω and the mad family M.

The set 𝔄 of all partitioners of M forms a subalgebra of P(ω) which is sometimes denoted by 𝔄_{M}. 𝔄/I_{M} is called the partitioner algebra of M. We also use P_{M} to denote the partitioner algebra of M. For any non trivial partitioner A of

22 2. A NON REPRESENTABLE BOOLEAN ALGEBRA

M, we use [A] to denote the corresponding element in P_{M}, i.e. the equivalence class containing A.

A Boolean algebra A is called representable if it is isomorphic to a partitioner algebra of some mad family on ω. In this case, we say that the mad family represents A.

Obviously, we can replace ω in the above definitions with any countably infinite set. Therefore when we mention a partitioner algebra we will not be concerned if the underlying set for the mad family is ω as long as it is a countable set.

2.2. Constructions

By Theorem 1.2.6, if there is a supercompact cardinal, then there is a model in which PFA^{+} and ♢(S_{0}^{2}) hold. Hence, from this chapterto chapter 5, we will assume that PFA^{+} and ♢(S_{0}^{2}) hold in V. Therefore by Theorem 1.2.13, 1.2.10 and 1.2.14 the following are true in our universe V:

(1) Chang’s Conjecture;

(2) 𝔠 = ω_{2};

(3) there is no weak Kurepa tree.

We now start the actual construction of the algebra 𝔅. We will spend considerable efforts to prove that it works. We will first construct a mad family M and 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 ideal on ω, i.e {ω \a_{α}: ∈ ω_{2}} is a base for an ultrafilter on ω. It is easy to see that P_{M} × P_{M} is isomorphic to the partitioner algebra of mad family M̃ defined as follows. M̃ = M ∪ M′, where M′ = {M × {0}: M ∈ M}. We will use P_{M} instead of P_{M} × P_{M}. 𝔅 will be, in some sense, generated by {a_{α}, a_{α} × {0}: α ∈ ω_{2}}. Formally we let 𝔅 be the set of equivalence classes in P_{M} of thoses partitioners x for which there in an α < ω_{2} such that either x or its complement contained in a_{α} ∪ α × {0}, i.e.

2.2. CONSTRUCTIONS 23

𝔅 = {[x] ∈ P_{M}: ∃α < ω_{2} x or (ω ∪ ω × {0}) \ x ⊆_{I_{M}} a_{α} ∪ a_{α} × {0}}. Obviously

ω is a nontrivial partitioner of M̃ which is not in 𝔅. Our motivation is to ensure that if 𝔅 is a partitioner algebra of some mad family on ω ∪ ω × {0}, then ω will actually be a non trivial partitioner of that mad family. This ensures that 𝔅 is not equal to the partitioner algebra. To ensure that 𝔅 is not representable, we will actually ensure that if a mad family N represents 𝔅, then some element analogous to [ω] will be a partitioner of N.

Let us first fix an enumeration, {C_{α} : α ∈ lim ω_{2}}, of [ω]^{ω} and choose a sufficiently large regular cardinal θ.

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}, where T_{α} is the tree consisting of all countable coherent sequences in 𝔐_{α}. The ordering on T_{α} is end extension, i.e. if B and A are in T_{α} and A = {(c_{ξ}, d_{ξ}) : ξ < β}, then B ≺ A iff for some γ ≤ β, B = {(c_{ξ}, d_{ξ}) : ξ < γ}. [T_{α}] is the set of all ℵ_{1} branches of T_{α}.

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

(4) For all α, α ∈ 𝔐_{α}, M_{α} ∈ 𝔐_{α}, a_{α} ∈ 𝔐_{α}, and if α < β, then M_{β} ∩ 𝔐_{α} = M_{α}.

(5) For limit α, M_{α} = ⋃_{β<α} M_{β} and

M_{α} = M_{α+1} = M_{α+2} and M_{α+n} = M_{α+3}(n ≥ 3)

and if a ⊆ ω and a ∈ 𝔐_{α+2}, then a is a partitioner of M_{α+3} \ M_{α}.

24 2. A NON REPRESENTABLE BOOLEAN ALGEBRA

(6) For limit α, 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 exists, 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 oft the conditions will be trivially satisfied. For example, to get condition (1) and (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) are, in some sense to guarantee, that the mad family ⋃_{α<ω_{2}} M_{α} and its partitioner algebra are complicated enough to make sure that 𝔅 is not representable. Conditions (1) - (5) are mainly technical ones which will be used in the next three chapters.

We will actually do 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 stage α and α is not a successor. Case 1: α = 0 Take any elementary submodel 𝔐 of H_{θ} as 𝔐_{0} such that 𝔐 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. 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 condition (1)-(7) are satisfied. Next we define, by induction on i, {𝔐_{α+i}, M_{α+i}, a_{α+i} : 0 < i ∈ ω} .

2.2. CONSTRUCTIONS 25

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 1.2.14, weak Kurepa Hypothesis fails, thus T_{α} has at most ℵ_{1} many ℵ_{1} branches, i.e. |[T_{α}]| ≤ ℵ_{1}. By the Proposition 1.2.15, 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}, 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, condition (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

26 2. A NON REPRESENTABLE BOOLEAN ALGEBRA

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}, no subset of ω in 𝔐_{α+2} splits M^{∗}. Clearly, M^{∗} is not in 𝔐_{α+2}.

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}.

2.3. Basic Facts and 𝔅

We first state some basic facts about what we have constructed.

Fact 2.3.1. M is a mad family.

Proof: This is obvious by (6). □

Fact 2.3.2. {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}.

2.3. BASIC FACTS AND 𝔅 27

Fact 2.3.3. (1) If a ∈ 𝔐_{α}, then a is a partitioner of M \ M_{α}.

(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. □

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}}.

Claim 2.3.1. 𝔅̃ forms a subalgebra of P(ω ∪ ω × {0}).

Proof: We verify that 𝔅̃ is closed under the Boolean operations. Obviously, a ∈ 𝔅̃ iff ω \ a ∈ 𝔅′. We check that it is closed under the intersection operation. Let a and b be in 𝔅̃. Let α and β witnesses that a and b are in 𝔅̃ and without loss of generality, assume that α < β. If a ⊂_{I_{M}} a_{α} ∪ a′_{α} or b ⊂_{I_{M}} a_{β} ∪ a′_{β}, then

a ∩ b ⊂_{I_{M}} a_{β} ∪ a′_{β}. Otherwise, ω \ a ⊂_{I_{M}} a_{α} ∪ a′_{α} and ω \ b ⊂_{I_{M}} a_{β} ∪ a′_{β}, thus

ω \ (a ∩ b) ⊂_{I_{M}} a_{β} ∪ a′_{β}. Therefore β witnesses that a ∩ b is in 𝔅̃. □

Let 𝔅 = 𝔅̃/I_{M}. Obviously, 𝔅 is the subalgebra of the partitioner algebra 𝔄 and we have the following.

28 2. A NON REPRESENTABLE BOOLEAN ALGEBRA

Fact 2.3.4. There does not exist a ∈ 𝔅̃ such that, for all α ∈ ω_{2}, a_{α} \ a ∈ I_{M}

and a′_{α} ∩ a ∈ I_{M} □

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 some notations in the following discussion.

Definition 2.3.5. Let A be a coherent sequence. We call A a good coherent sequence if A is of the following 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 use CSQ (GCSQ, RCSQ ) to denote the set of all the coherent sequences (good coherent sequences, reasonable sequences) with countable length.

2.4. 𝔅 is not representable

As the title indicates, we prove the following theorem in this section.

Theorem 2.4.1. The Boolean Algebra 𝔅 is not representable.

2.4. 𝔅 IS NOT REPRESENTABLE 29

Definition 2.4.2. 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 there is a isomorphism φ from 𝔅 to P_{N} such that for any partitioner of A of M′ with [A] ∈ 𝔅, F (A) is a non trivial partitioner of N and φ[A] = [F (A)], i.e, the following diagram commutes.

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 2.4.1 that the following Theorem essentially says that there is a coherent sequence A uniformized by some set X while the image of A under F can not be uniformized 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.

Theorem 2.4.3. If N is a mad family whose partitioner algebra is isomorphic to 𝔅, then there is 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 uniformized by some set X ⊆ ω. Moreover, there exists 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 holds

(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 following holds.

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

(2) ∀k ∈ ℕ,

{(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 uniformizes the image of A modulo I_{N}. In the theorem 2.4.4 below, we will obtain such a coherent sequence in some proper forcing extension. It turns out that it is very difficult to get such proper forcing notion. Then, since we are using PFA to produce such coherent sequence in the ground model, we are only allowed to use ω_{1} many dense subsets to determine the fact no suitable subsets of ω uniformizes the image of A modulo I_{N}. However, a Hausdorff-Luzin type family can be determined via ω_{1} many dense subsets. We need a notation to simplify the following discussion.

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 ∈

2.4. 𝔅 IS NOT REPRESENTABLE 31

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 the Theorem 2.4.3.

Theorem 2.4.4. If N is a mad family whose partitioner algebra is isomorphic to 𝔅, then there exist a function F which 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} ∈ RCSQ and A can be uniformized by some subset X of ω. (ii): F, A, C(A, F) and Σ satisfy the conclusions of Theorem 2.4.3.

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

We first complete the proof of Theorem 2.4.1 from Theorem 2.4.3. The proof of Theorem 2.4.3 from Theorem 2.4.4 is a standard dense sets argument. We will give that proof at the end of this chapter. The reader can choose to skip it, which will not result in any difficulty in reading other parts of this thesis. However, the proof of Theorem 2.4.4 will occupy from Chapter 3 to Chapter 5.

32 2. A NON REPRESENTABLE BOOLEAN ALGEBRA

Proof of Theorem 2.4.1: 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 Theorem 2.4.3. 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 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 2.3.3 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 uniformizes 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 uniformizes A and F induces an isomorphism from 𝔅 to P_{N},

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

Therefore, c_{α+ω} ⊆_{I_{N}} Y and e_{α+ω} ∩ Y =_{I_{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} ⊆_{I_{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 Theorem 2.4.3, {(c_{σ_{α}+k}, e_{σ_{α}+k}): α ∈ ω_{1}} is a Hausdorff-Luzin type family which is the contradiction we seek. □

2.4. 𝔅 IS NOT REPRESENTABLE 33

Finally we prove Theorem 2.4.3 from Theorem 2.4.4. We use PFA to show that there exist A, X, {σ_{α}: α ∈ ω_{1}} and a derived sequence {(c_{α+k}, e_{α+k}): α ∈ lim ω_{1} and 0 < i ≤ ω} of F and A, all in the ground model V, such that the conclusions of Theorem 2.4.4 are satisfied. The proof is tedious but standard.

Proof of Theorem 2.4.3: Let Ȧ, Ẋ, Σ̇, Ċ be the 𝔻-names for A, X, Σ and C such that 1 forces that Ȧ, Ẋ, Σ̇, and Ċ have the properties stated in the Theorem 2.4.4. Let Ṡ be a 𝔻-name for the sequence {Ȧ_{α}: α < ω_{1}}, i.e. Ṡ is a 𝔻-name for a function from ω_{1} to RCSQ. To simplify the notation we will write x in the place of x̌.

We assume

Ṡ = ⋃{{(α, S)} × πṠ ; α,S: α ∈ ω_{1}, S ∈ RCSQ and l(S) = α} ,

where πṠ ; α,S is a subset of 𝔻. We can assume that πṠ ; α,S is regular open, i.e. for q ∈ 𝔻, if πṠ ; α,S is predense below q, then q is in πṠ ; α,S. We can do so because if we replace πṠ ; α,S by {q: πṠ ; α,S is predense below q} we get a 𝔻-name Ṡ_{1} such that 1 ⊩ Ṡ_{1} = Ṡ.

Since 1 ⊩ “ Ṡ: ω_{1} → RCSQ and α < β → Ṡ(α) < Ṡ(β)”. Obviously we have (i): ∀α ∈ ω_{1} and S_{1}, S_{2} ∈ RCSQ

S_{1} ≠ S_{2} → πṠ ; α,S_{1}⊥πṠ ; α,S_{2} ,

(ii): ∀α,

DṠ ; α = ⋃{πṠ ; α,S: S ∈ RCSQ}

is dense in 𝔻, (iii): ∀α < β ∈ ω_{1},

EṠ ; α, β = ⋃{πṠ ; α,S ∩ πṠ ; β,S′: S ≺ S′ ∈ RCSQ}

34 2. A NON REPRESENTABLE BOOLEAN ALGEBRA

is dense in 𝔻. It is also obvious that conditions (i)-(iii) imply above forcing statement.

Similarly we can assume that

Σ̇ = ⋃{{(α, β)} × πΣ̇ ; α,β: α ∈ ω_{1} and β ∈ lim ω_{1}} ,

where each πΣ̇ ; α,β is regular open and

(1) ∀α < ω_{1}, β_{1}, β_{2} ∈ ω_{1},

β_{1} ≠ β_{2} → πΣ̇ ; α,β_{1}⊥πΣ̇ ; α,β_{2} ,

(2) ∀α,

DΣ̇ ; α = ⋃{πΣ̇ ; α,β: β ∈ lim ω_{1}}

is dense,

(3) ∀α < β ∈ ω_{1},

EΣ̇ ; α,β = ⋃{πΣ̇ ; α,ξ ∩ πΣ̇ ; β,η: ξ < η ∈ lim ω_{1}}

is dense in 𝔻. Let Ḣ be a name for a function from ω_{1} × ℕ × ℕ to [ω_{1}]^{<ω} such that, for all

(α, k, n) ∈ ω_{1} × ℕ × ℕ,

1 ⊩ “ Ḣ(α, k, n) = {β ∈ α: |ċ_{σ_{α}+k} ∩ ė_{σ_{β}+k}| ≤ n} is finite” .

We assume that

Ḣ = ⋃{{(t, J)} × πḢ ; t,J: t ∈ ω_{1} × ℕ × ℕ, J ∈ [ω_{1}]^{<ω_{1}}} ,

where each πḢ ; t,j is a regular open subset of 𝔻. Again, we have that

(i): ∀t ∈ ω_{1} × ℕ × ℕ, ∀J_{1}, J_{2} ∈ [ω_{1}]^{<ω},

J_{1} ≠ J_{2} → πḢ ; t,J_{1}⊥πḢ ; t,J_{2} .

2.4. 𝔅 IS NOT REPRESENTABLE 35

(ii): ∀t ∈ ω_{1} × ℕ × ℕ,

DḢ ; t = ⋃{πḢ ; t,J: J ∈ [ω_{1}]^{<ω}}

is dense in 𝔻, If S = {(b_{β}, a_{μ(β)}): β ≤ ξ + ω} ∈ RCSQ with length ξ + ω + 1 and η + ω ≤ ξ,

we say that (η, ξ, S) is (k, n)-big if

|c_{ξ+k} ∩ e_{η+k}| > n .

where

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

For α, ξ, β, η ∈ ω and S ∈ RCSQ, let E(α, ξ, β, η, S) be

πΣ̇ ; α,ξ ∩ πΣ̇ ; β,η ∩ πṠ ; ξ+ω+1,S .

Let D_{β}(α, k, n, J) be the following set

(πḢ ; (α,k,n),J)^{⊥} ∪ ⋃{E(α, ξ, β, η, S): η < ξ,

S ∈ RCSQ, l(S) = ξ + ω + 1 and (ξ, η, S) is (k, n)-big }

Claim: For all α ∈ ω_{1}, k, n ∈ ℕ, J ∈ [ω_{1}]^{<ω}, if β ∈ α \ J, then

D_{β}(α, k, n, J) is dense in 𝔻 .

Proof of the claim: Let p be an arbitrary element of 𝔻. Let t = (α, k, n). Since DḢ ; t is dense, there exists q ∈ DḢ ; t such that q ≤ p. Now choose J′ ∈ [ω_{1}]^{<ω} such that q ∈ πḢ ; t,J′. If J′ ≠ J, then q⊥πḢ ; t,J, (hence q ∈ D_{β}(α, k, n, J)) otherwise, by the choice of q,

q ⊩ “ J = Ḣ(α, k, n)” .

