Canonical Ramsey Theory
on Polish Spaces
Vladimir Kanovei
Institute for the Information Transmission Problems
Moscow Institute for Transport Engineering
Marcin Sabok
Academy of Sciences, Poland
Jindřich Zapletal
University of Florida
Academy of Sciences, Czech Republic
Marcin Sabok was supported through AIP project MEB051006.
Jindřich Zapletal was partially supported through the following grants: AIP
project MEB051006, AIP project MEB060909, Institutional Research Plan No.
AV0Z10190503 and grant IAA100190902 of Grant Agency of the Academy of
Sciences of the Czech Republic, NSF grant DMS 0801114, and INFTY, the
ESF research networking programme.
vii
Contents
1 Introduction
1.1 Motivation . . . .
1.2 Outline of results .
1.3 Outline of concepts
1.4 Navigation . . . . .
1.5 Notation . . . . . .
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1
1
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2 Background facts
2.1 Descriptive set theory . . . . .
2.2 Invariant descriptive set theory
2.3 Forcing . . . . . . . . . . . . .
2.4 Shoenfield absoluteness . . . .
2.5 Generic ultrapowers . . . . . .
2.6 Idealized forcing . . . . . . . .
2.7 Other tools . . . . . . . . . . .
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11
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3 Tools
3.1 Integer games connected with σ-ideals . . . . . .
3.2 The selection property . . . . . . . . . . . . . . .
3.3 A Silver-type dichotomy for a σ-ideal . . . . . . .
3.4 Equivalence relations and models of set theory .
3.4a The model V [x]E . . . . . . . . . . . . . .
3.4b The model V [[x]]E . . . . . . . . . . . . .
3.4c Comparison with the symmetric models of
3.5 Associated forcings . . . . . . . . . . . . . . . . .
3.5a The poset PIE . . . . . . . . . . . . . . . .
3.5b The reduced product posets . . . . . . . .
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ZF .
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29
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4 Classes of equivalence relations
4.1 Smooth equivalence relations . . . . . . . . . . .
4.2 Countable equivalence relations . . . . . . . . . .
4.3 Equivalences classifiable by countable structures
4.4 Orbit equivalence relations . . . . . . . . . . . . .
4.5 Hypersmooth equivalence relations . . . . . . . .
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73
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vi
5 Classes of σ-ideals
5.1 σ-ideals σ-generated by closed sets . . . . . . .
5.1a Calibrated ideals . . . . . . . . . . . . .
5.1b The σ-compact ideal and generalizations
5.1c The meager ideal . . . . . . . . . . . . .
5.2 Porosity ideals . . . . . . . . . . . . . . . . . .
5.3 Halpern-Läuchli forcing . . . . . . . . . . . . .
5.4 The E0 -ideal and its products . . . . . . . . . .
5.5 The E1 ideal . . . . . . . . . . . . . . . . . . .
5.5a The σ-ideals and their dichotomies . . .
5.5b Canonization results . . . . . . . . . . .
5.5c Borel functions on pairs . . . . . . . . .
5.6 The E2 ideal . . . . . . . . . . . . . . . . . . .
5.7 Products of perfect sets . . . . . . . . . . . . .
5.8 Silver cubes . . . . . . . . . . . . . . . . . . . .
5.9 Ramsey cubes . . . . . . . . . . . . . . . . . . .
5.10 Weak Milliken cubes . . . . . . . . . . . . . . .
5.11 Strong Milliken cubes . . . . . . . . . . . . . .
5.12 Products of superperfect sets . . . . . . . . . .
5.13 Laver forcing . . . . . . . . . . . . . . . . . . .
5.14 Fat tree forcings . . . . . . . . . . . . . . . . .
5.15 The Lebesgue null ideal . . . . . . . . . . . . .
5.16 Finite product . . . . . . . . . . . . . . . . . .
5.17 The countable support iteration ideals . . . . .
5.17a Iteration preliminaries . . . . . . . . . .
5.17b Weak Sacks property . . . . . . . . . . .
5.17c The preservation result . . . . . . . . .
5.17d A few special cases . . . . . . . . . . . .
CONTENTS
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89
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191
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199
Chapter 1
Introduction
1.1
Motivation
The Ramsey theory starts with a classical result:
Fact 1.1.1. For every partition of pairs of natural numbers into two classes
there is a homogeneous infinite set: a set a ⊂ ω such that all pairs of natural
numbers from a belong to the same class.
It is not difficult to generalize this result for partitions into any finite number
of classes. An attempt to generalize further, for partitions into infinitely many
classes, hits an obvious snag: every pair of natural numbers could fall into its
own class, and then certainly no infinite homogeneous set can exist for such a
partition. Still, there seems to be a certain measure of regularity in partitions
of pairs even into infinitely many classes. This is the beginning of canonical
Ramsey theory.
Fact 1.1.2 ([11]). For every equivalence relation E on pairs of natural numbers
there is an infinite homogeneous set: a set a ⊂ ω on which one of the following
happens:
1. pEq ↔ p = q for all pairs p, q ∈ [a]2 ;
2. pEq ↔ min(p) = min(q) for all pairs p, q ∈ [a]2 ;
3. pEq ↔ max(p) = max(q) for all pairs p, q ∈ [a]2 ;
4. pEq for all pairs p, q ∈ [a]2 .
In other words, there are four equivalence relations on pairs of natural numbers
such that any other equivalence can be canonized : made equal to one of the
four equivalences on the set [a]2 , where a ⊂ ω is judiciously chosen infinite set.
It is not difficult to see that the list of the four primal equivalence relations is
irredundant: it cannot be shortened for the purposes of this theorem. It is also
1
2
CHAPTER 1. INTRODUCTION
not difficult to see that the usual Ramsey theorems follow from the canonical
version.
Further generalizations of these results can be sought in several directions.
An exceptionally fruitful direction considers partitions and equivalences of substructures of a given finite or countable structure, such as in [38]. Another
direction seeks to find homogeneous sets of larger cardinalities. In set theory
with the axiom of choice, the search for uncountable homogeneous sets of arbitrary partitions leads to large cardinal axioms [25], and this is one of the central
concerns of modern set theory. A different approach will seek homogeneous
sets for partitions that have a certain measure of regularity, typically expressed
in terms of their descriptive set theoretic complexity in the context of Polish
spaces. This is the path this book takes. Consider the following classical result:
Fact 1.1.3 ([51]). For every partition [ω]ℵ0 = B0 ∪ B1 into two Borel pieces,
one of the pieces contains a set of the form [a]ℵ0 , where a ⊂ ω is some infinite
set.
Here, the space [ω]ℵ0 of all infinite subsets of natural numbers is considered
with the usual Polish topology which makes it homeomorphic to the space of
irrational numbers. This is the most influential example of a Ramsey theorem
on a Polish space. It deals with Borel partitions only as the Axiom of Choice can
be easily used to construct a partition with no homogeneous set of the requested
kind.
Are there any canonical Ramsey theorems on Polish spaces concerning sets
on which Borel equivalence relations can be canonized? A classical example of
such a theorem starts with an identification of Borel equivalence relations Eγ
on the space [ω]ℵ0 for every function γ : [ω]<ℵ0 → 2 (the exact statement and
definitions are stated in Section 5.9) and then proves
Fact 1.1.4 ([41, 36]). If f : [ω]ℵ0 → 2ω is a Borel function then there is γ and
an infinite set a ⊂ ω such that for all infinite sets b, c ⊂ a, f (b) = f (c) ↔ b Eγ c.
Thus, this theorem deals with smooth equivalence relations on the space [ω]ℵ0 ,
i.e. those equivalences E for which there is a Borel function f : [ω]ℵ0 → 2ω such
that b E c ↔ f (b) = f (c), and shows that such equivalence relations can be
canonized to a prescribed form on a Ramsey cube. Other similar results can be
found in the work of Otmar Spinas [57, 56, 34].
1.2
Outline of results
This book extends this line of research in two directions. Firstly, we consider
canonization properties of equivalence relations more complicated than smooth
in the sense of Borel reducibility complexity rating of equivalence relations [27,
14]. It turns out that distinct complexity classes of equivalence relations possess
various canonization properties, and most of the interest and difficulty lies in the
non-smooth cases. Secondly, we consider the task of canonizing the equivalence
1.2. OUTLINE OF RESULTS
3
relations on Borel sets which are large from the point of view of various σideals. Again, it turns out that σ-ideals commonly used in mathematical analysis
greatly differ in their canonization properties, and a close relationship with their
forcing properties, as described in [67], appears.
Given a σ-ideal I on a Polish space X, the strongest canonization result
one can obtain is the total canonization for analytic equivalence relations: for
every I-positive Borel set B ⊂ X and every analytic equivalence relation E on
B, there is an I-positive Borel set C ⊂ B consisting either only of pairwise
inequivalent elements, or only of pairwise equivalent elements.
Theorem 1.2.1. The following σ-ideals have the total canonization for analytic
equivalence relations:
1. (Theorem 5.1.6) all Π11 on Σ11 σ-ideals on compact metrizable spaces which
are σ-generated by compact sets and are calibrated;
2. (Corollary 5.2.14) all metric porosity σ-ideals on compact metric spaces;
3. (Solecki-Spinas [55], Theorem 5.1.15) the σ-ideal on ω ω σ-generated by
compact sets.
The class considered in the first item contains many σ-ideals commonly studied
in abstract analysis; the σ-ideal σ-generated by closed sets of measure zero,
the σ-ideal σ-generated by closed sets of uniqueness, or the σ-ideal of sets of
extended uniqueness are just three of an army of examples.
In all cases named above, we get an even stronger result modeled after
the classical Silver dichotomy for coanalytic equivalence relations: if B is an
I-positive Borel set and E is a Borel equivalence relation on B, either B decomposes into countably many equivalence classes and an I-small set, or it contains
a Borel I-positive subset consisting of pairwise inequivalent elements.
In most cases, such a strong canonization result is either not available or we
do not know how to prove it. One can then attempt to canonize only equivalence
relations of limited Borel reducibility complexity:
Theorem 1.2.2. The following σ-ideals have the total canonization for equivalence relations classifiable by countable structures:
1. (Corollary 4.3.10) the σ-ideal σ-generated by sets of continuity of a Borel
function;
2. (Corollary ??) Laver σ-ideal on ω ω ;
3. (Corollary 4.3.10) the σ-ideal σ-generated by sets of finite Hausdorff measure of a fixed dimension, on a compact metric space;
4. (Corollary 5.6.4) the σ-ideal of Borel sets on which the equivalence relation
E2 is classifiable by countable structures.
Theorem 1.2.3. The following σ-ideals have the total canonization for equivalence relations Borel reducible to equality modulo an analytic P -ideal on ω:
4
CHAPTER 1. INTRODUCTION
1. the σ-ideal σ-generated by sets of continuity of a Borel function;
2. (Michal Doucha) the σ-ideal of nondominating sets on ω ω .
In other cases still, the total canonization may fail because there are clearly
identifiable obstacles to it. There, we strive to canonize up to the known obstacles:
Theorem 1.2.4 (Theorem 5.4.2). Let I be the σ-ideal on 2ω generated by Borel
sets on which the equivalence relation E0 is smooth. If B is a Borel I-positive
set and E is an analytic equivalence relation, then there is a Borel I-positive
set C ⊂ B such that either C consists of pairwise inequivalent elements, or C
consists of pairwise equivalent elements, or E C = E0 .
Theorem 1.2.5 (Corollary 5.5.11). Let I be the σ-ideal on (2ω )ω generated by
Borel sets on which the equivalence relation E1 is Borel reducible to E0 . If B is a
Borel I-positive set and E is an a hypersmooth equivalence relation, then there is
a Borel I-positive set C ⊂ B such that either C consists of pairwise inequivalent
elements, or C consists of pairwise equivalent elements, or E C = E1 .
Theorem 1.2.6 (Theorem 5.16.2). Let n ∈ ω, let {Ii : i ∈ n} be σ-ideals on
respective compact spaces {Xi : i ∈ n}, each Π11 on Σ11 , σ-generated by compact
sets, and with covering property. If {Bi :Qi ∈ n} are Borel Ii -positive sets and
E is an analytic equivalence relation on i Bi , there
Q is a set a ⊂ n and Borel
I-positive sets {Ci ⊂ Bi : i ∈ n} such that E i Ci =equality on indices in
the set a.
In a similar vein, we achieve a number of ergodicity results. There, an obvious obstacle to canonization is identified and all equivalence relations containing
this obstacle and simpler than it are shown to have a large equivalence class.
Theorem 1.2.7 (Hjorth, Kechris ??? and Theorem 5.15.3). Let E be a Borel
equivalence relation on 2ω ; the space 2ω is equipped with the usual topology
and a Borel probability measure. If E2 ⊆ E and E is classifiable by countable
structures then E has a comeager class as well as a conull class.
Theorem 1.2.8 (Theorems 5.1.27 and 5.15.9). Let E be a Borel equivalence
relation on (2ω )ω ; the space (2ω )ω is equipped with the usual product topology
and the product Borel probability measure. If F2 ⊆ E and F2 is not Borel
reducible to E, then E has a comeager class as well as a conull class.
The weakest canonization results we obtain reduce the Borel reducibility
complexity of equivalence relations to a specified class. It should be noted
that no parallel to results of this kind exists in the realm of finite or countable
canonization results.
Theorem 1.2.9 (Theorems 5.7.5 and 5.7.7). Let E be an equivalence relation on
(2ω )ω which is classifiable by countable structures or Borel reducible to equality
modulo an analytic P-ideal
on ω. Then there are nonempty perfect sets hPn :
Q
n ∈ ωi such that E n Pn is smooth.
1.3. OUTLINE OF CONCEPTS
5
Theorem 1.2.10 (Mathias, Theorem 5.9.5). Let E be an essentially countable
Borel equivalence relation on [ω]ℵ0 . Then there is an infinite set a ⊂ ω such
that E [a]ℵ0 is Borel reducible to E0 .
There is a number of anticanonization results. Again, the classification of
analytic equivalence relations via their Borel reducibility complexity allows us
to prove theorems that have no immediate counterpart in the finite or countable
realm:
Theorem 1.2.11. There is a Borel equivalence relation on ω ω bireducible with
EKσ such that for every Borel dominating set B ⊂ ω ω , E B is still bireducible
with EKσ .
Many results in the book either connect canonization properties of σ-ideals
with the forcing properties of the associated quotient forcing PI of Borel Ipositive sets ordered by inclusion as studied in [67], or use that connection in
their proofs. This allows us to tap a wealth of information amassed about such
forcings by people otherwise not interested in descriptive set theory. A sampler
of results of this kind:
Theorem 1.2.12. Let I be a σ-ideal on a Polish space X such that the quotient
poset PI is proper.
1. (Proposition 4.1.2) PI adds a minimal real degree if and only if I has total
canonization for smooth equivalence relations;
2. (Corollary 4.2.6) if PI is nowhere c.c.c. and adds a minimal forcing extension then it has total canonization for equivalence relations classifiable
by countable structures;
3. (Theorem 4.3.8) if PI adds only finitely many real degrees then every equivalence relation classifiable by countable structures simplifies to an essentially countable equivalence relation on a Borel I-positive set.
1.3
Outline of concepts
The central canonization notion of the book is the following.
Definition 1.3.1. A σ-ideal I on a Polish space X has total canonization for
a class of equivalence relations if for every Borel I-positive set B ⊂ X and
every equivalence relation E on B in this class there is a Borel I-positive set
C ⊂ B consisting only of pairwise E-inequivalent elements or only of pairwise
E-equivalent elements.
The classes of equivalence relations considered in this book are nearly always
closed under Borel reducibility; the broadest class would be that of all analytic
equivalence relations. The total canonization is closely related to an ostensibly
stronger notion:
6
CHAPTER 1. INTRODUCTION
Definition 1.3.2. A σ-ideal I on a Polish space X has the Silver property for
a class of equivalence relations if for every Borel I-positive set B ⊂ X and every
equivalence relation E on B in this class, either there is a Borel I-positive set
C ⊂ B consisting of pairwise E-inequivalent elements, or B decomposes into a
union of countably many E-equivalence classes and an I-small set.
Thus, the usual Silver theorem can be restated as the Silver property for coanalytic equivalence relations for the σ-ideal of countable sets. Unlike the total
canonization, the Silver property introduces a true dichotomy, as the two options presented cannot coexist for σ-ideals containing all singletons: the Borel
I-positive set C from the first option would have to have a positive intersection
with one of the countably many equivalence classes from the second option,
which is of course impossible. The Silver dichotomy also has consequences for
undefinable sets. If there is any I-positive set C ⊂ B consisting of pairwise
E-inequivalent elements, then there must be a Borel such set, simply because
the second option of the dichotomy is excluded by the same argument as above.
The Silver property certainly implies total canonization within the same
class of equivalence relations: either there is a Borel I-positive set B ⊂ C
consisting of pairwise inequivalent elements, or one of the equivalence classes
form the second option of the dichotomy must be I-positive and provides the
second option of the total canonization. On the other hand, total canonization
does not imply the Silver property as shown in Section 5.3. In many common
cases though, various uniformization or game theoretic properties of the σ-ideal
I in question can be used to crank up the total canonization to Silver property.
This procedure is described in Section 3.3 and it is the only way used to argue
for the Silver property in this book.
There are a number of tricks used to obtain total canonization for various
restricted classes of equivalence relations used in this book. However, the total
canonization for all analytic equivalence relations is invariably proved via the
following notion originating in Ramsey theory:
Definition 1.3.3. A σ-ideal I on a Polish space X has the free set property if
for every I-positive analytic set B ⊂ X and every analytic set D ⊂ B × B with
all vertical sections in the ideal I there is a Borel I-positive set C ⊂ B such
that (C × C) ∩ D ⊂ id.
The free set property immediately implies the total canonization for analytic
equivalence relations. If E is an analytic equivalence relation on an I-positive
Borel set B ⊂ X, then either there is an I-positive E-equivalence class, which
immediately yields the second option of total canonization. Or, the relation
E ⊂ B ×B has I-small vertical sections, and the E-free I-positive set postulated
by the free set property yields the first option of total canonization. It is not at
all clear how one would argue for the opposite implication though, and we are
coming to the first open question of this book.
Question 1.3.4. Is there a σ-ideal on a Polish space that has total canonization
for analytic equivalence relations, but not the free set property?
1.3. OUTLINE OF CONCEPTS
7
The free set property is typically verified through fusion arguments as in
Theorem 5.1.6, or through some version of the mutual generics property.
Definition 1.3.5. A σ-ideal I on a Polish space X has the mutual generics
property if for every Borel I-positive set B and every countable elementary
submodel M of a large enough structure containing I and B there is a Borel
I-positive subset C ⊂ B such that its points are pairwise generic for the product
forcing PI × PI .
This is the first place in this book where forcing makes explicit appearance. The
mutual generics property is somewhat ad hoc in that in many cases, the arguments demand that the product forcing is replaced by various reduced product
forcings, see Section 5.3. The resulting tools are very powerful and flexible, and
where we cannot find them, we spend some effort proving that no version of
mutual generics property can hold. The concept used to rule out all its versions
is of independent interest:
Definition 1.3.6. A σ-ideal I on a Polish space X has a coding function if
there are a Borel I-positive set B ⊂ X and a Borel function f : B × B → 2ω
such that for every Borel I-positive set C ⊂ B, f 00 (C × C) = 2ω .
Clearly, if a σ-ideal has a coding function then no I-positive Borel set below the
critical set B (which is always equal to the whole space in this book) can consist
of points pairwise generic over a given countable model M , since some pairs in
this set code via the function f for example a transitive structure isomorphic
to M . We will show that many σ-ideals have a coding function (Theorem 5.1.9
or 5.1.19) and that coding functions can be abstractly obtained from the failure
of canonization of equivalence relations in many cases (Corollaries 5.5.18, 5.6.12,
5.17.21).
In the cases where total canonization is unavailable, we can still provide
a number of good canonization results. In the spirit of traditional Ramseytheoretic notation, we introduce an arrow to record them:
Definition 1.3.7. Let E, F be two classes of equivalence relations, and let I
be a σ-ideal on a Polish space X. E →I F denotes the statement that for every
I-positive Borel set B ⊂ X and an equivalence E ∈ E on B there is a Borel
I-positive set C ⊂ B such that E C ∈ F.
The strongest anti-canonization results we can achieve will be cast in terms
of a spectrum of a σ-ideal. Note that it has no obvious counterpart in the
canonical Ramsey theory on finite structures.
Definition 1.3.8. An analytic equivalence relation E is in the spectrum of a
σ-ideal I on a Polish space X if there is a Borel set B ⊂ X and an equivalence
relation F on B which is Borel bireducible with E, and also for every I-positive
Borel set C ⊂ B, F C remains bireducible with E.
For example, the equivalence relations such as E0 and F2 are in the spectrum
of the meager ideal, and EKσ is in the spectrum of the ideals associated with
8
CHAPTER 1. INTRODUCTION
the Silver forcing and Laver forcing. From the point of view of Ramsey theory,
finding a nontrivial equivalence relation in the spectrum amounts to a strong
negative result. However, the results of this sort may be quite precious in
themselves and have further applications.
1.4
Navigation
Chapter 2 contains background material which has on its face nothing to do
with canonization of equivalence relations. Most results are standard and stated
without proofs, even though there are exceptions such as the proof of canonical interpretation of Π11 on Σ11 classes of analytic sets in generic extensions
(Theorem 2.4.7). Section 2.6 describes the basic treatment of quotient posets
as explored in [67]. For a σ-ideal I on a Polish space X we write PI for the
poset of Borel I-positive subsets of X ordered by inclusion, and outline its basic
forcing properties.
Chapter 3 introduces a number of general concepts useful in the treatment
of the particular canonization theorems later. Section 3.4 explains the main tool
underlying much of the current of thought contained in this book: the correspondence between analytic equivalence relations and models of set theory. It
enables proofs of canonization theorems essentially by a discussion of the possibilities for intermediate forcing extensions of the PI models; such possibilities
have been in many cases explored independently by mathematicians interested
purely in forcing. Section 3.1 axiomatizes the treatment of integer games associated with certain σ-ideals on Polish spaces. Many such games have been
presented in [67]; however, the treatment here points out the common features
as well as the connection with uniformization theorems for Borel sets. The integer games are ultimately used for two purposes in this book: first, to limit
the possibilities for intermediate forcing extensions as in Theorem 3.2.3, and
second, in the proofs that improve the total canonization of Borel equivalence
relations to the much more informative Silver property. This latter connection
is explained in Section 3.3.
Chapter 4 discusses several classes of equivalence relations and their impact
on the structure of forcing extensions. Thus, the smooth equivalence relations
(Section 4.1) correspond to models of the form V [x] where x is a real. The
countable Borel equivalence relations (Section 4.2) are associated with a very
specific intermediate extension containing no new reals. The equivalence relations classifiable with countable structures (Section 4.3) correspond to a (often
choiceless) model of the form V (a) where a is a hereditarily countable sets.
The hypersmooth equivalence relations (Section 4.5) are mirrored in infinite
descending sequences of real degrees. The precise evaluation of these relationships helps us to obtain several remarkable general theorems; for example, if the
quotient poset PI is proper and adds a minimal forcing extension then the σideal I has total canonization for equivalence relations classifiable by countable
structures–Theorem 4.3.8(3).
Chapter 5 contains the main contributions of this book. It goes from one
1.5. NOTATION
9
class of σ-ideals to another, with the intention of proving the strongest canonization theorem possible. The task at hand is enormous and a great number of
natural σ-ideals remains untouched.
1.5
Notation
We employ the set theoretic standard notation as used in [22]. If E is an
equivalence relation on a set X and x ∈ X then [x]E denotes the equivalence
class of x. If B ⊂ X is a set then [B]E denotes the saturation of the set B, the
set {x ∈ X : ∃y ∈ B x E y}. If E, F are equivalence relations on respective
Polish spaces X, Y , we write E ≤B F if E is Borel reducible to F , in other words
if there is a Borel function f : X → Y such that x0 F x1 ↔ f (x0 ) E f (x1 ). If
f : X → Y is a Borel function between two Polish spaces and F is an equivalence
relation on the space Y , then f −1 F is the pullback of F , the equivalence relation
E on the space X defined by x0 E x1 ↔ f (x0 ) F f (x1 ); so E ≤B F . If T is
a tree then [T ] denotes the set of all of its infinite branches. id is the identity
equivalence relation on any underlying set, ee is the equivalence relation making
every two points of the underlying set equivalent. If x ∈ 2ω and m ∈ ω then
x flip m is the element of 2ω that differs from x exactly and only at its m-th
entry. If h ∈ 2<ω is a finite binary sequence then x rew h is the sequence
in 2ω obtained from x by rewriting its initial segment of corresponding length
with h. If σ is a winning strategy for one of the players in an integer game and
y ∈ ω ω is a sequence of moves of the other player, the symbol σ ∗ y denotes the
sequence that the strategy σ produces in response to y. If P is a forcing, τ is a
P -name, and G ⊂ P is a generic filter, then τ /G is the valuation of the name
τ according to the generic filter G. If P has an obvious “canonical” generic
object x (typically, an element of a Polish space), we will abuse this convention
to write also τ /x for the valuation. If M is a model of set theory and x is a set,
we say that x is M -generic if x belongs to a (set) generic extension of M . If κ
is a cardinal then Hκ denotes the collection of sets whose transitive closure has
cardinality < κ. Several theorems in the book use a rather weak large cardinal
assumption stated as “ω1 is inaccessible to the reals”, which is to say that for
every x ∈ 2ω , the ordinal ω1 as computed in the model L[x], the class of sets
constructible from x, is countable.
Theorem is a self-standing statement ready for applications outside of this
book. Claim is an intermediate result within a proof of a theorem. Fact is a
result that has been obtained elsewhere and is not going to be proved in this
book; this is not in any way to intimate that it is an unimportant or easy or
peripheral result.
10
CHAPTER 1. INTRODUCTION
Chapter 2
Background facts
2.1
Descriptive set theory
Familiarity with basic concepts of descriptive set theory is assumed throughout.
[33] serves as a standard reference.
Definition 2.1.1. A collection I of subsets of a Polish space X is Π11 on Σ11 if
for every analytic set A ⊂ 2ω × X the set {y ∈ 2ω : {x ∈ X : hy, xi ∈ A} ∈ I} is
coanalytic.
Fact 2.1.2. [33, Theorem 28.10] If I is Π11 on Σ11 then every analytic set in I
has a Borel superset in I.
Fact 2.1.3. (Luzin–Novikov, [33, Theorem 18.10]) If A ⊂ X × Y is a Borel
set with countable vertical sections, then A is the countable union of graphs of
Borel functions from X to Y .
Fact 2.1.4. [33, Theorem 18.6] Let X, Y be Polish spaces. If A ⊂ X × Y is
a Borel set then the set B = {x : Ax is not meager} is Borel, and the set
A ∩ (B × Y ) has Borel uniformization.
Fact 2.1.5. (Arsenin–Kunugui, [33, Theorem 18.18]) Let X, Y be Polish spaces.
If A ⊂ X × Y is a Borel set with Kσ vertical sections, then A has a Borel
uniformization.
Fact 2.1.6. (Jankov–von Neumann, [33, Theorem 18.1]) Let X, Y be Polish
spaces. If A ⊂ X × Y is an analytic set then it has a σ(Σ11 )-measurable uniformization. In particular, the uniformization is Baire measurable.
Fact 2.1.7. (Luzin–Suslin, [33, Theorem 15.1]) A Borel one-to-one image of a
Borel set is Borel.
In several instances, we will need the tools of effective descriptive set theory
as described in [27, Chapter 2].
11
12
CHAPTER 2. BACKGROUND FACTS
Fact 2.1.8.
1. The set {(x, y) ∈ 2ω × 2ω : y ∈ ∆11 (x)} is Π11 ;
2. every Σ11 (x) subset of 2ω with a non-∆11 (x)-element is uncountable;
3. (effective boundedness) If A is a Σ11 (x) collection of wellfounded trees on
ω then there is a wellordering of ω recursive in x whose rank is larger than
ranks of all trees in A.
Let X be a recursively presented Polish space. The Gandy-Harrington forcing is the poset of nonempty Σ11 subsets of X, ordered by inclusion. The poset
has obvious relativizations for every real parameter.
T
Fact 2.1.9. If G is a Gandy-Harrington generic filter, then G = {xgen } for
some point xgen ∈ X, and G = {A ∈ Σ11 : xgen ∈ A}.
As an interesting application of the Gandy-Harrington forcing, we will prove
the following simple
Proposition 2.1.10. Let A ⊂ (2ω )ω be a Σ11 set containing a sequence x such
that x(n) ∈
/ ∆11 for every number n ∈ ω. Then for every countable set a ⊂ 2ω
there is a sequence x0 ∈ A such that rng(x0 ) ∩ a = 0.
Proof. Let A0 = {x ∈ A : ∀n x(n) ∈
/ ∆11 }; this is a nonempty Σ11 set. Let M be
a countable elementary submodel of a large enough structure containing the set
a, and let G be a Gandy-Harrington-generic filter over M containing A0 , with
the attendant generic point x ∈ A0 . We claim that rng(x) ∩ a = 0.
And indeed, for every point y ∈ 2ω and every number n ∈ ω, the set D =
{B ⊂ A0 : for some basic open neighborhood O ⊂ 2ω of y, for every z ∈ B,
z(n) ∈
/ Oy } is open dense in the Gandy-Harrington forcing below A0 . To see this,
0
let B ⊂ A0 be a nonempty Σ11 set. Consider its projection into n-th coordinate;
this is a nonempty Σ11 subset of 2ω containing only non-∆11 -elements; therefore,
it is uncountable and it must contain some element distinct from y, separated
from y by some basic open neighborhood O. Then B = {x ∈ B 0 : x(n) ∈
/ O}
is the desired nonempty Σ11 set in D below B 0 . The rest follows from standard
density arguments.
2.2
Invariant descriptive set theory
If E, F are Borel or analytic equivalence relations on Polish spaces X, Y then
E ≤B F (E is Borel reducible to F ) denotes the fact that there is a Borel
function f : X → Y such that x0 E x1 ↔ f (x0 ) F f (x1 ). Bireducibility is an
equivalence relation on the class of all Borel equivalence relations. ≤B turns
into a complicated ordering on the bireducibility classes. There are several
equivalences occupying an important position in this ordering:
Definition 2.2.1. id is the identity equivalence relation; ee, or the full equivalence relation, is the equivalence relation that makes all elements of its domain
2.2. INVARIANT DESCRIPTIVE SET THEORY
13
equivalent. E0 is the equivalence on 2ω defined by x E0 y ↔ the symmetric
difference x∆y is finite. E1 is the equivalence relation on (2ω )ω defined by
~x E1 ~y ↔ {n ∈ ω : ~x(n) 6= ~y (n)} is finite. E2 is the equivalence relation on 2ω
defined by x E2 y ↔ Σ{1/(n + 1) : x(n) 6= y(n)} < ∞. F2 is the equivalence
relation on (2ω )ω defined by ~x F2 ~y ↔ rng(~x) = rng(~y ). EKσ is the equivalence
on Πn (n + 1) defined by x EKσ y ↔ ∃m∀n |x(n) − y(n)| < m. Ec0 is the
equivalence on [0, 1]ω defined by x Ec0 y ↔ lim(x(n) − y(n)) = 0.
Definition 2.2.2. An equivalence E on a Polish space X is classifiable by
countable structures if there is a countable relational language L such that E is
Borel reducible to the equivalence relation of isomorphism of models of L with
universe ω.
Definition 2.2.3. An equivalence is smooth if it is Borel reducible to the identity. An equivalence is hypersmooth if it is an increasing union of a countable
sequence of smooth equivalence relations.
Definition 2.2.4. An equivalence E on a Polish space X is essentially countable
if it is Borel reducible to a countable Borel equivalence relation: one all of whose
classes are countable.
The countable Borel equivalences are governed by the Feldman–Moore theorem:
Fact 2.2.5. ([13] [14, Theorem 7.1.4]) For every countable Borel equivalence
relation E on a Polish space X there is a countable group G and a Borel action
of G on X whose orbit equivalence relation is equal to E.
Fact 2.2.6. (Silver dichotomy, [52], [27, Section 10.1]) For every coanalytic
equivalence relation E on a Polish space X, either X is covered by countably
many equivalence classes or there is a perfect set of mutually E-inequivalent
points.
Corollary 2.2.7. For every coanalytic equivalence relation E on a Polish space
X and every analytic set A ⊂ X, either A intersects only countably many
E-equivalence classes, or it contains a perfect set of pairwise E-inequivalent
elements.
Proof. Let f : ω ω → X be a continuous function such that rng(f ) = A, and let
F = f −1 E; thus, F is a coanalytic equivalence relation on ω ω and the Silver
dichotomy can be applied to it. The corollary follows immediately.
Fact 2.2.8. (Glimm-Effros dichotomy, [18], [27, Section 10.4]) For every Borel
equivalence relation E on a Polish space, either E ≤B id or E0 ≤B E.
Fact 2.2.9. [19] If E ≤B E2 is a Borel equivalence relation then either E is
essentially countable, or E2 ≤B E.
Fact 2.2.10. [29], [27, Section 11.3] For every Borel equivalence relation E ≤ E1
on a Polish space, either E ≤B E0 or E1 ≤B E.
14
CHAPTER 2. BACKGROUND FACTS
Fact 2.2.11. [44], [27, Section 6.6] Every Kσ equivalence relation on a Polish
space is Borel reducible to EKσ .
Fact 2.2.12. [44], [27, Chapter 18] For every Borel equivalence relation E there
is a Borel ideal I on ω such that E ≤B =I .
Here, =I is the equivalence on 2ω defined by x =I y ↔ {n ∈ ω : x(n) 6= y(n)} ∈
I. It is not difficult to construct an Fσ -ideal I on a countable set such that =I
is bireducible with EKσ , and such an ideal will be useful in several arguments in
the book. Just let dom(I) be the countable set of all pairs hn, mi such that n ∈ ω
and m ≤ n and let a ∈ I if there is a number k ∈ ω such that for every number
n ∈ ω, there are at most k many numbers m with hn, mi ∈ a. This ideal is
clearly Fσ and so =I ≤B EKσ by the above fact. On the other hand, EKσ ≤=I ,
as the reduction f : Πn (n + 1) → dom(I) defined by f (x) = {hn, mi : m ≤ x(n)}
shows.
2.3
Forcing
Familiarity with basic forcing concepts is assumed throughout the book. If
hP, ≤i is a partial ordering and Q ⊂ P then Q is said to be regular in P if every
maximal antichain of Q is also a maximal antichain in P ; restated, for every
element p ∈ P there is a pseudoprojection of p into Q, a condition q ∈ Q such
that every strengthening of q in Q is still compatible with p.
Regarding extensions of models of set theory, we will use the standard
nomenclature of [22, Theorem 13.22 ff]. If V is a transitive model of ZF set
theory containing all ordinals and a (perhaps a ∈
/ V ) is a set, then V (a) is the
smallest transitive model of ZF containing V as a subset and a as an element,
and V [a] is the smallest model W of ZF containing V as a subset and such that
a ∩ W ∈ W . The two models are formed by transfinite induction in parallel to
the L(a) and L[a] constructions, except at successor stages of the induction, all
subsets of the constructed part which belong to V are added. If a is in addition a generic filter over V , the models V [a] and V (a) coincide and there is an
alternative description of them as a forcing extension: V (a) = V [a] = {τ /a : τ
is a name in V }. The basic development of forcing up to the forcing theorem
and product forcing theorem works in the context of ZF set theory without
choice, and there are several opportunities for considering forcing extensions of
choiceless models.
Whenever κ is a cardinal, Coll(ω, κ) is the poset of finite partial functions
from ω to κ ordered by reverse extensions. If κ is an inaccessible cardinal,
then Coll(ω, < κ) is the finite support product of the posets Coll(ω, λ) for all
cardinals λ ∈ κ. The following fact sums up the homogeneity properties of these
partial orders.
Fact 2.3.1. [22, Corollary 26.10] Let κ be a cardinal, P a poset of size < κ,
and G ⊂ Coll(ω, κ) be a generic filter. In V [G], for every V -generic filter
H ⊂ P there is a V [H]-generic filter K ⊂ Coll(ω, κ) such that V [G] = V [H][K].
2.3. FORCING
15
Identical statement holds true if κ is an inaccessible cardinal and Coll(ω, κ) is
replaced by the poset Coll(ω, < κ).
In several places, we need to invoke a forcing factorization theorem, and the
following is the exact statement that we will use:
Fact 2.3.2. (Forcing factorization theorem) If P, Q are posets and τ is a P name for a Q-generic filter, then there is a condition q ∈ Q such that in a
large collapse extension, for every filter H ⊂ Q generic over V containing the
condition q there is a filter G ⊂ P generic over V such that H = τ /G.
Proof. Let κ be a cardinal larger than |P(P )|. Consider the statement φ of the
Q-forcing language saying “Coll(ω, κ) there is a V -generic filter G ⊂ P̌ such
that Ḣ = τ̌ /G”; here, Ḣ is the Q-name for the generic filter on Q. Note that
the above Fact 2.3.1 implies that the largest condition in Coll(ω, κ) decides the
statement on the right of the forcing relation, as all its parameters are in the
model V [H].
First argue that there must be a condition q ∈ Q that forces φ. If this failed
then Q ¬φ. Let G ⊂ P be a filter generic over V , and let K ⊂ Coll(ω, κ) be
a filter generic over V [G]. Let H = τ /G. Now, Fact 2.3.1 shows that V [G][K]
is a Coll(ω, κ)-extension of the model V [H]. By the forcing theorem and the
failure of φ, there should be no V -generic filter on P in the model V [G][K] that
gives H. However, there is such a filter there, namely G; a contradiction.
Thus, fix a condition q ∈ Q forcing φ; we claim that q works as required. Let
K ⊂ Coll(ω, κ) be a filter generic over V , and let H ⊂ Q be a V -generic filter in
V [K], containing the condition q. By Fact 2.3.2, V [K] is a Coll(ω, κ) extension
of V [H], and the forcing theorem applied to φ and H shows that there is a filter
G ⊂ P generic over V in the model V [K] such that H = τ /G.
The following definition sums up the common forcing properties used in this
book:
Definition 2.3.3. A forcing P is proper if for every condition p ∈ P and
every countable elementary submodel M of large enough structure containing
P, p there is a master condition q ≤ p which forces the generic filter to meet
all dense subsets in M in conditions in M . The forcing is bounding if every
function in ω ω in its extension there is a ground model function with larger
value on every entry. The forcing preserves outer Lebesgue measure if every
ground model set of reals has the same outer measure whether evaluated in the
ground model or in the extension. The forcing preserves Baire category if every
ground model set of reals is meager in the ground model if and only if it is
meager in the extension. The forcing does not add independent reals if every
sequence in 2ω in the extension has an infinite ground model subsequence.
The phrase “countable elementary submodel of a large enough structure” appears in essentially every paper dealing with proper forcing. The large enough
structure should be interpreted as the structure with universe Hκ , the collection
16
CHAPTER 2. BACKGROUND FACTS
of sets of hereditary cardinality < κ for some cardinal κ such that Hκ contains
every object named in the proof before the phrase occurs and its powerset. The
structure has a binary relation ∈ and a designated unary predicate for every
object named in the proof before the phrase occurs.
If M is a model of some portion of set theory and P ∈ M is a poset, we
will say that a filter G ⊂ P is P -generic over M if for every dense set D ⊂ P
in M there is a condition p ∈ D ∩ M ∩ G. If the poset P adds a “canonical”
object nominally different from a generic filter (such as some point ẋgen in a
Polish space from which the generic filter is recoverable), we will extend this
usage to say that x is P -generic over M if the filter on P recovered from x is.
In contradistinction, the phrase “a is M -generic” will mean that a belongs to
some generic extension of the transitive isomorph of M ; it will be used only if
M is wellfounded.
We will need several facts about the interplay between elementary submodels
and generic extensions. If M is an elementary submodel of a large structure,
P ∈ M is a poset and G ⊂ P is a filter generic over V , let M [G] = {τ /G : τ ∈
M }.
Proposition 2.3.4. Let P be a partial order, let κ > |P |, let M ≺ Hκ , and let
G ⊂ P be a generic filter over V .
V [G]
1. hM [G], ∈i ≺ hHκ [G], ∈i = hHκ
, ∈i;
2. if G is moreover generic over M then hM [G], ∈, M i ≺ hHκ [G] ∈i;
3. if, in addition, Q ∈ M is another poset and K ∈ M [G] is a filter Q-generic
over both M and V , then M [K] = M [G] ∩ Hκ [K].
Proof. The first item is proved by a straighforward application of the Tarski
criterion. Let φ be a formula of set theory and let τi : i ∈ n be Q-names in
the model M . Piecing together various names, there is a name σ in Hκ such
that P if Hκ [Ġ] |= ∃x φ(x, τi : i ∈ n) then Hκ [G] |= φ(σ, τi : i ∈ n). By the
elementarity of the model M , there must be such a name σ in the model M .
Thus, Hκ [G] |= ∃x φ(x, τi /G : i ∈ n) → φ(σ/G, τi /G : i ∈ n) and the Tarski
criterion follows.
For the second item observe that M [G] ∩ HκV = M . To see this, if τ ∈ M
is a P -name, then the set D = {p ∈ P : either p τ ∈
/ HκV or there exists
V
a ∈ Hκ such that p τ = ǎ} is open dense subset in P , in the model M by
the elementarity of M . Since the filter G is generic over M , there must be some
condition p in the intersection G ∩ M ∩ D. If p τ ∈
/ HκV then τ /G ∈
/ V . If, on
the other hand, there is a ∈ Hκ such that p τ = ǎ, then by the elementarity
of the model M it is the case that there must be such an a in the model M ,
and then τ /G = a ∈ M .
The third item follows from the second. If τ ∈ M is a Q-name then τ /K ∈
M [G] since M [G] is an elementary submodel of Hκ [G] containing both τ and
K. On the other hand, if a ∈ M [G] ∩ Hκ [K] then there is a Q-name τ ∈ Hκ
such that a = τ /K, and by the elementarity in (2) such a name must exist in
M.
2.4. SHOENFIELD ABSOLUTENESS
2.4
17
Shoenfield absoluteness
Nearly every argument in this book uses the following classical fact:
Fact 2.4.1. (Shoenfield, [22, Theorem 25.20]) Let x ∈ X be an element of a
Polish space, and φ a Π12 formula with one variable. Then the truth value of
φ(x) is the same in all forcing extensions.
Shoenfield absoluteness is necessary to establish the existence of canonical
interpretations of many notions of descriptive set theory in forcing extensions,
a fact consistently underappreciated in existing literature, such as [67]. Here we
provide the necessary proofs.
Theorem 2.4.2. Every Polish space has a canonical interpretation in every
forcing extension.
The precise definition of the canonical interpretation is to a great extent a
matter of taste, and as such is likely to generate endless scholastic discussions on
the imaginary virtues and advantages of one approach over another. We pick the
completion approach, since it was the first reaction of all set theorists contacted
in an ad hoc survey and since it shows the role of Shoenfield absoluteness most
clearly. We will also show that another perfectly good approach using Gδ subsets
of the Hilbert cube is equivalent.
Proof. Let X be a Polish metric space, with a complete compatible metric d.
Let V [G] be a generic extension of V , and consider the completion Xd in V [G]
of the metric space hX, di there. This will be our canonical interpretation of
X in V [G]. We must show that this interpretation does not depend on the
choice of the complete metric d in the sense that if d, e are two compatible
complete metrics on X in the ground model, then the two spaces Xd and Xe are
homeomorphic in V [G]. What is more, if id : X → Xd and ie : X → Xe are the
two canonical metric embeddings, then there is a homeomorphism h : Xd → Xe
such that ie = h ◦ id . Such a homeomorphism must necessarily be unique.
Return to the ground model. Let A be a countable dense subset of X and
consider the usual equivalence relation Ed on d-Cauchy sequences and Ee the
similar equivalence on e-Cauchy sequences.
Claim 2.4.3. Every Ed -equivalence class intersects exactly one Ee -equivalence
class. Moreover, this is a Π12 statement.
Proof. First observe that for every point x ∈ X there is a sequence of points in
A converging to x that is simultaneously e-Cauchy and d-Cauchy. To find such
a sequence, note that the 2−n -balls around x in the sense of the metric e and d
are both open in the topology of the space X, and pick the n-th point on the
sequence in the open intersection of these two balls.
18
CHAPTER 2. BACKGROUND FACTS
The first sentence of the claim now immediately follows: every Ed -equivalence
class consists of d-Cauchy sequences converging to some fixed point x, and as
such may intersect only the unique Ee -class consisting of e-Cauchy sequences
converging to the same point. It does intersect this unique Ee -class in a nonempty
set by the previous paragraph. The complexity evaluation is also nearly trivial.
The first sentence of the claim says that for every d-Cauchy sequence there is
an Ed -equivalent e-Cauchy sequence, and moreover, if two sequences are both
e-Cauchy and d-Cauchy then they are Ed -equivalent if and only if they are
Ee -equivalent.
In the generic extension V [G], the first sentence of the claim remains in
force by Shoenfield’s absoluteness. The d-completion Xd of X is equal to the
d-completion of A and similarly on the e-side. The map h that sends every
d-equivalence class to the unique e-equivalence class that it intersects has the
required properties.
Another approach to a canonical interpretation of Polish spaces would use
the basic fact that every Polish space is homeomorphic to a Gδ subset of the
Hilbert cube, and that there is a good candidate for a canonical interpretation
of any Gδ subset of the Hilbert cube in a forcing extension. The Shoenfield
absoluteness will again appear in this approach at the point of showing that Gδ
sets homeomorphic in the ground model are still homeomorphic in the extension, and the homeomorphism in the extension can be chosen so that it extends
the homeomorphism in the ground model. We will only show that this approach
is equivalent in that the interpretation of the Gδ set in the extensionTis homeomorphic to the completion interpretation obtained above. Let B = n On be
a Gδ subset of [0, 1]ω , with a countable dense subset A ⊂ B, and a compatible
complete metric d. The following claim is nearly trivial.
Claim 2.4.4. For every d-Cauchy sequence of elements of A converges to some
point in B. Vice versa, every point in B is a limit of some d-Cauchy sequence
of elements of A. Moreover, this is a Π12 statement.
Now move to any generic extension. The first two sentences of the claim
remain in force by Shoenfield absoluteness. The map h from the d-completion
of the set B (as interpreted in V ) to B as interpreted in V [G] mapping each
Ed -class of elements of A to the point of B to which it converges has the required
properties.
We will use the above theorem in the following way. For any Polish space
X and any forcing P , the symbol Ẋ will denote the P -name for the canonical
interpretation of the space X in the generic extension. We may drop the dot if
the symbol stands on the right hand side of the forcing relation; in such a case,
the ground model version of X will be denoted by X ∩ V or X V . We will always
abuse the notation by considering the ground model version of X as a subset of
its canonical interpretation in the extension, and the canonical interpretations
of such simply definable spaces as 2ω or ω ω will be the spaces with the same
definition in the extension.
2.4. SHOENFIELD ABSOLUTENESS
19
Theorem 2.4.5. Analytic subsets of Polish spaces have canonical interpretation
in any generic extension.
Proof. First, the canonical interpretation of closed sets is rather obvious. If
C ⊂ X is a closed subset of a Polish space, then the closure of C in the canonical
interpretation of X in the generic extension is the canonical interpretation of C.
If A ⊂ X is an analytic set, then it is a projection of a closed set C ⊂ X × ω ω .
The canonical interpretation of A in the generic extension will be the projection
of the interpretation of C into the interpretation of X. We must show that this
definition does not depend on the choice of the closed set C. Let D be another
closed subset of X × ω ω projecting into A. Let U be a countable dense subset
of X × ω ω . The following nearly trivial claim is left to the reader.
Claim 2.4.6. For every sequence of elements of U converging to an element of
C there is a sequence of elements of U converging to an element of D with the
same first coordinate, and vice versa. Moreover, this is a Π12 statement.
By Shoenfield absoluteness, the first sentence of the claim remains in force
in any generic extension, and so the projection of the interpretation of C is the
same as the projection of the interpretation of D.
Notationally, this feature is exploited in the following way. If A ⊂ X is an
analytic (or coanalytic) subset of a Polish space, the same letter A is used to
denote the canonical interpretation of A in all forcing extensions.
The reader as well as the authors may equally wish to leave this seemingly
barren subject already. However, there is a much finer canonical interpretation
problem that must be resolved–the question of interpretation of Π11 on Σ11 classes
of analytic sets in generic extensions. It turns out that the verification of the
validity of the obvious candidate definition is a highly nontrivial matter that
uses the methods of effective descriptive set theory; it is certainly missing in
[67].
Theorem 2.4.7. Every Π11 on Σ11 class of analytic sets has a canonical Π11
on Σ11 interpretation in every generic extension. If the class is a σ-ideal of
analytic sets in the ground model, then its canonical interpretation in the generic
extension is again a σ-ideal of analytic sets.
Proof. Let I be a Π11 on Σ11 collection of analytic sets on a Polish space X.
For simplicity we assume that the space X is recursively presented; otherwise,
replace it with its homeomorph, a Gδ -subset of the recursively presented space
[0, 1]ω , and insert the code for the Gδ subset in appropriate places in the argument.
Let A ⊂ 2ω × X be a good universal Σ11 set in the sense of ???. The
set C = {y ∈ 2ω : Ay ∈ I} is coanalytic; therefore, both A and C have a
canonical interpretation in any forcing extension. The collection {Ay : y ∈ C},
as interpreted in the generic extension, is a natural candidate for the canonical
interpretation of I in the generic extension. It is necessary to check that in
the generic extension, this is a Π11 on Σ11 collection of analytic sets, and the
20
CHAPTER 2. BACKGROUND FACTS
interpretation does not depend on the choice of the good universal analytic set
A. These features will both follow very quickly as soon as we check that in
the generic extension, the set C still defines a collection of analytic sets, in the
sense that the sentence φ = ∀y, z ∈ 2ω (Ay = Az ) → (y ∈ C ↔ z ∈ C) holds in
the generic extension. The sentence φ holds in the ground model; the challenge
resides in the fact that φ appears to be more complex than Π12 , and therefore it is
impossible to use Shoenfield absoluteness to transfer it to the generic extension.
We need to resort to a patch that includes the use of effective descriptive set
theory. Assume for simplicity that the set C is lightface Π11 ; otherwise insert
the parameter used in its definition in the appropriate spots in the argument
below.
Fix a standard parametrization of Borel subsets of X such as in ???. That
is, there are Π11 sets D ⊂ 2ω and F ⊂ 2ω × X such that for every y ∈ D the
equality Az = Fz holds, and for every parameter z ∈ 2ω and every ∆11 (z) set
G ⊂ X there is y ∈ ∆11 (z) such that Fz = G. Moreover, this property of the
sets A, D, and F is provable in ZFC.
Claim 2.4.8. For every z ∈ C there is y ∈ ∆11 (z) such that y ∈ D ∩ C and
Az ⊆ Ay . Moreover, this is a Π12 statement.
Proof. The first sentence is just a restatement of the usual effective version of
the first reflection theorem. For the evaluation of the complexity, note that
y ∈ D ∧ Az ⊆ Ay is a Π11 statement about z and y: for y ∈ D, Az ⊆ Ay is
equivalent to ∀x x ∈
/ Az ∨ x ∈ Gy . The complexity evaluation then follows from
the well-known fact that the quantifier ∃y ∈ ∆11 (z) translates into a universal
quantifier ???
Claim 2.4.9. For every z ∈ 2ω and every y ∈ D ∩ C, if Az ⊆ Ay then z ∈ C.
Moreover, this is a Π12 statement.
Proof. The first sentence just says that the collection I is closed under taking
an (analytic) subset. This statement has an easy folklore proof: assume that
P ⊂ R ⊂ X are analytic sets and R ∈ I. Choose an analytic non-Borel set
S ⊂ 2ω and consider the analytic set T = (S × R) ∪ (2ω × P ) ⊂ 2ω × X. The
set U = {y ∈ 2ω : Ty ∈ I} is coanalytic as I is Π11 on Σ11 . The vertical sections
Ty for y ∈ S are equal to R, so S ⊆ U . Since the set S is not coanalytic, there
must be a point y ∈ U \ S. For such a point y, Ty = P and we conclude that
P ∈ I as well.
For the complexity computation, recall from the previous proof that y ∈
D ∧ Az ⊆ Ay is a Π11 statement about y and z, and rewrite the first sentence of
the claim as ∀z∀y (y ∈
/ D ∩ C) ∨ ¬(y ∈ D ∧ Az ⊆ Ay ) ∨ (z ∈ C). The formula
to the right of the universal quantifiers is a disjunction of Π11 and Σ11 formulas,
therefore Π12 , and the complexity estimate follows.
2.5. GENERIC ULTRAPOWERS
21
The sentence φ formally follows from the conjunction of the two statements
in the claims: If z0 , z1 ∈ 2ω are points such that Az0 ⊆ Az1 and z1 ∈ C, then
the first claim provides a point y ∈ D ∩ C such that Az1 ⊆ Ay , and the second
claim applied to Y0 , z then shows that z0 ∈ C. The statements in the claims are
Π12 and therefore will hold in any forcing extension by Shoenfield absoluteness;
so even φ will hold in every forcing extension.
It follows immediately that the canonical interpretation does not depend on
the choice of the analytic set A ⊂ 2ω × X. Let A∗ ⊂ 2ω × X be an arbitrary
analytic set and C ∗ = {y ∈ 2ω : A∗y ∈ I}. We will show that in every forcing
extension, for every y ∈ 2ω , y belongs to (the interpretation of) C ∗ if and only
if A∗y belongs to the interpretation of I described above. Since the set A is good
universal, there is a continuous function f : 2ω → 2ω such that for every y ∈ 2ω ,
A∗y = Af (y) ; moreover, for every y ∈ 2ω , y ∈ C ∗ ↔ f (y) ∈ C by the definition
of the set C ∗ . This feature of the function f is Π12 , and therefore persists to the
generic extension. Thus, even in the generic extension C ∗ = {y ∈ 2ω : A∗y ∈ I}.
The (canonical interpretation of the) collection I remains Π11 on Σ11 in every
generic extension, again due to the choice of the good universal set A ⊂ 2ω × X.
In the extension, if A∗ ⊂ 2ω × X is an analytic set, then there must be a
continuous function f : 2ω → 2ω such that for every y ∈ 2ω , A∗y = Af (y) . It
follows that the set C ∗ = {y ∈ 2ω : A∗y ∈ I} is coanalytic, since it is simply the
f -preimage of the (interpretation of) the set C.
Finally, if I is a σ-ideal of analytic sets in the ground model, then this is
easily restated as a Π12 statement using the good universal set A, and therefore
it persists to all forcing extensions.
We exploit the previous theorem in the following way. If I is a Π11 on Σ11 σ-ideal,
we use the same letter I to denote its canonical interpretation in every generic
extension.
As a final remark, in this book we are interested in many properties of Π11
on Σ11 σ-ideals, most of which seem to be impossible to restate as Π12 formulas,
and their absoluteness into forcing extension remains open to question. Sabok
[47] proved that the statement “I is σ-generated by closed sets” is absolute for
such ideals. The following remains open.
Question 2.4.10. Let I be a Π11 on Σ11 σ-ideal on a Polish space. If I is c.c.c.,
is (the canonical interpretation of) I c.c.c. in every forcing extension? If the
quotient forcing PI is proper, is it again proper in every forcing extension?
2.5
Generic ultrapowers
In several arguments throughout the book, we will exploit the technique of (illfounded) generic ultrapower outlined in [22, Section 35]. Let X be an arbitrary
set and I be an ideal on it, and consider the quotient poset RI = P(X) modulo
I. Whenever G ⊂ RI is a generic filter, one can form in the extension V [G] the
generic ultrapower, the model N of equivalence classes of ground model functions with domain X, where f is equivalent to g if {x ∈ X : f (x) = g(x)} ∈ G,
22
CHAPTER 2. BACKGROUND FACTS
and [f ] ∈N [g] if {x ∈ X : f (x) ∈ g(x)} in G. A modified proof of Loś’s theorem
shows that j : V → N defined by j(a) = [fa ], where fa is the constant function
with the single value a, is an elementary embedding. The model N may not be
wellfounded. However, for every N -ordinal α which is wellfounded in V [G], we
will always identify the wellfounded relation hVαN , ∈N i with its transitive isomorph. If I is a σ-ideal, then ω N is wellfounded, and (2ω )N , (ω ω )N etc. become
subsets of (2ω )V [G] , (ω ω )V [G] through this identification.
We will use the ultrapower technique exclusively through the following fact.
Fact 2.5.1. For every uncountable cardinal λ, in some generic extension there
is a generic ultrapower j : V → N such that
1. j 00 Hλ exists in N and it is countable there;
2. for every analytic set A coded in the ground model, j(A) ⊂ A, and for
every Borel set B coded in the ground model, j(B) = B ∩ N .
Note that both properties of the embedding j will survive into any further
generic extension of the ground model. It may happen that there is an analytic
set A such that j(A) 6= A ∩ N —for example if there is an illfounded countable
ordinal of N and A is the set of illfounded linear orders on ω. In such a case,
the model N may be incorrect about such items as codes for Borel sets etc.
Proof. The argument is entirely standard. Let I be the σ-ideal on nonstationary
subsets of [Hλ ]ℵ0 , and let RI be the quotient poset. Let G ⊂ RI be a generic
filter over V and form the generic ultrapower j : V → N . We claim that this
embedding works.
The first item follows from the normality of the nonstationary ideal on
[Hλ ]ℵ0 . Consider the equivalence class [id] ∈ N , and observe that [id] = j 00 Hλ .
First of all, if a ∈ Hλ then for a club subset of b ∈ [Hλ ]ℵ0 it is the case that
a ∈ b; in other words, the values of function fa constantly equal to a belong to
the values of the identity function everywhere but on a nonstationary set. By
Loś’s theorem, N |= j(a) = [fa ] ∈ [id]. Second, if S N |= [f ] ∈ [id], then for
all but nonstationarily many b ∈ S it is the case that f (b) ∈ b, and the normality
of the nonstationary ideal yields a stationary subset T ⊂ S and a ∈ Hλ such
that ∀b ∈ T f (b) = a. Clearly, T [f ] = j(a). Finally, N |= M = j 00 Hλ by
Loś’s theorem again, since the values of the id function are all countable sets.
Since M ∈ N has an initial segment of its ordinals isomorphic to λ, it
follows that λ is isomorphic to an initial segment of ordinals of the model N . In
particular, the wellfounded (transitive) part of the model N includes (Vω+3 )N
and for any Polish space X ∈ (Vω+3 )V it is the case that j(X) ⊂ X V [G] .
Suppose that A ⊂ X is an analytic set coded in the ground model, say A =
rng(f ) for some continuous function f : ω ω → X. Then, as the model N has
standard ω, j(f ) ⊂ f , and so j(A) = rng(j(f )) ⊂ rng(f ) = A. If B ⊂ X is
a Borel set, then both B, ¬B are analytic and respective ranges of continuous
function f, g : ω ω → X. By elementarity of the embedding j, rng(j(f )) and
rng(j(g)) are complementary subsets of j(X) in N , and as in the previous
2.6. IDEALIZED FORCING
23
sentence j(B) = rng(j(f )) ⊂ rng(f ) = B and j(¬B) = rng(j(g)) ⊂ rng(g) =
¬B. The only way that this can happen is that j(B) = B ∩ N .
2.6
Idealized forcing
The techniques of the book depend heavily on the theory of idealized forcing as
developed in [67]. Let X be a Polish space and I be a σ-ideal on it. The symbol
PI stands for the partial ordering of all Borel I-positive subsets of X ordered by
inclusion; we will alternately refer to it as the quotient forcing of the σ-ideal I.
Its separative quotient is the σ-complete Boolean algebra of Borel sets modulo
I. The forcing extension is given by a unique point xgen ∈ X such that the
generic filter consists of exactly those Borel sets in the ground model containing
xgen as an element. One can also reverse this operation and for every forcing P
adding a generic point ẋ ∈ X, one may form the σ-ideal J associated with the
forcing P , generated by those Borel sets B ⊂ X such that P ẋ ∈
/ Ḃ.
If I is a σ-ideal on a Polish space X, x ∈ X is a point and M is an elementary
submodel of a large structure such as Hκ , the collection of all sets of hereditary
cardinality < κ for some cardinal κ, and I ∈ M , we will say that the point
x is PI -generic over M if for every open dense set D ⊂ PI , D ∈ M , there is
B ∈ D ∩ M with x ∈ B. The set g ⊂ PI ∩ M consisting of those B ∈ PI ∩ M for
which x ∈ B is an M -generic filter on PI ∩ M . We then consider M [x] as the
transitive model obtained from the transitive isomorph of M by adjoining the
filter g. If τ ∈ M is a PI -name then τ /x denotes the evaluation of the transitive
collapse image of τ according to the filter g. Similar terminology will be applied
in case that M is instead some transitive model of set theory and M |= I is a
σ-ideal on a Polish space X.
Fact 2.6.1. [67, Proposition 2.2.2] Let I be a σ-ideal on a Polish space X. The
following are equivalent:
1. PI is proper;
2. for every Borel I-positive set B ⊂ X and every countable elementary
submodel M of a large structure with I ∈ M , the set C ⊂ B of all points
in B that are P -generic over the model M is I-positive.
There are many consequences of this characterization:
Fact 2.6.2. [67, Section 2.3] Let I be a σ-ideal on a Polish space X such that
the quotient PI is proper. Then
1. V [xgen ]∩2ω = {f (ẋgen ) : f is a Borel function coded in the ground model};
2. for every Borel set B ∈ PI and every analytic set A ⊂ B × ω ω with
nonempty vertical sections, there is a Borel I-positive set C and a Borel
function f : C → ω ω such that f ⊂ A;
24
CHAPTER 2. BACKGROUND FACTS
3. for every Borel set B ∈ PI and every Borel function f : B → ω ω there is
a Borel I-positive set C ⊂ B such that f 00 C is Borel.
4. whenever y ∈ V [xgen ] is a real then V [y] ∩ 2ω = {f (y) : f is a ground
model Borel function with y ∈ dom(f ) and rng(f ) ⊂ 2ω }.
The following simple fact is used throughout the book:
Fact 2.6.3. [67, Theorem 3.3.2] Suppose that I is a σ-ideal on a Polish space
X such that the quotient poset PI is proper. The following are equivalent:
1. PI is bounding;
2. every Borel I-positive set has a compact I-positive subset, and every Borel
function on an I-positive domain is continuous on a compact I-positive
set.
The following two properties deal with σ-ideals without any reference to
forcing; nevertheless, they are particularly useful when idealized forcing is considered.
Definition 2.6.4. A pair of σ-ideals I, J on respective Polish spaces X, Y has
the Fubini property if whenever B0 ⊂ X and B1 ⊂ Y are I- and J-positive Borel
sets respectively, then the product B0 × B1 cannot be covered by the union of
two Borel subsets of X × Y , one with all vertical sections in J and the other
with all horizontal sections in I.
Definition 2.6.5. A pair of σ-ideals I, J on respective Polish spaces X, Y has
the rectangular Ramsey property if whenever B0 ⊂ X and B1 ⊂ Y are I- and
J-positive
S Borel sets respectively and the product B0 × B1 is covered by the
union n Dn of countably many Borel subsets of X × Y , then there are Borel
respectively I-and J-positive sets C0 ⊂ B0 and C1 ⊂ B1 such that the product
C0 ×C1 is a subset of one of the sets forming the union. The definition naturally
extends to finite or countable collections of σ-ideals.
The main purpose of the rectangular Ramsey property in [67] is the computation
of the σ-ideal associated with the product PI × PJ . If the property holds, then
the collection K of all Borel subsets of X ×Y containing no Borel rectangle with
I- and J-positive sides respectively is a σ-ideal of Borel sets, and the product
is naturally isomorphic to a dense subset of the quotient PK . In this book, the
rectangular property will be also used to rule out various ergodicity phenomena.
Among the quotient forcings, the definable c.c.c. ones form a very special
class that has been investigated repeatedly. In several places in this book, we
will need to discuss the global properties of this class. The following fact records
the features important here. If one is willing to make large cardinal assumptions,
it will hold true of much larger class of σ-ideals than just Π11 on Σ11 .
Fact 2.6.6. Let I be a c.c.c. Π11 on Σ11 σ-ideal on a Polish space. Then
1. if PI is bounding, then it is a Maharam algebra;
2.6. IDEALIZED FORCING
25
2. if PI is not bounding, then it adds a Cohen real;
3. if I contains all singletons and not the whole space, and PI preserves Baire
category, then it is equivalent to adding one Cohen real.
Proof. This remarkable statement is in fact just a concatenation of various theorems scattered throughout the literature. If PI is bounding, then Player II has
a winning strategy in the bounding game [67, Theorem 3.10.7]. Fremlin [23,
Theorem 7.5] proved that the winning strategy yields a continuous submeasure
φ on the Polish space such that I = {A : φ(A) = 0}. If PI adds an unbounded
real, then it in fact adds a Cohen real by a result of Shelah [3, Theorem 3.6.47].
One has to adjust the final considerations of that proof very slightly to conclude
that it works for all Π11 on Σ11 c.c.c. σ-ideals. The third item is in essence [67,
Corollary 3.5.6] or [?].
Corollary 2.6.7. Let I be a c.c.c. Π11 on Σ11 σ-ideal on a Polish space. The
quotient forcing adds an independent real.
Proof. Maharam algebras add independent reals by [1, Lemma 6.1], and a Cohen
real is an independent real in its usual presentation. The rest is handled by the
first two items of the above Fact.
Corollary 2.6.8. Let I, J be c.c.c. Π11 on Σ11 σ-ideals on Polish spaces. The
pair I, J does not have the rectangular Ramsey property.
Proof. The first two items of the Fact again show that we only have to consider
the meager ideal, and the null ideals for the various continuous submeasures.
As shown in [12], pairs of these σ-ideals even fail to have the Fubini property
unless they are two null ideals obtained from Borel probability measures or two
meager ideals obtained from some Polish topologies. In these two cases, the
rectangular property fails again; the simplest way to see it is to consider the
equivalence E0 on 2ω and the usual Borel probability measure or topology on
2ω , and apply the Steinhaus theorem to see that there cannot be a product of
two Borel positive sets which is either disjoint from E0 or a subset of E0 .
The following humble theorem will be useful as well.
Theorem 2.6.9. Let I be a Π11 on Σ11 σ-ideal on a Polish space X such that
the quotient forcing PI is proper. Every c.c.c. intermediate extension of the PI
extension either is equal to the ground model or contains a new real.
In other words, the quotient does not add new branches through Suslin trees
and Suslin algebras. Note that the intermediate c.c.c. extension may not be
given by a single real; posets such as adding ℵ1 Cohen reals with finite support
may embed to PI under suitable circumstances.
Proof. Suppose for contradiction that the intermediate extension is effected by
a c.c.c. forcing Q, a regular subforcing of PI which does not add reals. Let M be
a countable elementary submodel of a large enough structure containing Q and
26
CHAPTER 2. BACKGROUND FACTS
I. Since no real is added by Q, it forces that the intersection of its generic filter
Ġ with the model M belongs to the ground model. Fix a maximal antichain
of conditions in the poset Q deciding the value of M ∩ Ġ. The antichain is
countable, and so we can list all possible values for the intersection as {gn : n ∈
ω}. Note that each of these filters in fact Q-generic over M : every antichain
of Q is countable, therefore if it is in M then it is a subset of M and so Q
forces the filter Ġ to contain an element of such an antichain which is also in
M . Let κ = |P(PI )|, let h ⊂ Coll(ω, κ) ∩ M be a filter in V , generic over
all the models M [gn ]; such a filter exists as the models contain together only
countably many antichains of the poset
T SColl(ω, κ) ∩ M . The Borel set B of
PI -generic reals over M is equal to { D ∩ M : D ∈ M is an open dense
subset of PI }; therefore, it has a Borel code in M [h] as the model M [h] can
see that the unions and intersections forming it are countable. The set B is
I-positive in V by properness of PI and Fact 2.6.1, this positivity is an analytic
statement by the complexity assumption on I, and therefore it transfers to the
wellfounded model M [h]. Let f : B → P(Q) ∩ M be the function defined by
f (x) = {q ∈ Q ∩ M : ∃C ∈ PI ∩ M x ∈ C ∧ C forces q to belong to the Q-generic
filter}. This is a Borel function with code in the model M [h]. In that model,
there are obviously two cases: either the f -preimage of every subset of Q ∩ M
is in the ideal I, or there is a subset of Q ∩ M with Borel I-positive preimage
C ⊂ B. In the first case, the property of B and f is a coanalytic statement
since the ideal I is Π11 on Σ11 , therefore transfers to the model V , and there B
forces that the intersection of the Q-generic filter with M is not in the ground
model, contradicting the assumptions. In the latter case, C the intersection
of the Q-generic filter with M is in the model M [h] and therefore not on the
list {gn : n ∈ ω}, reaching a contradiction again.
2.7
Other tools
In several places, we will make good use of the concentration of measure phenomenon on simple finite metric spaces. The following is the simplest context:
Definition 2.7.1. A Hamming cube of dimension n is the set 2n equipped with
the normalized counting measure and the normalized Hamming distance, i.e.
d(x, y) = |{m ∈ n : x(m) 6= y(m)}|/n.
Fact 2.7.2. For every ε, δ > 0 there is n ∈ ω such that in the Hamming cube
of dimension n, the ε-neighborhood of any set of mass ≥ 1/2 has mass ≥ 1 − δ.
Note that it immediately follows that 2ε neighborhood of a set of mass ≥ ε must
have a mass 1 − ε. This yields an attractive corollary:
Corollary 2.7.3. For every ε > 0 and every m > n there is n ∈ ω such that
whenever ai : i ∈ m are subsets of the Hamming cube of dimension n of mass
≥ ε then there is an ε-ball intersecting them all.
Observe that if c ⊂ n is a set and πc : 2n → 2n is the map flipping the bits
of its entries on the set c, then πc preserves both the Hamming measure and
2.7. OTHER TOOLS
27
the Hamming metric. Thus, replacing some of the sets ai above with their πc
images, one can realize all various metric configurations between the elements
of the sets ai with ε-precision.
We will also need the following more precise calculation:
Q
Fact 2.7.4. [40, Theorem 4.2.5] Let X = i∈n Mi be a product of finite metric
spaces, each with metric di of diameter ai and a probability measure µi . Equip
X with the product measure µ and the l1 -sum d of the metrics di . For every
ε > 0 and every set of µ-mass at least 1/2, its d − ε-neighborhood has µ-mass
2
2
at least 1 − 2e−ε /16Σi ai .
Let (ω)ω be the space of all infinite sequences a of finite sets of natural
numbers with the property that min(an+1 ) > max(an ) for every number n ∈ ω.
Define a partial ordering ≤ on this space by b ≤ a if every set on b is a union of
several sets on a.
Fact 2.7.5. (Bergelson-Blass-Hindman [61, Corollary 4.49]) For every partition
(ω)ω = B0 ∪ B1 into Borel sets there is a ∈ (ω)ω such that the set {b ∈ (ω)ω :
b ≤ a} is wholly included in one of the pieces.
Suppose that han , φn : n ∈ ωi is a sequence of finite sets and submeasures on
them such that the numbers φn (an ) increase to infinity very fast. The necessary
rate of increase is irrelevant for the applications in this book; suffice it to say
that it is primitive recursive in n and the sizes of the sets {am : m ∈ n}.
Fact 2.7.6. [50] For every partition Πn an × ω = B0 ∪ B1 into two Borel pieces,
one of the pieces contains a product Πn bn × c where each bn ⊂ an is a set of φn
mass at least 1 and c ⊂ ω is infinite.
28
CHAPTER 2. BACKGROUND FACTS
Chapter 3
Tools
3.1
Integer games connected with σ-ideals
Many of the σ-ideals considered in this book have integer games associated with
them. As a result, they satisfy several interconnected properties, among them
the selection property that will be instrumental in upgrading the canonization
results to Silver-style dichotomies for these σ-ideals.
The subject of integer games and σ-ideals was treated in [67] rather extensively, but on a case-by-case basis. In this section, we provide a general framework, show that it is closely connected with uniformization theorems, and prove
a couple of dichotomies under the assumption of the Axiom of Determinacy.
The following definition will be central:
Definition 3.1.1. Let I be a σ-ideal on a Polish space X. We say that I is a
game ideal if there is
1. a Borel basis B ⊂ ω ω × X for I–a Borel set all of whose vertical sections
are in I and such that every set in I is a subset of one of vertical sections
in B;
2. a scrambling tool –a Borel function f : ω ω → X × ω ω
such that for every closed set C ⊂ X × ω ω , Player I has a winning strategy in
the game G(C) if and only if the projection of the set C to the space X is in
I. Here, in the infinite game G(C), Players I and II alternate to play points
y, z ∈ ω ω respectively and Player II wins if f (z) ∈ C and the first coordinate of
the point f (z) does not belong to the vertical section By .
Note that the games G(C) as in the definition above are all Borel and therefore
determined by [35]. The notion is going to be used only for σ-ideals in this
book, but it appears to be useful even in the more general context of hereditary
collections of subsets of Polish spaces.
The meaning of the definition must be illustrated on examples. The existence
of a Borel basis for a σ-ideal is typically easy to verify. Thus, the σ-ideal of
29
30
CHAPTER 3. TOOLS
countable subsets of X = 2ω has a Borel basis B naturally indexed by the space
(2ω )ω ), and hy, xi ∈ B ↔ ∃n x = y(n). To produce a Borel basis B for the σideal of meager subsets of X = 2ω , note that the collection Z of closed nowhere
dense subsets of 2ω is Gδ in K(2ω ) and therefore Polish, index the basis by the
space Z ω , and let hy, xi ∈ B ↔ ∃n x ∈ y(n). There are many σ-ideals of Borel
sets without a Borel basis. One way how that can happen is when the σ-ideal
in question is not σ-generated by Borel sets of fixed Borel rank. Such is the
case of the E0 -ideal of Section 5.4. A more subtle failure may occur in the case
when I is σ-generated by Borel sets of a fixed Borel rank (e.g. closed sets), but
the generators cannot be organized into a Borel collection. Such is the case of
the σ-ideal generated by closed sets of uniqueness by a deep theorem of [8].
The scrambling tool is a more sophisticated concept. To motivate it, suppose
that A ⊂ X is a set, and attempt to design an integer game which detects the
membership of A in the σ-ideal I: Player I produces a set in the basis, and
Player II produces a point in X. Player II wins if his point is in the set A
and outside of the set Player I played. We would like to set the game up
so that Player I has a winning strategy if and only if A ∈ I. For that, we
must make the game as easy for Player II as possible; this is accomplished by
applying a scrambling tool to his moves so that during the play, Player I cannot
infer virtually anything about Player II’s resulting point from his moves. The
scrambling tool is typically obtained after a suitable basis is identified, and it
will have higher Borel rank than the basis.
For many σ-ideals–such as those investigated in [67, Chapter 4]–an integer
game has been manually produced independently of the concerns of the present
book, and these games can be trivially restated in the language of a basis and a
scrambling tool. We will show how this is done in the typical case of the Laver
ideal of Section 5.13, and leave other similar examples to the reader.
Theorem 3.1.2. The Laver ideal I on ω ω is a game ideal.
Proof. The σ-ideal I is generated by sets Ag = {x ∈ ω ω : ∃∞ n x(n) ∈
/ g(x n)}
ω <ω
as g varies over all functions from ω
. This obviously yields a Borel basis for
I: just find a bijection between ω and ω <ω , extend it to a continuous bijection
<ω
h between ω ω and ω ω , and let B ⊂ ω ω × ω ω be described by the formula
By = Ah(y) .
Now revisit the Laver game exhibited for example in [67, Section 4.5.2].
Suppose that C ⊂ ω ω × ω ω is a closed set, and consider the game G ∗ (C) in
which Player II first indicates a finite sequence t ∈ ω ω , and then in each round
i Player I plays ni ∈ ω, Player II responds with mi > ni , and at some rounds
of his choice Player II also produces another natural number ki . Player II wins
if h(ta m0 , m1 , . . . ), (k0 , k1 , . . . )i ∈ C. [65] shows that Player I has a winning
strategy in the game G∗ (C) if and only if the projection of C is I-positive.
To construct the scrambling tool, just ???
However, there are several general theorems that produce the scrambling
tools for various ideals just from various abstract descriptive properties of the
3.1. INTEGER GAMES CONNECTED WITH σ-IDEALS
31
ideals. Surprisingly enough, the construction of games is closely connected with
Borel uniformization theorems. The first example we give is associated with
Arsenin-Kunugui theorem [33, Theorem 18.18].
Theorem 3.1.3. Suppose that I be a σ-ideal on a Polish space X which has
a Borel basis consisting of Gδ -sets and moreover, if C ⊂ X × ω ω is a closed
set with I-positive projection then C has a compact subset with an I-positive
projection. Then I is a game ideal.
