CLUB GUESSING SEQUENCES NATURAL STRUCTURES IN SET THEORY

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1 Unspecified Journal Volume 00, Number 0, Pages S????-????(XX) 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 system 2 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, Mathematics Subject Classification. Primary 03E04, 03E05, 03E35. This material is based upon work supported by the National Science Foundation under Grant No The term Canonical structures is also used to describe the same notion. 2 In 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. 1 c 0000 (copyright holder)

2 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 theory 3 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 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 2 ℵ 0 = ℵ 1 under Axiom of Choice. The assertion 2 ℵ 0 = ℵ 1 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, 2 κ = 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 ℵ 0, the powers of larger regular cardinals have been investigated. The following facts were known. Fact 1.1. (1) κ < λ implies 2 κ 2 λ. (2) (König s lemma) 5 cf(2 κ ) > 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 4 As 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. 5 This 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.

3 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 ℵ ω is ℵ 0 since {ℵ n : n < ω} is an unbounded subset of ℵ ω. Therefore, ℵ ω 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 2 λ = λ + for every infinite cardinal λ < κ, then 2 κ = κ +. In fact, the assumption of the previous theorem can be weakened to {λ < κ : 2 λ = 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, 2 ℵ n = ℵ n+1 and 2 ℵ ω = ℵ ω+2. Therefore, in this model, ℵ ω 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 2 ℵ ω 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. ℵ ω ℵ 0 < max{ℵ ω4, (2 ℵ0 ) + }. 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. 6 For 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 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

5 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 {D α : α < µ} is a family of club subsets of δ and µ < cf(δ), then their intersection α<µ D α is also club in δ. Proof. It is trivial that α<µ D α 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 D α such that γ ζ. By induction, we shall define an increasing sequence γ n : n < ω in δ. Let γ 0 = ζ. Suppose that γ n has been defined. Since D α is unbounded in δ for every α < µ, there exists a ξ n,α D α such that ξ n,α > γ n. {ξ n,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 γ n+1 < δ such that ξ n,α < γ n+1 for every α < µ. Let γ = sup n<ω γ n. Since cf(δ) is uncountable, we have γ < δ. For every α < µ, clearly D α γ is unbounded in γ. Since D α is closed, we have γ D α. So, we have γ α<µ D α. Since γ γ 0 = ζ, the lemma was proved. The previous lemma does not hold when cf(δ) = ω. For example, if δ = ω, then D 1 = {2n : n < ω} and D 2 = {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 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 NS κ. 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 X α : α < κ a sequence of subsets of κ. (1) The diagonal intersection α<κ X α of X α : α < κ is the subset of κ defined as follows: α<κ X α = {γ < κ : α < γ(γ X α )}. (2) The diagonal union α<κ X α of X α : α < κ is the subset of κ defined as follows: α<κ X α = {γ < κ : α < γ(γ X α )}. These two are dual notions, and the following fact holds. κ \ α<κ X α = α<κ (κ \ X α ). Lemma 2.6. Let κ be an uncountable regular cardinal. Then, for every sequence D α : α < κ of club subsets of κ, its diagonal intersection α<κ D α is club in κ. Proof. Let D = α<κ D α. 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 γ n : n < ω in κ as follows. Let γ 0 = ζ. We shall describe how to define γ n+1 assuming γ n has been defined. By Lemma 2.2, α γ n D α is club in κ. Therefore, we can find a γ n+1 α γ n D α with γ n+1 > γ n. 8 The 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.