36 2. A NON REPRESENTABLE BOOLEAN ALGEBRA

Therefore,

q ⊩ “ |ċ_{σ_{α}+k} ∩ ė_{σ_{β}+k}| > n” .

Since EΣ̇ ; β,α is dense, choose r_{0} ≤ q such that r_{0} ∈ EΣ̇ ; β,α. Therefore there are η < ξ ∈ lim ω_{1}, such that

r_{0} ∈ πΣ̇ ; α,ξ ∩ πΣ̇ ; β,η .

Since DṠ ; ξ+ω+1 is dense , there is an r below r_{0} such that r ∈ DṠ ; ξ+ω+1. Fix S ∈ RCSQ such that r ∈ πṠ ; ξ+ω+1,S. Since πΣ̇ ; α,ξ, πΣ̇ ; β,η are both open,

r ∈ πΣ̇ ; α,ξ ∩ πΣ̇ ; β,η ∩ πṠ ; ξ+ω+1,S = E(α, ξ, β, η, S) .

By the choice of r, we have

r ⊩ “ σ̇_{α} = ξ and σ̇_{β} = η and Ȧ_{ξ+ω+1} = S” .

Let S = {(b_{ρ}, a_{μ(ρ)}): ρ ≤ ξ + ω}. Since

q ⊩ “ |ċ_{σ_{α}+k} ∩ ė_{σ_{β}+k}| > n” ,

we have

r ⊩ “ |ċ_{σ_{α}+k} ∩ ė_{σ_{β}+k}| > n” .

However, r ⊩ “ ċ_{σ_{α}+k} = c_{ξ+k} and ė_{σ_{β}+k} = e_{η+k}”. Where c_{ξ+k} and e_{η+k} are from the derived sequence of F and S. Therefore

|c_{ξ+k} ∩ e_{η+k}| > n .

This completes the proof that (ξ, η, S) is (k, n)-big. Therefore, r ∈ D_{β}(α, k, n, J) and finishes the proof of the claim.

We assume Ẋ = {{n} × πẊ ; n : n ∈ ω} and again we assume that each πẊ ; n is regular open. Therefore, p ⊩ “ n ∈ Ẋ” if and only if p ∈ πẊ ; n. For each n let D_{n} = πẊ ; n ∪ (πẊ ; n)^{⊥}. Let E_{n} = ⋃_{i≥n} πẊ ; i. Obviously each D_{n} is dense in 𝔻. Since 𝔻 forces that Ẋ is infinite, each E_{n} is dense in 𝔻.

2.4. 𝔅 IS NOT REPRESENTABLE 37

Let ḣ be a 𝔻-name for a function from ω_{1} to ω such that for each α ∈ ω_{1},

1 ⊩ “ (Ẋ ∩ ȧ_{μ(α)})△ḃ_{α} ⊆ ḣ(α)” .

As usual, we assume that

ḣ = ⋃{{(α, n)} × πḣ ; α,n: α ∈ ω_{1}, n ∈ ω} ,

where each πḣ ; α,n is regular open and

(i): ∀α ∈ ω_{1}, n_{1}, n_{2} ∈ ω,

n_{1} ≠ n_{2} → πḣ ; α,n_{1}⊥πḣ ; α,n_{2} ,

and (ii): ∀α ∈ ω_{1},

Dḣ ; α = ⋃{πḣ ; α,n: n ∈ ω_{1}} is dense .

Obviously for any p ∈ 𝔻, α ∈ ω_{1} and n ∈ ω,

p ⊩ “ ḣ(α) = n” ↔ p ∈ πḣ ; α,n .

Let

D_{1} ={DṠ ; α: α ∈ ω_{1}} ∪ {EṠ ; α,β: α < β ∈ ω_{1}}
D_{2} ={DΣ̇ ; α: α ∈ ω_{1}} ∪ {EΣ̇ ; α,β: α < β ∈ ω_{1}}

\begin{matrix}D_{3} = & \{D_{β}(α, k, n, J): α ∈ ω_{1}, k, n ∈ ω, J ∈ [ω_{1}]^{<ω}, β ∈ α \ J\} \\ ∪ & \{DḢ ; t: t ∈ ω_{1} × ℕ × ℕ\} \end{matrix}

D_{4} ={Dḣ ; α: α ∈ ω_{1}}
D_{5} ={E_{n} : n ∈ ω}
D_{6} ={D_{n} : n ∈ ω}

Let D = {⋂ D′: D′ ∈ [⋃_{i=1}^{6} D_{i}]^{<ω}}. By PFA, there is a filter G ⊆ 𝔻 such that G meets every dense set in D.

38 2. A NON REPRESENTABLE BOOLEAN ALGEBRA

It is easy to see that val(Ṡ, G) is an increasing sequence in (RCSQ, ≺). Let val(Ṡ, G) be {A_{α}: α ∈ ω_{1}} and A = ⋃_{α∈ω_{1}} A_{α} = {(b_{α}, a_{μ(α)}) : α < ω_{1}}. Obviously A is a good coherent sequence. Since G is D_{2} generic, val(Σ̇, G) is an increasing sequence in lim ω_{1} with length ω_{1}, so let val(Σ̇, G) be {σ_{α}: α ∈ ω_{1}}. Let X = val(Ẋ, G) and h = val(ḣ, G). Finally let C = {(c_{α+k}, e_{α+k}): α ∈ lim ω_{1}, 0 < k ≤ ω} be the derived sequence of F and A_{ω_{1}}. It is easy to see that X is an infinite subset of ω and h: ω_{1} → ω.

We have to show that X uniformizes A and that A, C, Σ satisfies (1) and (2) in the Theorem 2.4.3.

First we prove that X uniformizes A. Let α ∈ ω_{1}, we have to prove that X ∩ a_{μ(α)} =^{∗} b_{α}.

Since G is a D-generic, there exists p ∈ G ∩ Dḣ ; α ∩ DṠ ; α+ω+1. It is then obvious

that p ∈ πṠ ; α+ω+1,S, where S = A_{α+ω+1}and so p ⊩ “ ȧ_{μ(α)} = a_{μ(α)} and ḃ_{α} = b_{α}”.

Since p ∈ Dḣ ; α, there exists a unique n ∈ ω such that p ∈ πḣ ; α,n. Obviously h(α) = n. Since p ⊩ “ ḣ(α) = n”, we have that p ⊩ “ (Ẋ ∩ a_{μ(α)})△b_{α} ⊆ n”. Therefore, for all m ∈ a_{μ(α)} \ n, m ∈ b_{α} if and only if p ⊩ “ m ∈ Ẋ”, i.e. p ∈ πẊ ; m. Since any q ≤ p forces what p forces, it follows that, for the above m, m ∈ b_{α} if and only if q ⊩ “ m ∈ Ẋ”. Thus m ∉ b_{α} will imply p⊥πẊ ; m. Therefore p ∈ πẊ ; m if and only if G ∩ πẊ ; m is not empty and this is equivalent to m being in X. Thus we proved that (X ∩ a_{μ(α)})△b_{α} ⊆ n.

Next we check that the properties (1) and (2) in Theorem 2.4.3 hold. (1) follows immediately from the definition of C. We are only left to verify (2). Let α ∈ ω_{1} and k ∈ ℕ; we have to show that, for all n ∈ ω,

{β < α: |c_{σ_{α}+k} ∩ e_{σ_{β}+k}| ≤ n} is finite .

Let t = (α, k, n). Since DḢ ; t ∩ G is not empty, pick J ∈ [ω_{1}]^{<ω} and q ∈ G ∩ πḢ ; t,J. Now for any β ∈ α \ J, pick r ∈ G ∩ D_{β}(α, k, n, J). Since G ∩ πḢ ; t,J is not empty,

2.4. 𝔅 IS NOT REPRESENTABLE 39

r is compatible with some element in πḢ ; t,J. Therefore, there exists η < ξ ∈ lim ω_{1} and S ∈ RCSQ such that (ξ, η, S) is (k, n)-big, and r ∈ E(α, ξ, β, η, S). Since r ∈ G and by the definition of E(α, ξ, β, η, S), it must be the case that ξ = σ_{α}, η = σ_{β} and S = A_{ξ+ω+1}. Consequently, {(c_{γ+i}, e_{γ+i}) ∈ C : γ + i ≤ ξ + ω + 1} is the derived sequence of F and S. Now it is easy to see that

|c_{σ_{α}+k} ∩ e_{σ_{β}+k}| = |c_{ξ+k} ∩ e_{η+k}| > n .

Thus we proved that {β < α: |c_{σ_{α}+k} ∩ e_{σ_{β}+k}| ≤ n} is contained in the finite set J. We are done. □

For a quick reading of the proof for Theorem 2.4.3, the reader can skip next two chapters and read Chapter 5 directly. However, the theorems in Chapter 3 and Chapter 4 are not just purely technical, they actually reflect the motivations behind the proof for Theorem 2.4.3.

40 2. A NON REPRESENTABLE BOOLEAN ALGEBRA

CHAPTER 3 Separating Mad Families

In this chapter we will prove Theorem 3.0.1, which will be the foundation for the whole proof of Theorem 2.4.3.

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 3.0.1. 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} ≠^{∗} ∅.

3.1. REARRANGING THE MAD FAMILY N 41

(iii): for all α ∈ ω_{2}, for all m ∈ ω, F (a_{α}) ∩ J_{m} =^{∗} ∅; and

(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.

3.1. Rearranging 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) =_{I_{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 the Theorem 3.0.1 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 P_{M}). Recall we have PFA^{+} and ♢(S_{0}^{2}) available.

Lemma 3.1.1. There is a function F : [ω ∪ (ω × {0})]^{ω} → [ω]^{ω} which induces φ for which the following hold.

42 3. SEPARATING MAD FAMILIES

(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 1.2.22 and Lemma 1.2.21, 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}.

3.1. REARRANGING THE MAD FAMILY N 43

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 1.2.22 and Lemma 1.2.21, 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.

44 3. SEPARATING MAD FAMILIES

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}}. By Theorem 1.2.10, Chang’s Conjecture holds and by Lemma 1.2.12, 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}. By Lemma 1.2.17, 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}

3.1. REARRANGING THE MAD FAMILY N 45

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 exists 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}. 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}) ⊆_{I_{N}} P and f(a′_{α}) ∩ P =_{I_{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 3.1.1.

Next section is a proof for Theorem 3.0.1.

46 3. SEPARATING MAD FAMILIES

3.2. Proof of Theorem 3.0.1

Let F, P, {Y_{n}, Y′_{n} : n ∈ ω} satisfies the Lemma 3.1.1 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 ≠ ∅}. By Lemma 1.2.17, K_{0} is an open coloring for some separable metric topology on T . Claim 2: There is no uncountable 0-homogeneous set in T

Proof of Claim 2. The proof is almost identical to the proof of Claim 3 in the proof of Lemma 3.1.1. We give a sketch of the proof. Suppose {(c_{α}, d_{α}) : α ∈ ω_{1}} is an uncountable 0-homogeneous subset of T. For all α ∈ ω_{1}, let ξ_{α} ∈ ω_{2} be such

3.2. PROOF OF THEOREM 3.0.1 47

that c_{α} ⊆^{∗} F (a_{ξ_{α}}) . By Lemma 3.1.1, there is a λ > sup{ξ_{α} : α ∈ ω_{1}} such that

⋃_{α<ω_{1}} X_{ξ_{α}} \ X_{λ}

is countable. There is a finite subset N′ of N and an uncountable subset Γ of ω_{1}, such that, for all α ∈ Γ , X_{ξ_{α}} \ X_{λ} = N′. Therefore, F (a_{λ}) ∪ ⋃ N′ will separate {c_{α} : α ∈ Γ} and {d_{α} : α ∈ Γ}. Again this will contradict the fact that {(c_{α}, d_{α}) : α ∈ Γ} is a 0-homogeneous set.

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 exists 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}) ⊆_{I_{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

48 3. SEPARATING MAD FAMILIES

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 3.1.1, 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) =_{I_{N}} Q. Note that since F (A) =_{I_{N}} Q, it follows that for all α ∈ ω_{2}, a_{α} ∩ A ∈ I_{M} and a′_{α} \ A ∈ I_{M}. Therefore, by Fact 2.3.4, we have that Q is not a partitioner of N. On the other hand, fix any 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 3.0.1. Since Ñ and J̃ are countable, we will assume, for convenience, that they are

members of 𝔐_{λ} for all λ.

CHAPTER 4 Extending Coherent Sequences

4.1. An Extension Lemma

Recall that (See page 12) 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 chapter, we prove a preparatory lemma which will allow us to construct a good coherence sequence in the next chapter.

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)}) =_{I_{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 sequence can be extended while keeping certain commitments.

In order to ensure that, in V^{ℙ_{A}}, no partitioner of N will uniformize, 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)}) ≠_{I_{N}} F (b_{α+1}) \ F (b_{α})” .

50 4. EXTENDING COHERENT SEQUENCES

In constructing the coherent sequences A_{ω_{1}}, we will handle one name τ at a time. Lemma 4.1.1 will guarantee that, under certain situations, we can extend A_{α} to A_{α+1} and take care of name τ_{α} from a suitable list.

Lemma 4.1.1. 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 g is ℙ_{A}-generic over 𝔐_{μ} we will not verify the 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)}) ≠_{I_{N}} F (b_{α+1}) \ F (b_{α})” .

It will be clear when applying this lemma in the next chapter 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 first prove Lemma 4.1.2 from Lemma 4.1.3 and then prove Lemma 4.1.1 from Lemma 4.1.2. Finally we give, in

4.1. AN EXTENSION LEMMA 51

the next section, the longest proof, that of Lemma 4.1.3, which, using the structures from Theorem 3.0.1, basically tells us why F can not induce an isomorphism from B to the partitioner algebra of N.

Lemma 4.1.2. Let A, μ, τ be as in the Lemma 4.1.1. 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 4.1.3. Let A, μ, τ be as in Lemma 4.1.1. 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})

Proof of Lemma 4.1.2 from Lemma 4.1.3: Let μ′ ≥ μ+2. Let g ∈ 𝔐_{μ+1}, p ∈ ℙ_{A} be chosen by Lemma 4.1.3. We will actually prove that there are cub many λ in ω_{2} which satisfy the property (†). Assume that this is not the case, i.e. there are stationary many λ such that (†) is false for λ. Then for stationary many λ, there are ρ_{λ}, γ_{λ}, p_{λ} ∈ ℙ_{A}, N_{λ} ∈ [N]^{<ω} and m_{λ} ∈ ω such that ρ_{λ} < λ ≤ γ_{λ}, p_{λ} is compatible with g ↾a_{γ} ∪ p and

g ↾a_{γ_{λ}} ∪ p ∪ p_{λ} ⊩_{ℙ_{A′}} “ (τ△F (g^{−1}(1))) ∩ (F (a_{λ}) \ F (a_{ρ_{λ}})) ⊆ m_{λ} ∪ ⋃ N_{λ} ”,

where A′ = A(g, a_{γ_{λ}}). By the Pressing Down Lemma, there exist ρ and a stationary set S such that, for all λ ∈ S, ρ_{λ} is equal to ρ. Since the cardinality of ℙ_{A}

52 4. EXTENDING COHERENT SEQUENCES

is less then ω_{2}, we can also assume that there are q and m such that p_{λ} = q and m_{λ} = m for all λ ∈ S. Let λ_{0} = λ_{0}(ρ, q) be obtained by applying Lemma 4.1.3 to ρ and q. Then, for λ ∈ S \ λ_{0}, according to Lemma 4.1.3 we have

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

where A′ = A(g, a_{γ_{λ}}). This contradicts the choice of λ, γ_{λ}, q, m and N_{λ}. □ Now we prove Lemma 4.1.1 Proof of Lemma 4.1.1 from Lemma 4.1.2: Let p, g be as the Lemma

4.1.2. Let μ′ mu + 2. Take any increasing sequence {λ_{n} : n ∈ ω} such that λ_{0} > μ′ and each λ_{n} satisfies (†) in Lemma 4.1.2. 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′}) =_{I_{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.

