In this paper we review Shelah's strong covering property and its applications. We also extend some of the results of Shelah and Woodin on the failure of equation image by adding a real.

Elekes proved that any infinite-fold cover of a σ-finite measure space by a sequence of measurable sets has a subsequence with the same property such that the set of indices of this subsequence has density zero. Applying this theorem he gave a new proof for the random-indestructibility of the density zero ideal. He asked about other variants of this theorem concerning I-almost everywhere infinite-fold covers of Polish spaces where I is a σ-ideal on the space and the set of (...) indices of the required subsequence should be in a fixed ideal ${{\mathcal{J}}}$ on ω. We introduce the notion of the ${{\mathcal{J}}}$ -covering property of a pair ${({\mathcal{A}}, I)}$ where ${{\mathcal{A}}}$ is a σ-algebra on a set X and ${{I \subseteq \mathcal{P}(X)}}$ is an ideal. We present some counterexamples, discuss the category case and the Fubini product of the null ideal ${\mathcal{N}}$ and the meager ideal ${\mathcal{M}}$ . We investigate connections between this property and forcing-indestructibility of ideals. We show that the family of all Borel ideals ${{\mathcal{J}}}$ on ω such that ${\mathcal{M}}$ has the ${{\mathcal{J}}}$ -covering property consists exactly of non weak Q-ideals. We also study the existence of smallest elements, with respect to Katětov–Blass order, in the family of those ideals ${\mathcal{J}}$ on ω such that ${\mathcal{N}}$ or ${\mathcal{M}}$ has the ${\mathcal{J}}$ -covering property. Furthermore, we prove a general result about the cases when the covering property “strongly” fails. (shrink)

A subset of the Cantor cube is null-additive if its algebraic sum with any null set is null. We construct a set of cardinality continuum such that: all continuous images of the set into the Cantor cube are null-additive, it contains a homeomorphic copy of a set that is not null-additive, and it has the property $\unicode{x3b3} $, a strong combinatorial covering property. We also construct a nontrivial subset of the Cantor cube with the property (...) $\unicode{x3b3} $ that is not null additive. Set-theoretic assumptions used in our constructions are far milder than used earlier by Galvin–Miller and Bartoszyński–Recław, to obtain sets with analogous properties. We also consider products of Sierpiński sets in the context of combinatorial covering properties. (shrink)

Suppose that there is no transitive model of ZFC + there is a strong cardinal, and let K denote the core model. It is shown that if $\delta$ has the tree property then $\delta^{+K} = \delta^+$ and $\delta$ is weakly compact in K.

