We show that it is relatively consistent with ZFC to have PCF structures of heightδ, for all ordinalsδ<ω3.

We start by studying the relationship between two invariants isolated by Shelah, the sets of good and approachable points. As part of our study of these invariants, we prove a form of “singular cardinal compactness” for Jensen's square principle. We then study the relationship between internally approachable and tight structures, which parallels to a certain extent the relationship between good and approachable points. In particular we characterise the tight structures in terms of PCF theory and use our characterisation to prove (...) some covering results for tight structures, along with some results on tightness and stationary reflection. Finally, we prove some absoluteness theorems in PCF theory, deduce a covering theorem, and apply that theorem to the study of precipitous ideals. (shrink)

Since the work of Gödel and Cohen, which showed that Hilbert's First Problem was independent of the usual assumptions of mathematics, there have been a myriad of independence results in many areas of mathematics. These results have led to the systematic study of several combinatorial principles that have proven effective at settling many of the important independent statements. Among the most prominent of these are the principles diamond and square discovered by Jensen. Simultaneously, attempts have been made to find suitable (...) natural strengthenings of ZFC, primarily by Large Cardinal or Reflection Axioms. These two directions have tension between them in that Jensen's principles, which tend to suggest a rather rigid mathematical universe, are at odds with reflection properties. A third development was the discovery by Shelah of "PCF Theory", a generalization of cardinal arithmetic that is largely determined inside ZFC. In this paper we consider interactions between these three theories in the context of singular cardinals, focusing on the various implications between square and scales, and on consistency results between relatively strong forms of square and stationary set reflection. (shrink)

The tree property at κ+ states that there are no Aronszajn trees on κ+, or, equivalently, that every κ+ tree has a cofinal branch. For singular strong limit cardinals κ, there is tension between the tree property at κ+ and failure of the singular cardinal hypothesis at κ; the former is typically the result of the presence of strongly compact cardinals in the background, and the latter is impossible above strongly compacts. In this paper, we reconcile the two. We prove (...) from large cardinals that the tree property at κ+ is consistent with failure of the singular cardinal hypothesis at κ. (shrink)

We study a generalization of the splitting number s to uncountable cardinals. We prove that $\mathfrak{s}(\kappa) > \kappa^+$ for a regular uncountable cardinal κ implies the existence of inner models with measurables of high Mitchell order. We prove that the assumption $\mathfrak{s}(\aleph_\omega) > \aleph_{\omega + 1}$ has a considerable large cardinal strength as well.

Let S be the group of all permutations of the set of natural numbers. The cofinality spectrum CF(S) of S is the set of all regular cardinals λ such that S can be expressed as the union of a chain of λ proper subgroups. This paper investigates which sets C of regular uncountable cardinals can be the cofinality spectrum of S. The following theorem is the main result of this paper. Theorem. Suppose that $V \models GCH$ . Let C be (...) a set of regular uncountable cardinals which satisfies the following conditions. (a) C contains a maximum element. (b) If μ is an inaccessible cardinal such that $\mu = \sup(C \cap \mu)$ , then μ ∈ C. (c) If μ is a singular cardinal such that $\mu = \sup(C \cap \mu)$ , then μ + ∈ C. Then there exists a c.c.c. notion of forcing P such that $V^\mathbb{P} \models CF(S) = C$ . We shall also investigate the connections between the cofinality spectrum and pcf theory; and show that CF(S) cannot be an arbitrarily prescribed set of regular uncountable cardinals. (shrink)

§1. Introduction. Among the most remarkable discoveries in set theory in the last quarter century is the rich structure of the arithmetic of singular cardinals, and its deep relationship to large cardinals. The problem of finding a complete set of rules describing the behavior of the continuum function 2ℵα for singular ℵα's, known as the Singular Cardinals Problem, has been attacked by many different techniques, involving forcing, large cardinals, inner models, and various combinatorial methods. The work on the singular cardinals (...) problem has led to many often surprising results, culminating in a beautiful theory of Saharon Shelah called the pcf theory. The most striking result to date states that if 2ℵn < ℵω for every n = 0, 1, 2, …, then 2ℵω < ℵω4.In this paper we present a brief history of the singular cardinals problem, the present knowledge, and an introduction into Shelah's pcf theory. In Sections 2, 3 and 4 we introduce the reader to cardinal arithmetic and to the singular cardinals problems. Sections 5, 6, 7 and 8 describe the main results and methods of the last 25 years and explain the role of large cardinals in the singular cardinals problem. In Section 9 we present an outline of the pcf theory.§2. The arithmetic of cardinal numbers. Cardinal numbers were introduced by Cantor in the late 19th century and problems arising from investigations of rules of arithmetic of cardinal numbers led to the birth of set theory. The operations of addition, multiplication and exponentiation of infinite cardinal numbers are a natural generalization of such operations on integers. Addition and multiplication of infinite cardinals turns out to be simple: when at least one of the numbers κ, λ is infinite then both κ + λ and κ·λ are equal to max {κ, λ}. In contrast with + and ·, exponentiation presents fundamental problems. In the simplest nontrivial case, 2κ represents the cardinal number of the power set P, the set of all subsets of κ. (Here we adopt the usual convention of set theory that the number κ is identified with a set of cardinality κ, namely the set of all ordinal numbers smaller than κ. (shrink)