4.2. PROOF OF THE MAIN LEMMA 53

Corollary 4.1.4. If {R_{n} : n ∈ ω} satisfies (1) and (2) of Lemma 4.1.1, 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}). □

4.2. Proof of the main lemma

We need some preliminaries to prove Lemma 4.1.3.

Lemma 4.2.1. 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 4.2.2. 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)\} .

54 4. EXTENDING COHERENT SEQUENCES

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

Fact 4.2.3. E(n) is dense in ℙ_{A} if and only if E(π) is dense in ℙ_{A} .

Proof: It is easy to see that E(n)(n) = E(n) and E(π)(n) = E(π). It is also easy to see that E(π)(π^{−1}) = E(n). Therefore, it suffices to prove the “if” part. So assume that E(n) is dense and let r ∈ ℙ_{A}. Without loss of generality, we assume that n ⊆ dom r. Let q = π(r ↾ n) \overline{∪} r. Since E(n) is dense, choose q′ ∈ E(n) such that q′ ≤ q. Let r′ = r ↾n \overline{∪} q′, then r′ ∈ E(π) and r′ ≤ r. □

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.

Lemma 4.2.4. 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: We first show that (1) implies (2). Suppose (1) is true. Let E ∈ D and p ∈ ℙ_{A(g,d)}. We have to find an e ∈ E such that e and p are compatible. By the definition of ℙ_{A(g,d)}, we can assume that p = q \overline{∪} g ↾ d, where q ∈ ℙ_{A}. Let g′ = q \overline{∪} g. By (1), g′ is also D-generic. Therefore, there is an e ∈ E such that e ⊆ g′. Obviously, e is compatible with q \overline{∪} g ↾d = p. Next we prove (1).

4.2. PROOF OF THE MAIN LEMMA 55

Let g′ be any finite modification of g, i.e. there is an n ∈ ω such that g and g′ agree on ω \ n. Let E ∈ D. We have to prove that there is an e ∈ E such that e ⊆ g′. Let π be the bijection {0, 1}^{n} such that π(g ↾n) = g′ ↾n. Since D is closed, E(π) is in D. Therefore, since g is a D-generic, there is an r ∈ E(π) such that r ⊆ g. By the definition of E(π), π(r ↾n) \overline{∪} r ∈ E(n), i.e. g′ ↾n \overline{∪} r ∈ E(n). Since g ↾ω \ n = g′ ↾ω \ n and r ⊆ g, g′ ↾n \overline{∪} r ⊂ g′. But there is some e ∈ E such that e ⊆ g′ ↾n \overline{∪} r. Hence e ⊆ g′. □

The following corollary collects these results together.

Corollary 4.2.5. 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 contain each d_{α}. □

We will also need the following easy lemma.

Lemma 4.2.6. 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: Let A′ be A(g, d). Since A_{s} ∪ B_{s} is predense in ℙ_{A′}, it is easy to see that for any p ∈ ℙ_{A}, A_{s} ( respectively B_{s}) is predense below p in ℙ_{A} iff A_{s} ( respectively B_{s}) is predense below p in ℙ_{A′}. However, it is also easy to see

56 4. EXTENDING COHERENT SEQUENCES

that p ⊩_{ℙ_{A}} “ s ∈ S” if and only if A_{s} is predense below p in ℙ_{A} and p ⊩_{ℙ_{A′}} “ s ∈ S” if and only if A_{s} is predense below p in ℙ_{A′}. Similarly, we have that p ⊩_{ℙ_{A}} “ s ∉ S” if and only if B_{s} is predense below p in ℙ_{A} and that p ⊩_{ℙ_{A′}} “ s ∉ S” if and only if B_{s} is predense below p in ℙ_{A′}. We are done. □

Now we are ready to prove Lemma 4.1.3

Proof of Lemma 4.1.3: Let μ, A and τ be as given in the Lemma 4.1.1. Recall that we have fixed Ñ, J̃ and F satisfying the conclusion of Theorem 3.0.1 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}) Case 1: {N_{i,j} : i, j ∈ ω} satisfies (i) and (ii) of the Theorem 3.0.1. Subcase 1. There is a p ∈ ℙ_{A} such that

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

Let g_{0} = g_{0}^{μ} ∈ 𝔐_{μ+1} be the Cohen real over 𝔐_{μ} defined in the construction of Chapter 2 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 4.2.1, 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}. Let h ∈ 𝔐_{μ+2} ∩^{ω}ω be such that h dominates all reals in 𝔐_{μ+1}, i.e.

∀f ∈ ^{ω}ω ∩ 𝔐_{μ+1}, f <^{∗} h .

4.2. PROOF OF THE MAIN LEMMA 57

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 2.3.3, g^{−1}(1) is a partitioner of M\M_{μ+2}. But since g^{−1}(1)∩a_{μ} = g^{−1}(1)∩a_{μ(β)} =^{∗} b_{β}, g^{−1}(1) is also a partitioner of M_{μ} = M_{μ+2}. Therefore, 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^{∗} > 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 4.2.6,

r ⊩_{ℙ_{A′}} “ l ∉ τ” .

Therefore,

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

58 4. EXTENDING COHERENT SEQUENCES

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 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 .

Let h ∈ ^{ω}ω ∩ 𝔐_{μ+2} dominate ^{ω}ω ∩ 𝔐_{μ+1}. Take an h^{∗}, greater than both h and f_{0}, such that for all i ∈ ω,

{N_{i,j} : j ∈ ω} ∩ N (A) ⊆ {N_{i,j} : j < h^{∗}(i)}, (2)

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 3.0.1, 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.

For all λ ≥ λ_{0}, γ ≥ λ, m ∈ ω and N′ a finite subset of N, let i > k be such that N = N_{i,h^{∗}(i)} ∉ N′ and N ∈ X_{λ}. Since i > k, τg̃ ∩ N is infinite. By (2), N ∩ (F (g^{−1}(1)) ∪ F (a_{ρ})) is finite. Hence,

N ∩ [(τg̃ \ F (g^{−1}(1))) ∩ (F (a_{λ}) \ F (a_{ρ})) \ ⋃ N′] is infinite .

Let l > m be in the above set. Then since l ∈ τg̃, there exists r ⊆ g̃ such that

r ⊩_{ℙ_{A}} “ l ∈ τ”

4.2. PROOF OF THE MAIN LEMMA 59

Since ℙ_{A} and τ are in 𝔐_{μ} and g is a ℙ_{A}-generic over 𝔐_{μ}, again by Lemma 4.2.6,

r ⊩_{ℙ_{A(g,aγ)}} “ l ∈ τ”.

Hence

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

But r ⊆ g̃ = q \overline{∪} g and q is compatible with p. Therefore if q ∪ p ∪ g ↾ a_{γ} is a function, then r is compatible with it. Therefore either q ∪ p ∪ g ↾ a_{γ} is not a function or

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

Case 2 {J_{n} : n ∈ ω} satisfies the (iii) to (v) of Theorem 3.0.1.

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 the property (iv), there are at most finitely many i ∈ ω such that N ∩ J_{i} is not empty. Subcase 1 ∃p ∈ ℙ_{A}, p ⊩ “ ∀^{∞}n J_{n} ⊆^{∗} τ”

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 ⊆_{I_{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.

60 4. EXTENDING COHERENT SEQUENCES

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 3.0.1, 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 3.0.1, 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 is 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′” .

4.2. PROOF OF THE MAIN LEMMA 61

Subcase 2 ∃p ∈ ℙ_{A} p ⊩ “ ∃^{∞}n J_{n} ⊈^{∗} τ”.

Take a ℙ_{A}-generic g over 𝔐_{μ} such that p ⊆ g, g^{−1}(0) ⊆ a_{μ+1} and g ∈ 𝔐_{μ+1}. Let ρ ≥ μ, q ∈ ℙ_{A}. We assume, for the non-trivial case, q and p are compatible. Let h in ^{ω}ω ∩ 𝔐_{μ+2} dominate ^{ω}ω ∩ 𝔐_{μ+1}. For similar reasons as in subcase 1, we can find a f_{0} ∈ ^{ω}ω and k ∈ ω , such that for all n > k,

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

and f_{0} > h. Therefore,

F (g^{−1}(1)) ⊇ ⋃_{n>k} J_{n} \ f_{0}(n) .

Let g̃ = q \overline{∪} g; since q and p are compatible, p ⊆ g̃, and so there are \{n_{j} : j ∈ ω} ∈ 𝔐_{μ+1} such that J_{n_{j}} \ τg̃ is infinite. Using the method used in subcase 1, we can find {A_{i}, N_{i}, β_{i} : i ∈ ω} such that

(1) A_{i} ∩ J_{n_{j}} is finite for all i, j ∈ ω,

(2) A_{i} ⊆ ⋃_{j∈ω} J_{n_{j}} \ (τg̃ ∪ f_{0}(n_{j}) ∪ ⋃_{j<i} F (a_{β_{j}})),

(3) N_{i} ∩ A_{i} is infinite,

(4) N_{i} ⊆ F (a_{β_{i}}).

Consequently, there is a λ_{0} such that, for λ > λ_{0}, {N_{i} : i ∈ ω} ∩ X_{λ} is infinite. Now it is obvious that for each N_{i}, N_{i} \ τg̃ is infinite and, if N_{i} ∈ X_{λ}, then N_{i} ∩ (F (g^{−1}(1)) ∩ (F (a_{λ}) \ F (a_{ρ}))) is infinite. Being a partitioner of M, F (g^{−1}(1)) ∩ (F (a_{λ}) \ F (a_{ρ})) almost contains N_{i}. The rest of the proof is identical to the previous subcase. □

62 4. EXTENDING COHERENT SEQUENCES

CHAPTER 5 Collapsing ℵ_{2} and a Good Coherent Sequence

In this chapter, we will find a proper forcing notion 𝔻 whose extension will contain a coherent sequence with the properties stated in the Theorem 2.4.4. At the end of this chapter we will summarize the results and complete the proof of Theorem 2.4.4.

5.1. Collapsing ℵ_{2}

Recall that Fn(ω_{1}, ω_{2}, ℵ_{1}) denotes the set of all partial functions from ω_{1} to ω_{2} with countable domain. Let ℙ = Fn(ω_{1}, ω_{2}, ℵ_{1}) with the reverse inclusion as its partial order, i.e. for p and q in ℙ, p ≤ q if p extends q. This poset ℙ is also often denoted by Coll(ω_{2}, < ω_{1}) because it will collapse ω_{2} to ω_{1}.

We will prove several results to get ready to prove Theorem 2.4.4. The following theorem is the culmination of this Chapter.

Theorem 5.1.1. 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,

5.1. COLLAPSING ℵ_{2} 63

(b): ℙ_{A} forces that for all Y ⊆ ω,

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

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

In this section, we prove some lemmas before we move to next section to prove Theorem 5.1.1.

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 tha 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 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.

64 5. COLLAPSING ℵ_{2} AND A GOOD COHERENT SEQUENCE

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.

Lemma 5.1.2. 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}.

Proof: Since our construction in chapter 2 guaranteed that 𝔐_{0} contains a ♢(S_{0}^{2}) sequence, by a standard argument, we can fix, in 𝔐_{0}, a sequence J = {J_{α} : α ∈ S_{0}^{2}} such that, for each α, J_{α} = (X_{α}, Y_{α}, Z_{α}, W_{α}) ∈ (P(α))^{4} and for all X, Y, Z and W ⊆ ω_{2}, there are stationarily many α such that X ∩ α = X_{α}, Y ∩ α = Y_{α}, Z ∩ α = Z_{α} and W ∩ α = W_{α}. Let p_{0} ∈ ℙ and suppose that Ẋ, Ċ and ḟ are ℙ-names such that

p_{0} ⊩ “ Ẋ ⊂ ω_{1} and Ċ is a cub on ω_{1} and ḟ : Ẋ → ω̌_{2}” .

We will find q ≤ p_{0} and α ∈ ω_{1} such that

q ⊩ “ Ẋ ∩ α = ṡ_{α} and ḟ↾ṡ_{α} ∈ 𝔐_{δ̇_{α}+1} and α ∈ Ċ” . (†)

Let χ(Ẋ) be the ℙ-name for the characteristic function of Ẋ. For convenience, we assume that the name χ(Ẋ) is of the form {(ξ_{j}, i_{j}, p_{j}) : j ∈ I} where for any p ∈ ℙ and ξ ∈ ω_{1},

p ⊩ “ ξ ∈ Ẋ” → (ξ, 1, p) ∈ χ(Ẋ)

5.1. COLLAPSING ℵ_{2} 65

and

p ⊩ “ ξ ∉ Ẋ” → (ξ, 0, p) ∈ χ(Ẋ) .

To simplify the notation, we identify Ẋ with its characteristic function χ(Ẋ). We adopt the same convention on Ċ and ḟ. Thus we will have, for example,

p ⊩ “ ξ ∈ Ẋ” ↔ (ξ, 1, p) ∈ Ẋ

and

p ⊩ “ ḟ(α) ≠ β” ↔ (α, β, 0, p) ∈ ḟ .

Using the fact that 𝔠 = ℵ_{2}, we fix, in 𝔐_{0}, bijections h_{0}, h_{1}, h_{2} each with the same range, ω_{2}, and domains ω_{1}×{0, 1}×ℙ, ω_{1}×ω_{2}×{0, 1}×ℙ and ℙ respectively.

We intend to identify ℙ-names Ẋ, Ċ, ḟ and ℙ as subsets of ω_{2} so that we can use the diamond sequence J. Let X = h0^{′′} Ẋ, Y = h0^{′′} Ċ, Z = h1^{′′} ḟ and W = h2^{′′}ℙ respectively.

Take a sufficiently large cardinal κ and a continuous ∈-chain {N_{α} : α ∈ ω_{2}} of elementary submodels of H_{κ} such that N_{0} contains {ℙ, ℚ, p_{0}, J, Ẋ, Ċ, ḟ, ġ, e, c, 𝔐_{0}} and each N_{α} has cardinality ℵ_{1}. Note that N_{0} also contains 𝔐_{0} as a subset and so h_{0}, h_{1} and h_{2} are all in N_{0}. Now take δ such that cf δ = ω, δ = N_{δ} ∩ ω_{2} and

J_{δ} = (X ∩ δ, Y ∩ δ, Z ∩ δ, W ∩ δ) .

Since J and δ are in 𝔐_{δ+1}, J_{δ} ∈ 𝔐_{δ+1}. Therefore X ∩δ, Y ∩δ, Z ∩δ and W ∩δ are elements of 𝔐_{δ+1}. Let Ẋ_{δ} = Ẋ ∩ N_{δ}, Ċ_{δ} = Ċ ∩ N_{δ}, ḟ_{δ} = ḟ ∩ N_{δ} and ℙ_{δ} = ℙ ∩ N_{δ}. Claim 1 Ẋ_{δ} = Ẋ ∩ N_{δ}, Ċ_{δ} = Ċ ∩ N_{δ}, ḟ_{δ} = ḟ ∩ N_{δ} and ℙ_{δ} = ℙ ∩ N_{δ} are in 𝔐_{δ+1}.

Proof of Claim 1: We only prove that Ẋ_{δ} = Ẋ ∩ N_{δ} ∈ 𝔐_{δ+1} since the proofs for the others are similar. Since X ∩ δ and h_{0} are in 𝔐_{δ+1}, it is sufficient to show that Ẋ_{δ} = h_{0}^{−1}(X ∩ δ), i.e. h0^{′′}(Ẋ ∩ N_{δ}) = (h0^{′′} Ẋ) ∩ δ. However, for any (ξ, i, p) ∈ Ẋ, since h_{0} ∈ N_{δ}, (ξ, i, p) ∈ N_{δ} iff h_{0}((ξ, i, p)) ∈ N_{δ}. But N_{δ} ∩ ω_{2} = δ. Hence (ξ, i, p) ∈ N_{δ} iff h_{0}((ξ, i, p)) ∈ (h0^{′′} Ẋ) ∩ δ. We are done.