For example, the σ-ideal of φ-null sets for any subadditive outer regular capacity
φ on a Polish space is a game ideal by the Choquet’s theorem [33, Theorem 30.13]
and the above theorem.
Proof. Let B ⊂ ω ω × X be a Borel set with I-small Gδ vertical sections such
that every I-small sets is a subset of a vertical section of B. Let W be the space
of all compact subsets of X × ω ω . Use Arsenin-Kunugui uniformization theorem
[33, Theorem 18.18] to find a Borel function g : ω ω × W → X × ω ω such that
whenever hy, wi ∈ ω ω × W is a pair such that By does not cover the projection
of w then g(y) ∈ w is a point whose first coordinate is not in By . Let S be the
space of all strategies for Player I in integer games, and let h : ω ω → S × W be
a Borel bijection. Finally, let f (z) = g(σ ∗ z, w) for every z ∈ ω ω , where w is
such that h(z) = hσ, wi. We claim that f works as required.
Indeed, if C ⊂ X × ω ω is a closed set, then either its projection is in the
σ-ideal I, in which case Player I can win G(C) just by producing y ∈ ω ω such
that p(C) ⊂ By . Or, the projection of C is I-positive, and by the assumptions C
has a compact subset w with I-positive projection. In such a case, no strategy
σ can be winning for Player I in the game G(C): if z ∈ ω ω is a point such that
h(z) = hσ, wi, then the definition of the function f shows that z is a winning
counterplay for Player II against the strategy σ.
The uniformization theorem for sets with nonmeager sections [33, Theorem
18.6] yields the following:
Theorem 3.1.4. Let I be a σ-ideal on a Polish space X with a Borel basis.
If I is the intersection of meager ideals for some collection of Polish topologies
yielding the same Borel structure, then I is a game ideal. Similarly, if I is the
intersection of null ideals for some collection of Borel probability measures, then
I is a game ideal.
Proof. Both the meager and null case are similar. We start with the meager
case.Let B ⊂ ω ω × X be the Borel basis for the σ-ideal I. Let W be the
space of all partial continuous maps from ω ω to X × ω ω with Gδ domain with a
standard Borel structure on it as in [33, Definition 2.5, Proposition 2.6]. Use the
nonmeager uniformization theorem [33, Theorem 18.6] to find a Borel function
g : ω ω ×W → ω ω such that whenever hy, wi ∈ ω ω ×W is a pair such that the set
{u ∈ ω ω : the first coordinate of w(u) is not in By } is nonmeager in ω ω , then the
functional value g(y, w) picks an element from this nonmeager set. Let S be the
32
CHAPTER 3. TOOLS
space of all strategies for Player I in integer games, and let h : ω ω → S × W be
a Borel bijection. Let f (z) = g(σ ∗ z, w) for every z ∈ ω ω , where h(z) = hσ, wi.
The function f is the scrambling tool we need.
To see this, suppose that C ⊂ X × ω is a closed set whose projection is
not in I, and suppose that σ ∈ S is a strategy for Player I; we must produce
a winning counterplay for Player II in the game G(C). The assumption on the
σ-ideal I yields a topology t on the space X with the same Borel structure such
that the σ-ideal It of t-meager sets is a superset of I, and proj(C) ∈
/ It . The
set proj(C) is analytic, it has the Baire property in the topology t, and so it
is comeager in some nonempty t-open set O. There is a Borel isomorphism
k : ω ω → O which carries the meager ideal on ω ω to It . The Jankov-von
Neumann uniformization theorem yields a partial Baire-measurable function
l : ω ω → ω ω such that whenever u ∈ ω ω is a point with k(u) ∈ proj(C) then
l(u) is defined and hk(u), l(u)i ∈ C. There is a dense Gδ set P on which both
k, l are continuous maps (the continuity of k is considered with respect to the
original topology on X). Let w ∈ W be the function u 7→ hk(u), l(u)i restricted
to P . Find z ∈ ω ω such that h(z) = hσ, wi and show that z is the desired
winning counterplay for Player II. Indeed, the set Bσ∗z is in the σ-ideal I, so in
It , the set {u ∈ ω ω : g(u) ∈
/ Bσ∗z } is comeager in P , and therefore the functional
value f (z) is defined and picks a winning point in C.
The null case is similar, using the isomorphism theorem for measures ???
The class of ideals that fit into the scheme of Theorem 3.1.4 is much wider
than the class satisfying the assumption of Theorem 3.1.3. Remember though
that σ-ideals which are intersection of meager ideals or intersections of null
ideals may not have a Borel basis and so can fail to be game ideals on that
account.
Theorem 3.1.5. The following σ-ideals are intersections of meager ideals:
1. every Π11 on Σ11 σ-ideal I such that the quotient poset PI is proper and
preserves Baire category;
2. every hypergraph ideal;
3. the σ-ideal σ-generated by the classes of a given Borel equivalence relation.
Checking whether a given σ-ideal is an intersection of null ideals is much more
demanding, and also more informative. Such ideals are commonly referred to as
polar ideals in the literature. Choquet [9] proved that every strongly subadditive
capacity is the supremum of some collection of Borel probability measures, and
therefore its null ideal is a polar ideal. [67, Example 3.6.4] shows that for every
dimension h > 0, the σ-ideal generated by sets of finite h-dimensional Hausdorff
measure in a Euclidean space is polar. On the other hand, [20] works hard to
prove that the σ-ideal of σ-porous sets on the real line is not a polar σ-ideal.
Theorem 3.1.6. The following σ-ideals are intersections of null ideals:
1. the σ-ideal of sets of zero mass for a strongly subadditive capacity;
3.1. INTEGER GAMES CONNECTED WITH σ-IDEALS
33
2. the σ-ideal σ-generated by sets of finite Hausdorff measure of any fixed
dimension, on any fixed Euclidean metric space.
Faced with such a powerful machinery to produce integer games, it appears
hard to identify a σ-ideal with a Borel basis that is not a game ideal, and we
have no such an example. An obvious complexity computation shows that σideals with Borel basis are in general Π12 on Σ11 , while the game ideals are a tad
simpler:
Definition 3.1.7. A σ-ideal I on a Polish space X is ∆12 on Σ11 if for every
analytic set A ⊂ 2ω × X the collection {y ∈ 2ω : Ay ∈ I} is ∆12 .
Theorem 3.1.8. All game ideals are ∆12 on Σ11 .
Proof. Let A ⊂ 2ω × X be an analytic set, with C ⊂ 2ω × X × ω ω a closed set
projecting to it. Then, for every z ∈ 2ω , the vertical section Ay is in the ideal
I if and only if Player I has a winning strategy in the game G(Cz ) if and only
if Player II has no winning strategy in the game G(Cz ). These are a Σ12 and a
Π12 formulas respectively.
Question 3.1.9. Is every σ-ideal with a Borel basis a game ideal? Is every ∆12
on Σ11 σ-ideal with Borel basis a game ideal? Is there a simple uniformizationtype property of a σ-ideal that is equivalent to an existence of a game for I?
In the rest of this section, we will use the games to derive a couple of interesting corollaries in the context of the Axiom of Determinacy.
Theorem 3.1.10. (ZF+Axiom of Determinacy) Let I be a game ideal on a
Polish space X. If every coanalytic set is either in I or contains an analytic
subset not in I, then every set is either in I or contains an analytic subset not
in I.
The theorem has a substantial weakness in that for all game ideals we know, the
integer games can be readjusted to prove the assumption of the theorem under
AD. In the special but frequent case of σ-ideals such that the quotient forcing is
proper and preserves Baire category, the assumption on coanalytic sets can be
proved to hold from AD as shown in [7]. However, we do not know for example
whether for every outer regular capacity φ, every coanalytic subset of φ-positive
capacity contains an analytic subset of positive capacity.
Proof. Let X be the Polish domain of the σ-ideal I, let B ⊂ ω ω × X be a Borel
basis of I and f a scrambling tool as in Definition 3.1.1. Let A ⊂ X be an
arbitrary set, consider the set C = A × ω ω , and the associated game G(C). By
the determinacy assumptions, either Player I has a winning strategy or Player
II does.
If Player II has a winning strategy σ, let A0 = {x ∈ X : ∃y, z, v ∈ ω ω z = σ∗y
and f (z) = (x, v)}. This is clearly an analytic set, and since the strategy σ
is winning, it must be a subset of the set A. It also must be I-positive, since
whenever By is a generator of the σ-ideal I for some y ∈ ω ω , the first coordinate
34
CHAPTER 3. TOOLS
of the point f (σ ∗ y) is in the set A0 and not in the set By . Thus, the set A
contains an I-positive analytic subset.
If Player I has a winning strategy σ, let A00 = {x ∈ X : ∀y, z, v ∈ ω ω y = σ ∗z
and f (z) = (x, v) implies x ∈ By }. This is a coanalytic set, and certainly
A ⊂ A00 since σ was a winning strategy for Player I. We claim that that A00 ∈
I, and so A ∈ I. Otherwise, by the assumption of the theorem, A00 would
contain an analytic I-positive subset, which would be a projection of a closed
set D ⊂ X × ω ω . Player II would have a winning strategy τ in the game G(D).
The struggle between the strategies σ and τ would result in a point x ∈ A00
contradicting the definition of the set A00 .
In both cases, we verified the conclusion of the theorem.
Corollary 3.1.11. (ZF+DC+Axiom of Determinacy) Let E be a Borel equivalence relation on a Polish space X. Whenever A ⊂ X is a set, then either A is
covered by countably many E-equivalence classes or A contains a perfect subset
of pairwise E-inequivalent elements.
Note that some assumptions beyond ZF+DC are necessary to obtain this
already for coanalytic sets, since in the constructible universe L, there is an
uncountable coanalytic set without a perfect subset, and so the statement will
fail already for E = id. Unlike the Silver dichotomy, the Corollary fails for
coanalytic equivalence relations. Consider the space X of all trees on ω, and
let E be the coanalytic relation defined by x E y if and only if x = y or
(there is no level preserving homomorphic embedding of x into any of the trees
y t where t ∈ y is a node distinct from the root, and vice versa, there is
no such embedding of y into any of the trees x t). It is not difficult to see
that this is an equivalence relation that connects wellfounded trees with the
same rank, and illfounded trees form singleton equivalence classes. The set A of
wellfounded trees is coanalytic and it contains uncountably many equivalence
classes. Moreover, every perfect subset of it meets only countably many of
the classes by the boundedness theorem, and so it cannot consist of pairwise
inequivalent elements.
To derive the corollary, we have to deal with the case of analytic and coanalytic sets separately. This is in itself a quite interesting case that deserves
separate label as a theorem. It is a special case of forthcoming work of [7].
Theorem 3.1.12. Suppose that ω1 is inaccessible to the reals. Then every
Σ12 set either intersects only countably many equivalence classes of E, or else
contains a nonempty perfect subset of pairwise inequivalent elements.
Proof. We will start with the case of a Π11 set A ⊂ X. Fix a real parameter
z ∈ 2ω such that A ∈ Π11 (z) and let ≤ be a Π11 (z) rank on A. For every
countable ordinal α consider the Coll(ω, α)-extension of the model L[z] given
by a function y ∈ αω ; such a function exists by theassumption of inaccessibility
of ω1 . The model L[z][y] contains a Borel code for the Borel set Aα of all points
of A of ≤-rank < α. By the Silver dichotomy applied in the model L[x][y] to the
equivalence relation E on Aα , we see that either L[x][y] |= Aα intersects only
3.1. INTEGER GAMES CONNECTED WITH σ-IDEALS
35
countably many E-equivalence classes, or L[z][y] |= Aα contains a perfect set
of pairwise inequivalent elements. If the former case prevails, then the perfect
set of pairwise inequivalent elements retains its salient properties in V by the
coanalytic absoluteness between V and L[z][y], and we are done. Thus, we may
assume that for every α, the set Aα is forced by Coll(ω, α) to intersect only
countably many equivalence classes, for all α ∈ ω1 .
For every α ∈ ω1 fix names hτn,α : n ∈ ω in the model L[z] such that
S
Coll(ω, α) forces the set Aα to be covered by the set Ḃα = n [τn,α ]E . Note
that if E is a Borel equivalence relation of rank some β, then Ḃα is a name for a
Borel set of rank at most β +1 (independently of α). Consider the set Cα = {x ∈
X : L[z][x] |= Coll(ω, α) x̌ ∈ Ḃα }. By the work of Jacques Stern [60], this is a
Borel set and moreover, a code for it exists in every extension L[x][g] that makes
L[x]
iβ+1 countable. Such an extension exists by the inaccessibility assumption of
ω1 , and in it there are only countably many codes for Borel sets, while there are
uncountably many countable ordinals. Thus, there is a cofinal set R ⊂ ω1 and
a fixed Borel set C such that for all ordinals α ∈ R, it is the case that Cα = C.
We will show that A ⊂ C and C intersects only countably many E-classes. This
will conclude the proof for the case of a coanalytic set A.
First of all, whenever x ∈ A, then there is an ordinal α ∈ R such that
the ≤-rank of x is smaller than α. Whenever y is a Coll(ω, α)-generic filter
over L[z][x], then L[z][y] |= every element of Aα is equivalent to one of τn,α /y,
by the coanalytic absoluteness this also holds in L[z][x][y], and therefore x is
equivalent to one of τn,α /y. As y was arbitrary generic, this means that x ∈ Cα
and so x ∈ C.
To prove that the set C intersects only countably many equivalence classes,
assume for contradiction that it does not, and use the Silver dichotomy to
produce a perfect subset P ⊂ C of its pairwise inequivalent elements. Fix α
such that C = Cα , consider the Polish space Y = αω where α has the discrete
topology and Y has the product topology, and the maps f : Y → X given
by f (y) = τn,α /y. Thus, the maps fn are defined and continuous on a dense
Gδ subset of the space Y . Consider also the set D ⊂ Y × X of those points
hy, xi such that ∃n fn (y) E x. A brief review of the definitions will reveal
that Cα = {x : Dx is comeager}, while for every y ∈ Y , the vertical section
Dx ∩ P can be at most countable. Therefore the set D ∩ Y × P contradicts the
Kuratowski-Ulam theorem for the product Y × P .
This completes the proof for a coanalytic set A. If A is instead Σ12 , then it
is the image of some coanalytic set A0 under a continuous function f : ω ω → X.
The conclusion then follows from the application of the coanalytic case to the
pullback equivalence relation E 0 = f −1 E and the coanalytic set A0 .
Now we are ready for the proof of the corollary. By the Silver dichotomy,
it is enough to show that every IE -positive set contains an IE -positive analytic
subset. Since the σ-ideal IE is a game ideal by Theorem 3.1.4, it is enough to
verify this for coanalytic sets A ⊂ X. However, this isexactly the contents of
the previous theorem.
36
CHAPTER 3. TOOLS
Corollary 3.1.13. (ZF+Axiom of Determinacy) Let φ be a subadditive outer
regular capacity on a Polish space X. If every coanalytic φ-positive set contains an analytic φ-positive set, then every φ-positive set contains an analytic
φ-positive set.
Proof. This is just an application of Theorem 3.1.3 and the previous theorem.
Note that the collection of φ-null sets is a σ-ideal and in fact a game ideal, since
it satisfies the assumptions of Theorem 3.1.3 by Choquet’s theorem [33, Section
30.C].
3.2
The selection property
The most important application of game ideals and integer games in this book
is the following concept. It will lead towards many Silver-type dichotomies for
various σ-ideals.
Definition 3.2.1. A σ-ideal I on a Polish space X has the selection property
if for every analytic set A ⊂ 2ω × X with I-positive sections there is a partial
Borel map g : 2ω → X such that g ⊂ A and rng(g) ∈
/ I.
Theorem 3.2.2. All game ideals have the selection property.
Proof. Let A ⊂ 2ω × X be an analytic set with all vertical sections I-positive,
A equal to the projection of some closed set C ⊂ 2ω × X × ω ω . We will first find
a perfect set Z ⊂ Y and a continuous function τ with domain Z, assigning each
element of Z a winning strategy τ (z) for Player II in the game G(Cz ), where
Cz ⊂ X × ω ω is the vertical section of C associated with z.
To do this, consider the Sacks forcing P adding the generic point żgen to
2ω . Since the statement “Player I has no winning strategy in neither game Cz
for z ∈ 2ω ” is Π12 and true in the ground model by the assumptions, it is also
true in the Sacks extension by the Shoenfield absoluteness 2.4.1. In particular,
P Player I has no winning strategy in the game Cżgen . Since the game is
determined, there is a name τ̇ for a winning strategy for Player II in this game.
By the continuous reading of names for Sacks forcing, this name below some
condition reduces to a continuous function. Let M be a countable elementary
submodel of a large enough structure, and let Z ⊂ 2ω be a perfect set consisting
of M -generic Sacks reals below this condition only; thus, the name τ̇ can be
viewed as a continuous function e on the set Z. To see that e, Z work as
required, let z ∈ Z and look in the model M [z]. There, by the forcing theorem,
e(z) is a winning strategy for Player II in the game G(Cz ). This is a coanalytic
statement about z, e(z) and therefore is absolute between the wellfounded model
M [z] and V . Thus, e(z) indeed is a winning strategy for Player II in the game
G(Cz ).
Now, find any Borel bijection h : Z → ω ω and let g : Z → X be the
function defined in the following way: g(z) is the first coordinate of the point
f (e(z) ∗ h(z)). We claim that the function g is as required. It is certainly Borel.
For every z ∈ Z, the strategy e(z) is winning for Player II and therefore it
3.2. THE SELECTION PROPERTY
37
produces a point in Az , so g ⊂ A. Finally, rng(g) ∈
/ I, since whenever By is
some generator of the σ-ideal I and y = h(z) for some z ∈ Z, the point g(z) ∈ X
belongs to rng(g) and not to By .
The main application of the selection property in this book–cranking up the
total canonization to the Silver property for various σ-ideals–is postponed to
Section 3.3. Here, we will show that the selection property can be used to prove
total canonization itself in a certain context. The argument is very abstract and
uses a number of forcing tricks. First, in a rather surprising turn of events, we
show that the selection property rules out most nontrivial intermediate extensions of the PI extension. Then, in another unexpected twist, the trichotomy
theorem 3.4.4 shows that equivalence relations are directly connected to such intermediate extensions, and we conclude that the lack of such extensions implies
the total canonization.
In the early versions of this book, this was the path through which total
canonization was proved for a great number of σ-ideals. Later, in many cases
we found direct combinatorial arguments that avoid this path and shed more
light on the whole context. In the present version, this path is used to prove
total canonization only for the metric porosity σ-ideals of Section 5.2, reflecting
the lack of understanding of the combinatorial structure of these σ-ideals.
Theorem 3.2.3. Suppose that I is a σ-ideal on a Polish space X such that
the quotient forcing PI is < ω1 -proper in all σ-closed extensions. If I has the
selection property, then every intermediate forcing extension of the PI extension
is either a c.c.c. extension of the ground model, or it is equal to the whole generic
extension.
The wording of this theorem must be parsed carefully. First of all, σ-closed
forcing extensions add no new reals, and therefore the space X, σ-ideal I and
the poset PI remain the same in such extension. Thus, we do not need any
definability of the σ-ideal I to interpret it in σ-closed forcing extensions. Second,
a forcing P is < ω1 -proper if for every countable ordinal α and every ε-tower
hMβ : β ∈ αi of countable elementary submodels of a large structure such as
hVκ , εi for a sufficiently large ordinal κ, with P ∈ M0 , for every condition p ∈ P ∩
M0 there is a strengthening q ≤ p which is simultaneously Mβ -master for each
ordinal β ∈ α. This is a technical strengthening of properness due to Shelah; it is
satisfied for all σ-ideals in this book as proved by a standard transfinite induction
on α in each case. Moreover, in each case the proofs just use the definition of
the σ-ideals which remains the same in any σ-closed forcing extension, and
therefore yield < ω1 -properness in all σ-closed extensions. Ishiu [21] proved
that < ω1 -properness of a poset is equivalent to the Axiom A property of a
partial order in forcing sense equivalent to the original poset. For the purposes
of the theorem, this property does not seem to be replaceable by anything
descriptive set theoretic in nature. The conclusion cannot be strengthened to
rule out intermediate c.c.c. extensions as such examples as the Cohen forcing
(associated with the meager ideal, which is a game ideal and therefore does have
the selection porperty) will show.
38
CHAPTER 3. TOOLS
Proof. Let λ = c. Let A be a nowhere c.c.c. complete subalgebra of the
completion of PI ; we must show that the generic filter on PI can be recovered from its intersection with A. Note that A contains a term for a function
f : λ → λ such that for every condition p ∈ PI there is δ ∈ λ such that the
set {γ ∈ λ : ∃q ≤ p f˙(δ) = γ} is uncountable; here λ = cV . To see this, let
hpδ : δ ∈ κi be an enumeration of PI , let Aδ be an uncountable antichain of
conditions in A still compatible with pδ with some fixed enumeration, and let
f˙(δ) = γ if the generic filter contains the γ-th element of Aδ , and f˙(δ) =trash if
the generic filter contains no elements of Aδ . We will show that PI forces that
ẋgen can be recovered from f˙; this will be enough. The rest of the argument
does not use the complete algebra A anymore.
Move to any σ-closed generic extension V [G] in which the continuum hyV [G]
pothesis holds; so |λ| = ℵ1 . The poset PIV = PI
is < ω1 -proper and the
σ-ideal I still has the selection property. The rest of the proof takes place in
this extension.
Let M be a countable elementary submodel of a large structure containing
I, f˙. Write BM for the Borel set of all points in X PI -generic over M , and write
FM : BM → (λ ∩ M )λ∩M for the function defined by FM (x) = {hδ, γi : δ ∈
λ ∩ M, γ ∈ λ ∩ M, and for some condition B ∈ PI ∩ M such that B f˙(δ) = γ),
x ∈ B}. Note that FM (x) really is a function from λ ∩ M to itself since x is a
PI -generic point over M . Topologizing λ ∩ M with the discrete topology and
(λ ∩ M )λ∩M with the product topology, it is clear that FM is a Borel function.
Note that M ∩ V ∈ V by the σ-closedness of the extension V [G], and therefore
BM , FM ∈ V and the definition of FM above is a definition from parameters
in V . Also, if N ∈ M are two elementary submodels and x ∈ X is a point
PI -generic over both N and M , it is the case that FN (x) = FM (x) N .
We will produce a countable elementary submodel M of a large structure
and an I-positive Borel set B ⊂ BM such that FM B is one-to-one. This
will conclude the proof. For suppose that x ∈ B is a PI -generic point over
V [G]. In the model V [x] the Borel function FM (meaning the function with
the same definition as FM from the parameter M ) is still one-to-one by coanalytic absoluteness between V [G] and V [x], FM (x) = f˙/x M . The coanalytic
absoluteness between the models V [f˙/x] and V [x] will yield a point y ∈ B in
V [f˙/x] such that FM (y) = f˙/x M . Since the function FM B is one-to-one,
we conclude that y = x and so V [f˙/x] = V [x] as desired.
The set B is obtained by an application of the selection property. We will
find a Borel set C ⊂ 2ω ×X such that the vertical sections of C are I-positive and
consist of M -generic points for PI only, such that for distinct vertical sections
of C have disjoint FM -images. Once this is done, the selection property yields
a partial Borel function g : 2ω → X, whose range is I-positive. Since the
vertical sections of the set C are pairwise disjoint, the function g is injective,
and therefore its range B is a Borel I-positive set. The set B obviously has the
desired property.
Note that (regardless of the selection property) the vertical sections of C
form a perfect collection of pairwise disjoint conditions in PI . This should
3.2. THE SELECTION PROPERTY
39
be compared with the work of [?, Theorem 6.1], where a non-c.c.c. suitably
definable poset is produced which has no perfect antichain, and the construction
can produce a δ-proper poset for any ordinal δ ∈ ω1 fixed beforehand. The rest
of our argument uses only the < ω1 -properness of the poset PI in the σ-closed
forcing extension V [G]. The following claim is key:
~ 0 and M
~ 1 of
Claim 3.2.4. For every a ∈ Hc+ , there are countable towers M
~ 0)
countable elementary submodels of Hc+ with respective largest models max(M
~
and max(M1 ) such that
~ 0 (0), M
~ 1 (0), λ ∩ max(M
~ 0 ) = λ ∩ max(M
~ 1 );
1. a ∈ M
~ 0 and ev2. for every x ∈ X which is PI -generic over each model on M
~ 1 , it is the case that
ery y ∈ X which is generic over each model on M
Fmax(M
~ 0 ) (x) 6= Fmax(M
~ 1).
Note that the functions Fmax(M
~ 0 ) (x), Fmax(M
~ 1 ) have the same domain, so the
claim really ascertains that they have a disagreement in the values.
The construction of the required model M and the Borel set C follows im~ n,0
mediately from the claim. By induction on n ∈ ω build countable towers M
~
and Mn,1 such that the pair has the property from the claim and the tow~ n−1,0 , M
~ n−1,1 belong to the first models on the n-indexed towers. Let
ers M
M be the union of all models on the towers; this is an increasing union of
countable elementary submodels of Hc+ , and so is elementary as well. Define C ⊂ 2ω × X by hy, xi ∈ C if x is PI -generic over every model on all
~ n,y(n) : n ∈ ω. This is a Borel set. By the < ω1 -properness,
the towers M
the vertical sections of the set C are I-positive. The claim shows that vertical sections of C have disjoint FM -images as desired. To see this, let y0 6= y1
and x0 , x1 ∈ X be such that hy0 , x0 i, hy1 , x1 i ∈ C and n ∈ ω is such that
y0 (n) = 0 6= 1 = y1 (n), then Fmax(M
~ 0 ) (x0 ) ⊂ FM (x0 ), Fmax(M
~ 1 ) (x1 ) ⊂ FM (x1 ),
n
n
Fmax(M
~ 0 ) (x0 ) 6= Fmax(M
~ 1 ) (x1 ), and so FM (x0 ) 6= FM (x1 ).
n
n
~ 0 be any ω1 tower of countable elementary
For the proof of the claim, let M
0
submodels of some large structure such that the first model contains the set a,
~ 1 be a tower of length 1, containing just one model, which in turn
and let M
~ 0 . Let δ ∈ ω1 be the intersection of this single model with
contains the tower M
0
0
~0 = M
~ δ. We claim that the two towers M
~ 0, M
~ 1 work as required
ω1 and let M
0
in the claim.
~ 0 (α) ∩ λ : α ∈ ω1 }. This is a closed unbounded set of
Indeed, let D = {M
0
countable subsets of λ. The choice of the name f˙ shows that there must be an
element of D which is not closed under f˙. The single model on the sequence
~ 1 contains the set D as an element. Thus, if x ∈ X is a point generic for PI
M
~ 1 , there is α ∈ δ such that the set M
~ 0 (α)
for the single model on the sequence M
0
is not closed under the function f˙/x. On the other hand, if x is a point generic
~ 1 , all the sets M
~ 0 (α) : α ∈ δ are
for PI for all the models on the sequence M
closed under the function f˙/x. This concludes the proof of the claim and of the
theorem.
40
CHAPTER 3. TOOLS
Theorem 3.2.5. Suppose that I is a Π11 on Σ11 σ-ideal on a Polish space X
such that the quotient forcing PI is < ω1 -proper in all σ-closed extensions. If
I has the rectangular and selection properties, then I has total canonization for
analytic equivalence relations.
The Π11 on Σ11 part of the assumptions is used only to weed out possible
c.c.c. intermediate extensions of the PI -extension in the application of the
trichotomy theorem. The selection property assumption allows for cranking up
the conclusion to the much stronger Silver property of the σ-ideal I, as spelled
out in Section 3.3.
Proof. First note that the assumptions imply that there is no model of ZFC
strictly between the ground model and the PI -extension of it. First, Theorem 3.2.3 shows that such an intermediate model would have to be c.c.c. and
this possibility is ruled out essentially by the dicussion of all the (very unlikely)
cases. A nontrivial c.c.c. intermediate extension must contain a new real by
Theorem 2.6.9. Thus, let B ∈ PI be a condition forcing that f (ẋgen ) is such a
real for some fixed Borel function f : X → Y . Let J be a σ-ideal on the Polish
space Y defined by A ∈ J ↔ f −1 A ∈ I. Since f (ẋgen ) is forced to be a c.c.c.
real, falling out of all J-small sets, it must be the case that it is PJ generic. If
the ideal I is Π11 on Σ11 , then so is J. However, the rectangular property rules
out all nontrivial possibilities here, as Fact 2.6.6 shows.
Now, let B ⊂ X be a Borel I-positive set and E an analytic equivalence
relation on it. Let x ∈ B be a V -generic point for the poset PI and look at
the trichotomy theorem 3.4.4. The intermediate model case is ruled out by the
previous paragraph. If V [x] = V [x]E then there is an I-positive Borel set C ⊂ B
consisting of pairwise inequivalent points. On the other hand, if V [x]E = V ,
then there is a Borel I-positive set C ⊂ B on which E is ergodic. If every class
of E C was in the σ-ideal I, the rectangular property would have to produce
I-positive Borel sets C0 , C1 ⊂ C such that C0 ×C1 ∩E = 0; this would contradict
the ergodicity. Thus, in this case there has to be a Borel I-positive subset of C
consisting of pairwise E-equivalent elements, and the theorem follows.
Reviewing the assumptions of the previous theorem, it becomes clear that
very many interesting ideals are Π11 on Σ11 , < ω1 -proper, and have the selection
property by virtue of being game ideals. Such ideals appear in all the main
sections of [67, Chapter 4], and include, among others, the σ-ideals generated
by Borel collection of closed sets, σ-ideals generated by the sets of finite mass
for a fixed Hausdorff measure, or the σ-ideals of sets of zero mass for a fixed
pavement submeasure. The main problem on the path to total canonization for
these σ-ideals appears to be the verification of the rectangular property. Alas,
this is a difficult problem which in all the interesting borderline cases is open,
without any clear guidelines as to how it should be resolved.
Question 3.2.6. What is the status of the rectangular Ramsey property for
the following σ-ideals?
1. the σ-ideal σ-generated by sets of finite mass for a fixed Hausdorff measure;
3.3. A SILVER-TYPE DICHOTOMY FOR A σ-IDEAL
41
2. the σ-ideal σ-generated by sets of continuity of the Pawlikowski function;
3. the σ-ideal of sets of Newtonian capacity zero.
However, the connection between the rectangular Ramsey property and canonization is not as tight as the previous theorem may suggest. Total canonization may hold even where the rectangular property fails–such is the case of the
σ-ideal σ-generated by closed sets of zero Lebesgue mass, see Theorem 5.1.9.
Rectangular property and total canonization may fail and still there may be
useful canonization features–such is the case for the Laver ideal of Section 5.13.
3.3
A Silver-type dichotomy for a σ-ideal
The main purpose of the selection property and the integer games in this book
is that they can be used to upgrade total canonization results to Silver-style
dichotomies, which are significantly more striking. Recall from the introduction:
Definition 3.3.1. A σ-ideal I on a Polish space X has the Silver property for
a class E of equivalence relations if for every I-positive analytic set B ⊂ X and
every equivalence relation E ∈ E on the set B, either B can be decomposed
into countably many equivalence classes and an I-small set or there is a Borel
I-positive set C ⊂ B such that E C = id. If E is equal to the class of all Borel
equivalence relations, it is dropped from the terminology and we speak of the
Silver property of I instead.
This should be compared with the classical Silver dichotomy 2.2.6, which
establishes the Silver property for the ideal of countable sets. Observe that
unlike total canonization, the Silver property introduces a true dichotomy: the
two options cannot coexist. It also has consequences for undefinable sets: if an
equivalence relation E ∈ E has an I-positive set A ⊂ X consisting of pairwise
E-inequivalent points, then it has an I-positive Borel set B ⊂ X consisting of
pairwise E-inequivalent points, simply because the first clause of the dichotomy
cannot hold in such circumstances.
Let us offer a perhaps artificial reading of the dichotomy which nevertheless
fits well with the techniques developed in this book or [67]. Given a σ-ideal I
on a Polish space X and a Borel equivalence E on X, consider the ideal I ∗ ⊃ I
σ-generated by E-equivalence classes and sets in I. Then either I ∗ is trivial,
containing the whole space, or else the quotient forcing PI ∗ is equal to PI below
some condition.
Proposition 3.3.2. Suppose that I is a Π11 on Σ11 σ-ideal on a Polish space
X such that every analytic I-positive set contains a Borel I-positive subset and
the quotient poset PI is proper. The following are equivalent:
1. I has total canonization and the selection property;
2. I has the Silver property;
42
CHAPTER 3. TOOLS
3. Iˆ has the Silver property, where Iˆ is the σ-ideal on X × ω ω consisting of
the sets with I-small projection.
If the σ-ideal is ∆12 on Σ11 , then the conclusion of the proposition remains in
force under the additional assumption that ω1 is inaccessible to the reals.
Proof. First, the (1)→(3) direction. Let E be a Borel equivalence relation on
X × ω ω , let A ⊂ X × ω ω be an analytic set and consider the set C = {x ∈ A :
ˆ this set is again analytic by the definability assumption on I, or
[x]E ∩ A ∈
/ I};
if the σ-ideal I is merely ∆12 on Σ11 , then the set C is ∆12 .
Case 1. Either there are only countably many equivalence classes in the set
C. In this case, the set C is even relatively Borel in A and A \ C is analytic. If
A \ C ∈ Iˆ then we are in the first clause of the Silver property. On the other
hand, if A \ C ∈
/ Iˆ then the properness assumption shows that there is a Borel
I-positive set B ⊂ p(A \ C) and a Borel function f : B → A \ C such that for
every x ∈ X, x is the first coordinate of f (x). Let F be the equivalence relation
on B given by x F y iff f (x) E f (y). The equivalence F has I-small classes
by the definition of the set C; the total canonization yields a Borel I-positive
subset B 0 ⊂ B consisting of pairwise F -inequivalent elements. We may arrange
that f 00 B 0 is Borel by thinning out B 0 if necessary; the set f 00 B 0 ⊂ A witnesses
the second clause of the Silver property.
Case 2. If Case 1 fails, then the classical Silver dichotomy [52] in the case
of a Π11 on Σ11 σ-ideal, or Theorem 3.1.12 in the case of a ∆12 on Σ11 σ-ideal,
provides a perfect set P ⊂ C of pairwise inequivalent points. Let D ⊂ P × X be
the set defined by hy, xi ∈ D ↔ Ax ∩ [y]E 6= 0; this is a Borel set with I-positive
vertical sections. The selection property of the σ-ideal I yields a partial Borel
function f : P → X such that ∀y ∈ dom(f ) hy, f (y)i ∈ D and rng(f ) ∈
/ I. The
properness assumption applied through Fact 2.6.2 yields a Borel I-positive set
B ⊂ rng(f ) and a Borel function g : B → A such that for every x ∈ dom(g),
the point g(x) ∈ Ax belongs to some [y]E such that x = f (y). Thinning out B
if necessary, we may assume that the set g 00 B is Borel. It witnesses the second
clause of the Silver property!
The implication (3)→(2) is immediate from the definitions; thus, we are
left with the (2)→(1) implication. The Silver property clearly implies total
canonization. If E is a Borel equivalence relation on X and A ⊂ X is an Ipositive analytic set, then either A decomposes into countably many equivalence
classes and a set in I, in which case one of the equivalence classes in A must be
I-positive, or A contains a Borel I-positive subset consisting of pairwise inequivalent elements. In both cases, the total canonization follows. The verification of
the selection property is significantly harder. Let A ⊂ 2ω × X be an analytic set
with I-positive vertical sections. Consider the set B = {x ∈ X :the horizontal
section Ax is uncountable}; this is an analytic set. There are two cases.
Case 1. Either, B ∈ I. By the first reflection theorem 2.1.2, there is a
Borel set C ⊂ X such that B ⊂ X \ C ∈ I. By the first reflection theorem
again, there is a Borel set A0 ⊃ A ∩ (2ω × C) with nonempty and countable
horizontal sections. By the Novikov uniformization theorem 2.1.3, there are
3.3. A SILVER-TYPE DICHOTOMY FOR A σ-IDEAL
43
Borel functions fn : C → 2ω for n ∈ ω such that the inverses of the graphs
of fn cover the set A0 . Let Fn be the equivalence relation on C ⊂ X defined
by x Fn y if fn (x) = fn (y). Let Cn = {x ∈ C : hfn (x), xi ∈ A}. The Silver
dichotomy for the σ-ideal I yields a split into two subcases.
Case 1a. Either there is a number n ∈ ω and an I-positive Borel set
D ⊂ Cn such that D consists of fn -inequivalent elements. In that subcase, the
properness of the quotient forcing PI can be used to thin out D if necessary to
get rng(fn D) to be a Borel set as in Fact 2.6.2(3), and the inverse of fn D
will be the function required in the selection property.
Case 1b. Or, for every number n ∈ ω, the set Cn decomposes into countably
many Fn -equivalence classes (sets {x ∈ Cn : fn (x) = ynm } for some points
ynm ∈ 2ω ) and an I-small set Dn ⊂ ω. This subcase is impossible, though:
S let
z ∈ 2ω \ {ynm : n, m ∈ ω} be an arbitrary point, and let x ∈ Az ∩ C \ n Dn
be another arbitrary point. The ordered pair hz, xi must then both belong to
A and not belong to it, which yields a contradiction.
Case 2. Or, B ∈
/ I. Thinning out the set B, we may assume that it is
Borel. By a Shoenfield absoluteness argument, B forces in PI the set Aẋgen
to be uncountable. By a result of Velickovic and Woodin, [62, Theorem 3], in
proper extensions adding new reals the set of ground model points does not
contain a nonempty perfect subset. Therefore, the proper forcing PI forces that
the analytic set Aẋgen contains a point ẋ which is not in the ground model.
Thinning out the set B further if necessary, we may assume that there is a
Borel function f : B → X such that B ẋ = f˙(ẋgen ). Since ẋ is forced not
to belong to the ground model, preimages of singletons under the function f
must be I-small, and another application of the Silver dichotomy for I shows
that we may thin out B such that f B is one-to-one. Thinning out further if
necessary, we may assume that for every x ∈ B, hf (x), xi ∈ A and rng(f B)
is Borel. It is clear that the inverse of f B yields a function as required in the
definition of the selection property.
A quick perusal of the argument shows that the proposition in fact holds even
if restricted to any class of Borel equivalence relations closed under reduction.
Thus for example the ideal I(F in × F in) introduced in Section 5.1b is Π11
on Σ11 , has the selection property, and has canonization for equivalences below
EKσ , even though it perhaps does not have total canonization. The proof shows
that in fact I(F in × F in) has the Silver property for the class of equivalences
reducible to EKσ .
The Silver property of a σ-ideal can be abstractly cranked up to an ostensibly
stronger statement recently introduced by Ben Miller:
Definition 3.3.3. Let I be a σ-ideal on a Polish space X. We say that I has
the finite multiple Silver property if for every analytic set A ⊂ X, every n ∈ ω,
and every collection {Fi : i ∈ n} of Borel equivalence relations on X, either
A can be covered by contably many equivalence classes of the various equivalences and an I-small set, or it contains a Borel I-positive subset consisting of
elements pairwise simultaneously Fi -inequivalent for all i ∈ n. The σ-ideal I
44
CHAPTER 3. TOOLS
has the infinite multiple Silver property if this holds even for countably infinite
collections of Borel equivalence relations.
Ben Miller proved what amounts to the infinite multiple Silver property of
the σ-ideal of countable sets. Here, we will show that the simple Silver property
automatically yields the finite multiple Silver property for many σ-ideals I.
Theorem 3.3.4. Suppose that ω1 is inaccessible to the reals. If I is ∆12 on Σ11
σ-ideal on a Polish space X with the Silver property, then it also has the finite
multiple Silver property.
The proof relativizes to yield the same result restricted to any class of Borel
equivalence relations closed under reducibility. The complexity computations do
not seem to yield a general ZFC result even for the class of Π11 on Σ11 σ-ideals.
Note that all game σ-ideals are ∆12 on Σ11 .
Proof. Let I be a ∆12 on Σ11 σ-ideal on a Polish space X with the Silver property.
Let E be a Borel equivalence relation on X and consider the σ-ideal IE σgenerated by I and all equivalence classes of E. We will show that IE is again
a ∆12 on Σ11 σ-ideal with the Silver property.
The theorem immediately follows. Suppose that {Fi : i ∈ n} is a finite list
of Borel equivalence relations and A ⊂ X is an analytic set. Let Ji be the
σ-ideal σ-generated by I and the equivalence classes of the various equivalences
{Fj : j ∈ i}, for all i ≤ n. A repetitive use of the previous paragraph shows that
all of these σ-ideals have the Silver property. Now, there are two cases. Either,
A ∈ Jn ; then set An = A and by downward induction on i ∈ n find countable
S
collections Ci of equivalence classes of S
Fi such
S that the set Ai = Ai+1 \ Ci is
in the σ-ideal Ji . In the end, A ⊂ A0 ∪ i<n Ci is covered by countably many
classes of the various equivalence relations and an I-small set. Or, A ∈
/ Jn ; then
set An = A and by downward induction on i ∈ n use the Silver property of Ji
to find a Ji -positive Borel set Ai ⊂ Ai+1 consisting of pairwise Fi -inequivalent
elements. In the end, the set A0 is a Borel I-positive set consisting of elements
pairwise simultaneously inequivalent for all of the equivalence relations on the
finite list.
Now back to the σ-ideal IE . First observe that it is indeed a ∆12 on Σ11
σ-ideal. If A ⊂ X is an analytic set, the following are equivalent:
• A ∈ IE ;
• there is a countable set D ⊂ X such that A \ [D]E ∈ I;
• every analytic I-positive subset of A contains two distinct E-equivalent
points.
The equivalence of the first two items is just the definition of IE , and the
equivalence of the other two is exactly the Silver property of I. The second
item is obviously Σ12 by the complexity assumption on I. The third item is Π12 :
it asserts that for every analytic set B ⊂ X, either A ∩ B ∈ I or there are points
3.3. A SILVER-TYPE DICHOTOMY FOR A σ-IDEAL
45
x0 6= x1 in A ∩ B with x0 E x1 , and the complexity again follows from the
complexity assumption on I. (It is exactly this complexity computation that
seems to fail if we attempt to prove a ZFC theorem for Π11 on Σ11 σ-ideals only.)
The main point of the proof is the verification of the Silver property of the
σ-ideal IE . So let F be a Borel equivalence relation on the space X and let
A ⊂ X be an analytic set. Consider the set C = {x ∈ A : [x]F ∩ A ∈
/ IE }.
This is a ∆12 set by the previous paragraph. By Theorem 3.1.12, either C is
covered by countably many F -equivalence classes, or E contains a perfect set
of F -inequivalent elements.
Case 1. If C is covered by countably many F -equivalence classes, then it is
relatively Borel in A and the set A \ C is still analytic. By the Silver property of
I, there are two subcases. Either A \ C ∈ IE , and the verification of the Silver
property for IE , A, and F is complete. Or, A \ C contains a Borel I-positive
subset D consisting of pairwise E-inequivalent elements. Now, the definition of
the set C implies that every F -equivalence class in the set D is in I, and we
can apply the Silver property of the σ-ideal I to find a Borel I-positive subset
D0 ⊂ D consisting of F -inequivalent elements. This set must be IE -positive, and
so the verification of the Silver property of IE in this subcase is again complete.
Case 2. There is a perfect set D0 ⊂ C consisting of pairwise F -inequivalent
elements. We will now work towards a Borel I-positive set D0 ⊂ C consisting
of pairwise simultanously E- and F -inequivalent elements. This will be the IE positive Borel set witnessing the Silver property of IE in this case. There are
again two (not mutually exclusive) subcases.
Case 2a. There is a perfect subset D1 ⊂ D0 such that the set [D1 ]F ∩ A
contains only countably many I-positive E-classes. In this subcase, the set
Z ⊂ D1 × X given by hy, xi ∈ Z if x ∈ A, y F x, and [x]E ∩ [D1 ]F ∩ A ∈ I is
analytic with pairwise disjoint I-positive vertical sections: each vertical section
Dy is obtained from the IE -positive set [y]F ∩ A by removing at most countably
many classes, and therefore it must be I-positive. The Silver property (more
precisely, the selection property, which is implied by the Silver property by
Proposition 3.3.2) of the σ-ideal I shows that there must be a Borel I-positive
set D2 which meets each vertical section of the set Z in at most one point, and
every point of D2 belongs to some vertical section of the set Z. Now, such a set
D2 consists of pairwise F -inequivalent elements, and the E-classes in it must
belong to I. The Silver property of I applied again provides an I-positive Borel
subset of D2 consisting of pairwise E-inequivalent elements, and we are done in
this subcase.
Case 2b. In the case that no such perfect subset D1 ⊂ D0 exists, we will
use two claims of independent interest.
Claim 3.3.5. Suppose that ω1 is inaccessible to the reals. Suppose that G is
a Borel equivalence relation on a Polish space Y and U ⊂ 2ω × Y is a ∆12 set
whose vertical sections cannot be covered by countably many G-classes. Then
there is a perfect set P ⊂ 2ω and an analytic set U 0 ⊂ U such that none of
its vertical sections indexed by elements of P can be covered by countably many
G-classes.
46
CHAPTER 3. TOOLS
Proof. We will repeatedly use the following simple fact: if Z is Polish and
C ⊂ 2ω × Z is a coanalytic set with nonempty vertical sections, then there is a
nonempty perfect set P ⊂ 2ω and a continuous map g : P → Z such that g ⊂ C.
To see this, first use the Kondo uniformization to find a coanalytic total map
h : 2ω → Z such that h ⊂ C. Such a map is Baire measurable, since preimages
of open sets are ∆12 and therefore have the Baire property by the assumption
about ω1 . Therefore h will be continuous on a dense Gδ set. Let P be any
perfect subset of this dense Gδ set, and let g = h P .
Towards the proof of the claim, choose a coanalytic set C ⊂ 2ω × Y × ω ω
which projects to U . By the determinacy assumption and Theorem 3.1.12,
for every y ∈ 2ω there is a nonempty compact perfect set K ⊂ Y which is
a subset of Uy and consists of pairwise G-inequivalent points. Applying the
first paragraph, thinning out the set K if necessary, we may also produce a
continuous function h : K → ω ω such that the formula φ(y, K, h) =“for every
x ∈ K, hy, x, h(x)i ∈ C (and K consists of pairwise G-unrelated points)” holds.
Note that the formula φ(y, K, h) is coanalytic. Applying the first paragraph
again, produce a nonempty perfect set P ⊂ 2ω and a continuous map g : P →
K(X × ω ω ) such that for every y ∈ P , g(y) is a graph of some function h with
domain K such that φ(y, K, h) holds. Let U 0 = {hy, xi : x is in the domain of
g(y)}.
Claim 3.3.6. Suppose that G is a Borel equivalence relation on a Polish space
Y and U ⊂ 2ω × Y is an analytic set such that vertical sections of U cannot
be covered by countably many G-classes. Then there is a partial continuous
injection f : 2ω → Y with perfect domain such that f ⊂ U and rng(f ) consists
of pairwise G-unrelated points.
Proof. Note that the σ-ideal J σ-generated by E-equivalence classes is a game
ideal by the remarks after Theorem 3.1.4. Therefore, there is a partial Borel
selection g ⊂ U whose range is J-positive as in Theorem 3.2.2. The set rng(g)
is analytic and cannot be covered by countably many equivalence classes. By
Silver dichotomy or Corollary 2.2.7, there is a perfect set P ⊂ rng(g) consisting
of pairwise inequivalent elements. The Jankov–von Neumann uniformization
theorem 2.1.6 yields a Baire measurable inverse g −1 of g on P , which is then
continuous on a dense Gδ subset P 0 ⊂ P . Let Q be the image of P 0 under g −1 ,
let f = g Q and if necessary thin out Q to a nonempty perfect subset on which
f is continuous.
Now, choose any Borel bijection π : 2ω × 2ω → D0 , and define the set U ⊂
2ω × X by hy, xi ∈ U if there is z ∈ 2ω such that x F π(y, z) and the set
[x]E ∩ [π 00 ({y} × 2ω )]F is I-positive. This is a ∆12 set, its vertical sections are
pairwise F -unrelated, if hy, xi ∈ U then [x]E ∩ Uy is I-positive and moreover, no
vertical section of U is covered by countably many E-equivalence classes since
Subcase 2a fails. Use the first claim above to find a perfect set P ⊂ 2ω and an
analytic subset U 0 ⊂ U such that for every y ∈ P , the vertical section Uy0 cannot
be covered by countably many E-classes. The second claim yields a perfect set
Q ⊂ P and a continuous function f : Q → X such that f ⊂ U and the range
3.4. EQUIVALENCE RELATIONS AND MODELS OF SET THEORY
47
of f consists of pairwise E-inequivalent elements. Let D ⊂ Q × X be the
analytic set defined by hy, xi ∈ D if hy, xi ∈ U 0 and x E f (y). By the selection
property of the σ-ideal I again, there is a partial Borel uniformization of the
set D whose range is I-positive. The range is the desired analytic I-positive set
which consists of pairwise E- and F -unrelated elements.
3.4
Equivalence relations and models of set theory
Let I be a σ-ideal on a Polish space X. The key idea underlying much of the
current of thought in this book is the correspondence between Borel equivalence
relations on X and intermediate forcing extensions of the generic extension given
by the quotient poset PI of Borel I-positive sets ordered by inclusion. The
correspondence is easiest to illustrate on smooth equivalence relations. If E is a
smooth equivalence on X and f : X → 2ω is a Borel function reducing it to the
identity, then the model V [f (ẋgen )] depends only on E and not on the choice
of the reduction. However, we need to find a sufficiently general definition that
covers the nonsmooth case, where most of interest and difficulty lies.
3.4a
The model V [x]E
If x ∈ X is a V -generic point, we consider a ZFC model V [x]E that depends on
the E-equivalence class of x, not on x itself. The correct formalization:
Definition 3.4.1. Let E be an analytic equivalence relation on a Polish space
X. Let x ∈ X be a V -generic point. Let κ be a cardinal greater than (2|P | )V
where P ∈ V is a poset from which the generic extension is derived, and let
H ⊂ Coll(ω, κ) be a V [x]-generic filter. The model V [x]E is the collection of all
sets hereditarily definable from parameters in V and from the equivalence class
[x]E in the model V [x][H].
It is clear that V [x]E is a model of ZFC by basic facts about HOD type models
[22, Theorem 13.26]. The basic observation:
Theorem 3.4.2. Let X be a Polish space, E an analytic equivalence relation
on it, and let x ∈ X be a V -generic point.
1. V [x]E ⊂ V [x];
2. the definition of V [x]E does not depend on the choice of the cardinal κ or
the filter H;
3. if y ∈ X is a V -generic point such that x E y, then V [x]E = V [y]E .
The correct way to interpret (3) is that V is an inner model of some larger
universe containing the points x, y. We only need V to be a transitive model
of some large fragment of ZFC; in particular, the case where V is perhaps
48
CHAPTER 3. TOOLS
countable in the larger universe is allowed. A typical context may arise when
V is the transitive collapse of some countable elementary submodel of a large
structure.
Proof. Fix P , a P -name τ ∈ V for the real x and a cardinal κ > 2|P | . Let
φ(~v , [x]E , α) be a formula with parameters ~v ∈ V defining a set of ordinals in the
Coll(ω, κ) extension: a = {α ∈ β : φ(~v , [x]E , α)}. Homogeneity arguments such
as Fact 2.3.1 show that a = {α ∈ β : V [x] |= Coll(ω, κ) φ(~v , [x]E , α̌)} ∈ V [x].
As φ was arbitrary, the first item follows.
~ κ, [x]E , α) with
For the second item, we will find a different formula θ(D,
parameters in the ground model and another parameter [x]E , which defines a in
any other transitive model extending V containing x. This will show that the
Coll(ω, κ) extension of V [x] has the smallest possible HOD among all transitive
models extending V containing x. As κ > 2|P | was arbitrary, this will prove the
second item.
~ = hDα : α ∈ βi be the sequence of subsets of P in V defined by p ∈ Dα
Let D
~ [x]E , α) saying
if V |= p Coll(ω, κ) φ(~v , [τ ]E , α). Consider the formula ψ(D,
“there exists a V -generic filter g ⊂ P such that τ /g ∈ [x]E and g ∩ Dα 6= 0”.
~ κ, [x]E , α) says “Coll(ω, κ) ψ(D,
~ [x]E , α)”; we claim that θ
The formula θ(D,
works as desired.
First, observe that in any transitive model extending V , containing x, and
~ [x]E , α)
containing an enumeration of P(P )∩V in ordertype ω, the formula ψ(D,
is a statement about illfoundedness of a certain tree for any fixed ordinal α.
Namely, let {Cn : n ∈ ω} enumerate all open dense subsets of P in V , let
A ⊂ X × X × ω ω be a closed set coded in V that projects to the equivalence
E, and let T be the tree of all finite sequences hpi , Oi : i ∈ ji such that the
conditions pi ∈ P form a decreasing sequence such that p0 ∈ Dα , pi ∈ Ci , and
Oi ⊂ X ×X ×ω ω form a decreasing sequence of basic open neighborhoods coded
in V of radius < 2−i with nonempty intersection with A, such that x belongs to
the projection of Oi to the first coordinate, and pi τ belongs to the projection
~ [x]E , α) is equivalent
of Oi in the second coordinate. It is immediate that ψ(D,
to the statement that T contains an infinite branch.
Second, observe that the validity of the illfoundedness statement does not
depend on the particular enumeration of open dense subsets of P chosen–if one
gets an infinite branch with one enumeration, then a subsequence of the conditions in P used in this branch will yield a branch for a different enumeration.
~ [x]E , α) is absolute between all transitive models of
Therefore, the formula ψ(D,
set theory containing V , x satisfying ”P(P )∩V is countable”. Ergo, the formula
~ [x]E , α) is absolute between all transitive models containing V and x.
θ(D,
Lastly, the formula θ does define the set a in all such models. It is enough
~ [x]E , α) is satisfied: if
to check that this is so in V [x]. If α ∈ a then θ(D,
H ⊂ Coll(ω, κ) is a V [x]-generic filter, then ψ will be witnessed by the V generic filter on the poset P obtained from the point x. On the other hand, if
~ [x]E , α) holds, then for every V [x]-generic filter H ⊂ Coll(ω, κ) there is a
θ(D,
V -generic filter g ⊂ P in V [x][H] such that τ /g E x and g∩Dα 6= 0, by Fact 2.3.1
there is a V [g]-generic filter K ⊂ Coll(ω, κ) such that V [g][K] = V [x][H], and
3.4. EQUIVALENCE RELATIONS AND MODELS OF SET THEORY
49
by the forcing theorem applied in V [g], the model V [g][K] = V [x][H] satisfies
φ(~v , [τ /g]E , α), which is the same as φ(~v , [x]E , α). Thus α ∈ a as required!
For the third item, start with the case when the pair hx, yi is V -generic. Let
κ be a cardinal larger than the density of the poset in the ground model that
yields the pair hx, yi, let H ⊂ Coll(ω, κ) be a V [x, y]-generic filter, and let W
be the class consisting of all sets hereditarily definable from parameters in V
and the additional parameter [x]E = [y]E in the model V [x, y][H]. Standard
homogeneity arguments using Fact 2.3.1 show that the model V [x, y][H] is a
Coll(ω, κ)-extension of both V [x] and V [y]. By the second item above, W =
V [x]E = V [y]E .
Now look at the situation where x, y are each V -generic but perhaps the
pair hx, yi is not. Let α ∈ V be an ordinal. We will show that Vα ∩ V [x]E =
Vα ∩ V [y]E ; as α ∈ V is arbitrary, this will yield the desired conclusion. Let κ
be a cardinal in V much larger than α and the posets P, Q ∈ V that are used to
obtain the points x, y respectively; the necessary size of κ is determined below.
Let G be a Coll(ω, κ)-generic filter over V such that x ∈ V [G]. Let ẏgen ∈ V
be the name for the Q-generic point. The model V [G] satisfies the following
sentence φ: for every Q-generic filter H ⊂ Q over V such that ẏgen /H E x, it is
the case that Vα ∩ V [y]E = Vα ∩ V [x]E . We will show that this is a coanalytic
statement taking as parameters x, a real z coding a large part of V countable in
V [G], and a code for the equivalence relation E. By the coanalytic absoluteness
between the transitive model V [G] and the outer universe containing x, y, G, φ
will hold in the outer universe as well. In particular, Vα ∩ V [x]E = Vα ∩ V [y]E
as desired.
To find the complexity of φ, recall the usual rank on Q-names: the canonical
name for the empty set has rank 0, and a Q-name σ has rank ≤ β if all names
mentioned in σ have rank < β. It is a standard fact of forcing theory that Q
forces every set in V̇α to be a realization of a name of rank < α. The collection
of all Q-names of rank < α forms a set Nα ∈ V , the cardinal κ must be chosen
so large that Nα is countable in V [G], and the real z must code all of Nα , Q,
and P(Q) ∩ V . Now in the model V [G], the sentence φ says that for every filter
H ⊂ Q meeting all dense subsets of Q in V such that ẏgen /H E x there is a
map f : Nα → (Vα ∩ V [x]E ) ∪ {trash} such that
• f (τ ) =trash iff there is a condition q ∈ H forcing τ ∈
/ V [ẏgen ]E ;
• if f (τ ), f (σ) ∈ V [x]E then f (τ ) = f (σ) iff there is a condition q ∈ H
forcing τ = σ, and similarly for f (τ ) ∈ f (σ);
• Vα ∩ V [x]E ⊆ rng(f ).
Such a map is obtained as a consequence of the equality V [ẏgen /H]E = V [x]E
proved previously. Note that given a filter H ⊂ Q such that ẏgen /H E x, the
map f is unique corresponding to the uniqueness of ∈-isomorphism between the
(equal) transitive sets Vα ∩V [x]E and Vα ∩V [ẏgen /H]E . The coanalyticity of the
sentence φ is now an immediate consequence of Lusin theorem: the quantifier
”there exists exactly one” yields coanalytic sets [33, Theorem 18.11].
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CHAPTER 3. TOOLS
The definition and properties of the model V [x]E may be somewhat mysterious. Several more or less trivial examples will serve as a good illustration of
what can be expected. Let E = E0 and consider the cases of the Sacks real, the
Cohen real, and the Silver real. In the case of Sacks real x ∈ 2ω , a simple density argument will show that there will be a perfect tree T in the Sacks generic
filter whose branches are pairwise E0 -inequivalent. Then, x can be defined from
[x]E0 as the only point of [T ] ∩ [x]E0 and therefore V [x] = V [x]E0 . In the case of
Cohen generic real x ∈ 2ω , the rational translations of 2ω that generate E0 can
transport any Cohen condition to any other one, and the equivalence class [x]E0
is invariant under all of them, leading to the conclusion that V [x]E0 = V . In the
case of Silver generic real x, the model V [x]E0 is a nontrivial σ-closed generic
extension of V generated by the quotient of Silver forcing in which E0 -equivalent
conditions are identified, roughly speaking–Theorem 5.8.4. On general grounds
[22, Lemma 15.43], V [x]E is a generic extension of V , but it is not so easy to
compute the forcing that induces it. In a typical case, it is the poset of Borel
E-invariant sets positive with respect to some σ-ideal as in Theorem 3.5.9.
The importance of the model V [x]E is clarified in the main result of this
section, a trichotomy theorem. Its wording requires a definition:
Definition 3.4.3. Let E be an equivalence relation on a Polish space X, let I
be a σ-ideal on X, and let B ⊂ X be a set. We say that E B is I-ergodic (or
just ergodic if the σ-ideal is clear from the context) if for all Borel I-positive
sets C, D ⊂ B there are E-equivalent points x ∈ C, y ∈ D. The relation E B
is nontrivially I-ergodic if it is I-ergodic and its equivalence classes are in the
σ-ideal I.
Theorem 3.4.4. Suppose that X is a Polish space, I is a σ-ideal on it such
that the quotient forcing PI is proper, and E is an analytic equivalence relation
on X. Let V [G] be a PI -generic extension of V and xgen ∈ X its associated
generic point. One of the following holds:
1. (ergodicity) either V = V [xgen ]E , and then there is a Borel I-positive set
B ⊂ X such that E B is ergodic;
2. (intermediate extension) or the model V [xgen ]E is strictly between V and
V [G];
3. (canonization) or V [xgen ]E = V [G] and then there is a Borel I-positive
set B ⊂ X such that E B = id.
The ergodicity is fairly close to saying that the E-saturation of every Borel
set either has an I-small intersection with B, or contains all of B up to an Ismall set. This is indeed equivalent to (1) if every coanalytic set is either in I or
contains a Borel I-positive set, or if every positive Borel set has a positive Borel
subset whose E-saturation is Borel. Both of these assumptions frequently hold:
the former perhaps on the basis of the first dichotomy of the ideal I [67] or a
3.4. EQUIVALENCE RELATIONS AND MODELS OF SET THEORY
51
redefinition of the ideal I, and the latter perhaps on the basis of the quotient
PI being bounding and the equivalence E being Kσ .
A trivial way to satisfy the first item is to find an I-positive equivalence class
of E. A typical special case in which the ergodicity case holds nontrivially is that
of Laver forcing and the EKσ equivalence relation, see Section 5.13. The Laver
forcing generates a minimal forcing extension, prohibiting the second item from
ever occurring, but still there is an equivalence relation that cannot be simplified
to id or ee on any Borel I-positive set. Another interesting special case is the
Cohen forcing and the F2 equivalence relation, see Theorem 5.1.29. There,
the ergodicity is not at all surprising, but we get more information looking at
choiceless intermediate models of ZF as in Section 4.3.
In the second case we get an intermediate forcing extension strictly between
V and the PI -extension V [G]. For many σ-ideals, the existence of such an
intermediate model is excluded purely on forcing grounds, such as the ideal of
countable sets or σ-compact subsets of ω ω , or the more general Theorem 3.2.5.
In such cases, we get a neat dichotomy. It is not true that every intermediate
extension is necessarily obtained from a Borel equivalence relation in this way,
as Section 5.3 shows. In special cases though (such as the countable length
α ∈ ω1 iteration of Sacks forcing), the structure of intermediate models is wellknown (each of them is equal to V [xγ : γ ∈ β] for some ordinal β ≤ α) and
they exactly correspond to certain critical equivalence relations (such as the
relations Eβ on (2ω )α connecting two sequences if they are equal on their first β
entries). We cannot compute the forcing responsible for the extension V [xgen ]E
in a fully general situation, but in all specific cases we can compute, it is the
poset of E-invariant I-positive Borel sets ordered by inclusion, as described
in Theorem 3.5.9 and its section. In the common situation that the model
V [xgen ]E does not contain any new reals, we can compute the forcing leading
from V [xgen ]E to V [G]–it is a quotient poset of the form PI ∗ for a suitable σideal I ∗ ⊃ I which has the ergodicity property of the first item of the theorem.
As Asger Tornquist remarked, in this context the theorem can be viewed as
a general counterpart to Farrell-Varadarajan ergodic decomposition theorem
[31, Theorem 3.3] for actions of countable groups by measure preserving Borel
automorphisms.
Proof ofTheorem 3.4.4. Consider the first case, V [xgen ]E = V . In this case,
consider the collection K of those Borel sets C ⊂ X coded in V such that
C ∩ [xgen ]E 6= 0. This collection is certainly in the model V [xgen ]E as we have
just defined it from the class [xgen ]E , and therefore it is in V . There must be a
ˇ If there were
condition B ∈ PI and an element J ∈ V such that B K̇ = J.
two Borel I-positive sets C0 , C1 ⊂ B with [C0 ]E ∩ [C1 ]E = 0, then only one,
say C0 of these sets can belong to the set L. But then C1 C1 ∈ K̇ \ Ľ! This
contradiction verifies the conclusion of the first case of the theorem.
The second possibility is that V ⊂ V [xgen ]E ⊂ V [G] and these inclusions are
proper. In this case, we are content to fall into the second item of the theorem.
Lastly, assume that V [xgen ]E = V [G]; in particular, xgen ∈ V [xgen ]E . Then,
return to the ground model V and pick large enough cardinals λ ∈ κ. There
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CHAPTER 3. TOOLS
must be a condition B ∈ PI forcing ẋgen ∈ V̇ [xgen ]E ; so, in the product
PI × Coll(ω, κ), the condition hB, 1i forces ẋgen to be the only element x ∈ X
satisfying φ(x, [ẋgen ]E , ~v ) for some sequence of parameters ~v ∈ V and a formula φ. We must find an I-positive Borel set C ⊂ B consisting of pairwise
E-inequivalent points.
Let H be a Coll(ω, κ)-generic filter over V ; the model V [H] contains a generic
extension used in Fact 2.5.1 to produce an elementary embedding j : V → N
to a possibly illfounded model N such that M = j 00 Hλ is a countable set in N .
Recall that the wellfounded part of the model N is identified with its transitive
isomorph and with this identification, for every analytic set A coded in the
ground model, j(A) ⊆ AV [H] .
Working in N , let C 0 = {x ∈ j(B) : x is j(PI ) = Pj(I) -generic over M }.
This is a j(I)-positive Borel set by Fact 2.6.1 applied in the model N : the
set M = j 00 Hλ is a countable elementary submodel of j(Hλ ) by the Tarski
criterion, and the poset j(PI ) = Pj(I) is proper in N . We shall show that
N |= C 0 consists of pairwise j(E)-inequivalent points. The desired set C ∈ V
will then be obtained through elementarity of the embedding j.
Suppose N |= x 6= y ∈ C 0 . Both points x, y are PI -generic over the
model V : for every maximal antichain A ⊂ PI in V , j(A) is a maximal antichain of j(PI ), and since the points x, y are PjI -generic over M , there must
be sets Dx , Dy ∈ A such that N |= x ∈ j(Dx ), y ∈ j(Dy ). However, the
model N evaluates the membership in ground model Borel sets correctly, so
x ∈ Dx , y ∈ Dy . Both points x, y also belong to B V [H] and V [H]. Find filters
Hx , Hy ⊂ Coll(ω, κ) generic over the models V [x] and V [y] respectively such
that V [x][Hx ] = V [y][Hy ] = V [H]. If the points x, y ∈ X were E-equivalent in
V [H], then the formula φ(·, [x]E , ~v ) = φ(·, [y]E , ~v ) would have to define both x
and y in the model V [H] by the forcing theorem, and this is impossible. Thus,
V [H] |= ¬x E y. Since j(E) ⊂ E ∩ V [H], we conclude that N |= ¬x j(E) y as
desired.
Theorem 3.4.5. Suppose that X is a Polish space, I a σ-ideal on X such
that the quotient forcing PI is proper. Let x be a PI -generic point. Then,
2ω ∩ V [x]E = {f (x) : f : X → 2ω is a partial Borel function coded in V which
is E-invariant on its domain}.
Proof. For the first part, if f : B → 2ω is a Borel function which is constant on
each E-equivalence class, then certainly B f˙(ẋgen ) ∈ V [x]E . Namely, f˙(ẋgen )
will be definable from [ẋgen ]E as the unique value of f on Ḃ ∩ [ẋgen ]E . Note
that the constant property of f survives to every forcing extension by coanalytic
absoluteness. This proves the right-to-left inclusion.
For the opposite inclusion, proceed as in the third case of the trichotomy
theorem. Let λ ∈ κ be large regular cardinals. Suppose that hB, pi ∈ PI ×
Coll(ω, κ) is a condition that forces ẏ ∈ V [ẋgen ]E ∩ 2ω . Strengthening the
condition we may assume that there is a formula φ and a sequence ~v ∈ V of
parameters such that it is forced that ẏ is the unique element of 2ω satisfying
φ(ẏ, [ẋgen ]E , ~v ). Since V [xgen ]E ⊂ V [xgen ], we may also strengthen the condition
3.4. EQUIVALENCE RELATIONS AND MODELS OF SET THEORY
53
if necessary to find a Borel function f : X → 2ω such that it is forced that ẏ =
f˙(ẋgen ). We will now find a Borel I-positive set C ⊂ B on which the function g
is E-invariant. This will certainly complete the proof of the proposition, since
then hC, pi forces that ẏ is the image of the generic real by an E-invariant
function.
Let H ⊂ Coll(ω, κ) be a generic filter, and in V [H] find the elementary
embedding j : V → N as in Fact 2.5.1; in particular, M = j 00 Hλ is a countable
set in the model N . Working in the model N , let C 0 = {x ∈ j(B) : x is Pj(I) generic over M }; this is a Borel j(I)-positive set. We will show that in N , the
function j(f ) is j(E)-invariant on the set C 0 ; the desired set C is then obtained
by the elementarity of the embedding j. Thus, let x0 , x1 ∈ C 0 be j(E)-equivalent
points in the model N . The two points are E-equivalent in V [H] as j(E) ⊂ E,
and they are also PI -generic over V . Find filters H0 , H1 ⊂ Coll(ω, κ) generic
over V [x0 ] and V [x1 ] respectively such that V [H] = V [x0 ][H0 ] = V [x1 ][H1 ].
By the forcing theorem, it must be the case that the formula φ(·, [x0 ]E , ~v ) =
φ(·, [x1 ]E , ~v ) must define a point y ∈ 2ω which is the joint value of f (x0 ) and
f (x1 ) in the model V [H]. Now, by the definitions in the model N , the points
x0 , x1 are in j(B) and therefore in the domain of the function j(f ). Since
j(f ) ⊂ f V [H] , it follows that y must be the unique value of j(f )(x0 ) and j(f )(x1 )
in the model N . We conclude that in the model N , the function j(f ) is j(E)invariant on the set C 0 , and the theorem follows.
Let I be a σ-ideal on a Polish space X such that the quotient poset PI is
proper. Let x ∈ X be a V -generic point for PI . On general forcing grounds [22,
Lemma 15.43], it must be the case that V [x]E is a generic extension of V , and
V [x] is a generic extension of V [x]E . We do not know which poset is responsible
for the former extension, even though there is a canonical candidate isolated in
the next section which works in all cases computed in this book. In the common
case where V [x]E has the same reals as V , we can compute the poset responsible
for the latter extension.
Definition 3.4.6. Let I be a σ-ideal on a Polish space X, let E be an analytic
equivalence relation, and let x ∈ X be a PI -generic point. In the model V [x],
let I x be the collection of ground model coded Borel sets B such that for every
cardinal κ, Coll(ω, κ) forces that Ḃ contains no point that is simultaneously
PI -generic over V and E-equivalent to x.
It should be clear that I ⊂ I x , I x ∈ V [x]E , and if V [x]E contains no new reals
then I x is a σ-ideal of Borel sets there. Moreover, if 2ω ∩ V [x]E = 2ω ∩ V , the
ideal I x has the ergodicity property in V [x]E : any pair of I x -positive Borel sets
contains a pair of E-connected points. This follows from the fact that in some
large collapse, each of the sets in the pair contains some points equivalent to x;
in particular, such points are equivalent to each other, and analytic absoluteness
transports this feature back to V [x]E .
Theorem 3.4.7. Suppose that X is a Polish space, I a σ-ideal on X such that
the quotient forcing PI is proper. Let x be a PI -generic point. If 2ω ∩ V =
2ω ∩ V [x]E , then V [x]E = V [I x ] and V [x] is a PI x -extension of V [x]E .
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CHAPTER 3. TOOLS
Proof. We will first show that x is a PI x generic point over V [x]E . Move back
to the ground model, and assume for contradiction that a condition hB, pi ∈
PI × Coll(ω, κ) forces that Ḋ ⊂ PI x is an open dense set in the model V [xgen ]E
such that the point ẋgen does not belong to any of its elements. Strengthening
the condition if necessary, we may find a formula φ(u, [ẋgen ]E , ~v ) with parameters
in V ∪ {[ẋgen ]E } defining the set Ḋ: Ḋ = {C ∈ PI ẋgen : φ(C, [ẋgen ]E , ~v )}. Of
course, the condition hB, pi forces B ∈ PI ẋgen , so there must be a strengthening
B 0 ⊂ B, p0 ≤ p and a Borel set C ⊂ B such that hB 0 , p0 i C ∈ Ḋ. Note that
C ∈ PI must be an I-positive Borel set in the ground model.
Find mutually generic filters G ⊂ PI , H ⊂ Coll(ω, κ) with B 0 ∈ G, p0 ∈ H,
let xgen ∈ X be the point associated with the filter G, use the definition of
the ideal I xgen to find a point y ∈ C ∩ V [G, H] which is PI -generic over V and
equivalent to xgen , and use the homogeneity features of the poset Coll(ω, κ) as in
Fact 2.3.1 to find mutually generic filters Ḡ ⊂ PI , H̄ ⊂ Coll(ω, κ) such that y is
the generic point associated with Ḡ (so C ∈ Ḡ), p ∈ H̄, and V [G, H] = V [Ḡ, H̄].
Since the points xgen , y are E-equivalent, the definition of the set Ḋ is evaluated in the same way with either xgen or y plugged into the definition. Applying
the forcing theorem to the filters Ḡ, H̄, there must be a condition in the filter
Ḡ × H̄ stronger than hB, pi which forces C ∈ Ḋ and ẋgen ∈ C. However, this
directly contradicts the statement forced by the weaker condition hB, pi!
To show that V [x]E = V [I x ], note that I x ∈ V [xgen ]E , so the right to left
inclusion is immediate. For the opposite inclusion, assume that a ∈ V [x]E
is a set of ordinals, perhaps defined as a = {α : φ(α, ~v , [x]E )} in the model
V [x, H], where ~v are parameters in the ground model and H ⊂ Coll(ω, κ). Let
/ I x B Coll(ω, κ) φ(α̌, ~v , [xgen ]E )} ∈ V [I x ] and show
b = {α : ∃B ∈ PIV , B ∈
that a = b. It is immediate that a ⊂ b since the filter defined by the PI -generic
point x ∈ X has empty intersection with I x . On the other hand, if α ∈ b as
witnessed by a set B then there is a PI -generic point y ∈ B ∩ [x]E over V , by
the usual homogeneity argument there is a V [y]-generic filter Hy ⊂ Coll(ω, κ)
such that V [y][Hy ] = V [x][H], by the forcing theorem V [y][Hy ] |= φ(α, ~v , [x]E ),
and therefore α ∈ a. The theorem follows.
3.4b
The model V [[x]]E
The model V [x]E has a larger, less well understood, possibly choiceless companion V [[x]]E . In spirit, this larger model is the intersection of all models
containing V and a point E-equivalent to x. In formal language,
Definition 3.4.8. Let E be an analytic equivalence relation on a Polish space
X. Let x ∈ X be a V -generic point. The model V [[x]]E consists of all sets
a ∈ V [x] such that the model V [x] satisfies: for every ordinal κ, Coll(ω, κ)
forces that for every y ∈ X with x E y, a ∈ V [y].
Theorem 3.4.9. Let E be an analytic equivalence relation on a Polish space
X, and let x ∈ X be a V -generic point.
3.4. EQUIVALENCE RELATIONS AND MODELS OF SET THEORY
55
1. V [[x]]E is a model of ZF with V [x]E ⊆ V [[x]]E ⊆ V [x];
2. if x, y ∈ X are V -generic E-equivalent points, then V [[x]]E = V [[y]]E ;
Proof. Start with the second item and the special case where the pair hx, yi is
V -generic. Let φ(x) be the definition of the model V [[x]]E as described above.
We will argue that for every ordinal κ, Coll(ω, κ) forces that for every point
y ∈ X which is E-equivalent to x, the formula φ(y) defines the same class in
V [y] as φ(x) defines in V [x]. We will show that {a ∈ V [x] : V [x] |= φ(a, x)} ⊂
{a ∈ V [y] : V [y] |= φ(a, y)}; the other inclusion has the same proof. Suppose
that a ∈ V [y] and V [y] |= φ(a, y). Then a ∈ V [x], since otherwise V [x] forms a
counterexample to V [y] |= φ(a, y) inside some large common generic extension
of both models V [y] and V [x]. Also, V [x] |= φ(a, x), because otherwise in some
common collapse forcing extension of V [x] and V [y], there is a counterexample
to φ(a, x) which is also a counterexample to φ(a, y). The general case of (2)
now follows just as in the proof of Theorem 3.4.2(3).
To show that V [[x]]E is a model of ZF, argue that L(V [[x]]E ) = V [[x]]E .
Indeed, for every ordinal κ it is the case that Coll(ω, κ) forces that L(V [[x]]E )
is contained in every model V [y] where y E x, since as argued in the previous
paragraph, the models V [y] all contain V [[x]]E as a class. But then, every
element of L(V [[x]]E ) belongs to V [[x]]E by the definition of V [[x]]E .
To show that V [x]E is a subset of V [[x]]E , let a ∈ V [x]E be a set of ordinals
and let V [G] be a generic extension of V inside a generic extension of V [x]
containing a point y ∈ X equivalent to x; we must argue that a ∈ V [G]. Let P
be a poset generating that extension, and let κ = |P(P )|. There is a common
Coll(ω, κ) extension V [H] of both V [G] and V [x]. Then a is definable in V [H]
from parameters in V and perhaps the additional parameter [x]E = [y]E . Since
V [H] is a homogeneous extension of the model V [G] and all parameters of the
definition are themselves definable from parameters in V [G], it must be the case
that a ∈ V [G] as desired, and the proof is complete.