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:The Causation Debate in Modern Philosophy: 1637–1739J. A. CoverKenneth Clatterbaugh. The Causation Debate in Modern Philosophy: 1637–1739. New York and London: Routledge, 1999. Pp. xi + 239. Cloth, $75.00. Paper, $21.00.Over the scholastics and earliest moderns, Hume had an advantage of hindsight in declaring that "There is no question, which on account of its importance, as well as difficulty, has caus'd more disputes both among ancients and modern (...) philosophers, than this concerning the efficacy of causes, or that quality which made them be followed by their effects" (Treatise I.iii.14). Between the scholastics and Hume fell (i) Descartes, (ii) classical mechanists such as Hobbes and Gassendi, (iii) the Cartesians Le Grand and Malebranche, (iv) the explanatory rationalists Spinoza and Leibniz, (v) later experimental mechanists such as Boyle, Rohault and Newton, and (vi) the so-called empiricists Locke and Berkeley. This book presents an accessible and admirably broad overview of the modern debate about causation, without sacrificing serious attention to textual details. With two chapters on (i) and one apiece on (ii) - (vi), Clatterbaugh earns for himself and his reader a helpfully synoptic picture of philosophical shifts in the causation debate in the hundred years separating Descartes' Discourse and Hume's Treatise. By the end of the (modern) day, (1) the four-fold causes of Aristotelian-Scholasticism have given way to efficient causation alone; (2) no pre-existing forms in efficient causes are to be found, nor indeed any substantial active powers whatever; (3) there is no deep sense in which causes communicate reality to effects; (4) necessary connection has gone by the board; (5) divine causation has no place in a natural philosophy of causation. No surprises there, but Clatterbaugh's effort reminds us afresh of the varied and torturous routes to such shifts, helping advanced students better (or newly) appreciate them in the comfortable space of 200 pages.Of those pages, a third are devoted to Descartes and the Cartesians. That's not inappropriate, given the importance and difficulty of Descartes's response to Aristotelian-Scholastic accounts of causation, and its influence on occasionalist sentiments on the Continent. The tenor and value of this book may be illustrated by reflecting on the first of these—a rich and frustrating business, both in Descartes's texts and in Clatterbaugh's book. Suppose we grant Descartes the back of his hand to substantial forms and active formal causes as at best "unexplanatory" (counseling Regius): his "at-least-as-much-reality-in-the-cause-as-in-the-effect" principle might be treated as the basis of "an alternative" metaphysical account of causation (19-32). Well, it might be so-treated: but three versions of the principle later, the reader will be excused for judging it little more than a necessary condition for some event-pair x,y counting as a cause-effect pair. If the scholastic appeal to active forms is unexplanatory, then Descartes's [End Page 600] principle—which isn't Descartes's principle at all but a scholastic commonplace—can scarcely fill the explanatory gap. Glossed as a predictable consequence of some strong form of influxus physicus, it is embarrassed by Foucher's objection (that mind bears no likeness to body) and by the weight of texts urging that accidents are individual to their bearers. Having offered us no deep metaphysical story about what in the world answers to 'vis,' Descartes is at best a bungling "interactionist" (here appropriating Clatterbaugh's word for someone whose apparent talk of secondary causes we are asked to respect as expressing a strict and considered philosophical commitment). One can only side with Voetius in reckoning Descartes's replacement of forms by purely quantitative, mechanical principles of motion as expressing a mere "disposition of the movable to move... not an activity of an efficient cause, but merely a necessary condition and a causa sine qua non" (my emphasis: Selectarum Disputationum I [1648-59], 873). Descartes has no deep theory about the nature of causation.Is that too harsh? Had this book a different target, one could fairly complain that the author too infrequently steps away from the texts and commentators to offer and defend a... (shrink)

