Unspecified Journal
Volume 00, Number 0, Pages 000–000
S ????-????(XX)0000-0
CLUB GUESSING SEQUENCES
— NATURAL STRUCTURES IN SET THEORY —
TETSUYA ISHIU
Abstract. Natural structures in set theory played key roles in the rapid
growth of this field since late 1980’s. In this article, we survey the club guessing
sequence, which is one of the most successfully used natural structures.
1. Introduction
1.1. Natural structures in set theory. One of the new trends in the research of
set theory from the late 1980’s is the use of more “natural structures”1. This term
was introduced by M. Foreman and a few other researchers (See [7], [8]). Since
1960’s, the methods to build models of set theory with various properties, such as
forcing, inner model theory, and large cardinals, were applied in various forms and
developed. These methods led the rapid growth of set theory, and are essential
in this area. On the other hand, it was discovered last two decades that we can
prove unexpected theorems by using the structures that can be shown to exist just
from ZFC and their strengthening. These structures were collectively called natural
structures.
Set theory was founded by G. Cantor. Axiomatic set theory is an axiomatized
version of the theory. There are several systems of axioms for set theory, but the
most standard one is ZFC, which is made by modifying and formalizing Zermelo’s
system2 As the system of axioms in use became explicit, we can think about models
of set theory. Without using vague notions such as ‘all sets’, we can investigate
set theory by using its model as long as it satisfies ZFC. By the way, Gödel’s
incompleteness theorem implies that ZFC does not prove the existence of a model
of ZFC, it is common to consider a model of a finite but sufficiently large fragment
of ZFC.
A new era of (axiomatic) set theory began by the construction of the class L of
constructible sets by K. Gödel [17], and the development of the forcing method by
P. Cohen [4], [5]. Both are the ways to, assuming there is a model of ZFC, build
new models. It would not have been possible without axiomatization. One one
hand, Gödel’s construction was the beginning of inner model theory, in which we
shrink a given model to obtain a smaller model. On the other hand, the forcing
Received by the editors May 1, 2011.
2000 Mathematics Subject Classification. Primary 03E04, 03E05, 03E35.
This material is based upon work supported by the National Science Foundation under Grant
No. 0700983.
1
The term “Canonical structures” is also used to describe the same notion.
2In this article, we just say set theory to mean axiomatic set theory using ZFC as axioms.
Some other systems are also considered, but no other systems are as widely used as ZFC.
c
⃝0000
(copyright holder)
1
2
TETSUYA ISHIU
method discovered by Cohen is a very effective way to expand a given model of
ZFC. By using these two methods combined with large cardinals, so many models
with various properties were constructed. Thus, in 1970’s, it seems that almost all
non-trivial set-theoretic propositions were predicted to be independent of ZFC.
However, a variety of results against this prediction were obtained since mid 80’s.
The most significant examples are the PCF theory3 by S. Shelah, and the theory of
minimal walks developed by S. Todorcevic. Both theories not only produced many
theorems that can be proved just from ZFC, but also provided new viewpoints to
models of set theory. And they were both established by the wide use of natural
structures.
It becomes clearer that natural structures are powerful tools not only in proofs
under ZFC, but also in consistency proofs. As forcing is woven into set theory
in various ways and becomes an indispensable technique, the author believes that
arguments using natural structures will be more and more widely used in this field.
The theme of this article is club guessing sequences, which were defined in the
PCF theory and now became one of the most successful natural structures. The
author shall explain how natural structures changed set theory with club guessing
sequences as an example.
1.2. Powers of singular cardinals. The results of S. Shelah about the powers
of singular cardinals are one of the most successful examples of the methods to
investigate models of set theory through natural structures. Here, the author will
briefly go over the history toward his results, and explain why they were so shocking.
The author also refer the readers to [38], which is a very interesting article written
by S. Shelah about his own theorems with many personal comments.
The Continuum Problem was first considered by G. Cantor, who started the field
of set theory. This problem, which is Hilbert’s first problem, had been considered
one of the most important subject since then. The problem asks if every set of
reals is either countable or equinumerous to the set of all reals. In modern terms,
it can be rephrased as 2ℵ0 = ℵ1 under Axiom of Choice. The assertion 2ℵ0 = ℵ1 is
often called the Continuum Hypothesis (CH). The assertion that can be obtained
by extending this to all cardinals κ, i.e. “for every cardinal κ, 2κ = κ+ ” is called
the Generalized Continuum Hypothesis (GCH).
This problem was in a sense settled because CH was shown to be independent of
ZFC. K. Gödel [17], showed the consistency of ZFC + CH and P. Cohen [4] proved
the consistency of ZFC + ¬CH.4
In addition to the power of ℵ0 , the powers of larger regular cardinals have been
investigated. The following facts were known.
Fact 1.1.
(1) κ < λ implies 2κ ≤ 2λ .
(2) (König’s lemma)5cf(2κ ) > κ.
3‘PCF’ is also written without capitalization (‘pcf’), but it seems to be getting more common
to use capital letters particularly to describe the theory
4As K. Gödel argued in [18], the independence of CH does not necessarily “solve” the Continuum Problem. In fact, researchers including W. H. Woodin continue deep research on CH.
5This lemma was proved by Julius König. But his son Dénes König was also a mathematician,
and his result in graph theory is also often called König’s lemma.
CLUB GUESSING SEQUENCES
3
The cofinality cf(δ) of a limit ordinal δ is the least cardinality of unbounded
subsets of δ. In fact, for every limit ordinal δ, we can find an unbounded subset
X of δ such that the order type otp(X) of X is cf(δ)6. If a cardinal κ satisfies
cf(κ) = κ, then we say that κ is regular. Otherwise, κ is said to be singular. It is
known that cf(δ) is regular for every limit ordinal δ.
It is also known that for every cardinal κ, its successor κ+ is regular. Meanwhile,
the cofinality of ℵω is ℵ0 since {ℵn : n < ω} is an unbounded subset of ℵω .
Therefore, ℵω is singular. A regular limit cardinal is called an weakly inaccessible
cardinal, and its existence cannot be proved from ZFC.
W. B. Easton [12] proved that for regular cardinals, no stronger proposition
than Fact 1.1 can be proved from ZFC. It was expected that a similar result can be
proved about singular cardinals, but it was denied by the following theorem proved
by J. Silver [40].
Theorem 1.2 (J. Silver [40]). Suppose that κ is a singular cardinal of uncountable
cofinality. If 2λ = λ+ for every infinite cardinal λ < κ, then 2κ = κ+ .
In fact, the assumption of the previous theorem can be weakened to “{λ < κ :
2λ = λ+ } is a club subset of κ”.
Thus, for singular cardinals of uncountable cofinality, there is a much stricter
restriction than Fact 1.1. For singular cardinals of countable cofinality, M. Magidor
obtained the following result, which implies that Theorem 1.2 does not hold for
them.
Theorem 1.3 (M. Magidor [31]). Assuming the consistency of a certain large
cardinal, we can build a model in which for every n < ω, 2ℵn = ℵn+1 and 2ℵω =
ℵω+2 .
Therefore, in this model, ℵω is the least cardinal that does not satisfy GCH. It
is known that we need some assumption beyond the consistency of ZFC to build
such a model. The effort to weaken the large cardinal assumption or make 2ℵω a
bigger cardinal is still going on.
However, S. Shelah [36] established the theory called “PCF theory” and used it
to prove the following theorem.
Theorem 1.4. ℵω ℵ0 < max{ℵω4 , (2ℵ0 )+ }.
S. Shelah introduced the following quote of L. Harrington in [38] that was said
to himself: “Cardinal arithmetic? Yes, it had been a great problem, but now ...”
This feeling seemed to be shared by many set theorists at that time. This theorem,
proved under such circumstances, surprised many people.
What is as wonderful as the theorems he proved including Theorem 1.4 is that
S. Shelah proved the existence of many beautiful structures just from ZFC. One
of such structures is the club guessing sequence, which is the main subject of this
article. J. Cummings covered many such structures in [6]. The following is listed
as the themes of the article.
(1) The universe of set theory V is surprisingly L-like in the sense that weak
versions of Jensen’s combinatorial principles, diamond and square, are provable outright as theorems of ZFC.
6For a set X of ordinals, the order type otp(X) of X is the ordinal so that there exists a
one-to-one onto function f : X → otp(X).
4
TETSUYA ISHIU
(2) The extent to which L-like combinatorial principles hold in V can be measured by constructing certain “canonical invariants” which are typically
ideals or stationary sets; examples which are important in these notes include the ideal I[λ] and the stationary set of good points (qv). Understanding these invariants is the key to many combinatorial problems, especially
those involving singular cardinals and their successors.
(3) PCF theory had its origins in questions involving the Singular Cardinals
Hypothesis. However, the theory has much broader applicability, in particular, PCF is a fertile source of the sort of canonical invariants discussed
above.
(4) There is a tension in set theory between “compactness” and “incompactness”. If the universe is sufficiently L-like then there are many examples of
incompactness, such as nonreflecting stationary sets or κ-Aronszajn trees.
By contrast, in the presence of large cardinals or strong forcing axioms
there are typically fewer examples of incompactness; moreover, for a regular uncountable cardinal κ compactness statements such as “there are
no κ-Aronszajn trees” can have a high consistency strength, for example,
when κ is the successor of a singular cardinal or we demand the statement
be true for several successive values of κ at once. The canonical invariants
are especially useful in exploring the tension between compactness and incompactness
In this article, we shall begin with basic definitions and give examples that
demonstrate these items, particularly (1) and (3). The author tried his best to
make it self-contained. However, the least knowledge about ordinals and cardinals
is assumed. The readers are referred to standard textbooks on set theory such as
K. Kunen [29] and T. Jech [23]. In addition, the theorems and proofs introduced
here were chosen to demonstrate central ideas as concisely as possible. Hence, the
statements of the cited theorems may not be the strongest ones that are known.
One interesting point in the arguments with natural structures is that many profound and intriguing theorems can be proved without techniques of mathematical
logic. To demonstrate it, the author avoided such techniques such as Skolem hulls.
However, it should be noted that they are great tools in set theory and necessary
in proofs of some of the theorems and lemmas that were omitted in this article.