66 5. COLLAPSING ℵ_{2} AND A GOOD COHERENT SEQUENCE

Take, in 𝔐_{δ+1}, an increasing sequence {η_{n} : n ∈ ω} converging to δ. Claim 2: There exist {p_{n}: n ∈ ω} ⊆ ℙ_{δ} , {γ_{n} : n ∈ ω} increasing in ω_{1}, {x_{n} : n ∈ ω} and {r_{n} : n ∈ ω} such that the following hold:

(0) r_{n} is a function from x_{n} to ω_{2}.

(1) (γ_{n+1}, 1, p_{n+1}) ∈ Ċ_{δ};

(2) x_{n+1} ⊆ γ_{n} and for all ξ < γ_{n}, ξ ∈ x_{n+1} ↔ (ξ, 1, p_{n+1}) ∈ Ẋ_{δ} and ξ ∉ x_{n+1} ↔ (ξ, 0, p_{n+1}) ∈ Ẋ_{δ};

(3) for all α ∈ x_{n+1}, (α, r_{n}(α), 1, p_{n+1}) ∈ ḟ_{δ}

(4) dom p_{n} ∈ ω_{1}, γ_{n} < dom p_{n} < γ_{n+1}, η_{n} < p_{n+1}(dom p_{n}) < δ and

sup ran p_{n} < p_{n+1}(dom p_{n}). We first show that Claim 2 will give us q and α which satisfies (†). By elementarity, we can assume the sequences {p_{n} : n ∈ ω}, {γ_{n} : n ∈ ω}, {x_{n} : n ∈ ω} and {r_{n} : n ∈ ω} are in 𝔐_{δ+1}. Let p = p_{ω} = ⋃_{n∈ω} p_{n}, α = sup_{n} γ_{n}. Let r = ⋃_{n∈ω} r_{n} and x = ⋃_{n∈ω} x_{n}. Of course, x and r are in 𝔐_{δ+1}.

(1) implies that, for all n, p ⊩ “ γ_{n+1} ∈ Ċ”. Therefore p ⊩ “ α ∈ Ċ”. (4) implies that p ⊩ “ δ = δ̇_{α}”. (2) implies p ⊩ “ Ẋ ∩ α = x”. (3) implies that p ⊩ “ ḟ ↾ x = r”. The only thing left to do is to find a q below p such that q ⊩ “ ṡ_{α} = x”.

Recall π_{α} is a fixed bijection from α to ω. Define d : ω → {0, 1} such that d(n) = 1 iff π_{α}^{−1}(n) ∈ x. Let η ∈ ω_{2} be the unique ordinal such that d_{η} = d. Let q = p ∪ {(α, η)}. By the definition of s_{α} it is easy to see that

q ⊩ “ṡ_{α} = {ξ < α : e(ġ)(α, ξ) = 1}
= {ξ < α : d_{ġ(α)}(π_{α}(ξ)) = 1}
= {ξ < α : d_{η}(π_{α}(ξ)) = 1}
= {ξ < α : ξ ∈ x} = x”

Since q ≤ p, it is clear that q and α satisfies (†). Next we prove Claim 2.

5.1. COLLAPSING ℵ_{2} 67

We will construct the sequences by induction on ω. To keep the induction going, we add one more condition in the induction hypothesis.

(5) x_{n}, r_{n} ∈ N_{δ}. Let γ_{0} = 0, x_{0} = ∅, r_{0} = ∅. At stage n, we need to find p_{n+1}, γ_{n+1}, x_{n+1} and r_{n+1}. Since p_{n} ∈ N_{δ}, α = dom p_{n} < ω_{1} and β = sup ran p_{n} < δ. Let p′_{n} = p_{n}^{⌢}(α, max{η_{n} + 1, β + 1}) ∈ N_{δ}. Since

p′_{n} ⊩ “ Ċ is a cub ” ,

there exists p′′_{n} ≤ p′_{n} and γ′_{n} > α such that

p′′_{n} ⊩ “ γ′_{n} ∈ Ċ \ α” .

By elementarity, we can assume that p′′_{n} and γ′_{n} are in N_{δ}. Since ℙ is countably closed, there exists p̃_{n} ≤ p′′_{n} such that dom p̃_{n} is an ordinal and p̃_{n} decides Ẋ ∩ γ_{n} and ḟ↾(Ẋ ∩ γ_{n}). So there are x_{n+1} and r_{n+1} such that

p̃_{n} ⊩ “ Ẋ ∩ γ_{n} = x_{n+1}”

and

p̃_{n} ⊩ “ ḟ↾x_{n+1} = r_{n+1}” .

By elementarity, we can assume that x_{n+1}, r_{n+1} and p̃_{n} are all in N_{δ}.

Now let p_{n+1} = p̃_{n} and γ_{n+1} = γ′_{n}. We prove (1)-(5) holds at this stage. Since p_{n+1} ⊩ “ γ_{n+1} ∈ Ċ”, our assumption on the name Ċ implies (γ_{n+1}, 1, p_{n+1}) ∈ Ċ. But γ_{n+1} and p_{n+1} are in N_{δ}. Therefore (γ_{n+1}, 1, p_{n+1}) ∈ Ċ_{δ}. Thus (1) is true. Since p_{n+1} ⊩ “ Ẋ ∩ γ_{n} = x_{n+1}”, for ξ < γ_{n}, ξ ∈ x_{n+1} implies that p_{n+1} ⊩ “ ξ ∈ Ẋ” and ξ ∈ γ_{n} \ x_{n+1} implies that p_{n+1} ⊩ “ ξ ∉ Ẋ”. Similar to the proof of (1) we get ξ ∈ x_{n+1} implies (ξ, 1, p_{n+1}) ∈ Ẋ_{δ} and ξ ∉ x_{n+1} implies (ξ, 0, p_{n+1}) ∈ Ẋ_{δ}. Therefore we can get (2). (3) is proved in a same way we proved (2). (4) and (5) are obvious. We are done □

68 5. COLLAPSING ℵ_{2} AND A GOOD COHERENT SEQUENCE

The following Corollary is what will actually be used the the proof of Theorem 5.1.1.

Corollary 5.1.3. 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}

Proof: Since there is a bijection in 𝔐_{0} between ω_{2}^{V} and Z, we can assume that Z = ω̌_{2}. Let {s_{α} : α ∈ ω_{1}, cf α = ω} be the ♢-sequence in the Lemma 5.1.2. Let p_{0} denote the projection from ω_{1} × ω_{1} to ω_{1} defined by the formula p_{0}(α, β) = α. In 𝔐_{0}, take a bijection π : ω_{1} × ω_{1} → ω_{1}. Let C be the cub {α < ω_{1} : π′′α × α = α}. For all α ∈ C with cf α = ω, let S_{α} = p′′_{0}(π^{−1}(s_{α})∩ω_{1}×{0}) and T_{α} = p′′_{0}(π^{−1}(s_{α})∩ ω_{1} × {1}). We check that {(S_{α}, T_{α}) : α ∈ C and cf α = ω} works. Let X, Y ⊆ ω_{1} and f_{0} : X → ω̌_{2} and f_{1} : Y → ω̌_{2}. Let B = X × {0} ∪ Y × {1} and let A = π′′B. Now define F : A → ω̌_{2} as follows,

F (ξ) = \begin{cases}f_{0}(α) & ξ = π(α, 0) for some α in X \\ f_{1}(α) & ξ = π(α, 1) for some α in Y \end{cases}

Let E be all α ∈ C such that A ∩ α = s_{α} and F ↾ s_{α} ∈ 𝔐_{δ_{α}+1}. Of course, E is stationary. For any α ∈ E, it is easy to see that S_{α} = X ∩ α and T_{α} = Y ∩ α. To see that f_{0} ↾ S_{α} is in 𝔐_{δ_{α}+1}, let σ : X ∩ α → (X ∩ α) × {0} be such that σ(α) = (α, 0). It is easy to see that S_{α} = X ∩ α ∈ 𝔐_{δ_{α}+1}. Indeed

X ∩ α = p′′_{0}(π^{−1}(s_{α}) ∩ (ω_{1} × {0})) ∈ 𝔐_{δ_{α}+1} .

Therefore, σ ∈ 𝔐_{δ_{α}+1}. Observe that

f_{0} ↾S_{α} = (F ↾s_{α}) ∘ π ∘ σ .

Therefore, f_{0} ↾S_{α} ∈ 𝔐_{δ_{α}+1}. Similarly, f_{1} ↾T_{α} ∈ 𝔐_{δ_{α}+1}. □

5.1. COLLAPSING ℵ_{2} 69

We will also need the following notion.

Definition 5.1.4 (page 245 E1, [Kun80]). A poset (P, <) is called almost homogeneous if, for all p, q in P, there exists an automorphism i of P such that i(p) and q are compatible

We prove the following easy lemma.

Lemma 5.1.5. (1) If P is an almost homogeneous poset, then for any formula φ(x_{1}, …, x_{n}) with all free variables shown, for any u_{1}, …, u_{n} in the ground model, either 1 ⊩_{P} “ φ(ǔ_{1}, …, ǔ_{n})” or 1 ⊩_{P} “ ¬φ(ǔ_{1}, …, ǔ_{n})”.

(2) Suppose A = {(c_{α}, d_{α}) : α < λ} is a coherent sequence and each d_{α} is co-infinite. Then ℙ_{A} is an almost homogeneous poset.

Proof: (1) is well-known. We prove (2). Let p and q be in ℙ_{A}. There are α, β < λ and p′, q′ ∈ Fn(ω, 2) and m ∈ ω such that p = p′ ∪ Ch_{d_{α}\m}^{c_{α}\m} and q = q′∪Ch_{d_{β}\m}^{c_{β}\m}. Without loss of generality, we can assume that α ≤ β, c_{α}\m ⊆ c_{β}, d_{α} \ m ⊆ d_{β} and dom p′ ∪ dom q′ = m . Take a finite subset H of

ω \ (dom p ∪ dom q)

such that H has cardinality m. Let j : m → H be a bijection. Now let i : ℙ_{A} → ℙ_{A} be defined as follows. For any r ∈ ℙ_{A}, dom i(r) = (dom r\(m∪H))∪{n ∈ m∪H : j(n) ∈ dom r or j^{−1}(n) ∈ dom r}. And

i(r)(n) = \begin{cases}r(n) & if n ∈ dom r \ (m ∪ H) \\ r(j(n)) & if n ∈ m and j(n) ∈ dom r \\ r(j^{−1}(n)) & if n ∈ H and j^{−1}(n) ∈ dom r \end{cases}

Obviously, i ∘ i = id_{P}, hence i is a bijection. It is also easy to see that i is orderpreserving . Hence i is an automorphism of P. Observe that dom i(p)∩q = d_{α} \m and the restrictions of i(p) and q on d_{α} \ m are all equal to Ch_{d_{α}\m}^{c_{α}\m}. Hence i(p) and q are compatible. □

70 5. COLLAPSING ℵ_{2} AND A GOOD COHERENT SEQUENCE

5.2. Forcing a Good Coherent Sequence

We list some notations and properties in the next lemma to make the proof of Theorem 5.1.1 shorter.

Lemma 5.2.1. 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” ,

5.2. FORCING A GOOD COHERENT SEQUENCE 71

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 p ⊩_{ℙ_{Aα}} “ ∃^{∞}iR_{i}∩τ ≠^{∗} ∅” and q is compatible with p, 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 5.1.1: 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

72 5. COLLAPSING ℵ_{2} AND A GOOD COHERENT SEQUENCE

Corollary 5.1.3. 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 5.2.1 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}.

5.2. FORCING A GOOD COHERENT SEQUENCE 73

(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 an 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 5.2.1). Let f : ω_{1} → D defined by f(α) = f_{α}. Then by Corollary 5.1.3, 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}.

74 5. COLLAPSING ℵ_{2} AND A GOOD COHERENT SEQUENCE

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. By Lemma 5.1.5, we have 1 forces that

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

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

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

This will contradict the fact 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 5.2.1 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

5.2. FORCING A GOOD COHERENT SEQUENCE 75

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 5.2.1, 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.

76 5. COLLAPSING ℵ_{2} AND A GOOD COHERENT SEQUENCE

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 chapter2, 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)}. Observe that for any M ∈ M∩𝔐_{μ+2} = M_{μ}, there is a ξ < α such that M ⊆^{∗} a_{μ(ξ)} and that for such ξ, g_{β}^{−1}(1)∩a_{μ(ξ)} =^{∗} b_{ξ}. Since b_{ξ} is a partitioner of M, g_{β}^{−1}(1) does not split M. Therefore g_{β}^{−1}(1) does not split any member of M ∩ 𝔐_{μ+2}. On the other hand, g_{β}^{−1}(1) is in 𝔐_{μ+2}. By Fact 2.3.3, g_{β}^{−1}(1) does not split any member of M \ 𝔐_{μ+2}. Therefore, 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 4.2.4, 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

5.2. FORCING A GOOD COHERENT SEQUENCE 77

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 2.3.3, 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

78 5. COLLAPSING ℵ_{2} AND A GOOD COHERENT SEQUENCE

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 4.1.1, 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};

5.2. FORCING A GOOD COHERENT SEQUENCE 79

(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 nice ℙ_{A_{β}}-name, then τ′_{α} is defined as in Lemma 5.2.1. It is easy to see that

1 ⊩_{ℙ_{A′}} “ τ′_{β} =_{I_{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 5.1.1, 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

80 5. COLLAPSING ℵ_{2} AND A GOOD COHERENT SEQUENCE

in the forcing extension. We fix this in the rest of the Chapter.

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. The next theorem produce a better good coherent sequence which enable us to find the ccc forcing notion ℚ.

Theorem 5.2.2. Let G be a ℙ generic filter over V. Suppose A = {(b_{α}, a_{μ(α)}) : α ∈ ω_{1}} satisfies the conclusion of Theorem 5.1.1. 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): ℙ_{A_{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): ℙ_{A_{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)})) ≠_{I_{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}) \ F (b_{ξ})] ∉ I_{N} } ,

5.2. FORCING A GOOD COHERENT SEQUENCE 81

Γ_{α}^{1} = {ξ ∈ Γ_{α} : [F (b_{ξ+1}) \ F (b_{ξ})] \ [Y_{α} ∩ (F (a_{ξ+1}) \ F (a_{ξ}))] ∉ I_{N} } , Obviously,

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

and

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}. If i(α) = 0. Then for any η < ξ ∈ C_{α}, take β ∈ Γ_{α}^{0} such that η < β < ξ, we have

Y_{α} ∩ [F (a_{μ(β+1)}) \ (F (a_{μ(β)}) ∪ F (b_{β+1}))] ∉ I_{N} .

Since F (a_{μ(η)}) ⊆_{I_{N}} F (a_{μ(β)}), F (a_{μ(β+1)}) ⊆_{I_{N}} F (a_{μ(ξ)}) and F (b_{β+1}) =_{I_{N}} F (a_{μ(β+1)} ∩ b_{ξ}). We have

F (a_{μ(β+1)})\(F (a_{μ(β)}) ∪ F (b_{β+1}))
⊆_{I_{N}} F (a_{μ(β+1)}) \ (F (a_{μ(η)}) ∪ F (a_{μ(β+1)} ∩ b_{ξ}))
=_{I_{N}} F (a_{μ(β+1)}) \ (F (a_{μ(η)}) ∪ F (b_{ξ}))
⊆_{I_{N}} F (a_{μ(ξ)}) \ (F (a_{μ(η)}) ∪ F (b_{ξ}))

Therefore,

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} ,

82 5. COLLAPSING ℵ_{2} AND A GOOD COHERENT SEQUENCE

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} = ℙ_{A} and ℙ_{A} is ccc. We only need to prove (ii). Since ℙ_{A} is almost homogeneous we only need to prove that property stated in (ii) holds in V [G][H].