Many areas of genetic research, genetic forensics and genetic essentialism are treated in public sphere debate as suspicious and problematic or are subject to waves of moral panic. In cultural theory, likewise, strong critiques of the genetic essentialism emerge as part of a broader critical assessment of the discourses of the biological sciences and the assertion of a connection between genes and human behavior. However, the scientific and popular claim to the existence of a “gay gene” is not treated (...) in the same way, and has been widely celebrated both within lesbian/gay culture and more broadly in liberal-humanist speech. This essay suggests that it is not enough to presume that this is a result of the extension of essentialism into new popular understandings of human biology. Rather, this essay examines five reasons which have made the popular acceptance of a “gay gene” tenable: the dominance of “gay personage” arguments, the continued influence of early sexology, the suspicion of psychological and constructionist approaches to lesbian/gay sexualities, the effectiveness of genetic essentialism in“ethnic rights” political approaches and the continued centrality of “fixed subjecthood” in contemporary western culture. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Leibniz & Clarke: A Study of Their Correspondence by Ezio VailatiJan A. CoverEzio Vailati. Leibniz & Clarke: A Study of Their Correspondence. New York and Oxford: Oxford University Press, 1997. Pp. xii + 250. Cloth, $45.00.When Leibniz received the 1710 issue of the Royal Society’s Philosophical Transactions in early 1711, he read John Keill’s public charge that he had stolen the calculus from Newton. Leibniz twice sought amends (...) from the Society in the form of a retraction, in response to which a committee was formed to investigate and issue a report. The resulting Commercium epistolicum, written by the good Isaac Newton himself and condemning Leibniz, was published in early 1713. In Vienna at the time, Leibniz learned of its contents in a June letter from Johann Bernoulli, who complained about this “scarcely civilized way” of holding what was in effect a tribunal from which the chief witness was absent. By year’s end Leibniz had enlisted Wolff to see into print Leibniz’s anonymous statement—overstatement, in fact—of his case, and nine months later Keill replied with an equally overstated rejoinder. Late autumn of 1714 found Leibniz back in Hannover, suffering from gout and the tiresome pressure to advance his work on the history of the house of Brunswick-Lüneburg. If only he felt better and had more free time, he could collect his mathematical papers and set an end to the dispute by writing his own Commercium epistolicum: as it was (Leibniz told Wolff in the spring of 1715), he had no intention of giving Newton’s cronies the luxury of his time.The Leibniz-Clarke exchange began in the shadows of this dispute, contributing to Leibniz’s already established distaste for parts of British philosophy. Already in 1690 he was describing (to Huygens and others) Newton’s gravitational attraction as “inexplicable.” Soon after Leibniz was surprised to discover that Locke had accepted Newtonian attraction in the second edition of the Essay—surprised, at least, until he came to see it as part and parcel of the dangerous idea that God could attach any old property whatever to matter if He chose, including thought. Materialism and frequent miracles [End Page 533] were just around the corner, and from there it was a short distance to the mortality of the soul, and a mean view of God’s power; the decay of natural theology itself could only follow. Leibniz’s misgivings didn’t see the light of day in the early 1700s, when he penned them in the Preface to the Nouveaux Essais; but he went out of his way to work an explicit complaint about Mr. Newton and Mr. Locke into the Théodicée of 1710 (§19 of the Preliminary Discourse). The calculus dispute followed. By the time Leibniz got round to writing Princess Caroline about the dispute in a letter of May 1715, he was insisting that the Newtonians deserved public censure, and that the Britons deserved Leibniz as historiographer of England. For her own part, Caroline now wondered if Samuel Clarke, who had translated Newton’s Opticks into Latin, would in the end be unbiased enough to translate Leibniz’s Théodicée into English. Moreover Caroline doubted that Clarke’s notion of the soul was theologically innocent. Replying in November, Leibniz summarized his view that natural religion was in great decline in England, and the famous exchange began.Despite there being no monograph devoted to it until now, the Correspondence ıs (and was) famous. If a book-length study is overdue, so perhaps are three reminders—that a significant portion of the Correspondence is devoted to issues other than space and time and gravitation, that the depth to which its issues are pursued was compromised by impatience on the side of both parties, and that Clarke’s philosophical views aren’t so well-understood as Leibniz’s own. Nowhere does Vailati set out to make these points explicitly, but they emerge over the course of his book, and together with exegetical soundness are its chief contributions. While this volume makes no historical or philosophical waves, it indirectly (perhaps unwittingly) explains why none are readily forthcoming in a book... (shrink)