Let λ ≤ κ be infinite cardinals and let Ω be a set of cardinality κ. The bounded permutation groupBλ(Ω), or simplyBλ, is the group consisting of all permutations of Ω which move fewer than λ points in Ω. We say that a permutation groupGacting on Ω is asupplement of BλifBλGis the full symmetric group on Ω.In [7], Macpherson and Neumann claimed to have classified all supplements of bounded permutation groups. Specifically, they claimed to have proved that a groupGacting on (...) the set Ω is a supplement ofBλif and only if there exists Δ ⊂ Ω with ∣Δ∣ < λ such that the setwise stabiliserG{Δ}acts as the full symmetric group on Ω ∖ Δ. However I have found a mistake in their proof. The aim of this paper is to examine conditions under which Macpherson and Neumann's claim holds, as well as conditions under which a counterexample can be constructed. In the process we will discover surprising links with cardinal arithmetic and Shelah's recently developedpcftheory. (shrink)

In this paper we will present a simple condition for an ideal to be nowhere precipitous. Through this condition we show nowhere precipitousness of fundamental ideals onPkλ, in particular the non-stationary idealNSkλunder cardinal arithmetic assumptions.In this sectionIdenotes a non-principal ideal on an infinite setA. LetI+=PA/I(ordered by inclusion as a forcing notion) andI∣X= {Y⊂A:Y⋂X∈I}, which is also an ideal onAforX∈I+. We refer the reader to [8, Section 35] for the general theory of generic ultrapowers associated with an ideal. We recallIis said (...) to be precipitous if ⊨I+“Ult(V, Ġ) is well-founded” [9].The central notion of this paper is a strong negation of precipitousness [1]:Definition.Iis nowhere precipitous ifI∣Xis not precipitous for everyX∈ I+i.e., ⊨I+“Ult(V, Ġ) is ill-founded.”It is useful to characterize nowhere precipitousness in terms of infinite games (see [11, Section 27]). Consider the following gameG(I) between two players, Nonempty and Empty [5]. Nonempty and Empty alternately chooseXn∈I+andYn∈I+respectively so thatXn⊃Yn⊃n+1. After ω moves, Empty wins the game if⋂n<ωXn=⋂n<ωYn= Ø.See [5, Theorem 2] for a proof of the following characterization.Proposition.I is nowhere precipitous if and only if Empty has a winning strategy in G(I). (shrink)

We present several forcing posets for adding a non-reflecting stationary subset of Pω1, where λ≥ω2. We prove that PFA is consistent with dense non-reflection in Pω1, which means that every stationary subset of Pω1 contains a stationary subset which does not reflect to any set of size 1. If λ is singular with countable cofinality, then dense non-reflection in Pω1 follows from the existence of squares.

We prove equiconsistency results concerning gaps between a singular strong limit cardinal κ of cofinality 0 and its power under assumptions that 2κ=κ+δ+1 for δ<κ and some weak form of the Singular Cardinal Hypothesis below κ. Together with the previous results this basically completes the study of consistency strength of the various gaps between such κ and its power under GCH type assumptions below.

We prove that a combinatorial consequence of the negation of the PCF conjecture for intervals, involving free subsets relative to set mappings, is not implied by even the strongest known large cardinal axiom.

LetSbe the group of all permutations of the set of natural numbers. The cofinality spectrumCF(S)ofSis the set of all regular cardinalsλsuch thatScan be expressed as the union of a chain ofλproper subgroups. This paper investigates which setsCof regular uncountable cardinals can be the cofinality spectrum ofS. The following theorem.is the main result of this paper.Theorem.Suppose that V ⊨ GCH. Let C be a set of regular uncountable cardinals which satisfies the following conditions.(a)C contains a maximum element.(b)Ifμis an inaccessible cardinal such (...) thatμ= sup(C∩μ),thenμ∈C.(c)ifμis a singular cardinal such thatμ= sup(C∩μ),thenμ+∈C.Then there exists a c.c.c. notion of forcing ℙ such that Vℙ⊨ CF(S) = C.We shall also investigate the connections between the cofinality spectrum andpcftheory; and show thatCF(S)cannot be an arbitrarily prescribed set of regular uncountable cardinals. (shrink)

We investigate pseudopowers of singular cardinals and deduce some consequences for covering numbers at singular cardinals of uncountable cofinality.

We study the role of meeting numbers in pcf theory. In particular, Shelah's Strong Hypothesis is shown to be equivalent to the assertion that for any singular cardinal σ of cofinality ω, there is a size collection Q of countable subsets of σ with the property that for any infinite subset a of σ, there is a member of Q meeting a in an infinite set.

We show that o = k++ is necessary for ¬SCH. Together with previous results it provides the exact strenght of ¬SCH.