Theorem 3.4.10. Let I be a σ-ideal on a Polish space X such that the quotient
forcing PI is proper. Let E be an analytic equivalence relation on X, and let
x ∈ X be a PI -generic point. Then 2ω ∩ V [[x]]E = {f (x) : f is a Borel function
coded in V which on its domain attains only countably many values on each
equivalence class}.
An important observation complementary to the wording of the theorem: the
statement that a given Borel function f attains only countably many values on
each E-equivalence class is Π11 in the code for the function and the equivalence
relation, and therefore it is invariant under forcing by Shoenfield’s absoluteness.
To see this, note that every E-equivalence class is analytic, its image under f
is analytic, and so if countable it contains only reals hyperarithmetic in any
parameter that can define it as an analytic set. Thus, f attains only countably
many values on each E-equivalence class if and only if for every pair of points
x, y ∈ dom(f ), either x, y are not equivalent or f (x) is hyperarithmetic in y, the
code for f and the code for E. This is a coanalytic formula by the usual coding
of ∆11 sets as in [27, Theorem 2.8.6].
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CHAPTER 3. TOOLS
Proof. First, observe that if f : B → 2ω is a Borel function attaining only
countably many values on each equivalence class, and x ∈ X is any point in a
forcing extension of the ground model V , then the model V [x] contains some
enumeration of the countable set f 00 ([x]E ∩ B) by Shoenfield absoluteness, and
therefore contains all elements of this countable set. This proves the right-to-left
inclusion.
For the opposite inclusion, suppose that B ⊂ X is an I-positive Borel set
forcing ẏ ∈ V [[ẋgen ]]E ∩ 2ω . Let M be a countable elementary submodel of
a large structure containing B and the name ẏ, and let C ⊂ B be the Borel
I-positive set of all PI -generic points over M . Let f : C → 2ω be the function
defined by f (x) = ẏ/x; thus f is a Borel function and C f˙(ẋgen ) = ẏ. We will
prove that f attains only countably many values at every E-equivalence class
in C. By Theorem 3.4.9 applied to the transitive isomorph of M , whenever
M [x ]
M [x ]
x0 , x1 ∈ C are E-related points, then V [[x0 ]]E 0 = V [[x1 ]]E 1 . The model
M [x0 ]
V [[x0 ]]E
is countable, and it must contain the real ẏ/x1 . Thus the countable
M [x ]
model V [[x0 ]]E 0 contains all reals ẏ/x where x ∈ C is a point E-related to
x0 , and the proof is complete.
The axiom of choice may fail in the model V [[x]]E , and the study of its
failure may shed light on the canonization properties of the equivalence relation
E. The previous claim puts a slight restriction on the type of choice failure one
can get:
Corollary 3.4.11. Let I be a σ-ideal on a Polish space X such that the quotient
poset PI is proper, let E be an analytic equivalence relation on X, and let x ∈ X
be a PI -generic point over V . Then V [[x]]E satisfies the following sentence:
every set A ⊂ ω × 2ω with nonempty vertical sections has a subset A0 ⊂ A with
nonempty vertical sections such that ℵ1 6≤ |A0 |.
Example 3.4.22 provides a model of ZF where the above sentence fails.
Proof. Let B ∈ PI force Ȧ ⊂ ω × 2ω is a set in V [[ẋgen ]]E with nonempty
vertical sections. Let M be a countable elementary submodel of a large structure
containing B, and let C be the set of PI -generic points over M in B. A genericity
argument together with the previous claim will show that for every number
n ∈ ω, the set C is covered by a countable collection of Borel sets {Din :
i ∈ ω} such that every Din is in the model M and there is a Borel function
fin : Din → 2ω which takes at most countably many values at each E-class and
Din hn, f˙in (ẋgen )i ∈ Ȧ. Let Ȧ0 be the name for the set of all pairs hn, yi such
that y = fin (x) for some i ∈ ω and some point x E ẋgen . It is immediate that
A0 ∈ V [[x]]E . Also, the set A0 is countable in V [ẋgen ] and so V [[ẋgen ]]E cannot
satisfy ℵ1 ≤ |A0 |.
The structure of the model V [[x]]E turns out to be much easier to understand
in the case where the equivalence E is reducible to the orbit equivalence of a
Polish group action. If the acting group is a closed subgroup of S∞ , then
the model is in fact equal to V (a) for some hereditarily countable and fairly
3.4. EQUIVALENCE RELATIONS AND MODELS OF SET THEORY
57
canonically chosen set a as proved in Theorem 4.3.11. In the case of a general
Polish group action, we have less information: the model V [[x]]E is a HOD
type model like V [x]E , and the difference between the two models resides in the
hereditarily countable sets. Let x ∈ X be a PI -generic point over V , and let Z
be the set of all sets in V [[x]]E that are hereditarily countable in V [x]. Let κ be
a suitably large cardinal, let H ⊂ Coll(ω, κ) be a filter generic over V [x], and
in the model V [x][H] form the model V [[[x]]]E of the sets hereditarily definable
from parameters in V ∪ Z and from the parameter [x]E and Z.
Proposition 3.4.12. The set Z is definable in V [x][H] from the parameter [x]E .
Moreover, the model V [[[x]]]E does not depend on the choice of the cardinal κ
or the generic filter H.
Proof. For the first sentence, work in the model V [x][H] and recall the definition
of the set Z: z ∈ Z iff z ∈ V [[x]]E and z is hereditarily countable in V [x]. The
first conjunct depends only on the equivalence class [x]E by Theorem 3.4.9(2).
The second conjuct is equivalent to φ(z) =“for every y ∈ [x]E , z is hereditarily
countable in V [y]”, which again clearly depends only on [x]E as required. However, the equivalence of φ(z) with hereditary countability of z in V [x] needs an
argument.
Clearly, φ(z) implies the hereditary countability of z in V [x]. On the other
hand, assume that z ∈ V [[x]]E is a set hereditarily countable in V [x]; we must
verify that z is hereditarily countable in every model V [y] where y E x. As x
is PI -generic over V , a density argument will show that there must be a Borel
I-positive set B ⊂ X coded in V , a Borel function f : B → 2ω×ω coded in
V such that trcl(z) is equal to the transitive isomorph of the relation f (x),
and B the transitive isomorph of f˙(ẋgen ) ∈ V [[ẋgen ]]E , and there must be a
countable elementary submodel M ∈ V of some large structure containing f, B
such that x is a PI -generic point over M . Now let y ∈ X be an arbitrary point
E-related to X. By the analytic absoluteness between V [y] and V [x][H], there
must be a point x0 ∈ V [y] which is in B E-related with y and PI -generic over
M –in the model V [x][H], x0 = x is such a point. Now, the models M [[x]]E and
M [[x0 ]]E coincide by Theorem 3.4.9(2) applied to the model M . The transitive
closure of z (as the transitive isomorph of f (x)) belongs to M [[x]]E and therefore
to M [[x0 ]]E , and the model M [[x0 ]]E belongs to V [y] and is countable there–it
is a subset of the model M [x0 ] and a generic extension of a countable model is
again countable. All in all, V [y] |= z is hereditarily countable.
The second sentence is proved in the same way as Theorem 3.4.2.
Theorem 3.4.13. Let I be a σ-ideal on a Polish space X. Let E be an analytic
equivalence relation on the space X, Borel reducible to an orbit equivalence of a
Polish group action. Let x ∈ X be a PI -generic point over V . Then V [[x]]E =
V [[[x]]]E .
The argument will go through for equivalence relations somewhat more complicated than just orbit equivalence relations, such as E1 . For equivalence relations
of the complexity EKσ and beyond, the argument breaks down entirely and we
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CHAPTER 3. TOOLS
must replace it by ad hoc fixes specific to the given pair I, E. We do not have
an example of a situation where the conclusion of the theorem fails.
Corollary 3.4.14. If in addition to the assumptions of Theorem 3.4.13, 2ω ∩
V [[x]]E = 2ω ∩ V [x]E holds, then V [[x]]E = V [x]E .
Proof. If V [[x]]E and V [x]E contain the same reals, then they also contain the
same hereditarily countable-in-V [x] sets. To see this, assume for contradiction
that a ∈ V [[x]]E is a set of minimal rank with countable transitive closure such
that a ∈
/ V [x]E . Then a ⊂ V [x]E , there is b ∈ V [x]E which is countable in
V [x]E such that a ⊂ b and there is an enumeration π : ω → b in V [x]E ; the
set π −1 a is in V [[x]]E and therefore in V [x]E , and so a ∈ V [x]E , contradicting
the assumption. Therefore, V [[[x]]]E = V [x]E and by the theorem, V [[x]]E =
V [[[x]]]E = V [x]E as desired.
Corollary 3.4.15. If in addition to the assumptions of Theorem 3.4.13, classifiable by countable structures→I smooth holds, then V [[x]]E = V [x]E .
Proof. By virtue of the previous corollary, it is enough to show that V [[x]]E
and V [x]E agree on the reals. Suppose that B ∈ PI is a condition forcing
ẏ ∈ V [[ẋgen ]]E ∩ 2ω . Let M be a countable elementary submodel of a large
structure containing B, and let F be the equivalence relation on the Borel Ipositive set C ⊂ B of all PI -generic points over M in B defined by x0 F x1 if
V [[x0 ]]M [x0 ] = V [[x1 ]]M [x1 ] . It is not difficult to verify that this is an equivalence
relation classifiable by countable structures. By Theorem 3.4.9(2) applied to
the transitive isomorph of M , it is even the case that E ⊆ F . Now use the
assumption of the corollary to find a Borel I-positive set D ⊂ C on which
F is smooth, reduced to the identity by some Borel function g : D → 2ω .
Since E ⊆ F , the function g is constant on E-equivalence classes and so D ġ(ẋgen ) ∈ V [ẋgen ]E by Theorem 3.4.5. Now, if x ∈ D is a PI -generic point over
V , the model V [x]E contains g(x), and by an absoluteness argument contains
M [x]
also V [[x]]E , which in turn contains the point ẏ/x.
Proof. For the proof of Theorem 3.4.13, let x be a PI -generic point over V and
let W = V [[[x]]]E .
For the inclusion W ⊂ V [[x]]E , assume for contradiction that there is a set
a ∈ W which does not belong to V [[x]]E , and let a be an ∈-minimal such set.
We will reach a contradiction by showing that a belongs to every model V [y]
where y ∈ X is a V [x]-generic point E-related to x; this implies that a ∈ V [[x]]E
by the definitions. Let φ be a formula defining the set a in some large collapse
extension of V [x] from some parameters ~v ∈ V and [x]E ; we can in fact find
such a formula that defines a in every large collapse extension of V [x] just as in
the proof of Theorem 3.4.2. Let λ be a cardinal larger than the size of the poset
adjoining the point y over the model V [x]. Let b ∈ V [y] be the set of all sets c
such that V [y] |= Coll(ω, λ) φ([y]E , ~v , c); we will show that b = a. And indeed,
let Hx , Hy ⊂ Coll(ω, κ) be filters generic over V [x] and V [y] respectively such
that V [x][Hx ] = V [y][Hy ]. In these two equal models, the formula φ must define
3.4. EQUIVALENCE RELATIONS AND MODELS OF SET THEORY
59
the same set and the set defined is a. It follows that b = a ∩ V [y]. However, all
elements of a are in V [[x]]E by the minimality choice of a, so they are in V [y].
It follows that b = a ∩ V [y] = a as desired.
For the other inclusion, suppose that a ∈ V [[x]]E is a subset of W ; we must
show that a ∈ W . By a genericity and properness argument, there must be in
V a countable elementary submodel M of a large structure such that x is PI generic over M and M contains a name for the set a. Let M [x] = {σ/x : σ ∈ M
is a PI -name}. We have to show that the intersection M [x] ∩ V [[x]]E does not
depend on the particular choice of x in its equivalence class:
Claim 3.4.16. In every forcing extension of the model V [x], whenever x0 ∈
X is a point PI -generic over M and V , E-related to x, it is the case that
M [x] ∩ V [[x]]E = M [x0 ] ∩ V [[x]]E .
Proof. Suppose first that there is a poset Q ∈ M and a filter H which is Qgeneric over both M and V such that x, x0 both belong to M [H]. Then by
Proposition 2.3.4(3), M [H] ∩ V [x] = M [x] and M [H] ∩ V [x0 ] = M [x0 ]. Now,
since the model V [[x]]E = V [[x0 ]]E is in both models V [x] and V [x0 ], this means
that M [x] ∩ V [[x]]E = M [H] ∩ V [[x]]E = M [x0 ] ∩ V [[x]]E as required.
Now, in the case that the pair hx, x0 i is not as assumed in the first paragraph,
we will find another point x00 ∈ X which is E-equivalent to both x and x0 and
such that each of the pairs hx, x00 i and hx0 , x00 i is as in the first paragraph.
Then the previous argument shows that M [x] ∩ V [[x]]E = M [x00 ] ∩ V [[x]]E =
M [x0 ] ∩ V [[x]]E as required. The existence of x00 will be obtained from the
assumption that E is Borel reducible to an orbit equivalence of a Polish group
action. On the other hand, if EKσ were Borel reducible to E then such a point
x00 may not exist.
To find the point x00 , let f : X → Y be a Borel map reducing the equivalence
relation E to the orbit equivalence relation of a continuous action of some Polish
group G, all of this in V and M . Let h ∈ G be a group element in V [x, x0 ] such
that h · f (x) = f (x0 ). Let g ∈ G be a PJ -generic point over the model V [x, x0 ],
where J is the meager ideal on the Polish group G. Then gh is a PJ -generic point
over V [x, x0 ] as well, since the right multiplication by h is a selfhomeomorphism
of the group G. Since the notion of a maximal antichain in PJ is absolute
between wellfounded models of set theory, the point gh is PJ -generic over both
V [x] and M [x], and the point g is PJ -generic over both V [x0 ] and M [x0 ]. Let
y = gh · f (x) = g · f (x0 ); so y ∈ M [x][gh] ∩ M [x0 ][g]. By coanalytic absoluteness
between the models V [x], V [y], and M [y], there must be a point x00 ∈ X ∩ M [y]
such that f (x00 ) is orbit equivalent to y. This point x00 works: it is E-equivalent
to both x and x00 , and each of the pairs hx, x00 i and hx0 , x00 i belongs to some
PI × PJ extension of both M and V as required in the first paragraph.
Claim 3.4.17. M [x] ∩ W ∈ W and a ∩ M [x] ∈ W .
Proof. Let N = M [x] ∩ W . Move to some large collapse extension V [H] of the
model V [x]. As per the claim and the inclusion W ⊂ V [[x]]E , N is the unique
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CHAPTER 3. TOOLS
possible value for M [x0 ] ∩ W as x0 ranges over all points in V [H] that are PI generic over M and V and E-equivalent to x. This provides a definition of the
set N that will put it into W .
Let π : N̄ → N be the inverse of the Mostowski collapse of the model N .
Clearly, N̄ , π ∈ W and since W ⊂ V [[x]]E , it is the case that both of these
objects are in V [[x]]E . Thus, π −1 a ∈ V [[x]]E . Moreover, N̄ , π −1 a are both
hereditarily countable sets in the model V [x] and therefore they belong to the
set Z, and to the model W . Thus, the set π 00 π −1 a = a ∩ N belongs to the set
N.
Now, by the elementarity of the model M [x], there is only one subset of W
which is an element of M [x] and whose intersection with M [x] ∩ W is a ∩ M [x],
and it is the set a. The claim now shows that in every forcing extension of
V [x], the set a is the unique subset c ⊂ W such that for every point x0 PI generic over M and PI , E-related to x, it is the case that c ∈ M [x0 ] and
c ∩ M [x0 ] = a ∩ M [x]. This yields a definition of the set a from the parameters
M ∈ V and a ∩ M [x] ∈ W , and so a ∈ W .
One consequence of the proof is that in the model V [x], whenever a ∈ V [[x]]E
is a set then the set [a]ℵ0 ∩ V [[x]]E ⊂ [a]ℵ0 is stationary. This does not hold
for general intermediate choiceless extensions, as the model in Example 3.4.22
shows.
3.4c
Comparison with the symmetric models of ZF
Let I be a σ-ideal on a Polish space X, and let G be a Borel group acting in a
Borel I-preserving way on the space X. Then for every Borel set B ⊂ X and
every g ∈ G the set g · B = {g · x : x ∈ B} is Borel again, since it is a Borel
injective image of B, and B ∈ I ↔ g · B ∈ I. It follows that the group acts on
the poset PI by automorphisms. The action then can be immediately extended
to the Boolean model V PI : by induction on the rank of the PI -name τ , define
g · τ as the set {hg · B, g · σi : hB, σi ∈ τ }.
In such a situation, one can define symmetric models of ZF (typically without
the axiom of choice) as in [22, Lemma 15.51 ff]. Suppose that F is a normal
filter of subgroups of G: a collection of subgroups of G closed under superset,
finite intersection, and conjugation. For a name τ , define supp(τ ), the support
of τ , to be the stabilizer of τ , say that the name is F -symmetric if its support
is in F , and hereditarily F -symmetric if the supports of all names mentioned in
τ are symmetric. Then, for a PI -generic point x ∈ X over V , one can form the
symmetric model V [x]G,F = {τ /x : τ is a hereditarily F -symmetric PI -name}.
We have
Fact 3.4.18. [22, Lemma 15.51] The symmetric model is a model of ZF.
In this section we address the question whether the models V [x]E , V [[x]]E introduced in the previous section, where E is the orbit equivalence relation on X
3.4. EQUIVALENCE RELATIONS AND MODELS OF SET THEORY
61
generated by the action of the group G, have any relationship with the symmetric models obtained from the action. The general answer appears to be highly
structured.
Theorem 3.4.19. Let G be a closed subgroup of S∞ , acting continuously on a
Polish space X. Let F be the filter generated by clopen subgroups of G, and let I
be the σ-ideal of meager subsets of X. Then PI forces V [[ẋgen ]]E = V [ẋgen ]G,F .
Note that the closed subgroups of S∞ are characterized as those Polish groups
possessing a neighborhood basis around the unit consisting of clopen subgroups
[14, Theorem 2.4.1]. Note also that every continuous group action is an action
by homeomorphisms, and thefore preserves the σ-ideal I. A number of classical choiceless models of set theory are obtained in a way compatible with the
theorem.
Example 3.4.20. [22, Example 15.52] Let S∞ act on (2ω )ω by composition.
The resulting model V [[ẋgen ]]E is the classical Cohen model for the failure of
the axiom of choice. The prime ideal theorem holds in that model by the work
of Halpern and Levy; in particular, there is an ultrafilter on ω in the model.
Example 3.4.21. Let K be the countable group of al eventually constant infinite binary sequences that are eventually constant, with coordinatewise binary addition and discrete topology. Let G = K ω act on (2ω )ω by coordinatewise addition. The resulting model V [[ẋgen ]]E does not contain any
nonprincipal ultrafilters on ω. Consider the ω-sequence a of sets given by
a(n) = {[xgen ]E0 , [1 − xgen ]E0 }. An elementary symmetricity argument shows
that a ∈ V [[xgen ]]E and at the same time there is no choice function f ∈ V [[xgen ]]
that would choose one element from each pair a(n). It follows that V [[xgen ]]E
has no nonprincipal ultrafilter on natural numbers–such an ultrafilter would
immediately yield a choice function.
Example 3.4.22. Jech [22, Example 15.59] provides a different symmetric submodel of the Cohen extension in which there is no nonprincipal ultrafilter on ω.
Let G be the group (2ω )ω acting on X = G by translation, and let F be the filter
of those subgroups of G containing some Gn = {x ∈ G : x n is a constant zero
function}. The symmetric model M certainly does not fall into our context–
note that the orbit equivalence relation has only one class, and the groups in
the filter F are closed and not open. In fact, the model M is not of the form
V [[x]]E for any equivalence relation; it is an example of a (choiceless) submodel
of the Cohen forcing extension which is not of the form V (a) for some countable
transitive set a. To see that, consider the set A ⊂ ω × 2ω , A = {hn, yi : y is a
translate of the n-th entry of the generic sequence by a ground model element
of 2ω }. The set A belongs to the model M , and whenever B ⊂ A is a set in
M with nonempty vertical sections, a simple symmetricity argument shows that
one of its sections Bn is relatively open in An . The set of ground model elements
of 2ω can be injectively mapped into any nonempty relatively open subset of
An , and so ℵ1 ≤ |B| in the model M . Finally, compare this observation with
Corollary 3.4.11.
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CHAPTER 3. TOOLS
Proof. We shall first show that the models V [[ẋgen ]]E and V [ẋgen ]G,F are the
same as far as transitive sets countable in V [ẋgen ] are concerned. Since V [[ẋgen ]]E =
V (a) for some transitive countable set a ∈ V [x] as proved in Theorem 4.3.11,
this shows that V [[ẋgen ]]E ⊆ V [ẋgen ]G,F . We will conclude the argument by
showing that every intermediate model W of ZF, V [[ẋgen ]]E ⊆ W ⊆ V [ẋgen ],
is either equal to V [[ẋgen ]]E or else contains some transitive set countable in
V [ẋgen ] which is not in V [[ẋgen ]]E .
The comparison in the case of countable transitive sets passes through a
descriptive set theoretic characterization. If R ⊂ ω × ω is a wellfounded relation
satisfying the axiom of extensionality, let most(R) be the unique countable
transitive set such that the membership relation on it is isomoprhic to R.
Claim 3.4.23. Let B ⊂ X be a Borel nonmeager set. Let f : B → P(ω × ω)
be a Borel function such that for every x ∈ B, the relation f (x) is wellfounded
and satisfies the axiom of extensionality. The following are equivalent:
1. there is a Borel I-positive set C ⊂ B and a clopen subgroup H ⊂ G such
that for every x ∈ C and every h ∈ H, either hx ∈
/ C or most(f (x)) =
most(f (gx));
2. there is a Borel I-positive set C ⊂ B forcing most(f˙(ẋgen )) ∈ V [[ẋgen ]]E .
3. there is a Borel I-positive set C ⊂ B forcing most(f˙(ẋgen )) ∈ V [ẋgen ]G,F .
Proof. (1) certainly implies (2). As H has countable index in G, for every V generic point x ∈ X, the model V [x] contains a countable set a ⊂ [x]E ∩ C that
contains representatives of all H-suborbits of [x]E intersecting the set C, and so
it contains all the countably many hereditarily countable sets most(f (y)) : y ∈
[x]E ∩ C. Thus, the set most(f˙(ẋgen )) belongs to every forcing extension of V
containing an E-equivalent of ẋgen , and so belongs to V [[ẋgen ]]E .
(2) implies (1). Consider the poset PJ where J is the meager ideal on
the Polish space X × G. The poset PJ adds points ẋgen ∈ X and ġgen ∈ G;
the point ẋgen ∈ X is PI -generic over V by the Kuratowski-Ulam theorem.
Consider the condition D0 = C × G ∈ PJ . Since D0 f˙(ẋgen ) ∈ V [[ẋgen ]]E ,
there must be a Borel function f¯ and a stronger condition D1 ⊂ D0 forcing in
PJ that most(f˙(ẋgen )) = most(f¯(ġgen · ẋgen )). Let M be a countable elementary
submodel of a large structure and let D2 be the set of PJ -generic points over
M in the set D1 ; this is still a nonmeager Borel set, and the forcing theorem
shows that for every hx, gi ∈ D2 it is the case that most(f (x)) = most(f¯(g · x)).
Let U ⊂ X and V ⊂ G be basic open neighborhoods such that D is comeager
in U × V . Since the acting group G is a closed subgroup of S∞ , the unit has
a neighborhood basis consisting of clopen subgroups, and so there must be a
point g0 ∈ G and a clopen subgroup H ⊂ G such that Hg0 ⊂ V . We claim that
the set C 0 = {x ∈ X: for comeagerly many g ∈ V , hx, gi ∈ D} together with
the subgroup H will work as in (1).
First of all, the set C 0 ⊂ C is nonmeager by Kuratowski-Ulam theorem.
Since the meager ideal is Borel on Borel, the set C 0 is also Borel–[33, Theorem
3.4. EQUIVALENCE RELATIONS AND MODELS OF SET THEORY
63
16.1]. Now suppose x ∈ C 0 and h1 ∈ H are such that y = h1 · x ∈ C 0 . Since
the sets A0 = {h ∈ H : hx, hg0 i ∈ D} and A1 = {h ∈ H : hh1 · x, hg0 i ∈ D}
are comeager in H, there must be a group element h ∈ A0 ∩ A1 · h1 . Then,
both pairs hx, g0 hi as well as hy, g0 hh−1
1 i are in the set D. This means that
most(f (x)) is isomorphic to most(f¯(g0 h · x)) and most(f (y)) is isomorphic to
−1
−1
most(f¯(g0 hh−1
1 · y). However, g0 hh1 · y = g0 hh1 h1 · x = g0 h · x and so
most(f (x)) and most(f (y)) are isomorphic as desired.
To see why (3) implies (1), let τ be a symmetric PI -name for a countable
transitive set with support containing some clopen subgroup H ⊂ G. Find
a Borel function fτ : X → P(ω × ω) such that PI forces τ to be equal to
most(fτ (x)). For every group element h ∈ H it is the case that h · τ = τ and
so the set {x ∈ X : most(f¯τ (x)) = most(fτ (h · x))} contains a comeager set.
Thus, the Kuratowski-Ulam theorem implies that the Borel set D = {hx, hi :
most(fτ (x)) = most(fτ (h · x))} is comeager in X × H and the set Cτ = {x ∈
X : Dx is comeager in H} is comeager. As in the previous proof then, for every
x ∈ Cτ and every h ∈ H, either h · x ∈
/ Cτ or else most(fτ (x)) = most(fτ (h · x)).
Now let C, f be as in (3). Strengthen C if necessary to find a symmetric
name τ such that C most(f˙(ẋgen )) = τ and let H be the support of τ . The
set {x ∈ C : most(f (x) = most(fτ (x))} ∩ Cτ is Borel, nonmeager, and it has the
properties required in (1).
To see the final implication (1) to (3), note that by thinning out the set
C as in (1) we may assume that the rank of most(f (x)) is the same for all
x ∈ C; call this ordinal α(f, C). By transfinite induction on α(f, C), for all f, C
simultaneously, argue that (1) implies that there is a hereditarily symmetric
name τ of rank α(f, C) such that C τ = most(f˙(ẋgen )). Suppose this has
been proved for all ordinals less than α(f, C). Let τ be the set of all pairs
hD, σi such that there is an ordinal β ∈ α(f, C) such that σ is a hereditarily
symmetric name of rank ≤ β and D forces σ to be some element of the unique
set f˙(ẏ) where ẏ is some element of the set C in the H-orbit of ẋgen of rank
≤ β. It is clear that all names appearing in τ are hereditarily symmetric, and
moreover the name τ itself is symmetric with support H. Thus, τ is hereditarily
symmetric, and it is enough to show that C τ = most(f˙(ẋgen )). Suppose for
contradiction that some condition C0 ⊂ C forces the inequality. Let χ be a
name for some element of the set most(f˙(ẋgen )) \ τ . Since this is a name for
an element of a set in the model V [[ẋgen ]]E , it is itself forced to be an element
of that model of rank β < α(f, C). Since (2) implies (1), there is a Borel
function fχ , a condition C1 ⊂ C0 and a clopen subgroup H1 ⊂ H as in (1), such
that C1 χ = most(f˙χ (ẋgen )). The induction hypothesis yields a hereditarily
symmetric name ψ of rank ≤ β which is forced by C1 to be equal to χ. The
pair hC1 , ψi belongs to τ , and this contradicts the choice of χ.
It follows that the models V [[ẋgen ]]E and V [ẋgen ]G,F agree on transitive sets
countable in V [x]. To obtain the full agreement, we have to study the models
of ZF intermediate between V [[x]]E and V [x]. First, we verify that V [x] is a
forcing extension of V [[x]]E and compute the poset.
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CHAPTER 3. TOOLS
Fix a PI -name ȧ for some transitive countable set such that V [[ẋgen ]]E =
V (ȧ). Fix a Borel function f : X → P(ω × ω) such that it is forced that
ȧ = most(f˙(ẋgen )) Let x ∈ X be a PI -generic point over V . Let Q be the poset
of all pairs hB, f i such that B ⊂ X is a basic open set, e is a finite function
from ω to ȧ/x, and in some large forcing extension there is a PI -generic point
y ∈ X over V such that y ∈ B, most(f˙(y)) = ȧ/x, and the unique Mostowski
isomorphism extends e. The ordering on Q is coordinatewise extension.
Claim 3.4.24. The point x together with the unique isomorphism between f (x)
and the set a are Q-generic objects over V [a].
Proof. An important part of the proof is checking that the poset Q is in fact
nonempty. Choose a large enough cardinal κ and look at Coll(ω, Vκ ) extension
V (a)[z] of the model V (a). The model V (a)[z] satisfies the axiom of choice:
¯ ] where ∈
¯ is the relation on ω defined
since a ∈ Vκ , it is a model of the form V [∈
¯ m if z(n) ∈ z(m), and adjoining a single real to a model of choice
by n ∈
produces again a model of choice. By analytic absoluteness between V (a)[z]
and V [x] conclude that V (a)[z] contains a PI -generic point x0 ∈ X over V such
that a = most(f (x0 )), since the model V [x] contains such a point, namely x.
Thus indeed, hX, 0i ∈ Q.
First check that the intersection of the first coordinates of conditions in a
Q-generic filter over V (a) is forced to be a PI -generic point; let ẏgen be the
name for this point. To see this, let hB, ei ∈ Q be a condition and C ⊂ X be a
closed nowhere dense set; it will be enough to find an extension of the condition
hB, ei whose first coordinate is disjoint from C. This is simple though; just find
a large forcing extension of V (a) such that it contains a PI -generic point x0 ∈ B
over V as in the definition of the poset Q, find a basic open set B 0 ⊂ B such
that x0 ∈ B and B 0 ∩ C = 0, and consider the condition hB 0 , ei ≤ hB, ei.
The check that Q forces the union of the second coordinates in the generic
filter to be the unique isomorphism between a and most(f (ẏgen )) is similar.
Now, V (a) |= Q V [ẏgen ] = V (a)[ẏgen ] and by the forcing theorem, back in
V there must be a condition in PI that forces the generic ẋgen to be Q̇-generic
over the model V (most(f˙(ẋgen ))). This must be the largest condition: if some
condition in PI forced the opposite, then the whole argument above could be
repeated below it, reaching a contradiction. By the forcing theorem again, the
point x is indeed Q-generic over V (a) as desired.
Let σ ∈ V [[x]]E be a Q-name for a subset of V [[x]]E and a condition q ∈ Q
forcing σ ∈
/ V [[x]]E . We will find a set N ∈ V [[x]]E satisfying the axiom of
extensionality, countable in V [x], such that q σ ∩ Ň ∈
/ V [[x]]E . Thus, writing
π : N̄ → N for the inverse of the Mostowski collapsing map of N to a transitive
/ V [[x]]E ; note that π −1 σ is a
set N̄ in V [[x]]E , it is the case that q π −1 σ ∈
subset of a transitive set countable in V [x]. It will then follow from the claim
and the forcing theorem immediately that whenever b ⊂ V [[x]]E is a set in
V [x]E which does not belong to V [[x]]E , then the model V [[x]]E [b] contains
some transitive set countable in V [x] that does not belong to V [[x]]E . The
theorem will follow as per the first paragraph of this proof.
3.5. ASSOCIATED FORCINGS
65
Back in the ground model, choose a countable elementary submodel M of a
large structure containing a PI -name for the name σ. Write M [x] = {τ /x : τ is a
PI -name in the model M }; this is a countable set in the model V [x], containing
a and Q as subsets and σ as an element. The following claim is proved as in
Corollary 3.4.17.
Claim 3.4.25. M [x] ∩ V [[x]]E ∈ V [[x]]E .
Now, let N = M [x] ∩ V [[x]]E ; this is the set we wanted. Suppose for contradiction that there is a condition r ∈ Q, r ≤ q, forcing σ ∩ Ň ∈ V [[x]]E .
Strengthening r further if necessary, we may assume that there is b ∈ V [[x]]E
such that r σ ∩ Ň = b̌. Now, r ∈ N . There must be a set c ∈ V [[x]]E such
that r does not decide the membership of č in σ. By elementarity of the model
M [x] inside the model V [x], such a set c should exist in M [x] and therefore in
N . Now both c ∈ b and c ∈
/ b are impossible, the final contradiction.
3.5
Associated forcings
3.5a
The poset PIE
In all special cases investigated in this book, the intermediate extension V [xgen ]E
of Definition 3.4.1 is generated by a rather natural regular subordering of PI ,
denoted by PIE .
Let X be a Polish space and I a σ-ideal on it. We would like to define
PIE to be the partial ordering of I-positive E-saturated Borel sets ordered by
inclusion. The problem here is that E-saturations of Borel sets are in general
analytic, and in principle there could be very few Borel E-saturated sets. There
are two possibilities for fixing this. The preferred one is to work only with σideals I satisfying the third dichotomy [67, Section 3.9.3]: every analytic set is
either in the ideal I or it contains an I-positive Borel subset. Then the quotient
PI is dense in the partial order of analytic I-positive sets, and we can define PIE
as the ordering of analytic E-saturated I-positive sets.
Another possibility is to prove that in a particular context, E-saturations
of many Borel sets are Borel. This is for example true if E is a countable
Borel equivalence relation. It is also true if the poset PI is bounding and the
equivalence E is reducible to EKσ . There, if B ⊂ X is a Borel I-positive set
and f : B → Πn (n + 1) is the Borel function reducing E to EKσ , the bounding
property of PI can be used to produce a Borel I-positive compact set C ⊂ B
on which the function f is continuous, and then [C]E is Borel. However, the
question whether every analytic I-positive E-saturated set has a Borel I-positive
E-saturated subset seems to be quite difficult in general, even if one assumes
that every I-positive analytic set contains a Borel I-positive subset.
Definition 3.5.1. (The correct version) Let X be a Polish space and I a σ-ideal
on it such that every analytic I-positive set contains a Borel I-positive subset.
PIE is the poset of I-positive E-saturated analytic sets ordered by inclusion. If
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CHAPTER 3. TOOLS
C ⊂ X is a Borel I-positive set then PIE,C is the poset of relatively E-saturated
analytic subsets of C ordered by inclusion.
By a slight abuse of notation, if every I-positive analytic set contains a Borel
I-posiive set, we will write PI for the poset of all analytic I-positive sets ordered
by inclusion, so that PIE ⊂ PI . The previous discussion seems to have little to
offer for the solution of the main problem associated with the topic of this book:
Question 3.5.2. Is PIE,C a regular subposet of PI for some Borel I-positive
set C ⊂ X?
Note that there is a natural candidate for the projection from the poset
PI to PIE , namely the E-saturation. In all specifica cases investigated in this
book, the answer to the previous question turns out to be positive, the set C is
invariably equal to the whole space, and the saturation is the projection on a
dense set of conditions in PI . We will include one rather simple case criterion
for regularity that will be used throughout the book.
Definition 3.5.3. Let I be a σ-ideal on a Polish space X, E an analytic equivalence relation on X, and let B ⊂ X be an analytic set. We say that B is an
E-kernel if for every analytic set C ⊂ B, [C]E ∩ B ∈ I implies [C]E ∈ I.
Theorem 3.5.4. Let I be a σ-ideal on a Polish space X such that every Ipositive analytic set has a Borel I-positive subset, and let E be an analytic
equivalence relation on X. The poset PIE is regular in PI if and only if every
Borel I-positive set has an analytic I-positive subset which is also an E-kernel.
The main point here is that in the cases encountered in this book, the kernels
are typically very natural closed sets. Observe that in the case that saturations
of sets in I are still in I, every analytic set is a kernel:
Corollary 3.5.5. Suppose that I is a σ-ideal on a Polish space X such that
every I-positive analytic set has a Borel I-positive subset, and E is an analytic
equivalence relation on X. If E-saturations of sets in I are still in I, then the
poset PIE is regular in PI .
Proof. Suppose that every I-positive Borel set contains a Borel I-positive kernel;
we must show that PIE is regular in PI . Let A ⊂ PIE be a maximal antichain;
we must show that A is also maximal in PI . Suppose not, and let B ∈ PI be
an analytic set incompatible with all elements of the antichain A, and thin it
out if necessary to a kernel. Then, for every set C ∈ A, B ∩ C ∈ I, and so
[B ∩ C]E ∈ I by the assumptions. However, [B ∩ C]E = [B]E ∩ C since the set C
is E-invariant, and so [B]E is a condition in PIE incompatible with all elements
of the antichain A, contradicting its maximality in PIE .
On the other hand, suppose that PIE is regular in PI , and B ⊂ X is a Borel Ipositive set. To produce an I-positive E-kernel in it, consider the open dense set
D ⊂ PIE consisting of those analytic E-invariant sets with I-small intersections
with B and those sets which do not have an E-invariant I-positive analytic
subset with I-small intersection with B. Since the poset PIE is regular in PI ,
3.5. ASSOCIATED FORCINGS
67
there must be a set B 0 ∈ D compatible with B, so B 0 ∩ B ∈
/ I. Note that this
means that no E-invariant I-positive subset of B 0 has I-small intersection with
B. We claim that B ∩ B 0 is the requested E-kernel. And indeed, if C ⊂ B ∩ B 0
is an analytic set such that [C]E ∩ B ∩ B 0 ∈ I, then [C]E ⊂ B 0 has I-small
intersection with the set B and therefore has to be I-small!
The poset PIE is frequently ℵ0 -distibutive, and yields the model V [ẋgen ]E .
The following two theorems provide criteria for this to happen.
Definition 3.5.6. The equivalence E on a Polish space X is I-dense if for every
I-positive Borel set B ⊂ X there is a point x ∈ B such that [x]E ∩ B is dense
in B.
Proposition 3.5.7. If the poset PI has the continuous reading of names, PIE
is a regular subposet of PI , and E is I-dense, then PIE is ℵ0 -distributive.
Proof. Since PI is proper, so is PIE , and it is enough to show that PIE does
not add reals. Suppose that B ∈ PI forces that τ is a name for a real in the
PIE -extension. Let M be a countable elementary submodel of a large enough
structure, and let C ⊂ B be the I-positive set of all M -generic points for PI .
If x ∈ C is a point, write Hx ⊂ PIE for the M -generic filter of all sets D ∈ PIE
such that [x]E ⊂ D. It is clear that the map x 7→ Hx is constant on equivalence
classes, and so is the Borel map x 7→ τ /Hx . Thin out the set C if necessary
to make sure that the map x 7→ τ /Hx is continuous on C. By the density
property, there is a point x ∈ C such that [x]E ∩ C is dense in C. Thus, the map
x 7→ τ /Hx is continuous on C and constant on a dense subset of C, therefore
constant on C. The condition C clearly forces τ to be equal to the single point
in the range of this function.
Note that typically the poset PIE is fairly simply definable. If suitable large
cardinals exist, then the descending chain game on the poset PIE is determined.
Thus, when the poset is ℵ0 -distributive, Player I has no winning strategy in
this game, so it must be Player II who has a winning strategy and the poset
is strategically σ-closed. It is not entirely clear whether there may be cases in
which PIE is ℵ0 -distributive but fails to have a σ-closed dense subset. In many
situations, one can find forcing extensions in which PIE is even ℵ1 -distributive
or more.
In all situations described in the later chapters of this book, the forcing PIE
generates the V [ẋgen ]E extension of Theorem 3.4.4. To prove the appropriate
general theorem, we need a definition:
Definition 3.5.8. Let I be a σ-ideal on a Polish space X and let E be an
equivalence relation on X. We say that E is I-homogeneous if for every pair
of I-positive analytic sets B, C ⊂ X, either there is a Borel I-positive subset
B 0 ⊂ B such that [B 0 ]E ∩ C ∈ I, or there is a Borel I-positive subset B 0 ⊂ B
and an injective Borel map f : B 0 → C such that f ⊂ E and f preserves the
ideal I: images of I-small sets are I-small, and images of I-positive sets are
I-positive.
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CHAPTER 3. TOOLS
This property may not always be completely straightforward to verify, but it is
satisfied in all natural cases; in particular, it holds whenever E is a countable
Borel equivalence relation.
Theorem 3.5.9. Suppose that I is a σ-ideal on a Polish space X such that the
quotient forcing PI is proper and every analytic I-positive set contains a Borel
I-positive subset. If E is I-homogeneous, then the model V [ẋgen ]E is forced to
be equal to the PIE -extension.
Proof. Let G ⊂ PI be a generic filter and H = G ∩ PIE . It is clear that
V [H] ⊂ V [ẋgen ]E , since H is definable from [ẋgen ]E in every further forcing
extension as the collection of those Borel I-positive E-saturated sets coded in
the ground model containing the calss [ẋgen ]E as a subset. The other inclusion
is harder.
Suppose that φ(α, ~v , [ẋgen ]E ) is a formula with an ordinal parameter, other
parameters in V , and another parameter [ẋgen ]E , and let κ be a cardinal. A
density argument shows that there must be a condition B ∈ PI such that the
condition hP, 1i ∈ PI × Coll(ω, κ) decides the truth value of φ, and H contains a
condition below the pseudoprojection of B to PIE . We will show that for every
two such conditions, the decision of the truth value must be the same. Thus, the
set of ordinals defined by φ in the PI × Coll(ω, κ) must be already in the model
V [H], defined as the set of those ordinals α such that for some condition B ∈ PI
whose pseudoprojection to PIE belongs to the filter H, hB, 1i φ(α, ~v , [ẋgen ]E ).
So suppose for contradiction that there are conditions C0 , C1 ∈ PI and
B ∈ PIE such that B is below the pseudoprojection of both C0 , C1 into PIE , and
hC0 , 1i φ(α, ~v , [ẋgen ]E ) and hC1 , 1i ¬φ. Since B is below the pseudoprojections, the first item of the assumptions is excluded for B, C in both cases C = C0
or C = C1 ; indeed, the set [B 0 ]E would be an extension of B in PIE incompatible
with the condition C0 or C1 . So the second case must happen, and strengthening twice, we get a Borel I-positive set B 0 ∈ PI below B and Borel injections
f0 : B 0 → C0 and f1 : B 0 → C1 as postulated in the assumptions. Since these
functions preserve the ideal I and are subsets of the equivalence E, B 0 forces
in the forcing PI that the points ẋgen ∈ Ḃ 0 , f˙0 (ẋgen ) ∈ Ċ0 and f˙1 (ẋgen ) ∈ Ċ1
are E-equivalent V -generic points for the poset PI , giving the same generic extension. Thus, if G ⊂ PI is a V -generic filter containing the condition B 0 and
K ⊂ Coll(ω, κ) is a V [G]-generic filter, the forcing theorem applied to f0 (xgen )
and f1 (xgen ) implies that V [G, K] |= φ(α, ~v , [f0 (xgen )]E ) ∧ ¬φ(α, ~v , [f1 (xgen )]E ),
which is impossible in view of the fact that f0 (xgen ) E f1 (xgen ).
As a final remark regarding the poset PIE , its definition also called for the
consideration of relativized posets PIE,C for Borel I-positive sets C. The reason
is that even in quite natural cases, one has to pass to a condition in order to find
the requested regularity, as the following example shows. Compare this also to
Theorem 4.2.1 and its proof.
Example 3.5.10. Consider X = ω ω × 2 and the σ-ideal I generated by sets
B ⊂ X such that {y ∈ ω ω : hy, 0i ∈ B} is compact and {y ∈ ω ω : hy, 1i ∈ B}
3.5. ASSOCIATED FORCINGS
69
is countable. The quotient forcing is clearly just a disjoint union of Sacks and
Miller forcing. Consider the equivalence relation E on X defined as the equality
on the first coordinate. Then E has countable classes, indeed each of its classes
has size 2, and PIE is not a regular subposet of PI . To see that, consider a
maximal antichain A in Sacks forcing consisting of uncountable compact sets,
and let Ā = {C × 2 : C ∈ A}. It is not difficult to see that Ā is a maximal
antichain in PIE , but the set ω ω × 1 ⊂ X is Borel, I-positive, and has an I-small
intersection with every set in Ā.
3.5b
The reduced product posets
Another forcing feature of the arguments in this book is the use of a reduced
product of quotient posets. There is a general definition of such products:
Definition 3.5.11. Let I be a σ-ideal on a Polish space X, and let E be a
Borel equivalence relation on X. The reduced product PI ×E PI is defined
as the poset of those pairs hB, Ci of Borel I-positive sets such that for every
sufficiently large cardinal κ, in the Coll(ω, κ) extension there are E-equivalent
points x ∈ B and y ∈ C, each of them V -generic for the poset PI . The ordering
is that of coordinatewise inclusion.
The left coordinates of conditions in a reduced product generic filter form a
PI -generic filter, and similarly for the right coordinates. To see this, assume that
hB, Ci is a condition in PI ×E PI and D ⊂ PI is a dense set. The poset Coll(ω, κ)
forces that there exists a pair of V -generic E-equivalent points ẋ ∈ Ḃ, ẏ ∈ Ċ.
Find a condition p ∈ Coll(ω, κ) and conditions B 0 ⊂ B and C 0 ⊂ C in the dense
set D such that p ẋ ∈ B 0 , ẏ ∈ C 0 and observe that hB 0 , C 0 i is still a condition
in the reduced product PI ×E PI and it clearly forces that both filters on the
left hand side and the right hand side meet the dense set D. Thus, the reduced
product generic filter is given by two points ẋlgen , ẋrgen ∈ X, each of which is
V -generic for the poset PI . These two points are typically not E-related. The
general definition of the reduced product is certainly puzzling, and in all special
cases, a good deal of effort is devoted to the elimination of the Coll(ω, κ) forcing
relation from it. The reduced products are always used in the intermediate case
of the trichotomy 3.4.4:
Proposition 3.5.12. If I is a σ-ideal on a Polish space X such that the quotient
forcing PI is proper, and E is a Borel equivalence relation on X which is Iergodic, then PI ×E PI = PI × PI .
Proof. Let B ⊂ X be a Borel I-positive subset such that every two Borel Ipositive subsets C0 , C1 ⊂ B contain a pair of equivalent points. We will show
that the reduced product below the condition hB, Bi consists of all pairs of
Borel I-positive subsets of B. In other words, for every pair hC0 , C1 i of Borel Ipositive subsets of B there is a forcing extension in which there are E-equivalent
points x0 ∈ C0 , x1 ∈ C1 , both V -generic for the poset PI . Proceed as in the
third case of the trichotomy theorem 3.4.4 to find a generic extension V [K] in
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CHAPTER 3. TOOLS
which there is an elementary embedding j : V → N into a possibly illfounded
model N which nevertheless contains M = j 00 Hc+ as a countable set. Working
in N , let D0 = {x ∈ jC0 : x is M -generic for Pj(I) }, and similarly for D1 . In
the model N , these are Borel j(I)-positive sets by Fact 2.6.1, they are both
subsets of jB, and so they must contain j(E)-related points x0 ∈ D0 , x1 ∈ D1
by the ergodicity. Now, the model V [K] sees x0 ∈ C0 , x1 ∈ C1 , and these two
points are PI -generic over V . Moreover, since j(E) ⊂ E V [K] , the model also
sees x0 E x1 and we are finished.
Proposition 3.5.13. Let I be a σ-ideal on a Polish space X and E is an
analytic equivalence relation on X.
1. The reduced product PI ×E PI forces V [ẋlgen ] ∩ V [ẋrgen ] = V [ẋlgen ]E =
V [ẋrgen ]E ;
2. if the model V [ẋlgen ]E contains no new reals then the pair hẋlgen , ẋrgen i is
forced to be PI ẋlgen × PI ẋrgen -generic over V [ẋlgen ]E .
Proof. First argue that the models V [ẋlgen ]E and V [ẋrgen ]E are forced to be
equal. Indeed, if φ is a formula with a ground model parameter p, α is an
ordinal, κ is a large enough cardinal and hB, Ci is a condition in the reduced
product, it cannot be the case that B Coll(ω, κ) φ(α̌, p̌, [ẋgen ]E ) and C Coll(ω, κ) ¬φ(α̌, p̌, [ẋgen ]E ): in a Coll(ω, κ) extension, there would be V generic points x ∈ B and y ∈ C with x E y by the definition of the reduced
product, and homogeneity arguments would provide Coll(ω, κ)-generic filters
Gx , Gy such that V [G] = V [x][Gx ] = V [y][Gy ]. Then, the forcing theorem on
the x and y sides would indicate that V [G] |= φ(α, p, [x]E ) and φ(α, p, [y]E ),
which is certainly impossible as x E y. Thus, the reduced product forces that
the same formulas must define the same sets of ordinals in the models V [ẋlgen ]E
and V [ẋrgen ]E –so these two models must be the same.
The previous paragraph shows that the model V [ẋlgen ]E = V [ẋrgen ]E is a
subset of V [ẋlgen ] ∩ V [ẋrgen ]. To see the opposite inclusion, suppose that a
condition hB0 , C0 i forces some set τ of ordinals to be in the intersection of
V [ẋlgen ] and V [ẋrgen ]; we must show that it belongs to the model V [ẋlgen ]E .
Strengthening the condition if necessary, one can find PI -names τl and τr such
that hB0 , C0 i τ = τl /ẋlgen = τr /ẋrgen . Since the poset PI ×E PI below the
condition hB0 , C0 i forces the left generic to be PI -generic over V , there must be
a Borel I-positive set B1 ⊂ B0 such that in a large collapse extension, whenever
x ∈ B1 is a PI -generic point over V then there is a filter H ⊂ PI ×E PI generic
over V containing hB0 , C0 i and such that x is the left half of the pair given by
the filter H.
Claim 3.5.14. In a large collapse extension, whenever x, y are PI -generic
points over V , both in B1 and E-related, then τl /x = τl /y.
Proof. Move to a large collapse extension V [G] and suppose for contradiction
that x, y ∈ B1 are two E-related PI -generic points over V such that τl /x 6= τl /y.
Choose an ordinal α such that α ∈ τl /x \ τl /y.
3.5. ASSOCIATED FORCINGS
71
Back to V . By the forcing theorem, there must be conditions Bx , By ⊂ B1
in PI such that x ∈ Bx , y ∈ By , Bx α̌ ∈ τl and By α̌ ∈
/ τl . Moreover,
Bx can be chosen to force that large collapse forces that there is a PI -generic
point over V in the set By which is E-related to ẋgen . It follows that whenever
C2 ⊂ C0 is a Borel I-positive set such that hBx , C2 i ∈ PI ×E PI , then even
hBy , C2 i ∈ PI ×E PI . To see this, by the definitions in the collapse extension
V [G] there must be points x0 ∈ Bx , z ∈ C1 which are both PI -generic over V
and E-related. By the choice of the condition Bx , there must be also a point
y 0 ∈ By in V [G] which is PI -generic over V and E-related to x0 . Then the pair
hy 0 , zi ∈ E witnesses the fact that hBy , C2 i ∈ PI ×E PI .
In V [G] again, find a filter H ⊂ PI ×E PI generic over V such that x is its
left half and hB0 , C0 i ∈ H. There must be a condition hB2 , C2 i ∈ H such that
B2 ⊂ Bx and C2 α ∈ τr . But then, hBy , C2 i is a condition in the reduced
product such that By α ∈
/ τl and C2 α ∈ τr , contrary to hB0 , C0 i forcing
τl /ẋlgen = τr /ẋrgen .
Now, any condition hB2 , C2 i in the reduced product forcing ẋlgen ∈ B1
also forces τl /ẋlgen ∈ V [ẋlgen ]E : the set τl /ẋlgen is defined in a large collapse
extension as the unique set of the form τl /x where x ∈ B1 ∩ [ẋlgen ]E is a point
PI -generic over V . We conclude that V [ẋlgen ]∩V [ẋrgen ] ⊂ V [ẋlgen ]E as required.
To prove (2), assume that the model V [ẋlgen ] = V [ẋrgen ] does not contain
any new reals. The first paragraph of the present proof shows that the ideals
I ẋlgen and I ẋrgen are forced to be equal and we will denote them by I ∗ . Note
that any condition hB, Ci forces itself into the product PI ∗ × PI ∗ since it forces
ẋlgen ∈ B and ẋrgen ∈ C. Thus the generic filter on PI ×E PI is also a filter
on PI ∗ × PI ∗ . We will show that it is in fact a PI ∗ × PI ∗ -generic filter over
V [ẋlgen ]E .
Suppose for contradiction that hB0 , C0 i ∈ PI ×E PI is a condition forcing
the filter on PI ∗ ×PI ∗ not to be V [ẋlgen ]E -generic, missing a dense set Ḋ defined
by a formula φ. Since V [ẋlgen ]E is forced to contain no new reals or Borel sets,
there must be a condition hB1 , C1 i ≤ hB0 , C0 i in the reduced product forcing
some pair hB2 , C2 i ≤ hB0 , C0 i into the product PI ∗ × PI ∗ and the dense set Ḋ.
We will reach a contradiction by producing, in some large collapse extension,
pairs hx1 , y1 i and hx2 , y2 i, both reduced product generic over V , one of them
meeting the condition hB1 , C1 i, the other meeting the condition hB2 , C2 i, such
that x1 E x2 . The definition of the set D yields the same set whether defined
from [x1 ]E or [x2 ]E , and so the filter associated with hx2 , y2 i does meet the set
Ḋ (in the condition hB2 , C2 i), contrary to what was forced by the condition
hB0 , C0 i.
Let κ be a suitably large cardinal. Choose (PI ×E PI ) × Coll(ω, κ)-names
σ and τ such that hB1 , C1 , 1i forces σ, τ to be PI -generic points over V in the
sets Ḃ2 , Ċ2 respectively, E-equivalent to ẋlgen . By the forcing factorization,
Fact 2.3.2, there must be an I-positive Borel set B3 ⊂ B2 such that in a large
collapse extension, for every point x ∈ B3 PI -generic over V there is a filter
G ⊂ (PI ×E PI ) × Coll(ω, κ), generic over V , such that x = τ /G. This means
that for every condition B4 ⊂ B3 , it is the case that the pair hB4 , C2 i belongs to
72
CHAPTER 3. TOOLS
the reduced product: in a large collapse extension, one can pick a point x ∈ B4
PI -generic over V , then a filter G ⊂ (PI ×E PI ) × Coll(ω, κ) generic over V such
that τ /G = x, and then σ/G is a point in C2 which is PI -generic over V and Eequivalent to x. Since the reduced product forces that the left coordinate of the
generic pair is a PI -generic point over V , the forcing factorization, Fact 2.3.2,
applied again shows there must be an I-positive Borel set B5 ⊂ B3 such that
in a large collapse extension, for every PI -generic point x ∈ B5 over V there is
a filter reduced product generic over V containing the condition hB5 , C2 i such
that the left coordinate of the generic pair is exactly x.
Move to a large collapse forcing extension of V and pick a point x2 ∈ B5
PI -generic over V . There is a point y2 ∈ X such that the pair hx2 , y2 is PI ×E PI generic over V . There is also a filter G ⊂ (PI ×E PI ) × Coll(ω, κ)-generic over V
such that τ /G = x2 . Let hx1 , y1 i be the pair associated with G. By the choice
of the name τ , it is the case that x1 E x2 , and we are done.
Chapter 4
Classes of equivalence
relations
4.1
Smooth equivalence relations
In the case that the equivalence relation E is smooth, the model V [xgen ]E as
described in Theorem 3.4.4 takes on a particularly simple form.
Proposition 4.1.1. Let I be a σ-ideal on a Polish space X such that the quotient forcing PI is proper, and suppose that E is a smooth equivalence relation
on the space X. Then V [xgen ]E = V [f (ẋgen )] for every ground model Borel
function f reducing E to the identity.
Proof. Choose a Borel function f : X → 2ω reducing the equivalence E to the
identity. Let G ⊂ PI and H ⊂ Coll(ω, κ) be mutually generic filters, and let
xgen ∈ X be a point associated with the filter G. First of all, f (xgen ) ∈ 2ω is
definable from the equivalence class [xgen ]E : it is the unique value of f (x) for all
x ∈ [xgen ]E . Thus, V [f (xgen )] ⊂ V [xgen ]E . On the other hand, the equivalence
class [xgen ]E is also definable from f (xgen ), the model V [G, H] is an extension
of V [f (xgen )] via a homogeneous notion of forcing Coll(ω, κ), and therefore by
[22, Lemma 26.7] V [xgen ]E ⊂ V [f (xgen )].
The total canonization of smooth equivalence relations has an equivalent restatement with the quotient forcing adding a minimal real degree. It is necessary
though to discern between the forcing adding a minimal real degree, and the
forcing producing a minimal extension. In the former case, we only ascertain
that V [y] = V or V [y] = V [xgen ] for every set y ⊂ ω in the generic extension.
In the latter case, this dichotomy in fact holds for every set y of ordinals in the
extension. For example, the Silver forcing adds a minimal real degree, while it
produces many intermediate σ-closed extensions, as shown in Section 5.8.
Proposition 4.1.2. Let X be a Polish space and I be a σ-ideal on X such that
the quotient forcing PI is proper. The following are equivalent:
73
74
CHAPTER 4. CLASSES OF EQUIVALENCE RELATIONS
1. PI adds a minimal real degree;
2. I has total canonization of smooth equivalence relations.
Proof. First, assume that the canonization property holds. Suppose that B ∈ PI
forces that τ is a new real; we must show that some stronger condition forces
V [τ ] = V [ẋgen ]. Thinning out the set B if necessary we may find a Borel
function f : B → 2ω such that B τ = f˙(ẋgen ). Note that singletons must
have I-small f -preimages, because a large preimage of a singleton would be a
condition forcing τ to be that ground model singleton. Consider the smooth
equivalence relation E on B, E = f −1 id. Since it is impossible to find a Borel
I-positive set C ⊂ B such that E B = ee, there must be a Borel I-positive set
C ⊂ B such that E C = id, in other words f C is an injection. But then,
C V [τ ] = V [ẋgen ] since ẋgen is the unique point x ∈ C such that f (x) = τ .
Now suppose that PI adds a minimal real degree, and let B ∈ PI be a
Borel I-positive set and E a smooth equivalence relation on B with I-small
classes. We need to produce a Borel I-positive set on which E is equal to the
identity. Let f : B → 2ω be a Borel function reducing E to the identity. Note
that since f -preimages of singletons are in the ideal I, f˙(ẋgen ) is a name for
a new real and therefore it is forced that V [ẋgen ] = V [f˙(ẋgen )]. Let M be
a countable elementary submodel of a large structure, and let C ⊂ B be the
Borel I-positive set of all M -generic points in B for the poset PI . The set f 00 C
is Borel. Consider the Borel set D ⊂ f 00 C × C defined by hy, xi ∈ D if and only
if f (x) = y. The set D has countable vertical sections: If f (x) = y then by the
forcing theorem applied in the model M it must be the case that M [x] = M [y]
and thus the vertical section Dy is a subset of the countable model M [y]. Use
the uniformization therem to find countably many graphs of Borel functions
gn : n ∈ ω whose union is D. The sets rng(gn ) : n ∈ ω are Borel since the
functions gn are Borel injections, and they cover the whole set C. Thus there is
a number n ∈ ω such that the set rng(gn ) is I-positive. On this set, the function
f is an injection as desired.
4.2
Countable equivalence relations
Borel countable equivalence relations are a frequent guest in mathematical practice. Their treatment in this book is greatly facilitated by essentially complete
analysis of the associated forcing extensions.
Theorem 4.2.1. Suppose that I is a σ-ideal on a Polish space X such that
the quotient forcing is proper. Suppose that E is a countable Borel equivalence
relation on X. Then for every Borel I-positive set B ⊂ X there is a Borel Ipositive set C ⊂ B such that PIE,C is a regular subposet of PI below C. Moreover,
C forces the model V [ẋgen ]E to be equal to the PIE,C extension. The remainder
forcing PI C/PIE,C is weakly homogeneous and c.c.c.
4.2. COUNTABLE EQUIVALENCE RELATIONS
75
Here, PIE,C is the poset of I-positive Borel subsets of C which are relatively
E-saturated inside C. The set C ⊂ X may not be the whole space, as Example 3.5.10 shows.
Proof. Use the Feldman–Moore theorem 2.2.5 to find a Borel action of a countable group G on B whose orbit equivalence relation is exactly E.
Claim 4.2.2. There is a Borel I-positive set C ⊂ B such that the E-saturation
of any I-small Borel subset of C has I-small intersection with C.
Proof. Let M be a countable elementary submodel of a large structure and let
C be the set of all M -generic elements of the set B for the poset PI . We claim
that this set works.
Suppose for contradiction that there is an I-small Borel subset D of C whose
E-saturation has an I-positive intersection with C. Note that [D]E is a Borel
set again as E is a countable Borel equivalence relation. Thinning out the set
C ∩ [D]E we may find a Borel I-positive subset C 0 and a group element g ∈ G
such that gC 0 ⊂ D.
Now consider the set O = {A ∈ PI : if there is a Borel I-positive subset of A
whose shift by g is I-small, then A is such a set}. Clearly, this is an open dense
subset of PI below B in the model M , and therefore there is A ∈ M ∩ O such
that A ∩ C 0 ∈
/ I. Now g(A ∩ C 0 ) ⊂ D and therefore this set is in the ideal I, and
by definition of the set O it must be the case that gA ∈ I. But as gA ∈ I ∩ M ,
it must be the case that C ∩ gA = 0, which contradicts the fact that g(A ∩ C 0 )
is a nonempty subset of C.
Fix a set C ⊂ B as in the claim. The following is an immediate consequence
of Theorem 3.5.5 and the previous claim.
Claim 4.2.3. PIE,C is a regular subposet of PI below C.
Claim 4.2.4. The remainder forcing PI C/PIE,C is weakly homogeneous and
c.c.c.
Proof. Let G ⊂ PI be a generic filter containing the condition C with its associated generic point xgen , and let H = G ∩ PIE,C . Note first that the remainder
forcing consists of exactly those Borel sets D ⊂ C such that their E saturation
belongs to the filter H, or restated, those Borel sets containing some element of
the equivalence class of xgen .
For the c.c.c. suppose that {Dα : α ∈ ω1 } is a putative antichain in the
remainder poset in the model V [H]. Since the equivalence class of xgen is countable and ℵ1 is preserved in V [G], there must be α 6= β ∈ ω1 such that Dα , Dβ
contain the same element of the equivalence class. It follows that Dα ∩ Dβ is a
lower bound of Dα , Dβ in the remainder forcing.
For the weak homogeneity, return to the ground model. Suppose that
D0 , D1 , D2 ⊂ C are Borel I-positive sets, D2 is E-saturated and forces in PIE,C
both D0 , D1 to the remainder forcing. By the countable additivity of the ideal
I there must be an element g ∈ G and a positive set D̄0 ⊂ D0 ∩ D2 such that
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CHAPTER 4. CLASSES OF EQUIVALENCE RELATIONS
g D̄0 ⊂ D1 –if for each g ∈ G the set Ag = {x ∈ D0 ∩ D2 : gx ∈ D1 } was
in the σ-ideal S
I, so would be their union, and the saturation of the I-positive
set D0 ∩ D2 \ g Ag below D0 would force D2 not to belong to the remainder
forcing. The map π defined by π(A) = gA preserves the ideal I, preserves the
E-saturations and therefore the condition [D̄0 ]E ∈ PIE,C forces that it induces
an automorphism of the remainder poset moving a subset of D0 to a subset of
D1 . The homogeneity of the quotient forcing follows.
The fact that PIE,C generates the V [ẋgen ]E model now follows directly from
Theorem 3.5.9; the group action provides the necessary automorphisms of the
poset PIE,C .
Inspecting the proofs in the previous chapter, the fact that the model V [ẋgen ]E
is obtained by the forcing PIE are invariably proved in a significantly tighter fashion: it is proved that in the PI extension, the equivalence class of the generic
real is the only one which is a subset of all E-invariant Borel sets in the generic
filter. This means that, in the notation of the previous proof, the filter H is
interdefinable with [xgen ]E in the PI -extension V [G] and they therefore have
to define the same HOD model. This, however, does not have to be the case
in general even in the very restrictive case that the forcing PI adds a minimal real degree. Consider the case where PI is the Silver forcing and E is the
equivalence relation on 2ω defined by x E y if x, y differ on finite set of entries,
and moreover, the set of their differences has even cardinality. Comparing E
with E0 , we see that E ⊂ E0 and every E0 equivalence class decomposes into
exactly two E classes. Note that the E0 -saturation of any Silver cube is equal
to its E-saturation, and therefore the posets PIE0 and PIE are co-dense. If G
is the Silver generic filter and H = G ∩ PIE then the E0 -class of the Silver real
belongs to all sets in H, and it splits into two E-classes, each of which has the
property that it is a subset of all sets in the PIE -generic filter. Moreover, neither
of these classes is ordinally definable from H or parameters in V . To see this,
suppose that a condition p in the Silver forcing forces that a formula φ defines
the E-equivalence class of ẋgen from the parameters Ġ ∩ PIE0 and ~v ∈ V . Let
x ∈ 2ω be a Silver real meeting the condition p, and let x̄ be equal to x except
for one entry that is not in the domain of p. Then x̄ is again a Silver real, the
generic filters associated with x and x̄ are different, but still they yield the same
extension and the same intersection with PIE0 . Thus, the formula φ applied to
either of them yields the same set of reals. This is impossible though as ¬x E x̄.
Let us now return back to the general treatment of Borel countable equivalence relations. In the common case that the forcing PI adds a minimal real
degree, there is much more information available about the V [ẋgen ]E model.
Theorem 4.2.5. Suppose that I is a σ-ideal on a Polish space X such that the
quotient forcing is proper and adds a minimal real degree. Suppose that E is an
essentially countable equivalence relation on X which does not simplify to smooth
on any Borel I-positive set. Then the model V [ẋgen ]E is forced to contain no
4.2. COUNTABLE EQUIVALENCE RELATIONS
77
new countable sequences of ordinals, and it is the largest intermediate extension
in the PI -extension.
Proof. Let f : X → Y be a Borel reduction of E to a Borel countable equivalence
relation F on a Polish space Y . Let B ⊂ X be an I-positive Borel set. By the
total canonization of smooth equivalences we may pass to a Borel I-positive set
C on which the function is injective and therefore E itself has countable classes,
and by the previous theorem we may assume that the poset PIE,C is a regular
subposet of PI below C.
We will first argue that the poset PIE,C is ℵ0 -distributive. The forcing is
a regular suposet of a proper forcing, therefore proper, and so it is enough to
show that it does not add reals. Suppose that τ is a PIE,C -name for a real, and
D ⊂ C is a Borel I-positive set. We must produce a Borel I-positive subset of
D forcing a definite value to τ . Let M be a countable elementary submodel of a
large enough structure, and let A ⊂ D be the Borel I-positive subset consisting
of all PI -generic points over M . The map x 7→ Kx assigning to each generic
point the corresponding PIE,C generic filter for M , is E-invariant: Kx is the set
of all relatively E-saturated Borel subsets containing x, which depends only on
the E-equivalence class of a point x ∈ C. Thus the map x 7→ τ /Kx is Borel and
also E-invariant. Since the forcing PI adds a minimal real degree, there is a
Borel I-positive set A0 ⊂ A such that this map is either one-to-one or constant
on A0 . The former possibility cannot occur–since the function is E-invariant,
it would have to be the case that E A0 = id, contradicting the assumption
that E cannot be reduced to a smooth equivalence on a positive Borel set. The
latter possibility forces τ to be equal to the constant value of the function on
A0 , as desired.
Now we need to argue that the model V [xgen ]E is the largest intermediate
model between V and V [G]; in other words, if U is a model of ZFC such that
V ⊆ U ⊆ V [G], then either U = V [G] or U ⊆ V [xgen ]E . Suppose that U = V [Z]
is an intermediate model of ZFC, for a set Z ⊂ κ for some ordinal κ. Fix a
name Ż for the set Z. If there is a countable set a ⊂ κ in V such that Z ∩ a ∈
/V
then V [G] = V [Z] by the minimal real degree assumption. Suppose then that
for every countable set a ∈ V , it is forced that Ż ∩ ǎ ∈ V . Consider the names
for countably many sets {Ż/y : y ∈ [ẋgen ]E and y is PI -generic over V }. There
must be a countable set b ⊂ κ such that if any two of these sets differ at some
ordinal, then they differ at an ordinal in b. By properness, this set b has to be
covered by a ground model countable set a ⊂ κ. Suppose that D ⊂ C in PI is a
condition which forces this property of the set a, and moreover decides the value
of Ż ∩ ǎ to be some definite set c ∈ V . Then D forces that Z is definable from
[xgen ]E , D, c, and Ż in the model V [G] by the formula ”Z is the unique value
of Ż/y for all V -generic points y ∈ [xgen ]E ∩ D such that Ż/y ∩ a = c”. Note
that the same formula defines Z in every further generic extension of V [G]: the
equivalence class [xgen ]E is countable in V [G] and therefore evaluated the same
in V [G] as in the further extension by coanalytic absoluteness, and therefore
the quantifiers in φ range over the same set whether applied in V [G] or further
extension. Thus, V [Z] ⊂ V [xgen ]E !
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CHAPTER 4. CLASSES OF EQUIVALENCE RELATIONS
An important heuristic consequence of the last sentence of this theorem: if
the poset PI adds a minimal real degree, we cannot use the model V [ẋgen ]E
to discern between the various nonsmooth countable equivalence relations on
the underlying Polish space X, because they all happen to generate the same
model. This is apparently not to say that every two such countable equivalence
relations must be similar or of the same complexity at some I-positive Borel
set.
Corollary 4.2.6. If the forcing PI is proper, nowhere c.c.c. and adds a minimal
forcing extension, then I has total canonization for essentially countable Borel
equivalences.
Proof. Looking at the previous two theorems, if the poset PI adds a minimal real
degree and E is an essentially countable equivalence relation on the underlying
Polish space, then PI decomposes into an iteration of an ℵ0 -distibutive and a
c.c.c. forcing. If PI is nowhere c.c.c. then the first extension of the iteration
must be nontrivial, contradicting the assumptions.
In the next section, we will be able to extend the conclusion of the corollary to
total canonization of equivalences classifiable by countable structures, but the
treatment of the essentially countable equivalences is a necessary preliminary
step towards that goal.
The final observation in this section concerns the simplification within the
class of countable equivalence relations. The class of Borel countable equivalence
relations is very complex and extensively studied. It is possible that many
forcings have highly nontrivial intersection of their spectrum with this class.
In the fairly common case that the forcing preserves Baire category, all that
complexity boils down to E0 though.
Theorem 4.2.7. Whenever I is a σ-ideal on a Polish space X such that the
forcing PI is proper and preserves Baire category, then every essentially countable Borel equivalence relation is reducible to E0 on a positive Borel set.
Proof. Suppose B ∈ PI is an I-positive Borel set and f : B → Y is a Borel
map into a Polish space with a countable equivalence relation E on it. In [31,
Theorem 12.1], Miller and Kechris show that there is a Borel set D ⊂ ω ω ×Y ×Y
such that for every z ∈ ω ω the section Dz is a Borel hyperfinite equivalence
relation included in E, and for every y ∈ Y the set Ay = {z ∈ ω ω : [y]E = [y]Dz }
is comeager in ω ω . Now the set D0 ⊂ B ×ω ω of all pairs x, z such that z ∈ Af (x) ,
has comeager vertical sections. Since the forcing PI preserves the Baire property,
it must be the case that there is a point z ∈ ω ω such that the corresponding
horizontal section C ⊂ B is I-positive. But then, Dz = E on f 00 C and therefore
the pullback f −1 E is on C reducible to a hyperfinite equivalence relation Dz
and therefore to E0 .
4.3. EQUIVALENCES CLASSIFIABLE BY COUNTABLE STRUCTURES79
4.3
Equivalences classifiable by countable structures
This is a class of equivalence relations very common in mathematical practice.
Definition 4.3.1. An equivalence relation E on a Polish space X is classifiable
by countable structures, cbcs if there is a Borel map f : X → Πn P(ω n ) such
that elements x, y ∈ X are E-related if and only if f (x), f (y) are isomorphic as
relational structures on ω.
An isomorphism of structures on ω is naturally induced by a permutation of
ω, and therefore every equivalence relation classifiable by countable structures
is Borel reducible to an orbit equivalence generated by a Polish action of a
closed subgroup of the infinite permutation group S∞ . The converse implication
also holds by a result of Becker and Kechris [27, Theorem 12.3.3]: every orbit
equivalence relation generated by a Polish action of the infinite permutation
group or its closed subgroups is classifiable by countable structures.
The behavior of Borel equivalence relations classifiable by countable structures is best analyzed via the notion of Friedman-Stanley jump. If E be an
equivalence relation on a Polish space Y , then E + is the equivalence relation on
Y ω defined by ~x E + ~y if the E-saturations of rng(~x) and rng(~y ) are the same. If
E is a Borel equivalence relation then E + is again Borel and it is strictly higher
than E in the Borel reducibility hierarchy. By induction on α ∈ ω1 define Borel
equivalence relations Fα by setting F1 = id on the Cantor space, Fα+1 = Fα+
and Fα is the disjoint union of Fβ : β ∈ α for limit ordinals α.
Fact 4.3.2.
1. Every cbcs Borel equivalence relation is Borel reducible to
some Fα for some α ∈ ω1 .
2. For every ordinal α ∈ ω1 , Fα is Borel reducible to equality of transitive
countable sets of rank exactly ω + α.
Proof. The first item can be found for example in [27, Theorem 12.5.2]. To
interpret the second item correctly, consider the Polish space X of all binary
relations on ω with the relation of isomorphism E and consider the Borel set
Bα = {x ∈ X : x satisfies the axiom of extensionality and x has rank exactly
ω + α}. The second item then says that Fα is Borel reducible to E Bα . The
proof of the second item proceeds by induction on α, and the entirely routine
argument is left to the reader.
The equivalence F2 occupies a central position among the cbcs relations. It
is the equivalence relation of (2ω )ω that connects two sequences x, y if rng(x) =
rng(y). While it is not necessarily the case that every cbcs equivalence relation
can be simplified to one reducible to F2 on a positive set, still the canonization properties of F2 extend to the whole class of equivalences classifiable by
countable structures:
Theorem 4.3.3. Let I be a σ-ideal on a Polish space X such that the quotient
poset PI is proper. Then
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CHAPTER 4. CLASSES OF EQUIVALENCE RELATIONS
1. if ≤ F2 →I essentially countable then cbcs→I essentially countable;
2. if ≤ F2 →I smooth then cbcs→I smooth;
3. if I has total canonization for equivalences Borel reducible to F2 then it
has total canonization for equivalence relations classifiable by countable
structures.
Proof. Since essentially countable equivalence relations are Borel reducible to
F2 , the second and third item abstractly follow from the third. The proof of
the first item uses two claims of independent interest.
Claim 4.3.4. Assume that ≤ F2 →I essentially countable. Then, for every
Borel equivalence relation E on a Polish space Y , ≤B E →I essentially countable
implies ≤B E + →I essentially countable.
Proof. Let B ⊂ X be a Borel I-positive set, and let F be a Borel equivalence
relation on B reducible to E + by a Borel function f : B → Y ω . Let M be a
countable elementary submodel of a large structure and let C = {x ∈ B : x is
M -generic for the poset PI }. This is a Borel I-positive set; we will show that
F C ≤B F2 . Since equivalence relations reducible to F2 can be simplified to
essentially countable ones on a Borel I-positive set by the assumption, this will
finish the proof of the claim.
For every n ∈ ω, let Gn be the equivalence relation defined by x0 Gn x1 if
f (x0 )(n) E f (x1 )(n). The set Dn = {A ⊂ B : A is a Borel I-positive set such
that F n A is essentially countable} is dense in PI below B. Enumerate the
countable set Dn ∩ M as {Am
n : m ∈ ω}, for each n, m find a Borel reduction
m
fnm ∈ M of Gn Am
n to a countable Borel equivalence relation En on a Polish
m
m
space Yn . It is easy to arrange that S
Yn are pairwise disjoint subspaces of 2ω .
m 00
m
Note that for every number n, C ⊂ m Am
n , and the sets (fn ) (An ∩ C) are
all Borel by Fact 2.6.2(3).
Let R ⊂ C × 2ω be the set of all pairs hx, yi such that there are numbers
m
m and for all (some) z ∈ A
m, n, k such that y ∈ [(fnm )00 (Am
n ∩ C
n ∩ C)]En
m
m
such that fn (z) En y, it is the case that f (z)(n) = f (x)(k). This is a Borel
set with nonempty countable vertical sections, so it is the union of graphs of
countably many Borel functions gn : C → 2ω . Let g : C → (2ω )ω be defined
by g(x)(n) = gn (x). It will be enough to show that for all points x0 , x1 ∈ C,
x0 F x1 if and only if rng(g(x0 )) = rng(g(x1 )). This is immediate, however.