In lieu of an abstract, here is a brief excerpt of the content:BOOK REVIEWS 273 Verene, Donald Phillip. Vico's Science of Imagination. Ithaca, New York: Cornell University Press, 1981, Pp. 227. $19.5o. In Chapter 1 (Introduction: Vico's Originality), Verene announces two principal concerns, a two-fold approach, and the predominant contention of his study.. 1. Principal concerns: "to consider the philosophical truth of Vico's ideas themselves, rather than to examine their historical character" (p. 19); to consider "the importance of Vico's conception (...) of barbarism for understanding present society" (p. 28). 2. Two-fold approach: "to interpret the central theses of Vico's thought" from the inside in the spirit of the whole and, at the same time, "to develop the problems of Vico's philosophy themselves" (pp. 19--21 ). 3. Predominant contention: "that Vico's ideas constitute a philosophy of recollective universals which generates philosophical understanding fronl the image, not the rational category. This approach regards imagination as a method of pfiilosophical thought" (p. 19) and the imaginative universal as "a key to his conception of knowledge" (p. 67). Vico's philosophy is depicted and characterized by Verene as fi)llows. Beginning with the imagination (fantasia) as an original and independent power of mind, Vico builds his philosophy on the image, the imaginative universal, thereby creating a position outside Western philosophy as traditionally understood. His New Science is a process of recollective fantasia, a kind of memory, and the system built constitutes a system of memory for all of human reality. By means of his concept of memory, Vico works his way back to the world of original thought, the mythicopoeticfantasia of the first men. His discovery of the imaginative universal as their way of thinking and their kind of knowledge establishes a new genetic origin for philosophical thought. Recollective fantasia constitutes the subject-matter and form of Vico's science itself', i.e., the term designates a type of nlemory through which the New Science is acconlplished as well as the type of thought from which it must be understood. "The New Science is a metaphysical fable, h creates a vera narratio of the recollectire imagination. The storia ideale eterna is an ideal trnth, but not an a priori or fictional one. As the master image of the recollective mind, it precisely contains all of the human event. Metaphysical truth, achieved through the exercise of.[antasia.., is generated out of the mind itself.., as images that actually contain the providential structure of reality" (p. 125). Vico's per|ornlative science "is man's true production of a knowledge of" his own nature" (p. 133). "The proof of the science is found through the reader making it for himself" (p. 156). What is the "recollective imaginative universal"? (a term coined by Verene) as distinguished from the universale fantastico? Vico identifies the latter quite simply. Poetic imagination, he tells us, was active in the first men who, "not being able to form intelligible class concepts of things, had a natural need to create poetic characters ; that is, imaginative class concepts or universals, to which, as to certain models or ideal portraits, to reduce all the particular species that resembled them." (The New Science of Giambattista Vico, trans. Thomas Goddard Bergin and Max Harold Fisch. Cornell University Press, 1968; orig. pub. 1948, w ) Thus in the theological age, 274 HISTORY OF PHILOSOPHY "Jove was born naturally in poetry as a divine character or imaginative universal, to which everything having to do with the auspices was referred by all the ancient gentile nations, which must therefore all have been poetic by nature" (9 381). In the age of heroes, Achilles embodies the idea of valor common to all strong men and Ulysses the idea of prudence common to all wise men (9 403). The myths, fables, and allegories brought to birth by poetic imagination were true stories for their makers. Imaginative universals or genera, "as the human mind later learned to abstract forms and properties from subjects, passed over into intelligible genera, which prepared the way for philosophers.... " (9 934)- Abstracting, categorizing, and predicating on the basis of properties in common, the philosophers in their intelligible class concepts no hmger think in the same manner as the first men did... (shrink)

Interpretations of Nietzsche, particularly about politics, cover an exceptionally wide range. Additionally, Nietzsche is often said to commit “rhetorical excesses.” I argue and show that Nietzsche consciously crafted his published works to allow this range of interpretations, that he did this for critical purposes, and that his so-called rhetoric is there to serve this purpose.

The situated potentials for action between material things in the world and the interactional processes thereby afforded need to be seen as not only constituting the possibility of agency, but thereby also comprising it. Eo ipso, agency must be de-fused from any local, "contained" subject and be understood as a situational property in which subjects and objects can both participate. Any technological artifact should thus be understood as a complex of agential capacities that function relative to any number of (...) social and material factors. Keeping in mind that we are co-constituted by webs of relations involving increasingly complex collections of artefacts, networks, niches, and communities of practice, our investigation will be guided by interrogating the functional potential of a thing that in the last fifteen years has seamlessly worked its way into the everyday life of millions of human agents. This "thing," the smartphone, is merely a nodal point in a highly complex network. Recognizing this massive "collective," we nevertheless want to show some of the ways in which something as seemingly mundane as a smartphone can reflexively alter the range of actions available to a cognitive agent (Latour 1994). Specifically, we hope to shed light on some of the social-cognitive consequences of technological mediation by looking at the complementary, if not mutually implied, domains of memory and knowledge. (shrink)

The President’s Council on Bioethics has addressed the moral status of human preembryos in its reports on stem cell research and human therapeutic cloning. Although the Council has been criticized for being hand-picked to favor the right-to-life viewpoint concerning human preembryos, it has embraced the idea that the right-to-life position should be defended in secular terms. This is an important feature of the Council’s work, and it demonstrates a recognition of the need for genuine engagement between opposing sides in the (...) debate over stem cell research. To promote this engagement, the Council has stated in secular terms several arguments for the personhood of human preembryos. This essay presents and critiques those arguments, and it concludes that they are unsuccessful. If the best arguments in support of the personhood of human preembryos have been presented by the Council, then there are no reasonable secular arguments in support of that view. (shrink)