7 CLUB GUESSING SEQUENCES 7 Let γ = sup n<ω γ n. We shall show that γ D. By the definition of the diagonal intersection, it suffices to show that for every α < γ, γ D α. Fix α < γ. Since γ = sup n<ω γ n, there exists an m < ω such that α < γ m. By the definition of γ n, for every n < ω with n > m, we have γ n D α. Thus, γ is a limit point of D α. Since D α is closed, we have γ D α. To show that D is closed, let δ be a limit point of D. It suffices to show that for every α < δ, δ D α. First, observe D \ (α + 1) D 9 α. Hence, δ is a limit point of D α. Since D α is closed, we have δ D α. For every α < κ, if we define D α = κ \ α, then α<κ D α =. Thus, there is no club subset of κ that is contained in D α for every α < κ. Therefore, Lemma 2.2 cannot be extended to κ-many club subsets. However, Lemma 2.6 says that if {D α : α < κ} is a set of club subsets of κ, then there exists a club subset of κ that is almost contained in D α 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 {Y α : α < λ} I, we have α<λ Y α I. We say that I is normal if and only if for every sequence X α : α < κ in I, α<κ X α I. The following lemma can be proved by Lemma 2.2 and Lemma 2.6. Lemma 2.8. For every uncountable regular cardinal κ, NS κ is κ-complete and normal. Moreover, in the following sense, NS κ 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, NS κ I 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 C δ : δ S is called a fully club guessing sequence on S if and only if (1) for every δ S, C δ is an unbounded subset of δ, and (2) for every club subset D of κ, there exists a δ S such that C δ D. A sequence C δ : 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 C δ D. For example, if A δ : δ S is a κ (S)-sequence, then A δ : δ S is a fully club guessing sequence on S where S = {δ S : A δ 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 C δ D. However, if C = {C δ : δ S} is a fully club guessing sequence on S, for 9 Note D \ (α + 1) = {γ D : γ > α} since α + 1 is the set of all ordinals α.

8 8 TETSUYA ISHIU every club subset D of κ, S = {δ S : C δ D} is stationary in κ. To see this, suppose that S is non-stationary. Then, there exists a club subset D of κ such that S D =. However, since D D is a club subset of κ and C is a fully club guessing sequence, there exists a δ S such that C δ D D. Since C δ 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 C δ : δ S is a fully club guessing sequence, then C δ {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 C δ : δ S is a tail club guessing sequence on S, then there exists an ordinal ζ < κ such that C δ \ 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 Suppose that θ and κ are uncountable regular cardinals with θ + < κ. Let S be a stationary subset of κ Cof(θ) and C δ : δ S a sequence such that for every δ S, C δ is a club subset of δ with C δ = θ. Then, there exists a club subset E of κ such that C δ 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 D α : α θ + of club subsets of κ. Let D 0 = κ. Suppose that D β has been defined for every β < α and define D α. If α is a limit ordinal, let D α = β<α D β. By Lemma 2.2, D α is club in κ. If α is a successor ordinal, let β be so that α = β + 1. Then, for every δ S acc(d β ), C δ and D β δ are both club in δ. Since cf(δ) = θ is uncountable, C δ D β is also club in δ. By assumption, C δ D β : δ S acc(d β ) is not a fully club guessing sequence. Therefore, there exists a club subset D β+1 of κ such that for every δ S acc(d β ), C δ D β D β+1. Without loss of generality, we may assume D β+1 D β. Suppose that D α : α θ + is defined and let δ S acc(d θ +). Consider C δ D α : α < θ +. For every α < θ +, by the definition of D α+1, we have C δ D α+1 C δ D α. Let γ α be an element of C δ (D α \D α+1 ). Then, γ α : α < θ +