The vertical sections Rx depend only on the [rng(f (x))]E , and this proves the
left-to-right implication. For the opposite implication, note that if ¬x0 F x1
and n ∈ m is such that f (x0 )(n) ∈
/ [rng(f (x1 )]E , then for any number m such
m
m
that x0 ∈ Am
it
is
the
case
that
f
/ Rx1 .
n
n (x0 ) ∈ Rx0 while fn (x0 ) ∈
Claim 4.3.5. Given a countable collection of analytic equivalence relations on
Borel domains, every equivalence relation on a Borel I-positive set Borel reducible to their disjoint sum is reducible to one of them on an I-positive set.
4.3. EQUIVALENCES CLASSIFIABLE BY COUNTABLE STRUCTURES81
Proof. This is just a countable additivity argument.
Now, by induction on α ∈ ω1 argue that ≤B Fα →I essentially countable,
using the previous two claims on successor and limit steps respectively. If E
is an arbitrary (possibly analytic) cbcs equivalence relation on an I-positive
Borel set B, first use Theorem 4.4.1 to simplify E to Borel on a Borel I-positive
subset, use Fact 4.3.2 to argue that E then must be reducible to some Fα for
a countable ordinal α, and use the first sentence of this paragraph to complete
the argument.
As in the previous sections, we want to connect canonization properties of
cbcs equivalence relations with forcing properties of the quotient posets. For
the first theorem, the following new forcing property will be instrumental.
Definition 4.3.6. A poset P has the separation property if it forces that for
every countable set a ⊂ 2ω and every point y ∈ 2ω \ a there is a ground model
coded Borel set B ⊂ 2ω separating y from a: y ∈ B and a ∩ B = 0.
Theorem 4.3.7. Suppose that I is a σ-ideal on a Polish space X such that PI
is proper.
1. If PI has the separation property then cbcs→I smooth;
2. if I has total canonization for essentially countable Borel equivalence relations, then PI has the separation property.
Proof. For the first item, the previous theorem shows that it is enough to show
that the separation property implies that ≤B F2 →I smooth. Let B ⊂ X be a
Borel I-positive set and let E be a Borel equivalence relation on B reducible
to F2 by a Borel function f : B → (2ω )ω . By the separation property, each
element of the set rng(f˙(ẋgen )) is separated from the rest by a ground model
coded set. Let M be a countable elementary submodel of a large structure, let
C = {x ∈ B : x is M -generic for PI }, let {Dn : n ∈ ω} be an enumeration
of Borel sets in M , and let g : C → 2ω be the function defined by g(x)(n) =
1 if rng(f (x)) ∩ Dn contains exactly one element. We claim that g reduces
E C to the identity. Indeed, if x0 E x1 , then rng(f (x0 )) = rng(f (x1 ))
and g(x0 ) = g(x1 ). On the other hand, if ¬x0 E x1 then there will be a
number m such that f (x0 )(m) ∈
/ rngf (x1 ), and a number n ∈ ω such that
rng(f (x0 )) ∩ Dn = {f (x0 )(m)}; so g(x0 )(n) = 1. Now, either g(x1 )(n) = 0 and
then we are done. Or, g(x1 )(n) = 1, the set rng(f (x1 ))∩Dn contains exactly one
element necessarily distinct from f (x0 )(m). In this case, find a basic open set
O separating f (x0 )(m) from this unique element, let n0 be a number such that
Dn0 = Dn ∩ O, and conclude the proof by observing that g(x0 )(n0 ) 6= g(x1 )(n0 ).
For the second item, suppose that the assumptions hold and B ȧ ⊂ 2ω
is a countable set. We will find a stronger condition D ⊂ B and a countable
ground model list of Borel sets {Bn : n ∈ ω} such that D forces {Bn : n ∈ ω}
to separate the set ȧ in the sense that every element of ȧ belongs to at least
one Borel set on the countable list while no Borel set on the list contains more
82
CHAPTER 4. CLASSES OF EQUIVALENCE RELATIONS
than one element of ȧ. This certainly implies the separation property. Without
loss of generality we may assume that V ∩ ȧ = 0 is forced as well. (If not, use
properness to enclose the countable set ȧ ∩ V into a ground model countable set,
and separate it from the rest by a countable collection of singletons.) Thinning
out the set B we may assume that there are Borel functions {fn : n ∈ ω} on the
set B such that B ȧ = {f˙n (ẋgen ) : n ∈ ω}. Let M be a countable elementary
submodel of a large enough structure, and let C ⊂ B be the I-positive Borel
set of all M -generic points for PI in the set B.
Observe that each of the functions fn is countable-to-one on the set C:
since the poset PI adds a minimal real degree by Proposition 4.1.2, the set
Dn ⊂ PI consisting of those conditions
on which the function fn is one-toS
one is dense in PI , and C ⊂ (M ∩ Dn ). This means that the equivalence
relation E generated by the relation x0 Ex1 ↔ ∃n0 ∃n1 fn0 (x0 ) = fn1 (x1 ) has
countable equivalence classes; i t is easily verified that it is Borel as well. By
the canonization assumption, there must be then a Borel I-positive set D ⊂ C
such that E D = id. This means that for distinct points x0 , x1 ∈ D the
countable sets {fn (x0 ) : n ∈ ω} and {fn (x1 ) : n ∈ ω} are disjoint;
S in particular,
the functions fn are all injections on D. Let f¯n = (fn D) \ m∈n fm and let
Bn = rng(f¯n ). The sets Bn are ranges of Borel injections and as such they are
Borel, and for every point x ∈ D their list separates the set {fn (x) : n ∈ ω}. It
follows that D the sets in {Bn : n ∈ ω} separate the set a.
Theorem 4.3.8. Let I be a σ-ideal on a Polish space X such that the poset PI
is proper.
1. If PI adds only a finite collection of real degrees then cbcs→I essentially
countable;
2. if I has total canonization for essentially countable Borel equivalence relations then I has total canonization for equivalence relations classifiable
by countable structures;
3. if PI is nowhere c.c.c. and adds a minimal forcing extension then I has
total canonization for equivalence relations classifiable by countable structures.
The assumption of the first item is to be interpreted in the obvious way: if
G ⊂ PI is a generic filter then in the extension V [G], there is a finite set b ⊂ 2ω
such that for every point y ∈ 2ω there is z ∈ b such that V [y] = V [z].
Question 4.3.9. Can the assumption in (1) be weakened to “if PI adds only a
countable collection of real degrees”?
The main reason for asking this question are the results of Section 5.17. A
number of quotient forcings in this book are shown to add a single real degree.
The quotient forcings adding a finite set of real degrees of size greater than
one typically come from finite products (Section 5.4 or 5.16), and therefore
form quite a small class. On the other hand, the countable support iteration of
4.3. EQUIVALENCES CLASSIFIABLE BY COUNTABLE STRUCTURES83
countable length commonly adds only countable collection of real degrees, one
for each ordinal in the length of the iteration, by Corollary 5.17.14. Thus, the
positive answer to the question would yield a stronger canonization theorem for
the countable support iteration ideals.
Proof of Theorem 4.3.8. For the first item, by Theorem 4.3.3, it is enough to
show ≤B F2 →I essentially countable. Let B ⊂ X be a Borel I-positive set and
let E be a Borel equivalence relation on B reducible to F2 by a Borel function
f : B → (2ω )ω . Let ȧ be the name for the set rng(f˙(ẋgen )). It is clear from
the definitions that ȧ ∈ V [[ẋgen ]]E ; in fact, the following theorem will show that
V [ȧ] = V [[ẋgen ]]E .
The first observation: the degree assumption shows that V [[ẋgen ]]E contains
an enumeration of the setSȧ in ordertype ω. To see this, note that there is a finite
set b ⊂ ȧ such that ȧ ⊂ z∈b V [z], each model V [z] contains a wellordering of
its 2ω , and these wellorderings can be combined to give a wellordering of the set
ȧ. Since the set ȧ is countable in the PI -extension and the poset PI preserves
ω1 , this wellordering will have ordertype that is countable in V , and therefore
can be permuted into an ordering of ordertype ω in the model V [[x]]E .
Let ż ∈ (2ω )ω be a name for an enumeration of the set ȧ in the model
V [[ẋgen ]]E . Thinning out the set B if necessary, we may find a Borel function
g : B → (2ω )ω such that B ż = ġ(ẋgen ). Using Theorem 3.4.10, we may thin
out the set B further to ensure that the function g attains only countably many
values at each E-class. Use Fact 2.6.2 to thin out the set B further to make
sure that rng(f (x)) = rng(g(x)) for all x ∈ B, and that g 00 B is a Borel set. The
definitions immediately imply that g reduces E to F2 g 00 B and that F2 g 00 B
has countable classes. This concludes the proof of the first item.
The second item follows immediately from the first one; note that the total canonization for essentially countable equivalence relations contains total
canonization for smooth equivalence relations in its statement, which implies
that PI adds a single real degree by Proposition 4.1.2. The third item is the
conjunction of the second and Corollary 4.2.6.
Corollary 4.3.10. The following σ-ideals have total canonization for equivalence relations classifiable by countable structures:
1. the Laver ideal on ω ω ;
2. the σ-ideal generated by sets of finite Hausdorff measure of a fixed dimension, on every compact metric space X;
3. the σ-ideal generated by sets of continuity for a fixed Borel function on a
Polish space X;
4. the σ-ideal of sets of zero capacity for a fixed subadditive stable Ramsey
capacity.
84
CHAPTER 4. CLASSES OF EQUIVALENCE RELATIONS
Proof. We must verify that the quotient posets are proper and generate a minimal forcing extension. This follows from work done previously and independently elsewhere.
The first case, the Laver ideal, is separate from the others in that the associated σ-ideal is not Π11 on Σ11 [67, Proposition 3.8.15]. However, the quotient
poset is isomorphic to Laver forcing of Section 5.13 and the Laver forcing generates a minimal forcing extension by a theorem of Groszek [16].
All the named σ-ideals have been studied in the corresponding sections of
[67, Chapter 4]. They are game ideals, the quotient forcings are also < ω1 proper, and with the exception of the nondominating ideal they are Π11 on Σ11 .
Therefore, by Theorem 3.2.3 and Theorem 4.3.8, it is only necessary to verify
that they generate no intermediate c.c.c. extensions; Theorem 2.6.9 shows that
it is just enough to rule out definable c.c.c. reals in the extension. This must
be verified for each case separately.
In the case of the Hausdorff measure ideals and Ramsey capacity ideals, the
quotient poset PI does not add any independent reals by [67, Theorem 4.3.25]
and [67, Theorem 4.4.8]. This rules out any c.c.c. reals by Fact 2.6.6. In the
case of the σ-continuity ideal, the poset PI preserves Baire category, therefore
the only intermediate c.c.c. real that comes into question is a Cohen real by
Fact 2.6.6 again. The Cohen real is impossible though since the poset preserves
outer Lebesgue measure [67, Section 4.2.3]. Steprans showed earlier and in a
different language that the poset PI has the Laver property and therefore does
not add Cohen reals [59].
While the model V [x]E can acquire many forms in the case of equivalence
relations classifiable by countable structures, the model V [[x]]E is easy to compute.
Theorem 4.3.11. Let I be a σ-ideal on a Polish space X such that the quotient
poset PI is proper, and let E be a cbcs equivalence relation on X. Then PI forces
V [[ẋgen ]]E = V (ȧ) for some hereditarily countable set ȧ.
Proof. Let B ⊂ be a Borel I-positive set. Since E is reducible to an orbit
equivalence relation induced by a Polish action of S∞ , Theorem 4.4.1 shows
that there is a Borel I-positive subset C ⊂ B such that E C is a Borel
equivalence relation. By Fact 4.3.2, E C is Borel reducible to the equality of
countable transitive sets of some fixed countable rank α. Let Y be the space of
binary relations on ω, let F be the isomorphism relation on X, let D ⊂ Y be
the Borel set of those relations that satisfy the axiom of extensionality and are
wellfounded of rank α. Let f : C → D be a Borel reduction of the equivalence
relation E C to F D.
Let ȧ be the PI -name for the transitive isomorph of the relation f (ẋgen ),
and argue that C V [[ẋgen ]]E = V (ȧ). It is clear that the set ȧ belongs
to every model V [y] if y is some V [ẋgen ]-generic element of X E-related to
ẋgen : the model V [y] contains some element of the set C E-related to y by
analytic absoluteness between the models V [y] and V [x][y], and writing z ∈ V [y]
4.4. ORBIT EQUIVALENCE RELATIONS
85
for this element it must be the case that the transitive isomorph of f (z) is
isomorphic, and therefore equal, to the transitive isomorph of f (ẋgen ) = ȧ.
Thus, V (ȧ) ⊆ V [[ẋgen ]]E .
For the opposite inclusion, in the model V (a) consider the poset P that adds
a bijection between ω and HcV+ ∪ a with finite approximations. Let H ⊂ P be a
filter generic over V [xgen ]. The model V (a)[H] satisfies the axiom of choice; it is
equal to V [z] for some binary relation on ω that is isomorphic to the ∈ relation
on HcV+ ∪ a. By the analytic absoluteness between the models V [xgen ][H] and
V (a)[H], there is a point y ∈ V (a)[H] such that y is PI -generic over V and
the transitive isomorph of f (y) is exactly a. The forcing theorem shows that
there must be a P -name ẏ ∈ V (a) for such a point. Now let H0 , H1 ⊂ P be
mutually generic filters over V (a). A mutual genericity argument shows that
V (a) = V (a)[H0 ] ∩ V (a)[H1 ], and both of the models on the right side of this
equality contain points E-related to xgen , namely ẏ/H0 and ẏ/H1 . It follows
that V [[xgen ]]E ⊆ V (a)[H0 ] ∩ V (a)[H1 ] = V (a).
4.4
Orbit equivalence relations
Orbit equivalence relations of Polish group actions form a quite special subclass
of analytic equivalence relations. In our context, they can be simplified to Borel
equivalence relations by passing to a Borel I-positive Borel set:
Theorem 4.4.1. Suppose that X is a Polish space, I a σ-ideal on it such that
the quotient poset PI is proper, B ∈ PI an I-positive Borel set, and E an
equivalence relation on the set B reducible to an orbit equivalence relation of
a Polish action. Then there is a Borel I-positive set C ⊂ B such that E C
is Borel. If the ideal I is c.c.c. then in fact the set C can be found so that
B \ C ∈ I.
Note that the equivalence E may have been analytic non-Borel in the beginning.
Proof. Suppose that a Polish group G acts on a Polish space Y , and f : B → Y
is a Borel reduction of E to the orbit equivalence relation EG of the action. By
[4, Theorem 7.3.1], the space Y can be decomposed into a union of a collection
Aα : α ∈ ω1 of pairwise disjoint invariant Borel sets such that EG Aα is Borel
for every ordinal α; moreover, this decomposition is absolute between various
transitive models of ZFC with the same ℵ1 .
Let M be a countable elementary submodel of a large enough structure
and let C ⊂ B be the Borel I-positive subset of all M -generic points for PI
in the set B. This is a Borel I-positive set by the properness of the quotient
poset PI , and ifSI is c.c.c. then in fact B \ C ∈ I. For every point x ∈ C,
M [x] |=Sf (x) ∈ {Aα : α ∈ M ∩ ω1 }. Absoluteness considerations imply that
f 00 C ⊂ α∈ω1 ∩M Aα . The equivalence relation EG restricted on the latter set
is Borel, and so E C is Borel.
This feature is quite special to orbit equivalence relations, and elsewhere it
may not hold. Consider the following example:
86
CHAPTER 4. CLASSES OF EQUIVALENCE RELATIONS
Example 4.4.2. Let K be an analytic non-Borel ideal on ω and the equivalence
relation EK on X = (2ω )ω defined by ~xEK ~y if {n ∈ ω : ~x(n) 6= ~y (n)} ∈ K.
Consider the ideal I on X associated with the countable support product of
Sacks forcing. Then the equivalence EK is non-Borel, and since EK ≤ EK B
for every Borel I-positive set B ⊂ X, it is also the case that EK B is non-Borel.
4.5
Hypersmooth equivalence relations
Equivalence relations reducible to E1 are connected to decreasing sequences of
real degrees in the generic extension.
Theorem 4.5.1. Suppose that X is a Polish space and I is a σ-ideal such that
the forcing PI is proper. The following are equivalent:
1. E1 is in the spectrum of I;
2. some condition forces that in the generic extension there is an infinite
decreasing sequence of real degrees;
3. some condition forces that in the generic extension there is a strict infinite
decreasing sequence of real degrees.
Here, a decreasing sequence of V -degrees is a sequence hxn : n ∈ ωi of infinite
binary sequences such that whenever m ∈ n then V [xn ] ( V [xm ]. A strict decreasing sequence is a decreasing sequence such that moreover for every number
m, the sequence hxn : m ∈ ni belongs to V [xm ].
Proof. Suppose first that E1 is in the spectrum of I, as witnessed by a Borel
I-positive set B ⊂ X an equivalence relation E on B, and a Borel reduction
f : B → (2ω )ω of E to E1 . We claim that B forces that in the generic extension,
for every number n ∈ ω there is m ∈ ω such that f (xgen ) [n, ω) ∈
/ V [f (xgen ) [m, ω)]; so (2) holds.
Suppose for contradiction that some condition C ⊂ B forces the opposite
for some definite number n ∈ ω. Let M be a countable elementary submodel
of a large enough structure containing C and f , and let D ⊂ C be the Borel
I-positive set of all M -generic points in the set C. Consider the Borel set
Z ⊂ (2ω )ω\n defined by z ∈ Z if and only if for every number m > n there is a
Borel function g ∈ M such that g(z [m, ω)) = z. It is not difficult to see that
E1 restricted to Z has countable classes, since there are only countably many
functions in the model M . Now, whenever x ∈ D then f (x) [n, ω) ∈ Z by the
forcing theorem, and for x, y ∈ D, f (x) E1 f (y) if and only if f (x) [n, ω) E1
f (y) [n, ω). This reduces E D to a Borel equivalence relation with countable
classes, which contradicts the assumption that E1 is embeddable into it.
Now, in the generic extension V [x] define a function h ∈ ω ω by induction:
h(0) = 0, and h(n + 1) is the least number m ∈ ω such that f (x) [h(n), ω) ∈
/
V [f (x) [m, ω)]. The function h clearly belongs to all models V [f (x) [m, ω)],
4.5. HYPERSMOOTH EQUIVALENCE RELATIONS
87
and therefore the sequence f (x) [h(n), ω) : n ∈ ω forms the required strictly
decreasing sequence of real degrees.
To show (2) implies (3), suppose that some condition forces hẋn : n ∈ ωi to
be the decreasing sequence in 2ω . The strict decreasing sequence is obtained
by a simple adjustment. Just replace hxn : n ∈ ωi by a decreasing sequence
hx̄n : n ∈ ωi such that V [x̄n ] |= x̄n+1 is the ≤n -least point below which there
is an infinite decreasing sequence of degrees and V [x̄n+1 ] 6= V [xn ]. Here, ≤n is
any well-ordering of 2ω ∩ V [x̄n ] simply definable from a wellordering of V and
x̄n ; for example, let Q be the least poset of size ≤ c in V for which x̄n is generic,
and then turn any well-order of Q-names for reals into a well-order of V [x̄n ].
To prove (3) implies (1), Fact 2.6.2 yields a Borel I-positive set B and Borel
functions fn : B → 2ω , for each n ∈ ω, representing the points on the strict
decreasing sequence. We claim that the product of these functions, g : B →
(2ω )ω defined by g(x)(n) = fn (x), has the property that E1 is Borel reducible
to the pullback E = g −1 E1 restricted to any Borel I-positive set. This will
prove the theorem. Suppose for contradiction that there is a Borel I-positive
set C ⊂ B such that E1 6≤ E B; by the dichotomy 2.2.10 it must be the case
that E C ≤ E0 via some Borel map g : C → 2ω . Move to the forcing extension
V [xgen ] for some PI -generic point xgen ∈ C.
Claim 4.5.2. For every number n ∈ ω, g(xgen ) ∈ V [fn (xgen )].
Proof. Working in V [fn (xgen )], we see that there is a point x ∈ C such that
hfm (x) : m > niE1 hfm (xgen ) : m > ni, since such an x exists in the model
V [xgen ] and analytic absoluteness holds between the two models. Note that
the sequence hfm (xgen ) : m > ni is in the model V [fn (ẋgen )]. Now all such
numbers x must be mapped by g into the same E0 -equivalence class. This class
is countable and contans g(xgen ). The claim follows.
Now in the model V [g(xgen )], there is a point x ∈ C such that g(x) = g(xgen ),
since there is such a point x in the model V [xgen ] and analytic absoluteness holds
between the two models. The sequence hfn (x) : n ∈ ωi must be E1 -equivalent to
the sequence hfn (xgen ) : n ∈ ωi; let us say that they are equal from some m on.
But then, fm (x) = fm (xgen ) ∈ V [g(xgen )] ⊂ V [fm+1 (xgen )], contradiction!
Corollary 4.5.3. If the forcing PI adds just a finite set of real degrees then E1
does not belong to the spectrum of I.
This corollary applies immediately to many σ-ideals, among others the smooth
σ-ideal, the ideal associated with the product of two Miller forcings and the
Laver ideal.
88
CHAPTER 4. CLASSES OF EQUIVALENCE RELATIONS
Chapter 5
Classes of σ-ideals
5.1
σ-ideals σ-generated by closed sets
Following the exposition of [67], the σ-ideals that should be easiest to deal with
are those σ-generated by closed sets. Regarding the canonization properties of
σ-ideals in this class, on one hand there is the σ-ideal of countable sets which has
the Silver property as an immediate corollary of Silver’s theorem; on the other
hand, there is the σ-ideal of meager sets, which has a very complicated spectrum.
Thus, the treatment of this class must make finer distinctions between various
σ-ideals. Still, there is a wealth of relevant general information available:
Theorem 5.1.1. Suppose that I is a σ-ideal on a Polish space X, σ-generated
by closed sets. Then
1. every analytic I-positive set contains a Gδ I-positive subset;
2. every Borel function on an analytic I-positive domain is continuous on a
Gδ I-positive subset;
3. the σ-ideal I is Π11 on Σ11 if and only if the collection of closed sets in I is
coanalytic in the standard Borel space F (X) if and only if I is σ-generated
by a coanalytic collection of closed sets.
4. if I is a Π11 on Σ11 then I has the selection property;
5. the poset PI is < ω1 -proper and preserves Baire category;
6. moreover, if I is Π11 on Σ11 then every intermediate extension of the PI
extension is generated by a single Cohen real.
Proof. (1) is a result of Solecki [53]. To prove (2), choose a Borel function f on
an analytic I-positive set A ⊂ X. First, use (1) to thin out the set A to a Gδ
set whose intersections with open sets are either I-positive or empty. It is not
difficult to see that every closed set C ∈ I must be relatively meager in such
89
90
CHAPTER 5. CLASSES OF σ-IDEALS
A. Then, use a classical theorem [33, Theorem 8.38] to find a dense Gδ subset
B ⊂ A on which f is continuous and argue that such B must be I-positive by
the previous sentence.
For (3), it is clear that if I is Π11 on Σ11 then the set of closed sets in I is
coanalytic. On the other hand, if I is σ-generated by a coanalytic collection
of closed sets then it is Π11 on Σ11 by [33, Theorem 35.38]. (5) is contained
in [67, Theorem 4.1.2] and (6) is in [67, Theorem 4.1.7]. That leaves (4) to
be proved. The selection property follows from Theorem 3.1.4 if the σ-ideal is
generated by a Borel collection of closed sets. However, there are important
σ-ideals generated by closed sets without such a Borel basis, and the argument
from Theorem 3.1.4 must be restated to work for them.
Let D ⊂ 2ω ×X be an analytic set all of whose vertical sections are I-positive.
We will show
Claim 5.1.2. There is a perfect set P ⊂ 2ω and a continuous function g :
P × ω ω → X such that for every z ∈ P , the range of the function gz = g(z, ·) :
ω ω → X is a subset of Dz , and the images of nonempty open sets under this
function are I-positive.
Suppose that the claim has been obtained. Choose an Fσ -universal Fσ set
F ⊂ P × X, and use the nonmeager section uniformization theorem to find a
Borel function h : P → ω ω such that for every z ∈ P , if gz−1 Fz ⊂ ω ω is meager
then h(z) ∈
/ gz−1 Fz . Let f : P → X be the function defined by f (z) = g(z, h(z)).
This function is certainly Borel, and for each z ∈ P the value f (z) belongs to
Dz . We must show that rng(f ) ∈
/ I. Indeed, S
suppose that C ∈ I is a set, and
cover S
it by the union of closed sets in I, C ⊂ n Cn . There is z ∈ Z such that
Fz = n Cn . Note that the preimages gz−1 Cn ⊂ ω ω are closed sets with empty
interior by the choice of the function g, so they
S aremeager and so is their union
gz−1 Fz . Then the point f (z) is in rng(f ) \ n Cn , showing that the set rng(f )
is I-positive.
The proof of the claim is standard. Let k : ω ω → 2ω × X be a continuous
function whose image is the analytic set D. For every z ∈ 2ω let Az = f −1 Dz ;
remove from Az all the relatively open sets with I-small k-image, obtaining a
closed set Âz = [Tz ] for some pruned tree Tz ⊂ ω <ω . Note that the f -images of
nonempty relatively open subsets of Âz are I-positive, and for every t ∈ ω <ω ,
the statement t ∈ Tz is analytic since I is Π11 on Σ11 . The map z 7→ Tz is σ(Σ11 )measurable, therefore Baire measurable, and so continuous on a dense Gδ -set.
For every z there is a map π : ω <ω → ω <ω such that π preserves extension and
length of sequences and for every t ∈ ω <ω , π 00 {u : t ⊂ u} = {v ∈ Tz : π(t) ⊂ v}.
An application of a Jankov-von Neumann uniformization theorem 2.1.6 yields
a σ(Σ11 )-measurable map z 7→ πz , which is then continuous on further dense
Gδ subset. Let P ⊂ 2ω be a perfect subset of S
this dense Gδ set, and let g :
P × ω ω → X be the map defined by g(z, y) = f ( n πz (y n)). It is immediate
to check that the function g has the desired properties.
5.1. σ-IDEALS σ-GENERATED BY CLOSED SETS
91
The classification of possible intermediate forcing extensions of the PI extension yields an immediate attractive corollary:
Corollary 5.1.3. If I is a Π11 on Σ11 σ-ideal σ-generated by closed sets, then
the following are equivalent:
1. PI does not add Cohen reals;
2. I has total canonization for equivalence relations classifiable by countable
structures;
3. I has the Silver property for Borel equivalence relations classifiable by
countable structures.
Proof. (1) implies (2) by Theorem 4.3.8 and the fact that the only possible intermediate forcing extension is a single Cohen real extension. Note that the
poset PI must be nowhere c.c.c.; if it had a c.c.c. part, then that part would be
in fact isomorphic to Cohen forcing by Fact 2.6.6(3) (2) implies (3) by Proposition 3.3.2 and the selection property proved above. Finally, (3) implies (1):
if PI did add Cohen reals, all of the equivalence relations in the spectrum of
the meager ideal would also show up in the spectrum of I, and this includes
equivalence relations such as E0 . This of course immediately contradicts the
Silver property for such equivalence relations.
The corollary suggests an obvious question:
Question 5.1.4. Let I be a Π11 on Σ11 σ-ideal σ-generated by closed sets, such
that the quotient poset PI does not add Cohen reals. Does it follow that I has
total canonization for analytic equivalence relations?
The work in this section should be viewed as an effort to give a positive
answer to this question by considering certain special classes of σ-ideals first.
We will show that calibrated σ-ideals and certain other σ-ideals generalizing the
σ-ideal generated compact subsets of the Baire space have the free set property,
and therefore do have total canonization for analytic equivalence relations. Furthermore, Corollary 5.2.12 in Section 5.2 shows that if a σ-ideal σ-generated by
closed sets does not add an infinitely equal real (a condition slightly stronger
than not adding a Cohen real) then it has total canonization and Silver property
for the equivalence relations reducible to EKσ or =J for an analytic P-ideal J.
This still leaves a good number of interesting open special cases.
Now, the spectrum of the meager ideal is extremely complicated and it is
treated as a separate case in Section 5.1c.
5.1a
Calibrated ideals
In this section, we will prove a general canonization theorem for a class of σideals widely encountered in abstract analysis:
92
CHAPTER 5. CLASSES OF σ-IDEALS
Definition 5.1.5. [30] A σ-ideal on a Polish space X is calibrated if for every
closed I-positive set C ⊂ X and countably many closed sets {Dn : n ∈ ω} in
the ideal I there is a closed set C 0 ⊂ C disjoint from all the sets {Dn : n ∈ ω}.
One wide class of calibrated σ-ideals generated by compact sets consists
of those σ-ideals with the covering property, i.e. those where every I-positive
analytic set contains a compact I-positive subset. This includes the σ-ideal
of countable sets; the σ-ideal generated by sets of finite packing measure of
some fixed dimension [67, Section 4.1.6]; the σ-ideal generated by the sets of
monotonicity of a given continuous function [66, Lemma 2.3.7]; the σ-ideal generated by H-sets [42], [69]; or the σ-ideal generated by compact sets of extended
uniqueness [28].
Another class of calibrated σ-ideals consists of those σ-ideals σ-generated by
all closed sets in a fixed larger σ-ideal with the covering property. This includes
the σ-ideal σ-generated by closed Lebesgue measure zero sets; the σ-ideal σgenerated by the compact sets on which E0 is smooth; the σ-ideal σ-generated
by closed σ-porous sets; or the σ-ideal σ-generated by closed sets of capacity 0
for a fixed subadditive capacity.
Other σ-ideals σ-generated by closed sets may satisfy calibration for more
sophisticated reasons, such as the σ-ideal σ-generated by closed sets of uniqueness [8]. Some σ-ideals may fail calibration but satisfy some weakenings of it
which still allow our argument to go through without significant changes. Such
is the case of stratified calibration and the σ-ideal σ-generated by closed sets of
finite dimension in the Hilbert cube.
A simple example of a σ-ideal which is not calibrated is the meager ideal.
A more sophisticated class of non-calibrated σ-ideals are the ones discussed in
the next section. They in fact remain non-calibrated no matter what the choice
of Polish topology on the underlying space as long as the Borel structure is
preserved.
All the σ-ideals discussed above have a coanalytic set of generators and
therefore are Π11 on Σ11 .
Theorem 5.1.6. Let I be a Π11 on Σ11 σ-ideal on a compact metrizable space X,
σ-generated by compact sets. If I is calibrated, then I has the free set property.
Corollary 5.1.7. For such a calibrated σ-ideal I, the total canonization for
analytic equivalence relations and the Silver property for Borel equivalence relations both hold.
Proof of Theorem 5.1.6. Let B ⊂ X be a Borel I-positive set and let D ⊂ B ×B
be an analytic set with I-small vertical sections. We must find a Borel I-positive
set C ⊂ B such that D ∩ (C × C) ⊂ id.
Use the Burgess–Hillard theorem [33, Theorem 35.43] to cover the set D
with countably many Borel sets with I-small compact vertical sections. The
functions fn : B → K(X) for n ∈ ω assigning to each point its corresponding
vertical section are Borel
S by [33, Theorem 28.8]–so we have that for every x ∈ B,
fn (x) ∈ I and Dx ⊂ n fn (x). Thinning out the set B if necessary, we may
5.1. σ-IDEALS σ-GENERATED BY CLOSED SETS
93
assume that the functions fn are all continuous–Theorem 5.1.1 (2). Further
thinning out the set B, we may assume that it is a Gδ set such that intersections
with T
open sets are either I-positive or empty by Solecki’s theorem [53]. Let
B = n On for some open sets On ⊂ X.
Construct a suitable tree representation of the set B. By tree induction
on t ∈ ω <ω , build open sets Ot ⊂ X such that s ⊂ t implies Ōt ⊂ Os , the
diameters of the sets Ot tend to zero in some fixed complete compatible metric
for the space X, Ot ⊂ O|t| , Ot ∩ B 6= 0, and for every neighborhood O with
nonempty intersection with Ot ∩ B Tthere is m ∈ ω such that Ota m ⊂ O. For
every point y ∈ ω ω , the intersection n Oyn is a singleton and its single element
g(y) ∈ X belongs to the set B. The function g : ω ω → X is easily checked to
be continuous.
Pick a dense set a ⊂ ω ω . Use the calibration assumption to find compact
sets Kt included in both Ot and the closure of B that are I-positive and disjoint
from all sets {fn (g(y)) : n ∈ ω, y ∈ a}. We will find a certain tree T ⊂ ω <ω
which splits densely often and
(*) for every splitnode t ∈ T , every x ∈ Kt and every neighborhood P of x
there is m such that ta m ∈ T and Ota m ⊂ P .
Claim 5.1.8. Whenever T satisfies (*) then g 00 [T ] is an I-positive set.
Proof. Let {Hn : n ∈ ω} be compact subsets
S of X in the σ-ideal I; we must
produce a point y ∈ ω ω such that g(y) ∈
/ n Hn . To do that, by induction
on n ∈ ω build an increasingSsequence htn : n ∈ ωi of splitnodes tn such that
Otn+1 ∩ Hn = 0; then y = n tn will be as required. Now suppose that tn
has been constructed. Since Ktn is not a subset of Hn , there must be a point
x ∈ Ktn \ Hn . Use (*) to find a number m such that Ota m ⊂ X \ H and let
tn+1 be some splitnode of T extending ta
n m. The induction can proceed.
Of course, the point of the construction of T will be that the set g 00 [T ]
consists of pairwise E-independent elements. Then, it is an I-positive analytic
set which has an I-positive Gδ -subset C by a theorem of Solecki [53]. The set
C will be as required.
Let hhti , Pi i : i ∈ ωi be an enumeration of the product ω <ω × O, where
O is some fixed countable basis of the space X, with infinite repetitions. By
induction on i ∈ ω build finite sets Ti ⊂ ω <ω and open sets Ui ⊂ ω ω so that
I1 T0 ⊂ T1 ⊂ . . . and U0 ⊃ U1 ⊃ . . . . The set Ti is closed under the
operation of taking the longest common initial segment. Moreover, Ti+1
is an end-extension of Ti , and for every t ∈ Ti+1 \ Ti it is the case that
every infinite extension of t in T belongs to Ui ;
I2 for every t ∈ Ti , for every neighborhood P with nonempty intersection
with Kt there is m ∈ ω and y ∈ Ui such that ta m ⊂ y and Ota m ⊂ P ;
I3 if (a) ti ∈ Ti and Pi has nonempty intersection with Kti then Ti+1 =
Ti ∪ {s} for some node s ∈ ω <ω such that some m ∈ ω, ta
i m ⊂ s and
Ota m ⊂ Pi . Otherwise (b) Ti+1 = Ti ;
94
CHAPTER 5. CLASSES OF σ-IDEALS
<ω
I4 if I3(a) holds, then if y,
is a sequence with s ⊂ y
Sz ∈ Ui+1 and s ∈ ω
and s 6⊂ z then g(z) ∈
/ n∈i fn (g(y)).
let S be the downward closure of
S After the induction has been performed,
T
T
.
The
first
item
shows
that
[S]
⊆
U
;
in
fact, an equality can be arranged
i i
i i
S
here. It also implies that i Ti is exactly the set of splitnodes of the tree S, and
the third item implies that S satisfies (*). Now suppose that y, z are two distinct
branches through S and n ∈ ω, and argue that g(z) ∈
/ fn (g(y)). There has to
be i ∈ ω larger than n and such that the longest initial segment of y in Ti is ti ,
I3(a) holds, the node s ∈ Ti+1 \ Ti is still an initial segment of y, and moreover
ti is not an initial segment of z. But then, I4 shows that g(z) ∈
/ fn (g(y)) as
desired!
The induction is started with T0 = {0} and U0 = [T ]. Suppose Ti , Ui have
been obtained. If I3(b) holds then let Ti+1 = Ti and Ui+1 = Ui . If I3(a) occurs,
first use I2 to find a point y ∈ Ui such that ta
i m ⊂ y for a number m which is not
used on any of the sequences in the set Ti and such that Ota m ⊂
S Pi . Adjusting
y slightly, we may find it in the dense set a. The compact set n∈i fn (g(y)) is
disjoint from all the compact sets Kt for t ∈ Ti , and therefore it has an open
neighborhood P such that its closure is still disjoint from all of them. Use the
continuity of the functions fn and g to find an initial segment
S s of y such that
every infinite extension y 0 of s belongs to the set Ui , and n∈i fn (g(y 0 )) ⊂ P .
Let Ti+1 = Ti ∪ {s} and Ui+1 = {z ∈ Ui : s ⊂ z or g(z) is not in the closure
of P }. It is not difficult to verify that the induction hypotheses continue to
hold.
The σ-ideals with the covering property form a very special subclass of the
calibrated ideals. For them, we can even prove the mutual genericity property
as well as the rectangular Ramsey property, and this is utilized in Section 5.16
dealing with their products. The total canonization of analytic equivalence
relations could be proved in this subclass simply by quoting Theorem 3.4.4
and observing that the rectangular Ramsey property prevents the nontrivial
ergodicity in Case 1 of that theorem from ever occurring. The general calibrated
σ-ideals have much more complicated behavior, and typically both the mutual
genericity and the rectangular property will fail. In one especially common case,
we can find a Borel function on pairs that on any Borel positive rectangle covers
a large set.
Theorem 5.1.9. Let µ be the usual Borel probability measure on 2ω , and let
I be the σ-ideal σ-generated by the closed sets of µ-mass 0. There is a Borel
function f : (2ω )2 → 2ω such that for every pair B, C ⊂ 2ω of analytic I-positive
sets, the range f 00 (B × C) contains a nonempty open set.
Proof. Use the concentration of measure, Fact 2.7.2, to find numbers {ni : i ∈ ω}
such that for every i ∈ ω, ni < ni+1 and, considering the Hamming cube 2ni+1 \ni
with the normalized counting measure µi and the normalized Hamming distance
di , the di − 1/8-neighborhood of any set of µi -mass > 2−i−1 has µi -mass greater
than 1/2. Extend the definition of the distance di to 2ω and finite binary
5.1. σ-IDEALS σ-GENERATED BY CLOSED SETS
95
sequences of length greater than ni+1 by setting di (s, t) = di (s [ni , ni+1 ), t [ni , ni+1 )). For points x, y ∈ 2ω , let hij : j ∈ ωi be an increasing enumeration
of the numbers i such that x is di − 1/4-close to y. Define f (x, y)(j) = 0 if
i2j+1 = i2j + 1. We claim that this function f : (2ω )2 → 2ω works.
In order to show this, the following claim will be useful:
Claim 5.1.10. Let i ∈ ω, let s ∈ 2ni be a sequence and let B ⊂ 2ω be a Borel
set of relative mass > 2−i in [s].
1. if j > i and u ∈ 2nj \ni is a binary sequence, then there is an extension
s0 ⊃ s such that s0 ∈ 2nj , B has relative mass at least 2−j inside [s0 ], and
for every i ≤ k < j, dk (s0 , u) > 1/4;
2. if t ∈ 2ni is another sequence and C ⊂ 2ω is a Borel set with relative
mass > 2−i in [t], j > i and z ∈ 2j\i is a binary sequence, then there are
extensions s0 ⊃ s and t0 ⊃ t of length nj such that for every i ≤ k < j,
dk (s0 , t0 ) < 1/4 if and only if z(k) = 0.
Proof. Both parts of the claim are obtained by induction on j − i, and it is
clearly enough to resolve the case j = i + 1. For (1), note that the set b = {v ∈
2ni+1 \ni : B has relative mass > 2−i in [sa v]} has µi -mass at least 2−i by the
Fubini theorem. The 1/4-neighborhood of b and the 1/4-neighborhood of the
flip of b–both of these sets have µi -mass > 1/2, therefore they have a nontrivial
intersection. If w is a point in the intersection, there is a point v0 ∈ b within
1/4-distance of w and a point v1 ∈ b within 1/4-distance of the flip of w. Such
points v0 , v1 must have distance at least 1/2, so at least one of them is di -farther
than 1/4 from the sequence u; say it is v0 . The sequence s0 = sa v0 ∈ 2ni+1 is
as required.
For (2), consider the sets b = {v ∈ 2ni+1 \ni : B has relative mass > 2−i−1 in
a
[s v]} and c = {v ∈ 2ni+1 \ni : C has relative mass > 2−i−1 in [sa v]}; both of the
have µi -mass at least 2−i−1 . There are sequences v0 ∈ b and v1 ∈ c of di -distance
> 1/4, proved just as in the previous paragraph. There are also sequences v0 ∈ b
and v1 ∈ c of di -distance < 1/4, since the di − 1/8-neighborhoods of b, c have
both mass greater than 1/2 and therefore intersect. The conclusion of item (2)
immediately follows.
Now let B, C ⊂ 2ω be analytic I-positive sets. Thinning out the sets B, C if
necessary, we may assume that both are Gδ T
and their intersections
with open sets
T
are either empty or I-positive; write B = n On and C = n Pn , intersections
of open sets. The closures of B and C have positive µ-mass, and by the Lebesgue
density theorem, there must be finite binary sequences s0 , t0 of equal length such
that the closures of the sets B and C have relative masses > 1/2 in the respective
open neighborhoods [s0 ], [t0 ]. Let hij : j ∈ j0 i be an increasing enumeration of
the numbers i such that t0 is di − 1/4-close to s0 . To simplify the notation
assume that j0 is an even number. Let u0 : j0 /2 → 2 be the function defined
by u(j) = 0 if i2j+1 = i2j + 1. We will show that for every z ∈ 2ω extending u
there is x ∈ B ∩ [s0 ] and y ∈ C ∩ [t0 ] such that f (x, y) = z.
By induction on k ∈ ω build finite binary sequences sk , tk so that
96
CHAPTER 5. CLASSES OF σ-IDEALS
I1 s0 ⊂ s1 ⊂ . . . and similarly on the t side. Moreover, [sk+1 ] ⊂ Ok and
[tk+1 ] ⊂ Pk ;
I2 there are numbers lk such that sk , tk are both of length lk , and the closures
of the sets B and C are sets of relative mass > 2−j0 −k in the respective
open neigborhoods [sk ], [tk ];
I3 there are numbers i2(j+k) and i2(j+k)+1 such that they are the only numbers i between lk and lk+1 such that sk+1 is di − 1/4-close to tk+1 , and
z(j0 + k) = 0 iff these numbers are successive.
S
S
In the end, we will set x = k sk and y = k tk . I1 will imply that x ∈ B and
y ∈ C, and I3 will imply that f (x, y) = z as required.
To perform the induction, suppose that sk , tk have been found. First, use the
Lebesgue density theorem to find l0 > lk and an extension s0 ⊃ sk in 2nl0 such
0
that [s0 ] ⊂ Ok and B̄ has relative mass in [s0 ] larger than 2−l . Use Claim 5.1.10
0
(1) to find an extension t0 ⊃ tk in 2nl0 such that C̄ ∩ [t0 ] has relative mass > 2−l
and for every l ≤ i < l0 it is the case that di (s0 , t0 ) > 1/4. Now switch the
roles of s, t: find a number l00 > l0 and extensions t00 ⊃ t0 and s00 ⊃ s0 in 2nl00
00
such that both sets B, C have relative mass > 2−l in [s00 ], [t00 ] respectively,
[t00 ] ⊂ Pk , and for every lk ≤ i < l00 it is the case that di (s00 , t00 ) > 1/4. Finally,
use Claim 5.1.10 (2) on s00 , t00 to find their extensions sk+1 , tk+1 that satisfy the
induction hypothesis at k + 1.
5.1b
The σ-compact ideal and generalizations
The calibration feature of a σ-ideal I is highly dependent on the choice of the
topology for the underlying Polish space X. Moreover, passing to an I-positive
Borel set B ⊂ X, it may be difficult to recognize the calibration just from
the structure of the σ-ideal below B–even though the total canonization or
Silver property certainly survive such a restriction. In this section, we will deal
with a class of σ-ideals σ-generated by closed sets that cannot be presented
as calibrated σ-ideals restricted to a condition, no matter what the choice of
the topology or the Borel positive set. The most prominent representative of
this class is the σ-ideal I on ω ω σ-generated by compact sets. Its dichotomy
properties are well known, and its canonization properties have been thoroughly
studied by Otmar Spinas.
Fact 5.1.11. (Hurewitz, [32]) Whenever A ⊂ ω ω is an analytic set, exactly one
of the following is true: either A ∈ I or A contains all branches of a superperfect
tree. The ideal I is Π11 on Σ11 .
Fact 5.1.12. [55] The ideal I has the free set property. [57] The ideal I has
total canonization for Borel graphs.
As every Π11 on Σ11 σ-ideal σ-generated by closed sets, the ideal I has the
selection property, its free set property implies total canonization, and so in
conjunction with Proposition 3.3.2 this yields
5.1. σ-IDEALS σ-GENERATED BY CLOSED SETS
97
Corollary 5.1.13. The ideal I has the Silver property. Restated, for every
Borel equivalence relation E on ω ω , either the space ω ω is covered by countably
many E-equivalence classes and countably many compact sets, or it has a closed
subset homeomorphic to ω ω consisting of E-inequivalent elements.
We will now generalize Spinas’s results to partially ordered sets of infinitely
branching trees with various measures of the size of branching. For an ideal
K on a countable set a let P (K) be the poset of all trees T ⊂ a<ω such that
every node of T extends to a splitnode of T , and every splitnode t ∈ T the set
{i ∈ a : ta i ∈ T } is not in the ideal K. Thus for example the Miller forcing is
P (Frechet ideal on ω). The computation of the ideal I(K) associated with the
forcing gives a complete information. Let X = aω with the product topology,
where a is taken with the discrete topology. For every function g : a<ω → K
consider the closed set Ag = {x ∈ aω : ∀n x(n) ∈ g(x n)} ⊂ X, and the
σ-ideal I(K) on the space X σ-generated by all these closed sets. The following
is obtained by a straightforward generalization of the proof of Fact 5.1.11.
Fact 5.1.14. Let A ⊂ X be an analytic set. Exactly one of the following
happens:
• A ∈ I(K);
• there is a tree T ∈ P such that [T ] ⊂ A.
Moreover, if the ideal K is coanalytic then the σ-ideal I(K) is Π11 on Σ11 .
Thus the map T 7→ [T ] is a dense embedding of the poset P (K) into PI(K) . It
is clear now that the poset P (K) is proper and preserves Baire category since
the corresponding ideal is generated by closed sets [67, Theorem 4.1.2]. If the
ideal K is nonprincipal then the forcing adds an unbounded real. Other forcing
properties depend very closely on the position of the ideal K in the Katětov
ordering. The poset may have the Laver property (such as with K equal to the
Frechet ideal), or it may add a Cohen real (such as in K equal to the nowhere
dense ideal on 2<ω ). Similar issues are addressed in [48, Section 3]. We first
look at a very well behaved special case, generalizing the result of [55].
Theorem 5.1.15. Suppose that K is an Fσ ideal on a countable set. Then
I(K) has the free set property.
Proof. Fix an Fσ -ideal K on ω and use Mazur’s theorem [37] to find a lower
continuous submeasure φ on P(ω) such that K = {a ⊂ ω : φ(a) < ∞}. The
argument now proceeds with a series of claims.
Claim 5.1.16. The forcing P (K) has the weak Laver property.
Here, a forcing P has the weak Laver property if for every condition p ∈ P ,
every name f˙ for a function in ω ω forced to be dominated by a ground model
function there is a condition q ≤ p, an infinite set b ⊂ ω, and sets {cn : n ∈ b} of
the respective size n such that q ∀n ∈ b̌ f˙(n) ∈ cn . This is a forcing property
appearing prominently in connection with P-point preservation [68].
98
CHAPTER 5. CLASSES OF σ-IDEALS
Proof. Let p ∈ P (K) be a condition, g ∈ ω ω a function and f˙ a P (K)-name for
a function in ω ω pointwise dominated by g. By induction on n ∈ ω construct
trees Un ∈ P (K) and their finite subsets un ⊂ Un , numbers mn and finite sets
cn ⊂ ω so that
I1 T = U0 ⊃ U1 ⊃ . . . and {t} = u0 ⊂ u1 ⊂ . . . where t is the shortest
splitnode of T ;
I2 un is an inclusion initial set of splitnodes of Un and for every node t ∈ un ,
the set of its immediate successors that have some successor in un+1 has
φ-mass at least n;
I3 Un+1 f˙(mn ) ∈ čn and |cn | ≤ mn .
The induction is not difficult to perform. Given Un , un , let mn = |un | and
for every splitnode t ∈ un and every immediate successor s of t that has no
successor in un , thin out the tree Un s to decide the value of f˙(mn ). Since this
value must be smaller than g(mn ), there are only finitely many possible values
for this decision and so, thinning out the set of immediate successors further if
necessary, the decision can be assumed to be the same for all such immediate
successors of t, yielding a number kt ∈ ω. Let cn = {kt : t ∈ un } and let Un+1
be the thinned out tree.
T Finally, find un+1 ⊂ Un+1 satisfying I2.
In the end, U = n Un is a tree in the poset P (K) forcing ∀n f˙(mn ) ∈ č(n)
as desired.
Claim 5.1.17. Whenever a ⊂ ω is a K-positive subset and T ∈ P (K) forces
ḃ ⊂ ω is an element of K̇, then there is a condition S ≤ T and a K-positive
ground model set c ⊂ a such that S č ∩ ḃ = 0.
Proof. This is in fact true for any forcing with the weak Laver property in place
of P (K). Thinning out the condition T if necessary we may find a number m ∈ ω
such that T φ(ḃ) < m̌. Find disjoint finite sets an ⊂ a with φ(an ) > 2mn,
and use the weak Laver property to find a condition S ≤ T , an infinite set
b ⊂ ω and sets Bn ⊂ P(an ) for every n ∈ b such that all sets in Bn have φ-mass
< m, |Bn |S< n, and S ḃ ∩ ǎn ∈ B̌n . Then for every number n ∈ b, the set
cn = an \ Bn has S
mass at least nm, and it is forced by S to be disjoint from
ḃ. Thus the set c = n∈b cn works as desired.
Now, suppose that D ⊂ ω ω × ω ω is a Borel set with I(K)-small vertical
sections, and T ∈ P (K) is a tree. We must find a tree S ⊂ T such that
[S]×[S]∩D ⊂ id. Thinning out the tree T if necessary we may assume that there
is a continuous function f : [T ]×ω <ω → K such that for every branch x ∈ [T ] the
section Dx is included in the I(K)-small set {y ∈ ω ω : ∀∞ n y(n) ∈ f (x)(y n)}.
Let {tn : n ∈ ω} be an enumeration of ω <ω with infinite repetitions, and by
induction construct trees Un ∈ P (K) and their finite subsets un ⊂ Un so that
I1 T = U0 ⊃ U1 ⊃ . . . and {t} = u0 ⊂ u1 ⊂ . . . where t is the shortest
splitnode of T ;
5.1. σ-IDEALS σ-GENERATED BY CLOSED SETS
99
I2 un is an inclusion initial set of splitnodes of Un ;
I3 if tn ∈ un then un+1 = un together with some set of splitnodes extending
tn . All the new splitnodes differ from all the old ones and among each other
already at their |tn |-th entry. Moreover the set {i ∈ ω : ∃t ∈ un+1 ta
n i ⊂ t}
has φ-mass at least n. If tn ∈
/ un then un+1 = un ;
I4 if tn ∈ un and t 6= tn is another node in un , then for every node s ∈ un+1 \u
and every x ∈ [Un+1 s] it is the case that f (x)(t) is disjoint from the set
{i ∈ ω : ta i ∈ Un+1 but no node of un extends ta i}.
This is not difficult to do using the previous claim. In the end,
T
S the tree S =
U
belongs
to
P
(K),
and
its
set
of
splitnodes
is
exactly
n
n
n un . We must
show that [S] × [S] ∩ D ⊂ id. To see this, suppose for contradiction that
x 6= y ∈ [S] are two branches and y ∈ Dx . This means that there is a number
m0 such that for every m > m0 , y(m) ∈ f (x)(y m). Find a number n ∈ ω
such that tn ∈ un is an initial segment of x, un+1 \ un contains still longer initial
segment of x, and the longest node t ∈ un which is an initial segment of y is of
length greater than m0 . Then I4 shows that y(|t|) ∈
/ f (x)(t), contradicting the
choice of m0 !
Corollary 5.1.18. For Fσ ideals K, the ideal I(K) has the Silver property.
The previous result cannot be generalized to much more general ideals. A
more or less canonical ideal on a countable set which is not a subset of an Fσ ideal is K = F in × F in, the ideal on ω × ω generated by vertical sections and
sets with all vertical sections finite. To simplify notation, restrict the σ-ideal
I(K) to the closed subset X ⊂ (ω × ω)ω of those functions x such that for every
n < m, the numbers in the pair x(n) are smaller than both numbers in the pair
x(m).
Theorem 5.1.19. The ideal I(K) does not have the free set property. There
is a Borel function f : X × X → 2ω such that for Borel I(K)-positive sets
B, C ⊂ X, the image f 00 (B × C) has nonempty interior.
This theorem would not be particularly exciting if the poset PI(K) added a
Cohen real. However, it is not difficult to show that it preserves outer Lebesgue
measure. The results of the following section then imply that the σ-ideal I(K)
has the Silver property for equivalences Borel reducible to EKσ , Ec0 , or for Borel
equivalences classifiable by countable structures.
Proof. Say that points x, y ∈ X are entangled at n, m if the pairs x(n), y(m)
are not comparable in the natural coordinatewise ordering of pairs of natural
numbers. Note that if x, y are entangled at n0 , m0 and at n1 , m1 , then either
n0 ∈ n1 , m0 ∈ m1 or n1 ∈ n0 , m1 ∈ m0 ; so the locations where the pair x, y is
entangled are naturally linearly ordered. Now let f (x, y)(i) = 0 if at the i-th
location n, m where x, y are entangled, x(n)(0) > y(m)(0), and let f (x, y) = 1
otherwise.
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CHAPTER 5. CLASSES OF σ-IDEALS
To see that the function f works as required in the statement of the theorem,
let B, C ⊂ X be I(K)-positive subsets, and let S, T ⊂ (ω × ω)<ω be trees such
that [S] ⊂ B, [T ] ⊂ C, and K-positively splitting nodes are cofinal in both S
and T . Let s0 , t0 be the first K-positively splitting nodes in S, T respectively, let
a be the finite set of positions at which they are entangled, and let u : |a| → 2
be the binary sequence defined by u(j) = 0 if at the j-th position (n, m) of
entanglement of s0 , t0 , it is the case that s0 (n)(0) > t0 (m)(0). We will show
that whenever v ∈ 2ω extends u then there are points x ∈ [S], y ∈ [T ] such that
f (x, y) = v.
By induction on i ∈ ω build splitnodes si , ti of S, T respectively so that
I1 s0 ⊂ s1 ⊂ . . . and t0 ⊂ t1 ⊂ . . . ;
I2 si , ti are entangled exactly at pairs in a and the pairs (|sj |, |tj |) for j ∈ i
I3 v(|a| + i) = 0 if and only if si+1 (|si |)(0) > ti+1 (|ti |)(0).
S
S
If this succeeds, the points x = i si ∈ [S] and y = i ti ∈ [T ] will have
f (x, y) = v as required. The induction itself is easy. Suppose that si , ti have
been constructed, and for simplicity assume that v(|a| + i) = 0. Use the fact
that ti is K-positively splitting node of the tree T to find a number k ∈ ω such
that there are infinitely many l such that ta
i hk, li ∈ T . Let si+1 be any splitnode
of S extending si such that si+1 (|si |)(0) > k. Let ti+1 be any splitnode of the
tree T extending ti such that ti+1 (|ti |)(0) = k and ti+1 (|ti |)(1) is greater than
all numbers appearing in the pairs in rng(si+1 ). This concludes the induction
step.
For the failure of the free set property, let D = {hx, yi : f (x, y) ends with a
tail of 1’s}. The vertical sections of D are I-small: whenever x ∈ X is a point,
let A ⊂ ω × ω be the set of all pairs hk, li such that hn, mi ∈ rng(x) and n > k
implies m > l. The set A has finite vertical sections and as such is in the ideal
K. The vertical section Dx consists of those y ∈ Y such that for all but finitely
many i ∈ ω, y(i) ∈ A, and therefore Dx ∈ I(K). No I(K)-positive Borel set
B ⊂ X can be free for D, since the set f 00 (B × B) has a nonempty interior, and
as such must contain a pair of points hx, yi such that f (x, y) end with a tail of
1’s.
It is not too difficult to show that the quotient poset PI preserves outer
Lebesgue measure, and so by the results of Section 5.2, it has total canonization at least below EKσ , =J for analytic P-ideals J, and below equivalences
classifiable by countable structures.
5.1c
The meager ideal
The spectrum of the meager ideal is quite complicated; we only point out several
fairly simple features.
Proposition 5.1.20. E0 is in the spectrum of the meager ideal.
5.1. σ-IDEALS σ-GENERATED BY CLOSED SETS
101
No other countable Borel equivalence relations belong to the spectrum as by
Theorem 4.2.7, essentially countable→I reducible to E0 .
Proof. It is enough to show that for every Borel nonmeager set B ⊂ 2ω , E0 ≤
E0 B. By the Baire category theorem,
s ∈ 2<ω such
T there is a finite sequence
ω
that B is comeager in Os , B ⊂ Os ∩ n Pn where Pn ⊂ 2 is open dense. By
induction on n ∈ ω build pairs {t0n , t1n } of distinct finite binary sequences of the
same length such that for every number n ∈ ω, writing t for the concatenation
i(0)
i(1)
i(n)
sa (t0 )a (t1 )a . . .a (tn ), we have Ot ⊂ Pn no matter what the choices of
i(0), i(1), . . . i(n). Now let f : 2ω → B be the continuous function for which
x(n)
f (x) is defined as the concatenation of s with all tn : n ∈ ω. It is not difficult
to see that f is the required reduction of E0 to E0 B.
Proposition 5.1.21. E2 is in the spectrum of the meager ideal.
Proof. Consider the E2 equivalence on 2ω . The σ-ideal J σ-generated by Borel
grainy subsets of 2ω consists of meager sets only, as shown in Claim 5.6.10.
Thus, whenever B ⊂ 2ω is a Borel non-meager set then B ∈
/ J and therefore E2
is Borel reducible to E2 B by Fact 5.6.1. The proposition follows.
Proposition 5.1.22. The spectrum of Cohen forcing is cofinal in ≤B and it
includes EKσ .
Proof. In the Section 5.7 on the infinite countable support product of Sacks
forcing, we prove that its associated ideal on (2ω )ω consists of meager sets, and
its spectrum is cofinal in ≤B . Thus, the equivalence relations exhibited in that
proof will also yield the same feature for Cohen forcing. These equivalence
relations include EKσ , among others.
To place F2 in the spectrum of Cohen forcing, we will make a couple of
preliminary remarks. Let X = (2ω )ω and let I be the meager ideal on X. It is
well known that the poset P = PI is in the forcing sense equivalent to the finite
support product of countably many Cohen forcings, adding a point ẋgen ∈ X.
The following definition and claim are well-known and we omit the proofs.
Definition 5.1.23. Let M be a transitive model of set theory and a ⊂ 2ω be
a set. We will say that a is a Cohen-symmetric set for M if no finite tuple of
distinct elements of a belongs to any meager set coded in M , and moreover,
ω
a ⊂ 2ω is dense. A finite collection {ai : i ∈ n} of subsets
S of 2 is mutually
Cohen-symmetric over M if no finite tuple of elements of i ai belongs to any
meager set coded in M , and each of the sets is dense.
Claim 5.1.24. Let M be a countable transitive model of set theory, and let
{ai : i ∈ n} be a collection of subsets of 2ω . The following are equivalent:
1. the sets in {ai : i ∈ n} are mutually Cohen symmetric over M ;
102
CHAPTER 5. CLASSES OF σ-IDEALS
2. there is a sequence hxi : i ∈ ni of points in the space X which is P n -generic
over M and for every i ∈ n, rng(xi ) = ai .
Moreover, if (1) or (2) hold, if {yi : i ∈ n} are any one-to-one enumerations
of the sets ai and B ⊂ X n is a Borel nonmeager set coded in M , the set
n
{~g ∈ S∞
: the sequence hyi ◦ gi : i ∈ ni is P n -generic over M and belongs to B}
n
is nonmeager in the product S∞
.
Corollary 5.1.25. In the poset P n , the sentences of the forcing language mentioning only rng(ẋi ) : i ∈ n and some elements of the ground model are decided
by the largest condition.
Corollary 5.1.26. If M is a transitive model of set theory and bi : i ∈ nSare
nonempty subsets of X such that every finite tuple
S of distinct elements of i bi
is P n -generic over the model M , then the sets x∈bi rng(x) : i ∈ n are mutually
Cohen-symmetric.
Theorem 5.1.27. Let E be an analytic equivalence relation on X = (2ω )ω such
that F2 ⊂ E. Either E has a comeager equivalence class, or F2 ≤B E.
Proof. There are several distinct cases.
Case 1. Suppose first P ×P ẋ0 E ẋ1 . In such a case, let M be a countable
elementary submodel of a large structure and let B = {x ∈ X : x is M -generic
for P }. This is a Borel comeager subset of X. Any two elements of B must be
E-related. To prove this, for points x0 , x1 ∈ B pick a point y ∈ X such that y
is both M [x0 ]- and M [x1 ]- generic for the poset P , use the forcing theorem to
argue that M [x0 , y] |= x0 E y and M [x1 , y] |= x1 E y, use analytic absoluteness
for wellfounded models to conclude that x0 E y E x1 , and finally argue that
x0 E x1 by transitivity of E. The first case of the conclusion of the theorem
has just been verified.
Case 2. Suppose instead P ×P ×P rng(x0 )∪rng(x1 ) E rng(x0 )∪rng(x2 ).
Again, let M be a countable elementary submodel of a large structure and let
B = {x ∈ X : x is M -generic for P } and argue that any two elements of B must
be E-related. To see this, let x0 , x1 ∈ B and pick a point y ∈ X such that y is
both M [x0 ]- and M [x1 ]- generic for the poset P . Divide a = rng(x0 ) into two
dense sets a0 , a1 , similarly for b = rng(x1 ) = b0 ∪ b1 and c = rng(y) = c0 ∪ c1 .
By the claim and the forcing theorem applied in M , we can now argue that
a = a0 ∪ a1 E a0 ∪ c0 E c0 ∪ c1 E c0 ∪ b0 E b0 ∪ b1 = b, and by transitivity a E b
as desired. The first case of the conclusion of the theorem is satisfied.
Case 3. As another case, suppose that P ×P rng(x0 ) E rng(x0 )∪rng(x1 ).
In this case again, we get an E-comeager equivalence class similarly to the
previous argument.
Case 4. Now assume that all of the previous cases failed. Let M be a
countable elementary submodel of a large structure, and let f : 2<ω × 2ω → 2ω
be a continuous function such that s ⊂ f (s, y) and no finite tuple of distinct elements of rng(f ) belongs to any meager set in M . Such a function
is not difficult to construct. Now let g : X → X be the function defined by
g(x)(π(m, k)) = f (s(m), x(k)) where π : ω × ω → ω is a fixed bijection and
5.1. σ-IDEALS σ-GENERATED BY CLOSED SETS
103
hs(m) : m ∈ ωi is a fixed enumeration of 2<ω . We claim that the Borel function
g reduces F2 to E.
To see this, it is clear that the function g is an F2 -homomorphism, and
so F2 -equivalent points are sent to E-equivalent points. We must prove that
if x0 , x1 ∈ X are not F2 -equivalent then their g-images are not E-equivalent.
There are several cases depending on the inclusion comparison of rng(x0 ) and
rng(x1 ); we will handle the case when the intersection and rng(x0 ) \ rng(x1 ),
rng(x1 ) \ rng(x0 ) are all nonempty sets. In such a case, write a0 = {f (s, y) : y ∈
rng(x0 ) ∩ rng(x1 ), s ∈ 2<ω }, a1 = {f (s, y) : y ∈ rng(x0 ) \ rng(x1 ), s ∈ 2<ω } and
a2 = {f (s, y) : y ∈ rng(x1 ) \ rng(x0 ), s ∈ 2<ω }. These three sets as well as their
union are Cohen-symmetric over M , by Claim 5.1.24 and the failure of Claim 2
we see that ¬a0 ∪ a1 E a0 ∪ a2 , in other words ¬x0 E x1 . The other cases are
dealt with similarly.
Corollary 5.1.28. Whenever {Gn : n ∈ ω} is a collection of analytic equivalences on a fixed Polish space Y T
such that F2 is not Borel reducible to any of
them, then F2 is not reducible to n Gn either.
T
This should be compared with the case of E0ω = n E0n , where E0n is the equivalence on (2ω )ω connecting two points if their entries on the first n coordinates
are E0 connected. The equivalences forming the intersection are each reducible
to E0 while E0ω is known not to be even essentially countable.
ω ω
Proof. Suppose
T for contradiction that f : (2 ) → Y is a Borel function reducing F2 to n Gn . Write Ḡn for the pullback of Gn by the function f and
observe that these are all analytic equivalence relations containing F2 as a subset, to which F2 is not reducible. Thus, each of the equivalenceTrelations Ḡn
has a comeager equivalence class Cn ⊂ (2ω )ω . The set C = n Cn is still
comeager, and so it contains non-F2 -equivalent elements
x0 , x1 . Now, the points
T
f (x0 ), f (x1 ) are Gn -related for every n, and so T
G
-related.
This contradicts
n
n
the assumption that f was a reduction of F2 to n Gn .
Proposition 5.1.29. F2 is in the spectrum of the meager ideal.
Proof. Proceed as in the fourth case of the above proof with E = F2 . Clearly,
the first three cases are impossible. Let B ⊂ X be a Borel nonmeager set, and
let M be a countable elementary submodel of a large structure containing B.
Let f : X → X be a Borel function preserving F2 such that rng(f ) consists
of enumerations of Cohen-symmetric sets for M . Let D ⊂ X × S∞ be the set
{hx, gi : f (x)◦g ∈ B}. By Claim 5.1.24, the vertical sections of this Borel set are
nonmeager. By Fact 2.1.4, the set D has a Borel uniformization g : X → S∞ .
Plainly, the function x 7→ f (x) ◦ g(x) is a Borel reduction of F2 to F2 B.
The previous treatment also yields a useful uniformization-type theorem for
Borel reductions of F2 :
Corollary 5.1.30. Let E be a Borel equivalence relation on a Polish space Y ,
and let A ⊂ (2ω )ω × Y be an analytic set such that
104
CHAPTER 5. CLASSES OF σ-IDEALS
1. for every x ∈ (2ω )ω enumerating a dense subset of 2ω , the vertical section
Ax is nonempty;
2. whenever x0 , x1 ∈ (2ω )ω enumerate the same dense subset of 2ω and
y0 , y1 ∈ Y are points such that hx0 , y0 i, hx1 , y1 i ∈ A, then y0 E y1 ;
3. whenever x0 , x1 ∈ (2ω )ω enumerate distinct dense subsets of 2ω , then their
vertical sections are not E-related.
Then, F2 ≤B E.
Proof. The set A has a Baire measurable uniformization f : (2ω )ω → Y by
Jankov-von Neumann theorem 2.1.6. The function f is defined on a comeager
set by (1), and so it is continuous on a comeager subset B ⊂ (2ω )ω . We just
proved that F2 ≤B F2 B, and the function f witnesses F2 B ≤B E.
5.2
Porosity ideals
In [67, Section4.2], the class of porosity ideals was introduced and its main
properties investigated.
Definition 5.2.1. Let X be a Polish space, and let U be a countable collection
of its Borel subsets. An abstract porosity is a function por with domain P(U ) and
range consisting of Borel subsets of X, such that a ⊂ b implies por(a) ⊂ por(b).
Given an abstract porosity por,
S a set A ⊂ X will be called porous if it is covered
by a set of the form por(a)\ a for some a ⊂ U . The associated porosity σ-ideal
is the σ-ideal on X σ-generated by porous sets.
Example 5.2.2. Every σ-ideal I σ-generated by closed sets is a porosity ideal.
JustS
let U be a basis for the underlying Polish space, and for a ⊂ U , let f (a) =
X \ a if this is a set in I, and let f (a) = 0 otherwise.
Example 5.2.3. Let f : X → Y be a Borel function and let I be the σ-ideal
σ-generated by those sets A ⊂ X on which the function is continuous. Then
I is a porosity ideal. To see this, fix countable bases BX and BY for the two
Polish topologies on X and Y , let U = {u(O, P ) : O ∈ BX , P ∈ BY } where
u(O, P ) = {x ∈ O : f (x) ∈
/ P } and let por(a) = {x ∈ X : ∀O ∈ BX x ∈
O → ∃P ∈ BY f (x) ∈ P ∧ u(O, P ∈ a}. It is not difficult to see that I is
indeed the porosity σ-ideal associated with this porosity function. The forcing
properties of the poset PI do not really depend on the choice of the function
f . Either, the whole space X belongs to the σ-ideal I, and then the poset
PI is trivial. Or, X ∈
/ I, and then the poset PI is densely isomorphic to one
obtained from a specific function, the Pawlikowski function [6], [67, Section
4.2.3]. The combinatorial form of the resulting poset was investigated earlier
and independently by Steprans [59] and Sabok [45].
5.2. POROSITY IDEALS
105
Example 5.2.4. Let X be a compact metrizable space with a complete metric
d, and consider the σ-ideal I σ-generated by metrically σ-porous sets. There is
a great number of variations in its definition [63], but in all cases it is a porosity
ideal. To describe one version, for a set A ⊂ X and a point x ∈ X define the
porosity of A at x as the limsup of numbers r(A, x, δ)/δ as δ (a positive real
number) tends to zero, where r(A, x, δ) is the supremum of all radii of open balls
that are included in the open ball centered at x of radius δ, that are disjoint
from the set A. Call a set porous if it has nonzero porosity at all of its points,
and let I be the σ-ideal σ-generated by porous sets. It is not difficult to see
that this σ-ideal fits into our framework. Just let the set U consist of all open
balls of rational radius centered at a point in a fixed countable dense subset of
X, and for a ⊂ U let por(a) be the set of all points x ∈ X where the limsup of
numbers s(a, x, δ)/δ as δ tends to zero, is greater than zero. Here, s(a, x, δ) is
the supremum of the radii of balls in a which are subsets of the ball of radius δ
around the point x.
Fact 5.2.5. Let X be a Polish space and I be a porosity σ-ideal on it. Then
1. the quotient poset PI is < ω1 -proper and preserves Baire category;
2. if the generating porosity function is coanalytic then I is Π11 on Σ11 ;
3. if the generating porosity function is Borel then I is a game ideal;
4. if the generating porosity is coanalytic then every intermediate extension
of the PI -extension is given by a single Cohen real.
Here, to evaluate the complexity of the porosity function por, consider the
countable set U with discrete topology, the space P(U ) with the product topology, identify por with the set {ha, xi ∈ P(U ) × X : x ∈ por(a)} ⊂ P(U ) × X,
and let the words “Borel”, “coanalytic” etc. above refer to the complexity of
this set. Note that in Examples 5.2.4 and 5.2.3, the porosities are Borel.
Proof. (1) is proved in [67, Theorem 4.2.]. (2) is proved in [67, Theorem 3.8.7].
For the third item, the Borel basis for I is naturally indexed by the space
P(U )ω . The scrambling tool of Definition 3.1.1 is obtained either abstractly,
by an application of Theorem 3.1.4, or by a direct construction of the integer
game, as is done in the proof of [67, Theorem 4.2.3(3)]. (4) is in essence [67,
Theorem 4.2.7], and it will not be needed here.
By Example 5.2.2, the class of porosity σ-ideals includes the meager ideal,
and therefore we cannot expect a general canonization theorem for this class.
We will prove a canonization theorem showing that the Cohen real is the main
troublemaker here.
Theorem 5.2.6. Let I be a porosity σ-ideal on a Polish space X such that the
poset PI does not add an infinitely equal real. Then I has total canonization for
equivalence relations Borel reducible to EKσ , or =J for an analytic P-ideal J,
and for equivalence relations classifiable by countable structures.
106
CHAPTER 5. CLASSES OF σ-IDEALS
Here, an infinitely equal real is a point of ω ω in the generic extension which has
infinite intersection with every point in the ground model. Certainly, a Cohen
real is an infinitely equal real. It is a longstanding open problem in forcing
theory whether a forcing adding an infinitely equal real adds also a Cohen real;
in other words, if the hypothesis of our theorem is in fact equivalent to PI
not adding a Cohen real. On a heuristic level, every iterable forcing property
that rules out adding a Cohen real also rules out adding infinitely equal real.
For example, if a poset P is bounding then it does not add an infinitely equal
real (any point x ∈ ω ω in the extension is dominated by a ground model point
y ∈ ω ω and then x ∩ y = 0), and if P preserves outer Lebesgue measure then it
does not add an infinitely equal real (if an : n ∈ ω is a fast increasing sequence
of finite subsets
of ω, each equipped with uniform probability measures
Q
Q µn , the
product n an is equipped with the Borel probability measure
µ
=
n µn . If
Q
x ∈ ω ω is a point in the extension then the set Q
B = {y ∈ n an : x ∩ y 6= 0} is
a set of µ-mass < 1 and so its complement in n an contains a ground model
element y which then has an empty intersection with x).
The most efficient path towards the proof of Theorem 5.2.6 uses a Boolean
game.
Definition 5.2.7. Let I be a σ-ideal on a Polish space X. The symmetric
category game G (or G(I) if confusion is possible) between players I and II
proceeds in the following way. In the beginning, Player II indicates an initial
I-positive Borel set Bini ⊂ X. After that, the players alternate for ω many
rounds, in round n Player I plays a collection {Bin : i ∈ ω} of countably many
I-positive Borel sets, and Player II responds with a collection {Cin : i ∈ ω} of
I-positive Borel sets such that Cin ⊂ Bin for everyTi, S
n ∈ ω. Player I wins if in
the end, the result of the play, the set D = Bini ∩ n i Cin is an I-positive set.
The symmetric category game seems to be a close relative of the Boolean games
of [67, Section 3.10]. In contradistinction to the games described there, this
game does not seem to have a classical forcing equivalent. It is nevertheless
very useful for the present task.
Theorem 5.2.8. Let I be a σ-ideal on a Polish space X.
(1) If I is Π11 on Σ11 then the symmetric category game is determined;
(2) if Player I has a winning strategy in the game then the quotient poset PI
is proper.
If Player II has no winning strategy, then
(3) PI preserves Baire category;
(4) for every uncountable regular κ and every subset of κ in the PI -extension,
either the set contains a ground model stationary subset or its complement
contains a ground model infinite set;
(5) every intermediate extension is given by a single Cohen real;
5.2. POROSITY IDEALS
107
(6) if PI adds no infinitely equal real then I has total canonization for equivalence relations Borel reducible to EKσ or to =J for an analytic P-ideal J
on ω or classifiable by countable structures.
This greatly simplifies the proofs of [67, Theorem 4.1.6 and 4.1.7]. An attentive
reader will notice how Theorem 3.4.4 connecting equivalence relations and intermediate models, and the last two items above play together. If PI adds no
infinitely equal real then it does not add a Cohen real, by the fifth item there
are no nontrivial intermediate models between the ground model and the PI extension, and then Theorem 3.4.4 severely restricts the behavior or equivalence
relations on Borel I-positive sets. The last item then (partially) confirms the
resulting suspicions.
Proof. For item (1) of the theorem, under the assumption that the σ-ideal is
Π11 on Σ11 we will unravel the symmetric category game to a Borel game. To
this end, let A ⊂ 2ω × X be an analytic set universal for analytic subsets of
X, and let D ⊂ (2ω )ω ×Tω ω be a closed set such that its projection is the
analytic set {z ∈ (2ω )ω : n Az(n) ∈
/ I}. Now define the unraveled version Gu
of the symmetric category game G: Player II begins with a point zini ∈ 2ω such
that Azini is Borel and I-positive. After that, at round n Player I indicates a
countable list {Bin : i ∈ ω} of Borel I-positive sets, Player II responds with a
list {Cin : i ∈ ω} of their Borel I-positive subsets, and Player I reacts S
with a
point zn ∈ 2ω such that the set Az(n) is Borel, I-positive and a subset of i Cin .
At some rounds of his choice, Player I in addition indicates a natural number
m(n). In the end, Player I wins if he produced infinitely many of the additional
numbers, and the point hzini , z(n) : n ∈ ω, hm(n) : n ∈ ωii belongs to D.
The game Gu has a payoff set for Player I which is Gδ in the wide tree of
all possible plays: a point in the space (2ω )ω × ω ω is continuously produced
in the play and Player I must make sure that the production process will be
complete and ends with a point in a fixed closed set. Thus, by [35], the game
Gu is determined. The game Gu is obviously harder for Player I than G, so if
Player I has a winning strategy in Gu , then he has a winning strategy in G. We
must prove that a winning strategy σu for Player II in the game Gu yields a
winning strategy σ in G for Player II.