In V [G][H], let Y ⊆ ω and choose a β in ω_{1} such that Y = Y_{β}. Obviously if i(β) = 0 then for all ξ > η > β both μ(δ(η)) and μ(δ(ξ)) are in C \ β, hence

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

Similarly if i(β) = 1 then for all η > ξ both in C \ β,

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

From this point on we will only use the sequence Ã in Theorem 5.2.2, but for convenience we simply assume A = Ã.

Therefore we have the following

Theorem 5.2.3. In V^{ℙ}, there is a good coherent sequence A = {(b_{α}, a_{μ(α)}) : α ∈ ω_{1}} such that

(i): ∀α A_{α} ∈ RCSQ,

(ii): ℙ_{A} is ccc,

5.3. HAUSDORFF-LUZIN TYPE GAPS 83

(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.

We need more notions to define the poset ℚ. In the next section we will modify the usual techniques in order to add what might be called countably many simultaneous Hausdorff-Luzin type gaps.

5.3. Hausdorff-Luzin type gaps

Definition 5.3.1. 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 .

84 5. COLLAPSING ℵ_{2} AND A GOOD COHERENT SEQUENCE

Fact 5.3.2. 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 5.3.3. 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.

5.3. HAUSDORFF-LUZIN TYPE GAPS 85

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 a lot of Hausdorff-Luzin gaps as described. Let ℝ_{α} = {(H, n) ∈ ℝ, H ∩ α = ∅}.

We need the following 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}.

Suppose the Claim is false. Then for any α ∈ ω_{1}, there is a g(α) > α such that D_{g(α)}^{α} is not dense in ℝ_{α}. We can define an increasing sequence {ξ_{α} : α ∈ ω_{1}} and {(H_{α}, n_{α}) : α ∈ ω_{1}} such that ξ_{α+1} = g(ξ_{α}) and (H_{α}, n_{α}) witnesses that D_{ξ_{α+1}}^{ξ_{α}} is not dense in ℝ_{ξ_{α}}.

Since ℝ is ccc, there exists α < β such that (H_{α}, n_{α}) and (H_{β}, n_{β}) are compatible. By the choice of (H_{α}, n_{α}), this will imply that H_{β} ⊆ ξ_{α+1} ≤ ξ_{β}. However, H_{β} ∩ ξ_{β} is empty which is contradiction. Thus we finish proof the Claim 2.

Fix an α^{∗} which satisfies Claim 2. Let ℚ be ℝ_{α^{∗}}. Obviously ℚ is ccc. We check that ℚ also adds appropriate subsets I of ω_{1}. Suppose G be a ℚ-generic. Let I = ⋃{H : ∃n (H, n) ∈ G}. Since for all β > α, D_{β}^{α^{∗}} is dense in ℚ, I is uncountable. Let k ∈ ω, we prove that {(c_{α+k}, e_{α+k}) : α ∈ I} is a Hausdorff-Luzin

86 5. COLLAPSING ℵ_{2} AND A GOOD COHERENT SEQUENCE

type gap. Let α ∈ I and m ∈ ω. We have to show that {β ∈ I : β < α and |c_{α+k} ∩ e_{β+k}| < m} is finite. Pick (H, n) ∈ G such that α ∈ H and n > m + k. Take any β ∈ I ∩ α \ H. There is (K, l) ∈ G such that β ∈ K and (K, l) < (H, n) and so that |c_{α+k} ∩ e_{α+k}| > n ≥ m Thus {β ∈ I : β < α and |c_{α+n} ∩ e_{β+n}| < m} ⊆ H. We are done. □

Working in V^{ℙ}, for each limit α ∈ ω_{1} and k ∈ ℕ, let

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}

Recall we call sequence {(c_{α+k}, e_{α+k}): α ∈ lim ω_{1} and 0 < k ≤ ω} a derived sequence of F and A and it is clear that for all α ∈ lim ω_{1} and k ∈ ℕ, c_{α+k} ⊆ c_{α+ω} and e_{α+k} ⊆ e_{α+ω} and also that k < l ∈ ℕ implies c_{α+k}, e_{α+k}, c_{α+l} and e_{α+l} are mutually disjoint. Furthermore, each of the following holds:

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

Lemma 5.3.4. 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

5.3. HAUSDORFF-LUZIN TYPE GAPS 87

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} =_{I_{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} =_{I_{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 5.3.3, we have

Theorem 5.3.5. 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 5.2.3 and Theorem 5.3.5, we have now are ready to prove Theorem 2.4.4.

Proof of Theorem 2.4.4: Let Q̇ be the ℙ ∗ ℙ_{A}-name for the poset in Theorem 5.3.5 and let 𝔻 be the the poset ℙ ∗ ℙ_{A} ∗ Q̇. Obviously in V^{ℙ}, ℙ_{A} adds

88 5. COLLAPSING ℵ_{2} AND A GOOD COHERENT SEQUENCE

an X which uniformizes 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 5.3.4 and Σ be the sequence I ∈ V^{𝔻} obtained by Theorem 5.3.5. 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.

CHAPTER 6 Pseudoradialness under CH

As indicated in the introduction, we will now be discussing the relationship between the class of regular pseudoradial spaces and the spaces which are of the form ω ∪ {p} for some ultrafilter p on ω. Specifically the question of whether these spaces are sub pseudoradial.

In this chapterwe introduce the concept of pseudoradial space and discuss the sub pseudoradialness of space ω ∪ {p} under the assumption CH or 𝔭 = 𝔠.

6.1. Definition

Definition 6.1.1. A sequence {x_{α} : α < λ} of points in X is said to be convergent to x provided that for each neighbourhood U of x, there exists α < λ such that {x_{β} : β > α} ⊆ U. (We will call any transfinite sequence a sequence).

A space X is radial if for each subset A of X and x in Ā, there is a sequence in A convergent to x. The pseudoradial spaces are characterized by a natural relaxation of this property. This is analogous to, and motivated, by the relationship between sequential spaces and Frechet spaces.

Definition 6.1.2. A subset A of a space of X is said to be radially closed if no sequence in A is convergent to a point outside of A. A space X is said to be pseudoradial provided that any radially closed subset of X is closed in X.

90 6. PSEUDORADIALNESS UNDER CH

In [AIG86] the authors asked if ω ∪ {p} is a subspace of a pseudoradial space for p ∈ βω \ ω, where ω ∪ {p} takes the subspace topology in βω, i.e. p is the only non-isolated point and a neighbourhood of p is A ∪ {p} for A ∈ p. It is proved in [Zho94] that under the assumption 𝔭 = 𝔠, each space with countable tightness is a subspace of some pseudoradial space. It is essentially proved that, under the assumption 𝔭 = 𝔠, ω ∪ {p} is a subspace of a Hausdorff pseudoradial space for p ∈ βω \ ω. However the pseudoradial space constructed in [Zho94] is not regular. In a communication with the author, J. Vaughan asked if it is possible to make the pseudoradial space in [Zho94] regular. In [Nyi92] P. Nyikos asked whether each topological space can be embedded into a pseudoradial space (Problem 6.22 ). Our discussion will be focused on the class of regular spaces. We first prove in the next section that, under CH, ω ∪ {p} is a subspace of a regular pseudoradial space for each ultrafilter p on ω. However, the question of whether CH can be replaced by MA remains open.

6.2. Under CH

If p is a P_{𝔠} point on ω , let {A_{α} : ω ≤ α < 𝔠} be a strictly decreasing modulo finite base for p and let B be the Boolean Algebra generated by {A_{α} : α ∈ 𝔠} ∪ [ω]^{<ω}, then it is easy to see that the Stone space of B is just ω ∪ [ω, 𝔠 + 1] where [ω, 𝔠 + 1] has the usual ordinal topology and ω ∪ {𝔠} is homomorphic to ω ∪{p}. An immediate corollary is that , under CH, if p is a P -point, then ω ∪{p} can be embedded into a zero-dimensional Hausdorff compact pseudoradial space.

However, we have the following result about non P -points.

Lemma 6.2.1 (p=𝔠). If p is not a P -point, then ω ∪ {p} can be embedded in a zero-dimensional pseudoradial space.

Proof: Let p = {A_{α} : α < 𝔠} such that {A_{i} : i < ω} witnesses that p is not

6.2. UNDER CH 91

a P -point. First of all, we will construct a base {B_{α} : α < 𝔠} of p and an almost disjoint family {C_{α} : α < 𝔠} such that, for ξ, α ∈ 𝔠, the following are true:

(i)_{α} B_{α} ⊆ A_{α},

(ii)_{ξα} ξ ≤ α → C_{α} ⊆^{∗} B_{ξ},
(iii)_{ξα} α < ξ → C_{α} ∩ B_{ξ} =^{∗} ∅.

Without loss of generality, we can assume that {A_{n} : n ∈ ω} is strictly decreasing with empty intersection. We also assume that A_{0} = ω. It is easy to construct {B_{ξ}, C_{ξ} : ξ ≤ ω} with B_{n} = A_{n} for n ∈ ω. Suppose we have constructed {B_{ξ}, C_{ξ} : ξ < α} with α ≥ ω and such that (i)_{ξ}, (ii)_{ξη} and (iii)_{ξη} are true for each ξ, η < α. We have to define B_{α} and C_{α}. For ξ ∈ α and n ∈ ω, if ξ ≥ ω, then C_{ξ} ⊆^{∗} A_{n}. By 𝔭 = 𝔠, there is a D ⊆ ω such that D is a pseudo-intersection of {A_{n} : n ∈ ω} and, for ω ≤ ξ < α, C_{ξ} ⊆^{∗} D. Indeed, we can take, for ω ≤ ξ < α, an f_{ξ} ∈ ^{ω}ω such that for each n, C_{ξ} \ A_{n} ⊆ f_{ξ}(n). Since 𝔭 = 𝔠, 𝔟 = 𝔠 , hence we can choose f which is an upper bound of {f_{ξ} : ω ≤ ξ < α} in (^{ω}ω, <^{∗}). We assume f is strictly increasing. Let D = ⋃_{n∈ω}A_{n} ∩ f(n). It is not difficult to see that D works. By the choice of D, we know that D is not a member of p. Let B_{α} = A_{α} ∩B_{ω} \D. Let C_{α} be any pseudo-intersection of {B_{ξ} : ξ ≤ α}. It is easy to check that B_{α} and C_{α} satisfies the requirements.

Next we will use the family {B_{α}, C_{α} : α ∈ 𝔠} to construct a topology τ on X = ω∪((𝔠+1)×{0}) by assigning a neighbourhood system for each point. First of all, ω is an open discrete dense subset of ⟨X, τ⟩. For α < 𝔠, {(C_{α} − n)∪{ ⟨α, 0⟩} : n ∈ ω} is a neighbourhood base for ⟨α, 0⟩ . Let {(B_{α} − n) ∪ {⟨ξ, 0⟩ : α ≤ ξ ≤ 𝔠}: α ∈ 𝔠, n ∈ ω} be a neighbourhood base for ⟨𝔠, 0⟩ . It is easy to see that ⟨X, τ⟩ is a zero-dimensional Hausdorff pseudoradial space and the subspace ω ∪ {⟨𝔠, 0⟩} is homomorphic to ω ∪ {p}. □

Thus we proved

92 6. PSEUDORADIALNESS UNDER CH

Theorem 6.2.2 (CH). For every point p in ω^{∗}, ω ∪ {p} can be embedded in a zero-dimensional pseudoradial space. □

By Theorem 7.1.1 in the next chapter, the above lemma is not a consequence of ZFC.

CHAPTER 7 Pseudoradialness in the Cohen model

7.1. Main Theorem

In this chapter we prove the following

Theorem 7.1.1 (CH). If ℙ = Fn(ω_{2}, 2), then in V^{ℙ}, for every P -point p , the space ω ∪ {p} can not be embedded in any regular pseudoradial space.

Let us just remark that the Theorem remains valid if CH is dropped and κ many Cohen reals are added for any κ > 𝔠.

Suppose ṗ and τ̇ are ℙ-names, X is a ground model set and q ∈ ℙ such that

q ⊩ “ṗ is an ultrafilter on ω and ω ∪ {ṗ} is a subspace of regular pseudoradial space ⟨X, τ̇⟩” ,

We will prove

(∗) q ⊮ “ ṗ is a P-point ”

We will prove (∗) by contradiction. We will work in V^{Fn(λ,2)}, where λ is M ∩ω_{2} for some elementary submodel of H_{θ} for certain regular cardinal θ. We prove some preliminary facts in next sectionand give the proof of (∗) in section 7.3. In next two sections, we assume that CH is true.

94 7. PSEUDORADIALNESS IN THE COHEN MODEL

7.2. Some facts

Suppose we are given ℙ-names ṗ , τ̇ and a set X in the ground model such that

1 ⊩ “ṗ is an ultrafilter on ω and ω ∪ {ṗ} is a subspace of regular pseudoradial space ⟨X, τ̇⟩” ,

We furthermore assume that ṗ has the form ⋃{{π} × A_{π} : π ∈ dom(ṗ)}, where π is a nice name for a subset of ω. Let İ be a nice ℙ-name such that

1 ⊩ “ İ = {I ∈ [ω]^{ω} : I converges in ⟨X, τ̇⟩}”

We also fix a sufficiently large regular cardinal θ and an elementary submodel M of H_{θ} such that the following are true:

(i) M^{ω} ⊆ M;

(ii) {ℙ, X, ṗ, τ̇, İ} ⊂ M; and (iii) M has cardinality ℵ_{1}.

Let ℙ_{M} = ℙ ∩ M, X_{M} = X ∩ M and λ = M ∩ ω_{2}. For each ℙ-name Ẏ ∈ M, if 1 ⊩_{ℙ} “ Ẏ ⊆ X”, we can define a ℙ_{M}-name Ẏ_{M} as follows. For each x ∈ X_{M}, let A_{x} be a maximal antichain in {q ∈ ℙ_{M} : q ⊩_{ℙ} “ x ∈ Ẏ ”}. ℙ_{M} is actually Fn(λ, 2), therefore ℙ_{M} is completely embedded in ℙ. Hence A_{x} is also a maximal antichain in {q ∈ ℙ : q ⊩_{ℙ} “ x ∈ Ẏ ”}. Since M^{ω} ⊆ M and ℙ is ccc, A_{x} ∈ M. Let Ẏ_{M} be ⋃{{x} × A_{x}: x ∈ X_{M}}. If, on the other hand, Ẏ is a ℙ-name such that 1 ⊩_{ℙ} “ Ẏ ⊆ P(X)”, we define a ℙ_{M}-name Ẏ_{M} to be

{⟨Ẏ_{M}, 1⟩ : Ẏ ∈ M and 1 ⊩_{ℙ} “ Ẏ ∈ Ẏ”}

Fact 7.2.1. Suppose G is a ℙ-generic over V, and σ ∈ M such that 1 ⊩_{ℙ} “ σ ⊆ X”, then the following hold:

(i) if 1 ⊩_{ℙ} “ σ ⊆ ω”, then σ_{M} ∈ M, moreover, if σ is a nice name we can define σ_{M} to be σ; (ii) V [G] ⊧ “val(σ_{M}, G) = val(σ_{M}, M ∩ G) = val(σ, G) ∩ M”; and

(iii) if 1 ⊩_{ℙ} “ ṗ is a P-point ”, then, 1 ⊩_{ℙ_{M}} “ ṗ_{M} is an P-point ”.