9 CLUB GUESSING SEQUENCES 9 is a sequence of distinct element of C δ, which contradicts C δ = θ. This proof is easy, but has some remarkable points. Axiom of Choice is used to find D β+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 = ℵ 0, the previous proof does not work as is because if cf(δ) = ℵ 0, 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 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 C δ : δ S on S such that for every δ S, C δ Cof( λ). Proof. For each δ κ Lim, pick a club subset e δ of δ such that otp(e δ ) = cf(δ). Let D be a club subset of κ and δ S. By induction, we shall define f δ (D, n) and f δ (D, n) for every n < ω: f δ(d, 0) = e δ. f δ (D, n) = {sup(d ξ) : ξ f δ(d, n) and ξ > min(d)}. f δ(d, n + 1) = {e γ : γ f δ (D, n) Cof(<λ)}. Then, let f δ (D) = n<ω f δ(d, n). f δ (D) D. Since D is club, f δ (D, n) D and hence Claim If δ S acc(d), then f δ (D, 0) is unbounded in δ. So, f δ (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 e δ is unbounded in δ, there exists a ξ e δ such that ξ > γ. Then, we have ξ e δ = f δ (D, 0) and ξ > γ min(d). Hence, we have sup(d ξ) f δ (D, 0). We also have sup(d ξ) γ > ζ. Therefore, f δ (D, 0) is unbounded in δ. Claim For every δ S and n < ω, we have f δ (D, n) < λ and f δ(d, n) < λ Fix δ S, and go by induction on n < ω. First, note f δ (D, 0) = e δ = cf(δ) = θ < λ. If f δ (D, n) < λ, it is clear by definition that f δ (D, n) f δ (D, n) < λ. Now, we shall show that, assuming f δ (D, n) < λ, f δ (D, n + 1) < λ. For every γ f δ(d, n) Cof(<λ), we have e γ = cf(γ) < λ. Thus, f δ (D, n + 1) is the union of at most f δ(d, n) -many sets of cardinality <λ. Since f δ (D, n) < λ and λ is regular, we can see f δ (D, n + 1) < λ 10 The author learned this proof by the master dissertation of Y.Hirata [21], and consulted its presentation.

10 10 TETSUYA ISHIU Claim There exists a club subset D such that f δ (D) : δ S acc(d) is a fully club guessing sequence. Suppose that there is no such D. We shall define a decreasing sequence D α : α λ of club subsets of κ. Let D 0 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 D β is defined for every β < α, let D α = β<α D β. Assuming D α is defined, we shall define D α+1. By assumption, f δ (D α ) : δ S acc(d α ) is not a fully club guessing sequence. For every δ S acc(d α ), note that f δ (D α ) is club in δ. Hence, there exists a club subset D α+1 of κ such that for every δ S acc(d α ), f δ (D α ) D α+1. Without loss of generality, we may assume D α+1 D α. Fix a δ S acc(d λ ). By induction, we shall define an increasing sequence α n : n < ω in λ such that for every n < ω, if α n α < λ, then f δ (D α, n) = f δ (D αn, n). Suppose that α m has been defined for all m < n. First, we shall show that α n < λ such that α n α < λ implies f δ (D α, n) = f n δ (D α, n). Put α 0 = 0. Since f δ (D, 0) does not depend on D, the conclusion holds trivially. If n > 0, then let α n = α n 1. By inductive hypothesis, α n α < λ implies f δ (D α n, n 1) = f δ (D α, n 1). By the definition of f, clearly we have f δ (D α, n) = f n δ (D α, n). Suppose that there is no α n < λ such that α n α < λ implies f δ (D α, n) = f δ (D αn, n). Then, for every β < λ, there exists an α such that β α < λ and f δ (D α, n) f δ (D β, n). By using this fact, we can build an increasing sequence β ν : ν < λ such that for every ν < λ, f δ (D βν+1, n) f δ (D βν, n). For every ν < λ, by the definition of f δ and f δ (D β ν+1, n) = f δ (D β ν, n) = f δ (D α, n), there exists a n ξ ν f δ (D β ν, n) such that (1) min(d βν ) < ξ ν min(d βν+1 ) or (2) min(d βν+1 ) < ξ ν and sup(d βν+1 ξ ν ) < sup(d βν ξ ν ) Since f δ (D β ν, n) < λ, there exists a ξ f δ (D β ν, n) such that {ν < λ : ξ ν = ξ} is unbounded in λ. Let X = {ν < λ : ξ ν = ξ}. Suppose that there exists a ν X such that min(d βν ) < ξ ν min(d βν+1 ). By assumption, there exists a µ X such that ν < µ < λ. Then, since D βµ D βν+1, we have ξ µ = ξ min(d βµ ). This contradicts to the definition of ξ µ. Thus, for every ν X, min(d βν+1 ) < ξ and sup(d βν+1 ξ) < sup(d βν ξ) sup(d βµ ξ) sup(d βν+1 ξ) < sup(d βν ξ). If both ν < µ belong to X, then we have sup(d βµ ξ) sup(d βν+1 ξ) < sup(d βν ξ). So, sup(d βν ξ) : ξ X is an infinite decreasing sequence of ordinals, which contradicts to the well-foundedness of ordinals. So, we can find an α n that satisfies the inductive hypothesis. Let α ω = sup n<ω α n. Since λ is regular, we have α ω < λ. Then, for every β < λ and n < ω, by the definition of α n, β α ω implies f δ (D β, n) = f δ (D αω, n). Therefore, we have f δ (D β ) = f δ (D αω ). Thus, f δ (D αω +1) = f δ (D αω ) D αω This contradicts to the definition of D αω +1. Let S be the set of all δ S acc(d) such that f δ (D) acc(d) holds. Then, for each δ S, let C δ = nacc(f δ (D)). By Claim 2.17, S is stationary, and C δ : δ S is a fully club guessing sequence, Thus, it suffices to show the following claim. Claim For every δ S, we have C δ Cof( λ). Let δ S and γ C δ. Suppose cf(γ) < λ. We shall show that γ is a limit point of f δ (D). This contradicts to γ nacc(f δ (D)).