To describe the winning strategy σ in the game G, we will let Player II play
so that his first move is Azini , where zini is the first move of the strategy σu ,
and then in each round n he will on the side produce the following objects:
• for every function v : n + 1 → n + 1 ∪ {pass} whose numerical values stay
below the identity, a play pv ;
• a point z(n) ∈ 2ω ;
• pv is a play of the game Gu of n+1 rounds played according to the strategy
σu . In his last move in this play, Player I indicates the point z(n) and the
number (or pass) v(n). In the game G, Player II plays a list {Cin : i ∈ ω}
so that its union is equal to Az(n) ;
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CHAPTER 5. CLASSES OF σ-IDEALS
• u ⊂ v → pu ⊂ pv .
This is not difficult to do. Suppose that the play has proceeded to round n.
Player I makes his move, a countable list {Bin : i ∈ ω} of Borel I-positive
sets. Enumerate all functions v as in the first item above by hvk : k ∈ Ki
and by induction on k ∈ K + 1 construct a decreasing sequence of countable
lists {Bin (k) : i ∈ ω} so that Bin (0) = Bin , and the list {Bin (k + 1) : i ∈ ω} is
the answer of strategy σ after Player I extends the play pvk n withSthe move
{Bin (k) : i ∈ ω}. In the end, let zn ∈ 2ω be a point such that Azn = i Bin (K),
n
a
n
a
a
and let pvk = pa
vk n {Bi (k) : i ∈ ω} {Bi (k + 1) : i ∈ ω} vk (n) z(n).
In the end of a play of the game G played according to this strategy,
S S Player II
has to win. Suppose for contradiction that he lost, and so Cini ∩ n i Cin ∈
/ I;
then there must be a function y ∈ ω ω such that hzini , z(n) : n ∈ ω, yi ∈ D. Let
f : ω → ω ∪ {pass} be some function whose numerical values stay below the
identity and in their sequence
form the point y. A brief review of the definitions
S
shows that the play n pf n is a counterplay against the strategy σu in the
game Gu that Player I won, contradicting the choice of the presumably winning
strategy σu .
For the properness, suppose that Player I has a winning strategy σ in the
game G, B ∈ PI is an arbitrary condition and M is a countable elementary
submodel of a large structure containing B and σ. Consider a play agains the
strategy σ in which all moves of Player II are in the model M , Bini = B and
the sets Bin belong to the n-th open dense subset of PI in M in some fixed
enumeration. The outcome of the play is a master condition for the model M .
For the preservation of Baire category, it is enough to show that the poset PI
does not add an eventually different real by ???. Let B ∈ PI be an arbitrary
condition and ẏ a name for an element of ω ω . We will find a condition C ⊂ B
and a point z ∈ ω ω such that C ẏ ∩ ž is infinite. Consider the following
startegy for Player II. In the beginning, he indicates Bini = B. Fix a bijection
π : ω → ω × ω and, at round n, after Player I chooses his list {Bin : i ∈ ω},
Player II finds the sets Cin to decide the value of ẏ(π −1 (i, n)) to be some definite
number z(π −1 (i, n)), for each i, n. The outcome of the play is a condition C ⊂ B
with the required properties.
For item (4), let B ∈ PI be an arbitrary condition, κ an uncountable regular
cardinal, and ċ a PI -name for a subset of κ̌. Assume that no condition forces a
ground model stationary subset of κ to be a subset of ċ. Consider the following
strategy for Player II in the symmetric category game. He plays Bini = B. At
round n, after Player I provided a list {Bin : i ∈ ω} of Borel I-positive sets, the
assumptions imply that for each i ∈ ω the set bi = {α ∈ κ : ∃C ⊂ Bin C α̌ ∈
/ ċ}
must contain a club–otherwise its complement is a stationary
subset
forced
by
S
Bin to be a subset of ċ. Thus, there is an ordinal αn ∈ i bi larger than the
previously chosen ordinals αk : k ∈ n and conditions Cin ⊂ Bin , each forcing
α̌n ∈
/ ċ, and the list {Cin : i ∈ ω} will be Player II’s next move. Since this
strategy is not winning for Player II, there will be a counterplay against it whose
result C ⊂ B is I-positive. It is not difficult to see that C {αn : n ∈ ω} ⊂ ċ
is an infinite ground model set as desired.
5.2. POROSITY IDEALS
109
For item (5) of the theorem, suppose that A is a complete subalgebra of the
completion of the poset PI . We will show that if the weight of A is uncountable
below every condition, then the models V [H] and V [H ∩ A] are the same where
H is any generic filter on (the completion of) PI . Let B ∈ PI be an arbitrary
condition. Consider the strategy σ in the game G for Player II with the following
description. His first move is Bini = B. At round n, after Player I chooses his
list {Bin : n ∈ ω}, consider the projections {ani : i ∈ ω} of the sets on this list
to the complete Boolean algebra A. Since the weight of A is uncountable, this
collection has a disjoint refinement {bni : n ∈ ω} by a result of Balcar [2]. Let
Cin ⊂ Bin be any set forcing bni into the generic filter, with diameter < 2−n
in some complete compatible metric for the underlying Polish space X. Note
that the sets {Cin : i ∈ ω} must be pairwise incompatible as {bni : i ∈ ω} are.
The strategy σ given by this description cannot be winning for Player II, and
so there is a counterplay against it that Player I wins, and the outcome of the
play, a Borel set C ⊂ B, is I-positive. Now, the set C forces the following: for
every n ∈ ω there is exactly one i ∈ ω such that Cin ∈ Ḣ, and this is exactly
the only i ∈ ω such that bni ∈ Ḣ. Therefore, the collection {Cin : i, n ∈ ω} ∩ H
is in the model V [H ∩ A]. In the model V [H], its intersection is nonempty and
contains only the generic point xgen . By the analytic absoluteness between the
models V [H] and V [H ∩ A], the intersection must be nonempty in V [H ∩ A] as
well, and so xgen ∈ V [H ∩ A] as required.
For item (6) of the theorem, we will first treat the case of equivalence relations Borel reducible to EKσ . Recall the metric d on ω ω defined by d(x, y) =
supn |x(n) − y(n)|; this is a metric that can attain infinite values, and EKσ connects points with finite distance. The metric d will be naturally extended to
finite sequences of natural numbers. Let B be a Borel I-positive set and let E be
an equivalence relation on the set B, with a Borel reduction f : B → dom(EKσ )
of E to EKσ . The following claim is central.
Claim 5.2.9. Suppose that the equivalence classes of E are in the σ-ideal I.
Whenever n ∈ ω and {Bj : j ∈ ω} is a countable collection of Borel I-positive
subsets of B, then there are Borel I-positive sets {Cj : j ∈ ω} such that
1. Cj ⊂ Bj ;
2. if i 6= j are distinct natural numbers and x ∈ Ci , y ∈ Cj are points, then
d(f (x), f (y)) > n.
Once the claim is proved, the canonization for equivalence relations reducible
to EKσ follows immediately. Suppose that E has I-small equivalence classes;
we must produce a Borel I-positive selector. We will describe a strategy for
Player II: he starts with the set B = Bini and at round n answers with sets Cjn
of diameter < 2−n such that for i 6= j and x ∈ Cjn , y ∈ Cin it is the case that
d(f (x), f (y)) > n. This is possible by Claim 5.2.9. Since this is not a winning
strategy, there must be a counterplay by Player I in which the result C ⊂ B
is I-positive. This is the desired Borel I-positive set such that E C = id: if
110
CHAPTER 5. CLASSES OF σ-IDEALS
x 6= y are distinct points in the set C, it must be the case that d(f (x), f (y)) > n
for every number n and therefore ¬x E y.
To prove the claim, it is enough to show a weaker version in which i in the
second item is equal to 0 and then apply the claim repeatedly. For the weaker
version, first use the assumption that the equivalence relation E has small classes
to find, for every j > 0, countably many finite sequences {tkj : k ∈ ω} of natural
numbers such that for k 6= l, d(tkj , tlj ) > 2n and the set {x ∈ Bj : tkj ⊂ f (x)}
is I-positive. Now, if x ∈ B0 is a PI -generic point, for every j > 0 there can
be at most one k ∈ ω such that d(f (x), tkj ) ≥ n; let ġ be a name for a function
from ω \ {0} to ω that for each j identifies such k if it exists. Since the poset PI
is assumed not to add an infinitely equal real, there must be a Borel I-positive
set C0 ⊂ B and a ground model function h ∈ ω ω such that C0 ȟ ∩ ġ = 0;
thinning out the set C0 if necessary we may assume that for every j > 0 and
h(j)
every x ∈ C0 it is the case that d(f (x), tj ) > n. The weaker version of the
claim will be satisfied if we set Cj = {x ∈ Bj : tkj ⊂ f (x)}. This concludes the
argument in the case of equivalence relations Borel reducible to EKσ .
To prove the theorem for equivalence relations Borel reducible to =J for
an analytic P-ideal J on ω, first use a theorem of Solecki [54] to find a lower
semicontinuous submeasure φ on ω such that J = {a ⊂ ω : limk φ(a \ k) = 0}.
Let B be a Borel I-positive set, let E be an equivalence relation on B, and let
f : B → P(ω) be a Borel function reducing E to =J . There are two distinct
cases.
Case 1. For every ε > 0 and every Borel I-positive set B 0 ⊂ B there is a
Borel I-positive set C 0 ⊂ B 0 such that for every x, y ∈ C 0 , φ(f (x)∆f (y)) < ε.
In this case, we will find a Borel I-positive set C ⊂ B and an alternative Polish
topology τ on C such that E C is a closed relation in the topology τ × τ . By
???, the relation E D is smooth, and then the work below EKσ above will
conclude the canonization in this case.
To find the set C, consider the following strategy of Player II. He starts with
B = Bini , and at round n he plays sets Cjn so that for every x, y ∈ Cjn it is the
case that φ(f (x)∆f (y)) < 2−n . Since this is not a winning strategy, there must
be a counterplay whose result C ⊂ B is an I-positive set. Consider a Polish
topology τ on D which generates the same Borel structure as the original one,
and which makes all the sets Cjn ∩ C open. We claim that E C is a closed
relation in the topology τ × τ . Suppose that x, y ∈ C are E-unrelated points,
so there is n ∈ ω such that limk φ((f (x)∆f (y)) \ k) > 2−n . Let i, j be numbers
such that x ∈ Cin+2 , y ∈ Cjn+2 . The choice of the sets Cin+2 , Cjn+2 together
with the subadditivity of the submeasure φ shows that E ∩ Cin+2 × Cjn+2 = 0.
Since the sets Cin+2 ∩ C, Cjn+2 ∩ C are open in the new topology, we produced
an open neghborhood of the point hx, yi ∈
/ E which is disjoint from E, showing
that the complement of E D is open, and E D is closed in the new topology
as required.
Case 2. If Case 1 fails for some B 0 ⊂ B and ε > 0, proceed just like in
the argument for equivalence relations reducible to EKσ below the set B 0 . The
details are quite obvious and left to the reader.
5.2. POROSITY IDEALS
111
This concludes the argument for the equivalence relations Borel reducible to
=J for an analytic P-ideal J. The total canonization for equivalence relations
classifiable by countable structures, use Theorem 4.3.8. Note that total canonization for essentially countable equivalence relations follows from the EKσ
case, as essentially countable equivalence relations are Borel reducible to EKσ .
The previous theorem does allow for a uniform treatment of many forcing
notions, but in some sense it asks more questions than it answers.
Question 5.2.10. Is it true that if Player I has a winning strategy in the
symmetric category game then the σ-ideal I has the selection property? If
Player I has a winning strategy and PI adds no infinitely equal real, does it
follow that I has total canonization for analytic equivalence relations?
Proof of Theorem 5.2.6. In view of the previous theorem, it is enough to show
that if I is a a porosity σ-ideal then Player I has a winning strategy in the
symmetric category game G(I). Let U be a countable collection of Borel sets
and por the porosity function that define the σ-ideal I. Let Player I play so
that at round n+1, the collection {Bin+1 : i ∈ ω} includes some I-positive Borel
subset of each set of the form Cjn ∩ u where j ∈ ω, u ∈ U , and Cjn ∩ u S
∈
/ I,
and some I-positive Borel subset of each set of the form Cjn \ (por(a) ∪ a)
where a = {u ∈ U : Cjn ∩ u ∈S I}. Note that the sets mentioned last are
indeed I-positive: the set Cjn \ a is I-positive since it is obtained fromSCjn
n
by subtracting countably many I-small sets,
S and the set Cj \ (por(a)
S ∪ a)
n
is obtained from the I-positive set Cj \ a the generator por(a) \ a of the
σ-ideal I. As the play progresses, Player I will also create an increasing chain of
countable elementary submodels Mn of some large structure and he will make
sure that the sets Cjn are all elements of the model Mn and the sets Bjn belong
to the first n many open dense sets of PI in the models Mm : m ∈ n under some
fixed enumeration of the models Mm : m ∈ n.
To show that the previous paragraph indeed T
constitutes
a description of
S
a winning strategy for Player I, let D = Bini ∩ n i Cni be the result of a
play according to this strategy; we must show that D ∈
/ I. To this end, let
{an S
: n ∈ ω} be aScountable collection of subsets of U and find a point x ∈
D ∩ n (por(an ) \ an ). To find x, by induction on n find numbers jn ∈ ω so
that
I1 Bj00 ⊃ Bj11 ⊃ . . . ;
I2 Bjnn ∩ (por(an ) \
S
an ) = 0.
This is not difficult to do. At step n, let b = {u ∈ U : Cjnn ∩ u ∈ I}. If
an 6⊂ b, then find u ∈ an \ b, and choose jn+1 such that Bjn+1
⊂ Cjnn ∩ u–this is
n+1
possible by the description of the strategy–, otherwise find jn+1 so that Bjn+1
⊂
n+1
S
Cjnn \ (por(b) ∪ b–this is again possible by the description of the strategy. in
both cases, item I2 is satisfied, in the latter case because por(an ) ⊂ por(b)
112
CHAPTER 5. CLASSES OF σ-IDEALS
T
holds, as implied by an ⊂ b. In the end, the intersection n Bjnn is nonempty
and contains a single point which is generic over the union of the models Player I
played for the poset PI . The unique point x in that intersection has the desired
properties.
Corollary 5.2.11. If I is a porosity σ-ideal associated with a Borel porosity
function such that PI does not add an infinitely equal real, then it has the Silver
property for Borel equivalence relations in classes named in Theorem 5.2.6.
Corollary 5.2.12. If f : X → Y is a Borel function between Polish spaces
and I is the σ-ideal on X σ-generated by the sets on which f is continuous,
then I has the Silver property for Borel equivalence relations in classes named
in Theorem 5.2.6.
Theorem 5.2.13. Let I be a porosity σ-ideal associated with a Borel notion of
abstract porosity. If the quotient PI is bounding, then I has total canonization
for analytic equivalence relations, and the Silver property for Borel equivalence
relations.
Proof. [67, Theorem 5.2.6] shows that Π11 on Σ11 σ-ideals whose quotients are
proper, bounding, and preserve Baire category have the rectangular property.
Theorem 3.2.5 then shows that I has total canonization for analytic equivalence
relations. Finally, Proposition 3.3.2 yields the Silver property for I!
Corollary 5.2.14. The σ-ideal I of σ-porous sets has total canonization for
analytic equivalence relations, and the Silver property for all Borel equivalence
relations.
Proof. It is necessary to verify the bounding property for the σ-ideal I. [39]
showed that every analytic I-positive set contains a compact I-positive subset;
for the case of zero-dimensional spaces there is even a simpler determinacy
argument of Diego Rojas [43]. To get the bounding property, we must in addition
prove the continuous reading of names: every Borel function f : B → 2ω on an
I-positive Borel set can be restricted to an I-positive compact set on which it
is continuous. This was proved in [64] even though there is no statement of this
exact result in that paper. In order to extract it, consider Lemma 4.1 there and
use Theorem 4.5 to the σ-ideal of subsets of X × 2ω with σ-porous projection to
the X coordinate to find a compact subset of the graph of f with an I-positive
projection to X.
Note that the Silver property of I is a fairly short and quotable result.
However, its current proof is extremely heavy on technology of several sorts:
mathematical analysis and overspill techniques in the work of Zelený, Zajı́ček,
and Pelant, determinacy of Borel integer games, the original Silver dichotomy,
and a number of forcing arguments.
Question 5.2.15. Does the σ-ideal of σ-porous sets have the free set property?
Does it have a coding function?
5.3. HALPERN-LÄUCHLI FORCING
113
Question 5.2.16. Does the σ-continuity ideal have the rectangular Ramsey
property? How about the total canonization for analytic equivalence relations?
The free set property? Does it have a coding function?
5.3
Halpern-Läuchli forcing
The Halpern-Läuchli forcing P consists of those trees T ⊂ 2<ω such that there is
an infinite set aT ⊂ ω such that t ∈ T is a splitnode if and only if |t| ∈ aT . The
ordering is that of reverse inclusion. The terminology alludes to the HalpernLäuchli theorem, which for partitions of 2<ω provides homogeneous sets that
are exactly sets of splitnodes of trees in P . The poset P is quite similar to
the Sacks or Silver forcing notions, and the standard fusion arguments show
that it is proper and has the continuous reading of names. Consequently, the
computation of the associated ideal can be found in [67, Proposition 2.1.6]. Let
X = 2ω , and let I be the σ-ideal generated by those Borel sets A such that for
no tree T ∈ P , [T ] ⊂ A. Then I is a Π11 on Σ11 σ-ideal, every positive analytic
set contains a positive Borel subset, and the map T 7→ [T ] is a dense embedding
from P to PI .
This forcing serves as a good example refuting several natural conjectures.
It has the free set property, but not the mutual generics property. It has total
canonization, but not the Silver dichotomy. There is a natural σ-closed regular
subforcing, which then cannot be obtained as PIE from any Borel equivalence
relation E on X.
Proposition 5.3.1. I has the free set property.
Proof. We will need a sort of a reduced product of the Halpern-Läuchli forcing.
Let P ×r P be the poset of all pairs hT, U i ∈ P ×P such that aT = aU . Similarly
to the usual product, the reduced product adds points ẋlgen , ẋrgen ∈ X which are
the intersections of all trees on the left (or right, respectively) side of conditions
in the generic filter. Still, the difference between P ×r P and P × P should
be immediately apparent. A more detailed analysis will show that P forces
that the set {aT : T ∈ G} is a generic filter for the poset P(ω) modulo finite,
and the reduced product is equivalent to the two step iteration of P(ω) modulo
finite followed with the product of two copies of the remainder forcing P/(P(ω)
modulo finite).
We will need a computation of the σ-ideal associated with the product P ×r
P . Let I ×r I be the collection of all Borel subsets B ⊂ 2ω × 2ω such that there
is no pair hT, U i ∈ P ×r P such that [T ] × [U ] ⊂ B.
Claim 5.3.2. I ×r I is a σ-ideal of Borel sets, and the map hT, U i 7→ [T ] × [U ]
is a dense embedding of P ×r P into PI×r I .
In particular, if D ⊂ 2ω × 2ω is a Borel set with I-small vertical sections, it
cannot contain a reduced rectangle, therefore it is I ×r I-small, and the reduced
product forces its generic pair not to belong to the set D.
114
CHAPTER 5. CLASSES OF σ-IDEALS
S
Proof. Suppose that T, U ∈ P ×r P , and [T ] × [U ] = n Bn is a countable
union of Borel sets. We must find a pair of trees T̄ , Ū ∈ P ×r P such that
the set [T̄ ] × [Ū ] is a subset of one of the Borel sets in the union. Strengthen
the condition hT, U i if necessary to find a specific number n ∈ ω such that
hT, U i hẋlgen , ẋrgen i ∈ Ḃn . Let M be a countable elementary submodel of a
large enough structure, and let hDm : m ∈ ωi be an enumeration of all open
dense subsets of P ×r P in the model M . By induction on m ∈ ω build conditions
hTm , Um i ∈ M in the reduced product so that
I1 T0 = T , the conditions Tm ∈ M ∩ P form a decreasing sequence, and the
first m + 1 splitting levels of Tm are equal to the first m + 1 splitting levels
of Tm+1 , and the same on the U side;
I2 for every choice of nodes t ∈ Tm+1 , u ∈ Um+1 just past the m-th splitting
level, the condition hTm+1 t, Um+1 ui ∈ P ×r P is in the set Dm .
The induction is elementary, at each step making a pass through allTpairs ht, ui
as in the second item to handle them all. In the end, the pair hT̄ = m Tm , Ū =
T
n Um i ∈ P ×r P is a condition such that every pair hx, yi ∈ [T̄ ] × [Ū ] is M generic for the reduced product below the condition hT, U i. The forcing theorem
implies that M [x, y] |= hx, yi ∈ Bn , and by analytic absoluteness hx, yi ∈ Bn .
In other words, [T̄ ] × [Ū ] ⊂ Bn as desired.
An essentially identical argument yields
Claim 5.3.3. Suppose that M is a countable elementary submodel of a large
enough structure and T ∈ M ∩ P is a tree. There is a tree U ∈ P , U ⊂ T , such
that every two distinct elements of U are M -generic for the reduced product.
The proposition immediately follows. Let D ⊂ X × X be a Borel set with all
vertical sections in the ideal I; clearly, such a set belongs to the σ-ideal I ×r I.
Let T ∈ P . Let M be a countable elementary submodel of a large enough
structure containing both D and T . Let U ⊂ T be a tree in P such that every
two distinct points of [U ] are reduced product generic for M . An absoluteness
argument immediately shows that ([U ] × [U ]) ∩ D ⊂ id.
Proposition 5.3.4. I does not have the mutual generics property.
Proof. Suppose that M is a countable elementary submodel of a large structure,
and suppose for contradiction that there is a tree T ∈ P such that [T ] consists
solely of points mutually generic for P × P over M . For every point x ∈ [T ]
let gx ⊂ P ∩ M be the filter generic over M , generated by x: gx = {S ∈
P ∩ M : x ∈ [S]}. Similarly, let hx ⊂ (P(gw) modulo finite)∩M be the filter
generic over M generated by x: hx = {aS : S ∈ gx }. The key point: the set
B = {x ∈ [T ] : ∃S ∈ gx aT 6⊂ aS modulo finite} is relatively meager in [T ]. If
not, by the Baire category theorem there would be a tree S ∈ P ∩ M and a node
t ∈ T such that for comeagerly many points x ∈ [T ] passing through t, S ∈ gx
and aT \ aS contains some elements above |t|. Take two extensions t0 , t1 ∈ T of
5.4. THE E0 -IDEAL AND ITS PRODUCTS
115
the tree T that split at some level in aT \ aS and points x0 , x1 ∈ [T ] extending
them respectively such that S ∈ gx0 , gx1 . However, this means that t0 ∩ t1 is a
splitnode of S, contradicting the choice of the two nodes t0 , t1 .
Since the set B is meager, we can pick two distinct ponts x0 , x1 ∈
/ B and
observe that the set aT ⊂ ω diagonalizes both of the filters hx0 , hx1 . In particular, hx0 does not contain a set with finite intersection with some set in hx1 ,
as would be the case if the filters gx0 , gx1 were generic over M for the product
P × P!
Proposition 5.3.5. I does not have the selection property.
Proof. Find a Borel injection f : 2ω → [ω]ℵ0 whose range consists of pairwise
almost disjoint sets. Let D ⊂ 2ω × 2ω be the Borel set of all pairs hx, yi such
that y(n) = 0 whenever n ∈
/ f (x). It is fairly obvious that the vertical sections
of the set D are pairwise disjoint I-positive sets. It turns out that D is the
sought counterexample to the selection property.
Indeed, suppose that T ∈ P is a tree such that the closed set [T ] ⊂ 2ω is
covered by the sections of the set D, and visits each section in at most one
point. There must be distinct points y0 , y1 ∈ [T ] such that the set a = {n :
y0 (n) = y1 (n) = 1} is infinite. If x ∈ 2ω is such that y0 ∈ Dx , it must be the
case that a ⊂ f (x), and the same for y1 . However, since the sets in the range
of f are pairwise almost disjoint, there can be only one such point x ∈ 2ω , and
both y0 , y1 must belong to Dx , contradicting the choice of the tree T ∈ P .
Proposition 5.3.6. PI regularly embeds a nontrivial σ-closed forcing.
Proof. If G ⊂ PI is a generic filter, then H ⊂ P(ω) mod finite, H = {aT :
T ∈ G} is a generic filter as well. The function T 7→ aT is the associated
pseudoprojection from P to P(ω) mod finite.
5.4
The E0 -ideal and its products
Let E0 be the equivalence on 2ω defined by x E0 y iff x∆y is a finite set. This is
a basic example of a nonsmooth Borel equivalence relation. Let I be the σ-ideal
on the space X = 2ω generated by Borel partial E0 -selectors; this turns out
to be exactly the σ-ideal of Borel sets on which E0 is smooth. The purpose
of this section is to show that this ideal and its finite products possess a very
strong canonization property: while it is obviously not possible to canonize the
equivalence relation E0 on an I-positive set to anything simpler, this is the only
obstacle and all other analytic equivalence relations simplify to either id or ee
or E0 on a Borel I-positive set.
The ideal I and its quotient poset PI have been studied in [67]. In the
spirit of seeking connections between various fields of set theory, we will briefly
describe a creature forcing style presentation of the quotient forcing here. Define
a creature c = hc(0), c(1)i to be an ordered pair of distinct finite binary sequences
of the same length. A composition of a finite sequence of creatures hcj : j ∈ mi
is another creature, a pair of distinct binary sequences each of which is obtained
116
CHAPTER 5. CLASSES OF σ-IDEALS
by concatenation of all sequences cj (b(j)) : j ∈ m. for some function b ∈ 2m .
The partial order P consists of pairs p = htp , ~cp i where ~cp denotes an ω-sequence
of creatures and tp is a finite binary string. The order is defined by q ≤ p if there
is a decomposition of ω into finite consecutive intervals hin : n ∈ ωi such that
a a
tq = ta
cp (m) for every m ∈ i0 , and the creatures ~cq (n)
p s0 s1 . . . where sm ∈ ~
are obtained as a composition of creatures in the set {~cp (m) : in ≤ m < in+1 }.
For a condition p ∈ P the set [p] ⊂ 2ω consists of all concatenations of tp with
cp (j)(x(j)) for all j ∈ ω, as x varies over elements of 2ω .
Fact 5.4.1. [67, Section 4.7.1]
1. Every analytic subset of 2ω either is in the ideal I or contains a subset of
the form [p] for some p ∈ P , and these two options are mutually exclusive;
2. the σ-ideal I on 2ω is Π11 on Σ11 ;
3. I does not have the selection property;
4. the forcing PI is proper, bounding, preserves Baire category and outer
Lebesgue measure, and it adds no independent reals.
Thus, the quotient poset has a dense subset that is naturally isomorphic to the
forcing P described above. Note that for every condition p ∈ P there is a natural
homeomorphism f : 2ω → [p] defined by f (x) = the concatenation of tp with
the sequences cp (j)(x(j)) for all i ∈ ω. The homeomorphism f preserves the
E0 equivalence and therefore f -images and preimages of sets in I are again in
I. This proves the homogeneity of the poset PI and allows us to reduce various
jobs on an arbitrary Borel I-positive set to the same work on the whole space
2ω by pulling the context back along the natural homeomorphism.
Regarding the canonization properties of the σ-ideal I, the equivalence relation E0 clearly belongs to the spectrum of I essentially by its definition. We
will show that E0 is the only obstacle to canonization for I:
Theorem 5.4.2. Every Borel equivalence relation on an I-positive Borel set
simplifies to id or to ee or to E0 on I-positive Borel subset.
We will prove a canonization result for the E0 ideal in a much stronger
form, a form that deals with arbitrary finite products of the E0 ideal. The
computation of the product ideal is instructive and yields complete information.
Fix a number n ∈ ω and write Xn = (2ω )n .
Q
Definition 5.4.3. A set B ⊂ Xn is an E0 -box if it is of the form i∈n [pi ] where
each pi is a condition in the poset P . If this is the case, we write B(i) = [pi ].
Note that if B is an E0 -box then there is a natural homeomorphism of Xn
with B–namely the product of the natural homeomorphisms between 2ω and
B(i) for each i ∈ n. This homeomorphism preserves the E0 equivalence in each
coordinate. It will allow us to reduce work on an arbitrary E0 -box to work
on the whole space Xn by simply pulling back the context to Xn along the
homeomorphism.
5.4. THE E0 -IDEAL AND ITS PRODUCTS
117
Definition 5.4.4. The σ-ideal In on Xn is σ-generated by all analytic sets
A ⊂ X such that there is i ∈ n such that each section of A in the i-th coordinate
consists of pairwise E0 -inequivalent elements of 2ω .
Fact 5.4.5. Let n ∈ ω.
1. In is a Π11 on Σ11 σ-ideal consisting of meager sets;
2. every analytic subset of Xn either is in I or contains an E0 -box as a subset,
and these two options are mutually exclusive;
3. the quotient poset PIn is proper, bounding and preserves Baire category.
In particular, the quotient poset PIn has a dense subset naturally isomorphic
to the product of n copies of the poset P .
Proof. (1) immediately follows from the Kuratowski-Ulam theorem and the fact
that Borel E0 -independent subsets of 2ω are meager. (2) was proved in [66,
Lemma 2.3.58]. (3) follows from the fact that PIn is a product of n many copies
of the poset PI , that poset is proper, bounding, and preserves Baire category,
and the conjunction of these properties is preserved under products by [67,
Theorem 5.2.6]. A direct argument for (3) is of course possible as well.
Theorem 5.4.6. Let n ∈ ω be a number. Let Bi : i ∈ n be Borel subsetsQof 2ω
on which E0 is not smooth, and let E be a Borel equivalence relation on i Bi .
ThenQthere are
Q Borel sets Ci ⊂ Bi on which E0 is still not smooth, such that
E i Ci = i Fi where each Fi is one of id, ee, or E0 .
Proof. The strategy of the proof can be described as follows. First, we treat a
couple of special cases: the smooth equivalence relations, the equivalence relations that are a subset of the product of E0 ’s (this uses a partition theorem),
and the conjunction of these two cases can be cranked up to cover the case of
essentially countable equivalence relations. After that, a brief induction argument on the dimension of the product reduces the general case to the case of an
essentially countable equivalence relation, concluding the proof.
The following remnant of the free set property will be useful at two distinct
places in the proof:
Claim 5.4.7. Let n ∈ ω be a number.
1. Let R ⊂ (2ω )n × 2ω be a meager relation. Then there is a E0 -box C ⊂
(2ω )n such that for every x ∈ C and every y ∈ C(0), hx, yi ∈ R implies
x(0) E0 y.
2. Let B ⊂ (2ω )n be an E0 -box. Let R ⊂ B × B(0) be a relatively meager
relation. Then there is a E0 -box C ⊂ (2ω )n such that for every x ∈ C and
every y ∈ C(0), hx, yi ∈ R implies x(0) E0 y.
118
CHAPTER 5. CLASSES OF σ-IDEALS
Proof. We will start with the first item. Let Ok : k ∈ ω be a collection of open
dense subsets of (2ω )n × 2ω such that their intersection is disjoint from R. By
induction on l ∈ ω build finite binary sequences t(0, i, l) and t(1, i, l) for i ∈ n
so that
• for a fixed number l the sequences are pairwise distinct and have all the
same nonzero length;
• whenever u(i) : i ∈ n and v are finite binary sequences such that each u(i)
was obtained by concatenation of t(b(i, k), i, k) : k ∈ l for some function
b : n × l → 2 and v was obtained by concatenation of some t(c(k), 0, k) :
k ∈ l for some function c : l → 2 such that c(l − 1) 6= b(0, l − 1), then for
every x ∈ (2ω )n such that ∀i ∈ T
n u(i) ⊂ x and every y ∈ 2ω such that
v ⊂ y it is the case that hx, yi ∈ k∈l Ok .
This is easy to do. In the end, let pi ∈ PQbe the condition h0, ht(0, i, l), t(1, i, l)i :
l ∈ ωi. It is clear that the E0 -box C = i∈n [pi ] has the required properties.
The second item can be immediately reduced to the first. Let pi : i ∈ n
be the conditions in P such that [pi ] = B(i), let fi : 2ω → [pi ] be the natural
homomorphism, and apply the first item to the relation (f0 × f1 × . . . fn ×
f0 )−1 R ⊂ (2ω )n × 2ω .
Lemma 5.4.8. Let n ∈ ω be a number, let B ⊂ Xn be an In -positive Borel set,
and let E be a smooth equivalence relation on B. There is a set a ⊂ n and a
Borel I-positive set C ⊂ B such that E C = the equality of entries at the set
a.
Proof. The proof goes by induction on n ∈ ω. Let f : B → 2ω be a Borel
function reducing the equivalence relation E to the identity. Thinning out the
set B if necessary, we may assume that the function f is continuous. Inscribing
an E0 -box and pulling back along a canonical homeomorphism of Xn with the
box, we can assume that the set B is in fact equal to the whole space Xn . There
are two cases.
Case 1. There is a Borel In -positive set B 0 ⊂ B and an index i ∈ n such
that whenever x, y ∈ B 0 are two sequences whose entries are the same except
possibly at the entry i, then f (x) = f (y); without loss of generality assume that
i = n − 1. In such a case, the functional value of f on an entry from x ∈ B 0
depends only on x n − 1, and so f induces a smooth equivalence relation
E 0 on the projection of B 0 into the first n − 1 many coordinates. Inscribe an
E0 -box into B 0 and use the induction hypothesis to thin out the first n − 1
many coordinates of the box so as to canonize the equivalence relation E 0 to the
prescribed form. The equivalence relation E restricted to the resulting E0 -box
C ⊂ Xn is of the prescribed form.
Case 2. If Case 1 fails, then we will produce an E0 -box C ⊂ B such that
E C = id. By induction on l ∈ ω build finite binary sequences t(0, i, l) and
t(1, i, l) for i ∈ n so that
5.4. THE E0 -IDEAL AND ITS PRODUCTS
119
I1 for a fixed number l the sequences are pairwise distinct and have all the
same nonzero length; write d(i, l) for the set of all finite sequences u obtained by concatenation of t(b(i, k), i, k) : k ∈ l;
Q
I2 there is an injective function gl : i∈n d(i, l) → 2ml for some natural
number ml ∈ Q
ω such that for every choice of sequences u(i) ∈ d(i, l), it is
the case that i∈n [u(i)] ⊂ f −1 [gl (hu(i) : i ∈ ni].
The induction step is easy, using the case assumption and the continuity of
the function f repeatedly. In the end, let p(i) ∈ P be conditions for every i ∈ n
described by tp(i) = 0 and cp(i) (l) = ht(0, i, l), t(1, i, l)i for every l ∈ ω.QThe item
[I2] of the induction hypothesis immediately
Q implies that if x, y ∈ i∈n [p(i)]
are distinct points then f (x) 6= f (y). Thus i∈n [p(i)] is the desired E0 -box on
which E is equal to the identity.
Lemma 5.4.9. Let n ∈ ω be a number, and let E ⊂ E0n be a Borel equivalence
relation on an In -positive Borel set B ⊂ (2ω )nQ
. There is a set a ⊂ n and an
I-positive Borel set C ⊂ B such that E C = i∈n Fi , where Fi = id if i ∈ a,
and Fi = E0 if i ∈
/ a.
Proof. Most of the work necessary for the proof is performed in an abstract
and well known context of canonization on products of finite sets. We record
the finite results first. To compactify the notation, the following usage will
prevail. The number n ∈ ω denotes the dimension of the product. n sequences
will be adorned with an arrow (such as ~a) and simple set theoretic operations
and relations will be applied coordinatewise. Thus ~a ⊂ ~b means that for each
i ∈ n, ~a(i) ⊂ ~b(i) etc. If ~a is a sequence of sets, then ~x ∈ ~a means that
∀i
Q ∈ n ~x(i) ∈ ~a(i), in other words the sequence ~a is identified with the product
a(i).
i∈n ~
Claim 5.4.10. For every sequence ~k ∈ ω n there is a sequence ~l ∈ ω n such that
for every sequence ~b of finite sets of size ~l and every partition of the set [~b]2 into
two parts there is a sequence ~c ⊂ ~b of size ~k such that the set [~c]2 is a subset of
one piece of the partition.
Claim 5.4.11. For every sequence ~k ∈ ω n there is a sequence ~l ∈ ω n such that
for every sequence ~b of finite sets of size ~l and every equivalence relation E on ~b,
there is a sequence ~c ⊂ ~b of size ~k and a set a ⊂ n such that for every ~x, ~y ∈ ~c,
~x E ~y if and only if ∀i ∈ a ~x(i) = ~y (i).
Claim 5.4.12. For every sequence ~k ∈ ω n there is a sequence ~l ∈ ω n such
that for every sequence ~b of finite sets of size ~l and every equivalence relation
E on 2 × ~a there is a sequence ~c ⊂ ~b of size ~k such that either for every ~x ∈ ~c,
h0, ~xi E h1, ~xi holds (i.e. the equivalenceQrelation does not depend on the first
coordinate on 2 × ~c) or for every ~x, ~y ∈ ~c, ¬0a x E 1a ~y holds (i. e. the first
coordinate decomposes 2 × ~c into two E-unrelated blocks).
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CHAPTER 5. CLASSES OF σ-IDEALS
Proof. To establish a suitable terminology for the proof, a configuration is a pair
co = hco(0), co(1)i of sequences in 2n . If ~c is an n-sequence of pairs of natural
numbers, then co ◦~c is the pair h~x, ~y i of sequences defined by ~x(i) = the smaller
element of ~c(i) if co(0) = 0 and ~x(i) = the larger element of ~c(i) if co(0) = 1,
and similarly for ~y and co(1). A pair h~x, ~y i has configuration co if there is a
sequence ~c such that h~x, ~y i.
Choose ~l so large that the succession of the following homogenization arguments yields a sequence of size ~k. Let ~b be an n-sequence of finite sets of size
~l.
First, consider the partition p : ~b → 2 defined by p(~x) = 0 if 0a ~x E 1a ~x.
Shrinking the sequence ~b, we get a sequence which is entirely included in one
piece of the partition. If this partition piece has value 0 then we are in the
“either” clause of the claim, and we are finished. So assume that the partition
piece has value 1.
For every configuration co consider the partition pco : [~b]2 → 2 defined by
p(~x) = 0 if 0a (co ◦ ~x)(0) E 1a (co ◦ ~x)(1). Find a sequence ~c ⊂ ~b of size ~k such
that [~c]2 is homogeneous for all the 2n+1 many partitions obtained in this way.
We claim that the homogeneous value will be 1 for all the partitions. Then
clearly ~c is a sequence satisfying the “or” clause of the claim, completing the
proof.
Assume for contradiction that for some configuration co the homogenous
value is 0. Find sequences ~x, ~y ∈ ~c with this configuration such that for every
i ∈ n, if ~x(i) = ~y (i) then there are some elements of c(i) both below and above
~x(i), and if ~x(i) 6= ~y (i) then there are some elements of ~c(i) strictly between ~x(i)
and ~y (i). Find ~z ∈ ~c such that for every i ∈ n, if ~x(i) = ~y (i) then ~x(i) = ~z(i),
and if ~x(i) 6= ~y (i) then ~z(i) is strictly between ~x(i) and ~y (i). The homogeneity
implies that 0a ~x E 1a ~y , 0a ~x E 1a ~z, and 0a ~z E 1a y, since the ordered pairs
hx, yi, hx, zi and hz, yi all have the same configuration co. The transitivity of the
relation E implies that 0a ~z E 1a z. This, however, contradicts the homogeneity
assumption made in the third paragraph of the present proof.
Now we are ready to tackle the infinite situation. Let B ⊂ (2ω )n be an In positive Borel set, and let E be a Borel equivalence relation which is a subset of
<ω n
the product of E0 ’s. Let g : B → 2(ω ) be the function defined by g(~x)(~t) = 0
if ~x rew ~t E ~x. Shrinking the set B if necessary we may assume that the function
g is continuous. First, inscribing an E0 -box into B and pulling back through
the natural homeomorphism between (2ω )n and the box if necessary we may
assume that in fact the set B is the whole space (2ω )n . To cut the notational
clutter, below we identify 2ω with P(ω) via the characteristic functions.
By induction on j ∈ ω build n-sequences ~bj and ~ej of finite subsets of ω and
numbers mj , m̄j so that
I1 m0 < m̄0 < m1 < m̄1 < . . . the sets on the sequence ~bj are all subsets
of the interval (mj , m̄j ) and the sets on the sequence ~ej are all subsets of
the interval (m̄j , mj+1 );
5.4. THE E0 -IDEAL AND ITS PRODUCTS
121
I2 for any pair ~c, d~ of sequences of sets below m̄j either (a) for every ~x ∈
(P(ω)n such that each set on the sequence ~ej is an initial segment of the
corresponding set on ~x, ~c ∪ ~x E d~ ∪ ~x holds, or (b) for every such ~x,
~c ∪ ~x E d~ ∪ ~x fails. Thus the relation Fj on P(m̄j )n , connecting ~c and d~ if
(a) above occurs, is an equivalence relation;
I3 the sizes of the sets ~bj grow sufficiently fast as indicated by the needs of
the argument to follow.
For every sequence d~ of sets below mj , let Ed,j
~ be the equivalence relation on
Q~
bj defined by ~u Ed ~v iff d~ ∪ {~u} Fj d~ ∪ {~v }. By a repetitive use of the previous
two claims at every j ∈ ω, shrink the sets on the sequences ~bj if necessary so
that
Q~
I4 for every sequence d~ below mj there is a set ad,j
bi
~ ⊂ n such that Ed,j
~ is just the equality at the set ad~;
I5 for sequences ~c, d~ below mj , either (a) for every ~u ∈ bj ~c ∪ {~u} Fj d~ ∪ {~v }
holds, or (b) for every ~u, ~v ∈ bj ~c ∪ {~u} Fj d~ ∪ {~v } fails.
Q
For every j ∈ ω, let µj be the submeasure on the set ~[bj ]2Q
defined by µj (h) = 0
if h = 0, µj (h) = 1 if h 6= 0 contains no subset of the form [~cj ]2 for a sequence
of sets ~cj ⊂ ~bj each of size at least three, and µj (h) ≤ r if h can be decomposed
into r many sets of µj -mass The sequences ~bj must be chosen growing so fast
that
Q
I6 the numbers µi ( [~bj ]2 ) tend to infinity fast enough so that the partition
theorem 2.7.6 may be applied for partitions into 2n many pieces.
Q
This ends the inductive construction. Consider the partition p : j [~bj ]2 ×
ω → P(n) defined by p(k, hd~j : j ∈ ωi) = a ~ where d~ is the sequence of sets
d,k
defined as {max(d~0 )} ∪ ~e0 ∪ {max(d~1 (} ∪ · · · ∪ ~ej−1 . The partition theorem 2.7.6
yields sequences ~cj : j ∈ ω such that ~cj ⊂ ~bj is a sequence of sets of size at least
three, an infinite set w ⊂ ω and a set a ⊂ n such that whenever d~j : j ∈ ω are
sequences of pairs with d~j ∈ [~cj ]2 , and k ∈ w, then p(k, hd~j : j ∈ ωi) = a.
Finally, we are in a position to construct the sets Ci : i ∈ n such that each
Ci ⊂ P(ω) is aQ
closed set on which the equivalence relation E0 is nonsmooth, and
such that E i Ci connects pairs ~x and ~y just in case that ∀i ∈ a ~x(i) = ~y (i)
and ∀i ∈
/ a ~x(i) E0 ~y (i). Let Ci consist of all sets of the form {k0 } ∪ e0 ∪ {k1 } ∪
e1 ∪ . . . where kj is the largest element of the set ~cj (i) if j ∈
/ w, and kj is one of
the two largest elements of the set ~cj (i) if j ∈ w. We must verify the advertised
properties of the sets Bi : i ∈ n.
First of all, for each i ∈ n the equivalence relation E0 is not smooth on the
set Ci . To see this, consider the map h : 2ω → B defined by h(z) = the unique
element of the set B such that for every l ∈ ω, if j is l-th element of the set
122
CHAPTER 5. CLASSES OF σ-IDEALS
w ⊂ ω, h(z) contains the largest element of the set ~cj (i) if and only if x(l) = 1.
This map reduces E0 to E0 Ci , and therefore E0 Ci is not smooth.
Q To verify the canonical form of the equivalence relation E of the set C =
x ∈ C. The homogeneity properi∈n Ci , first choose an arbitrary n-sequence ~
ties of the collection h~cj : j ∈ ωi show that for every j ∈ w it is the case that
a~x∩mj ,j = a. This observation together with a simple induction on j ∈ w shows
that if sequences ~x, ~y ∈ C have the same entries on all i ∈ a, then for all j ∈ ω
~x ∩ m̄j Fj ~y ∩ m̄j . If, on the other hand, the sequences ~x, ~y differ at some entry
i ∈ a, then find the least j ∈ w such that the sequences ~x ∩ m̄j and ~y ∩ m̄j differ
at some entry i ∈ a, observe that this is also the least j ∈ w such that ~x ∩ m̄j
and ~y ∩ m̄j are not Fj -related, and use I5(b) to argue by induction on all larger
numbers k ∈ w that (~x ∩ m̄k ) Fk (~y ∩ m̄k ) fails.
Q
Now assume in addition that the sequences ~x, ~y ∈ i∈n Bi satisfy ∀i ∈
n ~x(i) E0 ~y (i). If for every i ∈ a, ~x(i) = ~y (i), then choose j ∈ w so large that
all differences between the sequences ~x and ~y occur below m̄j . The previous
paragraph together with the definition of the equivalence relation Fj shows that
~x E ~y holds. If, on the other hand, there is a number i ∈ a such that ~x(i) 6= ~y (i),
then again look at the same large j ∈ w. This time, the previous paragraph
and the definition of the relation Fj shows that ~x E ~y fails. The proof is
complete.
The previous lemmas can be easily cranked up to cover the general case of
analytic equivalence relations with countable classes, and then to the essentially
countable equivalence relations.
Lemma 5.4.13. Let n ∈ ω be a number, and let E be an analytic equivalence
relation with countable classes on an In -positive Borel set B ⊂ (2ω )nQ
. There
is a set a ⊂ n and an I-positive Borel set C ⊂ B such that E C = i∈n Fi ,
where Fi = id if i ∈ a, and Fi = E0 if i ∈
/ a.
Proof. For every i ∈ n let Ri ⊂ (2ω )n × 2ω be the relation consisting of all pairs
hx, yi such that x ∈ B and there is x0 ∈ B E-related to x such that x0 (m) =
y. These are analytic relations with countable vertical sections, and therefore
relatively meager in every E0 -box. A repeated application of Claim 5.4.7 then
yields an E0 -box B 0 ⊂ B such that for every x E x0 ∈ B 0 and every i ∈ n,
hx, x0 (i)i ∈ R implies x(i) E0 x0 (i). In other words, the equivalence E is a
subset of the product of E0 ’s on B 0 . Lemma 5.4.9 then yields the final Borel
In -positive Borel set C ⊂ B 0 .
Lemma 5.4.14. Let n ∈ ω be a number, and let E be an essentially countable
equivalence relation on an In -positive Borel set B ⊂ (2ω )n . There is a Q
partition
n = a ∪ b ∪ c and an I-positive Borel set C ⊂ B such that E C = i∈n Fi ,
where Fi = id if i ∈ a, Fi = E0 if i ∈ b, and Fi = ee if i ∈ c.
Proof. Let f : B → Y be a Borel function reducing E to some Borel equivalence
relation F on a Polish space Y with countable classes. First apply Lemma 5.4.8
5.4. THE E0 -IDEAL AND ITS PRODUCTS
123
to the smooth equivalence E 0 inducedQ
by f to find a Borel In -positive set B 0 ⊂ B
0
0
and a set c ⊂ n such that E B = i∈n Fi where Fi = id if i ∈
/ c and Fi = ee
if i ∈ c. This is to say that the values f (x) depend only on x n \ c and that in
an injective way. Ergo, the equivalence relation F induces a Borel equivalence
relation with countable classes on the set B 00 = the projection of B 0 into the
coordinates outside the set c. An application of Lemma 5.4.13 concludes the
proof.
Now we are ready to complete the proof of Theorem 5.4.6. By induction
on n ∈ ω we will prove the statement “for every m ∈ ω, for every analytic
equivalence relation E on a Borel Im+n -positive set B ⊂ (2ω )m × (2ω )n , there
is a Borel Im+n -positive subset C ⊂ B such that for every z ∈ (2ω )m , E Cz
is essentially countable”. The theorem will follow from the case m = 0 and
Lemma 5.4.14 above.
Suppose that m, n are given. Use the induction hypothesis to thin out B so
that for every index i ∈ n every z ∈ (2ω )m , and every y ∈ 2ω , the equivalence
relation E {x ∈ B : x m = z ∧ x(m + i) = y} is essentially countable. Let
B = B0 and consider the relation R0 on B0 × B0 (m) defined by hx, yi ∈ R0 if
there is x0 ∈ B0 such that x0 m = x m and x0 (m) = x(m).
Case 1. If the relation R0 is not meager, then use the Kuratowski-Ulam
theorem to find y ∈ B0 (m) such that the horizontal section R0y is nonmeager in
B0 –thus, R0y ⊂ B0 is an analytic Im+n -positive set. Use uniformization 2.6.2(3)
to find a Borel Im+n -positive set C ⊂ R0y and a Borel function f : C → B0
such that for every x ∈ C, f (x) m = x m, f (x)(m) = y, and f (x) E x.
Now, for every z ∈ (2ω )m , the function f reduces E Cz to the equivalence
relation E {x ∈ B : x m = z, x(m) = y}. This latter equivalence relation is
essentially countable and in this case we are done.
Case 2. If the relation R0 is meager, then by Claim 5.4.7 there is a box
B1 ⊂ B0 such that for every x ∈ B1 and every y ∈ B1 (m), hx, yi ∈ R0 implies
x(m) E0 y. Now repeat the whole argument at the coordinate m + 1 with the
relation R1 ⊂ B1 × B1 (m + 1) defined by hx, yi ∈ R1 if there is x0 ∈ B1 such
that x0 m = x m and x0 (m + 1) = x(m + 1), and by induction on i ∈ n
repeat it at every coordinate i. If at any stage of the induction we end up in
Case 1, then we are done. Otherwise, we end up with a box C = Bn ⊂ B such
that for every i ∈ n and every x0 , x1 ∈ Bn with x0 m = x1 m and x0 E x1 it
is the case that x0 (m + i) E0 x1 (m + i). This means that for every z ∈ (2ω )m
the equivalence relation E Cz is a subset of product of E0 ’s. The classes of Ez
are already countable, but we still have to deal with the possibility that Ez is
analytic and not Borel. An application of Lemma 5.4.9 shows that we can thin
out the box C even further so that for every z ∈ (2ω )m , E Cz takes one of the
finitely many prescribed Borel forms, and they all have countable equivalence
classes. The proof is complete.
124
5.5
CHAPTER 5. CLASSES OF σ-IDEALS
The E1 ideal
E1 is the equivalence relation on the Polish space X = (2ω )ω defined by x E1 y
if for all but finitely many n ∈ ω, x(n) = y(n). This is the canonical example of
a Borel equivalence relation not Borel reducible to an orbit equivalence relation
of a Polish group action. In view of the Kechris-Louveau dichotomy 2.2.10, it
is natural to define IE1 to be the σ-ideal on X σ-generated by Borel sets B
such that E1 B is essentially countable. Clearly, E1 cannot be canonized
to anything simpler on a Borel I-positive set. We will prove a canonization
theorem saying that among the hypersmooth equivalence relations this is the
only obstacle ???.
From the forcing point of view, the σ-ideal IE1 is somewhat awkward as
the quotient poset PIE1 is not proper [66, Lemma 2.3.33], [27, ???]. At the
same time, there is a closely related σ-ideal whose quotient poset is proper and
previously independently studied. It is the iteration of Sacks forcing along the
reverse ω ordering at which a number of authors arrived from different angles
and at different times [17], [26], [?]. The associated ideal was computed in [67,
Section 5.4]. The connection between the E1 equivalence and reverse iterations
should not be surprising in view of Theorem 4.5.1. Here, we show that there is
a natural collection of σ-ideals between the E1 ideal and the reverse iteration
ideal that allow uniform treatment.
5.5a
The σ-ideals and their dichotomies
Definition 5.5.1. Let S ⊂ [ω]ℵ0 be a nonempty analytic set. The σ-ideal IS
on X = (2ω )ω is σ-generated by the sets Az where z ranges over all elements of
2ω and Az = {x ∈ X : ∀s ∈ S ∃n ∈ s x(n) ∈ ∆11 (z, (ω, n)}.
Note that the generators of the σ-ideal IS are coanalytic. This is in contradistinction to other σ-ideals considered in this book, where the generators are
typically Borel or analytic, but it will cause only very minor annoyance. It
will turn out that IE1 = I[ω]ℵ0 , while the reverse Sacks iteration ideal is equal
to I{ω} . Before we can arrive at this conclusion though, we must show that
IS -positive analytic sets come in a specific form. That is the subject of the
following definition and theorem.
Definition 5.5.2. Let s ⊂ ω be an infinite set, with an increasing enumeration
π : ω → s. An s-keeping homeomorphism is a continuous injective map f :
X → X such that for every x0 , x1 ∈ X and every i ∈ ω, (a) if x0 (i) 6= x1 (i)
then f (x0 )(π(j)) 6= f (x1 )(π(j)) for all j ≤ i and (b) if x0 (j) = x1 (j) for all
j > i then f (x0 )(n) = f (x1 )(n) for all n > π(i). An s-cube is the range of a
strict s-keeping homeomorphism. If S ⊂ P(ω) is a collection of infinite sets, an
S-cube is a s-cube for some s ∈ S. A reverse cube is an ω-cube.
Note that if f is an s-keeping homeomorphism for an infinite s ⊂ ω then f is
a one-to-one continuous reduction of E1 to E1 rng(f ) by the two demands
(a) and (b). [26] used a definition of a projection-keeping homeomorphism that
5.5. THE E1 IDEAL
125
differs in an insignificant technical detail from our definition of an ω-keeping
homeomorphism.
Theorem 5.5.3. Let S ⊂ [ω]ℵ0 be a nonempty analytic set.
1. IS is a Π11 on Σ11 σ-ideal.
2. An analytic subset of X is IS -positive if and only if it contains a strict Scube. In fact, an analytic subset A ⊂ X × ω ω has an IS -positive projection
if and only if there is an S-cube C and a continuous function g : C → ω ω
such that g ⊂ A.
3. If A ⊂ X is an analytic IS -positive set and {An : n ∈ ω} are analytic
subsets of X, then there is an S-cube C ⊂ A such that for every n ∈ ω,
An ∩ C is a Borel set.
The quotient poset PIS is typically not proper. The statements (2) and (3)
above may be viewed as the remnants of the bounding property and properness,
respectively
Proof. The difficult part is inscribing an S-cube into an IS -positive analytic
set. After that, a couple of routine applications of Kuratowski-Ulam theorem
complete the proof.
So let S ⊂ [ω]ℵ0 be an analytic set and A ⊂ X × ω ω be an analytic set.
In order to promote the similarity with the canonization arguments in the next
section, we will divide the treatment into two cases, even though here one of
the cases is nearly trivial.
Case 1. There is a parameter z ∈ 2ω , a set s ∈ S and a point hx, yi ∈ A
such that S, A are both Σ11 (z) and for every n ∈ s, x(n) ∈
/ ∆11 (x (n, ω), z).
In this case, we will inscribe a strict S-cube C into p(A) and find a continuous
function g : C → ω ω such that g ⊂ A. The argument uses a version of GandyHarrington forcing. To simplify notation, assume that the parameter z is in fact
0; otherwise just insert the parameter in suitable places below.
First, a couple of simple difinitions regarding sets of finite sequences. If
~
k(i)
~k ∈ ω <ω write 2~k = Q
. The expression ~k ≤ ~l means that dom(~k) ⊆
i∈dom(~
k) 2
~
~
~
dom(~l) and ∀i ∈ dom(~k) ~k(i) ≤ ~l(i). If ~k ≤ ~l and σ ∈ 2l then σ 2k ∈ 2k is
the sequence hσ(i) ~k(i) : i ∈ dom(~k)i. Finally, define the poset Q to consist of
tuples q = h~k = kq , F = Fq , T = Tq i such that ~k ∈ ω <ω , T ⊂ S is a nonempty
~
Σ11 set, F is a function with dom(F ) = 2k and rng(F ) consisting of nonempty
Σ11 subsets of A and there is a witness to q ∈ Q. Here, a witness is a tuple
~
hs, hxσ , yσ i : σ ∈ 2k i such that
• s ∈ Tq with an increasing enumeration π : ω → s, hxσ , yσ i ∈ F (σ) and
~
whenever i ∈ dom(~k) and σ ∈ 2k then xσ (π(i)) ∈
/ ∆11 (xσ (π(i), ω));
~
• moreover, if σ, τ ∈ 2k are sequences and i ∈ dom(~k) is a number then (a)
if σ(i) 6= τ (i) then xσ (π(j)) 6= xτ (π(j)) for all j ≤ i and (b) if σ(j) = τ (j)
for all j > i then xσ (n) = xτ (n) for all n > π(i).
126
CHAPTER 5. CLASSES OF σ-IDEALS
~
The stem of the witness hs, hxσ , yσ i : σ ∈ 2k i is the common value of xσ ~
(π(max dom(~k)), ω) for σ ∈ 2k . The ordering on Q is defined by r ≤ q if
~kq ≤ ~kr , for every σ ∈ 2~kr Fr (σ) ⊂ Fq (σ 2~kq ), and finally Tr ⊂ Tq . Note that
Q is a countable notion of forcing.
Let M be a countable elementary submodel of a large structure containing
all the above objects, and let G ⊂ Q be a generic
T filter over the model M . There
will be a unique point sgen in the intersection q∈G Tq , and for each x ∈ X there
T
will be a unique point hf (x), h(x)i in the intersection q∈G Fq (x ~kq ). We will
show that f is a sgen -keeping homeomorphism, its range will be an s-cube C,
and the map g : C → ω ω defined by g = h ◦ f −1 is the desired continuous
uniformization of the set A on C. For all of this, several density arguments will
be necessary; they will be supported by the following series of claims.
Claim 5.5.4. Whenever q ∈ Q and D is a dense subset of the Gandy-Harrington
~
forcing on X × ω ω then there is r ≤ q such that for every σ ∈ 2kr , Fr (σ) ∈ D.
This is to assure that for every x ∈ X, the set {Fq (x ~kq ) : q ∈ G} is a GandyHarrington-generic filter over M , and therefore its intersection will contain a
unique point hf (x), h(x)i.
~
Proof. Enumerate 2kq as {σi : i ∈ n} and by induction on i ≤ n form a decreasing chain of conditions q = q0 ≥ q1 ≥ q2 ≥ . . . , all with the same ~k and T
coordinates, such that for every i ∈ n, Fqi+1 (σi ) ∈ D. The condition r = qn will
be as required.
Suppose that qi has been found. Consider the set B = {hx, yi ∈ Fqi (σi ) :
there is a witness to qi ∈ Q whose σi -th coordinate is hx, yi}. This is a nonempty
Σ11 set and therefore has a subset B 0 ⊂ B in the dense set D. Let Fqi+1 be equal
to Fqi except Fqi+1 (σi ) = B 0 . This completes the induction step.
The same type of argument, except simpler, yields
Claim 5.5.5. Whenever q ∈ Q and D is a dense subset of the Gandy-Harrington
forcing on [ω]ℵ0 , there is r ≤ q such that Tr ∈ D.
This is to assure that the filter {Tq : q ∈ G} is Gandy-Harrington-generic over
M and therefore its intersection will contain a unique point sgen ∈ S.
Claim 5.5.6. Whenever q ∈ Q and ~l ≥ ~kq , there is an extension r ≤ q such
that ~l = ~kr .
Proof. In order to give a concise proof of the claim, we first need to show that
any condition q ∈ Q has a sufficiently rich collection of witnesses. Let q ∈ Q
~
and write ~k = ~kq . Let hs, hxσ , yσ i : σ ∈ 2kq i be a witness to q ∈ Q with
stem x, let π : ω → s be an increasing enumeration, let i = max dom(~k), and
let a ⊂ 2ω be a countable set. We claim that there is an improved witness:
~
a tuple hs, hx0σ , yσ0 i : σ ∈ 2k i with the same stem as the original one such
that for every j ≤ i x0σ (π(j)) ∈
/ a. To see why this is true, note that the set
5.5. THE E1 IDEAL
127
~
~
{hx0σ (π(j)) : σ ∈ 2k , j ≤ ii : hs, x0σ : σ ∈ 2k i is a witness to q ∈ Q with stem
x} is a nonempty Σ11 (x) set: its definition does not depend on s as only the the
first dom(~k) many elements of s are used in the definition. The tuples in this
set contain no ∆11 (x) element, and apply Proposition 2.1.10.
Back to the proof of the claim. First of all, it is easy to extend q to r
such that dom(~kr ) is an arbitrary given number larger than dom(~k): simply
let ~kr = ~k concatenated with some zeroes and Fr (σ) = Fq (σ dom(~k)). So
we can assume that dom(~l) = dom(~k). It is also enough to consider the case
where there is a single number i ∈ dom(~k) such that ~l(i) = ~k(i) + 1 and the
other values of those sequences are equal. We must show that r = h~l, Fr , Tq i is
a condition in the poset Q, where Fr (σ) = Fq (σ ~k).
~
To prove this, we must find a witness for r ∈ Q. Choose hs, hxσ , yσ i : σ ∈ 2k i
Q
~
~
witnessing the fact that q ∈ Q. For every τ ∈ j>i 2k(j) and every u ∈ 2k(i)
let qτ,u be the condition defined by ~kqτ,u = ~k i, Tqτ,u = Tq and Fqτ,u (η) =
Fq (η a ua τ ). This is a condition in Q as witnessed by hs, hxηa ua τ , xηa ua τ i : η ∈
~
2ki i. Use the first paragraph to find, for every τ and u, two such witnesses
~
, yητ,ub i : η ∈ 2ki for qτ,u ∈ Q with the same stem and
, yητ,u0 i and hxτ,u1
hxτ,u0
η
η
the same s coordinate that use distinct points at coordinates π(j) : j ≤ i. For
~
, yσ = yητ,ub and check
σ ∈ 2l , σ = η a (ua b)a τ for some bit b ∈ 2 let xσ = xτ,ub
η
~
that hs, hxσ , yσ i : σ ∈ 2l i witness the fact that r ∈ Q.
Claim 5.5.7. For every q ∈ Q there is r ≤ q with ~kq = ~kr , and a map π :
dom(~kq ) → ω such that
1. π is an initial part of the increasing enumeration of every element of Tr ;
~
2. for every pair σ, τ ∈ 2k of sequences, every i such that σ(i) 6= τ (i) and
every j ≤ i, the projections of Fr (σ) and Fr (τ ) to the π(j)-th component
of their first coordinate are disjoint.
This is to assure that f has the property (a) of the definition of an sgen -keeping
homeomorphism.
~
Proof. Let ~k = ~kq . Pick a witness hs, hxσ , yσ i : σ ∈ 2k i to q ∈ Q. The condition
r ∈ Q will be constructed so that it has the same witness. Let π : ω → s
~
be the increasing enumeration, and for every σ ∈ 2k and every j ∈ dom(~k)
ω
find a basic open set Oσ,j ⊂ 2 containing xσ (π(j)) such that if xσ (j) 6= xτ (j)
then Oσ,j ∩ Oτ,j = 0. Improve the condition q ∈ Q to r so that ~kr = ~k,
T~r = {t ∈ Tq : π dom(~k) is an initial segment of the increasing enumeration
of t}, and Fr (σ) = {hx, yi ∈ Fq (σ) : ∀j ∈ dom(~k) x(π(j)) ∈ Oσ,j }. It is not
difficult to check that r ≤ q is a condition with the same witness as q, and it is
as required by the (a) definitory property of the witness.
128
CHAPTER 5. CLASSES OF σ-IDEALS
The rest of the proof of Case 1 is just a routine recitation of elementary density
arguments.
Case 2. If Case 1 fails, then let z ∈ 2ω be a parameter such that S, A are
both Σ11 (z) and note that then p(A) is a subset a generator of the σ-ideal IS ,
namely the set {x ∈ X : ∀s ∈ S ∃n ∈ s x(n) ∈ ∆11 (x (n, ω), z)}. We conclude
that p(A) ∈ IS .
The rest of the argument for Theorem 5.5.3 is now easy. For (2), suppose
that A ⊂ X × ω ω is an analytic set. If p(A) is IS -positive, then the above
argument provides an S-cube C and a continuous function g : C → ω ω such
that g ⊂ A. If, on the other hand, p(A) contains an S-cube C, a range of
an s-keeping homeomorphism f , then C and so p(A) must be IS -positive. To
see this, note that whenever z ∈ 2ω is a parameter and n ∈ s, then the set
Bn,z = {x ∈ X : f (x) ∈ ∆11 (f (x) (n, ω), z)} is meager by Kuratowski-Ulam–it
has countable sections in the ni-th coordinate where i is such that n is i-th
element of the set s. This means that the f -preimages of the generators of the
σ-ideal IS are meager S
(the generator associated with z has f -preimage which is
a subset of the union n Bn,z ) and so C = rng(f ) ∈
/ IS .
For (1), suppose that A ⊂ 2ω × X is an analytic set, and choose a parameter
z ∈ 2ω such that A, S are both Σ11 (z). The case 1 above then shows that for
every y ∈ 2ω , the set Ay is IS -positive if and only if there is x ∈ X and s ∈ S
such that hy, xi ∈ A and for every n ∈ s, x(n) ∈
/ ∆11 (x (n, ω), z, y). This is a
1
Σ1 (z) condition by ???.
Finally, for (3) suppose that A ∈
/ IS is an analytic set. Thinning out the
set A if necessary, we may assume that it is an S-cube, with an s-keeping
homeomorphism f : X → A for some s ∈ S. Now let {An : n ∈ ω} be countably
many analytic sets. Their f -preimages are still analytic, they have the Baire
property, and there is a dense Gδ subset B ⊂ X such that in it they are relatively
clopen. The proof of (2) above shows that f 00 B ⊂ A is an IS -positive set, and
in it all sets {An : n ∈ ω} are relatively clopen. (3) follows.
There are two interesting special cases that we will consider here. If S =
{ω} then the quotient poset PIS has a dense subset naturally isomorphic to
the reverse iteration of Sacks forcing, which is defined to consist of reverse
cubes ordered by inclusion. A more interesting case occurs when S = [ω]ℵ0 ; it
corresponds to the σ-ideal IE1 and the Kechris-Louveau dichotomy 2.2.10.
Corollary 5.5.8. Let S = [ω]ℵ0 . The ideals IS and IE1 contain the same
analytic sets. In other words, for every analytic set A ⊂ X, A ∈
/ IS iff E1 is
Borel reducible to E1 A iff E1 A is not Borel reducible to E0 .
Proof. Suppose that A ∈
/ IS ; we will conclude that E1 is Borel reducible to
E1 A. Use Theorem 5.5.3 to find an infinite set s ⊂ ω and an s-keeping
homeomorphism f : X → A whose compact range C is a subset of A. Clearly,
f is a continuous injective reduction of E1 to E1 C as required.
Suppose that A ∈ IS ; we must conclude that E1 B is Borel reducible to
E0 for some Borel superset B ⊃ A. For simplicity assume that A is Σ11 . Use
5.5. THE E1 IDEAL
129
the first reflection theorem 2.1.2 to find a Borel set B ⊃ A such that for every
x ∈ B the set {n : x(n) ∈
/ ∆11 (x (n, ω))} is finite. Use Kreisel selection [27,
Theorem 2.4.5] to find a Borel function h : B → ω such that this finite set
is always bounded by h(x). Let i : B → X be the Borel function assigning
each x ∈ B the sequence equal to x above h(x) and equal to the zero sequence
below it. Review the definitions to observe that the E1 -equivalence classes
on rng(i) are countable. Use the first reflection theorem to produce a Borel
superset C ⊃ rng(i) on which the E1 classes are countable. Now, E1 C is a
hypersmooth equivalence relation with countable classes, and by [27, Theorem
8.1.1(iii)], such an equivalence is Borel reducible to E0 via some Borel function
j : C → 2ω . The composition j ◦ i : B → 2ω reduces E1 B to 2ω as desired.
Corollary 5.5.9. (Kechris-Louveau) If E is a hypersmooth equivalence relation
on a Polish space Y then either E ≤B E0 or E1 ≤B E.
Proof. Fix Borel maps fn : n ∈ ω, fn : Y → 2ω such that for every y0 , y1 ∈ Y ,
y0 E y1 iff the set {n : fn (y0 ) = fn (y1 )} ⊂ ω is co-finite. Let g : Y → X be the
map defined by g(y) = hfn (y) : n ∈ ωi, and look at the set A = rng(g). The set
A ⊂ X is analytic; we discern the cases where A ∈ J or not.
If A ∈
/ J then by Theorem 5.5.3 there is an s-cube C ⊂ A for some infinite
set s ⊂ ω, a continuous map g : C → Y such that f ◦ g = id, and an s-keeping
homeomorphism h : X → C whose range C is. Review the definitions to see
that g ◦ h : X → Y is a continuous injective reduction of E1 to E A.
If A ∈ J then there is a Borel set B ⊃ A such that E1 B is Borel reducible
to E0 via a map h : B → 2ω . It is immediate that h ◦ f is a reduction of E to
E0 .
Corollary 5.5.10. Let S = {ω}. The poset PIS has a dense subset naturally
isomorphic to the Sacks iteration along ω with the inverted ordering.
Proof. Up to an insignificant technical detail, the iteration isolated in [26] consists of ω-cubes ordered by inclusion. Theorem 5.5.3 precisely states that this
is a dense subset of the ordering PIS .
5.5b
Canonization results
The canonization properties of reverse cubes are unclear in general. However,
the hypersmooth equivalences do canonize down to a simple irreducible set of
unavoidable obstacles.
Theorem 5.5.11. Let B ⊂ X = (2ω )ω be a Borel set such that E1 B is not
essentially countable. Let E be a hypersmooth equivalence relation on B. Then
there is a compact set C ⊂ B such that E1 C is still not essentially countable,
and E C equals to id or to ee or to E1 .
130
CHAPTER 5. CLASSES OF σ-IDEALS
Theorem 5.5.12. Let B ⊂ X = (2ω )ω be a reverse cube, and let E be a
hypersmooth equivalence relation on B. Then there is a reverse cube C ⊂ B
such that E C equals to id or to ee or to E1 , or to idn for some n ∈ ω.
Here, idn is the smooth equivalence relation
S connecting points x, y ∈ X if for
all m ≥ n, x(m) = y(m)–so that E1 = n idn . Theorem 5.5.11 is a formal
consequence of Theorem 5.5.12.
The arguments below follow the same pattern. There is a split into cases,
in one of the cases we use a Gandy-Harrington forcing just like in the previous
section, and in the other a fusion argument appears. Thus, we need to make
preliminary fusion definitions. Let X = (2ω )ω and let I be the σ-ideal of analytic
sets which contain no reverse cube.
Definition 5.5.13. Let B, C ⊂ X be reverse cubes and n ∈ ω. We write
C ≤n B if C ⊂ B and C is idn -invariant inside B. A fusionsequence is a
sequence hCj : j ∈ ωi of reverse cubes such that there are increasing numbers
hnj : j ∈ ωi such that Cj+1 ≤nj Cj .
PropositionT5.5.14. Suppose that hCj : j ∈ ωi is a fusion sequence of reverse
cubes. Then j Cj ∈
/ I.
T
Proof. It is clear that C = j Cj is a nonempty compact set. To show that
C∈
/ I, we will show that the generating sets of the ideal I are relatively meager
in C. Let z ∈ 2ω be an arbitrary parameter, let n ∈ ω and consider the coanalytic set B = {x ∈ C : x(n) ∈ ∆11 (x (n, ω), z)}. Let D ⊂ 2ω × (2ω )(n,ω) be the
nonempty compact projection of the set C into (2ω )[n,ω) . The horizontal sections
of the set D are all perfect: if nonempty, every such horizontal section is equal
to a corresponding section in the cube Cn since C ≤n Cn , and the section in the
cube Cn is perfect by the definition of a projection-keeping homeomorphism.
The horizontal sections corresponding to the set B are countable, and therefore meager in the horizontal sections of the set D. By the Kuratowski-Ulam
theorem, B is meager in C as desired.
As a useful warm-up, we will reprove the canonization of smooth equivalence
relations achieved in [26].
Theorem 5.5.15. For every reverse cube A ⊂ X and a smooth equivalence E
on A there is a reverse cube C ⊂ A such that E C = idn for some n ∈ ω or
E C = ee.
Proof. Let A ⊂ X be an I-positive Borel set and E a smooth equivalence
relation. Let f : A → 2ω be the Borel reduction of E to the identity. Passing
to an I-positive subset if necessary, we may assume that f A is continuous.
Case 1. Suppose that there exists a parameter z ∈ 2ω such that f ∈ Π01 (z)
and there is a set  ∈ Σ11 (z) with  ⊂ A, a point x ∈  and a number m ∈ ω
/ ∆11 (x [m, ω), z).
such that for every n ∈ ω, x(n) ∈
/ ∆11 (x (n, ω)), and f (x) ∈
In this case, we will use a variation of the Gandy-Harrington poset form the
previous section to find a reverse cube C ⊂ Â such that E C = idm−1 .
5.5. THE E1 IDEAL
131
Assume for simplicity z = 0 and choose x, m as described. Let Q be the
~
poset of all pairs q = h~k = ~kq , F = Fq i such that for every σ ∈ 2k , F (σ) is
a nonempty Σ11 subset of  and there is a witness to q ∈ Q. This is a tuple
~
~
hxσ : σ ∈ 2k i such that for every σ ∈ 2k , xσ ∈ F (σ) and for every n ∈ ω,
~
xσ (n) ∈
/ ∆11 (x (n, ω), and for every pair σ, τ ∈ 2k , (a) if σ(i) 6= τ (i) and
j ≤ i then xσ (j) 6= xτ (j), (b) if i is the largest number that σ(i) 6= τ (i) then
xσ (i, ω) = xτ (i, ω), and (c) if σ dom(~k) \ m 6= τ dom(~k) \ m, then
f (xσ ) 6= f (xτ ). The ordering on Q is defined by r ≤ q if ~kq ≤ ~kr and for every
~
σ ∈ 2kr , Fr (σ) ⊂ Fq (σ ~kq ). The case assumption yields the starting point:
Q 6= 0.
The argument now follows closely that of Theorem 5.5.3. Let M be a countable elementary submodel of a large enough structure and let G ⊂ Q be a filter
generic over M . For every x ∈ X consider the filter {Fq (x ~k) : q ∈ G} on
the Gandy-Harrington forcing on X. It turns out that this filter is GandyHarrington generic over M , and so its intersection contains a unique element
g(x) ∈ Â. The function g : X → A is a projection-keeping homeomorphism.
Moreover, if x0 , x1 ∈ X are points such that x0 [m, ω) 6= x1 [m, ω), then
f (x0 ) 6= f (x1 ). The whole treatment can be nearly literally copied from the previous section. Note the application of Proposition 2.1.10 in one of the density
arguments.
Let C = rng(g); this is a reverse cube such that for all x0 , x1 ∈ C with
x0 [m, ω) 6= x1 [m, ω) it is the case that f (x0 ) 6= f (x1 ). Now, if the
number m ∈ ω was chosen to be smallest possible, the reverse cube C is such
that for every x ∈ C, f (x) ∈ ∆11 (x [m, ω)). Now choose any reasonable
effective enumeration of ∆11 (·) singletons by natural numbers such as in [27,
Corollary 2.8.4]. For every natural number e, the set Be = {x ∈ C : e codes
a ∆11 (x S
[m, ω))-singleton and that singleton is equal to f (x)} is coanalytic,
and C ⊂ e Be . By Theorem 5.5.3 (3), we can thin out the cube C so that
all these sets Be have Borel intersection with C. By Theorem 5.5.3(2) and the
σ-additivity of the σ-ideal IS we can then further thin out C so that it is wholly
included in one of these sets. After that, E C = idm as required.
Case 2. If Case 1 fails, we will construct a fusion sequence C0 ≥0 C1 ≥
C2 ≥ . . . so that for every j ∈ ω, for every x ∈ CTj , the functional value f (x)
depends only on x (j, ω). In the end, let C = j Cj ; this is a reverse cube
such that f is constant on E1 classes of C. Since the equivalence relation E1 is
dense in C 2 and the function f is continuous, this means that f C is constant
and E C = ee as desired.
The construction of Cj proceeds by induction. Suppose that Cj has been
constructed. Find z ∈ 2ω such that Cj ∈ Σ11 (z). The case assumption shows
that for every x ∈ Cj such that for every n ∈ ω, x(n) ∈
/ ∆11 (x (n, ω)),
f (x) ∈ ∆11 (x [j + 1, ω), z). Just as above, one can use the Π11 coding of ∆11
sets and Theorem 5.5.3(3) to find a subcube Cj+1 so that f (x) depends only on
x [j + 1, ω). Since f (x) depends only on x [j, ω) for x ∈ Cj , we can find the
subcube Cj+1 so that Cj+1 ≤j Cj . This completes the induction step and the
proof.