Proof: (i) and (ii) are straightforward. For (iii), suppose that 1 ⊩_{ℙ} “ ṗ is a P-point ”. It is easy to see that 1 ⊩_{ℙ_{M}} “ ṗ_{M} is an ultrafilter on ω”. We prove that 1 ⊩_{ℙ_{M}} “ ṗ_{M} is countably complete ”. Suppose ḟ is a ℙ_{M}-name and q ∈ ℙ_{M} such that

q ⊩_{ℙ_{M}} “ ḟ is a function from ω into ṗ_{M}”

For each n ∈ ω, we can choose an antichain B_{n} = {q_{i}^{n} : i ∈ ω} which is maximal in {r ∈ ℙ_{M} : r ≤ q} and an associated set {π_{i}^{n} : i ∈ ω} ⊆ dom(ṗ_{M}) such that

q_{i}^{n} ⊩_{ℙ_{M}} “ ḟ(n) = π_{i}^{n}”

for each i ∈ ω. By (i) , dom(ṗ_{M}) ⊆ M. Therefore the ℙ_{M}-name

ρ = {⟨op⟨ň, π_{i}^{n}⟩, q_{i}^{n}⟩ : n, i ∈ ω}

is in M, where op⟨ň, π_{i}^{n}⟩ is a ℙ_{M}-name such that val(op⟨ň, π_{i}^{n}⟩, G) is ⟨n, val(π_{i}^{n}, G)⟩ (See [Kun80]). It is obvious that q ⊩ “ ḟ = ρ”.

Since q ⊩_{ℙ} “ ṗ is countably complete ”, we can choose a nice ℙ-name Ẏ ∈ M such that q ⊩_{ℙ} “ Ẏ is a pseudo-intersection of ran(ρ)” and 1 ⊩_{ℙ} “ Ẏ ∈ ṗ”. By (i), we can define Ẏ_{M} to be Ẏ. It is clear that q ⊩_{ℙ_{M}} “ Ẏ_{M} = Ẏ ”, hence q ⊩_{ℙ_{M}} “ Ẏ_{M} is a pseudo-intersection of ran(ḟ) = ran(ρ) and Ẏ_{M} ∈ ṗ_{M}” □

Note that ℙ_{M} is actually Fn(λ, 2). Let ℙ_{1} be ℙ_{M} and ℙ_{2} be Fn(ω_{2} \ λ, 2), then ℙ is isomorphic to ℙ_{1} × ℙ_{2}. For a ℙ-generic filter G, we will also refer to G ∩ M as G_{M} or as G_{1} and G_{2} will be G ∩ ℙ_{2} while V_{1} will be V [G_{1}]. To further simplify the notation, we will use Y to denote val(Ẏ , G) for a ℙ-name Ẏ. If Ẏ and Ẏ are in M, we will use Y_{M} and Y_{M} to denote val(Ẏ_{M}, G_{M}) and val(Ẏ_{M}, G_{M}) respectively.

Thus p, τ, p_{M} and τ_{M} etc. are well-defined when it is clear from context which ℙ-generic filter G is being used.

Fact 7.2.2. The following are true:

(i) 1 ⊩_{ℙ} “ ṗ_{M} = ṗ ∩ V_{1}, İ_{M} = İ ∩ V_{1}”; (ii) if ρ ∈ M is a ℙ-name such that 1 ⊩_{ℙ} “ ρ ⊆ P(ω)”, then 1 ⊩_{ℙ} “ ρ_{M} =

ρ ∩ V_{1}”; and (iii) 1 ⊩_{ℙ_{M}} “ İ_{M} = {a ∈ [ω]^{ω} : a converges in ⟨X_{M}, τ̇_{M}⟩}”.

Proof: (i) is a corollary of (ii). To prove (ii) , let ρ satisfy the conditions of (ii) and let G be a ℙ-generic filter over V. In V [G], take arbitrary a ∈ val(ρ, G) ∩ V_{1}. Let π ∈ dom(ρ) and σ be a nice ℙ_{M}-name such that a = val(π, G) = val(σ, G). Take a q in G which forces that π = σ ∈ ρ. Since σ, ρ are in M, q ∩ M ⊩_{ℙ} “ σ ∈ ρ”. Let Ẏ ∈ M such that 1 ⊩_{ℙ} “ Ẏ ∈ ρ and (σ ∈ ρ → Ẏ = σ)”, then val(Ẏ_{M}, G) ∈ val(ρ_{M}, G). But q ⊩_{ℙ} “ π = σ = Ẏ ”. Hence a = val(σ, G) = val(Ẏ , G) = val(Ẏ_{M}, G) ∈ val(ρ_{M}, G) Thus we proved 1 ⊩_{ℙ} “ ρ_{M} ⊇ ρ ∩ V_{1}”. By the definition of ρ_{M} , it is obvious that

1 ⊩_{ℙ} “ ρ_{M} ⊆ ρ ∩ V_{1}”.

Therefore 1 ⊩_{ℙ} “ ρ_{M} = ρ ∩ V_{1}”. We are left to prove (iii)

Let H be a ℙ_{M}-generic filter over V, there exists a ℙ-generic filter G over V such that H = G ∩ ℙ_{M}. We have to prove that, in V_{1} = V [G_{M}],

I_{M} = {a ∈ [ω]^{ω} : a converges in ⟨X_{M}, τ_{M}⟩}

“ ⊆ ”: Let a ∈ I_{M}. We have to prove that a converges in ⟨X_{M}, τ_{M}⟩. There exits Ẏ_{M} ∈ dom(İ_{M}), such that a = Y_{M}. Since a ⊆ ω, by (i) of Fact 7.2.1, Y_{M} = Y, hence a ∈ M[G]. In V [G], there exists x ∈ X, such that a converges to x in ⟨X, τ⟩. According to [She82], M[G] is an elementary submodel of H_{θ}^{V [G]}.

7.3. PROOF 97

By elementarity, we can assume x in X_{M}. Since a ⊆ ω ⊆ X_{M} and ⟨X_{M}, τ_{M}⟩ is a weaker topology than ⟨X_{M}, τ ↾ X_{M}⟩, a obviously converges to x in ⟨X_{M}, τ_{M}⟩.

“ ⊇ ”: Let a ∈ V_{1} ∩ [ω]^{ω} and for some x ∈ X_{M}, a converges to x in ⟨X_{M}, τ_{M}⟩. We have to show that a is in I_{M}. By (i), we only have to prove that a is in I, i.e.

V [G] ⊧ “a converges in ⟨X, τ⟩”.

By the facts that M^{ω} ⊆ M and that ℙ is ccc , any nice ℙ_{M}-name for a subset of ω is in M. In particular, a is in M[G]. By the choice of θ and the elementarity, it is sufficient to prove that

M[G] ⊧ “a → x in ⟨X, τ⟩” .

Let U be in τ ∩M[G] containing x. We have to show that a is almost contained in U. There exists Ẏ ∈ M such that U = val(Ẏ , G), we can assume 1 ⊩_{ℙ} “ Ẏ ∈ τ”, hence Ẏ_{M} ∈ dom(τ̇_{M}). Notice that U ∩ X_{M} = U ∩ M = Y_{M} is in τ_{M}. Therefore a is almost contained in U ∩ X_{M} therefore in U. □

7.3. Proof

Assume that q ⊩ “ ṗ is a P-point”. We can assume that q is the empty function, i.e. q = 1 ( otherwise we can work instead with ℙ_{q} = {r ∈ ℙ : r ≤ q} which is isomorphic to ℙ). Therefore we have

1 ⊩ “ṗ is a P-point and ω ∪ {ṗ} is a subspace of regular pseudoradial space ⟨X, τ̇⟩”.

Suppose θ, M, and I are obtained as in the previous discussions and G is a ℙgeneric filter over V. Let a ∈ V [G] be a Cohen real over V_{1} such that a is in p. Since, ⟨X, τ⟩ is regular, we can take two open neighbourhoods of p , say W and U such that cl(W) ⊆ U and a = U ∩ ω. Let b = W ∩ ω. Given any I in I_{M}, let x be in X_{M} such that I converges to x. By (i) of the Fact 7.2.2, I also converges

98 7. PSEUDORADIALNESS IN THE COHEN MODEL

to x in ⟨X, τ⟩. If b ∩ I is infinite , then x is in cl(W) ⊆ U. Therefore I is almost contained in a. This contradicts the fact that a is a Cohen real over V_{1}. Thus we proved that for each I in I_{M} , b ∩ I is finite. Let τ̇, İ, ȧ, U̇, Ẇ , ḃ and ṗ denote both the ℙ-names and ℙ_{2}-names for the corresponding objects in V [G]. Choose an r ∈ G_{2}, such that r forces the above fact over V_{1}.

Now work in V_{1}, let κ be a sufficiently large regular cardinal, let N be a countable submodel of H_{κ} such that N contains {τ̇, İ, ȧ, U̇, Ẇ , ḃ, ṗ, λ}. By (iii) of Fact 7.2.1, p_{M} is a P-point. We can take A in p_{M} such that for each B ∈ N ∩ p_{M}, A ⊆^{∗} B. Since in V [G], A is in p and ⟨X, τ⟩ is pseudoradial, there exists I ∈ I such that I ⊂ A. Notice that A ∈ M[G], and , according to [She82], M[G] is an elementary submodel of H_{θ}^{V [G]}. By elementarity, we can assume I ∈ M[G]. Hence I ∈ V_{1}. Therefore , by (i) of Fact 7.2.2. I ∈ I_{M}. By the choice of r, r ⊩_{ℙ_{2}} “ ḃ ∩ I is finite ”, let s ∈ ℙ_{2} and m ∈ ω, such that r ≥ s ⊩_{ℙ_{2}} “ ḃ ∩ I ⊆ m”. Claim: s ∩ N ⊩_{ℙ_{2}} “ ḃ ∩ I ⊆ m”. Otherwise, let k ∈ I \ m, such that, there is t ≤ s ∩ N, t ⊩_{ℙ_{2}} “ k ∈ ḃ”. Since k, s ∩ N and ḃ are in N, we can assume t ∈ N. But then t and s are compatible while they force the contradicting statements. Therefore we can assume s ∈ N. Let Z = {n : ∃t ≤ s t ⊩_{ℙ_{2}} “ n ∈ ḃ”} ∈ N. Since Z ⊇ b, Z ∈ p_{M}. Now I ⊆ A ⊆^{∗} Z, take k ∈ I ∩ Z \ m and t ≤ s such that t ⊩_{ℙ_{2}} “ k ∈ ḃ”. This contradicts the fact that s ⊩_{ℙ_{2}} “ ḃ ∩ I ⊆ m”. □

Remark: In a similar way, we can prove that in V^{Fn(ω_{2},2)}, for every ultrafilter p on ω, if p is a P -point limit of a sequence of P -points, then the space ω ∪ {p} can not be embedded in any regular pseudoradial space.

CHAPTER 8 Pseudoradialness for a Special Ultrafilter

In this chapter, we will show that if there exists a reflecting self-indexed stationary set on [ω_{2}]^{ω} and 𝔠 = ω_{2}, then for a special ultrafilter p constructed in chapter 9, ω ∪ {p} can be embedded into a zero-dimensional Hausdorff pseudoradial space. We will also prove that it is consistent that 𝔠 = ω_{2} while there is such a stationary set. We conclude that there is a forcing extension in which, for every P-point p, ω ∪ {p} can not be embedded in any regular pseudoradial space while there is an ultrafilter q on ω such that ω ∪ {q} is a subspace of a zero-dimensional Hausdorff pseudoradial space. We will have a detailed discussion for the special ultrafilter in the next chapter.

8.1. Reflecting Self-Indexed Stationary Set

A sequence {S_{α} : α < ω_{2}} is a reflecting self-indexed stationary set on [ω_{2}]^{ω} if for any λ < ω_{2} with cf(λ) = ω_{1}, {S_{ξ} : ξ < λ and sup S_{ξ} = ξ} is stationary on [λ]^{ω}.

The reflecting self-indexed stationary set is very similar to the notion of stationary coding set defined in [Zwi84].

Lemma 8.1.1. Let V ⊧ CH and ℙ = {p : p is countable functions such that dom(p) ⊆ ω_{2} and p(α) is a countable subset of α for α < ω_{2}}. Let ℙ have the inverse inclusion as its partial order. Then in V^{ℙ}, there is a reflecting self-indexed stationary set on [ω_{2}]^{ω} and CH is true.

Proof: First of all, ℙ is countably closed, hence CH preserves in the forcing extension. ℙ is also ℵ_{2}-cc, whence ℵ_{2} is preserved.

If we replace ω_{2} by ω_{1} in the definition of ℙ, then it is well-known that we get a diamond sequence. This is not much different.

Indeed, for ℙ-generic G, let S_{α} = G(α). Suppose that p_{0} ∈ ℙ, λ < ω_{2} has uncountable cofinality and Ċ be a name of a cub subset of [λ]^{ω} such that p_{0} ⊩ “ Ċ is cub on [λ]^{ω}”. Since ℙ is countably closed, it is straightforward to define an increasing sequence {C_{n} : n ∈ ω} in [λ]^{ω} and a descending sequence {p_{n} : n ∈ ω} in ℙ such that

(1) for each n, p_{n+1} ⊩ “ C_{n} ∈ Ċ”,

(2) for each n, dom p_{n} ∩ λ ⊆ C_{n} and C_{n} ∪ {sup C_{n} + 1} ⊆ dom p_{n+1}. Let p_{ω} = ⋃_{n∈ω} p_{n} and C = ⋃_{n∈ω}. Obviously p_{ω} ⊩ “ C ∈ Ċ” and α = sup C is a limit ordinal which is not in dom p_{ω}. Extend p_{ω} to some p such that p(α) = C. Then p ⊩ “ S_{α} = C ∈ Ċ and α = sup S_{α}”. □

Lemma 8.1.2. Reflecting self-indexed stationary set is preserved by ccc forcing

Proof: It is well known that after ccc forcing every cub subset of ω_{1} contains a ground model cub. The same proof gives that this also holds for cub subsets of [λ]^{ω} for each λ < ω_{2}. □

Remark: S. Todorcěvić informed us that the existence of a reflecting selfindexed stationary set follows from the existence of the ρ-function in [Bek91]. Indeed, one can easily produce such a stationary set directly from a □-sequence (See [Dow]).

8.2. A SPECIAL SPACE ω ∪ {P } 101

In the next section we show that with the presence of a reflecting self-indexed stationary set one can embed ω ∪ {p} into a zero-dimensional pseudoradial space for a special ultrafilter p which will be constructed in chapter 9.

8.2. A Special Space ω ∪ {p}

Definition 8.2.1 ( [BS89]). Given a family F of subsets of some set S, a family I is called a refinement of F if for each F in F , there is an I in I such that I ⊆ F. If I is an almost disjoint family we say I is an almost disjoint refinement of F.

The hypothesized ultrafilter in the following proposition is interesting enough for a separate discussion. We will give the proof in chapter 9.

Proposition 8.2.2. There is an ultrafilter p on ω which has an almost disjoint refinement I and a base B such that

(i) for each subset I′ ∈ [I]^{<𝔠} there is a member of p which is almost disjoint with each member of I′, and

(ii) for each B ∈ B and I ∈ I either B ∩ I is finite or B almost contains I and I is almost disjoint with all but finitely many members of B.

We have the following results about this ultrafilter.

Theorem 8.2.3. Suppose p is the ultrafilter stated in the Proposition 8.2.2. If 𝔠 = ω_{2} and there is a reflecting self-indexed stationary set in [ω_{2}]^{ω}, then the space ω ∪ {p} can be embedded in a zero-dimensional pseudoradial space.

Before we start to prove above theorem, we prove the following corollary which is one of our main results about the pseudoradial spaces.

102 8. PSEUDORADIALNESS FOR A SPECIAL ULTRAFILTER

Corollary 8.2.4. It is consistent that, for every P-point p, ω ∪ {p} can not be embedded into any regular pseudoradial space while there is an ultrafilter p on ω such that ω ∪ {p} is a subspace of a zero-dimensional pseudoradial space.

Proof: Let V be the model obtained through Lemma 8.1.1. Let Q = Fn (ω_{2}, 2). By Theorem 7.1.1, in V^{Q}, for every P-point p, ω ∪ {p} can not be embedded into any regular pseudoradial space.