11 CLUB GUESSING SEQUENCES 11 Let ζ < γ, and we shall show that there exists a ξ f δ (D) with ζ < ξ < γ. Since f δ (D) = n<ω f δ(d, n), there exists an n < ω such that γ f δ (D, n). Then, since γ f δ (D, n) Cof(<λ), we have e γ f δ (D, n). By the definition of S, we have f δ (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 e γ is unbounded in γ, we can pick a ξ e γ with ξ > ζ. Since e γ f δ (D, n), we have ξ f δ (D, n). By definition, we have sup(d ξ) f δ (D, n + 1) f δ (D). Since ζ D ξ, we have sup(d ξ) ζ > ζ. This proves that γ is a limit point of f δ (D). The assumption θ + < κ in Theorem 2.12 is shown to be necessary by A. Ros lanowski and S. Shelah in [33]. Moreover, we can prove the following corollary from Theorem Corollary If κ ℵ 2 is a regular cardinal with κ ℵ 2, then there exists a fully club guessing sequence on κ Lim. S. Shelah [37] showed that we need to assume κ ℵ 2 to prove this corollary 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 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 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 A δ : δ S such that (1) for every δ S, A δ δ (2) for every X κ, {δ S : X δ = A δ } is stationary in κ. κ(s) is the principle that asserts the existence of a sequence A δ : δ S such that (1) for every δ S, A δ P(δ) and A δ δ. (2) for every X κ, there exists a club subset D of κ such that for every δ D S, X δ A δ. In both cases, we may omit S when S = κ. Fact 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 ineffable 11. 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 ℵ An 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 12 TETSUYA ISHIU Theorem 2.23 (S. Shelah [39]). Suppose κ = λ + = 2 λ 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 ℵ 2, then 2 λ = λ + is equivalent to κ Note that the assumption κ ℵ 2 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 Let λ be an infinite cardinal. λ is the principle that asserts the existence of a sequence C α : α λ + Lim such that (1) for every α λ + Lim, C α is a club subset of α, and (2) for every α λ + Lim, cf(α) < λ implies otp(c α ) < λ, and (3) if β < α is a limit point of C α, then C β = C α β. Such a sequence C α : α λ + Lim is called a λ sequence. By (2) and (3), for every α λ + Cof(λ), we have otp(c α ) = 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 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 C α : α S such that (1) for every α S, C α is a club subset of α with otp(c α ) = θ, and (2) for every α, β S, if γ is a limit point of both C α and C β, we have C α γ = C β 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 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.