132
CHAPTER 5. CLASSES OF σ-IDEALS
S
Proof of Theorem 5.5.12. Let A ⊂ X be a reverse cube. Let E = Fn for some
inclusion increasing sequence of smooth equivalence relations, and let fn : A →
2ω be Borel functions reducing each Fn to the identity.
Case 1. There is a parameter z ∈ 2ω such that the functions fn are all Π01 (z),
a set  ∈ Σ11 (z) with  ⊂ A, a number m ∈ ω, and a point x ∈  such that
∀n x(n) ∈
/ ∆11 (x (n, ω), z) and ∀i fi (x) ∈
/ ∆11 (x (m, ω), z). We will assume
that m is minimal possible, we will make the harmless assumption that  = A
and that thinning out if necessary, that ∀x ∈ A ∀n x(n) ∈
/ ∆11 (x (n, ω), z)
holds. Here we use the Gandy-Harrington forcing to extract a reverse cube
C ⊂ A such that on it E C ⊆ idm , and then thin it out further so that on it
either E C = idm or E C = idm−1 .
Assume for simplicity that the parameter z is in fact 0. Let Q be the poset
of all pairs q = h~kq = ~k, Fq = F i, where ~k ∈ ω <ω and Fq is a map that assigns
~
each sequence σ ∈ 2k a nonempty Σ11 subset of A. Moreover, there must be a
~
~
witness to q ∈ Q, and this is a tuple hxσ : σ ∈ 2k i such that for every σ ∈ 2k
1
it is the case that xσ ∈ F (σ) and ∀n x(n) ∈
/ ∆1 (x (n, ω), z), and for every
~
pair σ, τ ∈ 2k , (a) if σ(i) 6= τ (i) and j ≤ i then xσ (j) 6= xτ (j), (b) if i is
the largest number that σ(i) 6= τ (i) then xσ (i, ω) = xτ (i, ω), and (c) if
σ (m, dom(~k)) 6= τ (m, dom(~k)) then fn0 (xσ ) 6= fn1 (xτ ) for every n0 , n1 ∈ ω.
~
The ordering on Q is the usual one: r ≤ q if ~kq ≤ ~kr and for every σ ∈ 2kr ,
Fr (σ) ⊂ Fq (σ ~kq ). The case assumption shows that Q 6= 0.
The treatment of the forcing follows the previous proofs in this section. In
the claim dealing with splitting of witnesses, the full strength of ??? is used.
Let M be a countable elementary submodel of a large enough structure and
let G ⊂ Q be a filter generic over M . For every x ∈ X consider the filter
{Fq (x ~k) : q ∈ G} on the Gandy-Harrington forcing on X. It turns out that
this filter is Gandy-Harrington generic over M , and so its intersection contains
a unique element g(x) ∈ A. The function g : X → A is a projection-keeping
homeomorphism, and whenever x0 , x1 ∈ X are points such that x0 [m, ω) 6=
x1 [m, ω) and n0 , n1 ∈ ω are natural numbers, then fn0 (x0 ) 6= fn1 (x1 ). So,
writing C = rng(g), we see that this is a reverse cube on which E ⊆ idm .
By the minimal choice of the number m, for every x ∈ C there is a number
n ∈ ω such that fn (x) ∈ ∆11 (x [m − 1, ω)). Use Theorem 5.5.3(3) to thin
out C if necessary so that the coanalytic sets On,k = {x ∈ C : is a code for a
∆11 (x [m − 1, ω)) singleton which is equal to fn (x)} are relatively Borel in C,
and then use Theorem 5.5.3(2) to thin out the reverse cube C further so that it
is a subset of one of these sets, say On0 ,k0 . It follows that idm−1 ⊆ E C. We
will now thin out C further to fully canonize E C.
Write Cm for the projection of C to (2ω )[m,ω) and similarly for Cm−1 ; these
are reverse cubes in the appropriate sense. Note that for all n > n0 and x ∈ C,
the functional value fn (x) depends on x [m − 1, ω) only, and write fn for the
function from Cm−1 to 2ω . Use Theorem 5.5.3(2) to thin out C so that the
functions fn : n ∈ ω are all continuous on Cm−1 .
Consider the sets D0 = {hx, P i : x ∈ Cm , P ⊂ 2ω is a nonempty perfect
set, {x} × P ⊂ Cm−1 , and for some n ≥ n0 , fn is constant on {x} × P } and
5.5. THE E1 IDEAL
133
D1 = {hx, P i : x ∈ Cm , P ⊂ 2ω is a nonempty perfect set, {x} × P ⊂ Cm−1 ,
and for every n ≥ n0 , fn is one-to-one on {x} × P }. Since the functions fn are
continuous, both of these sets are analytic in (2ω )[m,ω) × K(2ω ). Note that the
union of projections of D0 and D1 covers the whole cube Cm : for every x ∈ Cm ,
either there is n ≥ n0 such that the function fn has an uncountable fiber on
the set {y ∈ 2ω : y a x ∈ Cm } and then x belongs to the projection of D0 , or
else the fibers are countable and then there is a perfect set on which all of these
functions are injective, for example by Mycielski’s theorem ???. Thus, we can
thin out Cm into a reverse subcube if necessary and find a continuous function
g : Cm → K(2ω ) such that (a) g ⊂ D0 or (b) g ⊂ D1 . In either case, it is
possible to thin out C further so that for every x ∈ C, x(m − 1) ∈ g(x [m, ω)).
In case (a), we get E C = idm , in case (b) we get E C = idm−1 as desired.
Case 2. There is a reverse cube C0 ⊂ A, a parameter z ∈ 2ω and a number
n such that for every x ∈ C0 and every m ∈ ω, f (x) ∈ ∆11 (x (m, ω), z). In this
case, a fusion construction similar to Case 2 of the proof of Theorem 5.5.15 will
yield a subcube C ⊂ C0 such that fn C is constant, and therefore E C = ee.
Case 3. Suppose now that both Cases 1 and 2 fail. We will build a fusion
sequence whose limit, a reverse cube C ⊂ X, will have E C = E1 . By
induction on j ∈ ω build numbers mj , nj ∈ ω and cubes Cj so that m0 <
m1 <Tm2 < . . . , Cj ≥mj Cj+1 , and fnj Cj = idmj . If this succeeds, let
C = j Cj . This is a reverse cube; the assumptions immediately imply that
n0 < n1 < . . . and E C = E1 . To perform the induction, suppose Cj , mj , nj
have been constructed. Let z ∈ 2ω be such that Cj ∈ Σ11 (z). Since Case 1
fails, for every x ∈ Cj such that ∀i x(i) ∈
/ ∆11 (x (i, ω), z) there must be n
1
such that fn (x) ∈ ∆1 (x (mj , ω), z). Use Theorem 5.5.3(3) to thin out Cj so
that the coanalytic set {x : fn (x) ∈ ∆11 (x (mj , ω), z)} are relatively Borel for
each n ∈ ω, and then use Theorem 5.5.3(2) to thin out Cj to be a subset of one
of these sets, for n = nj+1 . These surgeries on the reverse cube Cj may have
resulted in a change of the parameter z from which Cj is defined, so find a new
parameter z 0 ∈ 2ω coding z and such that Cj is Σ11 (z 0 ). Since Case 2 fails, there
must be x ∈ Cj such that for every i ∈ ω, x(i) ∈
/ ∆11 (x (i, ω), z 0 ) and for some
1
0
m > mj , fnj+1 (x) ∈
/ ∆1 (x (m, ω), z ). Let mj+1 > mj be the least number
m for which such a point x exists. The proof of Case 1 of Theorem 5.5.15
shows that there is a subcube Cj+1 ⊂ Cj0 such that Enj+1 Cj+1 = idmj ;
since Enj ⊂ Enj+1 and Enj Cj = idmj , we can find such a subcube so that
Cj+1 ≤mj Cj . This concludes the induction step and the proof.
Proof of Theorem 5.5.11. Let A ⊂ X be an analytic set such that E1 A is
not essentially countable, and let E be a hypersmooth equivalence on A. Using
Theorem 5.5.3(2), thinning out the set A if necessary we may assume that A
is an s-cube for some infinite set s ⊂ ω, with an s-keeping homeomorphism
f : X → A. Let F = f −1 E be the (still hypersmooth) pullback of E, and
use the previous theorem to find a reverse cube D ⊂ X with a strict ω-keeping
homeomorphism g : X → D such that F D is either equal to E1 or to ee or to
idn for some n ∈ ω. In the first two cases, the composition f ◦ g is an s-keeping
homeomorphism and on its range C, the equivalence E is equal to either E1
134
CHAPTER 5. CLASSES OF σ-IDEALS
or to ee. In the last case, there is an obvious ω \ n-keeping homeomorphism
h : X → D, and then f ◦ g ◦ h is an s0 -keeping homeomorphism (for s0 = s\the
first n-many elements of s) such that on its range C, E C = id.
5.5c
Borel functions on pairs
The methods of this section make it possible to identify coding functions for the
σ-ideals discussed above. This makes it very unlikely that any reduced product
forcing type of argument could be used to produce further canonization results.
Theorem 5.5.16. Let I be the σ-ideal on X consisting of Borel sets with no
reverse subcube. There is a Borel function f : X 2 → 2ω such that for every
reverse cube C ⊂ X, f 00 C 2 = 2ω .
Proof. Let f (x, y)(m) = x(m)(k) where k is the least number such that x(m)(k) 6=
y(m)(k) if such exists; if such a k does not exist then let f (x, y)(m) =trash. This
function works.
Let C ⊂ X be a reverse cube and let z ∈ 2ω be an arbitrary point. We
have to produce points x, y ∈ C such that f (x, y) = z. Choose a parameter
z 0 ∈ 2ω coding both C and z and consider the following Gandy-Harrington type
forcing Q. A condition in Q is a pair q = hAq , Bq i of Σ11 (z 0 ) subsets of C such
that there is an witness to q ∈ Q: this is a pair hxq , yq i such that xq ∈q A,
yq ∈ Bq , for every n ∈ ω xq (n) ∈
/ ∆11 (z 0 , x (n, ω)) and similarly on y-side, and
finally, for some number n ∈ ω (the split point of the witness), for every m ≥ n
xq (m) = yq (m) holds and for every m < n, xq (m) 6= yq (m) and xq (m)(k) = z(k)
hold, where k is the least number such that xq (m)(k) 6= yq (m)(k). The ordering
is that of coordinatewise inclusion. Note that Q is a countable poset.
Let M be a countable elementary submodel of a large structure and let
G ⊂ Q is a filter generic over M . The intersection of the first coordinates of
conditions in the generic filter contains a unique point x ∈ X and the second
coordinates all intersect in a singleton as well, {y}. We will argue that f (x, y) =
z.
The arguments transfers from the previous subsections in all details, except
for one. For any condition q ∈ Q we must produce witnesses with arbitrarily
large split points. Suppose hxq , yq i is a witness with split point n; we must
produce a witness with split point n + 1. For every m ∈ n let k(m) be the least
number k such that xq (m)(k) 6= yq (m)(k) and consider the set D = {u ∈ 2ω :
there are points xu ∈ A, yu ∈ B such that for all m ∈ ω, xu (m) ∈
/ ∆11 (z 0 , xu (m, ω)), similarly on the yu side, for all m > n xu (m) = yu (m) = xq (m),
xu (n) = yu (n) = u, and for all m < n, xq (m) k(m) + 1 ⊂ xu (m) and
yq (m) k(m) + 1 ⊂ yu (m)}. This is a Σ11 (z 0 , xq [n + 1, ω) set containing a non∆11 (z 0 , xq [n + 1, ω) point, namely xq (n). Therefore, the set D is uncountable
and contains two distinct points u, u0 ∈ 2ω . Let k be the least number such that
u(k) 6= u0 (k); say u(k) = z(n). Use the definition of the set D to get points x0q
and yq0 such that for all m > n x0q (m) = yq0 (m) = xq (m), x0q (n) = u and yq0 (n) =
u0 , and for all m < n, xq (m) k(m) + 1 ⊂ x0q (m) and yq (m) k(m) + 1 ⊂ yq0 (m).
The pair hx0q , yq0 i is the desired witness with split point n + 1.
5.5. THE E1 IDEAL
135
Theorem 5.5.17. Let I be the σ-ideal on X consisting of Borel sets on which
E1 is reducible to E0 . There is a Borel function f : X 2 → ω ω such that for
every I-positive Borel set B ⊂ X, the set f 00 B 2 ⊂ ω ω is dominating.
Proof. Let f : X 2 → ω ω be the function defined by f (x, y)(n) = min{k : for
every m ∈ n either x(m) = y(m) or x(m) k 6= y(m) k}. This function works.
In order to prove this, let B ⊂ X be a Borel I-positive set, and z ∈ ω ω
be a function. We must find points x, y ∈ B such that f (x, y)(n) > z(n) for
all but finitely many n ∈ ω. In order to do so, use Corollary 5.5.8 to find an
infinite set s ⊂ ω with the increasing enumeration π : ω → s and a strict skeeping homeomorphism g : X → B. Let z 0 ∈ 2ω be a point coding z, s, and g.
Consider the poset Q of all pairs q = hAq , Bq i of Σ11 (z 0 ) sets such that there is
a witness for q ∈ Q, and this is a pair hx, yi such that x ∈ Aq , y ∈ Bq , for every
n ∈ ω g(x)(π(n)) ∈
/ ∆11 (z 0 , x (n, ω)), similarly on the y-side, and for some
number n ∈ ω, for all m ≥ n x(m) = y(m), and for all m < n, it is the case
that f (x)(π(m)) 6= f (y)(π(m)) and writing k for the least number such that
f (x)(π(m)(k) 6= f (y)(π(m))(k), k > z(l) for all l ≤ π(m + 1). The ordering on
Q is that of coordinatewise inclusion.
Just as in all the previous similar arguments, if M is a countable elementary
submodel of a large structure and G is a Q-generic filter over M , the first and
second coordinates of the conditions in the filter give some points x, y ∈ B. It
is then straightforward to show that f (x, y)(n) > z(n) for all n > π(0).
Corollary 5.5.18. Let J be a σ-ideal on a Polish space X such that the quotient
forcing PJ is proper. If E1 is in the spectrum of the σ-ideal J then J has a coding
function.
Proof. By Theorem 4.5.1, if E1 is in the spectrum then there is a condition
B ∈ PJ and names hẏn : n ∈ ωi for elements of the Cantor space such that
B ∀n yn ∈
/ V [hym : m > ni]. Using the properness of the quotient poset,
thinning out the condition B if necessary it is possible to find Borel functions
gn : B → 2ω representing the names ẏn for every n ∈ ω, and combine these
functions to g : B → (2ω )ω . Observe that if C ⊂ B is a J-positive Borel
set then the analytic set g 00 C ⊂ (2ω )ω is I-positive: whenever z ∈ 2ω is a
parameter, M is a countable elementary submodel of a large enough structure
containing z, J, C, and x ∈ C is a PI -generic point over M , it is the case for every
n ∈ ω that gn (x) ∈
/ M (hgm (x) : m > ni) by the forcing theorem. Therefore,
gn (x) ∈
/ ∆11 (z, gm (x) : m > n) and so g(x) does not belong to the generator
set of the σ-ideal J associated with the parameter z. Now, let h be the coding
function for J as isolated in Theorem 5.5.16. Clearly, h ◦ g is a coding function
for I.
136
5.6
CHAPTER 5. CLASSES OF σ-IDEALS
The E2 ideal
1
: x(n) 6=
E2 is the equivalence relation on X = 2ω defined by xE2 y if Σ{ n+1
y(n)} < ∞. It is a basic example of a turbulent equivalence relation, and therefore it is not classifiable by countable structures [27, Theorem 13.9.1]. There is
an associated σ-ideal and a quotient proper notion of forcing. For xE2 y define
1
: x(n) 6= y(n)}, otherwise let d(x, y) = ∞. Thus, d is a metric
d(x, y) = Σ{ n+1
on each equivalence class. Kanovei [27, Definition 15.2.2] defined the collection
of grainy subsets of X as those sets A ⊂ 2ω such that there is a real number
ε > 0 such that there is no finite sequence of elements of A, the successive
elements of which have d-distance at most ε and the two endpoints are at a
distance at least 1 from each other. Let I be the σ-ideal generated by the Borel
grainy sets.
Fact 5.6.1. [27, Section 15.2] The ideal I is Π11 on Σ11 , and every analytic
I-positive set has a Borel I-positive subset. A Borel set B ⊂ X is in the ideal
I if and only if the equivalence E2 B is essentially countable if and only if E2
is not reducible to E2 B.
The features of the quotient forcing were surveyed in [27, 67]. Here, we will
provide a more thorough treatment:
Theorem 5.6.2. The forcing PI is proper, bounding, preserves Baire category
and outer Lebesgue measure, and adds no independent reals.
The canonization properties of the ideal I seem to be difficult to isolate.
Clearly, the ideal was defined so as to have the equivalence relation E2 in the
spectrum. This is reflected in the following:
Theorem 5.6.3. E2 is in the spectrum of I. Moreover, PIE2 is a regular ℵ0 distributive subposet of PI , and it generates the model V [ẋgen ]E2 .
No other features of the spectrum are obvious.We will prove
Theorem 5.6.4. I has total canonization for equivalences classifiable by countable structures.
In particular, this shows that the forcing PI adds a minimal real, improving [27,
Theorem 15.6.3].
Question 5.6.5. Are there any equivalence relations in the spectrum of I other
than E2 ?
Proof of Theorem 5.6.2. We will need some preliminary notation and observations. Abuse the notation a little and include I-positive analytic sets in it; by
ω
Fact 5.6.1, Borel sets will be dense
P in1 it anyway. For points x, y ∈ 2 and a
number k ∈ ω write dk (x, y) = { n+1 : n ≥ k, x(n) 6= y(n)} < ε; this is a
notion of pseudodistance between sets satisfying triangle inequality which can
be infinite or zero for distinct points. For sets A, B ⊂ 2ω let dk (A, B) be the
associated Hausdorff distance: the infimum of all ε such that for every x ∈ A
there is y ∈ B such that dk (x, y) < ε and vice versa, for every x ∈ B there is
y ∈ A such that dk (x, y) < ε. The following claims will be used repeatedly:
5.6. THE E2 IDEAL
137
Claim 5.6.6. If A, B ⊂ 2ω are analytic sets such that A ∈
/ I and ∀x ∈ A ∃y ∈
B x E2 y then B ∈
/ I.
Proof. If B ∈ I, then B can be enclosed in a countable union B 0 of Borel grainy
sets, and E2 B 0 can be reduced to some countable equivalence relation E on
2ω via a Borel function f : B 0 → 2ω . The set A can be thinned out to a Borel
I-positive set A0 ⊂ A. Consider the relation R ⊂ A0 × 2ω where hx, yi ∈ R
if ∃z ∈ B 0 x E2 z ∧ f (z) = y. This is an analytic relation such that non-E2 equivalent points in A0 have disjoint vertical sections, and the union of vertical
sections of points belonging to the same fixed E2 class is countable. These two
are Π11 on Σ11 properties of relations, and so by the first reflection theorem, R
can be enclosed in a Borel relation R0 with the same properties. The Novikov
uniformization theorem for Borel sets with countable sections yields a Borel
uniformization g : A0 → 2ω of R0 . This is a Borel function under which distinct
E2 -classes have disjoint countable images. By a result of Kechris, [27, Lemma
7.6.1], E2 A0 must then be essentially countable.
Claim 5.6.7. If {Ai : i ∈ n} are I-positive analytic sets with dk (Ai , Ai+1 ) < ε
for all i ∈ n − 1 and D ⊂ PI is an open dense set, then there are analytic sets
{A0i : i ∈ n} such that A0i ⊂ Ai , A0i ∈ D, and dk (A0i , A0i+1 ) < ε for all i ∈ n − 1.
Note that in the conclusion, the dk -distances between the various sets {A0i : i ∈
n} must be smaller than (n − 1)ε.
Proof. By induction on i ∈ n find sets Aij ⊂ Aj : j ∈ n such that at stage i, the
dk -distances are still < ε and Aii ∈ D. The sets An−1
: j ∈ n will then certainly
j
work as desired. The induction is easy. Given the sets {Aij : j ∈ n}, strengthen
Aii+1 to a set B in the dense set D. Then, for every j ∈ n, write Ai+1
= {x ∈
j
ω
ω n
2 : ∃~y ∈ (2 ) ∀l ∈ n − 1 dk (~y (l), ~y (l + 1)) < ε and ∀l ∈ n ~y (l) ∈ Ail and
i+1
~y (i + 1) ∈ B and ~y (j) = x}. These are all analytic sets with dk (Ai+1
j , Aj+1 ) < ε,
i+1
Ai+1 = B, and by the previous claim they must all be I-positive since the set
B is. The induction can proceed.
Claim 5.6.8. Suppose that {Ai : i ∈ n} are I-positive analytic sets with ddistance < ε, and 1 > δ > 0, γ > 0 are real numbers. Then there are I-positive
analytic sets Abi : i ∈ n, b ∈ 2, binary sequences tbi ∈ 2<ω of the same length k
such that
1. |d(t0i , t1i ) − δ| < 2ε and d(t0i , t0j ), d(t1i , t1j ) are both smaller than 2ε;
2. Abi ⊂ Ai ∩ [tbi ], and the dk -distances of the sets {Abi : i ∈ n, b ∈ 2} are < γ.
Proof. For each choice of k and ~t = htbi : i ∈ n, b ∈ 2i as in the first item of the
claim let B~t ⊂ (2ω )n×2 be the analytic set of all tuples hxbi : i ∈ n, b ∈ 2i such
that tbi ⊂ xbi , xbi ∈ Ai , and the dk -distances between points on the tuple are < γ.
If for some ~t, the projection of B~t into the first coordinate is I-positive, then
the projections into any coordinate are I-positive analytic sets by Claim 5.6.6,
and they will work as {Abi : i ∈ n, b ∈ 2} required in the claim.
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CHAPTER 5. CLASSES OF σ-IDEALS
So it will be enough to derive a contradiction from the assumption that the
projections of the sets B~t into the first coordinate are always in the ideal I.
These are countably many I-small analytic sets, so they can be enclosed in a
Borel I-small set C ⊂ 2ω . The set A0 \ C is analytic and I-positive, and so it
contains an ε-walk whose endpoints have d-distance > 1, so it has to contain
two points x00 , x10 with |d(x00 , x10 ) − δ| < ε. Choose points xbi : i > 0, b ∈ 2 in
the sets Ai so that d(x00 , x0i ), d(x10 , x1i ) are always smaller than ε, find a number
k such that the dk -distance of these points is pairwise smaller than γ, and let
~t = hxbi k : i ∈ n, b ∈ 2i. This sequence should belong to the set B~t, but its
first coordinate should not belong to the projection of B~t. This is of course a
contradiction.
This ends the preliminary part of the proof. For the properness of PI , let M
be a countable elementary submodel of a large enough structure and let B ∈ PI
be a Borel I-positive set. We need to show that the set C = {x ∈ B : x is
M -generic} is I-positive. To do that, we will produce a continuous injective
embedding of the equivalence E2 into E2 C. Fix positive real numbers εn
whose sum converges. Enumerate the open dense subsets of PI in the model M
by {Dn : n ∈ ω}. By induction on n ∈ ω build functions πn : 2n → 2<ω as well
as sets {At : t ∈ 2n } so that
I1 if m ∈ n then πn extends πm in the natural sense: for every t ∈ 2m and
every s ∈ 2n , πm (t) ⊂ πn (s). Also As ⊂ At , and As ∈ PI ∩ M ∩ Dn is an
I-positive analytic set;
I2 all sequences in the range of πn have the same length kn ∈ ω, and
πn approximately preserves the d-distance: |d(u, v) − d(πn (u), πn (v))| <
Σm∈n εm ;
I3 the sets {Au : u ∈ 2n } have pairwise dkn -distance smaller than εn /2.
The induction step is handled easily using the previous two claims. In the
end, for every sequence y ∈ 2ω the sets {At : t ⊂ y} generate an M -generic
filter, so theySdo have a nonempty intersection which must contain a single
point π(y) = n πn (x n). It is immediate from I2 that the map π : 2ω → C is
a continuous injective reduction of E2 to E2 C and so C ∈
/ I as desired.
For the bounding part, we must show that for every I-positive Borel set
B ⊂ 2ω and every Borel function f : B → 2ω there is a compact I-positive
subset of B on which the function f is continuous. This is obtained by a
repetition of the previous argument. Just note that the image of the map π
obtained there is compact, and if the open dense set Dn consists of sets B 0 on
which the bit f (x)(n) is constant, then f rng(π) is continuous.
For the preservation of outer Lebesgue measure, it is enough to argue that
the ideal I is polar in the sense of [67, Section 3.6.1]; i.e., it is an intersection of
null ideals for some collection of Borel probability measures. In other words, we
need to show that every Borel I-positive set B ⊂ 2ω carries a Borel probability
5.6. THE E2 IDEAL
139
measure that vanishes on the ideal I. This uses a simple claim of independent
interest:
Claim 5.6.9. The usual Borel probability measure µ on 2ω vanishes on the ideal
I, meaning that every set in I has µ-mass zero.
Proof. Argue by repeatedly applying Steinhaus theorem [3, Theorem 3.2.10] to
the compact Cantor group h2ω , +i. Let B ⊂ 2ω be a Borel non-null set; we must
show that B is not grainy; in other words, given ε > 0, we must find an ε-walk
in B whose endpoints have distance at least 1. Fix ε > 0 and by induction on
i ∈ ω build Borel non-null sets Bi ⊂ 2ω and finite sets ai ⊂ ω so that
I1 B = B0 ⊃ B1 ⊃ . . . ;
I2 the sets
Pai are 1pairwise disjoint and for every i ∈ ω it is the case that
< ε;
ε/2 < n∈ai n+1
I3 Bi+1 ∆χai ⊂ Bi where χai is the characteristic function of ai .
To perform the induction step, suppose that the set Bi+1 and ai do not exist
to satisfy the requirements in I1, I2 and I3. This means that for every finite set
b ⊂ ω such that I2 is satisfied with b S
= ai , the set Cb = {x ∈ Bi : x∆χb ∈ Bi } is
Lebesgue null, and the set Bi0 = Bi \ b Cb is Borel and Lebesgue positive. Now
use Steinhaus theorem to argue that Bi0 ∆Bi0 contains an open neighborhood of
0, and so contains the characteristic function of some set b satisfying the second
item. This is of course a contradiction.
In the end, choose j ∈ ω such that jε
2 > 1 and a point x ∈ Bj+1 . The
points x, x∆χaj , x∆χaj ∆χaj−1 . . . form the required ε-walk whose endnodes
have distance greater than 1.
If B ⊂ 2ω is a Borel I-positive set and π : 2ω → B is a continuous one-to-one
reduction of E2 to E2 B then the transported measure µ̂ on B must vanish
on the sets in the ideal I as well.
For the preservation of Baire category, it is enough to argue that every Borel
I-positive set B ⊂ 2ω there is a Polish topology on B generating the same Borel
structure as the usual one, such that all sets in the ideal I are meager in this
topology. Then, a reference to Kuratowski-Ulam theorem and/or [67, Corollary
3.5.8] will finish the argument. Just as in the treatment of Lebesgue measure,
it is enough to show
Claim 5.6.10. The ideal I consists of meager sets.
The proof of the claim is the same as in the measure case, given that Steinhaus theorem holds for Baire category too.
Proof of Theorem 5.6.4. One can argue directly by a demanding fusion argument via the separation property of the poset PI , which was our original way
of dealing with the challenge. There is a simple proof that uses the existence
140
CHAPTER 5. CLASSES OF σ-IDEALS
of a forcing with E2 in the spectrum and total canonization for equivalences
classifiable by countable structures.
Let B ⊂ 2ω be a Borel I-positive set and E an equivalence on B that is
classifiable by countable structures. Fix a Borel function f with domain B
reducing E to isomorphism of countable structures. Fix a function g : 2ω → B
reducing E2 to E2 B. Fix a σ-ideal J on a compact space Y such that J has
total canonization for equivalences classifiable by countable structures and E2
is in the spectrum of J as witnessed by a Borel equivalence F on Y , reduced to
E2 by a Borel function h : Y → 2ω –see Theorem 5.14.4.
Consider the equivalence relation G on Y defined by y0 G y1 if the structures
f gh(y0 ) and f gh(y1 ) are isomorphic. Use the total canonization feature of the
σ-ideal J to find a Borel J-positive set D ⊂ Y such that the function g ◦ h is
one-to-one and the equivalence relation G D is either equal to identity or to
D2 . The set C = gh00 D ⊂ B is a one-to-one image of a Borel set and therefore
Borel. It is also I-positive, since E2 Borel reduces to F D which Borel reduces
to E2 C via g ◦ h. The equivalence relation E C is either equal to identity
or to C 2 depending on the behavior of G D. The theorem follows.
Proof of Theorem ??. The first sentence follows from the analysis of the ideal
I immediately. The Borel sets in I are exactly those sets B for which E2 B is
essentially countable, and so by the E2 dichotomy 5.6.1, E2 is Borel reducible
to E2 B for every Borel I-positive set B.
For the regularity of PIE2 in PI , just notice that E2 -saturations of analytic
I-small sets are I-small by Claim 5.6.6, and then apply Corollary 3.5.5. For the
ℵ0 -distributivity, first verify that E2 is I-dense and then apply Proposition 3.5.7.
To see the I-density, note that the fusion process from the previous proofs starts
with an arbitrary analytic I-positive set B and inscribes into it a compact Ipositive set C ⊂ B such that the equivalence class of any point in C is dense in
C.
To show that the poset PIE generates the V [ẋgen ]E2 model, first verify the
I-homogeneity of the equivalence relation E2 and then apply Theorem 3.5.9.
For the I-homogeneity, just observe that if B, C ⊂ 2ω are analytic I-positive
sets with the same E2 -saturation, then one can thin B if necessary to get a
function f : B → C whose graph is a subset of E2 . Then, Claim 5.6.6 shows
that the function f transports I-positive sets to I-positive sets, as required in
the definition of I-homogeneity.
Theorem 5.6.11. There is a coding function for the σ-ideal I.
Proof. The target space for the coding function will be R+ . Let f (x, y) =
Σ{1/(n + 1) : x(n) 6= y(n)} if the indicated sum is finite, and let f (x, y) =trash
otherwise. This function works.
For suppose that B ⊂ 2ω is a Borel I-positive set. A closer look at the
fusion argument in Theorem 5.6.2 shows that it yields numbers {ni : i ∈ ω} and
a function g : 2<ω → 2<ω such that
• 0 = n0 < n1 < n2 < . . . , for every i ∈ ω and every t ∈ 2i g(t) ∈ 2ni , and
t ⊂ s ∈ 2<ω implies g(t) ⊂ g(s);
5.7. PRODUCTS OF PERFECT SETS
141
• for every i ∈ ω and every s, t ∈ 2i+1 , write ε(s, t) = Σ{ /1 n + 1 : ni ≤ n <
ni+1 , g(s) 6= g(t)}. It is the case that |ε(s, t) − 1i | ≤ 2−i−2 whenever the
last entries on s, t differ, and ε(s, t) ≤ 2−i−2 if the last entries of s, t are
the same;
S
• for every x ∈ 2ω , g(x) = i g(x i) ∈ B.
Now suppose that r ∈ R is a positive real number. By induction on m ∈ ω
build numbers im ∈ ω and binary sequences sm , tm ∈ 2im so that
I1 sm ⊂ sm+1 , tm ⊂ tm+1 ;
I2 2−im−1 < r − d(g(sm ), g(tm )) < 2−im , except for 0 where d(g(s0 ), g(t0 )) <
r − 2−i0 .
In the end, let x = limm g(sm ) and y = limm g(tm ); it immediately follows from
I2 that d(x, y) = r. The induction starts with choosing i0 so that r > 2−i0
and letting s0 = t0 be any binary sequence of length i0 . Suppose that sm , tm
have been successfully found. Find infinite binary sequences z0 , z1 extending
sm such that d(z0 , z1 ) = r − 2−im −1 − d(g(sm ), g(tm )). Let z2 be an infinite
binary sequence extending tm such that for every j > im , z2 (m) = z1 (m).
It follows from the second property of the function g above that r − 2−im <
d(g(z0 ), g(z2 )) < r. Find im+1 so large that r − 2−im < d(g(z0 im+1 ), g(z2 im+1 )) < r − 2−im+1 , let sm+1 = z0 im+1 , tm+1 = z2 im+1 , and continue to
the next step of the induction!
Corollary 5.6.12. If E2 is in the spectrum of a σ-ideal J then there is a coding
function for J.
Proof. Let X be the Polish space on which J lives. Let B ⊂ X be a Borel
J-positive set such that there is an equivalence relation E on B Borel reducible
via some function g : B → 2ω to E2 , such that for every J-positive Borel set
C ⊂ B, E2 ≤B E C. Let h : (2ω )2 → 2ω be a coding function for the E2 -ideal
as found in Theorem 5.6.11; we claim that h ◦ g is a coding function for J.
Indeed, if C ⊂ B is a Borel J-positive set then g 00 C ⊂ 2ω is an analytic set on
which the equivalence relation E2 is not reducible to a countable one, therefore
the set g 00 C is positive in the sense of the E2 -ideal and the set h00 (g 00 C)2 has
nonempty interior. The corollary follows.
5.7
Products of perfect sets
In this section, we will provide several canonization and anticanonization results
for infinite products of perfect sets on the underlying Polish space X = (2ω )ω .
It is fairly clear that the equivalence relation E1 maintains its complexity on
every such infinite product. On the other hand, equivalence relations far away
from E1 –such as the equivalence relations classifiable by countable structures–do
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CHAPTER 5. CLASSES OF σ-IDEALS
simplify all the way to smooth by passing to a suitable product of perfect sets.
The whole situation is closely related to the forcing properties of the infinite
product of Sacks forcing.
The infinite product of Sacks forcing is the poset P of all ω-sequences p of
perfect binary trees, ordered by coordinatewise inclusion. The computation of
the associated ideal yields a complete information. For a condition p ∈ P write
[p] = {x ∈ X : for all n ∈ ω, x(n) is a branch of the tree p(n)}. Let I be the
collection of those Borel subsets of X = (2ω )ω which do not contain a subset
of the form [p] for some p ∈ P . The following Fact is a conjunction of the
rectangular property of Sacks forcing [67, Theorem 5.2.6] and [67, Proposition
2.1.6].
Fact 5.7.1. I is a Π11 on Σ11 σ-ideal of Borel sets. Every positive analytic set
contains a Borel positive subset.
In particular, the map p 7→ [p] is an isomorphism between P and a dense subset
of the poset PI . The forcing properties of the poset P are well known.
Fact 5.7.2. The poset P is proper, bounding, preserves Baire category and outer
Lebesgue measure, and it adds no independent reals.
The spectrum of the ideal I is very complicated. Already the understanding
of smooth equivalence relations (i.e. names for reals) seems to be out of reach.
We will include several basic results. First, consider the obvious equivalence E1
on X.
Theorem 5.7.3. E1 is in the spectrum of I. Moreover, the poset PIE1 is regular
in PI , it is ℵ0 -distributive, and its extension is equal to V [ẋgen ]E1 .
It can be shown that the reduced product PI ×E1 PI is proper, and it adds a
dominating real. The model V [[ẋgen ]]E1 is equal to V [ẋgen ]E1 .
Proof. To see that E1 is indeed in the spectrum, if B ⊂ X is an I-positive
set, use Fact 5.7.1 to find a sequence p ∈ P of perfect trees such that [p] ⊂ B
and let πn : 2ω → [p(n)] be a homeomorphism for every number n. Then
π : x 7→ hπn ~x(n) : n ∈ ωi is a Borel reduction of E1 to E1 B.
To prove that PIE1 is a regular ℵ0 -distributive subposet of PI , we will argue
that every set of the form [p] for p ∈ P is an E1 -kernel and then apply Theorem 3.5.4. Suppose that B ⊂ [p] is an analytic set such that [B]E1 ∈
/ I; we must
conclude that [B]E1 ∩ [p] ∈
/ I. Just find a condition q ∈ P so that [q] ⊂ [B]E1 ,
and use the σ-additivity of the σ-ideal I to thin out the trees on the sequence q if
necessary and find a number n such that ∀x ∈ [q] ∃y ∈ B x (n, ω) = y (n, ω).
But then, for every number i > n it must be the case that q(i) ⊂ p(i). Thus,
the set [B]E1 ∩ [p] contains the product Πi≤n p(i) × Πi>n q(i), and it is I-positive.
To show that the poset PIE1 yields the model V [ẋgen ]E1 , we must show that
for analytic I-positive sets B, C ⊂ X, either there is I-positive Borel B 0 ⊂ B
such that [B 0 ]E1 ∩ C ∈ I, or there is an I-positive Borel B 0 ⊂ B and an Ipreserving Borel injection f : B 0 → C with f ⊂ E1 . After that, Theorem 3.5.9
completes the proof.
5.7. PRODUCTS OF PERFECT SETS
143
Thus, fix the sets B, C and find a condition p ∈ P with [p] ⊂ B. If [p]E1 ∩C ∈
I, then we are done, otherwise find a condition q ∈ P with [q] ⊂ [p]E1 ∩C. There
must be a number n0 ∈ ω such that ∀n ≥ n0 q(n) ⊂ p(n). Consider the set
B 0 = Πn∈n0 [p(n)] × Πn≥n0 [q(n)], fix homeomorphisms πn : [p(n)] → [q(n)] for
every n ∈ n0 , and consider the map f : B 0 → C defined by f (x)(n) = πn (x)(n)
if n ∈ n0 and f (x)(n) = x(n) for n ≥ n0 . It is not difficult to see that the
function f is a Borel injection with the required properties. As an interesting
aside, note that in V [G], [ẋgen ]E1 is the only equivalence class contained in all
sets in the filter K and so K and [ẋgen ]E1 are interdefinable elements. This is to
say, in the ground model, if B ⊂ X is an I-positive Borel set and f : B → X is
a Borel function such that ∀x ∈ B ¬f (x) E1 x, then there is an I-positive Borel
set C ⊂ B such that [C]E1 ∩ f 00 C = 0. Since the poset PI is bounding, there are
increasing functions g, h ∈ ω ω and an I-positive Borel set C ⊂ B such that for
every point x ∈ C and every number i ∈ ω there are numbers n ∈ [g(i), g(i + 1))
and k ∈ h(i) such that f (x)(n)(k) 6= x(n)(k). Thinning the set C further if
necessary we may assume that C = [p] for some condition p ∈ P such that
whenever n ∈ [g(i), g(i + 1)) then the first splitnode of the tree p(n) is past the
level h(i). It is not difficult to see that such a set C ⊂ B works as desired.
Theorem 5.7.4. The spectrum of I is cofinal among the Borel equivalence
relations under ≤B . It contains EKσ .
Proof. Let J be a Borel ideal on ω, then the equivalence relation EJ , the modulo
J equality of sequences in (2ω )ω is a Borel equivalence, it is in the spectrum of
the ideal I for the same reason as E1 , and the equalities of this type are cofinal
in ≤B by Fact 2.2.12. There is an Fσ ideal J such that equality modulo J is
bireducible with EKσ , and so EKσ is in the spectrum of I as well.
The equivalences from the previous proof are never reducible to orbit equivalences of Polish group actions, which leaves open the possibility of a strong
canonization theorem for I and orbit equivalences. The following two canonization theorems should be viewed as important stepstones to that so far unattained
goal.
First, set up a fixed framework for fusion arguments. Let p ∈ P . A fusion
sequence below p is a sequence hun , pn : n ∈ ωi such that un is a sequence of
length n whose entries are finite binary trees with n splitting levels, pn is a
condition in P below p, and for every m ∈ ω the tree pn (m) end-extends un (m).
Moreover, we require that pn+1 ≤ pn and for every m ∈ n, the tree un+1 (m)
end-extends the tree un (m). It is clear that the conditions {pn : n ∈ ω} in a
fusion sequence S
share a lower bound, the condition q ∈ P such that for every
m ∈ ω, q(m) = m<n un (m). In this context, we will write [un ] for the set of
all sequences t of length n such that t(m) is an endnode of un (m). For every
t ∈ [un ] the condition pn t is defined as that sequence r such that for all m ∈ n,
r(m) =the set of all nodes of pn (m) compatible with t(m), and for all m ≥ n,
r(m) = pn (m). For every x ∈ [pn ] let x un be the unique element of un ] such
that x ∈ pn t.
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CHAPTER 5. CLASSES OF σ-IDEALS
Theorem 5.7.5. Equivalences classifiable by countable structures→I smooth.
In other words, for every equivalence relation on X classifiable by countable
structures there is an infinite product of perfect sets on which E is smooth.
Proof. We will start with a simple claim.
Claim 5.7.6. Let p ∈ P be a condition and {f, gm : m ∈ ω} be Borel functions
from [p] to 2ω . At least one of the following two options holds:
1. either there is a condition q ≤ p such that for every x, y ∈ [q], ∀n >
0 x(n) = y(n) implies f (x) = f (y);
2. or there is a condition q ≤ p such that for every x, y ∈ [q], x(0) 6= y(0)
implies f (x) 6= gm (y) for every m ∈ ω.
Proof. Thinning the condition p if necessary, we may assume that the functions
f, g are continuous. Suppose that the first item fails. Build a fusion sequence
hun , pn : n ∈ ωi below p such that for every n ∈ ω and every t, s ∈ [un ], if
00
t(0) 6= s(0) then f 00 [pn t] ∩ gm
[pn s] = 0 for all m ∈ ω. It is clear that in the
end, the lower bound q of the fusion sequence will be the condition required in
the second item.
To perform the induction step, find any sequence un+1 of finite trees below
pn that can serve as the next entry of the fusion sequence. We must show how
to thin out pn to pn+1 and preserve the induction hypothesis.
Suppose that t, s ∈ [un+1 ] are two sequences such that ~t(0) 6= ~s(0) and
choose a number m ∈ n. Use the assumption to find points x0 , x1 ∈ [pn ~t]
which differ at the first coordinate only, such that f (x0 ) 6= f (x1 ); perhaps
f (x0 )(i) = 0 while f (x1 )(i) = 1 for some i ∈ ω. Use the continuity of the
function f to find finite sequences t̂0 , t̂1 of the same length that differ only at
0-th coordinate, x0 ∈ [pn t̂0 ], every x ∈ [pn t̂0 ] has f (x)(i) = 0, and similarly
for subscript 1. Consider the sequence ŝ which is defined by ŝ(j) = s(j) if
t(j) 6= s(j) and ŝ(j) = t̂(j) otherwise; so ŝ ≥ s. The continuity of the function g
on [pn ] shows that strengthening ŝ if necessary, it must be the case that either
gm (x)(i) = 0 for all y ∈ [pn ŝ], or gm (x)(i) = 1 for all y ∈ [pn ŝ]. Suppose the
former case holds. Thinning out pn below s so that pn s = pn ŝ and below t
so that pn t ≤ pn t̂1 (which is possible to do simultaneously) we achieve that
00
[pn s] = 0.
f 00 [pn t] ∩ gm
The condition pn+1 ≤ pn is now obtained by repeating the thinning procedure in the previous paragraph for all pairs of sequences t, s ∈ [un+1 ] which
differ at the first coordinate and all numbers m ∈ n. The induction hypothesis
continues to hold.
The canonization of equivalences classifiable by countable structures is achieved
via the separation property of Definition 4.3.6 and Theorem 4.3.7. To prove the
separation property, suppose that p ∈ P is a condition forcing ż ∈ 2ω and
ȧ ⊂ 2ω is a countable set not containing ż. We must produce a condition q ≤ p
and a compact set C ⊂ 2ω such that q ż ∈ Ċ and ȧ ∩ Ċ = 0. Thinning out
the condition p if necessary, we can find continuous functions f, gn : [p] → 2ω
5.7. PRODUCTS OF PERFECT SETS
145
for n ∈ ω such that p ż = f˙(ẋgen ) and ȧ = {ġn (ẋgen ) : n ∈ S
ω}. It will be
enough to find a condition q ≤ p such that f 00 [q] is disjoint from n gn00 [q]; then,
q together with the set C = f 00 [q] will have the required properties.
To find the condition q, build a fusion sequence hun , pn : n ∈ ωi below p
such that
I1 for every t ∈ [un ], if there is a condition r ≤ pn+1 such that r t 6= 0 and
the value of f (x) does not depend on x(n) for x ∈ [r t], then pn+1 is
such a condition;
I2 for every t ∈ [un ], if there is a condition r ≤ pn+1 such that ∀s ∈ [un ] r s 6= 0 and for all x ∈ [r t] and every y ∈ [r], x(n) 6= y(n) implies
f (x) 6= fm (y) for all m ∈ ω, then pn+1 is such a condition.
The induction process is trivial. Let q ∈ P be a lower bound of the fusion
sequence, and let x, y ∈ [q]. We must prove that f (x) 6= gm (y) for all m ∈ ω.
There are two distinct cases.
Case 1. There is no number n ∈ ω such that, writing t = x un , it is
the case that x(n) 6= y(n) and the assumption of item (I1) fails for t and n. In
this case, write zn ∈ [q] for the sequence obtained from y by replacing its first
n entries with the corresponding entries of x and argue that the value f (zn )
does not depend on n. To show that f (zn ) = f (zn+1 ), discern two subcases. If
x(n) = y(n) then zn = zn+1 and f (zn ) = f (zn+1 ) certainly holds. On the other
hand, if x(n) 6= y(n), the case assumption shows that the assumption of item
(I1) holds and implies exactly that f (zn ) = f (zn+1 ). Now note that z0 = y and
limn zn = x. Since the function f is continuous, it follows that f (y) = f (x) and
f (x) = f (y) 6= gm (y) for all m ∈ ω by the initial assumptions on the functions
f, gm : m ∈ ω.
Case 2. If Case 1 fails then there must be the witnessing number n. Write
t = x un . Since the assumption of item (I1) fails for t, Claim 5.7.6 shows
that the assumption of item (I2) holds. Since x(n) 6= y(n), we conclude that
f (x) 6= gm (y) for all m ∈ ω as required. The proof is complete.
Theorem 5.7.7. Whenever J is an analytic P-ideal on ω, then ≤B =J →I smooth.
In other words, for every equivalence relation E on X Borel reducible to =J there
is an infinite product of perfect sets such that E is smooth on it.
Proof. Use a result of Solecki [54] to find a lower semicontinuous measure φ on
ω such that J = {a ⊂ ω : limn φ(a \ n) = 0}. The proof of the following claim
is parallel to Claim 5.7.6:
Claim 5.7.8. Let p ∈ P , let f, g : [p] → 2ω be Borel functions and let a ⊂ ω be
a finite set. At least one of the following happens:
1. either either for every ε > 0 there is a condition q ≤ p such that for all
x, y ∈ [q] with ∀n ∈
/ a x(n) = y(n) it is the case that φ(f (x)∆f (y) < ε;
2. or there is a condition q ≤ p and ε > 0 such that for every pair x, y ∈ [q]
such that ∀n ∈ a x(n) 6= y(n) it is the case that ¬f (x) =J f (y).
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CHAPTER 5. CLASSES OF σ-IDEALS
Let p ∈ P and let f : [p] → 2ω be a Borel function reducing E [p] to
=J . Thinning out the condition p we may assume that the function f is in
fact continuous. We will construct a fusion sequence hun , pn : n ∈ ωi below p
subject to a fairly nasty induction hypothesis. The aim of the construction is to
find equivalence relations Fn on [un ] for every n ∈ ω such that E restricted to
the lower bound
Q q of the fusion sequence is Borel
Q reducible to the equivalence
relation F = n Fn modulo finite on the set n [un ]. The equivalence relation
is a modulo finite product of equivalence relations on finite sets, and as such it
is Borel reducible to E0 . The previous theorem can then be applied to show
that thinning out q further if necessary, E q is smooth. The reduction of
E q to FQwill be attained through the natural continuous one-to-one function
g : [q] → n [un ] defined by g(x)(n) = x un
It is time to expose the induction hypothesis on the fusion sequence. For
every n ∈ ω there will be a number kn > n and a real number εn > 0 so that
I1 the numbers kn are increasing, the numbers εn decrease towards 0 with
εn+1 < εn /8;
I2 for every pair s, t ∈ [un ], writing a = {m ∈ n : t(m) = s(m)}, either (a)
for all x ∈ pn t and y ∈ pn s such that x a ∪ [n, ω) = y a ∪ [n, ω)
x E y fails, or (b) for all such x, y it is the case that x E y holds;
I3 for every pair s, t ∈ [un ] such that clause (I2)(a) holds, for all x, y as above
lim supk φ(f (x)∆f (y) \ k) > εn holds;
I4 for every pair s, t ∈ [un ] such that clause (I2)(b) holds, for all such x, y
φ(f (x)∆f (y) \ kn ) < εn /8 holds;
I5 for all t ∈ [un ] and all a ⊂ n, if there is a condition r ≤ pn t such that for
every x, y ∈ [r], x a ∪ [n, ω) = y a ∪ [n, ω) implies φ(f (x)∆f (y)) < εn /8
then pn t is such a condition;
I6 for all t ∈ [un ] and all a ⊂ n, if there is a condition r ≤ pn such that
for all s ∈ [un ], r s 6= 0 and such that for every x ∈ [r t], y ∈ [r],
∀m ∈ n \ a x(m) 6= y(m) implies ¬x E y, then pn t is such a condition;
I7 for all t ∈ [un ] and all a ⊂ n, if there is a condition r ≤ pn+1 t such
that for every x, y ∈ [r], x a ∪ [n + 1, ω) = y a ∪ [n + 1, ω) implies
φ(f (x)∆f (y)) < εn /8 then pn+1 t is such a condition;
I8 for all t ∈ [un ] and all a ⊂ n, if there is a condition r ≤ pn+1 such that
for all s ∈ [un ], r s 6= 0 and such that for every x ∈ [r t], y ∈ [r],
∀m ∈ (n + 1) \ a x(m) 6= y(m) implies ¬x E y, then pn+1 t is such a
condition.
While the list of induction hypotheses may sound daunting, it is in fact very
easy to get. Suppose that un , pn , kn , εn have been obtained. First, strengthen
pn to pn+1 ≤ pn satisfying the last two items. Let un+1 to be the sequence of
the finite trees recording pn+1 (m) up to n + 1-st splitting level, for m ∈ n + 1.
5.7. PRODUCTS OF PERFECT SETS
147
Now run through the list of all pairs s, t ∈ [un+1 ], using the σ-additivity of
the σ-ideal I each time to gradually strengthen pn+1 so as to satisfy item (I2).
Run through the list again, gradually thinning out pn+1 again and using the
σ-additivity of the σ-ideal to find real numbers εn+1 (s, t) > 0 such that if
I2(a) holds for s, t then item (I3) is satisfied with εn+1 (s, t) in place of εn . Let
εn+1 > 0 be some real number smaller than all the numbers thus obtained. Run
through the list again, thinning out pn+1 and using σ-additivity of the ideal I
again to find numbers kn+1 (s, t) such that if the claus (I2)(b) holds for s, t then
(I4) is satisfied with kn+1 (s, t) in place of kn . Let kn+1 be some number larger
than all numbers thus obtained. Finally, run through the list of all t ∈ [un+1 ]
and a ⊂ n + 1, gradually strengthening the condition pn+1 again so as to satisfy
(I5) and (I6). This completes the induction step.
Now suppose that the fusion has been performed. For each n ∈ ω, let Fn be
the relation on [un ] defined by s Fn t if (I2)(b) is satisfied for s, t. A brief check
is in order:
Claim 5.7.9. Fn is an equivalence relation.
Proof. The only nontrivial point is the transitivity. Let t0 Fn t1 Fn t2 ; we want
to show that t0 Fn t2 . Write a01 = {m ∈ n : t0 (m) = t1 (m) and similarly for
a02 and a12 . Choose points x0 ∈ [pn t0 ], x1 ∈ [pn t1 ] and x2 ∈ [pn t2 ] such
that x0 a01 ∪ [n, ω) = x1 a01 ∪ [n, ω) and x1 a12 ∪ [n, ω) = x2 a12 ∪ [n, ω).
By the assumptions, we have x0 E x1 E x2 . Now amend x0 on a02 \ a01 so
that x0 = x2 on this set; the assumptions of case I2(a) are still satisfied and
show that x0 E x1 . Similarly, amend x2 on a02 \ a12 so that x0 = x2 on this
set; it will be still the case that x1 E x2 . In conclusion, x0 E x2 , and since
x0 a02 ∪ [n, ω) = x2 a02 ∪ [n, ω), the pair x0 , x2 contradicts I2(a) for t0 , t2 .
It follows that I2(b) for t0 , t2 must occur, in other words x0 Fn x1 holds.
Let q be a lower bound of the fusion sequence and suppose x, y ∈ [q]; we
must show that x E y if and only if g(x) F g(y). There are several cases:
Case 1. There is a number n such that, writing t = g(x)(n) and a = {m ∈
n : x(m) = y(m)}, the assumption of case (I5) of the induction hypothesis fails
for t, a. In this case, we will conclude that both x E y and x F y fail. Indeed,
Claim 5.7.8 shows that then the assumption of (I6) is satisfied and therefore
x E y must fail. Also, for every sufficiently large number n̄ ∈ ω, writing
ā = {m ∈ n̄ : g(x)(n̄)(m) = g(y)(n̄)(m)}, it will be the case that (n \ a) ∩ ā = 0,
and so the clause (I2)(a) is satisfied for t̄ = g(x)(n̄) and s̄ = g(y)(n̄) by the
conclusion of (I6) at n, t, a. Thus, both g(x)(n̄) Fn̄ g(y)(n̄) and x F y must fail.
Case 2. Same as case 1, except about item (I7) of the induction hypothesis.
The argument and the conclusion are the same as in Case 1.
Case 3. If both Case 1 and Case 2 fail, then for every n let zn ∈ [q]
be the sequence obtained by replacing the first n entries of x with the first n
entries of y, and observe that φ(f (zn )∆f (zn+1 )) < εn /2. To see this, discern
two cases: if x(n) = y(n) then zn = zn+1 and the inequality certainly holds.
If x(n) 6= y(n) then, since Case 2 failed at n, it must be that item (I7) of the
induction hypothesis occurred for g(y)(n) and the inequality again follows. Note
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CHAPTER 5. CLASSES OF σ-IDEALS
that since limn zn = y, the function f is continuous, and the submeasure φ is
lower semicontinuous, it follows that φ(f (zn )∆f (y)) < Σm≥n εm /2 < εn .
We will now show that if g(x)(n) Fn g(y)(n) for infinitely many n, then
x E y, and if ¬g(x)(n) Fn g(y)(n) for some n then ¬x E y. This will prove the
reduction property of the function g on the pair x, y.
Case 3a. Suppose first that n is such that t = g(x)(n) Fn g(y)(n) = s.
Writing a = {m ∈ n : x(m) = y(m)} and b = {m ∈ n : g(x)(m) = g(y)(m)}, it
is obvious that a ⊆ b. Write zn0 for the sequence obtained from x by replacing the
entries in b\a with the corresponding entries of y. Since Case 1 fails, item (I5) of
the induction hypothesis must hold for t, a, and we have φ(f (x)∆f (zn0 )) < εn /4.
Since t Fn s, clause (I2)(b) shows that φ((f (zn0 )∆f (zn )) \ kn ) < εn /4. Moving
from f (x) to f (zn0 ) to f (zn ) to f (y), we have that φ((f (x)∆f (y)) \ kn < εn /4 +
εn /4 + εn /2 = εn . Now, if there are infinitely many numbers n such that
g(x)(n) Fn g(y)(n), we conclude that f (x) =J f (y) and x E y.
Case 3b. Suppose second that n is such that t = g(x)(n) Fn g(y) = s fails.
As in the previous subcase, write a = {m ∈ n : x(m) = y(m)} and b = {m ∈ n :
t(m) = s(m)}, and let zn0 ∈ [q] be the sequence obtained from x by replacing its
entries in the set b \ a with the corresponding entries of y. Since ¬t Fn s holds,
item (I3) shows that lim supk φ((f (zn0 )∆f (zn ) \ k) > εn . However, just as in the
previous subcase, we have φ(f (x)∆f (zn0 )) < εn /4 and φ(f (y)∆f (zn )) < εn /4 as
well. As φ is a submeasure, we conclude that lim supk φ((f (x)∆f (y)\k) > εn /2,
in particular ¬x E y.
The results of this section obviously suggest a positive answer to the following
question, which nevertheless remains open.
Question 5.7.10. Is it true that for every equivalence relation E on (2ω )ω ,
reducible to the orbit equivalence of a Polish group action, there are perfect
sets Pn ⊂ 2ω for each n ∈ ω such that E restricted to Πn Pn is smooth?
5.8
Silver cubes
In this section, we will consider the behavior of equivalence relations on Silver
cubes. Recall that Silver forcing is the poset P whose elements p are just partial
functions from ω to 2 with coinfinite domain. The ordering is that of reverse
inclusion. For p ∈ P , write [p] for the set of all totalizations of the function p;
thus [p] ⊂ 2ω is a compact set. A Silver cube is a set of the form [p] for some
p ∈ P.
The calculation of the associated σ-ideal gives complete information. Let
X = 2ω and let G be the Borel graph on X given by xGy if and only if there
is exactly one n such that x(n) 6= y(n). A set B ⊂ X is G-independent if no
two elements of B are G-connected. Let I be the σ-ideal generated by Borel
G-independent sets. We have
Theorem 5.8.1.
1. Whenever A ⊂ X is an analytic set, either A ∈ I or A
contains a Silver cube, and these two options are mutually exclusive;
5.8. SILVER CUBES
149
2. the ideal I is Π11 on Σ11 ;
3. the ideal I does not have the selection property;
4. the forcing P is proper, bounding, preserves outer Lebesgue measure and
Baire category, and adds an independent real.
Proof. (1) comes from [66, Lemma 2.3.37]. (2) follows from (1) and [67, Theorem
3.8.9], or one can compute directly. (4) is well-known; the independent real ȧ ⊂
ω is read off the generic real ẋgen ∈ 2ω as a = {n ∈ ω : {m ∈ n : ẋgen (m) = 1}
has even size}. Finally, the failure of the selection property is witnessed by the
set D ⊂ 2ω × X given by hy, xi ∈ D if x(3n) = x(3n + 1) = y(n).
The theorem shows that the map p 7→ [p] is an isomorphism between P and a
dense subset of PI . The forcing properties of Silver forcing are well-known.
In the following proofs, we will repeatedly use fusion type arguments, and
it will be useful to establish a fixed framework for them. A fusion sequence
below a condition p ∈ P is a sequence hin , pn i such that p = p0 ≥ p1 ≥ . . .
and him : m ∈ ni is an increasing enumeration of the first n many points of
ω \ dom(pn+1 ). It is clear S
that the conditions in a fusion sequence have a lower
bound, the condition q = n pn .
The canonization features of Silver cubes start with the total canonization
of smooth equivalence relations. This has been proved in a different language
by Grigorieff [15, Corollary 5.5] in a rather indirect fashion. We provide a direct
argument containing features useful later.
Theorem 5.8.2. The ideal I has total canonization for smooth equivalences.
Restated, every smooth equivalence on a Silver cube is equal to either the identity
or the full equivalence relation on a Silver subcube.
In other words, Silver forcing adds a minimal real degree.
Proof. The argument uses a claim of independent interest. Recall that if x ∈ 2ω
is an infinite binary sequence and m ∈ ω is a number, then x flip m is the
binary sequence obtained from x by flipping its m-th bit.
Claim 5.8.3. Whenever p ∈ P and {fj : j ∈ k} is a finite collection of continuous functions from p to 2ω such that preimages of singletons are I-small, then
there is an arbitrarily large number m ∈ ω and a cube q ≤ p such that for every
x ∈ [q], both x, x flip m ∈ [p] and fi (x) 6= fi (x flip m) for all i ∈ k.
Proof. This is proved by induction on k. The case k = 0 is trivial, since the
collection of continuous functions in question is empty. Suppose that the claim
has been established for some k and fails for some {fj : j ∈ k + 1} and p ∈ P .
We will reach a contradiction by producing a cube q ≤ p such that fk is constant
on [q].
Build a fusion sequence hin , pn i below p so that for every n ∈ ω, the value
of fk (x) does not depend on x an+1 for points x ∈ [pn+1 ], where an+1 = {il :
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CHAPTER 5. CLASSES OF σ-IDEALS
l ≤ n}. The continuity of the function fk implies that fk [q] must then be
constant for any lower bound q of the fusion sequence.
The fusion sequence is built by induction on n. Suppose that pn and in−1
have been found. By the induction hypothesis at k, there is a subcube p0n ≤
pn and a number in > in−1 such that for every x ∈ [p0n ] it is the case that
x flip in ∈ [pn ] and for all j ∈ k it is the case that fj (in ) 6= fj (in+1 ). The set
A = {x ∈ [p0n ] : fk (x) 6= fk (x flip in+1 )} is relatively open in [p0n ]. If this set
is nonempty, then it contains a Silver cube; however, this cannot happen since
the claim is supposed to fail for {fj : j ∈ k + 1} and p. Ergo, A = 0. Now let
pn+1 ∈ P be the function obtained from p0n by omitting the numbers i0 , . . . in
from the domain of p0n .
We claim that for x ∈ [pn+1 ], the value of fk (x) does not depend on x an .
To see this, argue that fk (x) = fk (y) for every y ∈ [p0n ] such that x agrees with
y everywhere off the set an+1 . Indeed, first amend y to y 0 by flipping the entry
y(in ) if necessary to get an agreement with x(in ) if necessary. By the definitions,
we still have f (y 0 ) = f (y) and since x, y 0 ∈ [pn ] and x, y 0 agree everywhere off an ,
the induction hypothesis on the fusion sequence shows that f (x) = f (y 0 ) = f (y)
as desired. This completes the induction step in the construction of the fusion
sequence and the proof.
Now suppose that B ⊂ X is an I-positive Borel set and f : B → 2ω is
a Borel function with I-small preimages of singletons; we must produce an Ipositive Borel subset of B on which f is one-to-one. Use the bounding property
of Silver forcing and Fact 2.6.3 to get a Silver cube p ∈ P such that [p] ⊂ B
and f [p] is continuous. Build a fusion sequence hin , pn i below pn such that,
writing an = {il : l ∈ n}, every two points x, y ∈ pn+1 that differ on the set
an also have distinct f -images. It is clear that for any lower bound q of such a
fusion sequence, the function f [q] is injective as required.
To perform the inductive step of the construction, suppose that in−1 , pn have
been found. For every h ∈ 2an let fh : [pn ] → 2ω be the function defined by
fh (x) = f (x rew h) where xh ∈ pn is the point obtained from x by rewriting the
entries of x an with h. These are continuous functions with I-small preimages
of singletons. By the claim, there is a cube p0n and a number in > in−1 such
that fh (x) 6= fh (x flip in ) for all h ∈ 2an and all x ∈ [p0n ]. By the continuity
of the functions fh , passing to a clopen subcube of p0n if necessary we can find
numbers mh ∈ ω and bits bh 6= ch such that for all x ∈ p0n , fh (x)(mh ) = bh and
fh (x flip in )(mh ) = ch . Now let pn+1 be the subcube of pn obtained from p0n
by omitting the numbers i0 , . . . in from its domain.
To verify the properties of the cube pn+1 , suppose that x, y ∈ [pn+1 ] are
points that differ on the set an+1 ; we must show that they have different functional values. If they differ already on the set an then we are done by the
induction hypothesis. If x an = y an = h for some h ∈ 2an , then it must be
the case that x(in ) 6= y(in ). Find points x0 , y 0 ∈ [p0n ] such that x = x0 rew h and
y = y 0 rew h and observe that f (x)(mh ) = fh (x0 )(mh ) 6= fh (x0 flip in )(mh ) =
fh (y 0 )(mh ) = f (y)(mh ). This completes the induction step and the proof.
5.8. SILVER CUBES
151
At the level of hyperfinite equivalence relations we hit the first nontrivial
feature of the spectrum of Silver forcing:
Theorem 5.8.4. E0 belongs to the spectrum of the Silver forcing. The poset
PIE0 is regularly embedded in PI , it is ℵ0 -distributive, and it yields the model
V [ẋgen ]E0 of Definition 3.4.1.
The decomposition of Silver forcing into PIE0 and the remainder forcing
obtained from this theorem and Theorem 4.2.1 is different from the widely used
Grigorieff decomposition of [15].
Proof. Look at the equivalence relation E0 on X. Whenever p ∈ P is a Silver
cube and π : ω → ω \ dom(p) is a bijection then the map g : X → [p] defined by
g(x) = f ∪ (x ◦ π −1 ) is a continuous reduction of E0 to E0 [p], and therefore
E0 is in the spectrum. The remainder of the theorem is a consequence of
Theorem 4.2.1 as soon as we prove that E0 -saturations of sets in I are still in
I. The easiest way to see that is to consider the action of the countable group
of finite subsets of ω with symmetric difference on the space X where a finite
set acts on an infinite binary sequence by flipping all entries of the sequence on
the set. This action generates E0 as its orbit equivalence relation, and it also
preserves the graph G and with it the σ-ideal I. Thus, E0 -saturations of I-small
sets are still I-small.
As an interesting extra bit of information, the equivalence class [xgen ]E0 is
the only one in the extension V [H] which is a subset of all sets in the filter K,
where H ⊂ PI is a generic filter and K = H ∩ PIE0 . In the ground model, this
is immediately translated to the following: If p ∈ P and f : [p] → X is a Borel
function such that ∀x ∈ [p] ¬f (x) E0 x, then there is a subcube q ≤ p such that
f 00 [q] ∩ [q]E0 = 0.
Since the forcing PI is bounding, there must be a partition of ω into a
collection {In : n ∈ ω} of intervals and a cube q ≤ p such that for every x ∈ C
and every n ∈ ω there is a number m ∈ In with f (x)(m) 6= x(m), and the
function f [q] is continuous. Thin out the cube q if necessary so that infinitely
many of the intervals do not intersect the set ω \ dom(q). The cube [q] works
as desired.
There are many nonsmooth subequivalences of E0 that do not canonize to
E0 or the identity on any Silver cube. This is in contradistinction to the features
of the smooth ideal or the weak Milliken cubes. The simplest such an example
is the equivalence relation E connecting x, y ∈ X if x, y differ on a finite set
of entries, and the set is of even size. There seems to be such a variety of
subequivalences of E0 that no nontrivial canonization theorem on Silver cubes
is possible for them. However, this is the only obstacle for canonization in at
least two directions:
Theorem 5.8.5. If E is an equivalence on a Silver cube classifiable by countable
structures, then there is a Silver subcube on which E ⊆ E0 or E equals to the
full equivalence relation.
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CHAPTER 5. CLASSES OF σ-IDEALS
Proof. Suppose that B ⊂ 2ω is a Borel I-positive set and E is an equivalence
on it classifiable by countable structures. Since the Silver forcing extension
contains a single minimal real degree, Theorem 4.3.8 implies that E simplifies
to an essentially countable equivalence relation on a positive Borel set. Since
the Silver forcing extension preserves Baire category, Theorem 4.2.7 shows that
E restricted to a further I-positive subset is reducible to E0 . Now we can deal
with this fairly simple situation separately.
Suppose that p ∈ P and E is an equivalence relation on [p] reducible to E0
via a Borel function f : [p] → 2ω . Passing to a subcube if necessary, we may
assume that f is continuous on [p]. Suppose that E has no I-positive equivalence
classes. We must produce a subcube q ≤ p such that E ⊆ E0 on [q].
By induction on n ∈ ω, build a fusion sequence hin , pn : n ∈ ωi so that for
every x, y ∈ [pn+1 ] such that x(in ) 6= y(in ) there is a number m > n such that
f (x)(m) 6= f (y)(m). It immediately follows that if q is a lower bound of such a
fusion sequence, then E [q] ⊆ E0 . Suppose that pn , in−1 have been successfully
found. Write an = {il : l ∈ n} and for every h ∈ 2a , let fh (x) = f (x rew h) as
in the previous proof. By the σ-additivity of the σ-ideal I, it is possible to find
a condition p0n ≤ pn such that
• for every h, k ∈ 2a , either (a) for every x ∈ [p0n ], fh (x) E0 fk (x) holds or
(b) for every x ∈ [p0n ], fh (x) E0 fk (x) fails;
• there is a number m0 > n such that if h, k are such that (a) above holds
then fh (x) = fk (x) on all entries above m0 for all x ∈ [p0n ]; and for every
h ∈ 2an , for all x ∈ [p0n ] the value of fh (x) m0 is the same;
• if h, k are such that (b) above holds then there is a number mhk > n and
distinct bits bhk 6= chk such that for all x ∈ [p0n ], fh (x)(mh k) = bhk and
fk (x)(mhk ) = chk .
Now let p00n ≤ p0n be a condition such that for some number in > in−1 ,
for every x ∈ [p00n ] and every h ∈ 2an , it is the case that x flip in ∈ [p0n ]
and fh (x) 6= fh (x flip in ). Such a subcube exists by Claim 5.8.3. Use the
continuity of the function f to pass to a relatively clopen subcube if necessary
and find numbers mh and distinct bits bh 6= ch such that for every x ∈ [p00n ] and
every h ∈ 2a it is the case that fh (x)(mh ) = bh and fh (x flip in )(mh ) = ch .
Let pn+1 be the condition obtained from p00n by removing the points {il : l ≤ n}
from its domain. We claim that pn+1 works as required.
For this, let x, y ∈ pn+1 be points such that x(in ) 6= y(in ); we must show
that f (x) and f (y) differ above n. Let h = x an and k = y an , and find
points x0 , y 0 ∈ [p00n ] such that x = x0 rew h and y = y 0 rew k; so f (x) = fh (x0 )
and f (y) = fk (y 0 ). There are two distinct cases according to whether (a) or
(b) above holds for h, k. If (b) holds then clearly fh (x0 )(mhk ) 6= fk (y 0 )(mhk )
and the conclusion follows. If, on the other hand, the clause (a) holds, then
fh (x0 )(mh ) 6= fh (x0 flip in )(mh ) = fk (x0 flip in )(mh ) = fk (y 0 )(mh ) by the
choice of the number mh . This completes the proof.
Regarding further canonization properties of Silver cubes, Michal Doucha proved
5.8. SILVER CUBES
153
Fact 5.8.6. [10] If E is an equivalence relation on a Silver cube, Borel reducible
to =J for some analytic P-ideal J, then there is a Silver subcube on which
E ⊆ E0 or E equals to the full equivalence relation.
Finally, there is a lack of canonization in the direction of general Kσ equivalence
relations, as witnessed by the following
Theorem 5.8.7. EKσ is in the spectrum of I. It is witnessed by a certain
Borel equivalence relation E on X such that the poset PIE is regular in PI , it is
ℵ0 -distributive and it yields the V [ẋgen ]E extension.
Proof. Let c : 2ω → ω ω be the continuous function defined by c(x)(n) = |{m ∈
n : x(m) = 1}| and let E be the Borel equivalence relation on 2ω connecting
x0 , x1 ∈ 2ω if the function |c(x0 ) − c(x1 )| ∈ ω ω is bounded. Clearly, the function
h reduces the equivalence relation E to EKσ . To show that EKσ ≤B E, choose
successive finite intervals In ⊂ ω of length 2n + 1 and for every function y ∈ ω ω
below the identity consider the binary sequence g(y) ∈ 2ω specified by the
demand that g(y) In begins with y(n) many 1’s, ends with n − y(n) many 1’s
and has zeroes in all other positions. It is not difficult to see that if y0 , y1 ∈ ω ω
below the identity are two functions then |c(g(y0 )) − c(g(y1 )) In | is bounded
by |y0 (n) − y1 (n)| and the bound is actually attained at the midpoint of the
interval In . Thus, the function g reduces EKσ to E. To show that E ≤ E [p]
for every p ∈ P let π : ω → ω \ dom(p) be the increasing bijection, and observe
that the map π̄ : 2ω → [p] defined by π̄(x) = p ∪ (x ◦ π −1 ) reduces E to E [p].
Thus, the equivalence relation E shows that EKσ is in the spectrum.
The following definition will be helpful for the investigation of the poset PIE .
Say that conditions p, q are n-dovetailed if between any two successive elements
of ω \ dom(p) there is exactly one element of ω \ dom(q) and min(ω \ dom(p)) <
min(ω \ dom(q)), and moreover, for every number m ∈ ω, the difference |{k ∈
m : p(k) = 1}| − |{k ∈ m : q(k) = 1| is bounded in absolute value by n. Two
conditions are dovetailed if they are n-dovetailed for some n. It is clear that if
p, q are dovetailed and f : ω \ dom(p) → ω \ dom(q) is the unique increasing
bijection, then the continuous function π : x 7→ h ∪ x ◦ f −1 from [p] to [q]
preserves the σ-ideal I, and its graph is a subset of E. Moreover, if p, q ∈ P are
arbitrary conditions and x ∈ [p] and y ∈ [q] are E-equivalent points such that
both x \ p, y \ q take both 0 and 1 value infinitely many times, then there are
dovetailed conditions p0 ≤ p and q 0 ≤ q: just let a, b ⊂ ω be infinite subsets of
ω \ dom(p) and ω \ dom(q) respectively such that between any two successive
elements of a there is at most one element of b and vice versa, and x a and y b
both return constantly zero value, and let p0 = x (ω \ a) and q 0 = y (ω \ b).
For the regularity of PIE , by Theorem 3.5.4 it will be enough to show that
every Silver cube is an E-kernel. So let [p] be a Silver cube for some p ∈ P
and let A ⊂ [p] be an analytic set such that [A]E ∈
/ I; we must conclude that
[A]E ∩ [p] ∈
/ I. Choose a Silver cube [q] ⊂ [A]E and a point y ∈ 2ω such that
q ⊂ y and on ω \ dom(q), y attains both values 0 and 1 infinitely many times.
Find x ∈ A such that x E y. Note that x must attain both values infinitely
many times on ω \ p: if it attained say 1 only finitely many times, there would
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CHAPTER 5. CLASSES OF σ-IDEALS
be no E equivalent point in [p] to the point y 0 ∈ [q] which extends q with only
zeroes. As in the previous paragraph, find dovetailed Silver cubes p0 ≤ p and
q 0 ≤ q, and conclude that [p0 ] ⊂ [p] ∩ [q]E as desired.
For the distributivity just note that E is I-dense and apply Proposition 3.5.7.
To show that PIE yields the V [xgen ]E extension, it is enough to prove that for
all conditions p, q ∈ Q, either there is a condition p0 ≤ p such that [p0 ]E ∩[q] ∈ I,
or there is a condition p0 ≤ p and an injective Borel map f : [p0 ] → [q] such
that f ⊂ E and f preserves the ideal I; and then apply Theorem 3.5.9. So fix
the conditions p, q ∈ P . If [p]E ∩ [q] ∈ I then we are done; otherwise thin out q
so that [q] ⊂ [p]E holds. As in the previous paragraph, there will be dovetailed
Silver cubes p0 ≤ p and q 0 ≤ q and a homeomorphism π : [p0 ] → [q 0 ] whose
graph is a subset of E and which preserves the ideal I. Thus, the assumptions
of Theorem 3.5.9 are satisfied and PIE yields the V [xgen ]E extension.
As an additional piece of information, in the PI -generic extension V [G] the
equivalence class [xgen ]E is the only one contained in every set in the filter
H = G ∩ PIE . In the ground model, this translates to the statement that
whenever p ∈ P and g : [p] → 2ω is a Borel function such that ∀x ∈ [p] ¬g(x) E
x then there is a condition q ≤ p such that [q]E ∩ g 00 [q] = 0. Since Silver
forcing is bounding, strengthening p if necessary we may assume that g [p] is
continuous and there are successive intervals In : n ∈ ω in ω such that for every
x ∈ [p] and every n ∈ ω there is m ∈ In such that |f (x)(m) − f (g(x))(m)| >
3n. Find a strengthening q ≤ p such that the set ω \ dom(q) visits every
interval in the collection {In : n ∈ ω} in at most one point. To see that q
works as required, pick x, y ∈ q and argue that for every n ∈ ω there is m
such that |f (y)(m) − f (g(x))(m)| > n. Just find a number m ∈ In such that
|f (x)(m) − f (g(x))(m)| > 3n and observe that |f (x)(m) − f (y)(m)| ≤ n since
the set ω \ dom(q) has at most n elements below m.
Theorem 5.8.8. There is a coding function on Silver cubes.
Proof. Consider the function f : (2ω )2 → 2ω defined by f (x, y) = x ◦ π −1 where
π is the increasing enumeration of the set {n : x(n) 6= y(n)} if this set is infinite,
and f (x, y) =trash if this set is finite. This function works.
Let p ∈ P be a condition. Whenever z ∈ 2ω is a binary sequence, let x, y ∈ 2ω
be the unique points such that g ⊂ x, y and z = x ◦ π −1 and 1 − z = y ◦ π −1
where π is the increasing enumeration of the complement of the domain of p.