On the other hand, by Lemma 8.1.2, in V^{Q}, there is a reflecting self-indexed stationary set on [ω_{2}]^{ω} and 𝔠 is ω_{2}. Therefore, by Theorem 8.2.3, there is an ultrafilter p on ω such that ω∪{p} is a subspace of a zero-dimensional pseudoradial space. □

We will need the following Lemma in the proof of Theorem 8.2.3.

Lemma 8.2.5. Suppose R is a subset of space X, x ∈ X, and U is a local subbase at x, i.e. the intersections of finitely many members of U constitute a neighbourhood base at x. Let κ be any cardinal number such that the following are true,

(i) for all U′ ∈ [U]^{<κ}, R ∩ ⋂ U′ is not empty ; (ii) for all y ∈ R, {U ∈ U : y ∉ U} has cardinality at most κ,

Then there is a sequence {y_{α} : α < κ} in R convergent to x.

Proof: For each y ∈ R, let {U_{α}^{y} : α < κ} enumerate, with possible repetition, the set {U ∈ U : y ∉ U}. It is straightforward to define, by induction, a sequence {y_{α} : α < κ} such that for all α < κ

y_{α} ∈ R ∩ ⋂{U_{η}^{y_{ξ}}: η < α, ξ < α} (∗)

We show that {y_{α}}_{α<κ} is convergent to x by showing that each member of U contains a tail of the sequence. Suppose U ∈ U. We have to show that there is a

8.2. A SPECIAL SPACE ω ∪ {P } 103

α < κ, such that , for all β ∈ κ, β > α implies y_{β} ∈ U. If {y_{α} : α < κ} is contained in U, we are done. Otherwise, choose any ξ such that y_{ξ} ∉ U. Take η < κ such that U = U_{η}^{y_{ξ}}. Let α = max{η, ξ}, then, for any β > α, by (∗), y_{β} ∈ U. We are done. □

The rest of the section is to prove Theorem 8.2.3

Proof of Theorem 8.2.3 : We first fix a reflecting self-indexed stationary set {S_{α}: α ∈ ω_{2}} in [ω_{2}]^{ω}. Let p = {A_{α}: α < ω_{2}} be the ultrafilter in Proposition 8.2.2. Let {I_{A}: A ∈ p} be the almost disjoint refinement of p such that I_{A} ⊆ A and B is the base with the properties in Proposition 8.2.2. By induction we can define an almost disjoint family {a_{α}: α < ω_{2}} and a base {F_{α}: α < ω_{2}} of p such that

(i) for all α, a_{α} ⊆ F_{α} ⊆ A_{α}, (ii) for all α , β α < β implies a_{α} ∩ F_{β} is finite, (iii) for all α there are only finitely many β such that a_{α} ∩ F_{β} is infinite and if

a_{α} ∩ F_{β} is infinite, then a_{α} ⊆ F_{β}, (iv) for all λ if cf(λ) is uncountable, then {F_{α}: α < λ} is a filter base. We construct {a_{α}, F_{α}: α < ω_{1}η} by induction on η. At stage η, the only non

trivial case is when η is a successor ordinal. Suppose η = ξ + 1. We define a_{ω_{1}ξ+α} and F_{ω_{1}ξ+α} for α ∈ ω_{1} by induction on ω_{1}. Let f: ω_{1} → [ω_{1}ξ + ω_{1}]^{<ω} be an onto mapping such that f(α) ⊆ ω_{1}ξ + α for each α ∈ ω_{1}. At stage α, take a B in B such that B is almost disjoint with a_{ζ} for each ζ < ω_{1}ξ + α. Now simply define F_{ω_{1}ξ+α} be A_{ω_{1}ξ+α} ∩ B ∩ ⋂{F_{ζ}: ζ ∈ f(α)} and let a_{ω_{1}ξ+α} to be I_{F_{ω_{1}ξ+α}}.

Let H = {b ⊆ ω: (∀α ∈ ω_{2})(a_{α} ⊆^{∗} b)}. Let A be the subalgebra of P(ω) generated by [ω]^{<ω} ∪ {a_{α}, F_{α}: α ∈ ω_{2}} ∪ H

We will define a subspace X of the Stone space St(A) such that X contains a copy of ω ∪ {p} and X is pseudoradial. X will be of the form ω ∪ {x_{α} : α ∈ Γ},

104 8. PSEUDORADIALNESS FOR A SPECIAL ULTRAFILTER

where Γ is a subset of ω_{2} + 1 and each x_{α} is an ultrafilter of A. We specify the set Γ and x_{α} for α ∈ Γ as follows.

For convenience, we define a subset x_{α}^{+} of {F_{γ} : γ < ω_{2}} for each α ∈ ω_{2}. For successor ordinal α + 1, x_{α+1}^{+} is the finite of F_{β}’s which contains a_{α}. For limit α with cf(α) = ω, x_{α}^{+} is {F_{γ}: γ ∈ S_{α}}, where S_{α} is from the reflecting self-indexed stationary set. For limit α with cf(α) = ω_{1}, x_{α}^{+} is {F_{γ}: γ < α}. We denote the set {ω \ F_{β} : F_{β} ∉ x_{α}^{+}} by x_{α}^{−}.

First of all, Γ contains all the successor ordinals and the ordinals with uncountable cofinality.

For each α, since no member of A splits a_{α}, the cofinite subsets of a_{α} generate an ultrafilter on A. Let x_{α+1} be this ultrafilter. The point x_{ω_{2}} will simply be the ultrafilter p ∩ A. Before we handle other ordinals, we prove the following.

Fact 8.2.6. If λ < ω_{2} has cofinality ω_{1}, then x_{λ}^{+} ∪ x_{λ}^{−} ∪ H is actually {F_{α}, ω \ F_{β}: α < λ and β ≥ λ} ∪ H and generates an ultrafilter x_{λ} on A. If α < ω_{2} has cofinality ω and x_{α}^{+} ∪ x_{α}^{−} ∪ H has the finite intersection property, then it generates an ultrafilter x_{α} on A.

Proof of Fact 8.2.6 If λ < ω_{2} has cofinality ω_{1}, then {F_{α}: α < λ} is a filter base and it is easy to see that {F_{α}, ω \ F_{β}: α < λ and β ≥ λ} ∪ H has the finite intersection property. To prove that it generates an ultrafilter, it is sufficient to show that for each β ∈ ω_{2}, ω \ a_{β} contains some member of the family. Indeed, if β < λ, then F_{β+1} is almost disjoint with a_{β} hence F_{β+1} ⊆^{∗} ω \ a_{β}. If β ≥ λ, then ω \ F_{β} is almost contained in ω \ a_{β} . The proof for the case when α has countable cofinality is virtually the same.

We resume the proof of the Theorem. For λ with cofinality ω_{1}, let λ be in Γ and x_{λ} be the ultrafilter defined Fact in 8.2.6. For an ordinal α with cf(α) = ω,

8.2. A SPECIAL SPACE ω ∪ {P } 105

α ∈ Γ if x_{α}^{+} ∪ x_{α}^{−} ∪ H generate a filter, we let x_{α} be the ultrafilter defined in Fact 8.2.6

By above discussion Γ is well defined and, for each α ∈ Γ, x_{α} is a point in the Stone space St(A). The following claim is trivial.

Claim 8.2.1. ω ∪ {x_{ω_{2}}} is homomorphic to ω ∪ {p}

For simplicity we will abuse the notation of Stone duality. If F ⊂ ω is a member of A, we also let F denote the set of the points of X which as an ultrafilter on A contain F as a member. Of course, F is clopen. Hence, to say x_{α} ∈ F is equivalent to saying F ∈ x_{α}.

Let X_{0} be all x_{α} with α a successor and X_{1} be all x_{α} with cf(α) = ω. Recall that x_{α+1} corresponds to a_{α}. Let X_{2} be all x_{α} such that α has uncountable cofinality.

Fact 8.2.7. The sets X_{1} ∪ X_{2} and X_{2} are closed in X. X_{2} is homomorphic to a subspace of ω_{2} +1, namely {α ≤ ω_{2} : cf(α) is uncountable } and so X_{2} is radial.

Proof of Fact 8.2.7: For the first statement of the claim, the only case that requires a proof is that a point x_{α} of X_{1} is not in the closure of X_{2}. If α is countable, then ω \ F_{α+1} is disjoint from X_{2}. On the other hand, if α is uncountable, then we can find γ < β < α, so that F_{β} ∈ x_{α}^{+} and F_{γ} ∉ x_{α}^{+}. Note that F_{β} \ F_{γ} is not a member of x_{λ} for any λ with uncountable cofinality. For the second statement, it is sufficient to see that, for any λ < ω_{2} with uncountably cofinality, the typical neighhourhood of x_{λ} in the subspace X_{2} is (F_{α} \F_{λ})∩X_{2} = {x_{ξ} ∈ X_{2} : α < ξ ≤ λ} for some α < λ. Similarly, a typical neighbourhood of x_{ω_{2}} is {x_{ξ} ∈ X_{2} : α < ξ ≤ ω_{2}}.

The following claim will finish the proof of Theorem 8.2.3.

Claim 8.2.2. X is a zero-dimensional pseudoradial space.

Proof of Claim 8.2.2: Since X is a subspace of a St(A), X is zero-dimensional. Suppose X is not pseudoradial. Let R be a radially closed subset of X which

106 8. PSEUDORADIALNESS FOR A SPECIAL ULTRAFILTER

is not closed. Let x_{α} ∈ cl R \ R and α is the minimal such index. Obviously x_{α} ∈ X_{1} ∪ X_{2} because other points have countable neighbourhood bases. We will prove x_{α} ∈ R to produce a contradiction. We will need the following fact.

Fact 8.2.8. x_{α} ∈ cl(R \ ω).

Proof of Fact 8.2.8: Suppose x_{α} ∉ cl(R \ ω). Then x_{α} ∈ cl(R ∩ ω) \ cl(R \ ω). Let U be a clopen neighbourhood of x_{α} such that U ∩ (R \ ω) = ∅. Therefore b = R ∩ U is a subset of ω. Since R is radially closed and U is closed , b is also radially closed. For any β ∈ ω_{2}, if a_{β} ∩ b is infinite, then x_{β+1} would be in b contradicting that b ⊆ ω. Hence, b is almost disjoint with each a_{β}, i.e. ω \ b is in H. Thus ω \ b is in the ultrafilter x_{α}, which contradicts that x_{α} is in the closure of b.

Since R \ ω is also radially closed in X, we can now assume R ∩ ω = ∅. By Fact 8.2.7, X_{2} is radial and closed so we can assume that R ∩ X_{2} = ∅ by again restricting to a neighbourhood of x_{α}. For the rest of the proof we assume that R ⊆ X_{0} ∪ X_{1}. Recall that x_{α} ∈ X_{1} ∪ X_{2}.

We claim that α ≠ ω_{2}. Indeed fix any M ≺ H(ω_{3}) such that |M| = ω_{1} and M contains all relevant sets such as R and X, and so that the cofinality of δ = M ∩ ω_{2} is uncountable. We show that x_{δ} is in the closure of R which will contradicts that R is assumed to be disjoint from X_{2} and that α is the minimal index of a limit point which is not in R. A typical neighbourhood of x_{δ} has the form H ∩ ⋂{F_{ξ} : ξ ∈ f} \ ⋃{F_{γ} : γ ∈ g}, where f ∈ [δ]^{<ω}, g ∈ [ω_{2} \ δ]^{<ω} and H ∈ H. However, ⋂{F_{ξ} : ξ ∈ f} is a neighbourhood of x_{ω_{2}} and is a member of M. Choose any β ∈ M such that x_{β} ∈ R ∩ ⋂{F_{ξ} : ξ ∈ f}. Clearly x_{β} ∉ F_{γ} for all γ > δ, hence x_{α} is in the above neighbourhood of x_{δ}.

Next we apply Lemma 8.2.5 to prove that there exists a sequence in R convergent to x_{α} and thus finish the proof.

8.2. A SPECIAL SPACE ω ∪ {P } 107

Case 1. cf(α) = ω

To apply Lemma 8.2.5, let κ be ω, U be x_{α}^{+} ∪ x_{α}^{−} ∪ H, and x be x_{α}. U is a local subbase of x. We show x, R and U satisfies the conditions in Lemma 8.2.5. Since x ∈ cl(R), the condition (i) is trivial. To verify the condition (ii), let x_{ξ} ∈ R. Then cf(ξ) ≤ ω. It is easy to see that

{U ∈ U: x_{ξ} ∉ U} ⊆ x_{α}^{+} ∪ {ω \ F: F ∈ x_{ξ}^{+}}.

But both x_{α}^{+} and x_{β}^{+} are countable. Therefore {U ∈ U: x_{ξ} ∉ U} is countable.

Case 2. cf(α) = ω_{1}.

We let κ be ω_{1}, and U be x_{α}^{+} ∪ x_{α}^{−} ∪ H, and x be x_{α}. Again U is a local subbase at x. The second condition of Lemma 8.2.5 is verified as in the Case 1. We are left to show that for any U′ ∈ [U]^{ω},

R ∩ ⋂ U′ ≠ ∅.

Since for each H in H, R ⊆ X \ ω ⊆ H, we can assume U′ = {F_{α_{i}}, ω \ F_{β_{i}} : i ∈ ω}. We take a countable elementary submodel M of H(ω_{3}), such that {X, R, U′, α} ⊆ M and M ∩ α = S_{γ} and γ = sup(M ∩ α), where S_{γ} is from the the reflecting self-indexed stationary set. Since S_{γ} = M ∩α and {F_{γ} : γ < α} is a filter base, x_{γ}^{+} is a filter base. It follows that x_{γ}^{+} ∪ x_{γ}^{−} ∪ H has the finite intersection property. To see this, it suffices to check that for H ∈ H, β ∈ S_{γ} and a finite subset g disjoint from S_{γ} the set H ∩ F_{β} \ ⋃_{δ∈g} F_{δ} is infinite. However, β ∈ M, the finite subset f of δ’s with a_{β} ⊆ F_{δ} is contained in M ∩ α = S_{γ}. Therefore, a_{β} ⊆ H ∩ F_{β} \ ⋃_{δ∈g} F_{δ}. Therefore it generates an ultrafilter on A. Therefore γ ∈ Γ, i.e. x_{γ} is a point of our X. Next we show that x_{γ} ∈ R ∩ ⋂ U′.

For i ∈ ω, since, U′ ∈ M, α_{i} ∈ M ∩ α = S_{γ}. Thus, F_{α_{i}} ∈ x_{γ}^{+} while F_{β_{i}} ∉ x_{γ}^{+} because β_{i} ≥ α > γ. This proves that x_{γ} ∈ ⋂ U′.

108 8. PSEUDORADIALNESS FOR A SPECIAL ULTRAFILTER

To prove x_{γ} ∈ R, it is sufficient to prove that x_{γ} ∈ cl(R) because α is the minimal index such that x_{α} ∈ cl(R) \ R. Take a neighbourhood W of x_{γ}, we have to show R ∩ W ≠ ∅. We can assume W = ⋂ F \ ⋃ G, where F and G are finite subsets of x_{γ}^{+} and {F_{γ} : γ ∈ ω_{2}} \ x_{γ}^{+} respectively. Obviously F and G ∩M are members of M. If F_{ξ} ∈ G ∩M, then by the definition of x_{γ}^{+} and the fact S_{γ} = M ∩α, ξ ≥ α. Therefore, (R∩⋂ F)\⋃(G ∩M) is a neighbourhood of x_{α} and it is in M. Take a x_{ξ} ∈ M ∩ R ∩ ⋂ F \ ⋃(G ∩ M). Since x_{ξ}^{+} is a countable member of M, x_{ξ}^{+} ⊆ M, therefore G \ M is disjoint with x_{ξ}^{+}. Therefore x_{ξ} ∉ ⋃(G \ M) and x_{ξ} ∈ R ∩ ⋂ F \ ⋃ G = R ∩ W. We are done. □

We finish this chapterwith the following question.