13 CLUB GUESSING SEQUENCES The non-stationary ideal on a regular cardinal ℵ 2 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 + AD R + Θ is regular, that it is consistent that NS ω1 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 ω 1, 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 14 TETSUYA ISHIU Theorem 3.6 (M. Gitik and S. Shelah [16]). (1) For every regular cardinal κ ℵ 2, NS κ is not saturated. (2) For every pair of regular cardinals θ and κ with θ + < κ, NS κ (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 ℵ Club guessing sequences play a key role in the proof. We shall only show that NS κ cannot saturated, but a similar argument also proves that NS κ (κ Cof(θ)) cannot be saturated. Definition 3.7 (S. Shelah [36]). Let C = C δ : δ S be a tail club guessing sequence on a stationary subset of an uncountable regular cardinal κ consisting of limit ordinals. Define the tail club guessing filter TCG( C) associated with C to be the set of all X κ such that there exists a club subset D of κ such that for every δ < κ, C δ D implies δ X. The dual ideal of TCG( C) is called the tail club guessing ideal associated with C and denoted by TCG( C). It is trivial that for every subset X of κ, X TCG( C) if and only if there exists a club subset D of κ such that for every δ X, we have C δ D. Lemma 3.8. Let C = C δ : δ S be a tail club guessing sequence on a stationary subset S of an uncountable regular cardinal κ. Then, TCG( C) is a κ-complete normal ideal and contains the set of all bounded subsets of κ. Proof. Let X be a bounded subset of κ and we shall show X TCG( C). Since X is bounded, there exists a ζ < κ such that X ζ. Since κ \ ζ is a club subset of κ, we have {δ S : C δ κ \ ζ} TCG( C). Clearly X {δ S : C δ κ \ ζ}, so we get X TCG( C). To see the κ-completeness, let {X α : α < µ} be a set of cardinality µ < κ such that X α TCG( C) for every α < µ. Then, for every α < µ, there exists a club subset D α of κ such that for every δ X α, C δ D α. Let D = α<µ D α. Since µ < κ, by Lemma 2.2, D is club. To see that X = α<µ X α belongs to TCG( C), it suffices to show that for every δ X, C δ D. Let δ X. Then, there exists an α < µ such that δ X α. By assumption, we have C δ D α. Since D D α, it implies C δ D. To see the normality, let X α : α < κ be a sequence in TCG( C). For each α < κ, there exists a club subset D α of κ such that for event δ X α, C δ D α. Let D = α<κ D α. By Lemma 2.6, D is club. To see X = α<κ X α TCG( C), we shall show that for every δ X, C δ D. By the definition of the diagonal union, there exists an α < δ such that δ X α. By the definition of D α, we have C δ D α. But as we saw in the proof of Lemma 2.6, we have D \ (α + 1) D α. Thus, we have C δ D. Lemma 3.9 (J. Baumgartner, A. Taylor, and S. Wagon [2]). Let I and J be κ- complete normal ideals. Suppose that I is saturated and I J. Then, there exists an X P(κ) \ I such that J = I X. 12 The case of κ = ℵ2 can be proved by a similar argument though some modification is necessary.