Clearly then, f (x, y) = z and f 00 [p]2 = 2ω .
5.9
Ramsey cubes
A central notion of Ramsey theory of Polish spaces is the following:
Definition 5.9.1. A Ramsey cube is a set of the form [a]ℵ0 where a ⊂ ω is
infinite. A dented Ramsey cube is the collection of all sets of the form c ∪ b
where c ⊂ ω is a fixed finite set and b ranges over all elements of a fixed Ramsey
cube.
5.9. RAMSEY CUBES
155
The partition properties of Ramsey cubes are governed by Silver and Prikry
theorems:
Fact 5.9.2. For every Borel partition of a Ramsey cube into finitely many
pieces, one of the pieces contains a Ramsey cube. For every Borel partition of
a Ramsey cube into countably many pieces, one of the pieces contains a dented
Ramsey cube.
We first introduce the poset associated with the notion of a Ramsey cube,
and its combinatorial presentation. Adrian Mathias defined the Mathias forcing
P as the poset of all pairs p = hap , bp i where ap ⊂ ω is finite and bp ⊂ ω is
infinite; thus, every condition p ∈ P is associated with a dented Ramsey cube
[p] = {ap ∪ b : b ∈ [bp ]ℵ0 . The ordering is defined by by q ≤ p if ap ⊂ aq , bq ⊂ bp ,
and aq \ ap ⊂ bp .
Let I be the σ-ideal on X = [ω]ℵ0 consisting of those sets A ⊂ X which are
nowhere dense in the quotient partial order P(ω) modulo the ideal of finite sets.
Fact 5.9.3.
1. [36] Every analytic or coanalytic subset of [ω]ℵ0 is either in
I or contains a dented Ramsey cube;
2. the map p 7→ [p] is an isomorphism of P with a dense subset of PI ;
3. [46] the σ-ideal I is Π12 on Σ11 and not ∆12 on Σ11 ;
4. the poset P is proper.
The canonization properties of Ramsey cubes are complex. The total canonization for smooth equivalence relations fails badly, as the Mathias forcing adds
many intermediate real degrees. Prömel and Voigt [41] gave an explicit canonization formula for smooth equivalence relations which can be also derived from
earlier work of Mathias [36, Theorem 6.1]. For a function γ : [ω]<ℵ0 → 2 let
Γγ : [ω]ℵ0 → [ω]≤ℵ0 be the function defined by Γγ (a) = {k ∈ a : γ(a ∩ k) = 0}
and Eγ the smooth equivalence relation on [ω]ℵ0 induced by the function Γγ .
Fact 5.9.4. [36, 41] For every smooth equivalence relation E on [ω]ℵ0 there is
a function γ : [ω]<ℵ0 → 2 and a Ramsey cube on which E = Eγ .
Here, we shall present a further canonization theorem, due to Adrian Mathias, regarding countable Borel equivalence relations. There is an obvious obstacle for canonization there, the relation E0 as well as some of its subequivalences.
It turns out that other countable equivalence relations can be simplified to these
on Silver cubes.
Theorem 5.9.5. (Mathias) For every countable Borel equivalence relation E on
[ω]ℵ0 , there is a Ramsey cube on which E ⊆ E0 . For every essentially countable
equivalence relation E on [ω]ℵ0 , there is a Ramsey cube on which E is reducible
to E0 .
Proof. The proof of Theorem 5.9.5 starts with two claims of independent interest.
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CHAPTER 5. CLASSES OF σ-IDEALS
Claim 5.9.6.
1. Let c ⊂ ω be an infinite set and f : [c]ℵ0 → P(ω) be a Borel
function such that for every e ∈ [c]ℵ0 , f (e) \ e is infinite. Then there exists
an infinite set d ⊂ c such that for every e ∈ [d]ℵ0 , f (e) \ d is infinite.
2. Let f : [ω]ℵ0 → [ω]ℵ0 be a Borel function. The Mathias forcing forces the
following: either f˙(ẋgen ) ⊂ ẋgen up to finitely many numbers, or there is
a ground model superset d ⊃ ẋgen such that f (ẋgen ) \ d is infinite.
Proof. For the first item, for an infinite set e ⊂ c, write eev for the set of its
elements that are indexed by even numbers in its increasing enumeration, and
eod for e \ eev . Partition [c]ℵ0 into two Borel parts: B = {e : f (eev ) ∩ eod is
infinite}, and C = {f (eev ) ∩ eod is finite}. By Galvin–Prikry theorem, there
must be a set d0 ⊂ c such that [d0 ]ℵ0 is either wholly contained in B or C. Let
d = d0ev ; we claim that the set d works as required.
Case 1. Suppose first that [d0 ]ℵ0 ⊂ B. Then for every infinite set e ⊂ d
there is e0 ⊂ d0 such that e0ev = e and e0od ⊂ d0od . Thus, f (e) ∩ d0od must be
infinite and so must be its superset, the set f (e) \ d. In this case, we are done.
Case 2. Now suppose that [d0 ]ℵ0 ⊂ C. Then for every infinite set e ⊂ d,
it must be the case that f (e) ∩ (d0 \ e) is finite. If this set was infinite, find a
set e0 ⊂ d0 such that e0ev = e and for any two successive elements of the set e,
if there is an element of the set f (e) ∩ (d0 \ e) between them then the unique
element of e0 between them belongs to f (e) ∩ (d0 \ e). Then clearly f (e0ev ) ∩ e0od
is infinite, contradicting the case assumption. Thus, the set f (e) \ d0 must be
infinite, as well as its superset f (e) \ d. The first item follows.
The second item is an elaboration of the first. Suppose B ∈ PI is a condition
forcing f˙(ẋgen ) \ ẋgen is infinite. The set C = {x ∈ B : f (x) \ x is infinite} is
still I-positive and Borel, so thinning it out if necessary we may assume that it
is a dented Ramsey cube. Using the first item, we can thin out this cube still
further to find d ⊂ ω such that all sets in the cube are subsets of d and their
f -images have infinitely many elements outside of d. This condition forces the
”or” clause in the second item.
Claim 5.9.7. Let M be a countable elementary submodel of a large structure
and let b ⊂ ω be an M -generic real for Mathias forcing. If e0 , e1 ⊂ b are infinite
sets then M [e0 ] = M [e1 ] if and only if the symmetric difference e0 ∆e1 is finite.
Proof. An infinite subset of a Mathias generic real is again Mathias generic
[36], so it is the case that both e0 , e1 are Mathias generic reals for the model
M . Clearly, if their symmetric difference is finite then they construct each other
and the models M [e0 ] and M [e1 ] are the same. For the opposite implication,
suppose that e0 \ e1 is infinite and argue that e0 ∈ M [e1 ] is impossible. Indeed,
if e0 ∈ M [e1 ], then e0 = f (e1 ) for some Borel function f ∈ M . The second item
of Claim 5.9.6 shows that there is an infinite set d ∈ M such that e1 ⊂ d and
f (e1 ) \ d is infinite. Since b is Mathias generic with infinite intersection with
d, all but finitely many of its elements belong to d. But then, the set f (e1 ) \ b
must be infinite, and therefore cannot be equal to the set e0 which is a subset
of b.
5.9. RAMSEY CUBES
157
Towards the proof of the theorem, suppose first that the Borel equivalence
relation E has in fact countable classes; it is then generated as an orbit equivalence relation of a Borel action of some countable group G. Let M be a countable
elementary submodel of a large structure, and let b ⊂ a be a Mathias generic
real for M . If e ⊂ b is an infinite set, then the model M [e] contains all Eequivalents of e, since they are simply the images of e under the Borel action of
G. Claim 5.9.7 then shows that the E-class of e is a subset of the E0 -class of e,
and as e ⊂ b was arbitrary, E [b]ℵ0 ⊆ E0 .
The case of the general essentially countable equivalence relation uses the
canonization of smooth equivalence relations, as in Fact 5.9.4. Suppose that
a ⊂ ω is an infinite set, F is a countable Borel equivalence relation on a Polish
space Y , f : [a]ℵ0 → Y is a Borel function, and E is the pullback of F by the
function f . Use the canonization of smooth equivalence relations, Fact 5.9.4, to
thin out the set a if necessary to find a function γ : [ω]<ℵ0 → 2 such that the
smooth equivalence relations associated with f and Γγ coincide at [a]ℵ0 . This
allows us to replace the general case with the situation where f = Γγ and F is
a Borel equivalence relation with countable classes on [ω]ℵ0 . Let C = Γ00γ [a]ℵ0 .
By Silver’s theorem 5.9.2, there is an infinite set c ⊂ a such that [c]ℵ0 is either
a subset of C or disjoint from C. The latter case cannot happen, since the set
Γγ (c) ⊂ c would contradict the homogeneity; thus, the former case rules. By
the previous paragraph, there must be an infinite set d ⊂ c such that F [d]ℵ0
is Borel reducible to E0 . It follows that E [d]ℵ0 is also reducible to E0 , since
the function Γγ Borel reduces it to F [d]ℵ0 .
Beyond the realm of essentially countable equivalence relations, there is a
rich and chaotic spectrum of Borel equivalences on [ω]ℵ0 which maintain their
complexity on every Ramsey cube.
Theorem 5.9.8. E0 is in the spectrum of Mathias forcing. The poset PIE0 is
regularly embedded in PI , it is ℵ0 -distributive and it yields the V [ẋgen ]E0 model.
Proof. Identify subsets of ω with their characteristic functions and use this identification to transfer E0 to the space X. The theorem follows from Theorem 4.2.1
once we prove that E0 saturations of Borel I-small sets are I-small. However, it
is clear that E0 -saturation of a set nowhere dense in the algebra P(ω) modulo
finite is again nowhere dense, because the saturation does not change the place
of the elements of the set in this algebra. The ℵ0 -distributivity follows from
proposition 3.5.7, since the E0 orbit of any element in a dented cube is dense in
the dented cube.
In fact, it is not difficult to see that the decomposition of Mathias forcing associated with the E0 equivalence is just the standard decomposition into P(ω)
modulo finite followed with a poset shooting a set through a Ramsey ultrafilter.
Note that in the Mathias extension, there are many E0 -classes which are subset
of every E0 -invariant set in the generic filter; every infinite subset of the generic
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CHAPTER 5. CLASSES OF σ-IDEALS
real will serve as an example. It turns out though that E0 is the only countable
equivalence relation in the spectrum of Mathias forcing.
Theorem 5.9.9. The spectrum of the σ-ideal I is cofinal among Borel equivalence relations in ≤B ; it includes EKσ .
Proof. Let K be a Borel ideal on ω containing all finite sets and let EK be
the equivalence relation on the space Y of increasing functions in ω ω defined
by y EK z if {n ∈ ω : y(n) 6= z(n)} ∈ K. For the cofinality of the spectrum,
by Fact 2.2.12, it is enough to show that EK is in the spectrum of Mathias
forcing. Look at E, the relation on [ω]ℵ0 connecting two points if their increasing
enumerations are EK -equivalent. Clearly, this is an equivalence relation Borel
reducible to EK . On the other hand, if B ⊂ X is a Borel I-positive set, then
EK is Borel reducible to E B. Just find a condition p in the Mathias forcing
such that [p] ⊂ B and let π : ω → ap ∪ bp be the increasing enumeration. To
each point y ∈ Y associate the set by = ap ∪ rng((π ◦ y) ω \ |ap |) and observe
that the map y 7→ by reduces EK to E B. If K is an Fσ -ideal on ω such that
=K is bireducible with EKσ , then also EK is bireducible with EKσ , showing
that EKσ is in the spectrum of I.
Theorem 5.9.10. F2 is in the spectrum of the Ramsey null ideal.
Proof. We will need a couple of definitions. For elements x, y ∈ [ω]ℵ0 , say that
x, y are dovetailed if they are disjoint and between any two successive elements
of x there is exactly one element of y and vice versa. Say that x, y are almost
dovetailed if some of their finite modifications are. If x, y, z ∈ [ω]ℵ0 are pairwise
almost dovetailed, say that x is between y and z if for all but finitely many
n ∈ y, the smallest element of x above n is smaller than the smallest element
of z above n. A set of pairwise almost dovetailed points is dense in itself if
between any two elements of it there is another one. The following is nearly
trivial.
Claim 5.9.11. Let x, y, z ∈ [ω]ℵ0 be pairwise almost dovetailed. Then
1. x is either between y and z, or between z and y;
2. if x is between y and z, then x is not between z and y, and z is between x
and y;
3. if x is between y and z, then the same holds of all the finite modifications
of the sets x, y, and z;
If a ⊂ [ω]ℵ0 is a countable collection of pairwise almost dovetailed points then
there is a map π from a to the unit circle such that x is between y and z if the
triple hy, x, zi goes in the counterclockwise direction.
Let us now introduce a Mathias name for a somewhat canonical countable
set of pairwise almost dovetailed points, dense in itself. Consider 2<ω ordered
with the dense lexicographic order <lex . Divide ω into successive intervals {In :
5.9. RAMSEY CUBES
159
n ∈ ω} and find a function χ : ω → 2<ω such that χ In is an order isomorphism
between In and (2≤n , <lex ). Finally, for every x ∈ [ω]ℵ0 and every s ∈ 2<ω , let
xs ⊂ x be the image of the set {m : χ(m) = s} under the increasing enumeration
of x. It is not difficult to see that the set {xs : s ∈ 2<ω } consists of pairwise
almost dovetailed points, and it is dense in itself.
To formulate the following claim, for sets y ⊂ x ⊂ ω say that y is thin in x
if for every k ∈ y the next 2k many elements of x are not in y.
Claim 5.9.12. Let x0 ∈ [ω]ℵ0 , and let x1 ⊂ x0 be a set thin in x0 . For every
countable set a consisting of pairwise almost dovetailed subsets of x1 , dense in
itself, there is a set x2 ⊂ x0 such that the sets {xs2 : s ∈ 2<ω } and a are equal
up to finite modifications of their elements.
Proof. First, use the density of the set a to build a bijection π : 2<ω → a
such that whenever s, t, u are three binary sequences listed in the lexicographic
order, then π(t) is between π(s) and π(u). We will find the point x2 ∈ [ω]ℵ0 so
that for every binary sequence s ∈ 2<ω , xs2 = π(s) up to a finite modification.
Find numbers {mn : n ∈ ω} such that for every triple s <lex t <lex u ∈ 2≤n ,
everything works out correctly above mn :
• the sets π(s), π(t), π(u) are pairwise disjoint above mn ;
• between any two successive elements of π(s) larger than mn there is exactly
one element of π(u), and the same for the other two pairs
• for every m > mn in π(s), the smallest element of π(t) above m is smaller
than the smallest element of π(u) above mn .
Find intervals {Jk : k ∈ ω} of natural numbers such that whenever min(Jk ) >
mn , the interval Jk contains exactly one point from each setSπ(s) : s ∈ 2≤n listed
in the lexicographic order, and for every s ∈ 2<ω , π(s) ⊂ k Jk ∪ m|s| . Now it
is not difficult to use the thinness of the set x1 in x0 to find a set x2 ⊂ x1 so
that, writing ξ : ω → x2 for its increasing enumeration, for every k ∈ ω, every
number l ∈ Ik and every s ∈ 2<ω , if min(Jk ) > m|s| and χ(l) = s then ξ(l) is
the unique point in π(s) ∩ Jk . The set x2 ⊂ x0 works as required.
Now let f : [ω]ℵ0 → ([ω]ℵ0 )ω be any Borel function such that f (x) enumerates the set {xs : s ∈ 2<ω } and all finite modifications of elements of this set.
Let E be the Borel equivalence relations connecting points x0 , x1 if f (x0 ), f (x1 )
enumerate the same set. It will be enough to show that F2 is reducible to E
restricted to any dented Ramsey cube. To simplify the notation, consider just
a Ramsey cube, a set [x0 ]ℵ0 for some infinite set x0 ⊂ ω. Let x1 ⊂ x0 be a set
thin in x0 . Choose a function π : 2<ω → x0 such that for finite binary sequences
s, t ∈ 2<ω |s| < |t| implies π(s) < π(t), and if |s| = |t| and s is lexicographically
less than t then again π(s) < π(t). The map π can be naturally extended to
π : 2ω → [ω]ℵ0 by π(y) = {π(y m) : m ∈ ω}. It is not difficult to see that the
range of π consists of pairwise almost dovetailed sets, and if a ⊂ 2ω is a dense
set consisting of sequences that are not eventually constant, then π 00 a is a set
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CHAPTER 5. CLASSES OF σ-IDEALS
dense in itself. Let B ⊂ (2ω )ω × [x0 ]ℵ0 be the set of all pairs hy, xi such that the
sets π 00 rng(y) and {xs : s ∈ 2<ω } are the same up to finite modifications of their
elements. If y ∈ (2ω )ω is an enumeration of a dense subset of 2ω consisting of
not eventually constant sequences, then the vertical section By is nonempty by
the previous claim. By Corollary 5.1.30, F2 is Borel reducible to E [x0 ]ℵ0 .
The whole treatment in the previous proof should be understood as a development of a symmetric Mathias extension, very much parallel to the symmetric
Cohen extension of Section 5.1c.
Definition 5.9.13. Let M be a transitive model of set theory. Say that a
set a ⊂ [ω]ℵ0 is a symmetric Mathias set for M if a is closed under finite
modifications, any two elements of a are either finite modifications of each other
or otherwise almost dovetailed, x ∪ y is a Mathias real for M for every x, y ∈ a,
and finally if x, y ∈ a then there is z ∈ a that is between x and y.
The point of this definition is that just as in the case of symmetric Cohen
or random extensions, the simple demands on the set a guarantee that the
choiceless model M (a) can be constructed as a submodel of a Mathias generic
extension, and its theory is an element of the model M . The arguments in the
previous proof fairly easily give the following:
Proposition 5.9.14. Let M be a transitive model of set theory and a ⊂ [ω]ℵ0
be a symmetric Mathias set for M . Then in some forcing extension there is a
point x ∈ [ω]ℵ0 , M -generic for Mathias forcing, such that the closure of the set
{xs : s ∈ 2ω } under finite modifications is equal to a.
5.10
Weak Milliken cubes
Definition 5.10.1. The symbol (ω)ω stands for the collection of all sequences
a of finite sets of natural numbers such that for every n ∈ ω, max(a(n))
<
S
min(a(n + 1). A weak Milliken cube is a set of the form [a] = { n∈b a(n) :
b ∈ [ω]ℵ0 } where a ∈ (ω)ω . A dented weak Milliken cube is a set of the form
{c ∪ b : b ∈ [a]} for some finite set c ⊂ ω and a ∈ (ω)ω .
The partition theory of weak Milliken cubes is governed by Fact 2.7.5, in
particular we have
Fact 5.10.2. For every Borel partition of a weak Milliken cube into finitely
many pieces, there is a weak Milliken cube wholly contained in one of the pieces
of the partition. For every Borel partition of a weak Milliken cube into countably
many pieces, there is a dented weak Milliken cube wholly contained in one piece
of the partition.
Regarding the canonization properties of weak Milliken cubes, one can immediately identify two obstacles to total canonization of Borel equivalence relations: the equivalence Emin which connects two sets of natural numbers if they
have the same minimum, and the equivalence E0 . It turns out that these two
obstacles are to a great extent unique:
5.10. WEAK MILLIKEN CUBES
161
Theorem 5.10.3. Let E be an equivalence relation on [ω]ℵ0 which is Borel
reducible to EKσ , or classifiable by countable structures, or Borel reducible to
=J for some analytic P-ideal J on ω. Then there is a weak Milliken cube C
such that E C is equal to one of the following: id, ee, E0 , Emin , E0 ∩ Emin .
Before we turn to the proof of the above canonization theorem, we take the
opportunity to introduce a propoer forcing associated with weak Milliken cubes
and its combinatorial presentation. Define the partial ordering P to consist of
all pairs p = htp , ap i where tp ⊂ ω is a finite set and ap is an infinite sequence of
finite subsets of natural numbers such that max(ap (n)) < min(ap (n + 1)) for all
n ∈ ω. Thus, every
S condition p ∈ P is associated with a dented weak Milliken
cube [p] = {tp ∪ n∈b ap (n) : b ∈ [ω]ℵ0 }. The ordering is defined by q ≤ p if
tp ⊂ tq and tq is the union of tp with some sets on the sequence ap , and all sets
on aq are unions of some sets on ap .
We will introduce a standard fusion-type terminology for the forcing P .
q ≤ p is a direct extension of p if tq = tp . If a ∈ (ω)ω then a fusion sequence
below a is a sequence hui , bi : i ∈ ωi such that u0 = 0, b0 = a, and for every
i ∈ ω, ui is a finite union of sets on the sequence bi and bi+1 ≤ bi is a member
of (ω)ω with min(bi+1 (0) > max(ui+1 . If a ∈ (ω)ω is a sequence and m ∈ ω
then a \ m is the sequence in (ω)ω obtained by erasing the first m entries on the
sequence a.
Let I be the collection of all analytic subsets of [ω]ℵ0 containing no dented
weak Milliken cube as a subset. We have
Theorem 5.10.4.
1. The collection I is a σ-ideal and the map p 7→ [p] is
an isomorphism of P with a dense subset of PI ;
2. Iis Π11 on Σ11 ;
3. I fails to have the selection property;
4. the forcing P is proper, preserves Baire category and outer Lebesgue measure, and it adds no independent reals. It adds an unbounded real.
Proof. The proof starts with the identification of basic forcing properties of the
poset P .
Claim 5.10.5. Let O ⊂ P be an open dense set and p ∈ P a condition. There is
a direct extension q ≤ p such that for every n ∈ ω, the condition htq ∪aq (n), aq \ni
belongs to the set O.
Proof. Set up a simple fusion process
S to find a sequence b ≤ a such that for
every finite set u ⊂ ω, either htp ∪ n∈u a(n), b \ max(u)i ∈ O or else none
of direct extensions of this condition belong to O. Applying Fact 2.7.5 to the
partition π : (ω)ω → 2 given by π(d) = 0 if the former case holds for d(0), we get
a homogeneous sequence c ⊂ b. The homogeneous color cannot be 1 since the
condition htp , ci has an extension in the open dense set O. If c is homogeneous
in color 0 then it confirms the statement of the claim with q = htp , ci.
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CHAPTER 5. CLASSES OF σ-IDEALS
Claim 5.10.6. If p ∈ P is a condition and [p] = B0 ∪ B1 is a partition into
two Borel sets then there is a direct extension q ≤ p such that [q] is contained
in one of the pieces of the partition.
Proof. This is an immediate consequence of Fact 2.7.5.
For the proof of properness of the poset P , let M be a countable elementary
submodel of a large enough structure, let p ∈ P ∩ M be a condition. We must
produce a master condition q ≤ p. Enumerate the open dense sets of P in
the model M by hOn : n ∈ ωi and use the claim and the elementarity of the
model M repeatedly to build a sequence p = p0 , p1 , p2 , . . . of direct extensions
of p in the model M such that for every n ∈ ω, apn n = apn+1 n and
for every
S finite set u ⊂ ω with max(u) > n it is the case that the condition
htp ∪ m∈u apn+1 (m), apn+1 \ max(u)i ∈ M belongs to the open dense set On .
The limit q ≤ p of the sequence of conditions hpn : n ∈ ωi will be the required
master condition.
In order to show that I is a σ-ideal, suppose that {Bn : n ∈ ω} are Borel sets
in the collection S
I and suppose for contradiction
S that there is a condition p ∈ P
such that [p] ⊂ n Bn . Clearly p ẋgen ∈ n Bn and so, strengthening p if
necessary, we may find a single natural number n ∈ ω such that p ẋgen ∈ Ḃn .
Let M be a countable elementary submodel of a large enough structure, and let
q ≤ p be an M -generic condition described in the previous
S proof. It is immediate
that for every point x ∈ [q] the conditions htq ∪ (q ∩ i∈n aq (i)), aq \ ni : n ∈ ω
generate an M -generic filter gx ⊂ P ∩ M . By the forcing theorem applied in the
model M , M [x] |= x ∈ Bn , and by Borel absoluteness, x ∈ Bn . Thus [q] ⊂ Bn ,
contradicting the assumption that Bn ∈ I.
In order to show that p 7→ [p] is an isomorphism between P and a dense
subset of PI , it is necessary to show that if p, q ∈ P are conditions such that
[q] ⊂ [p] then in fact q ≤ p. To simplify the discussion a bit, assume that
tp = tq = 0. Suppose that q ≤ p fails, and find a number n ∈ ω such that aq (n)
is not a union of sets on the sequence
ap . In such a case, either there is an
S
element i ∈ aq (n) such that i ∈
/ rng(a), in which case no set x ∈ [q] containing
i can be in [p]; or else there is a number m such that ap (m) ∩ aq (n) 6= 0 and
ap (m) \ aq (n) 6= 0. In the latter case find a point x ∈ [q] containing aq (n) as
a subset and disjoint from the set ap (m) \ aq (n); such a point cannot belong
to p] either. In both cases we have a contradiction with the assumption that
[q] ⊂ [p]!
To show that the ideal I is Π11 on Σ11 , let A ⊂ 2ω × [ω]ℵ0 be an analytic set,
a projection of a closed set C ⊂ 2ω × [ω]ℵ0 × ω ω . We must show that the set
/ I.
{y ∈ 2ω : Ay ∈ I} is coanalytic. Fix a point y ∈ 2ω and suppose that Ay ∈
This means that there is a condition p ∈ P such that [p] ⊂ Ay , by Shoenfield
absoluteness this will be true in the P -forcing extension as well, and so there
must be a P -name τ for a point in ω ω such that p hy̌, ẋgen , τ i ∈ Ċ. By a fusion
argument, there is a condition r ≤ p and a function g : [ω]<ℵ0 → ω <ω such that
the following holds. For every b ∈ [ω]<ℵ0 , write rb ≤ r to be that condition
whose trunk consists of tr and the union of i-th elements of the sequence ar for
5.10. WEAK MILLIKEN CUBES
163
all i ∈ b, and arb = ar \ max(b) + 1; we will have that for every b ∈ [ω]<ℵ0 ,
dom(g(b)) = max(b) + 1 and rb g(b) ⊂ τ . In particular, the following formula
φ(y, r, g) holds: the function g respects inclusion, dom(g(b)) = max(b) + 1 and
the basic open neighborhood of the space 2ω ×[ω]ℵ0 ×ω ω given by the first max(b)
elements of y in the first coordinate, the trunk trb in the second coordinate, and
the sequence g(b) in the third coordinate has nonempty intersection with the
closed set C. It is also clear that ψ(y, r, g) implies that [r] ⊂ Ay since the
function g yields the requisite points in the set C witnessing that elements of
the set [r] belong to Ay . We conclude that Ay ∈
/ I if and only if there is r ∈ P
and a function g : [ω]<ℵ0 → ω <ω such that ψ(y, r, g) holds, and this is an
analytic statement.
For the continuous reading of names, let p ∈ P and let f : [p] → ω ω be a
Borel function. Use Claim 5.10.5 to find a direct extension q ≤ p such that for
every number n ∈ ω and every
S set u ⊂ n + 1 with n ∈ u there is a number m
such that the condition htp ∪ i∈u aq (i), aq \ n + 1i forces f˙(ẋgen )(ň) = m̌. Let
M be a countable elementary submodel of a large structure and find a direct
extension r ≤ q as in the previous argument. The forcing theorem will then
show that f [r] is a continuous function.
For the preservation of Baire category, there is a simple fusion-type proof.
A more elegant argument will show that the ideal I is the intersection of a
collection of meager ideals corresponding to Polish topologies on the Polish
space [ω]ℵ0 that yield the same Borel structure, and use the Kuratowski-Ulam
theorem as in [67, Corollary 3.5.5]. This is the same as to show that for every
condition p ∈ P there is a Polish topology on [p] such that all sets in the ideal
I are meager with respect to it.
We will deal with the case of the largest condition p ∈ P ; there, tp = 0 and
ap (n) = {n}, so [p] = [ω]ℵ0 . We will show that every Borel nonmeager set in
that B ⊂ [ω]ℵ0 is
the usual topology on the space [ω]ℵ0 is I-positive. Suppose
T
such a nonmeager set, containing the intersection O ∩ On : n ∈ ω, where O is
a nonempty basic open set and On : n ∈ ω are open dense sets. Let r0 ∈ 2<ω
be a finite sequence such that whenever the characteristic function of a point
x ∈ [ω]ℵ0 contains r0 then x ∈ O. By a simple induction on n ∈ ω build
finite sequences r0 ⊂ r1 ⊂ r2 ⊂ . . . such that T
for every point x ∈ [ω]ℵ0 whose
characteristic function contains rn+1 \ rn , x ∈ m∈n Om holds. In the end, let
t = {i ∈ dom(r0 ) : r0 (i) = 1} and let a ∈ (ω)ω be the sequence defined by
a(n) = {i ∈ dom(rn+1 \ rn ) : rn+1 (i) = 1}, and observe that [t, a] ⊂ B and so
B∈
/ I as required.
For the preservation of outer Lebesgue measure, suppose that A ⊂ 2ω is a
set of outer Lebesgue mass > ε + δ and p ∈ P is a condition forcing Ȯ to be a
subset of 2ω of mass < ε. We must find a condition q ≤ p and a point z ∈ A
such that q ž ∈
/ Ȯ.
Find positive real numbers δu : u ∈ [ω]<ℵ0 that sum up to less than δ, and
find names Ȯj : j ∈ ω for basic open sets whose union Ȯ is. Using Claim 5.10.5
and a simple fusion process, find a direct extension q ≤ p such that for every
nonempty
set u ∈S [ω]<ℵ0 , there is an open set Pu such that the condition
S
tp ∪ i∈u aq (i) j∈max(u) Ȯj ⊂ Pu ⊂ Ȯ and the mass of Ȯ \ Pu is less than
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CHAPTER 5. CLASSES OF σ-IDEALS
S
δu . Clearly, the set R0 = {Pu \ Pu∩max(u) : u ∈ [ω]<ℵ0 , |u| > 1} has mass at
most δ, and the set R1 = lim sup{P{n} : n ∈ ω} has size at most ε. Find a
point z ∈ A \ (R0 ∪ R1 ), thin out the sequence q if necessary to make sure that
z∈
/ P{n} for any n ∈ ω and conclude that q ž ∈
/ Ȯ as desired.
Not adding independent reals is always a somewhat more delicate issue.
Suppose that p ∈ P is a condition and ẏ a name for an infinite binary sequence.
A simple fusion process using Corollary 5.10.6 repeatedly at each stage yields
a direct extension htSp , bi ≤ p such that for every n ∈ ω and every set u ⊂ n
the condition htp ∪ i∈u b(i), b \ n + 1i decides the value ẏ(min(b(n)). Define
a partition π : ([ω]<ℵ0 )2 → 2 by setting π(u, v) = 0 if the decision for u and
min(bmin(v) ) was 0. This is clearly a clopen partition and therefore there is a
homogeneous sequence d ∈ (ω)ω by Fact 2.7.5. Let us say that the
S homogeneous
color is 0. Consider the condition q ≤ p given by tq = tp ∪ i∈d(0) b(i), and
S
aq (j) = i∈d2j+1 b(i). It is not difficult to use the homogeneity of the sequence
d to show that the condition q forces ẏ {min(bmin(d(2j)) ) : j > 0} = 0.
Finally, P certainly adds an unbounded real: the increasing enumeration of
the generic subset of ω cannot be bounded by any ground model function. The
failure of the selection property of the ideal I is also simple. Let C ⊂ P(ω)
be a perfect set consisting of pairwise almost disjoint sets, and consider the set
D ⊂ C × [ω]ℵ0 consisting of all pairs hc, xi such that c ∈ C and x ⊂ c. It is clear
that D is a Borel set with I-positive vertical sections. If B ⊂ [ω]ℵ0 is a Borel set
covered by the vertical sections of the set D, visiting each vertical section in at
most one point, it must be the case that the elements of B are pairwise almost
disjoint, therefore B cannot contain a subset of the form [p] for any p ∈ P , and
so B ∈ I.
The investigation of the spectrum of the ideal I begins with identifying the
only obvious feature:
Theorem 5.10.7. E0 is in the spectrum of the σ-ideal I. The poset PIE0 is
regularly embedded in PI , it is ℵ0 -distributive, and it yields the model V [ẋgen ]E0 .
Proof. This is an immediate corollary of the work in Theorem 4.2.1 and the
simple observation that E0 -saturations of I-small sets are still I-small.
As an additional piece of information, the equivalence class [xgen ]E0 is forced
by PI to be the only E0 equivalence class in the intersection of all E0 -invariant
sets in the generic filter. Suppose that p is a condition and p ẏ ⊂ ω is an
infinite set not E0 related to ẋgen . Strengthening the condition p if necessary
we may find a continuous function f : [p] → P(ω) such that p ẏ = f˙(ẋgen ).
We must find a condition q ≤ p such that no element of [q] is E0 -equivalent to
/ [q]E0 and so the class [ẏ]E0 is not a subset of all
f 00 [q]; then clearly q ẏ ∈
E0 -invariant sets in the PI -generic filter.
To produce the condition q, first use a standard fusion
S process to strengthen
p so that for every finite set u ∈ ω, the condition htp ∪ n∈u ap (n), ap \ max(u)i
identifies some number mu > max(ap (max(u) − 1)) such that mu ∈ ẋgen ∆ẏ,
and some number ku > max(ap (max(u) − 1)) such that ku ∈ ẏ. Thinning out
5.10. WEAK MILLIKEN CUBES
165
the sequence ap if necessary, assume that the numbers {ku , mu : u ⊂ n} are all
smaller than min(ap (n)). It is now not difficult to verify that the thinned out
condition qSworks as desired; that is, whenever
u, v ⊂ ω are infinite sets, writing
S
xu = tp ∪ n∈u aq (n)) and xv = tq ∪ n∈v aq (n) we must show ¬f (xu ) E0 xv .
For every n ∈ u, we must produce a number k > max(aq (n − 1)) which belongs
to the symmetric difference of f (xu ) and xv . There are two cases:
• either n ∈
/ v. Then k = ku∩n+1 will work: as n ∈ u, k ∈ f (xu ), while
xv contains no numbers in the interval (max(aq (n − 1)), min(aq (n + 1))),
where k belongs;
• or n ∈ v. In this case, k = mu∩n+1 will work. Either it is in the set aq (n),
in which case it belongs to both xu , xv and does not belong to f (xu ), or
it is not in the set aq (n), in which case it does not belong to either xu , xv
and does belong to f (xu ).
The proof is complete.
Proof of Theorem 5.10.3. We will start with equivalence relations reducible to
EKσ . Recall that dom(EKσ ) is the collection of all functions in ω ω below the
identity equipped with the distance δ(x, y) = supn |x(n) − y(n)|; EKσ then
connects the points x, y if δ(x, y) is finite. Let a ∈ (ω)ω be a sequence, and let
E be an equivalence relation reducible to EKσ by a Borel function f : [a] →
dom(EKσ ). Thinning out the sequence a we may assume that the function f
is in fact continuous. Consider the partition (a)ω = B ∪ C placing a sequence
b ≤ a into the set B just in case that for every i there are points x, y ∈ [b(0), b\0]
such that δ(f (x), f (y)) ≥ i. Using Milliken theorem ??? we may thin out the
sequence a so that (a)ω ⊂ B or (a)ω ⊂ C. There is an appropriate split into
cases.
Case 1. Assume first that (a)ω ⊂ B. In this case, we will first thin out a
to c that E [c] ⊆ E0 , and then proceed with another partition argument. To
perform the thinning, by induction on i ∈ ω build a fusion sequence huS
i , ci : i ∈
ωi so that whenever j ∈ S
ω and t, s ⊂ j are two sets, then for every x ∈ [ i∈s ui ∪
uj , cj+1 ] and every y ∈ [ i∈t ui , cj+1 ] it is the case that δ(f (x), f (y)) ≥ j. It is
clear that the limit c of the fusion sequence will satisfy E [c] ⊆ E0 .
Suppose that ui : i ∈ j and cj have been constructed. Enumerate the set
P(j) × P(j) by hhsk , tk i : k ∈ 22j i, and by induction on k build finite sets ukj
and Milliken cubes ckj so that
I1 u0j = 0 and the set uk+1
is a union of ukj with some sets on the sequence
j
k
cj ;
I2 cj = c0j ≥ c1j ≥ . . . ;
S
I3 for every k there are numbers mk , lks , lkt so that for every x ∈ [ i∈sk ui ∪
S
uk+1
, ck+1
] and every y ∈ [ i∈tk ui , ck+1
] it is the case that f (x)(mk ) = lks ,
j
j
j
f (y)(mk ) = lkt , and |lks − lkt | > j.
166
CHAPTER 5. CLASSES OF σ-IDEALS
This is not difficult to do using the case assumption. Suppose that ckj , ukj have
S
been found. The case assumption yields x0 , x1 ∈ [ i∈sk ui ∪ uk+1
, ckj ] such that
j
d(f (x0 ), f (x1 )) > 2j. Find mk such that |f (xS
)(m
)
−
f
(x
)(m
)|
0
k
1
k > j. Find a
k
k
]
the
value
f (y)(mk ) is
such
that
for
every
y
∈
[
u
,
c
sequence ck+1
≤
c
j
j
i∈t i j
the same, equal to some rk . Then either f (x0 )(mk ) or f (x1 )(mk ) is farther than
j from f (y)(mk ), say that it is x0 , and put lk = f (x
the continuity
S0 )(mk ). Use
k
k
of the function f to find a set uk+1
such
that
x
∈
[
u
∪
u
,
0
j+1 cj ] and for all
j
i∈s i
points x in this dented cube it is the case that f (x)(mk ) = lk . This concludes
the induction step.
After the induction on k has been performed, let uj , cj be the last set and
the last sequence arrived at in it. This completes the construction of the fusion sequence, and we now have E (c)ω ⊂ E0 , where
S c is the limit of the
ω
0
0
0
fusion
sequence.
Let
(c)
=
B
∪
C
where
b
∈
B
if
i∈ω b(i) is equivalent to
S
i∈ω,i6=1 b(i). The Milliken theorem ??? can again be used to thin out c so that
either (c)ω ⊂ B 0 or (c)ω ⊂ C 0 . There are the corresponding two subcases.
Case 1a. Suppose that (c)ω ⊂ B 0 . In this case, first observe that E [c]
connects every pair of E0 -equivalent points with the same minimum. To see this,
use the case assumption to argue
set e ⊂ ω and every
S that for every infinite S
j > min(e) it is the case that i∈e c(i) is E-related to i∈e∪{j} c(i). Then,
whenever x, y ∈ [c] are E0 -related points with the same minimum, it is possible
to transform one into another in finitely many steps described in the previous
sentence, and the transitivity of the relation E shows that x and y are E-related.
ω
00
00
Consider
S the partition of (c) into
S two Borel pieces B and C , putting
b ∈ B 00 if i∈ω b(i) is E-related to i∈ω,i6=0 b(i).The Milliken theorem again
provides a split into cases.
Case 1aa. Either there is d ≤ c such that (d)ω ⊂ B 00 . In such a case,
E [d] = E0 : whenever x, y ∈ [d] are E0 -related, if their minimums are the
same then they are E-related by the preceding paragraph. If their minimums
are different, say min(x) < min(y), then there is aSsequence b ≤ d such that
min(b0 ) = min(x), min(b1 ) = min(y),Sand x E0 S i b(i). The homogeneity
assumption applied to b shows that x E i∈ω b(i) E i∈ω,i6=0 b(i) E y as desired.
Case 1ab. Or, there is d ≤ c such that (d)ω ⊂ C 00 . In such a case,
E [d] = E0 ∩ Emin by a similar homogeneity argument as in Case 1aa.
Case 1b. Suppose now that (c)ω ⊂ C 0 . In this case, we will thin out c
to d such that E [d] = id. By induction on i ∈ ω build a fusion sequence
hvi , di : i ∈ ωi S
so that whenever j ∈ ω and
S s, t ⊂ j are two distinct sets then
for every x ∈ [ i∈s vi , dj ] and every y ∈ [ i∈s vi , dj ] such that x \ min(dj ) =
y \ min(dj ) it is the case that x E y fails. The fusion d of this fusion sequence
will certainly have the required property that E [d] = id: if x, y ∈ [d] are not
E0 -related then they are not E-related by the initial work in Case 1, and if they
are E0 -related and distinct then they are not E-related by the assumption on
the fusion sequence past the last difference of the two points x, y.
Suppose that vi : i ∈ j and dj have been constructed. For every pair s, t
ω
of subsets of j define the partition
Bs,t and
S of (dj )S into two Borel pieces S
Cs,t by placing b ≤ dj into Bs,t if i∈s vi ∪ i∈ω b(i) is E-related to i∈s vi ∪
5.10. WEAK MILLIKEN CUBES
S
i∈ω,i6=0 b(i).
Bs,t : ???
167
Observe that there can be no sequence e ≤ dj such that (e)ω ⊂
Case 2. Suppose now that (a)ω ⊂ C. In this case, first observe that every
two points in [a] with the same minimum are E-related: if min(x) = min(y)
then the homogeneity assumption applied to the sequence a \ min(x) shows that
the distance d(f (x), f (y)) must be finite. Now let F be the equivalence relation
on ω relating i and j if for every x, y ∈ [c] such that min(x) = min(c(i)) and
min(y) = min(c(j)), x E y holds. There are two possibilities.
Case 2a. Either F has an infinite class, say e ⊂ ω. Let d ≤ c be the
sequence consisting only of those entries of c indexed by elements of the set
e ⊂ ω. It is clear that E [d] = ee.
Case 2b. Or F has an infinite set consisting of pairwise inequivalent elements, say e ⊂ ω. Again, let d ≤ c be the sequence consisting only of those
entries of c indexed by elements of the set e ⊂ ω. It is clear that E [d] = Emin .
This concludes the case of equivalence relations reducible to EKσ . Now
assume that a ∈ (ω)ω and E is an equivalence relation on [a] reducible to =J
for some analytic P-ideal J on ω. Use the theorem of Solecki ??? to find a
submeasure lower semicontinuous submeasure φ on ω such that J = {e ⊂ ω :
limi φ(e \ i) = 0}. Let f : [a] → P(ω) be a Borel function reducing E to =J ;
thinning out the sequence a we may assume that the function f is continuous.
For every sequence b ≤ a and every j ∈ ω let ε(j, b) be the supremum
of all
S
numbers of the form φ(f (x)∆φ(y)) as x, y vary over all points in [ i∈j b(i), b \
j]. Note that for a fixed sequence b, the numbers ε(j, b) cannot increase as
j increases, and so they converge to some real number ε(b) ≥ 0. Define the
partition (a)ω into two Borel pieces B and C by placing b ∈ B if ε(b) = 0. The
Milliken theorem gives us the first split into subcases:
Case 1. It is possible to thin out the sequence a so that (a)ω ⊂ B. In this
case, we will find c ≤ a so that E [c] becomes a relatively closed equivalence
relation. Closed equivalence relations are smooth by ???, and so a reference
to the previously treated case of equivalence relations reducible to EKσ will
complete the argument here. To find c, by induction on i ∈ ω find nonempty
finite sets ui such that min(ui+1 ) > max(ui ), each ui is a union of finitely many
entries on a, S
and for every j ∈ ω, every set s ⊂ j and every x, y ∈ [a] such
that the set i∈s ui ∪ uj is an initial segment of both x and y it is the case
that φ(f (x)∆f (y)) < 2−j . This is easily possible using the case assumption
repeatedly. Let c = hui : i ∈ ωi. To see that the complement of E is relatively
open in [c], let x, y ∈ [c] be E-unrelated points, and j ∈ ω be so large that
limi φ((f (x)∆f (y)) \ i) > 2−j . Let s, t be initial segments of x, y such that for
some numbers ks , kt > j it is the case that uks ⊂ s and ukt ⊂ t. We claim that
the relatively open sets [s] and [t] inside [c] are E-unrelated. Well, if x0 ∈ [s] ∩ [c]
and y 0 ∈ [t] ∩ [c], then φ(f (x)∆f (x0 )) < 2ks < 2j+1 and φ(f (y)∆f (y 0 )) < 2kt <
2j+1 . Since limi φ((f (x)∆f (y))\i) > 2−j , this means that limi φ((f (x0 )∆f (y 0 ))\
i) > 0 and ¬x0 E y 0 as required.
Case 2. It is possible to thin out the sequence a so that (a)ω ⊂ C. In this
case, we will thin out a further so that E [a] ⊂ E0 . The treatment will then
hook up into Case 1 of the equivalence relations reducible to EKσ . To perform
168
CHAPTER 5. CLASSES OF σ-IDEALS
the thinning, by induction on i ∈ ω build a fusion sequence hui , S
ci : i ∈ ωi so that
whenever j ∈ ωSand t, s ⊂ j +1 are two sets then for every x ∈ [ i∈s ui ∪uj , cj+1 ]
and every y ∈ [ i∈t ui , cj+1 ] it is the case that φ((f (x)∆f (y)) \ j) > 12 ε(hui : i ∈
tia cj ). Once such a construction is performed, writing c ∈ (ω)ω for its limit,
it will satisfy E [c] ⊂ E0 . To see this, let x, y ∈ [c] be E0 -unrelated points.
One of them, say x, must have infinitely many elements that do not occur in
the other. Let d ≤ c be the sequence enumerating all j such that uj ⊂ x
and note that ε(d) > 0 by the homogeneity assumption. Now there must be
infinitely many j ∈ ω such that uj ⊂ x and uj ∩ y = 0, and for each such j ∈ ω,
writing s = {i < j : ui ⊂ x} and t = {i < j : ui ⊂ y}, the construction yields
φ(f (x)∆f (y)) \ j) > ε(d). Therefore f (x)∆f (y) is not in the ideal J and x, y
are not E-related.
Suppose that ui : i ∈ j and cj have
S been constructed. Thin out cj so that
for every set s ⊂ j, for every x ∈ [ i∈s ui , cj ] the set f (x) ∩ k is the same.
Enumerate the set P(j) × P(j) by hhsk , tk i : k ∈ 22j i, and by induction on k
build finite sets ukj and Milliken cubes ckj so that
I1 u0j = 0 and the set uk+1
is a union of ukj with some sets on the sequence
j
ckj ;
I2 cj = c0j ≥ c1j ≥ . . . ;
I3 for every
is a number mk and setsSwks , wkt ⊂ mk so that for every
S k therek+1
x ∈ [ i∈sk ui ∪ uj , ck+1
] and every y ∈ [ i∈tk ui , ck+1
] it is the case that
j
j
s
f (x) ∩ mk = wk , f (y) ∩ vk = wks , and wks ∆wkt > 12 ε(hui : i ∈ tia ckj ).
This is not difficult to do using the case assumption. Suppose that ckj , ukj have
been found. Let d = hui : i ∈ sia hukj ia ckj . The definition of ε(d) yields points
S
x0 , x1 ∈ [ i∈sk ∪uk+1 ,ck ] such that φ(f (x0 )∆f (x1 )) > 21 ε(d). Let mk ∈ ω be a
j
j
number such that φ((f (x0 )∆f (x1 )) ∩ mk ) > 12 ε(d). Find a sequence ck+1
≤ ckj
j
S
such that for every y ∈ [ i∈tk ui , ckj ] the set f (y)∩mk is the same, equal to some
Wkt . Then either f (x0 )(mk ) or f (x1 )(mk ) is farther than j from f (y)(mk ), say
that it is x0 , and put wks = f (x0 ) ∩ mk . Use the continuity of the function f to
S
find a set uk+1
such that x0 ∈ [ i∈s ui ∪ uj+1
, ckj ] and for all points x in this
j
i
s
dented cube it is the case that f (x) ∩ mk = wk . This concludes the induction
step.
After the induction on k has been performed, let uj , cj be the last set and
the last sequence arrived at in it. This completes the construction of the fusion
sequence,
As a final remark in this section, canonization of objects more complicated
than equivalence relations must fail badly in the case of weak Milliken cubes
due to the following simple
Proposition 5.10.8. The σ-ideal I has a coding function.
5.11. STRONG MILLIKEN CUBES
169
Proof. Let x, y ∈ X be two points, not equal modulo finite. Consider all sets
u ⊂ x ∩ y which are an interval inside x and inside y, and are maximal with
respect to this property; list them in an increasing order as hun : n ∈ ωi.
Define f (x, y) ∈ 2ω by setting f (x, y)(n) = 1 if max((x ∪ y) \ min(un ) ∈ x and
f (x, y)(n) = 0 otherwise.
Suppose now that B ⊂ X is an I-positive set; it contains a dented cube of
the form [t, a] for some nonempty finite set t ⊂ ω and a ∈ (ω)ω . Let z ∈ 2ω be a
prescribed value. Let x be the union of t the sets a(2n + 1) for all n ∈ ω and the
sets a(2n) for all n with z(2n) = 1. Let y be the union of t the sets a(2n + 1)
for all n ∈ ω and the sets a(2n) for all n with z(2n) = 0. It is immediate that
f (x, y) = z.
5.11
Strong Milliken cubes
Definition 5.11.1. A strong Milliken cube is a set of the form {c ∈ (ω)ω : c ≤
a} for a sequence a ∈ (ω)ω . A dented strong Milliken cube is a set of the form
{ta c : c ≤ a} where t is a S
finite increasing sequence of finite subsets of ω and
a ∈ (ω)ω is such that max t < min(a(0)).
The σ-ideal on (ω)ω associated with the notion of strong Milliken cube is
isolated in the same way as in the previous sections. Let P be the poset of
all pairs p = htp , ap i such that tp is an increasing finite sequence of nonempty
finite subsets of ω and ap ∈ (ω)ω is a sequence of sets above tp . The ordering
is defined by q ≤ p if tp ⊂ tq , aq ≤ ap , and all sets on the sequence tq \ tp
are unions of some sets on ap . For a condition p ∈ P write [p] to denote the
dented cube obtained from tp , ap . Quite obviously, the poset P adds a generic
sequence ẋgen ∈ (ω)ω which is the union of the first coordinates of conditions
in the generic filter. To define the associated σ-ideal, let I be the collection of
Borel subsets of (ω)ω which do not contain a dented strong Milliken cube. The
following seems to be folklore knowledge:
Theorem 5.11.2. The poset P is proper. The collection I is a σ-ideal and the
map p 7→ [p] is an isomorphism between P and a dense subset of PI .
Proof. The argument is best channeled through the following useful fact.
Claim 5.11.3. Whenever M is a countable elementary submodel of a large
enough structure, a ∈ (ω)ω is a P -generic sequence over M and b ≤ a, then b
is a P -generic sequence over M as well.
Proof. We will show that for every condition p ∈ P and every open dense set
D ⊂ P , there is a condition q ≤ p with tq = tp such that for every c ≤ ta
q aq
there is a number n such that h(c n), aq \ ni ∈ D. The lemma will follow: if
D ∈ M then the sequence a must pass though a condition q ∈ M as above by
genericity, and then every sequence c ≤ a passes through a condition in D ∩ M
by the properties of q.
170
CHAPTER 5. CLASSES OF σ-IDEALS
To find the condition q ≤ p, assume for simplicity that tp = 0. Use a simple
fusion argument to find a sequence a0p ≤ ap such that for every n ∈ ω and every
sequence u ≤ a0p n, either (a) hu, a0p \ ni ∈ D or (b) there is no sequence c ≤ a0p
such that hu, bi ∈ D. Now let (a0p )ω = B0 ∪ B1 be the partition into two Borel
sets defined by c ∈ B0 if for some n ∈ ω, (a) above occurs for u = c n. The
Milliken theorem shows that one of the sets B0 , B1 contains a strong Milliken
cube. If aq ≤ a0p is such that (aq )ω ∈ B0 then the condition q = htp , aq i is
as required. On the other hand, the set B1 cannot contain a strong Milliken
cube: if aq ≤ a0p is such that (aq )ω ∈ B1 and r ≤ htp , aq i is a condition in the
open dense set D, then (a) must have occurred for u = tr \ tp , contradicting the
assumption that (aq )ω ⊂ B1 .
To finish the proof of the theorem, let J be the collection of those Borel sets
B such that P ẋgen ∈
/ Ḃ. The collection J is a σ-ideal; we will show that
J = I. It is clear that J ⊆ I: if B ∈
/ I then there is a condition p ∈ P such
that [p] ⊂ B, thus p ẋgen ∈ B and B ∈
/ J. To show that I ⊆ J, look at
an arbitrary Borel set B ∈
/ J and a condition p ∈ P forcing ẋgen ∈ Ḃ. Let M
be a countable elementary submodel of a large enough structure and let b be a
P -generic sequence over M meeting the condition p. Let q ≤ p be the condition
defined by tq = tp and aq = b \ tp ; we will show that [q] ⊂ B and therefore
B∈
/ I. Indeed, by the claim every sequence c ≤ b is generic over M , so every
sequence c ∈ [q] is generic over M meeting the condition p, so M [c] |= c ∈ B by
the forcing theorem and c ∈ B by the analytic absoluteness between the models
M [c] and V .
Thus, I is a σ-ideal; the map p 7→ [p] is a bijection between P and a dense
subset of PI that is also an isomorphism of ≤ and ⊆. Finally, the properness
of the poset P also follows abstractly from the claim. If B ∈ PI is an Ipositive Borel set and M is a countable elementary submodel of a large structure
containing the set B, we must show that the set {x ∈ B : x is PI -generic over
M } is I-positive. Find a condition p ∈ M such that [p] ⊂ B, and find a sequence
x ∈ (ω)ω generic over M which meets p. Let q = htp , x \ tp i. The claim shows
that [q] consists only of points generic over M and meeting the condition p. As
[q] ∈
/ I, the proof is complete.
The canonization properties of strong Milliken cubes are much more complicated than those of weak Milliken cubes. We will just describe the part of
the behavior which is parallel to the canonization properties of Ramsey cubes.
Regarding the smooth equivalence relations, Klein and Spinas proved a parallel of the Prömel-Voigt canonization theorem. They identified a well-organized
collection Γ of continuous maps from (ω)ω to Vω+1 such that, writing Eγ for the
equivalence relation {hx, yi ∈ (ω)ω × (ω)ω : γ(x) = γ(y)}, the following holds:
Fact 5.11.4. [34] For every a ∈ (ω)ω and every smooth equivalence relation E
on (a)ω there is b ≤ a and γ ∈ Γ such that E (b)ω = Eγ .
Among the countable equivalence relations on the space (ω)ω , the relation
Etail stands out. It connects two sequences a, b if they have the same tail,
5.11. STRONG MILLIKEN CUBES
171
that is, if there is an integer n such that for all but finitely many m ∈ ω,
a(m) = b(m + n). The first canonization theorem elaborates on the central
position of Etail .
Theorem 5.11.5. For every a ∈ (ω)ω and every countable equivalence relation
E on (a)ω there is b ≤ a such that E (b)ω ⊆ Etail .
Proof. The whole argument follows closely the proof of the first sentence of
Theorem 5.9.5, with one additional partition trick. For b, c ∈ (ω)ω write c ≤∗ b
if for some tail c0 of c, c0 ≤ b.
Claim 5.11.6. Whenever c ∈ (ω)ω and f : (c)ω → (ω)ω is a Borel function
such that for every e ≤ c, f (e) 6≤∗ e, then there is a sequence d ≤ c such that
for every e ≤ d, f (e) 6≤∗ d.
S
S
Proof. First, divide (c)ω to two Borel classes: e ∈ B0 if n f (e)(n) \ n e(n) is
a finite set, and e ∈ B1 otherwise. The Milliken theorem divides the treatment
into two cases.
Case 1. There is c0 ≤ c such that (c0 )ω ⊂ B1 . In this case, for every e ≤ c0 ,
write eev for the sequence of even entries of e, and eod for the sequence
of odd
S
0 ω
entries
of
e,
and
partition
(c
)
into
two
Borel
classes:
e
∈
C
if
f
(e
))(n)
∩
0
ev
n
S
e
(n)
is
finite
and
e
∈
C
otherwise.
The
remainder
of
the
argument
is
1
n od
identical with S
Claim 5.9.6; S
we will find d ≤ c0 such that for every e ≤ d it is
the case that n f (e)(n) \ n d(n) is an infinite set for all e ≤ d. Therefore,
f (e) 6≤∗ d as required.
Case 2. There is c0 ≤ c such that (c0 )ω ⊂ B0 . In this case, for every e ≤ c0
write eco for the sequence he(2n) ∪ e(2n + 1) : n ∈ ωi. Since f (eco ) 6≤∗ eco and
Case 2 prevails, it must be the case that for infinitely many n, u = f (eco )(n)
is a finite set that splits some v = eco (mn ), (meaning that v has nonempty
intersection with both u and its complement) and this can happen in one of
four ways. Writing v0 = e(2mn ) and v1 = e(2mn + 1) (so that v = v0 ∪ v1 ) it
must be that either (a) u splits v0 , or (b) u splits v1 , (c) v0 ⊂ u and v1 ∩ u = 0,
or (d) v0 ∩ u = 0 and v1 ⊂ u. one of these four cases will repeat infinitely many
times, and we partition (c0 )ω into corresponding four Borel pieces. The Milliken
theorem divides the treatment in four corresponding subcases, of which we treat
the third, the others being similar.
Case 2c. There is d0 ≤ c0 such that for every e ≤ d0 there are infinitely
many n for which there exists mn such that e(2mn ) ⊂ f (eco )(n) and e(2mn +
1) ∩ f (eco )(n) = 0. Let d = d0co ; we claim that d works as in the claim. Indeed,
if e ≤ d, then write ē ≤ d0 for the sequence defined by ē(2m + 1) = d0 (k) for
the largest k such that d0 (k) ⊂ e(m), and ē(2m) = e(m) \ ē(2m + 1); note
that m must be odd as e ≤ d. By the homogeneity of the sequence d0 there
are infinitely many n for which there is mn such that ē(2mn ) ⊂ f (e)(n) and
ē(2mn +1)∩f (e)(n) = 0. A review of the definitions will show that then f (e)(n)
splits the set d( m−1
2 ). Since this happens for infinitely many n, we conclude that
f (e) 6≤∗ d as required.
172
CHAPTER 5. CLASSES OF σ-IDEALS
The rest of the proof is a literal repetition of the argument for Theorem 5.9.5.
We just state the key claim and leave the rest to the reader.
Claim 5.11.7. Let M be a countable elementary submodel of a large structure
and a ∈ (ω)ω a P -generic sequence over M . For any two sequences b, c ≤ a,
M [b] = M [c] ↔ b Etail c.
5.12
Products of superperfect sets
As another application of Milliken theorem, we consider the two-dimensional
product of Miller forcing. Otmar Spinas [56] computed the associated σ-ideal
and proved a canonization theorem for smooth equivalence relations. We provide
simpler proofs of his results and show that E0 and EKσ are in the spectrum of
the associated σ-ideal.
Let I be the σ-ideal on ω ω σ-generated by compact sets, as in Section 5.1b.
Let J be the collection of Borel subsets B of the space X = ω ω × ω ω such that
B contains no rectangle C × D where C, D ⊂ ω ω are Borel I-positive sets.
Theorem 5.12.1. [56], [24]
1. The collection J is a σ-ideal of Borel sets.
2. The quotient PJ is proper and adds a dominating real.
3. The σ-ideal J is not Π11 on Σ11 .
4. If B ⊂ X is a Borel rectangle with I-positive sides and E a smooth equivalence relation on it, then there exists a Borel subrectangle C ⊂ B with
I-positive sides such that E C either is ee or vertical sections of C are
E-unrelated, or horizontal sections of C are E-unrelated.
The last item is a weak form of canonization for smooth equivalence relations,
and as such speaks about real degrees in the product extension. There, the
generic is given by a pair hxlgen , xrgen i ∈ ω ω × ω ω . Each of the two points is
Miller generic over the ground model, and therefore minimal over the ground
model. The last item says that every real in the extension is either in the ground
model or constructs one of the two coordinates in the generic pair. There are
uncountably many other degrees above the minimal pair, and they have been
classified in [56].
All the items in the theorem have been proved elsewhere. We will provide
Milliken-style proofs that avoid the excessive length of the original arguments.
The proof of (1) is due to Todorcevic.
Proof. Items (1) and (4) use the same trick recorded in the following claim.
ω
Define a function g : (ω)S
→ ω ω × ω ω by g(a) = hx, yi where x is the increasing
enumeration
of
the
set
i a(2i) and y is the increasing enumeration of the set
S
a(2i
+
1).
The
following
easy claim is the heart of our proofs:
i
5.12. PRODUCTS OF SUPERPERFECT SETS
173
Claim 5.12.2. Whenever B is a dented strong Milliken cube, then there are
superperfect trees S, T ⊂ ω <ω such that
1. [S] × [T ] ⊂ g 00 B;
2. for every n ∈ ω and every s, t ∈ S ∪ T with n ∈ rng(s) ∩ rng(t), both s, t
come from the same tree S or T , and they have a common initial segment
u such that n ∈ rng(u).
Proof. For simplicity
S of notation assume that B is a strong Milliken cube, B =
(a)ω . Let ω = s∈ω<ω ds be a partition into infinite sets and consider the
superperfect tree U ⊂ ω <ω consisting of those sequences t such that t(0) = 0
ω
and t(i) ∈ dti for all i ∈ dom(t). For
S each branch x ∈ [U ] let g(x) ∈ ω be
the increasing enumeration of the set i a(2t(i)) and find a superperfect tree S
ω
such that [S] = g 00 [U ]. For
S each branch x ∈ [U ] let h(x) ∈ ω be the increasing
enumeration of the set i a(2t(i) + 1) and find a superperfect tree T such that
[T ] = g 00 [U ]. These two trees work.
To see the first item, let y, z be some branchesSof S and T respectively.
By
\
S induction on j ∈ ω define sets b(2j)S= (rng(y) \ i<j b(2i)) ∩ min(rng(z)
S
b(2i+1)),
and
b(2j
+1)
=
(rng(z)\
b(2i+1))∩min(rng(z)\
b(2i)).
i<j
i<j
i≤j
It is not difficult to see that the sequence b belongs to (a)ω and g(b) = y, z. The
second item is obvious from the construction.
The first item
S of the theorem is now easy. Let S, T be superperfect trees
and [S] × [T ] = i Bi be a partition into countably many Borel sets. We must
inscribe a product of superperfect trees into one of the Borel sets in the partition.
Pulling back through the natural homeomorphisms of [S], [T ], and ω ω , we may
assume that in fact S = T = ω <ω . Now use Theorem 5.11.2 to find a number
i ∈ ω and a dented strong Milliken cube C such that g 00 C ⊂ Bi . It follows from
Claim 5.12.2 that Bi contains a product of superperfect trees as desired.
For the canonization, we will first introduce useful notation. (ω)<ω is the set
of all finite sequences t of nonempty finite subsets of ω such that max(t(n)) <
min(t(n + 1)) whenever n, n + 1 ∈ dom(t). If t ∈ (ω)<ω and a ∈ (ω)ω then
[t, a] = {x ∈ (ω)ω : t is an initial segment of x, and the rest of x is below a in
the Milliken order}. [t, a]1 = {x ∈ (ω)ω : t max(dom(t)) is an initial segment
of x, the next entry on x is the union of the last entry of t with some entries on
a, and the rest of x is below a in the Milliken order}. If x ∈ (ω)ω and n ∈ ω then
x ∩ n ∈ (ω)<ω is the sequence described in the following way: if there is m such
that x(m) ∩ n and x(m) \ n are both nonempty sets, then t = x ma hx(m) ∩ ni;
if no such m exists, then t includes all entries of x which are subsets of n. If
t, s ∈ (ω)<ω and a ∈ (ω)ω , the the expression s ≤ [t, a] denotes the fact that t is
an initial segment of s and the remaining entries of s are unions of some entries
of a.
Now let S, T ⊂ ω <ω be superperfect trees and E be a smooth equivalence
relation on [S] × [T ]. We must find superperfect subtrees s” ⊂ S and T 0 ⊂ T
such that E ([S 0 ] × [T 0 ]) is of the prescribed form. Pulling back the whole
174
CHAPTER 5. CLASSES OF σ-IDEALS
context through the canonical homeomorphisms between ω ω and [S], [T ], we
may assume that in fact S = T = ω <ω .
Let f : ω ω × ω ω → 2ω be a Borel function reducing the equivalence relation
E to the identity. Repeating standard homogenization arguments, we can find
a sequence a ∈ (ω)ω such that for every t ∈ (ω)<ω and every i ∈ ω there is a
tail b of a such that the bit f ◦ g(x)(i) is the same for all x ∈ [t, b], denoted
as π 0 (t)(m), and the bit f ◦ g(x)(i) is the same for all x ∈ [t, b]1 , denoted as
π 1 (t)(i). Thus, π 0 , π 1 are both functions from (ω)<ω to 2ω .
For every t ∈ (ω)<ω consider the equivalence relation Et on [ω]<ℵ0 defined
by u Et v if π(ta u) = π(a v). Thinning out the cube a repeatedly, we may
assume that for every t ≤ a, Et has one of the following four forms:
• u Et v ↔ u = v–we will say that t is id-like;
• u Et v ↔ min(u) = min(v)–we will say that t is min-like;
• u Et v ↔ max(u) = max(v)–we will say that t is max-like;
• u Et v ↔ min(u) = min(v) and max(u) = max(v)–we will say that t is
min max-like;
• u Et v ↔ u = v–we will say that t is ee-like.
Define partitions qod and qev : qod (b) = 0 if there are infinitely many odd
numbers i such that b i is not ee-like, and qod (b) = 1 otherwise; similarly
for qev . Thinning out the sequence a if necessary, we may assume that a is
homogeneous for both partitions.
Case 1. The homogeneous color for one of the partitions qev , qod is 1; for
definiteness assume that it is the partition qev . In this case, we will produce
superperfect trees S 0 and T 0 such that vertical sections of [S 0 ] × [T 0 ] are pairwise
E-unrelated. By induction on n ∈ ω, pick sets c(n) and numbers m(n) so that
I1 if t ≤ hc(i) : i ∈ ni and t is not ee-like then for every u ≤ hc(0), . . . c(n − 1)
the binary sequence π(ta c(n)) is not equal to either π 0 (u) or π 1 (u);
I2 if ta d ≤ hc(i) : i ∈ ni and t is not min or ee-like then same conclusion as
in the previous item;
I3 the number m(n) is so large that the differences indicated in the previous
two items appear already at entries smaller than m(n);
I4 for every sequence u ≤ hc(i) : i ≤ ni, for every x ∈ [u, b] π 0 (u) m(n) ⊂
f ◦ g(x), and for every x ∈ [u, b]1 π 1 (u) m(n) ⊂ f ◦ g(x).
This is not difficult to do. Consider the induction step to construct the
set c(n) and the number m(n). There are only finitely many functions on the
list {π0 (u), π1 (u) : u ≤ hc(i) : i ∈ ni}. On the other hand, the entries of the
sequence a have pairwise distinct minima and maxima, so for every t as in the
first (or similarly second) item, the functions π0 (ta u) : u ∈ rng(a) are pairwise
5.12. PRODUCTS OF SUPERPERFECT SETS
175
distinct, so there must be u ∈ rng(a) such that c(n) = a will satisfy I1 and I2.
Any sufficiently large number m(n) will then work for I3. Lastly, find a small
enough tail of the sequence a so that I4 is satisfied.
Now find superperfect trees S 0 , T 0 as in Claim 5.12.2. These trees work as
required. To verify this, let hx0 , y0 i, hx1 , y1 i ∈ [S 0 ] × [T 0 ] be two points in the
product such that x0 6= x1 ; we must check that f (x0 , y0 ) 6= f (x1 , y1 ). Find
sequences d0 , d1 ≤Sc such that
T hx0 , y0 i = g(e0 ) and hx1 , y1 i = g(e1 ). Since
x0 6= x1 , the set i d0 (2i) ∩ i d1 (i) is finite. The homogeneity property of
a implies that there is a finite sequence t, an initial segment of d0 of even
a
length,
S which is not ee-like, and writing w for the set such that t w ⊂ d0 ,
w ∩ i d1 (i) = 0. There are two distinct cases.
Case 1a. Suppose first that t is min-like. In such a case, let nS0 < n1 <
n2 < · · · < ni be an increasing sequence of numbers such that w = k≤i c(nk )
and look at the construction of the set c(n0 ). Let u = d1 min(cn0 ). Then
f (x1 , y1 ) begins either with π 0 (u) m(n0 ) or π 1 (u) m(n0 ) by I4 at n0 . On the
other hand, f (x0 , y0 ) begins with π 0 (ta w) m(ni ) by I4 at ni ; as t is min-like,
this is the same as π 0 (ta c(n0 )) m(ni ), which begins with π 0 (ta c(n0 )) m(n0 ),
and that sequence is distinct from both π 0 (u) m(n0 ) and π 1 (u) m(n0 ) by
I2 at n0 . Thus, f (x0 , y0 ) 6= f (x1 , y1 ), and the difference between these binary
sequences appears already before m(n0 ). We are done in this case. S
Case 1b. Suppose now that t is not min-like. Again, write w = k≤i c(nk )
and look at the construction of the set c(ni ). Let u = d1 min(c(ni )). As in
the previous item, f (x1 , y1 ) begins either with π 0 (u) m(ni ) or π 1 (u) m(ni ).
On the other hand, f (x0 , y0 ) begins with π 0 (ta w) m(ni ) by I4 at ni , and this
sequence is distinct from both π 0 (u) m(ni ) and π 1 (u) m(ni ) by I2 at ni .
Thus, f (x0 , y0 ) 6= f (x1 , y1 ), and the difference between these binary sequences
appears already before m(ni ). This completes the argument in Case 1.
Case 2. The homogeneous color of both partitions is 0. In this case, we will
present a dented cube on which the function f ◦ g is constant. An application
of Claim 5.12.2 will yield a product of superperfect trees on which E = ee,
completing the proof.
An application of Theorem 5.11.2 shows that there is a dented cube [t, b]
such that every s ≤ ht, bi is ee-like. To simplify the notation, assume t = 0.
The properness of the product was proved in ??? The dominating real is the
maximum of the left generic and the right generic ???
The complexity estimate in the third item follows from [67, Proposition
3.8.15]–if the quotient poset adds a dominating real then the σ-ideal cannot be
Π11 on Σ11 .
Theorem 5.12.3. E0 is in the spectrum of J.
Proof. For any two increasing functions x, y ∈ ω ω let f (x, y) = {n :there is an
element of rng(y) between x(n) and x(n + 1)} and let E be the equivalence
relation connecting hx0 , y0 i and hx1 , y1 i if their f -images have finite symmetric
difference. This is certainly a Borel equivalence relation reducible to E0 ; we will
show that it remains bireducible with E0 on every product of superperfect sets.
176
CHAPTER 5. CLASSES OF σ-IDEALS
Let S, T ⊂ ω <ω be superperfect trees consisting of increasing functions. We
must find a continuous reduction of E0 to E [S] × [T ]. By induction on n ∈ ω
build
• splitnodes sn of the tree S so that s0 ⊂ s1 ⊂ . . . ;
• splitnodes tv : v ∈ 2n of the tree T so that u ⊂ v → tu ⊂ tv ;
• numbers i(n) and j(n) such that i(0) = max(dom(s0 )) < j(0) < i(1) =
max(dom(s1 )) < j(1) < . . . , and for every n and every v ∈ 2n rng(tva 0 \tv )
is a nonempty subset of the interval (sn+1 (in ), sn+1 (in +1)), and rng(tva 1 \
tv ) is a nonempty subset of the interval (sn+1 (jn ), sn+1 (jn + 1)).
This is not difficult to do. Let s0 , t0 be arbitrary splitnodes of the respective trees
S, T . Suppose that sn , tv : v ∈ 2n have been found. Let in = max(dom(sn )).
For every v ∈ 2n find a splitnode tva 0 in the tree T extending tv such that the
number tva 0 (dom(tv )) is greater than all elements of rng(sn ). Find a splitnode
s0n of the tree S extending sn such that the number s0n (in + 1) is larger than
all elements of rng(tva 0 ) for all v ∈ 2n . Let j(n) = max(dom(s0n )). For every
v ∈ 2n find a splitnode tva 1 in the tree T extending tv such that the number
tva 1 (dom(tv )) is greater than all elements of rng(s0n ). Find a splitnode sn+1
of the tree S extending s0n such that the number sn+1 (j(n)) is larger than all
elements of rng(tva 0 ) for all v ∈ 2n . ThisScompletes
S the induction step.
ω
Now for every z ∈ 2 let h(z) = h n sn , n tzn i. It is not difficult to
see that f ◦ h(z) = {in : z(n) = 0} ∪ {jn : z(n) = 1} modulo finitely many
exceptions caused by the interplay between s0 and t0 . Thus, the continuous
function z reduces E0 to E [S] × [T ] as required.
Theorem 5.12.4. EKσ is in the spectrum of J.
5.13
Laver forcing
The Laver forcing P is the partial order of those infinite Laver trees T ⊂ ω <ω
such that there is a finite trunk and all nodes past the trunk split into infinitely
many immediate successors. The associated Laver ideal I on ω ω is generated
by all sets Ag where g : ω <ω → ω is a function and Ag = {f ∈ ω ω : ∃∞ n f (n) ∈
g(f n)}.
Theorem 5.13.1.
1. Every analytic subset of ω ω is either in I or it contains
all branches of a Laver tree, and these two options are mutually exclusive;
2. the forcing P is < ω1 -proper, preserves outer Lebesgue measure, and adjoins a minimal generic extension;
3. I is a game ideal.
5.13. LAVER FORCING
177
Proof. The first item comes from [5], [67, Proposition 4.5.14]. The minimality
of Laver extension is due to Groszek [16, Theorem 7]; it can be also derived
from the general treatment in [67, Theorem 4.5.13]. The Laver game was in its
simplest form presented in [67, Section 4.5].
There are standard corollaries and remarks. The function π : P → PI defined
by π(T ) = [T ], is a dense embedding of the poset P into PI . The study of the
ideal I is complicated by the fact that it is not Π11 on Σ11 . Laver forcing adds
a dominating real; no ideal whose quotient forcing adds a dominating real can
have such a simple definition [67, Proposition 3.8.15]. It is well-known that the
Laver forcing is nowhere c.c.c.: given a Laver tree T ⊂ ω <ω , for every splitnode
t ∈ T write bt ⊂ ω for the infinite set {n ∈ ω : ta n ∈ T }, fix an uncountable
collection {btα : α ∈ ω1 } of pairwise almost disjoint infinite subsets of ω, and let
Aα = {x ∈ [T ]: for every n ∈ ω such that x n is a splitnode of T , x(n) ∈ bxn
α }
for every α ∈ ω1 . The sets {Aα : α ∈ ω1 } are easily checked to be I-positive
and their pairwise intersections are in the σ-ideal I. The computation of the
spectrum of the Laver ideal is then facilitated by the general results such as
Theorem 4.3.8 and Theorem 4.3.7 to yield
Corollary 5.13.2. The σ-ideal I has total canonization for equivalences classifiable by countable structures.
Proposition 3.3.2 then yields
Corollary 5.13.3. Suppose that ω1 is inaccessible to the reals. Then the Laver
ideal has the Silver property for Borel equivalences classifiable by countable structures.
Further canonization properties of the Laver ideal were discovered by Michal
Doucha:
Fact 5.13.4. [10] The Laver ideal has total canonization for equivalence relations Borel reducible to =J , where J is any Fσ P-ideal on ω. If ω1 is inaccessible
to the reals, then the Laver ideal has the Silver property for these equivalence
relations.
Question 5.13.5. Does the Laver ideal have total canonization for equivalence
relations Borel reducible to orbit equivalence relations of Polish group actions?
Further inquiry into the spectrum identifies a nontrivial feature:
Theorem 5.13.6. EKσ is in the spectrum of the Laver forcing.
Proof. Consider the equivalence relation E on the set of increasing functions in
ω ω defined by x E y if and only if there is a number n ∈ ω such that for all m,
x(m) < y(m + n) and x(m) < y(m + n). In other words, a finite shift of the
graph of x to the right is below the graph of y, and vice versa, a finite shift of
the graph of x is below the graph of y.
It is not difficult to show that E ≤B E B for every Borel E-positive set.
Such positive set must contain all branches of some Laver tree T ; to simplify
178
CHAPTER 5. CLASSES OF σ-IDEALS
the notation assume that T has empty trunk. Find a level-preserving injection
π from the set of all finite increasing sequences of natural numbers into T so
that if the last number on s is less than the last number on t then also the last
number on π(s) is less than the last number on π(t). It is immediate that π
extends to a continuous map which embeds E to E B.