2. Club sets and club guessing sequence
2.1. Club sets. First of all, we shall define several basic notions, particularly closed
unbounded sets. We say that a subset X of a limit ordinal α is unbounded in α if
and only if for every β < α, there exists a γ ∈ X such that γ ≥ β. This coincides
with the definition of unboundedness for partially ordered sets. For a set X of
ordinals, if X ∩ δ is an unbounded subset of δ, then we say that δ is a limit point
of δ. Note X ∩ δ = {γ ∈ X : γ < δ} because in set theory, an ordinal δ is identified
with the set {γ : γ < δ} of all ordinals below δ. lim(X) denotes the set of all
limit points of X. We use this notation also when X is a proper class of ordinals.
sup(X) denotes the supremum of X and max(X) denotes the maximum element
of X if it exists. For sets X and Y of ordinals, if either X = ∅ or there exists a
ζ < sup(X) such that X \ ζ ⊆ Y , then we say that X is almost contained in Y and
write X ⊆∗ Y . This notion is usually used when X does not have the maximum
CLUB GUESSING SEQUENCES
5
element. In such a case, since we have X \ ζ = {α ∈ X : α ≥ ζ}, X ⊆∗ Y means
that a tail of X is a subset of Y . The negation of X ⊆∗ Y is denoted by X ⊈∗ Y .
Definition 2.1. Let δ be a limit ordinal. We say that a subset D of δ is closed
unbounded (club)7 in δ if D satisfies the following conditions.
(1) D is unbounded in δ, and
(2) for every limit point δ < γ of D, γ ∈ D.
(2) is equivalent to the property that D is closed in the order topology. For
example, if δ is a limit ordinal of uncountable cofinality, for every unbounded subset
X of δ, δ ∩ lim(X) is a club subset of δ.
We shall show several important properties of club sets.
Lemma 2.2. Let δ be a limit ordinal of uncountable cofinality. If ∩
{Dα : α < µ} is
a family of club subsets of δ and µ < cf(δ), then their intersection α<µ Dα is also
club in δ.
∩
Proof. It is trivial that α<µ Dα is closed. So, we shall show that it is unbounded
∩
in δ. It suffices to show that for a fixed ζ < δ, there exists a γ ∈ α<µ Dα such
that γ ≥ ζ.
By induction, we shall define an increasing sequence ⟨γn : n < ω⟩ in δ. Let
γ0 = ζ. Suppose that γn has been defined. Since Dα is unbounded in δ for every
α < µ, there exists a ξn,α ∈ Dα such that ξn,α > γn . {ξn,α : α < µ} is a subset of δ
whose cardinality is µ < cf(δ). Thus, by the definition of cf(δ), it is bounded in δ.
Therefore, for every α < µ, there exists a γn+1 < δ such that ξn,α < γn+1 for every
α < µ.
Let γ = supn<ω γn . Since cf(δ) is uncountable, we have γ < δ. For every α < µ,
clearly
∩ Dα ∩ γ is unbounded in γ. Since Dα is closed, we have γ ∈ Dα . So, we have
□
γ ∈ α<µ Dα . Since γ ≥ γ0 = ζ, the lemma was proved.
The previous lemma does not hold when cf(δ) = ω. For example, if δ = ω, then
D1 = {2n : n < ω} and D2 = {2n + 1 : n < ω} are club subsets of ω that have
empty intersection.
Definition 2.3. Let X be a set. A set F of subsets of X is called a filter on X if
and only if
(1) ∅ ̸∈ F and X ∈ F ,
(2) (closed under superset) for every A, B, if A ⊆ B ⊆ X and A ∈ F , then
B ∈ F , and
(3) (closed under intersection) for every A, B, if A ∈ F and B ∈ F , then
A ∩ B ∈ F.
For example, if (X, E, µ) is a complete probability space, F = {Y ∈ E : µ(Y ) =
1} is a filter on X. In this case, for a predicate φ(x), {x ∈ X : φ(x)} ∈ F means
that φ(x) is true for almost every x ∈ X. As this example demonstrates, elements
of F is considered as big subsets of X in a certain way.
Let δ be an limit ordinal of uncountable cofinality. Then, if we let
F = {Y ⊆ δ : Y contains a club subset of δ},
7“Club” is now a very common abbreviation for “closed unbounded”, but it was written as
“c.u.b.” in [29]
6
TETSUYA ISHIU
then Lemma 2.2 implies that F is a filter on δ.8 This filter is called the club filter.
The dual notion of filter is ideals.
Definition 2.4. A set I of subsets of X is called an ideal on X if and only if
(1) ∅ ∈ I and X ̸∈ I,
(2) for every A, B, if A ⊆ B ⊆ X and B ∈ I, then A ∈ I, and
(3) for every A, B, if A ∈ I and B ∈ I, then A ∪ B ∈ I.
Let (X, E, µ) be a complete probability space again. Then, I = {Y ∈ E : µ(Y ) =
0} is an ideal on X. Elements of I are considered as small subsets of X in a certain
sense.
If F is a filter on X, then I = {Y ⊆ X : X \ Y ∈ F } is an ideal on X.
This ideal is called the dual ideal of F . Similarly, if I is an ideal on X, then
F = {Y ⊆ X : X \ Y ∈ I} is a filter on X, which is called the dual filter of I. In
the example of a complete probability space, the dual ideal of the filter of sets of
measure 1 is the ideal of sets of measure 0, and vice versa.
The dual ideal of the club filter is called the non-stationary ideal, i.e. the nonstationary ideal on δ is the set of all subsets X of δ such that X ∩ D = ∅ for some
club subset D of X. A subset of δ that does not belong to the non-stationary ideal
is called a stationary subset of δ. That is, a subset S of δ is stationary if and only if
for every club subset D of δ, S ∩ D ̸= ∅. By Lemma 2.2, if S is a stationary subset
of δ and D is a club subset of δ, then S ∩ D is stationary.
As we wrote, the club filter can be defined on any limit ordinal of uncountable
cofinality, but particularly important are the club filter on an uncountable regular
cardinal κ and its dual ideal NSκ . The filter and ideal have a good property called
normality. To see this, we prepare the following definition.
Definition 2.5. Let κ be an uncountable regular cardinal, and ⟨Xα : α < κ⟩ a
sequence of subsets of κ.
(1) The diagonal intersection △α<κ Xα of ⟨Xα : α < κ⟩ is the subset of κ
defined as follows:
△α<κ Xα = {γ < κ : ∀α < γ(γ ∈ Xα )}.
(2) The diagonal union ▽α<κ Xα of ⟨Xα : α < κ⟩ is the subset of κ defined as
follows:
▽α<κ Xα = {γ < κ : ∃α < γ(γ ∈ Xα )}.
These two are dual notions, and the following fact holds.
κ \ ▽α<κ Xα = △α<κ (κ \ Xα ).
Lemma 2.6. Let κ be an uncountable regular cardinal. Then, for every sequence
⟨Dα : α < κ⟩ of club subsets of κ, its diagonal intersection △α<κ Dα is club in κ.
Proof. Let D = △α<κ Dα . To see that D is unbounded, let ζ < κ be fixed
and show that there is an element of D that is greater than ζ. We shall define an
increasing sequence ⟨γn : n < ω⟩ in κ as follows. Let γ0 = ζ. We
∩ shall describe how
to define γn+1 assuming γn has been defined. By Lemma 2.2, α≤γn Dα is club in
∩
κ. Therefore, we can find a γn+1 ∈ α≤γn Dα with γn+1 > γn .
8The set of all club subsets of δ is not a filter in general because it is easy to build a subset of
δ that is not closed but still contains a club subset of δ.
CLUB GUESSING SEQUENCES
7
Let γ = supn<ω γn . We shall show that γ ∈ D. By the definition of the diagonal
intersection, it suffices to show that for every α < γ, γ ∈ Dα . Fix α < γ. Since
γ = supn<ω γn , there exists an m < ω such that α < γm . By the definition of γn ,
for every n < ω with n > m, we have γn ∈ Dα . Thus, γ is a limit point of Dα .
Since Dα is closed, we have γ ∈ Dα .
To show that D is closed, let δ be a limit point of D. It suffices to show that for
every α < δ, δ ∈ Dα . First, observe D \ (α + 1) ⊆ Dα 9. Hence, δ is a limit point of
Dα . Since Dα is closed, we have δ ∈ Dα .
□
∩
For every α < κ, if we define Dα = κ \ α, then α<κ Dα = ∅. Thus, there is
no club subset of κ that is contained in Dα for every α < κ. Therefore, Lemma
2.2 cannot be extended to κ-many club subsets. However, Lemma 2.6 says that if
{Dα : α < κ} is a set of club subsets of κ, then there exists a club subset of κ that
is almost contained in Dα for every α < κ.
Definition 2.7. Let I be an ideal on X. We say∪that I is κ-complete if and only
if for every λ < κ and {Yα : α < λ} ⊆ I, we have α<λ Yα ∈ I.
We say that I is normal if and only if for every sequence ⟨Xα : α < κ⟩ in I,
▽α<κ Xα ∈ I.
The following lemma can be proved by Lemma 2.2 and Lemma 2.6.
Lemma 2.8. For every uncountable regular cardinal κ, NSκ is κ-complete and
normal.
Moreover, in the following sense, NSκ is the smallest κ-complete normal ideal on
κ that contain all bounded subsets of κ.
Lemma 2.9. Let κ be an uncountable regular cardinal and I a κ-complete normal
ideal on κ that contain all bounded subsets of κ. Then, NSκ ⊆ I.
2.2. Club guessing sequences. Let Lim denote the class of all limit ordinals.
When κ is a regular cardinal, Cof(κ) denotes the class of all ordinals of cofinality
κ. If λ < κ are both regular cardinals, then we can show that κ ∩ Cof(λ) is a
stationary subset of κ. Cof(≥κ) denotes the class of all ordinals whose cofinality is
equal to or greater than κ.
Definition 2.10 (S. Shelah [36]). Let κ be an uncountable regular cardinal and
S a stationary subset of κ consisting of limit ordinals. A sequence ⟨Cδ : δ ∈ S⟩ is
called a fully club guessing sequence on S if and only if
(1) for every δ ∈ S, Cδ is an unbounded subset of δ, and
(2) for every club subset D of κ, there exists a δ ∈ S such that Cδ ⊆ D.
A sequence ⟨Cδ : δ ∈ S⟩ is called a tail club guessing sequence on S if and only
if the sequence satisfies (1) and the following (2)’ instead of (2):
(2)’ for every club subset D of κ, there exists a δ ∈ S such that Cδ ⊆∗ D.
For example, if ⟨Aδ : δ ∈ S⟩ is a ♢κ (S)-sequence, then ⟨Aδ : δ ∈ S ′ ⟩ is a fully
club guessing sequence on S ′ where S ′ = {δ ∈ S : Aδ is unbounded in δ}. (See
Section 2.3 for ♢κ (S)).
The condition (2) of the previous definition requires at least one δ ∈ S such that
⃗ = {Cδ : δ ∈ S} is a fully club guessing sequence on S, for
Cδ ⊆ D. However, if C
9Note D \ (α + 1) = {γ ∈ D : γ > α} since α + 1 is the set of all ordinals ≤α.