15 CLUB GUESSING SEQUENCES 15 Particularly, when NS κ 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 = NS κ S. Let S be a stationary subset of κ. Then, it is easy to see that if {[X α ] NSκ S : α < µ} is an antichain in P(κ)/NS κ S, then {[X α S] NSκ : α < µ} is an antichain in P(κ)/NS κ. So, if NS κ is saturated, then so is its restriction NS κ 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 {[X α ] I : α < κ} be an antichain in P(κ)/I. Then, there exists a sequence X α : α < κ in P(κ) \ I such that (1) for every α < κ, [X α ] = [X α], and (2) for every β < α < κ, X β X α =. 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 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 ℵ 3. Proof(Theorem 3.6). Let κ ℵ 3 be a regular cardinal such that NS κ is saturated. Let S be the set of all γ κ Cof(ω) such that γ is a limit point of Cof( ℵ 2 ). By Theorem 2.14, for every stationary subset S of S, there exists a tail club guessing sequence C = C δ : δ S on S such that for every δ S, C δ Cof( ℵ 2 ). By Lemma 3.8, TCG( C) is a κ-complete normal ideal on κ that contains all bounded subsets of κ. So, by Lemma 2.9, we have NS κ TCG( C). Since NS κ is a saturated κ-complete normal ideal, by Lemma 3.9, there exists a stationary subset T of S such that TCG( C) = NS κ T. By Lemma 3.11, there exists a sequence S α : α < κ of stationary subsets of S. (1) {[S α ] NSκ S : α < κ} is a maximal antichain in P(κ)/NS κ S. (2) For every α < κ, there exists a tail club guessing sequence Cδ α : δ S α such that TCG( C δ α : δ S α ) = NS κ S α. Without loss of generality, we may assume otp(cδ α) = ω for everyδ S α. Since NS κ is saturated, so is NS κ S. By Lemma 3.10, there exists a sequence S α : α < κ of stationary subsets of κ such that for every α < κ, [S α ] NSκ S = [S α] NSκ S, and for every β < α < κ, S β S α =. Without loss of generality, we may assume S α S α for every α < κ. Since [S α ] NSκ S = [S α] NSκ S, S α S α, and TCG( C δ α : δ S α ) = NS κ S α, we have TCG( C δ α : δ S α ) = NS κ S α. Define a sequence C δ : δ S as follows. If there exists an α < κ such that δ S α, then C δ = Cδ α. Otherwise, let C δ be any unbounded subset of δ Cof( ℵ 2 ) such that otp(c δ ) = ω. From this definition, clearly we have for every δ S, otp(c δ ) = ω.

16 16 TETSUYA ISHIU This sequence C δ : δ S has the following stronger property than just being a tail club guessing sequence. Claim For every club subset D of κ, there exists a club subset E of κ such that for every δ S E, C δ D. Let D be a club subset of κ. Put S = {δ S : C δ D}. If S (κ \ S) contains a club subset of κ, then the conclusion is witnessed by the club subset. Suppose not, i.e. assume κ \ (S (κ \ S)) is stationary in κ. Note κ\(s (κ\ S)) = (κ\s) S = S \S. Thus, S \S is a stationary subset of S. Since {[S α] NSκ S : α < κ} is a maximal antichain in NS κ S, there exists an α < κ such that S α ( S \ S) is stationary. Since S α S, we have S α ( S \ S) = S α \ S. We have S α \ S = {δ S α : C δ D} (by the definition of S) = {δ S α : C α δ D} (since C δ = C α δ for every δ S α) So, by definition, this set belongs to TCG( C δ α : δ S α ). However, by assumption, TCG( C δ α : δ S α ) = NS κ S α. Thus, we have S α \ S NS κ S α. Since S α \ S S α. it means that S α \ S is non-stationary, which is a contradiction. We shall derive a contradiction from this claim. Define a decreasing sequence D α : α ω 1 of club subsets of κ as follows. Let D 0 be the set of all limit points of Cof( ℵ 2 ) that is less than κ. If α ω 1 is a limit ordinal and D β : β < α has been defined, let D α = β<α D β. We shall explain how to build D α+1 from D α. Since D α is a club subset of κ, by Claim 3.12, there exists a club subset D α+1 of κ such that for every δ D α+1 S, C δ D α. Without loss of generality, we may assume D α+1 acc(d α ). Let δ be the ω-th element of D ω1, i.e. the unique element of D ω1 such that otp(d ω1 δ) = ω. Then, since δ D 0, by the definition of D 0, δ is a limit point of Cof( ω 2 ). Hence, we have δ S. For every α < ω 1, since δ D α+1 S, we have C δ D α. So, there exists a ζ α < δ such that C δ \ ζ α D α. Without loss of generality, we may assume ζ α C δ. Since C δ = ℵ 0. there exists a ζ C δ such that {α < ω 1 : ζ α = ζ} is unbounded in ω 1. Then, for every β < ω 1, there exists an α < ω 1 such that β α and ζ α = ζ. Hence, we have C δ \ ζ D α D β. So, C δ \ ζ D ω1. Let γ C δ \ ζ. Since C δ Cof( ℵ 2 ), we have cf(γ) ℵ 2. Then, for every α < ω 1. we have γ D α+1 acc(d α ). Therefore, D α γ is a club subset of γ. Since cf(γ) ω 2 > ω 1, α<ω 1 (D α γ) is also a club subset of γ. In particular, otp ( α<ω 1 (D α γ) ) ω 2 Note (D α γ) = D α γ α<ω 1 α<ω 1 = D ω1 γ So, we have otp(d ω1 δ) otp(d ω1 γ) ω 2. This contradicts to the assumption that δ is the ω-th element of D ω1