The existence of exact upper bounds for increasing sequences of ordinal functions modulo an ideal is discussed. The main theorem gives a necessary and sufficient condition for the existence of an exact upper bound ƒ for a ¦A¦+ is regular: an eub ƒ with lim infI cf ƒ = μ exists if and only if for every regular κ ε the set of flat points in tf of cofinality κ is stationary. Two applications of the main Theorem to set theory (...) are presented. A theorem of Magidor's on covering between models of ZFC is proved using the main theorem : If V-W are transitive models of set theory with ω-covering and GCH holds in V, then κ-covering holds between V and W for all cardinals κ. A new proof of a Theorem by Cummings on collapsing successors of singulars is also given . The appendix to the paper contains a short proof of Shelah's trichotomy theorem, for the reader's convenience. (shrink)

We use pcf theory to prove results on reflection at singular cardinals.

We consider simplified representation theorems in pcf-theory and, in particular, we prove that if ${\aleph_{\omega}^{\aleph_{0}} > \aleph_{\omega_{1}}\cdot2^{\aleph_{0}}}$ then there are cofinally many sequences of regular cardinals such that ${\aleph_{\omega_{1}+1}}$ is represented by these sequences modulo the ideal of finite subsets, using a topological approach to the pcf-structure.

A self contained proof of Shelah's theorem is presented: If μ is a strong limit singular cardinal of uncountable cofinality and 2μ > μ+ then $\left( {\begin{array}{*{20}c} {\mu ^ + } \\ \mu \\ \end{array} } \right) \to \left( {\begin{array}{*{20}c} {\mu ^ + } \\ {\mu + 1} \\ \end{array} } \right)_{< cf\mu } $.

It is shown that the power set of $\kappa$ ordered by the subset relation modulo various versions of the non-stationary ideal can be embedded into the partial order of Borel equivalence relations on $2^\kappa$ under Borel reducibility. Here $\kappa$ is an uncountable regular cardinal with $\kappa^{<\kappa}=\kappa$.

In connection with some known results on uncountable cardinal sequences for superatomic Boolean algebras, we shall describe some open questions for superatomic Boolean algebras concerning singular cardinals.

In this paper we will present a simple condition for an ideal to be nowhere precipitous. Through this condition we show nowhere precipitousness of fundamental ideals onPkλ, in particular the non-stationary idealNSkλunder cardinal arithmetic assumptions.In this sectionIdenotes a non-principal ideal on an infinite setA. LetI+=PA/I(ordered by inclusion as a forcing notion) andI∣X= {Y⊂A:Y⋂X∈I}, which is also an ideal onAforX∈I+. We refer the reader to [8, Section 35] for the general theory of generic ultrapowers associated with an ideal. We recallIis said (...) to be precipitous if ⊨I+“Ult(V, Ġ) is well-founded” [9].The central notion of this paper is a strong negation of precipitousness [1]:Definition.Iis nowhere precipitous ifI∣Xis not precipitous for everyX∈ I+i.e., ⊨I+“Ult(V, Ġ) is ill-founded.”It is useful to characterize nowhere precipitousness in terms of infinite games (see [11, Section 27]). Consider the following gameG(I) between two players, Nonempty and Empty [5]. Nonempty and Empty alternately chooseXn∈I+andYn∈I+respectively so thatXn⊃Yn⊃n+1. After ω moves, Empty wins the game if⋂n<ωXn=⋂n<ωYn= Ø.See [5, Theorem 2] for a proof of the following characterization.Proposition.I is nowhere precipitous if and only if Empty has a winning strategy in G(I). (shrink)

There exists a family $\{B_\alpha\}_{\alpha of sets of countable ordinals such that: (1) max B α = α, (2) if α ∈ B β then $B_\alpha \subseteq B_\beta$ , (3) if λ ≤ α and λ is a limit ordinal then B α ∩ λ is not in the ideal generated by the $B_\beta, \beta , and by the bounded subsets of λ, (4) there is a partition {A n } ∞ n = 0 of ω 1 such that for (...) every α and every n, B α ∩ A n is finite. (shrink)

Let λ ≤ κ be infinite cardinals and let Ω be a set of cardinality κ. The bounded permutation group B λ (Ω), or simply B λ , is the group consisting of all permutations of Ω which move fewer than λ points in Ω. We say that a permutation group G acting on Ω is a supplement of B λ if B λ G is the full symmetric group on Ω. In [7], Macpherson and Neumann claimed to have classified (...) all supplements of bounded permutation groups. Specifically, they claimed to have proved that a group G acting on the set Ω is a supplement of B λ if and only if there exists $\Delta \subset \Omega$ with $|\Delta| such that the setwise stabiliser G {Δ} acts as the full symmetric group on $\Omega\setminus\Delta$ . However I have found a mistake in their proof. The aim of this paper is to examine conditions under which Macpherson and Neumann's claim holds, as well as conditions under which a counterexample can be constructed. In the process we will discover surprising links with cardinal arithmetic and Shelah's recently developed pcf theory. (shrink)