The last point is to show that EK
with E. To embed EKσ
Sσ is bireducible
S
in E, look at the two partitions ω = n Kn = n Ln into consecutive intervals
such that |Kn | = n + 1, |Ln | = 2(n + 1) and for a point y ∈ dom(EKσ ) let f (y)
be the function in ω ω specified by f (y)(min(Kn ) + i) = min(Ln ) + i + y(n) for
every n and i ∈ n. It is not difficult
to see that f reduces EKσ to E. To embed
S
E into EKσ , note that E = n Fn is a countable union of closed sets with
compact sections: just write Fn = {hx, yi: for every m ∈ ω, x(m) < y(m + n)
and y(m) < x(m + n). Viewing ω ω as a Gδ subset of a compact metric space
Y and consider the closures F̄n of Fn in Y × Y . Since the vertical sections of
Fn are compact, the closures F̄n have the same sections corresponding to the
points in ω ω as the sets Fn . Thus, Ē, the relation on the space Y defined by
y Ē z if either z, y ∈
/ ω ω or there is n such that hy, zi ∈ F̄n , is a Kσ equivalence
relation equal to E on ω ω . By a result of Rosendal [44], it is Borel reducible to
EKσ and the restriction of that Borel reduction reduces E to EKσ as desired.
Corollary 5.13.7. Whenever J is a σ-ideal on a Polish space X such that the
quotient forcing PJ is proper and adds a dominating real, then EKσ is in the
spectrum of the ideal J.
Proof. The assumptions imply that there is a Borel J-positive set B ⊂ X and
a Borel function f : B → ω ω such that for every point y ∈ ω ω , the set {x ∈ B :
f (x) does not modulo finite dominate y} is in the σ-ideal I. Let π : ω <ω → ω
be a bijection and let g : B → ω ω be the Borel function defined inductively by
g(x)(n) = f (x)(π(x n)).
Observe that g-preimages of sets in the Laver σ-ideal I are in J. For, if
h : ω <ω → ω is any function and Ah ⊂ ω ω is the associated generating set
for the Laver ideal, Ah = {y ∈ ω ω : ∃∞ n y(n) ∈ h(y n)}, then the set
Bh = {x ∈ B : f (x) ◦ π modulo finite does not dominate h} is I-small by the
properties of the function h, and g −1 Ah ⊂ Bh by the definitions.
Let F be the equivalence relation on B defined by x0 F x1 if and only if
g(x0 ) E g(x1 ), where E is the equivalence relation described in the previous
proof. We will show that E witnesses the fact that EKσ is in the spectrum of
J. It is clear that g reduces F to E, which is in turn reducible to EKσ . On the
other hand, if C ⊂ B is a Borel J-positive set, then its g-image is an analytic
I-positive set. A standard argument will then yield a Laver tree T ∈ P and
a continuous function k : [T ] → C such that [T ] ⊂ G00 C and g ◦ k = id. The
continuous function k reduces E [T ] to F C, and since EKσ is reducible
to E [T ] by the previous proof, we have that EKσ is reducible to F C as
desired.
5.14. FAT TREE FORCINGS
179
Theorem 5.13.8. The Laver ideal has a coding function.
Proof. Consider the function f : (ω ω )2 → 2ω defined by f (x, y)(n) = 1 ↔
x(n) > y(n). This function works.
Every I-positive Borel set B ⊂ ω ω contains all branches of some Laver tree
T with trunk of length n, and then it is easy to find, for every binary sequence
z ∈ ω ω , two branches x, y ∈ [T ] so that f (x, y) = z on all entries past n. So
f 00 B 2 contains a nonempty open set.
5.14
Fat tree forcings
The posets consisting of finitely branching trees with some condition enforcing very fast branching have been used in many independence results, such as
[3]. While they are not well known outside of the forcing community, one can
calculate their associated ideals and prove canonization and anticanonization
results.
Let Tini be a finitely branching tree without terminal nodes, and for every
node t ∈ Tini fix a submeasure φt on the set at of immediate successors of t in
Tini , such that limt φt (at ) = ∞. Consider the fat tree forcing P of those trees
T ⊂ Tini such that limt∈T φt (bt ) = ∞, where bt ⊂ at is the set of all immediate
successors of the node t in the tree T ; the ordering is that of inclusion.
This is a special case of the generalized Hausdorff measure forcings of [67,
Section 4.4] or the liminf ∞ tree forcings of [?]. The computation of the associated ideal gives a complete information. The underlying space is X = [Tini ]
and the σ-ideal I on X is generated by all sets Af , where f is a function with
domain Tini associating with each node t ∈ Tini a set of its immediate successors of φt -mass ≤ nf for some number nf ∈ ω. The set Af is defined as
{x ∈ X : ∃∞ n x(n) ∈ f (x n)}. The following sums up the work in [67]:
Theorem 5.14.1. Let φt : t ∈ Tini be submeasures such that limt φt (at ) = ∞.
Then
1. every analytic subset of X either is in the ideal I or contains all branches
of some tree T ∈ P , and these two options are mutually exclusive;
2. the ideal I is Π11 on Σ11 ;
3. I is a game ideal;
4. the forcing P is < ω1 -proper, bounding, adds no independent reals, and
adds a minimal forcing extension.
Proof. The first item comes from [67, Proposition 4.4.14], the second item comes
from [67, Theorem 3.8.9]. The integer game associated with the ideal I is described in [67, Proposition 4.4.14]. The first three assertions in the last item
follow from [67, Theorem 4.4.2, Theorem 4.4.8]. For the minimality of the extension, the selection property and < ω1 -properness imply that every intermediate
180
CHAPTER 5. CLASSES OF σ-IDEALS
extension is c.c.c. by Theorem 3.2.3. Potential c.c.c. intermediate extensions
without new reals are ruled out by Theorem 2.6.9, and intermediate c.c.c. extensions with new reals are ruled out by Fact 2.6.6 as the poset P adds no
independent reals. Thus all intermediate forcing extensions must be trivial!
The usual corollaries apply. The map π : T 7→ [T ] is an isomorphism between
the poset P and a dense subset of the quotient poset PI , and Theorem 4.3.8
and Proposition 3.3.2 yield
Corollary 5.14.2. The ideal I has the Silver property for Borel equivalence
relations classifiable by countable structures.
Further forcing and/or canonization properties very much depend on the choice
of the submeasures φt .
Theorem 5.14.3. There is a choice of submeasures such that the resulting ideal
I has the Silver property for all Borel equivalence relations.
Proof. For a submeasure φ on
S a finite set a let add(φ) be the minimum size of
a set B ⊂ P(a) such that φ( B) ≥ max{φ(b) : b ∈ B} + 1. Submeasures with
large add can be obtained for example by starting with the counting measure
and then inflicting on it the operation φ 7→ log(1 + φ) repeatedly many times.
Now choose a tree Tini with the submeasures φt : t ∈ ω so that there is a linear
ordering of Tini in type ω in which add(φt ) is much bigger than Πs |as | where s
ranges over all nodes of Tini preceding t in the linear order. We claim that this
choice of submeasures works.
In fact, we will show that the resulting σ-ideal I has the rectangular property.
The Silver property then follows by the usual chain of implications: total canonization for analytic equivalence relations is implied by Theorem 3.2.5 and the
Silver property for Borel equivalence relations follows by Proposition 3.3.2. An
alternative direct argument avoiding the use of Theorem
3.2.3 is also possible.
S
So let T, S ∈ P be fat trees and let [T ] × [S] = n Bn be a decomposition of
the rectangle into countably many Borel sets; we have to find fat trees T 0 ⊂ T
and S 0 ⊂ S such that the product [T 0 ] × [S 0 ] is wholly included in one piece of
the partition. Thinning out the trees T and S if necessary we may assume that
they have incomparable trunks. For every node t ∈ T ∪ S beyond the trunks let
bt be the set of immediate successors of t in T , equipped with the submeasure
φt . Consider the space Y = Πt∈T bt and the continuous map f : Y → [S] × [T ]
assigning to every point y ∈ Y the unique pair hx0 , x1 i ∈ X such that for all n
beyond the length of the trunk of the tree T , x0 (n) = y(x0 n), and similarly
on the x1 -side. The creature forcing arguments of [50] show that the collection
J of those Borel subsets of Y which do not contain a subset of the form Πt ct ,
where the sets ct ⊂ bt are nonempty and the numbers φt (ct ) tend to infinity, is a
σ-ideal of Borel sets. Therefore, there are nonempty sets ct ⊂ bt such that their
masses tend to infinity and the f -image of the product Πt ct is wholly contained
in some set Bn of the partition. Consider the trees S 0 ⊂ S and T 0 ⊂ T that
at each of their nodes t branch into immediate successors from the set ct . A
5.14. FAT TREE FORCINGS
181
review of the definitions will show that S 0 , T 0 are fat trees and [S 0 ] × [T 0 ] ⊂ Bn
as desired.
Theorem 5.14.4. There is a choice of submeasures such that E2 and EKσ are
in the spectrum of the associated σ-ideal.
In fact, it will be obvious from the construction that there are very many complicated equivalence relations on the underlying space X. For every Borel equivalence relation E there will be a Borel equivalence relation F on X such that
on every Borel I-positive set B ⊂ X, E ≤ F B holds. In order to see this,
just note that the equivalence relations =K , where K is a Borel ideal on ω and
x =K y if {n : x(n) 6= y(n)} ∈ K for x, y ∈ 2ω , are cofinal in the reducibility
ordering of Borel equivalence relations by Fact 2.2.12, and use these Borel ideals
in place of J in the proof below.
Proof. Use the concentration of measure as in Fact 2.7.2. Find a fast increasing
sequence of numbers ni : i ∈ ω so that in the Hamming cube h2ni , µi , di i, for
every subset of P(2ni ) of cardinality 2Πj∈i 2nj , consisting of sets of µni -mass
at least 1/i, there is a ball of radius 2−i intersecting them all. Consider the
tree Tini consisting of all finite sequences t such that ∀i ∈ dom(t) t(i) ∈ 2ni ,
and for every t ∈ T of length i let φt be the normalized counting measure on
2ni multiplied by i. We claim that the fat tree forcing given by this choice of
submeasures works.
To get EKσ in the spectrum, let J be an Fσ -ideal on 2ω such that =J
is bireducible with EKσ as in the construction after fact 2.2.12. Define an
equivalence E on [Tini ] by setting x E y if there is a set a ∈ J and a number
k ∈ ω such that for every i ∈ ω \ a, 2i dni (x(i), y(i)) < k. This is clearly Kσ , so
reducible to EKσ . We will show that =J is Borel reducible to E B for every
Borel I-positive set B ⊂ [Tini ].
So suppose B is such a Borel I-positive set, and inscribe a fat tree T into
it. Find a node t ∈ T such that past t, the nodes of the tree T split into a
set of immediate successors of mass at least 1. Let j = dom(t). The choice of
the numbers ni shows that for every i > j there is a point pi ∈ 2ni such that
its neigborhood of dni -radius 2−i intersects all sets of immediate successors of
nodes in T extending the node t of length i, and also their flips (a flip of a subset
of 2ni is obtained by interchanging 0’s and 1’s in all of its elements). Now it is
easy to find a continuous function h : 2ω → [T ] such that for every x ∈ 2ω and
every i > j, h(x)(i) is some point within dni -distance 2−i from pi if x(i) = 0,
and h(x)(i) is some point within dni -distance 2−i from the flip of pi if x(i) = 1.
Note that the dni -distance of pi from its flip is equal to 1. This clearly means
that for x, y ∈ 2ω , x =J y if and only if h(x) E h(y), and therefore EKσ is in
the spectrum.
To get E2 in the spectrum, consider the equivalence relation F on Tini defined
by x F y if Σ{1/j + 1 : for some i ∈ ω, ni ≤ j < 2ni and x(i)(j − ni ) 6=
y(i)(j − ni )} < ∞. This is clearly reducible to E2 ; we must show that E2 is
Borel reducible to F B for every I-positive Borel set B ⊂ [Tini ].
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CHAPTER 5. CLASSES OF σ-IDEALS
A direct construction of the embedding is possible by a trick similar to the
previous paragraph. We note that if E2 was not reducible to F B, then
F B would be essentially countable by the E2 dichotomy, the ideal I has
total canonization for essentially countable equivalences, and so there would be
a Borel I-positive subset consisting of either pairwise inequivalent or pairwise
equivalent elements. So it is enough to show that every Borel I-positive set
B contains a pair of equivalent elements, and a pair of inequivalent elements.
Inscribe a fat tree T into B, choose the node t and the points pi ∈ 2ni : i >
dom(t) as in the EKσ case. Note that u, v ∈ 2ni are sequences such that u is
2−i -close to pi and v is 2−i -close to the flip of pi , then Σ{1/j + 1 : ni ≤ j < 2ni
and u(i)(j − ni ) 6= v(i)(j − ni )} > 1/2 − 4 · 2−i , and if u, v are both close to pi
or both close to the flip of pi , then this sum is smaller than 4 · 2−i . Thus, with
the continuous function h constructed in the EKσ argument, h(x) F h(y) if and
only if x E0 y; in particular, there are both F -related and F -unrelated pairs in
[T ] ⊂ B.
5.15
The Lebesgue null ideal
Let λ be the usual Borel probability measure on 2ω , and let I = {B ⊂ 2ω :
λ(B) = 0}. The quotient forcing PI is the random forcing.
Proposition 5.15.1. E0 is in the spectrum of I.
Proof. The classical Steinhaus theorem [3, Theorem 3.2.10] shows that in every
Borel set of positive mass there are two distinct E0 -related points. Thus, Borel
partial E0 -selectors must have Lebesgue mass zero, and since every Borel set
on which E0 is smooth is a countable union of partial Borel E0 selectors, all
such sets must have Lebesgue mass zero. It follows that E0 is non-smooth on
every Borel set B ⊂ 2ω of positive mass, and by the Glimm-Effros dichotomy,
E0 ≤ E0 B as desired.
Proposition 5.15.2. E2 is in the spectrum of I.
Proof. This follows from the treatment of the ideal J associated with E2 of
Section 5.6. As Claim 5.6.9 shows, J ⊂ I, and so E2 is Borel reducible to
E2 B for every Borel non-null set B ⊂ 2ω .
Theorem 5.15.3. If E ⊃ E2 is an equivalence relation classifiable by countable
structures, then E has an equivalence class of λ-mass 1.
Proof. We have no direct proof of this remarkable statement. Rather, we will
find a fat tree forcing as in Section 5.14 such that the associated σ-ideal J
on 2ω is a subset of the null ideal, and the equivalence relation E2 is ergodic
on I. Then, the canonization of equivalence relations classifiable by countable
structures on I as in Corollary 5.14.2 will yield the theorem.
To see this abstract argument, let E ⊃ E2 be an equivalence relation classifiable by countable structures. Observe that E can have at most one J-positive
5.15. THE LEBESGUE NULL IDEAL
183
class. If there were two such classes, then as E2 is ergodic for the σ-ideal J, they
would have to contain E2 -connected points. Since E2 ⊂ E, these two points
would have to be also E-connected, which is impossible if they come from distinct E-classes. Let B be the complement of the single J-positive equivalence
class; we will argue that the set B has zero mass. First of all, it is a coanalytic
set and therefore measurable. If B had nonzero mass, it would contain a Borel
subset B 0 ⊂ B of nonzero mass. The set B 0 is J-positive and its E-equivalence
classes are all in J. Since I has total canonization for equivalence relations
classifiable by countable structures by Corollary 5.14.2, it must be the case that
B 0 contains an J-positive Borel subset B 00 on which E is the identity. Again,
this is impossible by the ergodicity of the equivalence relation E2 ⊂ E: dividing
B 00 into two disjoint Borel J-positive pieces, they should contain E2 -connected
points, which would be also E-connected. This contradiction will complete the
proof of the theorem.
S
For the construction of the σ-ideal J ⊂ I, let ω = n Jn be a partition
of ω into successive finite intervals so long that for every n ∈ ω, 1/n2 ≥
−n−1
/8kn+1
e−2
. Here, kn = Σj∈Jn 1/(j+1)2 . Such a partition exists since the sum
Σj∈ω 1/(j + 1)2 converges. Consider the set 2Jn equipped with the normalized
counting measure µn and the distance dn (x, y) = Σx(j)6=y(j) 1/j + 1. Fact 2.7.4
shows that every subset of 2J−n of µn -mass ≥ 1/n2 has 2−n -neighborhood
of µn -mass greater than 1/2. As in Corollary 2.7.3, if A, B ⊂ 2Jn are two
sets of normalized counting measure at least 1/n2 , then they contain points
x ∈ A, y ∈ B of dn -distance at most 2−n : the 2−n−1 -neighborhoods of the
sets A and B have normalized counting measure greater than 1/2 and therefore
they intersect. Replacing the set A with the flips of its points and repeating
this argument, we also find other points x̄ ∈ A, ȳ ∈ B of dn -distance at least
(Σj∈Jn 1/j + 1) − 2−n .
Consider the fat tree forcing on the initial tree Tini = Πn 2Jn , and the submeasure φn is simply the normalized counting measure on 2Jn multiplied by n2 .
Clearly, every branch through the tree Tini can be identified with the union of
the labels on its nodes, which is an element of 2ω . With this identification, the
generating sets of the associated σ-ideal I have µ-mass 0 by the Borel-Cantelli
lemma, since the sum of the numbers 1/n2 converges. And just as in the proof
of Theorem 5.14.4, one can show that every two trees T, S ∈ P contain branches
x ∈ [T ], y ∈ [S] which are E2 -related, and branches x̄ ∈ [T ] and ȳ ∈ [S] that are
E2 -unrelated.
Proposition 5.15.4. The spectrum of I is cofinal in the Borel equivalence
relations under the Borel reducibility order, and it includes EKσ .
Proof. Look at the space X = (2ω )ω with the product Borel probability measure
instead. Consider the ideal J associated with the product of Sacks forcing, as
in Section 5.7. Then J ⊂ I, the ideal J is suitably homogeneous, and so the
spectrum of J is included in the spectrum of I. This proves the proposition.
The equivalence F2 belongs to the spectrum of the null ideal, and there is an
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CHAPTER 5. CLASSES OF σ-IDEALS
ergodicity theorem parallel to Theorem 5.1.27. Consider the infinite product µ
of countably many copies of the measure λ; so µ is a Borel probability measure
on the space X = (2ω )ω . Let I be the σ-ideal on the space X consisting of sets
of µ-mass 0. The poset P = PI adds a generic point ẋgen ∈ X. Instead of the
product forcings P n for n ∈ ω we will utilize the posets P ⊗n = PI n where In is
the σ-ideal on X n consisting of the set of µn -mass zero, where µn is the product
measure on X n .
Definition 5.15.5. Let M be a transitive model of set theory and a ⊂ 2ω be
a set. We will say that a is a random-symmetric set for M if no finite tuple of
distinct elements of a belongs to any null set coded in M , and moreover, for
every finite set b ⊂ a, every m ∈ ω and every Borel set B ⊂ (2ω )m of nonzero
mass coded in the model M [b], there is an m-tuple c consisting of elements of
ω
a such that c ∈ B. A finite collection ai : i ∈ n of subsets
S of 2 is mutually
random-symmetric over M if no finite tuple of elements
of i ai belongs to any
S
null set coded in M , and for every finite set b ⊂ i ai , every m ∈ ω, every Borel
set B ⊂ (2ω )m of nonzero mass coded in the model M [b] and every i ∈ n, there
is an m-tuple c consisting of elements of ai such that c ∈ B.
In the study of random symmetric sets, the poset Qn is helpful. It is the direct
limit of the sequence hQin : i ∈ ωi, where Qin is the poset of Borel subsets of
(2ω )n×i of nonzero mass, where the measure on (2ω )n×i is just the usual product
measure of ni copies of λ. It is clear that Qn is a c.c.c. poset adding points
ẋj,i : j ∈ n, i ∈ ω such that any finite tuple among them is random generic over
the ground model. It is important to distinguish between the posets P ⊗n and
Qn .
Claim 5.15.6. Let M be a countable transitive model of set theory, and let
ai : i ∈ n be countable subsets of 2ω . The following are equivalent:
1. the sets ai : i ∈ n are mutually random symmetric over M ;
2. there is a sequence hxi : i ∈ ni of points in the space X which is Qn -generic
over M and for every i ∈ n, rng(xi ) = ai .
3. there is a sequence hxi : i ∈ ni of points in the space X which is P ⊗n generic over M and for every i ∈ n, rng(xi ) = ai .
Moreover, if (1) or (2) hold, if yi : i ∈ n are any one-to-one enumerations of
n
the sets ai and q ∈ Qn ∩ M is a condition, the set {~g ∈ S∞
: the sequence
hyi ◦ gi : i ∈ ni is Qn -generic over M and meets q} is nonmeager in the product
n
S∞
.
Corollary 5.15.7. In the poset P ⊗n , the sentences of the forcing language
mentioning only rng(ẋi ) : i ∈ n and some elements of the ground model are
decided by the largest condition.
Corollary 5.15.8. If M is a transitive model of set theory and bi : i ∈Sn are
nonempty subsets of X such that every finite tuple of distinct elements of i bi is
5.15. THE LEBESGUE NULL IDEAL
P ⊗n -generic over the model M , then the sets
random-symmetric.
185
S
x∈bi
rng(x) : i ∈ n are mutually
Theorem 5.15.9. Let E be an analytic equivalence relation on X = (2ω )ω such
that F2 ⊂ E. Either F2 ≤B E or else E has a co-null equivalence class.
Proof. The argument follows the lines of Theorem 5.1.27. There are cases 1
through 4 just as in that proof. The first three are deal with in exactly the
same fashion, replacing product of Cohen forcings with random algebras P ⊗n .
Case 4 (the negation of the first three cases) must be handled through the
following well known (?) claim:
Claim 5.15.10. For every countable elementary submodel M of a large enough
structure, there is a perfect subset of X such that every finite tuple of its pairwise
distinct points is P ⊗n -generic over M for the suitable n.
Proof. Let Q be the poset consisting of all Borel sets B such that for some
n
n = n(B), B ⊂ X 2 has nonzero product mass. Let C ≤ B if n(C) ≥ n(B) and
for every function π : 2n(B) → 2n(C) such that ∀s s ⊂ π(s), it is the case that
n
∀x ∈ C π −1 x ∈ B, where π −1 x = y if y ∈ X 2 is the unique element such that
y(s) = x(π(s)).
Let G be a Q-generic filter over the model M . For every y ∈ 2ω , let Gy
be the collection of those Borel sets B ⊂ X in M such that for some number
n
n ∈ ω, the condition {x ∈ X 2 : x(y n) ∈ B} belongs to the filter G. We must
argue that for distinct elements yk : k ∈ m of 2ω , the filters Gyk : k ∈ m are
P ⊗m -generic over M . Once this is done, the claim follows by considering the
set of all points in X corresponding to some filter Gy : y ∈ 2ω .
Let B ∈ Q be a condition, m ∈ ω be a number smaller than n = n(B),
and let D be a dense set in the random algebra on X m . We will produce a set
C ≤ B such that for every set a ⊂ 2n of size m and every map π : a → m,
writing Ca,π = {x ∈ X m : ∃y ∈ C∀s ∈ a y(s) = x(π(s))}, it will be the case
that Ca,π ∈ D. An obvious density argument will then finish the proof.
The construction of the set C ⊂ B is straightforward. List all the finitely
many possibilities for a, π as ai , πi : i ∈ j and by induction on i ∈ j + 1 build
a decreasing chain of conditions B i below B = B 0 such that n(B i ) = n and
∈ D. The condition C = B j will then be as desired. To obtain the set
Bai+1
i ,πi
n
i+1
B
from B i , let A = {x ∈ X m : the set {y ∈ X 2 \ai : y ∪ (x ◦ pii ) ∈ B i }
has nonzero measure}. This is a Borel set by [33, Theorem 17.25]; the Fubini
theorem shows that it has nonzero mass. Let A0 ⊂ A be a set in D and let
B i+1 = {y ∈ B i : y ◦ πi ∈ A0 }.
Now let M be a countable elementary submodel of a large structure, and
let P ⊂ X be a perfect set consisting of P -generic sequences over M . Let
D ⊂ P ω × X be the
S Borel set consisting of pairs hz, xi such that x enumerates
the countable set n rng(z(n)). We will verify for any two pairs hz0 , x0 i and
hz1 , x1 i in the set D, x0 E x1 if and only if rng(z0 ) = rng(z1 ). The proof is then
completed by a reference to Corollary 5.1.30.
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CHAPTER 5. CLASSES OF σ-IDEALS
Now, if rng(z0 ) = rng(z1 ) then x0 , x1 enumerate the same set, and since
F2 ⊂ E, they must be E connected as desired. On the other hand, assume that
rng(z0 ) 6= rng(z1 ); there are several cases depending on the relationship of the
two ranges. Assume for example that rng(z0 ) ∩ rng(z1 ), rng(z0 ) \ rng(z1 ), and
ω
rng(z1S) \ rng(z0 ) are all nonempty sets. Then the
S sets a0 , a1 , a2 ⊂ 2 defined by
a0 = {rng(y)
S : y ∈ rng(z0 ) ∩ rng(z1 )}, a1 = {rng(y) : y ∈ rng(z0 ) \ rng(z1 )}
and a2 = {rng(y) : y ∈ rng(z1 ) \ rng(z0 )} are mutually random symmetric,
and since Case 2 fails, they are not E-equivalent as desired.
Proposition 5.15.11. F2 is in the spectrum of the null ideal.
Proof. Consider the product measure µ on the space X = (2ω )ω . Proceed as in
the fourth case of the above proof with E = F2 . Clearly, the first three cases
are impossible. Let B ⊂ X be a Borel nonnull set, and let M be a countable
elementary submodel of a large structure containing B. Let f : X → X be
a Borel function preserving F2 such that rng(f ) consists of enumerations of
random symmetric sets for M . Let D ⊂ X × X be the set {hx, yi : y ∈ B and
rng(f (x)) = rng(y)}. Corollary 5.1.30 applied to the Borel set D shows that
F2 ≤B F2 B as required.
5.16
Finite product
???Needs a rewrite???
Fact 5.16.1. [67, Theorem 5.2.6] If the σ-ideals in the collection {In : n ∈ ω}
are Π11 on Σ11 , the quotient forcings are proper, bounding, and preserve the
Baire category, then the collection has the rectangular Ramsey property, and the
product ideal inherits all mentioned properties as well.
The canonization properties of the product ideals may be difficult to identify.
We will start with the finite products; the infinite products are much more
challenging, and they will be treated at the end of this section. Even in the case
of finite product of length n ∈ ω, there are obvious obstacles to canonization: if
Fi : i ∈ n are equivalence relations onQthe coordinates of the product that resist
further simplification, their product i Fi will be impossible to simplify further
on the product. The natural hope is that there are no other obstacles. In this
section, we will exhibit three cases in which this hope comes true.
Theorem 5.16.2. Let n ∈ ω and {Ii : i ∈ n} be Π11 on Σ11 σ-ideals on the
respective Polish spaces {Xi : i ∈ n} σ-generated by closed sets, with the covering
property. Let J denotes the product ideal on the product space. Then for every
sequence hBi : i ∈Qni of respectively Ii -positive Borel sets and every analytic
equivalence
Q
Q E on i Bi there are Borel Ii -positive sets Ci ⊂ Bi such that E i Ci =
i Fi where each Fi is either id or ee.
5.16. FINITE PRODUCT
187
Proof. First observe that the quotient posets are proper, preserve Baire category, and have the continuous reading of names by [67, Theorem 4.1.2]. In
this context, the covering property is equivalent to the bounding property [67,
Theorem 3.3.2], and so by the above Fact, the collection of σ-ideals has the
rectangular Ramsey property, and the product P is proper. Now, let a ⊂ n
be a set. The first step is to compute the reduced product associated with the
equivalence Ea connecting two sequences if they have the same entries on the
set a. Let Pa be the product of PIi : i ∈ a with PIi × PIi : i ∈
/ a. We will
organize the generic sequence into two, ~xlgen ∈ (2ω )n and ~xrgen ∈ (2ω )n which
use the PIi -generic point for i ∈ a, and the left or right PIi -generic point for
i∈
/ a; thus, ~xlgen Ea ~xrgen .
It is not difficult (and not necessary for the proof of the theorem) to show
that Pa is naturally isomorphic to a dense subset of the reduced product P ×Ea P
via the map π : r 7→ hhr(i) : i ∈ a, rlef t (i) : i ∈
/ ai, hr(i) : i ∈ a, rright (i) : i ∈
/ aii.
Note that the range of π is a subset of P ×Ea P and π preserves the ordering.
To prove the density of the range of π, note that if hp, qi ∈ P ×Ea P is a
condition (a pair of conditions in P such that some large forcing introduces a
P -generic sequence to each which are Ea -related), then for every i ∈ a the set
p(i) ∩ q(i) must be an Ii -positive Borel set, and so the sequence r given by
r(i) = p(i) ∩ q(i) if i ∈ a and rlef t (i) = p(i) and rright (i) = q(i) if i ∈
/ a is an
element of Pa mapped by π below the pair hp, qi.
The next step is a version of the mutual genericity for the product.
Lemma 5.16.3. Let M be a countable elementary submodel of a large structure,
and let {Bi : i ∈ n} be Ii -positive Borel sets in the model M . Let a ⊂ n be a
set. There are Ii -positive Borel subsets {Ci ⊂ Bi : i ∈ n} such that the product
Πi∈n Ci consists of pairwise reduced product generic sequences for the model M .
This is to say that any two sequences ~x, ~y ∈ Πi∈n Ci satisfying x(i) = y(i) ↔
i∈
/ a, are reduced product generic for the model M .
Proof. We must first introduce the game-theoretic characterization of the forcing properties of the quotient posets PI as described in [67, Section 3.10.10].
The game GI associated with I between Players I and II starts with Player
II indicating an initial I-positive Borel set Bini ; then the game has ω many
megarounds, and each of the megarounds consists of finitely many rounds, in
each of which Player I plays an I-positive Borel set and Player II answers with
its Borel I-positive subset. At some point of his choice, Player I decides to end
the megaround and pass to a next one. Given a play p of finite length at the
end of which Player I ends its last megaround, write [p] for the intersection of
Bini with all the sets obtained as unions of all Player II’s answers in a fixed
megaround of p; similar usage will prevail for infinite plays p. Player I wins an
infinite play p if he ended infinitely many megarounds and [p] ∈
/ I. In what
follows, the letters p, q, . . . will denote either infinite plays of this game or finite
plays at the end of which Player I decided to end the current megaround.
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CHAPTER 5. CLASSES OF σ-IDEALS
Fact 5.16.4. [67, Theorem 3.10.24] Suppose that the σ-ideal I is Π11 on Σ11
and the quotient poset is proper. The following are equivalent:
1. PI preserves Baire category and is bounding;
2. Player I has a winning strategy in the game G.
We conclude that for every i ∈ n Player I has a winning strategy σi in the game
GIi . This provides ammunition for various fusion arguments like the following
claim:
Claim 5.16.5. Let pi : i ∈ n be finite playsQaccording to the respective strategies
σi , and let D be an open
subset of i PIi . Then there are extensions p0i
Q dense
0
of pi such that the set i [pi ] is covered by finitely many elements of D.
S
Now, write n × ω <ω = m Tm as an increasing union of finite sets Tm such
that for every m, i the set {t : hi, ti ∈ Tm } is closed under initial segments. By
induction on m build functions fm with the domain Tm such that
I1 for every hi, ti ∈ Tm the value fm (i, t) is a finite play in the model M
respecting the strategy σi , and fm (i, 0) starts with Bi ;
I2 for m < m0 and hi, ti ∈ Tm , the play fm0 (i, t) extends fm (i, t);
I3 for if hi, ti ∈ Tm and hi, ta ji ∈ Tm+1 \ Tm and [fm (i, t)] has nonempty
intersection with the j-th open set in some fixed enumeration of a basis
for the space Xi , then [fm+1 (i, ta j)] is a subset of this intersection;
I4 whenever a ⊂ n is a set, and {hi, ti i : i ∈ a} ∪ {hi, t0i i, hi, t1i Q
i:i∈
/ a} are
0
1
nodes
of
T
such
that
t
=
6
t
for
all
i
∈
/
a,
then
the
product
[f
m
m (i, ti )] ×
i
i
Q
Q
0
1
[f
(i,
t
)]
×
[f
(i,
t
)]
is
covered
by
finitely
many
conditions
in
m
i
i
i∈a
/
i∈a m
the m-th dense subset of Pa in the model M under some fixed enumeration.
The induction process is trivial, using the claim repeatedly to assure that I4
holds. In the end, for every i ∈ n and every y ∈ ω ω look at the filter generated
by the conditions {[fm (i, y k)] : m is the least such that hi, y ki ∈ Tm }. This
is an M -generic filter for PI and so there is a unique point gi (y) ∈ Bi in it. We
will first argue that rng(gi ) is an Ii -positive set.
Indeed, let {Ak : k ∈ ω}Sbe closed sets in the σ-ideal Ii ; we must produce
y ∈ ω ω such that gi (y) ∈
/ k Ak . By induction build an increasing sequence
of nodes tk ∈ ω <ω of length k such that [fm (i, tk )] ∩ Ak = 0, where m is the
the least
S such that hi, tk i ∈ Tm . The induction step is arranged by noting
that [ m fm (i, tk )] is an Ii -positive set and therefore has Ii -positive intersection
with some basic open set disjoint from Ak (this is the only place where we use
the assumption that the generators of Ii are closed). If this basic open set is
a
enumerated by some j then the
S induction hypothesis at k + 1Swith tk+1 = tk j
will hold. In the end, let y = k tk and observe that gi (y) ∈
/ k Ak as desired.
Now let Ci be some Borel Ii -positive subset of the analytic Ii -positive set
rng(gi ). The induction hypothesis I4 above immediately implies that the sets
Ci have the required mutual genericity property.
5.17. THE COUNTABLE SUPPORT ITERATION IDEALS
189
The theorem immediately Q
follows from the lemma. Let E be a Borel equivalence relation on the space i Xi , and find an inclusion minimal set a ⊂ n
such that some condition in Pa forces ~xlgen E ~xrgen . If a condition p =
hBi : i ∈ (n \ a), Bi : i ∈ a, Ci : i ∈ ai in Pa forces this, then so does
q = hBi : i ∈ (n \ a), Bi : i ∈ a, Bi : i ∈ ai: whenever ~xlgen , ~xrgen are generic
sequences meeting the condition q, in a further generic extension one can find a
condition ~ygen such that the pairs ~xlgen , ~ygen and ~xrgen , ~ygen are generic sequences
meeting the condition q, so these pairs consist of E-equivalent sequences by the
forcing theorem, and consequently ~xlgen E ~xrgen by the transitivity of the relation E. Now let M be a countable elementary submodel of a large structure
and use the claim to find sets Ci ⊂ Bi : i ∈ n whose product consists of
pairwise reduced product generic sequences for the model M . We claim that
E Πi∈n Ci = Ea .
To show this, suppose that ~x, ~y ∈ Πi∈n Ci are sequences and let b = {i ∈ n :
~x(i) = ~y (i)}. Find sequences ~x0 , ~y 0 in the product such that a = {i ∈ n : ~x(i) =
~x0 (i)} = {i ∈ n : ~y (i) = ~y 0 (i)} and a ∩ b = {i ∈ n : ~x0 (i) = ~y 0 (i)}. The forcing
theorem applied to the reduced product Pa together with Borel absoluteness
implies that ~x E ~x0 and ~y E ~y 0 . Now if a ∩ b = a then the forcing theorem also
implies that ~x0 E ~y 0 and so ~x E ~y . on the other hand, if a ∩ b 6= a then the
minimal choice of the set a together with the forcing theorem applied to Pa∩b
give ¬~x0 E ~y 0 and ¬~x E ~y !
Things get complicated very quickly beyond the relatively simple case of
the previous theorem. We are still able to deal with the smooth σ-ideal of
Section 5.4.
5.17
The countable support iteration ideals
The countable support iteration is a central operation on forcings since the
pioneering work of Shelah [49]. It corresponds to a certain operation on σideals, the transfinite Fubini power, as shown in [67]. The natural question then
arises whether this operation preserves natural canonization properties of the
σ-ideals in question.
The question may be natural, but it seems to be also very hard, even for
iterations of length two in the general case. We will prove a rather restrictive
preservation result–Theorem 5.17.12. For Π11 on Σ11 σ-ideals I whose quotient
forcing PI has a rather strong preservation property (the weak Sacks property),
the equivalence relations reducible to EKσ in the countable support iteration
simplify down to an unavoidable and fairly simple collection of obstacles to canonization. The weak Sacks property may look quite restrictive, but in practice a
good number of forcings satisfy it, it is easy to check and some forcings with this
property have nontrivial features in the spectrum–the E0 , E2 , or Silver forcings
of previous sections do satisfy the weak Sacks property. We will also show that
190
CHAPTER 5. CLASSES OF σ-IDEALS
in a certain more general case, very complex equivalence relations arise already
at the first infinite stage of the iteration; on the other hand, in the countable
support iterations of Sacks forcing a much stronger canonization result holds.
5.17a
Iteration preliminaries
Let us first recall the analysis of the countable support iteration of definable
proper forcing as introduced in [67, Section 5.1]:
Definition 5.17.1. [67, Definition 5.1.1] Let X be a Polish space and I a σ-ideal
on X. Let α ∈ ω1 be a countable ordinal. The space X α of all α-sequences of
elements of X is equipped with the product topology. The α-th Fubini iteration
I α of I is the σ-ideal on the space X α generated by sets Af , where f : X <α → I
is an arbitrary function and Af = {~x ∈ X α : ∃β ∈ α ~x(β) ∈ f (~x β)}.
Definition 5.17.2. Let I be a σ-ideal on a Polish space X and J a σ-ideal on
a Polish space Y . The symbol I ∗ J stands for the Fubini product of I and J:
I ∗ J = {B ⊂ X × Y : {x ∈ X : {y ∈ Y : hx, yi ∈ B} ∈
/ J} ∈ I}}.
If I, J are Π11 on Σ11 σ-ideals such that the quotient forcings are proper
and every analytic positive set contains a Borel positive subset, then the twostep iteration of posets PI and PJ is naturally equivalent to the quotient PI∗J .
This pattern persists to transfinite iterations. We will need a canonical form of
I α -positive Borel set:
Definition 5.17.3. [67, Definition 5.1.5] An I, α-tree is a tree p ⊂ X <α such
that each node extends into an I-positive set of immediate successors and the
tree is closed under unions of inclusion-increasing sequences of nodes whose
height is bounded below α.
Fact 5.17.4. [67, Section 5.1.3] Suppose that
1. the σ-ideal I is Π11 on Σ11 and every analytic I-positive set contains an
I-positive Borel subset;
2. the quotient forcing PI is provably proper.
Then
1. the σ-ideal I α is Π11 on Σ11 , and every analytic I α -positive set contains an
I α -positive Borel subset;
2. the coutable support iteration PIα of PI of length α is in the forcing sense
equivalent to the quotient PI α ;
3. every I α -positive analytic set contains all branches of a Borel I, α-tree.
If suitable large cardinals exist, the above fact holds even for σ-ideals with definition more complicated than Π11 on Σ11 , obviously except for the part of the
conclusion that evaluates the complexity of the Fubini iteration ideal. This
5.17. THE COUNTABLE SUPPORT ITERATION IDEALS
191
makes it possible to apply the analysis to forcings such as Laver or Mathias,
whose associated ideals are more complicated. Large cardinals or other assumptions transcending ZFC are necessary for the argument to go through in the case
of Laver forcing. The countable support iterations in which the iterands vary
can be handled in a quite similar way, and we will not bother to explicitly state
the trivial adjustments necessary.
Let I be a σ-ideal on a Polish space X and let α ∈ ω1 be a countable
ordinal. There are obvious obstacles to canonization of analytic equivalence
relations in the iterated Fubini power I α on the product Polish space X α . Let
β ≤ α be an ordinal; define idβ to be the identity on X β and idβ × ee to be the
equivalence relation on X α that relates the sequences with identical initial βsegments. These are smooth equvalence relations, and their associated models
correspond to the initial segment of the iteration of length β. Also, if E is
an analytic equivalence relation on the space X which cannot be simplified
considerably on a Borel I-positive set, we must consider the equivalences idβ ×
E × ee for each β ∈ α. These are defined as ~x (idβ × E × ee) ~y if ~x β = ~y β
and ~x(β) E ~y (β). Other obstacles are not immediately apparent, but they
seem to be difficult to exclude even already in the case of two step iteration!
The main result of this section, Theorem 5.17.12, asserts that in a certain fairly
special case, there are no other obstacles to canonization of equivalence relations
reducible to EKσ .
5.17b
Weak Sacks property
In order to state and prove our main result, we must define a rather common
preservation property, provide its strategic characterizations, and apply them
to prove common preservation theorems.
Definition 5.17.5. A forcing P has the weak Sacks property if for every function f ∈ ω ω in the extension there is a ground model infinite set a ⊂ ω and a
ground model function g with domain a such that for every n ∈ a, |g(n)| ≤ 2n
and f (n) ∈ g(n).
This is the obvious weakening of the Sacks property that requires the infinite set a to be actually equal to ω [3, Definition 6.3.37]. It is not difficult
to show that a great number of posets have the weak Sacks property, such as
every quotient PI if I is σ-generated by a σ-compact family of compact sets of
the ambient space, the limsup creature forcings of [?] , the E0 forcing, the E2
forcing, or the Silver forcing. The weak Sacks property implies the bounding
property and preservation of Baire category, which provides a host of nonexample posets. The poset of σ-splitting sets [58] is proper, bounding, and preserves
Baire category, and still does not have the weak Sacks property. It follows from
[68] that if I is a Π11 on Σ11 σ-ideal such that the quotient PI is bounding, then
PI preserves P-points if and only if PI adds no independent reals and has the
weak Sacks property.
In order to state and prove our main result, we cannot avoid a closer study
of the weak Sacks property. Our central reference for definable forcing [67]
192
CHAPTER 5. CLASSES OF σ-IDEALS
does not mention it, since this is a new development; thus, we will have to
provide the basic proofs. As is the case with other standard forcing preservation
properties [67, Section 3.10], the treatment with the weak Sacks property hinges
on its strategic characterizations. We will introduce three different games which
characterize the weak Sacks property in definable forcing. The first one speaks
directly about posets and names for natural numbers, the last two use a quotient
form of the poset in question.
Definition 5.17.6. Let P be a partial order. The game G1 (P ) between players
I and II proceeds in infinitely many rounds in the following way. Player II first
indicates an initial condition pini ∈ P . After that, in round n Player I indicates
a natural number in ∈ ω, Player II plays a P -name τn for a natural number,
and Player I answers with a set an ⊂ ω of size in . In the end, Player I wins if
there is a condition q ≤ pini forcing τn ∈ ǎn for all n ∈ ω.
Definition 5.17.7. Let I be a σ-ideal on a Polish space X. The game G2 (I)
between Players I and II proceeds in infinitely many rounds. In the beginning,
Player II indicates an I-positive Borel set Kini ⊂ X. At round n, Player I plays
a finite set Kn ⊂ Kini and Player II answers with an open set On ⊂ X. The
players must
T observe the rules Kn ⊂ On and Kn+1 ⊂ On . Player I wins if the
set Kini ∩ n On is I-positive.
Definition 5.17.8. Let I be a σ-ideal on a Polish space X. The game G3 (I)
between Players I and II proceeds in the same way as G2 (I), except in round
n Player II first chooses a set An ∈ I, Player I answers with a finite set Kn ⊂
Kini \ An and finally Player II indicates an open set On ⊂ X.
In the last two games, Player II will certainly not hurt himself by playing
smaller open sets, so we may assume that his open sets come from some fixed
countable basis closed under finite intersections, and the closure of On+1 is a
−n
subset of On , and each point of On is within
T 2 -distance of some point of Kini .
That way, the result of the play, the set n On , is a compact subset of Kini .
Theorem 5.17.9. Suppose that I is a σ-ideal on a Polish space X such that
the quotient forcing PI is proper and bounding. The following are equivalent for
l = 1, 2, 3:
1. PI has the weak Sacks property;
2. Player II has no winning strategy in the game Gl (I).
Proof. Start with l = 1. The nonexistence of a winning strategy for Player II
in G1 (I) easily leads to the weak Sacks property; given a name f˙ for a function
in ω ω and a condition p ∈ PI , consider the strategy of Player II consisting
of playing pini = p and f˙(m) whenever Player I plays the natural number m.
This is not a winning strategy, and the resulting condition of any counterplay
winning for Player I will force the function f˙ into a tunnel of the required sort.
On the other hand, the weak Sacks property implies that Player II cannot have
5.17. THE COUNTABLE SUPPORT ITERATION IDEALS
193
a winning strategy in G1 (PI ). If σ were such a putative winning strategy with
an initial move pini ∈ P , consider the name f˙ which assigns each m the finite
sequence of all names the strategy σ answers to all finite plays of length ≤ m
in which all natural number moves of Player I were of size ≤ m and all of his
set moves were subsets of m. Use the weak Sacks property to find a condition
q ≤ pini , an infinite set a ⊂ ω and a ground model function h with domain a
such that for every m ∈ a, the size of h(m) is at most 2m and q f (m) ∈ h(m)
(as a finite sequence of natural numbers). Now by induction build counterplays
t0 ⊂ t1 ⊂ . . . against the strategy σ such that ti ends at round i with moves
mi , bi of Player I and τi of Player II, and q τi τi ∈ b̌i . This way, Player
II must win as witnessed by the condition q. The induction step is simple:
once ti has been constructed, find m ∈ a larger than the maximum of all sets
and numbers played along ti , let Player I extend ti by mi+1 = m, and when
the strategy σ answers with τi+1 , let bi+1 be the set of all numbers that the
function h permits as possible values of the name τi+1 . The set τi+1 is of size
at most 2m , and the induction can proceed.
The case l = 2 is strictly easier than l = 3, and the game G2 is mentioned
here only because it also appeared in the pivotal paper [69]. The game G3 seems
to be much more difficult for Player I than G2 ; essentially we are saying that
in game G2 Player I has very many winning moves in each round. Suppose for
contradiction that PI has the weak Sacks property and Player II has a winning
strategy σ in the game G3 , beginning with an initial move Kini . Let M be
a countable elementary submodel of a large structure containing all objects
mentioned so far, and let L ⊂ Kini be a compact I-positive set consisting of
M -generic points only, such that all sets in M are relatively clopen in L. Such a
set exists by the properness and boundedness assumption [67, Theorem 3.3.2].
Consider a natural number m ∈ ω and a finite partial play t ∈ M in which
Player II follows the winning strategy, ending with some open set On−1 and an
I-small set An . Since An ∈ M , any m-tuple from the compact set L ∩ On−1
is a legal move of Player I after t. A compactness argument with the space
(L ∩ On−1 )m shows that there is a finite collection R(t, m) consisting of basic
open subsets of X such that every 2m -tuple of L∩On−1 is contained in one of the
sets in R(t, m) and each of the sets in R(t, m) can be provoked as the next move
of the strategy σ after some m-tuple of points in L ∩ On−1 is played as the next
move of Player I. Let S(t, m) be the set of open sets with nonempty intersection
with L such that each set in R(t, m) is either disjoint from them or a superset
of them. Observe that whenever {Pi : i ∈ 2m } are sets in S(t, m) then there is
a move Kn of Player ISwhich provokes the strategy σ to answer with a set On
which is a superset of i Pi : just pick a point xi ∈ Li ∩ Pi for each i ∈ 2m , note
that there must be an open set O ∈ R(t, m) with {xi : i ∈ 2m } S
⊂ O, and since
∀i ∈ 2m O ∩ Pi 6= 0, it must be the case that ∀i ∈ 2m Pi ⊂ O, i Pi ⊂ O, and
O = On will work. By the elementarity of the model M , there must be aSplay
in the model M extending t by one round such that its last move covers i Pi .
Enumerate all finite plays in the model M as htk : k ∈ ωi, and let f˙ be
the name from ω to basic open subsets of X defined by f˙(m) =some sequence
194
CHAPTER 5. CLASSES OF σ-IDEALS
hPk : k ∈ mi such that ∀k Pk ∈ S(tk , m) ∧ ẋgen ∈ Pk . By the weak Sacks
property, there is a compact I-positive set L0 ⊂ L, a ground model infinite set
a ⊂ ω and a ground model function h such that for every m ∈ a, |h(m)| = 2m
and L0 f˙(m) ∈ h(m). We will show that every finite play t ∈ M following
the strategy σ whose last open set move covers L0 can be extended by one more
round to a play in M following σ such that the last open set move still covers
L0 . This will yield an infinite play against the strategy σ whose resulting set
covers L0 , so is I-positive, contradicting the assumption that σ was a winning
strategy. So, find a number m ∈ a larger than some k such that t = tk . The
collection A = {O ∈ S(tk , m)S: O is k-th element of some sequence in h(m)} has
size at most 2m , and L0 ⊂ A. As the previous paragraph shows, there is a
move in the model M that, S
after t has been played, provokes the strategy σ to
respond with a superset of A, and that is our required one round extension
of the play t.
Theorem 5.17.10. Suppose that I is a Π11 on Σ11 σ-ideal on a Polish space
X such that the quotient forcing PI is proper and bounding. Then the games
G1 (PI ), G2 (I), G3 (I) are all determined.
Proof. This is just a standard unraveling argument as those in [67, Section
3.10].
Thus, we can conclude that the weak Sacks property in suitably definable
forcing is equivalent to Player I having a winning strategy in any (or all) games
defined above. The first game leads to a rather standard corollary: the weak
Sacks property is preserved under the countable support iteration of definable
forcing. The third game will be used to provide a strong canonization result for
such iterations.
Theorem 5.17.11. Let I be a Π11 on Σ11 σ-ideal on a Polish space X such that
the quotient poset is provably proper and has the weak Sacks property. Then
the countable support iterations of PI of arbitrary length has the weak Sacks
property as well.
Proof. This is a simple strategy diagonalization argument completely parallel
to the proofs in [67, Section 6.3.5] and we omit it.
5.17c
The preservation result
It is now time to state the main result of this section.
Theorem 5.17.12. Let I be a σ-ideal on a compact metric space X such that I
is Π11 on Σ11 , every I-positive analytic set has a Borel I-positive subset, and the
quotient poset PI is provably proper and has the weak Sacks property. Let α ∈ ω1
be a countable ordinal. Then, for every Borel I α -positive Borel set B and every
equivalence relation E on B reducible to EKσ there is an ordinal β ≤ α and
an equivalence F on X reducible to EKσ and a Borel I α -positive subset C ⊂ B
such that on C, E = Eβ × F .
5.17. THE COUNTABLE SUPPORT ITERATION IDEALS
195
Proof. The proof of the theorem goes by induction on α, and starts with the
investigation of the successor step, where in fact most of the difficulty lies.
Claim 5.17.13. Let I, J be Π11 on Σ11 σ-ideals on Polish spaces X, Y such that
the quotients PI , PJ are proper and have the weak Sacks property. Write K for
the Fubini iteration of I followed with J. Let E be an equivalence relation on
X × Y reducible to EKσ , and let B ⊂ X × Y be a K-positive Borel set. There
is a Borel K-positive Borel set C ⊂ B such that
1. either there is an equivalence relation F on X, reducible to EKσ , such that
E C = F × ee;
2. or there is an equivalence relation G on Y , reducible to EKσ , such that
E C = id × G.
In the second case, no two points of C with distinct X coordinates are Eequivalent.
Proof. To simplify the notation, assume that both σ-ideals I, J live on the
Cantor space 2ω . Let B ⊂ 2ω × 2ω be a Borel I ∗ J-positive set, and let E
be a Borel equivalence relation on B reducible to EKσ ; let h : B → ω ω be the
Borel reducing functio;. To establish a suitable terminology for the equivalence
relation EKσ , define the notion of distance between elements y, z ∈ ω ω as the
supremum of {|y(n) − z(n)| : n ∈ ω}. The distance can be infinite, and EKσ
connects those points in ω ω with finite distance.
Since the iteration PI ∗ PJ is a bounding forcing, we can thin out B if
necessary to a compact I ∗J-positive set on which the reduction h is continuous.
There are two distinct cases.
Case 1. Suppose first that the set A = {x ∈ 2ω : ∃y ∈ 2ω φ(x, y)} is Ipositive, where φ(x, y) = hx, yi ∈ B ∧ {z ∈ 2ω : hx, zi ∈ B ∧ hx, yi E hx, zi} ∈
/ J;
Note that the set A is analytic since the σ-ideal J is Π11 on Σ11 . If A is I-positive,
then it has a Borel I-positive subset A0 . Now for every x ∈ A there is y ∈ 2ω such
that φ(x, y) holds, this is a Π12 formula and by Shoenfield’s absoluteness 2.4.1
A0 forces in PI that there is y ∈ 2ω such that φ(ẋgen , y); let τ be a PI -name
for such a y. Let M be a countable elementary submodel of a large structure
containing A0 and τ , and find an I-positive compact set A00 ⊂ A0 consisting of
M -generic points only such that the function g : x 7→ τ /x is continuous on it;
there is such a set since the poset PI is bounding. Note that for every x ∈ A00 ,
M [x] |= φ(x, g(x)) by the forcing theorem, and therefore φ(x, g(x)) holds in V by
the analytic absoluteness. Consider the set C = {hx, yi ∈ B : x ∈ A00 ∧ hx, yi E
hx, g(x)i}. This is clearly a Borel K-positive set and if hx0 , y0 i, hx1 , y1 i are two
points in C then they are E-related just in case hx0 , g(x0 )i, hx1 , g(x1 )i are. Item
(1) of the claim holds with the equivalence relation F connecting points x0 , x1
if hx0 , g(x0 )i E hx1 , g(x1 )i .
Case 2. Suppose now that Case 1 fails. Let B1 ⊂ 2ω be a Borel I-positive
set such that for every x ∈ B1 , Bx ∈
/ J. Since J is a Π11 on Σ11 σ-ideal, this is a
Π12 statement about the set B and therefore absolute for all forcing extensions.
196
CHAPTER 5. CLASSES OF σ-IDEALS
In particular, B1 as a condition in PI forces the PI -generic ẋgen into B1 and
so also forces Bẋgen ∈
/ J. A similar absoluteness argument also shows that
B1 forces that the Borel equivalence relation Eẋgen on Bẋgen , connecting points
y0 , y1 ∈ Bẋgen if hẋgen , y0 i E hẋgen , y1 i, has J-small classes.
Let τ̇ be a PI -name for a winning strategy for Player I in the unraveled
version of the game G3 (J, Bẋgen ). Note that we may assume that the strategy
produces a sequence of finite sets hLm : m ∈ ωi in Bẋgen such that whenever
n 6= m ∈ ω are distinct natural numbers then no point in Lm is Eẋgen -equivalent
to any point in Ln . At any rate, Player II can force the strategy
τ̇ to play in
S
this way at no cost to himself, just adding the J-small set [ m∈n Ln ]Eẋgen to
his move at round n.
Let M be a countable elementary submodel of a large structure containing
τ and let B2 ⊂ B1 be a Borel I-positive set consisting of M -generic points only;
we can also B2 to be compact and such that all sets in M are relatively clopen
in B2 by the bounding property of the forcing PI . For every point x ∈ B2 , look
into the model M [x], at the strategy τ̇ /x. Observe that the nonexistence of a
winning counterplay for Player II against τ̇ /x is a wellfoundedness statement in
the model M [x], and since the model M [x] is wellfounded, it holds even in V .
Therefore, every counterplay by Player II against τ̇ /x in which Player II uses
only moves from the model M [x], will result in a J-positive compact subset of
Bx . We will find a Borel I-positive set B3 ⊂ B2 and for every point x ∈ B3
a counterplay qx against the strategy τ̇ /x such that the x0 6= x1 ∈ B3 implies
that no point of the result of qx0 is equivalent to no point of the result of qx1 .
This will complete the proof.
Fix a winning strategy σ for Player I in the game G2 (I, B2 ). By induction
on n ∈ ω build
• finite counterplays p0 ⊂ p1 ⊂ . . . against σ. The play pn will last n
rounds and its last move will be played of Player II. There will be a finite
set an ⊂ 2<ω such that every t ∈ an has exactly one continuation in the
last move of Player I in the play pn , and vice versa, every point in the last
move of Player I will have exactly one intial segment in an . Also, every
sequence in an−1 has an extension in an and every sequence in an has an
initial segment in an−1 .
• maps fn with domain an such that for every t ∈ an , the value fn (t) is
a name for a finite couterplay against τ̇ . The name fn (t) will be in the
model M . It will be the case that, writing s for the unique initial segment
of t in an−1 , B2 ∩ [t] forces fn−1 (s) to be an initial segment of fn (t).
• There will also be maps gn with domain an such that for every t ∈ an ,
g(t) is a clopen subset of 2ω , B2 ∩ [t] forces fn (t) to have gn (t) as the last
move of Player II, and if t0 6= t1 ∈ an then the h-images of points in the
set B ∩ ([t0 ] × gn (t0 )) have distance at least n from the h-images of the
points in the set B ∩ ([t1 ] × gn (t1 )).
S
Suppose the induction succeeds. Let B3 be the result of the play n pn , so
B3 = {x ∈ 2ω : for every n, x has an initial segment in an }. For every point
5.17. THE COUNTABLE SUPPORT ITERATION IDEALS
197
S
x ∈ B3 , consider the play qx = {fn (t)/x : t is the unique initial segment of x
in an , n ∈ ω}. It follows from the second item that this is a play against the
strategy τ /x whose initial segments
belong to the model M [x], and as explained
T
earlier, its result (equal to {gn (t) : t is the unique initial segment of x in an ,
n ∈ ω}) is a J-positive compact subset of Bx . The last item then implies that
if x0 6= x1 ∈ B3 are distinct then no point of the result of qx0 is equivalent to no
point of the result of qx1 as desired. It should be noted that there is no reason
to believe that the play qx actually belongs to the model M [x].
The induction is elementary except for arranging the last item. Suppose
pn−1 , an−1 , fn−1 , gn−1 have been constructed. To extend the play pn−1 by one
more round, let first Player II play an empty set, and the strategy σ responds
with a finite set Kn . For every x ∈ Kn , look into the model M [x] and find
a tentative infinite extension rx ∈ M of the play fn−1 (s)/x (where s is the
unique initial segment of x in an−1 ) played in accordance with the strategy
σ. Denote the finite sets produced by the strategy in these various plays by
Lm,x : m ∈ ω, x ∈ Kn . Enumerate the set Kn as {xi : i ∈ j} and by induction
on i ∈ j find increasing numbers mi larger than the length of fn−1 (s)/xi such
that for every i0 ∈ i1 ∈ j, no point in {xi0 } × Lmi is E-related to any point in
{xi1 }×Lm for any m ≥ mi0 +1 . This is certainly possible since the set {xi0 }×Lmi
is finite, while the sets {xi1 } × Lm : m ∈ ω are finite and E-unrelated by the
choice of the (name for the) strategy τ . Thus, the sets {xi } × Lmi : i ∈ j are
pairwise E-unrelated, and by the continuity of the reduction h, there are clopen
sets Oi , Pi ⊂ 2ω for i ∈ j such that xi ∈ Oi , Lmi ⊂ Pi , and whenever i0 ∈ i1 ∈ j
are numbers then the h-images of points in B ∩ Oi0 × Pi0 have distance at least
n from the h-images of points in B ∩ Oi1 × Pi1 . Note that Player II can legally
use a subset of Oi as an answer after the play rxi mi . By the forcing theorem
applied to the model M [xi ], there must be a name żi , a clopen set Oi0 ⊂ Oi and
a condition Di ∈ M such that xi ∈ Di and Di żi is an extension of the play
g(s) (where s is the unique initial segment of xi in s) in which Player I uses Oi0 as
his last move. The set Di ∩ Pi ∈ M is relatively clopen in B3 , contains the point
xi , and by the choice of the set B3 there is an initial segment ti ⊂ xi such that
B3 ∩ [ti ] ⊂ Di ∩ Pi . Extend the initial segments ti : i ∈ j if necessary to make
them pairwise incompatible, let an = {ti : i ∈ j}, fn (ti ) = żi , gn (ti ) = Oi0 and
bow to the applause of the audience for the successfully completed induction
step!
The theorem is now fairly easy to prove by induction on α. The successor
step is immediately handled by the claim. For the limit step, we will consider
the iterations of length ω, and the rest will follow by the usual bootstrapping
argument. So assume that B ⊂ X ω is a Borel I ω -positive set and E is an
equivalence relation on it reducible to EKσ . For every n ∈ ω, let An = {~x ∈ X n :
there is an I β\n -positive set D ⊂ X ω\n such that the product {~x}×D is a subset
of the set B and of a single E-equivalence class}. This is an analytic set. If there
is n ∈ ω such that the set An ⊂ X n is I n -positive, then similarly to the first
case of the claim, there is a Borel set B 0 ⊂ B and an equivalence relation F on
198
CHAPTER 5. CLASSES OF σ-IDEALS
X n such that E B 0 is equal to F × ee. Then, it is possible to use the induction
hypothesis for the iteration of length n to get the desired simplification of the
equivalence relation E on some I ω -positive set C ⊂ B.
If, on the otherShand, the sets {An : n ∈ ω} are I n -small for every n ∈ ω,
then the set A = n (An × X ω\n ) ⊂ X ω is analytic and I ω -small. The first
reflection theorem implies that A has an I ω -small Borel superset A0 . Look at
the set B 0 = B \ A0 ; it is I ω -positive, and so contains an I ω -positive compact
subset B 00 ⊂ B 0 . We will find a Borel I-positive Borel set C ⊂ B 00 such that
E C = id.
Choose a condition hpn : n ∈ ωi in the iteration forcing the generic to belong
to the set B 00 , let n ∈ ω, and consider what happens after the first n stages of
the iteration. The remainder of the iteration is forced to be equal to the two step
iteration of PI and PI ω\n+1 under some condition, and that condition forces the
tail of the generic sequence to belong to some compact subset of X×X ω\n+1 such
that the E-equivalence classes in its vertical sections are I ω\n+1 -small (recall
the definitions of the sets An here). By the Claim applied in this situation to the
iteration of PI and PI ω\n+1 (note that the reminder of the iteration has the weak
Sacks property as well), the compact subset can be thinned out so that points
in its distinct vertical sections are E-unrelated. By induction on n ∈ ω, one can
then strengthen the condition hpn : n ∈ ωi so that for every n ∈ ω, p n forces
that there is a compact subset of X × X ω\n+1 such that points in its distinct
vertical sections are E-unrelated, and the remainder of the condition forces the
tail of the generic sequence to belong to this compact set. Now let M be a
countable elementary submodel of a large structure containing the condition
hpn : n ∈ ωi, and let C ⊂ X ω be the Borel I ω -positive set of all M -generic
sequences meeting this condition. By an absoluteness argument, C ⊂ B 00 . Also,
for any two distinct sequences ~x, ~y ∈ C, if n is the largest number such that
~x n = ~y n, the forcing theorem shows that there is a compact subset of
X × X ω\n+1 in the model M [~x n] in which distinct vertical sections contain
no E-related points, and the tails of the sequences belong to this compact set,
hnece ~x, ~y must be E-unrelated. Thus, E C = id as required.
Corollary 5.17.14. Under the assumption of the theorem, if in addition PI
adds a minimal real degree, then the real degrees in the extension by the iteration
of length α are well-ordered in ordertype α.
Corollary 5.17.15. Under the assumptions of the theorem, if F, G ≤B EKσ
and ≤B F →I ≤B G, then the same thing holds for I α .
In particular, E0 is not in the spectrum of the countable support iteration of
E2 forcing. E2 is not in the spectrum of the countable support iteration of Silver
forcing or the products of Sacks forcing, and in the countable support iteration
of E0 forcing, every equivalence relation ≤B EKσ simplifies to one reducible to
E0 .
5.17. THE COUNTABLE SUPPORT ITERATION IDEALS
5.17d
199
A few special cases
In a number of situations, much better canonization theorems can be proved
about the iteration σ-ideals. Here, we will show that in the very specific case of
the iteration of Sacks forcing, the canonization obtained in Theorem 5.17.12 in
fact holds for all Borel equivalence relations as opposed to just those reducible
to EKσ .
Theorem 5.17.16. Let I be the σ-ideal of countable subsets of 2ω . Let α ∈ ω1
be a countable ordinal. Then analytic→Iα {idβ × ee : β ≤ α}.
Proof. To simplify the notation, write X = 2ω , X α = (2ω )α , Pα to be the poset
of I α -positive analytic subsets of X α ordered by inclusion (note that compact
sets are dense in Pα as the iteration is bounding), and for a condition p ∈ Pα
and an ordinal β ∈ α write p β = {~x ∈ X β : {~y ∈ X α\β : ~xa ~y ∈ p} ∈
/ I α\β }.
α\β
1
1
β
Since the σ-ideal I
is Π1 on Σ1 , this is an I -positive analytic set and as
such a condition in the poset Pβ . For an ordinal β ∈ α, consider the reduced
product Pβα consisting of pairs hp, qi ∈ Pα × Pα for which p β = q β. The
reduced product adds sequences ~xlgen , ~xrgen ∈ (2ω )α which coincide on their
first β many coordinates. If M is a countable elementary submodel of a large
structure, we call sequences ~x, ~y ∈ (2ω )α reduced product generic over M if the
set β = {γ ∈ α : ~x(γ) = ~y (γ)} is an ordinal and the sequences are Pβα -generic
for the model M . As in the product case, there is a key lemma:
Lemma 5.17.17. Let M be a countable elementary submodel of a large enough
structure, and B ∈ PIα ∩ M is a condition. There is a Borel Iα -positive set
C ⊂ B consisting of pairwise reduced product generic sequences over M .
To prove the lemma, consider a fusion type poset Q for PIα . A condition
p ∈ Q is a triple p = hap , np , rp i where ap ⊂ α is a finite set, np ∈ ω, and rp is
a function from (2np )ap to PIα B such that for every s, t ∈ dom(rp ) and every
ordinal β ∈ ap , if s β = t β then rp (t) β = rp (s) β. If M is a countable
elementary submodel of a large structure
G ⊂ Q is a filter generic over M ,
T andS
then one can form the set C ⊂ (2ω )α = p∈G rng(rp ). This is the desired set.
Checking its required properties is easy and left to the reader; we only state the
central claim.
Claim 5.17.18. Let β ∈ α, p ∈ Q and let D ⊂ Pβα be an open dense set. There
is a condition q ≤ p such that β ∈ aq and for every pair s, t ∈ dom(rq ) such
that β is the smallest ordinal on which they differ, hrq (s), rq (t)i ∈ D.
Granted the lemma, the proof is completed as follows. Let E be an analytic
equivalence relation on a Borel Iα -positive set B ⊂ (2ω )α . Let β be the least
ordinal such that there is a condition hB0 , C0 i ∈ Pβα such that hB0 , C0 i ~xlgen E
~xrgen . Observe that then hB0 , B0 i ~xlgen E ~xrgen : given a Pβα -generic pair
h~x0 , ~x1 i below the condition hB0 , B0 i, a simple mutual genericity argument will
produce a sequence ~x2 such that both pairs h~x0 , ~x2 i and h~x1 , ~x2 i are Pβα -generic
below the condition hB0 , C0 i. By the forcing theorem then, ~x0 E ~x2 E ~x1 , and
by the transitivity of the relation E, ~x0 E ~x1 .
200
CHAPTER 5. CLASSES OF σ-IDEALS
Now let M be a countable elementary submodel of a large structure containing E and B0 and use the lemma to produce a Borel Iα -positive set C ⊂ B0
consisting of pairwise reduced product generic sequences. For any two sequences
~x, ~y ∈ C write γ(~x, ~y ) to denote the set of all ordinals on which their values agree.
If γ < β then M [~x, ~y ] |= ¬~x E ~y by the minimality of the ordinal β and the
forcing theorem; the analytic absoluteness between M [~x, ~y ] and V then shows
that ¬~x E ~y holds in V . If γ = β then by a similar argument ~x E ~y . If γ > β
then we must use the transitivity of the equivalence relation E again. Use a
mutual genericity argument to produce a sequence ~z ∈ B such that both pairs
h~x, ~zi and h~y , ~zi are Pβα -generic over M , use the forcing theorem and absoluteness to argue that ~x E ~z E ~y , and by transitivity ~x E ~y again. In conclusion,
E C = idβ × ee as desired.
On the other end of the complexity spectrum, if the σ-ideal I to be iterated
contains an arbitrary ergodic equivalence relation, the canonization situation
already at the first infinite stage of the iteration is so complex as to allow only
negative results.
Proposition 5.17.19. Suppose that I is a Π11 on Σ11 equivalence relation on
a Polish space X such that every analytic I-positive set has a Borel I-positive
subset and the quotient PI is provably proper. Suppose that E is a Borel equivalence relation on X with all equivalence classes in I, ergodic with respect to I.
Then the iteration ideal I ω has a coding function, and its spectrum is cofinal
among the analytic equivalence relations.
Proof. Let f (~x, ~y )(n) = 1 if ~x(n) E ~y (n). We claim that this is a coding function
for I ω . Suppose that B ⊂ X ω is a Borel I ω -positive set. By Fact 5.17.4, it
contains all branches of a Borel I-branching tree T ⊂ X <ω . Let z ∈ 2ω be an
arbitrary point. By induction on n ∈ ω, build nodes sn , tn ∈ T of length n such
that sn+1 (n) E tn+1 (n) iff z(n) = 1. The induction step uses the ergodicity of
a
the equivalence relation E: the sets {x ∈ X : sa
n x ∈ T } and {x ∈ X : tn x ∈ T }
are I-positive, and the ergodicity of E implies the existence of both an E-related
and E-unrelated pair whoseScomponents come
S from the respective two sets. In
the end, the sequences ~x = n sn and ~y = n yn satisfy f (~x, ~y ) = z as required.
For the spectrum, suppose that J is an analytic ideal on ω and let EJ be the
equivalence relation on X ω defined by ~x EJ ~y if the set {n ∈ ω : ¬~x(n) E ~y (n)}
belongs to the ideal J. It is not difficult to prove that =J is reducible to
EJ restricted to any Borel I-positive subset of X ω . Thus the complexity of
equivalence relations in the iteration grows to encompass a cofinal set of Borel
equivalence relations as well as a complete analytic equivalence relation.
We do not know anything tangible about the canonization properties of
iterations of posets that add unbounded reals. The following theorem shows
that the mutual genericity arguments from the iteration of Sacks forcings above
cannot work: there is a coding function on pairs. The argument is a slight
upgrade of the proof of Velickovic-Woodin coding on triples [62].
5.17. THE COUNTABLE SUPPORT ITERATION IDEALS
201
Theorem 5.17.20. Let I be the σ-ideal of σ-compact subsets of ω ω . There is
a Borel function f : ω ω × (ω ω × ω ω ) → 2ω such that for every I-positive Borel
set B ⊂ ω ω and every I ∗ I-positive Borel set C ⊂ ω ω × ω ω the image f 00 B × C
contains a nonempty open set.
Corollary 5.17.21. Let J be a Π11 on Σ11 σ-ideal on a Polish space X, such
that every analytic J-positive set contains a Borel J-positive subset. If the poset
PJ is proper and adds an unbounded real then the two step iteration ideal J has
a coding function.
To prove the corollary from the theorem, just find a Borel J-positive set
B ⊂ X and a Borel function g : B → ω ω such that preimages of I-small sets are
J-small, and consider the function h : B×B → 2ω given by h(hy0 , y1 i, hx0 , x1 i) =
f (g(y0 ), g(x0 ), g(x1 )). This is a coding function on the J ∗ J-positive set B × B.
Proof. To define the functional value f (y, x0 , x1 ), let hni : i ∈ ωi enumerate
those numbers n such that x0 (m) > y(m) for some m between x1 (n) and x1 (n +
1). Let f (y, x0 , x1 )(i) = 0 if in the interval [x1 (ni ), x1 (ni + 1)) the values of x, y
oscilate fewer than four times, and f (y, x0 , x1 )(i) = 1 otherwise. If some of the
numbers ni are undefined, let f (y, x0 , x1 ) =trash. This function works.
Suppose that B ⊂ ω ω and C ⊂ ω ω × ω ω are I-positive and I ∗ I-positive
Borel sets respectively. Let T be a superperfect tree such that [T ] ⊂ C. For
simplicity, assume that the trunk of T is 0; we will show that f 00 B × C = 2ω .
Let z ∈ 2ω be a prescribed value; we must find points y ∈ B and hx0 , x1 i ∈ C
such that f (y, x0 , x1 ) = z.
For the iteration of Miller forcing of length 2, write ẋ0gen for the first generic
point and ẋ1gen for the second. Find a condition p in the iteration which forces
the pair hẋ0gen , ẋ1gen i into Ċ. Find a countable elementary submodel M of a
large enough structure and by induction on i ∈ ω build finite integer sequences
si , ti , ui , numbers ni and conditions pi in the iteration such that
I1 the sequences si extend as i increases, the same for ti and ui . The length
|si | is smaller than the last entry on the sequence ui , the length of ti is
larger than the last entry of ui , and ni = |ui | − 1;
I2 si is a splitnode of the tree S, and pi is a condition of the form hTi , U̇i i
such that ti is a splitnode of the tree Ti and pi forces ui to be a splitnode
of the tree U̇i ;
I3 f (si , ti , ui ) = z i;
I4 the conditions pi decrease as i increases and pi ∈ M meets the i-th open
dense subset of the iteration in the model M in some fixed enumeration.
S
If this induction
S
S succeeds, then in the end consider the points y = i si , ωx0 =
i ti and x1 =
i ui . Clearly, f (y, x0 , x1 ) = z by item I3. The point y ∈ ω is a
branch through the tree S and therefore an element of the set B. Writing g for
the filter generated by the conditions {pi : i ∈ ω} in the two step iteration, it is
202
CHAPTER 5. CLASSES OF σ-IDEALS
obvious that g is M -generic and x0 = ẋ0gen /g, x1 = ẋ1gen /g and so hx0 , x1 i ∈ C
by the forcing theorem. This will conclude the proof.
To start the induction, just find a condition p0 ≤ p in the model M such
that p0 = hT0 , U̇0 i, t0 is the trunk of T0 and T0 forces some specific sequence u0
to be the trunk of U̇0 ; moreover, let s0 = 0. Now suppose si , ti , ui are given, and
assume that z(i) = 0 (the case z(i) = 1 is similar). Use the superperfect property
of the trees S, Ti to choose splitnodes si+1 ∈ S and t0 ∈ Ti such that |si+1 | < |t0 |,
si ⊂ si+1 , ti ⊂ t0 , and si+1 (|si |) > ti (ni ), and moreover, the sequences si+1 , t0
oscilate only twice on dom(si+1 ) \ ni . Find some condition T 0 ≤ Ti t0 in the
Miller forcing in the model M indicating a number m > dom(si+1 ) such that
0
0
u0 = ua
i m ∈ U̇i . Find some condition pi+1 ≤ hT , U̇i u i in the model M in the
i + 1-th open dense set and strengthen the first coordinate if necessary to decide
the first splitnode of U̇i+1 to be some definite ui+1 . Further, we can strengthen
the first coordinate if necessary to make sure that its trunk ti+1 is longer than
the last entry on ui+1 . This concludes the induction step.
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Index
< ω1 -proper, 37
E0 , 115
E2 , 136
PIE , 65, 136, 142, 151, 157
Axiom of Determinacy, 33
Coll(ω, κ), 14, 47, 54
Cohen, 100
capacity, 32, 40, 83
Gandy-Harrington, 12, 125, 126
coding function, 7, 94, 99, 134, 140,
Laver, 176
168, 179, 201
proper, 5, 15, 23, 161
concentration of measure, 26, 94, 181,
reduced product, 69, 113
183
fusion sequence, 130, 143, 149, 161
cube
s-cube, 124
game
dented strong Milliken, 169
boolean, 106
dented weak Milliken, 160
symmetric category, 106
Hamming, 26, 94, 181
generic ultrapower, 21
Ramsey, 154
reverse, 124, 130
ideal
Silver, 76, 148
σ-continuity, 40, 104
strong Milliken, 169
σ-finite Hausdorff mass, 3, 32, 40,
weak Milliken, 160
83
analytic P-ideal, 161
equivalence
c.c.c., 24
E0 , 164
calibrated, 92
E1 , 86, 124, 142
game, 29, 105
E2 , 101, 136, 144, 181, 182
generated by closed sets, 104
F2 , 103, 186
Laver, 3, 30, 83, 176
EKσ , 153, 158, 161, 177, 181, 183
meager, 100
Etail , 170
polar, 32
classifiable by countable structures,
porosity, 104, 112
13, 79, 136, 144, 151, 161, 177 inaccessible, 34, 177
countable, 74, 171
essentially countable, 5, 13, 78, 122 kernel, 66
hypersmooth, 13, 86, 124, 129
model
orbit, 85
V [[x]]E , 54, 136, 142
smooth, 13, 73, 118, 130
V [x]E , 47, 68, 136, 142, 151, 153,
forcing
164
E0 -box, 116
Π11 on Σ11 , 8, 11, 40, 41, 92, 105, 116,
125, 136, 142, 161, 194
208
INDEX
property
covering, 92, 94, 186
free set, 6, 92, 97, 99, 113
Fubini, 24
multiple Silver, 43
mutual generics, 114
rectangular Ramsey, 24, 25, 40,
112, 113
selection, 36, 89, 115, 161, 179
separation, 81, 144
Silver, 6, 41, 92, 97, 99, 180
weak Laver, 97
weak Sacks, 191
scrambling tool, 29
spectrum, 7, 101, 136, 142, 164, 177,
181
total canonization, 5, 41
209