17 CLUB GUESSING SEQUENCES 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 ℵ sequences were used to prove ℵ 0 ω < max{ℵ ω4, (2 ℵ 0 ) + }, 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 also 13. 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 properties 14, 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 = {ℵ n : n < ω or n = ω + 1} is an interval of regular cardinals. ℵ ω does not belong to A though ℵ 1 < ℵ ω < ℵ ω+1. It does not matter since ℵ ω is singular. 13 We say that a filter F on X is principal if and only if there exists an X0 X such that F = {Y X : X 0 Y }. 14 We 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 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 = {ℵ n : 0 < n < ω}. By Fact 4.4(1)(3) and Fact 4.6(1), we have ℵ ω+1 pcf(a). That is, there exists an ultrafilter U on A such that cf(πa, U) = ℵ ω+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)} < ℵ 0. i.e. f(κ) < g(κ) holds for all but finitely many κ A. Theorem 4.7. LetA = {ℵ n : 0 < n < ω}. Then, there exist B A and a sequence f α : α < ω ω+1 in ΠB such that (1) f α : α < ω ω+1 is < fin -increasing, and (2) {f α : α < ω ω+1 } is cofinal in (ΠB, < fin ), i.e. for every f ΠB, there exists an α < ω ω+1 such that f < fin f α. Such a sequence is called a scale in ΠA/fin of length ω ω+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 = {ℵ n : 0 < n < ω}. Then, pcf(a) < ℵ 4. Proof. Suppose pcf(a) ℵ 4. 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 ℵ 1 A pcf(a). Thus, every uncountable regular cardinal less than ℵ ω4 belongs to pcf(a). By Theorem 2.12, there exists a fully club guessing sequence C δ : δ ω 3 Cof(ω 1 ) on ω 3 Cof(ω 1 ). For every δ ω 3 Cof(ω 1 ), without loss of generality, we may assume C δ is club in δ and otp(c δ ) = ℵ 1. We shall define an increasing sequence κ α : α < ω 3 of cardinals less than ℵ ω4 as follows: Let κ 0 = ℵ 1. If α is a limit ordinal, and κ β has been defined for every β < α, let κ α = sup β<α κ β. Suppose that for every β α, κ β has been defined. For each δ ω 3 Cof(ω 1 ), let B α,δ be the set of all κ + β so that β nacc(c δ ) and β α. Set λ α,δ = 15 This 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 B 0 of B such that B 0 A and λ pcf(b 0 )