8
TETSUYA ISHIU
every club subset D of κ, S ′ = {δ ∈ S : Cδ ⊆ D} is stationary in κ. To see this,
suppose that S ′ is non-stationary. Then, there exists a club subset D′ of κ such
⃗ is a fully club
that S ′ ∩ D′ = ∅. However, since D ∩ D′ is a club subset of κ and C
guessing sequence, there exists a δ ∈ S such that Cδ ⊆ D ∩ D′ . Since Cδ ⊆ D, we
have δ ∈ S ′ . However, since δ is a limit point of D′ , we also have δ ∈ D′ . This
contradicts to the assumption S ′ ∩ D′ = ∅. A similar fact can be shown for a tail
club guessing sequence.
Every fully club guessing sequence is clearly a tail club guessing sequence, but
the converse is not true. For example, if ⟨Cδ : δ ∈ S⟩ is a fully club guessing
sequence, then ⟨Cδ ∪ {0} : δ ∈ S⟩ is a tail club guessing sequence, but not a fully
club guessing sequence. However, we can prove that it is essentially the only way
to obtain such an example.
Theorem 2.11 (Ishiu [22]). Let κ be an uncountable regular cardinal, and S a
stationary subset of κ consisting of limit ordinals. If ⟨Cδ : δ ∈ S⟩ is a tail club
guessing sequence on S, then there exists an ordinal ζ < κ such that ⟨Cδ \ ζ : δ ∈
S \ (ζ + 1)⟩ is a fully club guessing sequence on S \ (ζ + 1).
In particular, the existence of a tail club guessing sequence on S is equivalent to
the existence of a fully club guessing sequence on S.
As we pointed out, the existence of a fully club guessing sequence on any uncountable regular cardinal is consistent, but the following surprising theorem can
be proved just from ZFC.
Theorem 2.12 (S. Shelah [36]). Suppose that θ and κ are regular cardinals with
θ+ < κ. Then, for every stationary subset S of κ ∩ Cof(θ), there exists a fully club
guessing sequence on S.
First, we shall prove the theorem assuming θ is uncountable. Clearly, it suffices
to show the following lemma. For a set X of ordinals, let acc(X) denote the set of
all elements α of X such that X ∩ α is unbounded in α, i.e. acc(X) = X ∩ lim(X).
Define nacc(X) = X \ acc(X). If δ is a limit ordinal of uncountable cofinality, and
D is a club subset of D, then acc(D) is a club subset of δ.
Lemma 2.13. Suppose that θ and κ are uncountable regular cardinals with θ+ < κ.
Let S be a stationary subset of κ ∩ Cof(θ) and ⟨Cδ : δ ∈ S⟩ a sequence such that for
every δ ∈ S, Cδ is a club subset of δ with |Cδ | = θ. Then, there exists a club subset
E of κ such that ⟨Cδ ∩ E : δ ∈ S ∩ acc(E)⟩ is a fully club guessing sequence on κ.
Proof. Suppose that such an E does not exist. By induction, we shall define a
decreasing sequence ⟨Dα : α ≤ θ+ ⟩ of club subsets of κ. Let D0 = κ. Suppose that
Dβ has∩ been defined for every β < α and define Dα . If α is a limit ordinal, let
Dα = β<α Dβ . By Lemma 2.2, Dα is club in κ. If α is a successor ordinal, let β
be so that α = β + 1. Then, for every δ ∈ S ∩ acc(Dβ ), Cδ and Dβ ∩ δ are both
club in δ. Since cf(δ) = θ is uncountable, Cδ ∩ Dβ is also club in δ. By assumption,
⟨Cδ ∩ Dβ : δ ∈ S ∩ acc(Dβ )⟩ is not a fully club guessing sequence. Therefore, there
exists a club subset Dβ+1 of κ such that for every δ ∈ S ∩acc(Dβ ), Cδ ∩Dβ ⊈ Dβ+1 .
Without loss of generality, we may assume Dβ+1 ⊆ Dβ .
Suppose that ⟨Dα : α ≤ θ+ ⟩ is defined and let δ ∈ S ∩ acc(Dθ+ ). Consider
⟨Cδ ∩ Dα : α < θ+ ⟩. For every α < θ+ , by the definition of Dα+1 , we have
Cδ ∩Dα+1 ⊊ Cδ ∩Dα . Let γα be an element of Cδ ∩(Dα \Dα+1 ). Then, ⟨γα : α < θ+ ⟩
CLUB GUESSING SEQUENCES
is a sequence of distinct element of Cδ , which contradicts |Cδ | = θ.
9
□
This proof is easy, but has some remarkable points. Axiom of Choice is used to
find Dβ+1 inductively. In this sense, this proof is non-constructible. Moreover, it
does not cover all cases by, for example, a diagonal argument. It is interesting that
this proves the existence of a sequence that guesses all club subsets of κ despite
these facts. This is the characteristic property of natural structures that is pointed
out by M. Foreman and others many times (See [7] and [13]).
When θ = ℵ0 , the previous proof does not work as is because if cf(δ) = ℵ0 , then
the intersection of two club subsets of δ may not be club. To overcome this difficulty,
we need to use an argument using well-foundedness of ordinals. We shall prove the
following theorem proved by S. Shelah [36], which is stronger than Theorem 2.12.
This proof is given by M. Kojman [27]10.
Theorem 2.14 (S. Shelah [36]). Let θ, λ, κ be three regular cardinals such that
θ < λ < κ. Let S be a stationary subset of κ ∩ Cof(θ) such that every δ ∈ S is a
limit point of Cof(λ). Then, there exists a fully club guessing sequence ⟨Cδ : δ ∈ S⟩
on S such that for every δ ∈ S, Cδ ⊆ Cof(≥λ).
Proof. For each δ ∈ κ ∩ Lim, pick a club subset eδ of δ such that otp(eδ ) = cf(δ).
Let D be a club subset of κ and δ ∈ S. By induction, we shall define fδ′ (D, n) and
fδ (D, n) for every n < ω:
fδ′ (D, 0) = eδ .
fδ (D, n) = {sup(D ∩ ξ) : ξ ∈ fδ′ (D, n) and ξ > min(D)}.
∪
fδ′ (D, n + 1) = {eγ : γ ∈ fδ (D, n) ∩ Cof(<λ)}.
∪
Then, let fδ (D) = n<ω fδ (D, n). Since D is club, fδ (D, n) ⊆ D and hence
fδ (D) ⊆ D.
Claim 2.15. If δ ∈ S ∩ acc(D), then fδ (D, 0) is unbounded in δ. So, fδ (D) is
unbounded in δ.
⊢ Let δ ∈ S ∩acc(D) and ζ < δ. Since δ ∈ acc(D), D ∩δ is unbounded in δ. So,
there exists a γ ∈ D such that ζ < γ < δ. Since eδ is unbounded in δ, there exists
a ξ ∈ eδ such that ξ > γ. Then, we have ξ ∈ eδ = fδ′ (D, 0) and ξ > γ ≥ min(D).
Hence, we have sup(D ∩ξ) ∈ fδ (D, 0). We also have sup(D ∩ξ) ≥ γ > ζ. Therefore,
fδ (D, 0) is unbounded in δ.
⊣
Claim 2.16. For every δ ∈ S and n < ω, we have |fδ′ (D, n)| < λ and |fδ (D, n)| < λ
⊢ Fix δ ∈ S, and go by induction on n < ω.
First, note |fδ′ (D, 0)| = |eδ | = cf(δ) = θ < λ. If |fδ′ (D, n)| < λ, it is clear by
definition that |fδ (D, n)| ≤ |fδ′ (D, n)| < λ. Now, we shall show that, assuming
|fδ (D, n)| < λ, |fδ′ (D, n + 1)| < λ. For every γ ∈ fδ (D, n) ∩ Cof(<λ), we have
|eγ | = cf(γ) < λ. Thus, fδ′ (D, n + 1) is the union of at most |fδ (D, n)|-many sets of
cardinality <λ. Since |fδ (D, n)| < λ and λ is regular, we can see |fδ′ (D, n + 1)| < λ
⊣
10The author learned this proof by the master dissertation of Y.Hirata [21], and consulted its
presentation.
10
TETSUYA ISHIU
Claim 2.17. There exists a club subset D such that ⟨fδ (D) : δ ∈ S ∩ acc(D)⟩ is a
fully club guessing sequence.
⊢ Suppose that there is no such D. We shall define a decreasing sequence
⟨Dα : α ≤ λ⟩ of club subsets of κ. Let D0 be the set of all limit points of κ ∩ Cof(λ)
that are less than κ. It is easy to see that this set
∩ is club. If α ≤ λ is a limit ordinal
and Dβ is defined for every β < α, let Dα = β<α Dβ . Assuming Dα is defined,
we shall define Dα+1 . By assumption, ⟨fδ (Dα ) : δ ∈ S ∩ acc(Dα )⟩ is not a fully
club guessing sequence. For every δ ∈ S ∩ acc(Dα ), note that fδ (Dα ) is club in δ.
Hence, there exists a club subset Dα+1 of κ such that for every δ ∈ S ∩ acc(Dα ),
fδ (Dα ) ⊈ Dα+1 . Without loss of generality, we may assume Dα+1 ⊆ Dα .
Fix a δ ∈ S ∩acc(Dλ ). By induction, we shall define an increasing sequence ⟨αn :
n < ω⟩ in λ such that for every n < ω, if αn ≤ α < λ, then fδ (Dα , n) = fδ (Dαn , n).
Suppose that αm has been defined for all m < n. First, we shall show that αn′ < λ
such that αn′ ≤ α < λ implies fδ′ (Dα′n , n) = fδ′ (Dα , n). Put α0′ = 0. Since fδ′ (D, 0)
does not depend on D, the conclusion holds trivially. If n > 0, then let αn′ = αn−1 .
By inductive hypothesis, αn′ ≤ α < λ implies fδ (Dα′n , n − 1) = fδ (Dα , n − 1). By
the definition of f ′ , clearly we have fδ′ (Dα′n , n) = fδ′ (Dα , n).
