Mathematics

This is a paper in mathematics, specifically in set theory. On the example of the measure recognition problem (MRP), the paper highlights the phenomenon of the utility of a multidisciplinary mathematical approach to a single mathematical problem, in particular, the value of a set-theoretic analysis. MRP asks if for a given Boolean algebra, B, and a property, Phi, of measures, one can recognize by purely combinatorial means if B supports a strictly positive measure with property Phi. The most famous instance of this problem is MRP (countable additivity), and in the first part of the paper, we survey the known results on this and some other problems. We show how these results naturally lead to asking about two other specific instances of the problem MRP, namely MRP (non-atomic) and MRP (separable). Then, we show how our recent work gives an easy solution to the former of these problems and some partial information about the latter. The long-term goal of this line of research is to obt...

If κ is any strongly unfoldable cardinal, then this is preserved in a forcing extension in which ✸ κ (reg) fails. This result continues the progression of the corresponding results for weakly compact cardinals, due to Woodin, and for indescribable cardinals, due to Hauser.

We develop the framework ofnatural spacesto study isomorphic embeddings of Banach spaces. We then use it to show that a sufficient failure of the generalized continuum hypothesis implies that the universality number of Banach spaces of a given density under a certain kind of positive embedding (very positive embedding) is high. An example of a very positive embedding is a positive onto embedding betweenC(K)andCLfor 0-dimensionalKandLsuch that the following requirement holds for allh≠0andf≥0inC(K): if0≤Th≤Tf, then there are constantsa≠0andbwith0≤a·h+b≤fanda·h+b≠0.

This paper investigates a connection between the semantic notion provided by the ordering / * among theories in model theory and the syntactic (N)SOPn hierarchy of Shelah. It introduces two properties which are natural extensions of this hierarchy, called SOP2 and SOP1. It is shown here that SOP3 implies SOP2 implies SOP1. In Shelah's article (Ann. Pure Appl. Logic 80 (1996) 229) it was shown that SOP3 implies / *-maximality and we prove here that / *-maximality in a model of GCH implies a property called SOP 2. It has been subsequently shown by Shelah and Usvyatsov that SOP 2 and SOP2 are equivalent, so obtaining an implication between / *-maximality and SOP2. It is not known if SOP2 and SOP3 are equivalent. Together with the known results about the connection between the (N)SOPn hierarchy and the existence of universal models in the absence of GCH, the paper provides a step toward the classiÿcation of unstable theories without the strict order property.

Suppose that λ is the successor of a singular cardinal µ whose cofinality is an uncountable cardinal κ. We give a sufficient condition that the club filter of λ concentrating on the points of cofinality κ is not λ +-saturated. 2 The condition is phrased in terms of a notion that we call weak reflection. We discuss various properties of weak reflection. We introduce a weak version of the ♣-principle, which we call ♣ * − , and show that if it holds on a stationary subset S of λ, then no normal filter on S is λ +-saturated. Under the above assumptions, ♣ * − (S) is true for any stationary subset S of λ which does not contain points of cofinality κ. For stationary sets S which concentrate on points of cofinality κ, we show that ♣ * − (S) holds modulo an ideal obtained through the weak reflection.

Suppose that λ = λ <λ ≥ ℵ 0 , and we are considering a theory T. We give a criterion on T which is sufficient for the consistent existence of λ ++ universal models of T of size λ + for models of T of size ≤ λ + , and is meaningful when 2 λ + > λ ++. In fact, we work more generally with abstract elementary classes. The criterion for the consistent existence of universals applies to various well known theories, such as triangle-free graphs and simple theories. Having in mind possible applications in analysis, we further observe that for such λ, for any fixed µ > λ + regular with µ = µ λ + , it is consistent that 2 λ = µ and there is no normed vector space over Q of size < µ which is universal for normed vector spaces over Q of dimension λ + under the notion of embedding h which specifies (a, b) such that ||h(x)||/||x|| ∈ (a, b) for all x. 1 In the list of publications of S. Shelah, this is publication number 614. Both authors thank the United States-Israel Binational Science Foundation for a partial support and various readers of the manuscript. for their helpful comments.

We show the relative consistency of ℵ 1 satisfying a combinatorial property considered by David Fremlin (in the question DU from his list) in certain choiceless inner models. This is demonstrated by first proving the property is true for Ramsey cardinals. In contrast, we show that in ZF C, no cardinal of uncountable cofinality can satisfy a similar, stronger property. The questions considered by D. H. Fremlin are if families of finite subsets of ω 1 satisfying a certain density condition necessarily contain all finite subsets of an infinite subset of ω 1 , and specifically if this and a stronger property hold under MA + ¬CH. Towards this we show that if MA + ¬CH holds, then for every family A of ℵ 1 many infinite subsets of ω 1 , one can find a family S of finite subsets of ω 1 which is dense in Fremlin's sense, and does not contain all finite subsets of any set in A. We then pose some open problems related to the question.

This article extends a paper of Abraham and Bonnet which generalised the famous Hausdorff characterisation of the class of scattered linear orders. They gave an inductively defined hierarchy that characterised the class of scattered posets which do not have infinite incomparability antichains (i.e. have the FAC). We define a larger inductive hierarchy κℌ* which characterises the closure of the class of all κ-well-founded linear orders under inversions, lexicographic sums and FAC weakenings. This includes a broader class of “scattered” posets that we call κ-scattered. These posets cannot embed any order such that for every two subsets of size &lt; κ, one being strictly less than the other, there is an element in between. If a linear order has this property and has size κ it is unique and called ℚ(κ). Partial orders such that for every a &lt; b the set {x: a &lt; x &lt; b} has size ≥ κ are called weakly κ-dense, and posets that do not have a weakly κ-dense subset are called strongly κ...