Question 8.2.9. Does MA or 𝔭 = 𝔠 implies that for each ultrafilter p on ω, ω ∪ {p} is a subspace of a regular pseudo-radial space?

By Lemma 6.2.1, the answer is yes for the non P-point p.

CHAPTER 9 Another Ultrafilter On ω

In this chapter, we will give the proof of Proposition 8.2.2. We will restate here Proposition 8.2.2 as Proposition 9.0.1. It will follow that the ultrafilter in Proposition 9.0.1 has a strong Noetherian base. Therefore we give a positive answer to Malykin’s question.

Recall the following definition of refinement for some family (see page 101). Given a family F of subsets of some set S. A family I is called a refinement of F if for each F in F , there is an I in I such that I ⊆ F. If I is an almost disjoint family we say I is an almost disjoint refinement of F.

There has been many interesting results about almost disjoint refinements of filters on ω, see section 4 of [BS89] for a survey. Proposition 9.0.1 gives an ultrafilter which has an almost disjoint refinement that does not follow from the constructions in [BS89].

A Noetherian space is a space with a base B such that for each U ∈ B, the set {V ∈ B: U ⊆ V } is finite (see [Mal81]). A base B for an ultrafilter p on ω is a Noetherian base (strong Noetherian base ) if for each A ∈ B, the set {B ∈ B: A ⊆ B} ({B ∈ B: A ⊆^{∗} B}) is finite. Malykin has asked if there exists an ultrafilter with a Noetherian base. The base B in the ultrafilter p in

110 9. ANOTHER ULTRAFILTER ON ω

the following proposition is actually strong Noetherian. Thus we give a positive answer to Malykin’s question. In addition the property (i) of the filter p is quite unexpected.

Proposition 9.0.1. There is an ultrafilter p on ω which has an almost disjoint refinement I and a base B such that

(i) for each subset I′ ∈ [I]^{<𝔠} there is a member of p which is almost disjoint with each member of I′, and

(ii) for each B ∈ B and I ∈ I either B ∩ I is finite or B almost contains I and I is almost disjoint with all but finitely many members of B.

Remark 9.0.2. It is straightforward to see that the property (ii) in Proposition 9.0.1 implies the base B is a strong Noetherian base. A careful analysis of the construction in [BS89] will reveal that the almost disjoint refinements constructed by that method will fail to satisfy property (ii) for any base, i.e. for any base, there will be some member of the refinement almost contained in infinitely many members of the base.

The following Lemmas give the proof of the Proposition 9.0.1.

Lemma 9.0.3. If F is a filter base on ω such that, for each F ∈ F , there is an infinite I ⊆ F such that {F′ ∈ F : I ∩ F′is infinite} is finite, then there is an ultrafilter p on ω which extends F and satisfies the following property

\begin{matrix}F or each F ∈ p , there exists I ∈ [F]^{ω} \\ such that \{F′ ∈ F : I ∩ F′is infinite\} is finite \end{matrix} (†)

Proof: Let [ω]^{ω} = {Y_{α} : α < 𝔠}. We will build inductively an increasing sequence {F_{α} : α ≤ 𝔠} of filters on ω such that

9. ANOTHER ULTRAFILTER ON ω 111

(i) F ⊆ F_{0}

(ii) For each α ≤ 𝔠, F_{α} satisfies (†)( with p taken to be F_{α}).
(iii) For each α ∈ 𝔠, either Y_{α} ∈ F_{α+1} or ω \ Y_{α} ∈ F_{α+1},

Let F_{0} be the filter generated by F. Suppose we have constructed {F_{β} : β < α} which satisfies (i)-(iii). If α is a limit ordinal, let F_{α} = ⋃_{β<α} F_{β}. If α is the successor of β, we have to build F_{β+1} which extends F_{β} and satisfies (ii) and (iii).

We say a set Y ⊆ ω is F_{β}-acceptable if for each F ∈ F_{β} there exists an I ∈ [F ∩ Y]^{ω} such that {F′ ∈ F : F′ ∩ I is infinite} is finite.

Claim 9.0.3. There exists an Y ∈ {Y_{β}, ω \ Y_{β}} which is F_{β}-acceptable.

Proof of Claim 9.0.3: Suppose not. Then there exist F_{1} and F_{2} in F_{β} such that for each I ∈ [ω]^{ω}, if I ⊆ F_{1} ∩ Y_{β} or I ⊆ F_{2} \ Y_{β}, then {F′ ∈ F : F′ ∩ I is infinite} is infinite. Since F_{β} satisfies (†), there exists an I_{0} ⊆ F_{1} ∩ F_{2} such that {F′ ∈ F : F′ ∩ I_{0} is infinite} is finite. Without loss of generality, we can assume I_{0} ∩ Y_{β} is infinite. Then I_{0} ∩ Y_{β} ∈ [ω]^{ω} and I_{0} ∩ Y_{β} ⊆ F_{1} ∩ Y_{β} and {F′ ∈ F : F′ ∩ I ∩ Y_{β} is infinite} is finite. This contradicts the choice of F_{1}.

Now take an Y ∈ {Y_{β}, ω \ Y_{β}} which is F_{β}-acceptable. Let F_{β+1} be the filter generated by F_{β} ∪ {Y }. Obviously, F_{β+1} satisfies (ii) and (iii). Finally, we let p = F_{𝔠}. It is easy see that p works. □

Lemma 9.0.4. Under the condition of the Lemma 9.0.3, if |F| = 𝔠, and p is obtained from Lemma 9.0.3, then Proposition 9.0.1 holds for p.

Proof: Let p = {a_{α} : α < 𝔠} and F = {d_{α} : α < 𝔠}, such that if α ≠ β, then d_{α} ≠ d_{β}. For each F ∈ [ω]^{ω}, denote by I_{F} the set {I ∈ [F]^{ω} : {F′ ∈ F : F′ ∩ I is infinite} is finite}. Thus Lemma 9.0.3 says that for each member F of p, I_{F} is not empty. We will construct, by induction, {b_{α} : α < 𝔠} ⊆ p and {I_{α} : α < 𝔠} such that

112 9. ANOTHER ULTRAFILTER ON ω

(i) I_{α} ⊆ b_{α} ⊆ a_{α} ∩ d_{α}, (ii) ∀ξ < α, I_{ξ} ∩ b_{α} is finite, (iii) ∀α < 𝔠, {ξ < α : I_{α} ⊆ b_{ξ}} = {ξ < α : I_{α} ∩ b_{ξ} is infinite } is finite, (iv) ∀α < 𝔠, {F ∈ F : I_{α} ∩ F is infinite } is finite.

Suppose first that we have constructed the above sequences. Let I = {I_{α} : α < 𝔠} and B = {b_{α} : α < 𝔠}. Obviously, condition (i) and (ii) ensures that B is a base of p and I is an almost disjoint family. It is also straightforward to see that I and B satisfies the rest properties of Proposition 9.0.1. Let b_{0} be a_{0} ∩ d_{0} and I_{0} be any element of I_{b_{0}}. At stage α, we have to find b_{α} and I_{α} which satisfies (i) to (iv). By condition (iv) and the fact that |F| = 𝔠, we can find F ∈ F such that F ∩ I_{ξ} is finite for each ξ < α. Let b_{α} = a_{α} ∩ d_{α} ∩ F. Take any I ∈ I_{b_{α}}. Since for each ξ ∈ α, b_{ξ} ∩ I ⊆ d_{ξ} ∩ I, there are only finitely many ξ such that b_{ξ} ∩ I is infinite. It is easy to find a subset I_{α} of I such that, for each of those finitely many b_{ξ}’s which has infinite intersection with I, either it contains I_{α} or it is disjoint from I_{α}. □

Lemma 9.0.5. There exists a filter base, F, on ω such that |F| = 𝔠 and for each F ∈ F there exists an infinite set I ⊆ F such that {F′ ∈ F : F′ ∩ I is infinite} is finite.

Proof: We will identify ω with the set P = ⋃_{n∈ω} P(^{n}2). For each E ∈ [^{ω}2]^{<ω}, let X_{E} = ⋃_{n∈ω}{A ⊆ ^{n}2: E ↾n ⊆ A} and I_{E} = {E ↾n: n ∈ ω}.

For finite subsets E and E′, it is easy to see that I_{E} ⊆ X_{E} and X_{E∪E′} = X_{E} ∩ X_{E′}. Moreover, if E′ ⊈ E, then, there is an m such that for all n > m, E′ ↾ n ⊈ E ↾ n. Therefore, I_{E} ∩ X_{E′} ⊆ {E ↾ n: n ≤ m} is finite. Let F = {X_{E} ⊆ P : E ∈ [^{ω}2]^{<ω}}. Clearly F is a filter base on P with cardinality 𝔠 and for each X_{E} in F, I_{E} is an infinite subset of X_{E} which is almost disjoint from all but finitely many members of F. □

9. ANOTHER ULTRAFILTER ON ω 113

Remark 9.0.6. If we take p as in Proposition 9.0.1, the space ω ∪ I ∪ {p} with the usual topology is quite interesting: the first property ensures that it has tightness 𝔠 while the second property says that it is a 0-dimensional scattered space.

114 9. ANOTHER ULTRAFILTER ON ω

CHAPTER 10 Conclusions

In this chapter, we will try to give a summary of our results. As we have almost listed all the results in Chapter 1, we will rather focus on some aspects of our work which we think deserve further mention. We will also discuss the potential topics arising from the present work for further research. Finally we will give a list of open questions at the end of this chapter.

In the study of the Boolean algebra P(ω)/ Fin. One important technique frequently used in the iterative construction of models is that one creates a gap in P(ω)/ Fin in some intermediate stage and then freezes the gap in the further construction, i.e. no set will allowed to fill the gap in any extension of. For example, Shelah, used this technique in [She82] to construct a model in which there are only trivial automorphisms of the Boolean algebra P(ω)/ Fin. This result but even more so, the techniques and the idea involved have been widely and numerously applied. There are also other standard techniques in handling gaps in P(ω)/ Fin which have been widely exploited in both set theory and set-theoretic Topology. But progress has been much slower for gaps in P(ω)/I when I is a more general ideal. The gap structure is simply much more complicated. We believe that this is one of the reasons that there has been no single progress in

10. CONCLUSIONS 115

solving Baumgartner and Weese’s problem in twelve years.

We invented a technique to produce simultaneously infinitely many gaps in P(ω)/ Fin which we could relate to an unfilled gap in P(ω)/I_{M}. Our technique will ensure that these gaps will remained unfilled in any meaningful extension of the present model. In the construction of the counter example to Baumgartner and Weese’s question, we were adopting Shelah’s idea to prove that the algebra 𝔅 is not representable. But all the argument involve gaps in Shelah’s construction had to be carefully modified. . This technique helped us to overcome the difficulty to find a gap in our partitioner algebra. We think it is because of this technique that our proof is not merely an improvement of Shelah’s proof rather it in some sense shows how to transplant Shelah’s argument to a much broader area which is was not done before. We believe that this is the first technique in producing an unfilled gap in P(ω)/I when I is as complicated as our I_{M} (See Chapter 2). More to the point, this technique should have more applications in the study of representable algebras as well as the Boolean algebra of the form P(ω)/I. We will investigate if our technique can lead to the generalizations of Shelah’s results to many more Boolean algebras of the form P(ω)/I which is an important, interesting but largely unexplored area. In particular, we are optimistic that we can construct representable algebra which has only trivial automorphisms.

An immediate question following from our proof is that if the large cardinal is needed. In actual fact, the only place it is now necessary is the application of Chang’s conjecture in the theorem about strong chains in [ω_{1}]^{ω_{1}}. At other places the real strength we are able to restrict the application PFA to the posets with cardinality not bigger than ℵ_{2} which is consistent relatively to ZFC. However, we currently do not know how to prove the fundamental theorem (Theorem 3.0.1) without using Chang’s Conjecture.

116 10. CONCLUSIONS

As pointed out in the introduction the spaces of the form ω∪{p} are important in the study of sub pseudoradialness. Our results about the pseudoradial spaces shows that one can not expect to decide generally that, for ultrafilters p ∈ ω^{∗}, whether or not the space ω ∪ {p} is sub pseudoradial. By Theorem 6.2.2, we know it is impossible to find an ultrafilter p on ω, in ZFC, such that ω ∪ {p} is not a subspace of any regular pseudoradial spaces. It is still not known if, in ZFC, there exists an ultrafilter p on ω such that ω ∪ {p} is a subspace of a regular pseudoradial space. However, we think it is very likely that the ultrafilter constructed in Chapter 9 could be such an ultrafilter. At least our proof in Chapter 8 suggests that 𝔠 = ω_{2} could imply that ω∪{p} will be a subspace of a pseudoradial spaces. We wish we can resolve this problem.

Two other interesting things arising from our study of pseudoradial spaces are Lemma 8.2.5 and the notion of reflecting self-indexed stationary set on ω_{2}.

Two extreme cases for the conditions in the Lemma 8.2.5 are: (1) when U ∩ R is the family of co-finite subsets of R; (2) U ∩ R are well-ordered non empty set. These two cases are actually most frequently used to guarantee a convergent sequence to x. Lemma 8.2.5 offers an easy tool to find a convergent sequence in the situation which is neither as good as (1) nor as good as (2) but somehow a combination of them. It is quite surprising that Lemma 8.2.5 is so easy to prove but we believe the idea is quite new. We hope it will be really helpful in the studies of topological spaces where the question of existence of certain convergent sequence is involved.

The notion of a reflecting self-indexed stationary set on ω_{2} has been very interesting to quite a number of set theorists after we introduced it. Other applications have already been found (see [Dow]). We believe that it will also be an useful tool when the argument involves reflecting certain properties of a structure on ω_{2} to lower ordinals. It may be a theorem of ZFC that there exists a reflecting self-

10. CONCLUSIONS 117

indexed stationary set on ω_{2}. Certainly we can prove the existence in a very wide variety of models and that a large cardinal must be involved in any non-existence result. Of course we do not know if, in the Cohen model, we can embed ω ∪ {p} into a regular pseudoradial space, for the ultrafilter p constructed in Chapter 9, we need that there is a reflecting self-indexed stationary set on ω_{2} in the ground model. We hope we can answer whether ZFC ensures that there is a a reflecting self-indexed stationary set on ω_{2}.

Finally we give a list of problems 1. Is the large cardinal strength is really necessary in our construction of the non representable algebra 𝔅?

2. Does PFA or the existence of a supercompact cardinal implies that, for each Boolean algebra of the form P(ω)/I, each automorphism is basically induced by a bijection from a co-finite subset of ω to a co-finite subset of ω?

3. Does the reflecting self-indexed stationary set on ω_{2} exists in ZFC?

4. Does MA imply that, for each p ∈ ω^{∗}, the space ω ∪ {p} is a subspace of regular pseudoradial space?

5. Is there a model of MA and ¬ CH in which for each p ∈ ω^{∗}, the space ω ∪ {p} is a subspace of regular pseudoradial space. We expect the answer to this question is yes.

6. Does there exist, in ZFC, an ultrafilter p on ω such that ω ∪ {p} is a subspace of regular pseudoradial spaces?

7. Does 𝔠 = ω_{2} implies that for the ultrafilter p constructed in Chapter 9, ω ∪ {p} is a subspace of a regular pseudoradial spaces?

118 10. CONCLUSIONS

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Index

E_{1} ⊥ E_{2}, 11 S_{0}^{2}, 15 derived, 30 ♢(E), 15 λ-directed closed, 13 PFA^{+}, 14

ad family , 10

ccc, 13 Chang’s Conjecture, 15 co-infinite, 10 coherent sequence, 11 CSQ, 28 cub, 13

gap, 19 GCSQ, 28 good coherent sequence, 28

Hausdorff-Luzin type family, 19

independent, 17

mad family, 10

partitioner, 11, 21 partitioner algebra, 21 PFA, 14 proper, 14 pseudoradial, 89

radial, 89 RCSQ, 28 reasonable sequence, 28 representable, 22

strong chain, 16

uniformize, 11