Suppose that there is no αn < λ such that αn ≤ α < λ implies fδ (Dα , n) =
fδ (Dαn , n). Then, for every β < λ, there exists an α such that β ≤ α < λ and
fδ (Dα , n) ̸= fδ (Dβ , n). By using this fact, we can build an increasing sequence
⟨βν : ν < λ⟩ such that for every ν < λ, fδ (Dβν+1 , n) ̸= fδ (Dβν , n). For every ν < λ,
by the definition of fδ and fδ′ (Dβν+1 , n) = fδ′ (Dβν , n) = fδ′ (Dα′n , n), there exists a
ξν ∈ fδ′ (Dβν , n) such that
(1) min(Dβν ) < ξν ≤ min(Dβν+1 ) or
(2) min(Dβν+1 ) < ξν and sup(Dβν+1 ∩ ξν ) < sup(Dβν ∩ ξν )
Since |fδ′ (Dβν , n)| < λ, there exists a ξ ∈ fδ′ (Dβν , n) such that {ν < λ : ξν = ξ}
is unbounded in λ. Let X = {ν < λ : ξν = ξ}. Suppose that there exists a ν ∈ X
such that min(Dβν ) < ξν ≤ min(Dβν+1 ). By assumption, there exists a µ ∈ X such
that ν < µ < λ. Then, since Dβµ ⊆ Dβν+1 , we have ξµ = ξ ≤ min(Dβµ ). This
contradicts to the definition of ξµ . Thus, for every ν ∈ X, min(Dβν+1 ) < ξ and
sup(Dβν+1 ∩ξ) < sup(Dβν ∩ξ) sup(Dβµ ∩ξ) ≤ sup(Dβν+1 ∩ξ) < sup(Dβν ∩ξ). If both
ν < µ belong to X, then we have sup(Dβµ ∩ ξ) ≤ sup(Dβν+1 ∩ ξ) < sup(Dβν ∩ ξ).
So, ⟨sup(Dβν ∩ ξ) : ξ ∈ X⟩ is an infinite decreasing sequence of ordinals, which
contradicts to the well-foundedness of ordinals. So, we can find an αn that satisfies
the inductive hypothesis. Let αω = supn<ω αn . Since λ is regular, we have αω <
λ. Then, for every β < λ and n < ω, by the definition of αn , β ≥ αω implies
fδ (Dβ , n) = fδ (Dαω , n). Therefore, we have fδ (Dβ ) = fδ (Dαω ). Thus,
fδ (Dαω +1 ) = fδ (Dαω ) ⊆ Dαω
This contradicts to the definition of Dαω +1 .
⊣
Let S ′ be the set of all δ ∈ S ∩acc(D) such that fδ (D) ⊆ acc(D) holds. Then, for
each δ ∈ S ′ , let Cδ = nacc(fδ (D)). By Claim 2.17, S ′ is stationary, and ⟨Cδ : δ ∈ S ′ ⟩
is a fully club guessing sequence, Thus, it suffices to show the following claim.
Claim 2.18. For every δ ∈ S ′ , we have Cδ ⊆ Cof(≥λ).
⊢ Let δ ∈ S ′ and γ ∈ Cδ . Suppose cf(γ) < λ. We shall show that γ is a limit
point of fδ (D). This contradicts to γ ∈ nacc(fδ (D)).
CLUB GUESSING SEQUENCES
11
Let ζ < γ, and
∪ we shall show that there exists a ξ ∈ fδ (D) with ζ < ξ < γ.
Since fδ (D) = n<ω fδ (D, n), there exists an n < ω such that γ ∈ fδ (D, n). Then,
since γ ∈ fδ (D, n) ∩ Cof(<λ), we have eγ ⊆ fδ′ (D, n). By the definition of S ′ , we
have fδ (D) ⊆ acc(D), and hence γ ∈ acc(D). Thus, D ∩ γ is unbounded in γ.
Let ζ ′ ∈ D be so that ζ < ζ ′ < γ. Since eγ is unbounded in γ, we can pick
a ξ ∈ eγ with ξ > ζ ′ . Since eγ ⊆ fδ′ (D, n), we have ξ ∈ fδ′ (D, n). By definition, we have sup(D ∩ ξ) ∈ fδ (D, n + 1) ⊆ fδ (D). Since ζ ′ ∈ D ∩ ξ, we have
sup(D ∩ ξ) ≥ ζ ′ > ζ. This proves that γ is a limit point of fδ (D).
⊣
□
The assumption θ+ < κ in Theorem 2.12 is shown to be necessary by A. Roslanowski
and S. Shelah in [33]. Moreover, we can prove the following corollary from Theorem 2.12.
Corollary 2.19. If κ ≥ ℵ2 is a regular cardinal with κ ≥ ℵ2 , then there exists a
fully club guessing sequence on κ ∩ Lim.
S. Shelah [37] showed that we need to assume κ ≥ ℵ2 to prove this corollary.
2.3. The results similar to the existence of fully club guessing sequences.
It is known that, when θ and κ are regular cardinals satisfying θ+ < κ, κ ∩ Cof(θ)
has good properties other than Theorem 2.12. They are not directly related to club
guessing sequences, but let me introduce two typical examples.
First, we shall mention the following principle called diamond.
Definition 2.20. Let κ be an uncountable regular cardinal, and S a stationary
subset of κ. Then, ♢κ (S) is the principle that asserts the existence of a sequence
⟨Aδ : δ ∈ S⟩ such that
(1) for every δ ∈ S, Aδ ⊆ δ
(2) for every X ⊆ κ, {δ ∈ S : X ∩ δ = Aδ } is stationary in κ.
♢∗κ (S) is the principle that asserts the existence of a sequence ⟨Aδ : δ ∈ S⟩ such
that
(1) for every δ ∈ S, Aδ ⊆ P(δ) and |Aδ | ≤ |δ|.
(2) for every X ⊆ κ, there exists a club subset D of κ such that for every
δ ∈ D ∩ S, X ∩ δ ∈ Aδ .
In both cases, we may omit S when S = κ.
Fact 2.21. For every uncountable regular cardinal κ and its stationary subset S,
♢∗κ (S) implies ♢κ (S)
Theorem 2.22 (D. Jensen [25]). V = L implies that for every uncountable regular
cardinal κ, ♢∗κ holds if and only if κ is ineffable11. Particularly, for every infinite
successor cardinal κ, ♢∗κ holds. Hence, for every stationary subset S of κ, ♢κ (S)
holds.
By forcing, we can easily show that the assumption V = L is necessary in the
previous theorem. It is easy to see that ♢κ implies 2<κ = κ. However, S. Shelah
proved the following theorem, which implies that the converse also holds if κ ≥ ℵ2 .
11An uncountable regular cardinal κ is ineffable if and only if for every f : [κ]2 → 2, there
exists a stationary subset X of κ such that |{f ({a, b}) : a ̸= b ∈ X}| = 1. Here, [X]2 denotes the
set of all pairs of two distinct elements of X. It is known that every ineffable cardinal is weakly
inaccessible.
12
TETSUYA ISHIU
Theorem 2.23 (S. Shelah [39]). Suppose κ = λ+ = 2λ and let S be a stationary
subset of κ. If cf(α) ̸= cf(λ) for every α ∈ S, then ♢κ (S) holds.
Corollary 2.24 (S. Shelah [39]). If κ = λ+ and κ ≥ ℵ2 , then 2λ = λ+ is equivalent
to ♢κ
Note that the assumption κ ≥ ℵ2 is necessary in the previous corollary because
CH + ¬♢ω1 is consistent.
For ♢∗ , the following theorem holds. J. Gregory proved the case when λ is
regular, and S. Shelah proved the case when λ is singular.
Theorem 2.25 (J. Gregory [20], S. Shelah [34]). Suppose GCH. Assume κ = λ+
and let T = {α < κ : cf(α) ̸= cf(λ)}. Then, ♢∗κ (T ) holds.
Moreover, we can say a similar thing to the following principle called square.
Definition 2.26. Let λ be an infinite cardinal. □λ is the principle that asserts the
existence of a sequence ⟨Cα : α ∈ λ+ ∩ Lim⟩ such that
(1) for every α ∈ λ+ ∩ Lim, Cα is a club subset of α, and
(2) for every α ∈ λ+ ∩ Lim, cf(α) < λ implies otp(Cα ) < λ, and
(3) if β < α is a limit point of Cα , then Cβ = Cα ∩ β.
Such a sequence ⟨Cα : α ∈ λ+ ∩ Lim⟩ is called a □λ sequence.
By (2) and (3), for every α ∈ λ+ ∩ Cof(λ), we have otp(Cα ) = λ. By the
definition of λ+ , for every α ∈ λ+ ∩ Lim, we have cf(α) ≤ |α| ≤ λ, so there exists
an unbounded subset of α whose cardinality is λ. □λ says that we can assign such
an unbounded subset to each α ∈ λ+ ∩ Lim so that they are coherent in the sense
of (3).
This principle also follows from V = L.
Theorem 2.27 (D. Jensen [25]). V = L implies that for every infinite cardinal λ,
□λ holds.
S. Shelah considered the following weakening of □λ .
Definition 2.28. Let θ < λ be two regular cardinals, and S a subset of λ+ ∩Cof(θ).
We say that S has partial square if and only if there exists a sequence ⟨Cα : α ∈ S⟩
such that
(1) for every α ∈ S, Cα is a club subset of α with otp(Cα ) = θ, and
(2) for every α, β ∈ S, if γ is a limit point of both Cα and Cβ , we have
Cα ∩ γ = Cβ ∩ γ.
Such a sequence is called a partial square sequence.
If we assume □λ holds for every infinite cardinal λ, then by induction, we can
show that for every infinite cardinal λ and regular cardinal θ ≤ λ, λ+ ∩ Cof(θ) has
partial square. However, S. Shelah showed the following theorem just from ZFC.
Theorem 2.29 (S. Shelah [36]). Let θ < λ be both regular cardinals. Then, λ+ ∩
Cof(θ) can be expressed as the union of λ-many sets that have partial squares.
Unlike Theorem 2.25, which holds even when λ is singular, it is known that λ
must be regular in Theorem 2.29.
The reason why so various and strong results can be obtained on κ ∩ Cof(θ)
assuming θ+ < κ is not understood well yet as far as the author is concerned.
Nonetheless, it is true that many beautiful structures were discovered on this set.
CLUB GUESSING SEQUENCES
13
3. The non-stationary ideal on a regular cardinal ≥ℵ2 is not
saturated.
As the first example of applications of club guessing sequence, we shall introduce
the result of M. Gitik and S. Shelah about the saturatedness of the non-stationary
ideals. Their argument gave a simple solution to an important open problem at
that time by using club guessing sequences. It demonstrates how useful natural
structures are.
To describe the notion of saturatedness of ideals, prepare some definitions. Let
I be an ideal on an uncountable regular cardinal κ.
Definition 3.1. Define an equivalence relation ∼I on P(κ) as follows:
X ∼I Y ⇐⇒ (X \ Y ) ∪ (Y \ X) ∈ I.
Define P(κ)/I = {[X]I : X ∈ P(κ) \ I}. Here, [X]I is the equivalence class of X
under ∼I . Define a partial ordering ≤I on P(κ)/I as follows:
[X]I ≤I [Y ]I ⇐⇒ X \ Y ∈ I.
It is easy to see that this definition is well-defined.
Definition 3.2. Two elements [X]I and [Y ]I of P(κ)/I are said to be compatible
if and only if there exists a [Z]I ∈ P(κ)/I such that [Z]I ≤I [X]I and [Z]I ≤I [Y ]I .