The notion of stationary reflection is one of the most important notions of combinatorial set theory. We investigate weak reflection, which is, as its name suggests, a weak version of stationary reflection. Our main result is that modulo a large cardinal assumption close to 2hugeness, there can be a regular cardinal κ such that the first cardinal 691 revision:2002-03-01 modified:2002-03-02 weakly reflecting at κ is the successor of a singular cardinal. This answers a question of Cummings, Džamonja and Shelah. 1 1 This paper is numbered 691 (10/98) in Saharon Shelah's list of publications.

The paper investigates possible generalisations of Maharam's theorem to a classification of Boolean algebras that support a finitely additive measure. We prove that Boolean algebras that support a finitely additive non-atomic uniformly regular measure are metrically isomorphic to a subalgebras of the Jordan algebra with the Lebesgue measure. We give some partial analogues to be used for a classification of algebras that support a finitely additive non-atomic measure with a higher uniform regularity number. We show that some naturally induced equivalence relations connected to metric isomorphism are quite complex even on the Cantor algebra and therefore probably we cannot hope for a nice general classification theorem for finitely-additive measures. We present an example of a Boolean algebra which supports only separable measures but no uniformly regular one.

The notion of stationary re∞ection is one of the most important notions of combinatorial set theory. We investigate weak re∞ection, which is, as its name suggests, a weak version of stationary re∞ection. Our main result is that modulo a large cardinal assumption close to 2- hugeness, there can be a regular cardinalsuch that the flrst cardinal

We investigate various kinds of bases in infinite dimensional Banach spaces. In particular, we consider the complexity of Hamel bases in separable and non-separable Banach spaces and show that in a separable Banach space a Hamel basis cannot be analytic, whereas there are non-separable Hilbert spaces which have a discrete and closed Hamel basis. Further we investigate the existence of certain complete minimal systems in ∞ as well as in separable Banach spaces.

We consider various versions of the ♣ principle. This principle is a known consequence of ◊. It is well known that ◊ is not sensitive to minor changes in its definition, e.g., changing the guessing requirement form “guessing exactly” to “guessing modulo a finite set”. We show however, that this is not true for ♣. We consider some other variants of ♣ as well.

The paper is concerned with the existence of a universal graph at the successor of a strong limit singular μ of cofinality ℵ0. Starting from the assumption of the existence of a supercompact cardinal, a model is built in which for some such μ there are μ ++ graphs on μ + that taken jointly are universal for the graphs on μ +, while . The paper also addresses the general problem of obtaining a framework for consistency results at the successor of a singular strong limit starting from the assumption that a supercompact cardinal κ exists. The result on the existence of universal graphs is obtained as a specific application of a more general method.

We investigate properties of the class of compact spaces on which every regular Borel measure is separable. This class will be referred to as MS. We discuss some closure properties ofMS, and show that some simply defined compact spaces, such as compact ordered spaces or compact scattered spaces, are in MS. Most of the basic theory for regular measures is true just in ZFC. On the other hand, the existence of a compact ordered scattered space which carries a non-separable (non-regular) Borel measure is equivalent to the existence of a real-valued measurable cardinal c. We show that not being in MS is preserved by all forcing extensions which do not collapse !1, while being in MS can be destroyed even by a ccc forcing.

The notion of stationary reflection is one of the most important notions of combinatorial set theory. We investigate weak reflection, which is, as the name suggests, a weak version of stationary reflection. This sort of reflection was introduced in [DjSh:545] (math.LO/9601219), where it was shown that weak reflection has applications to various guessing principles, in the sense that if there is no weak reflection, than a guessing principle holds, and an application dealing with the saturation of normal filters. Further investigations of weak reflection were carried in [CDSh:571] (math.LO/9504221) and [CuSh:596] (math.LO/9703219). While various ZFC restrictions on the one hand, and independence results on the other, were discovered about the weak reflection, the question of the relative consistency of the existence of a regular cardinal kappa such that the first cardinal weakly reflecting at kappa is a successor of singular, remained open. This paper answers that question by proving ...

We give a construction under CH of a non-metrizable compact Hausdorff space K such that any uncountable semi-biorthogonal sequence in C(K) must be of a very specific kind. The space K has many nice properties, such as being hereditarily separable, hereditarily Lindelöf and a 2-to-1 continuous preimage of a metric space, and all Radon measures on K are separable. However K is not a Rosenthal compactum. We introduce the notion of a bidiscrete system in a compact space K. These are subsets of K 2 which determine biorthogonal systems of a special kind in C(K) that we call nice. We note that every infinite compact Hausdorff space K has a bidiscrete system and hence a nice biorthogonal system of size d(K), the density of K. 1

editor of the Notices, invited the editors of this article to prepare a collective remembrance celebrating Mary Ellen Rudin's life and work. In doing so, we asked several of Mary Ellen's colleagues, collaborators, and students to each contribute a short piece. The choice of contributors was not an easy one, since Mary Ellen had a very rich mathematical and social life, to which this article can only attest. Space limitations meant that we could not accommodate all who would have liked to pay tribute to her, but we hope that they will find themselves, at least to some extent, represented by the reminiscences appearing here. This article is far from being the only initiative to celebrate Mary Ellen Rudin, and we are particularly happy to mention two additional ones: the Mary Ellen Rudin Young Researcher Award Fund established by Elsevier and an upcoming special issue of Topology and Its Applications edited by Gary Gruenhage, Jan van Mill, and Peter Nyikos.