[X]I and [Y ]I are said to be incompatible if and only if they are not compatible.
Note that [X]I and [Y ]I are incompatible if and only if X ∩ Y ∈ I.
Definition 3.3. We say that A ⊆ P(κ)/I is an antichain in P(κ)/I if and only
if any two distinct elements of A are incompatible. We say that an antichain A
in P(κ)/I is maximal if and only if whenever A′ is an antichain in P(κ)/I with
A ⊆ A′ , we have A′ = A.
Note that an antichain A in P(κ)/I is maximal if and only if for every [X]I ∈
P(κ)/I, there exists an [A]I ∈ A that is compatible with [X]I .
Definition 3.4. We say that I is saturated if and only if every antichain A in
P(κ)/I has cardinality at most κ.
In other words, I is saturated if and only if P(κ)/I satisfies κ+ -chain condition.
Saturated ideals has many good properties, particularly related to the method
of generic embeddings, and various applications are known. The first among such
applications is the argument given by R. Solovay in [41] about real-valued measurable cardinals. M. Foreman [15] provides a comprehensive survey about generic
embeddings.
J. Steel and R. Van Wesep [42] proved, assuming the consistency of ZF + DC +
ADR + ‘Θ is regular’, that it is consistent that NSω1 is saturated. Later, M. Foreman, M. Magidor, and S. Shelah [14] proved this consistency from the consistency
of a supercompact cardinal. We will not give the definitions of these assumptions
because they are not essential in this article, but interested readers are referred to
A. Kanamori [26]. However, it can happen only for ω1 , i.e. the following theorem
holds.
Definition 3.5. The restriction I ↾ S of an ideal I on X to S is an ideal defined
as:
I ↾ S = {Y ⊆ X : Y ∩ S ∈ I}.
14
TETSUYA ISHIU
Theorem 3.6 (M. Gitik and S. Shelah [16]).
(1) For every regular cardinal κ ≥ ℵ2 , NSκ is not saturated.
(2) For every pair of regular cardinals θ and κ with θ+ < κ, NSκ ↾ (κ ∩ Cof(θ))
is not saturated.
Here, we shall introduce the proof by M. Gitik and S. Shelah in [16] to show
(1) when κ ≥ ℵ3 .12 Club guessing sequences play a key role in the proof. We
shall only show that NSκ cannot saturated, but a similar argument also proves that
NSκ ↾ (κ ∩ Cof(θ)) cannot be saturated.
⃗ = ⟨Cδ : δ ∈ S⟩ be a tail club guessing
Definition 3.7 (S. Shelah [36]). Let C
sequence on a stationary subset of an uncountable regular cardinal κ consisting of
⃗ associated with C
⃗ to be
limit ordinals. Define the tail club guessing filter TCG(C)
the set of all X ⊆ κ such that there exists a club subset D of κ such that for every
⃗ is called the tail club
δ < κ, Cδ ⊆∗ D implies δ ∈ X. The dual ideal of TCG(C)
⃗
˘
⃗
guessing ideal associated with C and denoted by TCG(C).
˘ C)
⃗ if and only if there exists
It is trivial that for every subset X of κ, X ∈ TCG(
a club subset D of κ such that for every δ ∈ X, we have Cδ ⊈∗ D.
⃗ = ⟨Cδ : δ ∈ S⟩ be a tail club guessing sequence on a stationary
Lemma 3.8. Let C
˘ C)
⃗ is a κ-complete
subset S of an uncountable regular cardinal κ. Then, TCG(
normal ideal and contains the set of all bounded subsets of κ.
˘ C).
⃗ Since
Proof. Let X be a bounded subset of κ and we shall show X ∈ TCG(
X is bounded, there exists a ζ < κ such that X ⊆ ζ.
˘ C).
⃗
Since κ \ ζ is a club subset of κ, we have {δ ∈ S : Cδ ⊈∗ κ \ ζ} ∈ TCG(
∗
˘ C).
⃗
Clearly X ⊆ {δ ∈ S : Cδ ⊈ κ \ ζ}, so we get X ∈ TCG(
To see the κ-completeness, let {Xα : α < µ} be a set of cardinality µ < κ such
˘ C)
⃗ for every α < µ. Then, for every α < µ, there exists a club
that Xα ∈ TCG(
subset Dα of ∩
κ such that for every δ ∈ Xα , Cδ ⊈∗ Dα .
Let D = α<µ Dα . Since µ < κ, by Lemma 2.2, D is club. To see that
∪
˘ C),
⃗ it suffices to show that for every δ ∈ X,
Xα belongs to TCG(
X =
α<µ
Cδ ⊈∗ D. Let δ ∈ X. Then, there exists an α < µ such that δ ∈ Xα . By
assumption, we have Cδ ⊈∗ Dα . Since D ⊆ Dα , it implies Cδ ⊈∗ D.
˘ C).
⃗
To see the normality, let ⟨Xα : α < κ⟩ be a sequence in TCG(
For each
α < κ, there exists a club subset Dα of κ such that for event δ ∈ Xα , Cδ ⊈∗ Dα .
Let D = △α<κ Dα . By Lemma 2.6, D is club.
˘ C),
⃗ we shall show that for every δ ∈ X, Cδ ⊈∗ D.
To see X = ▽α<κ Xα ∈ TCG(
By the definition of the diagonal union, there exists an α < δ such that δ ∈ Xα . By
the definition of Dα , we have Cδ ⊈∗ Dα . But as we saw in the proof of Lemma 2.6,
we have D \ (α + 1) ⊆ Dα . Thus, we have Cδ ⊈∗ D.
□
Lemma 3.9 (J. Baumgartner, A. Taylor, and S. Wagon [2]). Let I and J be κcomplete normal ideals. Suppose that I is saturated and I ⊆ J. Then, there exists
an X ∈ P(κ) \ I such that J = I ↾ X.
12The case of κ = ℵ can be proved by a similar argument though some modification is
2
necessary.
CLUB GUESSING SEQUENCES
15
Particularly, when NSκ is saturated, for every κ-complete normal I on κ that
contain all bounded subsets of κ, there exists a stationary subset S of κ such that
I = NSκ ↾ S.
Let S be a stationary subset of κ. Then, it is easy to see that if {[Xα ]NSκ ↾S :
α < µ} is an antichain in P(κ)/NSκ ↾ S, then {[Xα ∩ S]NSκ : α < µ} is an antichain
in P(κ)/NSκ .
So, if NSκ is saturated, then so is its restriction NSκ ↾ S to any stationary subset
S of κ.
Lemma 3.10 (folklore). Let I be a saturated κ-complete normal ideal on an uncountable regular cardinal κ. Let {[Xα ]I : α < κ} be an antichain in P(κ)/I. Then,
there exists a sequence ⟨Xα′ : α < κ⟩ in P(κ) \ I such that
(1) for every α < κ, [Xα ] = [Xα′ ], and
(2) for every β < α < κ, Xβ′ ∩ Xα′ = ∅.
This property is sometimes called disjointing property.
The following lemma is a special case of a more general one that says that for
every partially ordered set (P, ≤P ) and its dense subset D, there exists a maximal
antichain that is a subset of D. This lemma is frequently used in forcing.
Lemma 3.11. Let I be an ideal on an uncountable regular cardinal κ.
Let D ⊆ P(κ)/I be so that for every [X]I ∈ P(κ)/I, there exists a [D]I ∈ D such
that [D]I ≤I [X]I .
Then, there exists a maximal antichain A in P(κ)/I such that A ⊆ D.
Now, we are ready to prove Theorem 3.6(1) in case of κ ≥ ℵ3 .
Proof(Theorem 3.6). Let κ ≥ ℵ3 be a regular cardinal such that NSκ is saturated.
Let S̄ be the set of all γ ∈ κ ∩ Cof(ω) such that γ is a limit point of Cof(≥ ℵ2 ). By
Theorem 2.14, for every stationary subset S of S̄, there exists a tail club guessing
⃗ = ⟨Cδ : δ ∈ S⟩ on S such that for every δ ∈ S, Cδ ⊆ Cof(≥ℵ2 ). By
sequence C
˘ C)
⃗ is a κ-complete normal ideal on κ that contains all bounded
Lemma 3.8, TCG(
˘ C).
⃗ Since NSκ is a saturated
subsets of κ. So, by Lemma 2.9, we have NSκ ⊆ TCG(
κ-complete normal ideal, by Lemma 3.9, there exists a stationary subset T of S such
˘ C)
⃗ = NSκ ↾ T . By Lemma 3.11, there exists a sequence ⟨Sα : α < κ⟩ of
that TCG(
stationary subsets of S̄.
(1) {[Sα ]NSκ ↾S̄ : α < κ} is a maximal antichain in P(κ)/NSκ ↾ S̄.
(2) For every α < κ, there exists a tail club guessing sequence ⟨Cδα : δ ∈ Sα ⟩
α
˘
such that TCG(⟨C
δ : δ ∈ Sα ⟩) = NSκ ↾ Sα . Without loss of generality, we
may assume otp(Cδα ) = ω for everyδ ∈ Sα .
Since NSκ is saturated, so is NSκ ↾ S̄. By Lemma 3.10, there exists a sequence
⟨Sα′ : α < κ⟩ of stationary subsets of κ such that for every α < κ, [Sα ]NSκ ↾S̄ =
[Sα′ ]NSκ ↾S̄ , and for every β < α < κ, Sβ′ ∩ Sα′ = ∅. Without loss of generality, we
may assume Sα′ ⊆ Sα for every α < κ. Since [Sα ]NSκ ↾S̄ = [Sα′ ]NSκ ↾S̄ , Sα′ ⊆ Sα , and
′
′
α
α
˘
˘
TCG(⟨C
δ : δ ∈ Sα ⟩) = NSκ ↾ Sα , we have TCG(⟨Cδ : δ ∈ Sα ⟩) = NSκ ↾ Sα .
Define a sequence ⟨Cδ : δ ∈ S̄⟩ as follows. If there exists an α < κ such that
δ ∈ Sα′ , then Cδ = Cδα . Otherwise, let Cδ be any unbounded subset of δ ∩ Cof(≥ℵ2 )
such that otp(Cδ ) = ω. From this definition, clearly we have for every δ ∈ S̄,
otp(Cδ ) = ω.
16
TETSUYA ISHIU
This sequence ⟨Cδ : δ ∈ S̄⟩ has the following stronger property than just being
a tail club guessing sequence.
Claim 3.12. For every club subset D of κ, there exists a club subset E of κ such
that for every δ ∈ S̄ ∩ E, Cδ ⊆∗ D.
⊢ Let D be a club subset of κ. Put S = {δ ∈ S̄ : Cδ ⊆∗ D}. If S ∪ (κ \ S̄)
contains a club subset of κ, then the conclusion is witnessed by the club subset.
Suppose not, i.e. assume κ \ (S ∪ (κ \ S̄)) is stationary in κ.
Note κ\(S ∪(κ\ S̄)) = (κ\S)∩ S̄ = S̄ \S. Thus, S̄ \S is a stationary subset of S̄.
Since {[Sα′ ]NSκ ↾S̄ : α < κ} is a maximal antichain in NSκ ↾ S̄, there exists an α < κ
such that Sα′ ∩ (S̄ \ S) is stationary. Since Sα′ ⊆ S̄, we have Sα′ ∩ (S̄ \ S) = Sα′ \ S.
We have
Sα′ \ S = {δ ∈ Sα′ : Cδ ⊈∗ D}
= {δ ∈
Sα′
:
Cδα
∗
⊈ D}
(by the definition of S)
(since Cδ = Cδα for every δ ∈ Sα′ )
α
′
˘
So, by definition, this set belongs to TCG(⟨C
δ : δ ∈ Sα ⟩).
α
′
˘
However, by assumption, TCG(⟨Cδ : δ ∈ Sα ⟩) = NSκ ↾ Sα′ . Thus, we have
′
Sα \ S ∈ NSκ ↾ Sα′ . Since Sα′ \ S ⊆ Sα′ . it means that Sα′ \ S is non-stationary,
which is a contradiction.
⊣
We shall derive a contradiction from this claim. Define a decreasing sequence
⟨Dα : α ≤ ω1 ⟩ of club subsets of κ as follows. Let D0 be the set of all limit points
of Cof(≥ℵ2 ) that is less ∩
than κ. If α ≤ ω1 is a limit ordinal and ⟨Dβ : β < α⟩ has
been defined, let Dα = β<α Dβ . We shall explain how to build Dα+1 from Dα .
Since Dα is a club subset of κ, by Claim 3.12, there exists a club subset Dα+1 of κ
such that for every δ ∈ Dα+1 ∩ S̄, Cδ ⊆∗ Dα . Without loss of generality, we may
assume Dα+1 ⊆ acc(Dα ).
Let δ be the ω-th element of Dω1 , i.e. the unique element of Dω1 such that
otp(Dω1 ∩ δ) = ω. Then, since δ ∈ D0 , by the definition of D0 , δ is a limit point
of Cof(≥ω2 ). Hence, we have δ ∈ S̄. For every α < ω1 , since δ ∈ Dα+1 ∩ S̄ , we
have Cδ ⊆∗ Dα . So, there exists a ζα < δ such that Cδ \ ζα ⊆ Dα . Without loss
of generality, we may assume ζα ∈ Cδ . Since |Cδ | = ℵ0 . there exists a ζ ∈ Cδ such
that {α < ω1 : ζα = ζ} is unbounded in ω1 . Then, for every β < ω1 , there exists
an α < ω1 such that β ≤ α and ζα = ζ. Hence, we have Cδ \ ζ ⊆ Dα ⊆ Dβ . So,
Cδ \ ζ ⊆ Dω1 .
Let γ ∈ Cδ \ ζ. Since Cδ ⊆ Cof(≥ℵ2 ), we have cf(γ) ≥ ℵ2 . Then, for every
α < ω1 . we have γ ∈ Dα+1
∩ ⊆ acc(Dα ). Therefore, Dα ∩ γ is a club subset of γ.
Since cf(γ) ≥ ω2 > ω1 , α<ω1 (Dα ∩ γ) is also a club subset of γ. In particular,
(∩
)
otp α<ω1 (Dα ∩ γ) ≥ ω2 Note
∩
α<ω1
(Dα ∩ γ) =
∩
Dα ∩ γ
α<ω1
= Dω1 ∩ γ
So, we have otp(Dω1 ∩ δ) ≥ otp(Dω1 ∩ γ) ≥ ω2 . This contradicts to the assumption
that δ is the ω-th element of Dω1
□
CLUB GUESSING SEQUENCES
17
4. Club guessing sequences in PCF theory
PCF theory, established by S. Shelah in [36], surprised many people by its beauty
and strength. The key tools in it is the usage of various natural structures, including club guessing sequences. In this section, we shall describe how club guessing
sequences were used to prove ℵω ℵ0 < max{ℵω4 , (2ℵ0 )+ }, the most frequently cited
result of PCF theory.
First, we shall define ultrafilters.
Definition 4.1. A filter U on X is said to be an ultrafilter on X if and only if F is
a filter on X such that U ⊆ F , we have F = U . This is equivalent to the assertion
that for every Y ⊆ X, either Y ∈ U or X \ Y ∈ U
Let A be a set of ordinals. Let ΠA be the set of all functions f such that for
every α ∈ A, f (α) < α. When U is an ultrafilter on A, define two binary relations
=U and ≤U on ΠA as follows:
f =U g ⇐⇒ {α ∈ A : f (α) = g(α)} ∈ U.
f ≤U g ⇐⇒ {α ∈ A : f (α) ≤ g(α)} ∈ U.
Then, =U is an equivalence relation, and ≤U is a pseudo linear ordering.
Let X be a set, and ≤X a pseudo partial ordering on X. We say that a subset
Y of X is cofinal in (X, ≤X ) if and only if for every x ∈ X, there exists a y ∈ Y
such that x ≤X y. The cofinality cf(X, ≤X ) is defined to be the least cardinality
of subsets of X that is cofinal in (X, ≤X ). If ≤X is linear, cf(X, ≤X ) is always
regular.
Definition 4.2. For a set A of cardinals, define pcf(A) as follows:
pcf(A) = {cf(ΠA, ≤U ) : U is an ultrafilter on A}.
We often use the term ‘ultrafilter’ to mean only non-principal ones, but here it
includes principal ones also13.
pcf means ‘possible cofinality’ and is the reason why the theory is called PCF
theory. S. Shelah showed that the function pcf has very good properties14, and
close relationship with cardinal arithmetic.
To describe the properties of pcf, we need some definitions.
Definition 4.3.
(1) For every set A of cardinals, define A(+) = {κ+ : κ ∈ A}.
(2) A set A is said to be progressive if and only if |A| < min A
(3) A set A of cardinals is an interval of regular cardinals if and only if whenever
µ and κ both belong to A, and λ is a regular cardinal such that µ < λ < κ,
we have λ ∈ A
For example, A = {ℵn : n < ω or n = ω + 1} is an interval of regular cardinals.
ℵω does not belong to A though ℵ1 < ℵω < ℵω+1 . It does not matter since ℵω is
singular.
13We say that a filter F on X is principal if and only if there exists an X ⊆ X such that
0
F = {Y ⊆ X : X0 ⊆ Y }.
14We will not use it in this article, but for example, by using pcf as the closure operation, we
can define a topology on pcf(A). This topological space is compact, Hausdorff, 0-dimensional,
and scattered.
18
TETSUYA ISHIU
Fact 4.4. Let A be a progressive set of regular cardinals. Then, the following hold.
(1) A ⊆ pcf(A).
(2) A′ ⊆ A implies pcf(A′ ) ⊆ pcf(A).
(3) pcf(A) has the maximum element max pcf(A).
(4) There is no subset B of pcf(A) such that |B| = |A|+ and for every β ∈ B,
max pcf(B ∩ β) < β.15
Definition 4.5. When X is a set and κ is a cardinal, [X]κ denotes the set of all
subsets of X whose cardinality is κ.
Fact 4.6. Let A be a progressive interval of regular cardinals. Then, the following
hold.
(1) pcf(A) is an interval of regular cardinal.
(2) cf([sup A]|A| , ⊆) = max pcf(A).
It is not directly related to the topics in this article, but we shall mention scales,
one of the natural structures discovered by S. Shelah. For example, let A = {ℵn :
0 < n < ω}. By Fact 4.4(1)(3) and Fact 4.6(1), we have ℵω+1 ∈ pcf(A). That
is, there exists an ultrafilter U on A such that cf(ΠA, U ) = ℵω+1 . However, the
following much stronger theorem was proved by S. Shelah in [36]. For f, g ∈ ΠA,
let f <fin g denote |{κ ∈ A : f (κ) ≥ g(κ)}| < ℵ0 . i.e. f (κ) < g(κ) holds for all but
finitely many κ ∈ A.
Theorem 4.7. LetA = {ℵn : 0 < n < ω}. Then, there exist B ⊆ A and a sequence
⟨fα : α < ωω+1 ⟩ in ΠB such that
(1) ⟨fα : α < ωω+1 ⟩ is <fin -increasing, and
(2) {fα : α < ωω+1 } is cofinal in (ΠB, <fin ), i.e. for every f ∈ ΠB, there
exists an α < ωω+1 such that f <fin fα .
Such a sequence is called a scale in ΠA/fin of length ωω+1 .
Go back to the property of the pcf function.
Fact 4.8. Let µ be a cardinal of uncountable cofinality. Then, there exists a club
subset D such that µ+ = max pcf(D(+) ).
The following proof is based on the one in [1].
Theorem 4.9. Let A = {ℵn : 0 < n < ω}. Then, |pcf(A)| < ℵ4 .
Proof. Suppose |pcf(A)| ≥ ℵ4 . Since A is an interval of regular cardinals, by
Fact 4.6(1), pcf(A) is also an interval of regular cardinals. By Fact 4.4(1), we
have ℵ1 ∈ A ⊆ pcf(A). Thus, every uncountable regular cardinal less than ℵω4
belongs to pcf(A). By Theorem 2.12, there exists a fully club guessing sequence
⟨Cδ : δ ∈ ω3 ∩ Cof(ω1 )⟩ on ω3 ∩ Cof(ω1 ). For every δ ∈ ω3 ∩ Cof(ω1 ), without loss
of generality, we may assume Cδ is club in δ and otp(Cδ ) = ℵ1 .
We shall define an increasing sequence ⟨κα : α < ω3 ⟩ of cardinals less than ℵω4
as follows: Let κ0 = ℵ1 . If α is a limit ordinal, and κβ has been defined for every
β < α, let κα = supβ<α κβ .
Suppose that for every β ≤ α, κβ has been defined. For each δ ∈ ω3 ∩ Cof(ω1 ),
let Bα,δ be the set of all κβ + so that β ∈ nacc(Cδ ) and β ≤ α. Set λα,δ =
15This follows from the following intuitive property called localization: Let A be a progressive
set of regular cardinals, and B a progressive subset of pcf(A). Then, for every λ ∈ pcf(B), there
exists a subset B0 of B such that |B0 | ≤ |A| and λ ∈ pcf(B0 )
CLUB GUESSING SEQUENCES
19
max pcf(Bα,δ ) and λα = sup{λα,δ : δ ∈ ω3 ∩ Cof(ω1 ) and λα,δ < ℵω4 }. Since λα is
the supremum of a subset of ℵω4 whose cardinality is at most ℵ3 , we have λα < ℵω4 .
Let κα+1 = (max{κα , λα })+ .
Let D0 = {κα : α < ω3 }, and η = sup D0 . Then, D0 is a club subset of
η, and cf(η) = ω3 ≥ ω1 . By Fact 4.8, there exists a club subset D1 of η such
that max pcf(D1 (+) ) = η + . Let D2 = D0 ∩ D1 . Then, D2 is a club subset of η
with D2 ⊆ D0 . Moreover, by Fact 4.4(2), we have max pcf(D2 (+) ) ≤ η + . Since
D2 (+) is an unbounded subset of η, we have max pcf(D2 (+) ) ≥ η. However, since
max pcf(D2 (+) ) is regular, it cannot be η, and hence we have max pcf(D2 (+) ) ≥ η + .
So, we obtained max pcf(D2 (+) ) = η + .
Let E = {α < ω3 : κα ∈ D2 }. Then, E is a club subset of ω3 . Since ⟨Cδ : δ ∈
ω3 ∩ Cof(ω1 )⟩ is a fully club guessing sequence, there exists a δ ∈ ω3 ∩ Cof(ω1 )
such that Cδ ⊆ E. Let B = {κγ + : γ ∈ nacc(Cδ )}. We shall show that B does not
satisfy Fact 4.4(4), which will be a contradiction. It is trivial that |B| = ℵ1 = |A|+ .
Since every regular cardinal less than ℵω4 belongs to pcf(A), we have B ⊆ pcf(A).
So, it suffices to see that for every κ ∈ B, we have max pcf(B ∩ κ) < κ. For every
κ ∈ B, by the definition of B, there exists a γ ∈ nacc(Cδ ) such that κ = κγ + .
Let α = max(Cδ ∩ γ). Since γ ∈ nacc(Cδ ), we have α < γ.16 Then, we have
B ∩ κ = {κβ + : β ∈ nacc(Cδ ) and β ≤ α} = Bα,δ . Then,
λα,δ = max pcf(Bα,δ )
= max pcf(B ∩ κ)
≤ max pcf(B)
≤ max pcf(D2 (+) ) = η + < ℵω4
By construction, λα,δ ≤ λα < κα+1 ≤ κγ < κ. Therefore, we have max pcf(B ∩κ) <
κ.
□
Theorem 4.10. ℵω ℵ0 < max{ℵω4 , (2ℵ0 )+ }.
Proof. If 2ℵ0 ≥ ℵω , the theorem is trivial since
ℵω ℵ0 ≤ (2ℵ0 )ℵ0 = 2ℵ0
Suppose 2ℵ0 < ℵω . Let A = {ℵn : 0 < n < ω}. By Theorem 4.9 and Fact 4.6(2),
we have cf([ℵω ]ℵ0 , ⊆) = max pcf(A) < ℵω4 .
Let λ = cf([ℵω ]ℵ0 , ⊆), and ⟨Xα : α < λ⟩ a cofinal subset of ([ℵω ]ℵ0 , ⊆). Thus,
for every X ∈ [ℵω ]ℵ0 , there exists an α < λ such that X ⊆ Xα . So, we have
ℵω ℵ0 ≤ λ · 2ℵ0 = λ < ℵω4
□
By approaching through the pcf function and a club guessing sequence, this
surprising theorem can be proved by a simple argument. Of course, we need to
prove the facts about the pcf function that were admitted without proofs, but they
are easy to grasp intuitively. Personally, the author believe that the construction
of such beautiful structures just assuming ZFC is as an important contribution as
more concrete theorems.
16That is, α is the greatest element of C that is less than γ.
δ
20
TETSUYA ISHIU
For the readers who want to learn more about PCF theory, the author recommends [3] and [1] for example.
5. The Universality problem
One of the topics in which club guessing sequences are the most effectively used
is the universality problems, studied by M. Džamonja, M. Kojman, S. Shelah, and
others. We would like to describe it in this section.
For example, let A be a class of linearly ordered sets. For A, B ∈ A, we write
A ,→ B if there is an order-preserving injection from A into B. That is, A ,→ B
means that we can embed A into B. Then, ,→ is a pseudo ordering on A. Then,
we say that U ⊆ A is a universal family of A if and only if for every A ∈ A, there
exists a B ∈ U such that A ,→ B. The minimal cardinality of universal families is
called the universality number of A. Meanwhile, we say that B ⊆ A is a basis if
and only if for every A ∈ A, there exists a B ∈ B such that B ,→ A.
For example, let A be the class of all countably infinite linearly ordered sets.
G. Cantor proved that for every A ∈ A, A ,→ Q. Hence, {Q} is a universal family
of A, and the universality number of A is 1. When a singleton is a universal family
as in this example, its (unique) element is called a universal model. Moreover, it
is easy to see that for every A ∈ A, either ω ,→ A or (−ω) ,→ A holds. Here, −ω
is the reverse of ω, i.e. give an ordering on the set ω by n ≤(−ω) m ⇐⇒ m ≤ n.
Then, {ω, −ω} is a basis.
For another example, let A be a class of topological spaces. For A, B ∈ A,
define A ,→ B to mean that there exists a subspaces of B that is homeomorphic
to A. This can be also considered as an embeddability relation. We can define its
universal family and basis by the same way as for classes of linearly ordered sets.
These are considered as the measure of complexity of A. Universal families and
bases are defined for many other classes and investigated.
The proof of the following theorem regarding universality numbers of classes of
linearly ordered sets is one of the oldest applications of club guessing sequences.
Theorem 5.1 (M. Kojman, S. Shelah [28]). Let θ and κ be regular cardinals such
that θ < κ < 2θ . Suppose that one of the following two conditions holds.
(1) θ = ℵ0 , κ = ℵ1 , and there is a club guessing sequence on ω1 .
(2) θ+ < κ.
Then, the universality number of the class of all linearly ordered sets of cardinality
κ is at least 2θ .
By a classical result of model theory, we know that if 2<κ = κ, then the universality number of such a class is 1.17 Thus, it is impossible to show that the
universality number is big just from ZFC. However, by forcing, it is easy to build
a model of ZFC in which the universality number is big. The key point of Theorem 5.1 is that just from a cardinal arithmetic assumption (and the existence of
a club guessing sequence on ω1 in case of (1)), we can prove that the universality
number is big.
We shall outline the proof of this theorem. Suppose that either (1) or (2) holds.
Let S = κ ∩ Cof(θ). Since either (1) or (2) is satisfied, there exists a fully club
guessing sequence ⟨Cδ : δ ∈ S⟩ on S such that for every δ ∈ S, otp(Cδ ) = θ. In case
17In general, if 2<θ = θ, then for every countable first- order theory T , the universality number
of the class of all models of T of cardinality θ is 1
CLUB GUESSING SEQUENCES
21
of (2), we furthermore assume that this sequence is truly tight, i.e. there exists a
sequence ⟨Pξ : ξ < κ⟩ such that
(1) for every ξ < κ, Pξ ⊆ P(ξ) and |Pξ | < κ, ∪
(2) for every δ ∈ S and γ ∈ nacc(Cδ ), Cδ ∩ γ ∈ ξ<γ Pξ .
S. Shelah showed in [35] that θ+ < κ implies that there exists a truly tight fully
club guessing sequence.
Let Cδ (i) be an element γ ∈ Cδ such that otp(Cδ ∩ γ) = i. For example, Cδ (0)
is the least element of Cδ , and Cδ (1) is the next element of Cδ .
⃗ = ⟨Lγ : γ < κ⟩
Let ⟨L, ≤L ⟩ be a linearly ordered set of cardinality κ. Then, let L
be a sequence so that
(1) for every γ < κ, |Lγ | < κ, or
(2) for every γ < γ ′ < κ, Lγ ⊊ Lγ ′ , i.e, ⟨L
∪γ : γ < κ⟩ is an increasing sequence,
(3) for every limit ordinal γ < κ, Lγ = ξ<γ Lξ (This property is sometimes
called
continuity.).
∪
(4) γ<κ Lγ = L
Such a sequence is called a filtration of L.18
For δ ∈ S and x ∈ L, we define the invariant invL,
⃗ C
⃗ (δ, x) to be the set of all
′
i < θ such that there exists an x ∈ LCδ (i+1) \ LCδ (i) such that {y ∈ LCδ (i) : y <L
x′ } = {y ∈ LCδ (i) : y <L x}.
⃗ =
The invariant defined as above depends on the filtration. However, if both L
⃗ ′ = ⟨L′γ : γ < κ⟩ are filtrations of L, then {γ < κ : Lγ = L′γ }
⟨Lγ : γ < κ⟩ and L
contains a club subset of κ. In this sense, the invariants are almost independent of
the filtration.
The following two lemmas play key roles in the proof of Theorem 5.1.
Lemma 5.2 (Construction Lemma). For every A ⊆ θ, there exist a linearly ordered
⃗ of L such that {δ ∈ κ ∩ Lim : ∃x ∈
set ⟨L, ≤L ⟩ of cardinality κ and a filtration L
L(invL,
(δ,
x)
=
A)}
contains
a
club
subset
of
κ.
⃗ C
⃗
Lemma 5.3 (Preservation Lemma). Let ⟨L, ≤L ⟩ and ⟨L′ , ≤L′ ⟩ be two linearly or⃗ a filtration of L, L
⃗ ′ a filtration of L′ , and f : L → L′
dered sets of cardinality κ, L
an embedding, i.e. an order-preserving injection. Then, there exists a club subset E
such that for every δ ∈ S and x ∈ L, if Cδ ⊆ E, then invL,
⃗ C
⃗ (δ, x) = invL
⃗ ′ ,C
⃗ (δ, f (x)).
We will not present the proofs of these lemmas because they are too technical for
⃗ is a truly tight fully club
this article, but they effectively use the assumption that C
guessing sequence. The Construction Lemma says that for every A ⊆ θ, we can
⃗ such that for almost all δ (modulo
build a linearly ordered set L and a filtration C
club filter), there exists an x ∈ L such that invL,
⃗ C
⃗ (δ, x) = A. The Preservation
Lemma can be interpreted as when L is embedded into L′ , almost all invariants of
⃗ remain in L′ .
L (modulo TCG(C))
If we admit these lemmas, then Theorem 5.1 can be proved as follows.
Proof(Theorem 5.1). Let µ be the universality number and suppose µ < 2θ . By
definition, there exists a universal family U of cardinality µ. Let {Lα : α < µ} be
⃗ α of Lα .
an enumeration of U. For each α < µ, pick a filtration L
18In general, we usually require that for every γ < κ, L is a substructure of L. In case of
γ
linearly ordered sets, any subset forms a linearly ordered sets, so we can omit the requirement.
22
TETSUYA ISHIU
Define B = {invL
⃗ α ,C
⃗ (δ, x) : δ ∈ S, x ∈ Lα and α < µ}. Then, we have
|B| ≤ max{κ, µ} and B ⊆ P(θ). Since max{κ, µ} < 2θ by assumption, there exists
an A ⊆ θ with A ̸∈ B. By Lemma 5.2, there exist a linearly ordered set ⟨L, ≤L ⟩,
⃗ and a club subset D of κ such that for every δ ∈ S ∩ D, there exists
its filtration L,
an x ∈ L such that invL,
⃗ C
⃗ (δ, x) = A. Since U is a universal family, there exist an
α < µ and an order-preserving function f : L → Lα . Let E be a club subset of
κ that satisfies the conclusion of Lemma 5.3. Without loss of generality, we may
⃗ is a fully club guessing sequence, there exists a δ ∈ S
assume E ⊆ D. Since C
with Cδ ⊆ E. Clearly we have δ ∈ E ⊆ D. By assumption, there exists an x ∈ L
such that invL,
⃗ C
⃗ (δ, x) = A. Hence, we have invL
⃗ α ,C
⃗ (δ, f (x)) = invL,
⃗ C
⃗ (δ, x) = A.
Therefore, we obtained A ∈ B, which contradicts to the definition of A.
□
It is known that by applying the same idea, we can prove similar theorems about
many classes, and the further research is going on. Moreover, there is also a project
to classify classes of structures by their universality numbers, and partial success
was attained even on complicated classes that cannot be described by first-order
logic. The interested readers are referred to M. Džamonja [10] on this topic.
It does not have direct relationship to club guessing sequences, but we will mention bases. J. T. Moore proved the following theorem by expanding the notion of
minimal walk, first introduced by S. Todorcevic.
Theorem 5.4 (J. T. Moore [32]). Every basis of the class of uncountable regular
Hausdorff spaces has cardinality at least ℵ2 .
This theorem was proved at the same time as he solved the L-space problem,
which was one of the most important open problems in general topology. As well
as the results of PCF theory, these theorems proved by minimal walks suggest
that ZFC determines the properties of its models more than people thought. For
minimal walks, please refer to S. Todorcevic [43].
On the other hand, J. T. Moore also proved the following theorem.
Theorem 5.5. Assume Proper Forcing Axiom(PFA)19. Then, {R, ω1 , −ω1 , C, −C}
is a basis of the class of all uncountable linearly ordered sets. Here,
(1) R is an arbitrary ℵ1 -dense subset of R with |R| = ℵ1 20,
(2) ω1 is the ordinal ω1 with usual order,
(3) −ω1 is the reverse of ω1 ,
(4) C is an arbitrary Countryman line21, and
(5) −C is the reverse of C.
Thus, it is impossible by using any construction under ZFC to avoid these five
linearly ordered sets. On the other hand, B. Dushnik and E. W. Miller proved in
[9] that CH implies that every basis of the class of uncountable linearly ordered sets
has cardinality 2ℵ1 . Moreover, it is also well known that every basis must have at
19For PFA, please refer to T. Jech [23] for example.
20Let X be a subset of a linearly ordered set ⟨L, ≤ ⟩. We say that X is ℵ -dense if and only
1
L
if for every x, y ∈ X with x <L y, |{z ∈ X : x <L z <L y}| ≥ ℵ1 .
21We say that an uncountable linearly ordered set C is called a Countryman line if and only
if C 2 is the union of countably many chains. Here, X ⊆ C 2 is said to be a chain if and only if
when we give coordinate-wise partial ordering to C 2 , X is a linearly ordered subset, i.e. for every
(c1 , c2 ), (d1 , d2 ) ∈ X, either c1 ≤ d1 and c2 ≤ d2 , or d1 ≤ c1 and d2 ≤ c2 .
CLUB GUESSING SEQUENCES
23
least five elements. As well as the arguments on universality numbers, these results
are very interesting to consider the complexity of classes of structures.
6. A model with many precipitous ideals
So far, we have mainly discussed the theorems that are provable under ZFC.
However, there are some methods to prove consistency that would not be possible without natural structures. It suggests that the arguments by using natural
structures will not only be used in special situations, but also be combined with
other methods such as forcing and inner models and applied for every topic in set
theory. As an example, we will mention the construction of a model in which there
are many essentially different precipitous ideals, first discovered by the author. In
this section, we assume basic knowledge in forcing. For example, K. Kunen [29]
provides a good introduction to the method.
Let I be an ideal on an uncountable regular cardinal κ. Since P(κ)/I is a forcing
notion, we may consider the generic extension V [G]. Here, G is a generic filter on
P(κ)/I. Then, we can consider the ultrapower (V κ ∩ V )/G. If this ultrapower is
well-founded no matter what generic filter is used, then we say that I is precipitous.
For example, R. Solovay [41] essentially proved that every saturated ideal is
precipitous. By a standard argument, if I is a precipitous ideal and G is a generic
filter on P(κ)/I, then we can define an elementary embedding j : V → M ≃
(V κ ∩ V )/G.22 An elementary embedding that is built in this way is called a
generic elementary embedding.
If I is normal, then the critical point23 of the generic elementary embedding from
I is κ. It is known that there can be a precipitous ideal on a small cardinal. For
example, the following theorem was proved by T. Jech, M. Magidor, W. Mitchell,
and K. Prikry.
Theorem 6.1 (T. Jech, M. Magidor, W. Mitchell, K. Prikry [24]). The following
statements are equiconsistent, i.e. their consistency is equivalent.
(1) There is a measurable cardinal.
(2) Then exists a precipitous ideal on ω1 .
(3) NSω1 is precipitous.
This is very different from elementary embeddings built from large cardinals,
whose critical points are always measurable. The existence of a precipitous ideal
implies many properties of the model.
The precipitousness of natural ideals is more interesting by the following reasons.
Firstly, by using the definition of the ideal, we can prove various properties of the
⃗ = ⟨Cδ : δ ∈
generic elementary embedding from it. For example, suppose that C
κ ∩ Lim⟩ is a tail club guessing sequence on an uncountable regular cardinal κ
˘ C)
⃗ is precipitous. Let j : V → M ⊆ V [G] be the generic elementary
and TCG(
˘ C).
⃗ Then, there exists a subset C ∈ M of κ such that
embedding built from TCG(
for every club subset D ∈ V of κ, C ⊆∗ D. This phenomenon is called outside club
guessing, and investigated by M. Džamonja and S. Shelah [11].
22We say that j : V → M is an elementary embedding if and only if for every (first-order)
formula φ(v1 , v2 , . . . , vn ) and a1 , a2 , . . . , an ∈ V , whenever V ⊨ φ(a1 , a2 , . . . , an ), we have M ⊨
φ(j(a1 ), j(a2 ), . . . , j(an )).
23The critical point of a non-trivial elementary embedding j is the least κ such that j(κ) > κ.
24
TETSUYA ISHIU
Secondly, the definition of an ideal can be used to show precipitousness and
other properties. M. Foreman, M. Magidor, and S. Shelah [14] proved the following
theorem.
Theorem 6.2 (M. Foreman, M. Magidor, and S. Shelah [14]). Let κ be an uncountable regular cardinal, and λ > κ a supercompact cardinal24. Then, in the forcing
extension by the Levy collapse that forces λ to be κ+ , NSκ is precipitous.
To build a precipitous ideal, we often begin with an elementary embedding built
from a large cardinal, and apply forcing in such a way that the extension of the
embedding becomes the generic elementary embedding. However, the argument
used to prove Theorem 6.2 is totally different, and cleverly uses the properties of
the non-stationary sets.
By modifying this argument, the author obtained the following result.
Theorem 6.3 (Ishiu [22]). Let κ be an uncountable regular cardinal, and λ > κ a
Woodin cardinal. In the forcing extension by the Levy collapse to force λ to be κ+ ,
⃗ on κ, TCG(
˘ C)
⃗ is precipitous.
for every tail club guessing sequence C
In the model obtained in the proof of the previous theorem, there are κ+ -many
essentially different tail club guessing sequences. So, there are κ+ - many essentially
different precipitous ideals on κ in the model25. K. Kunen and J. Paris [30] showed
that if the existence of a measurable cardinal is consistent, then it is also consistent
κ
that there are 22 -many normal ultrafilters on a measurable cardinal κ, Since the
dual ideal of a normal ultrafilter is considered as a precipitous ideal, there are many
precipitous ideals on one fixed cardinal in this model. However, the model in which
there are many precipitous ideals on one cardinal that is not measurable was first
constructed in the proof of Theorem 6.3 in [22].
As this example shows, there are techniques to prove consistency that can be
applied only through natural structures.
7. Concluding remarks
For these few years, there have been some conferences that celebrate the sixtieth
birthdays of researchers who have been leading the field of set theory, including
S. Shelah. In January 2010, Very Informal Gathering26 to celebrate the sixtieth
birthday of J. Steel was held at UCLA. I wonder what the set theorists of this
generation, who were probably students when forcing was born, think about the
development of set theory since then in retrospect.
Though it may not be as fast as forcing, arguments by using natural structures
is changing set theory. In particular, the results about the powers of singular
cardinals obtained by S. Shelah (covered in Section 4), and the construction of Lspace assuming only ZFC given by J. T. Moore (mentioned in the second half of
24It was proved by N. Goldring [19] that a Woodin cardinal, which is weaker than supercompact, is sufficient.
25If I is a precipitous ideal on κ, then by taking its restrictions, we can obtain at least κ+ -many
precipitous ideals. However, since the generic elementary embedding obtained from a restriction of
I can be obtained from I, so we can say that there is no essential difference. ”κ+ -many essentially
different precipitous ideals” mean that there are κ+ - many precipitous ideals that cannot be
obtained by such a trivial method, but produce different generic elementary embeddings.
26Very Informal Gathering(VIG) is a conference that has been held at UCLA since 1976, and
this one was the fifteenth.
CLUB GUESSING SEQUENCES
25
Section 5) were against many people’s anticipation. I look forward to seeing the
development of set theory and its applications when these methods are more widely
understood and more frequently applied. I sincerely hope that some readers are
interested in these relatively new techniques, and will contribute its growth.
8. Acknowledgment
I am deeply thankful for the following people who gave me invaluable comments
to a draft of this article: Yoshihiro Abe, Katsuya Eda, Sakaé Fuchino, Hiroshi
Fujita, Yasushi Hirata, Daisuke Ikegami, Masaru Kada, Yo Matsubara, Masahiro
Shioya, Teruyuki Yorioka (in alphabetical order).
This material is based upon work supported by the National Science Foundation
under Grant No. 0700983.
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Department of Mathematics, Miami University, Oxford, OH 45056
E-mail address: [email protected]