We prove the following theorems: (1) If X has strong measure zero and if Y has strong first category, then their algebraic sum has property s 0 . (2) If X has Hurewicz's covering property, then it has strong measure zero if, and only if, its algebraic sum with any first category set is a first category set. (3) If X has strong measure zero and Hurewicz's covering property then its algebraic (...) sum with any set in APC ' is a set in APC '. (APC ' is included in the class of sets always of first category, and includes the class of strong first category sets.) These results extend: Fremlin and Miller's theorem that strong measure zero sets having Hurewicz's property have Rothberger's property, Galvin and Miller's theorem that the algebraic sum of a set with the γ-property and of a first category set is a first category set, and Bartoszynski and Judah's characterization of SR M -sets. They also characterize the property (*) introduced by Gerlits and Nagy in terms of older concepts. (shrink)

In a previous paper-[17]-we characterized strong measure zero sets of reals in terms of a Ramseyan partition relation on certain subspaces of the Alexandroff duplicate of the unit interval. This framework gave only indirect access to the relevant sets of real numbers. We now work more directly with the sets in question, and since it costs little in additional technicalities, we consider the more general context of metric spaces and prove: 1. If a metric space has a covering (...) property of Hurewicz and has strong measure zero, then its product with any strong measure zero metric space is a strong measure zero metric space (Theorem 1 and Lemma 3). 2. A subspace X of a σ-compact metric space Y has strong measure zero if, and only if, a certain Ramseyan partition relation holds for Y (Theorem 9). 3. A subspace X of a σ-compact metric space Y has strong measure zero in all finite powers if, and only if, a certain Ramseyan partition relation holds for Y (Theorem 12). Then 2 and 3 yield characterizations of strong measure zeroness for σ-totally bounded metric spaces in terms of Ramseyan theorems. (shrink)

There is a well-established correspondence between interpolation and amalgamation for algebraizable logics that satisfy certain additional assumptions. In this paper, we introduce the Robinson property of a logic and show that a conditionally algebraizable logic without any additional assumptions has the Robinson property if and only if the corresponding class of Lindenbaum–Tarski algebras has the amalgamation property. Moreover, we give the logical characterization of the strong amalgamation property, solving an open problem of Andréka–Németi–Sain. It is (...) also shown that given the mentioned extra assumptions the Robinson property implies the interpolation property. As conditionally algebraizable logics cover algebraizable logics as well as various quantifier logics such as classical first order logic, our results yield a generalization of some of the results concerning interpolation and amalgamation. (shrink)

Let T be a strongly minimal theory with quantifier elimination. We show that the class of existentially closed models of T{“σ is an automorphism”} is an elementary class if and only if T has the definable multiplicity property, as long as T is a finite cover of a strongly minimal theory which does have the definable multiplicity property. We obtain cleaner results working with several automorphisms, and prove: the class of existentially closed models of T{“σi is an automorphism”: (...) i=1,2} is an elementary class if and only if T has the definable multiplicity property. (shrink)

This article focuses on the analysis of the main positions of the Constitutional Court of the Republic of Lithuania in the cases of intellectual property law. In the article three judgments and the positions of the Constitutional Court extracted therefrom are analysed. The Constitutional Court has formed several important positions with reference to intellectual property law regarding usage of property protection norms for the protection of intellectual property, requirements of application of compensation as an alternative to (...) damages compensation and the calculation of criterions thereof as well as functional legal protection of trademarks. It should be noted that Article 23(1) of the Constitution of the Republic of Lithuania has extended protection to property and, as the Constitutional Court of the Republic of Lithuania has noted, the constitutional protection of property covers not only the protection of tangible, but also of intellectual property. In the same judgment, the Constitutional Court stated another important intellectual property related rule, namely that constitutionally established property protection rights have to be implemented regardless of whether a corresponding international treaty has not been ratified, as protection of authors’ rights is granted under Articles 23 and 42(3) of the Constitution of the Republic of Lithuania. However, the second position stating that the rights of foreign subjects are to be implemented even though a corresponding international treaty has not been signed is to be criticised as by coming to such a conclusion the Constitutional Court has ignored the otherwise generally recognised principle of territoriality of intellectual property rights. In two other cases the Constitutional Court has made important statements relevant to intellectual property law. First of all, the Court stated that compensation instead of claiming actual damages is constitutional. By analysing the positions formulated by the Court, an indirect conclusion can be made to the effect that compensation instead of claiming actual damages is necessary in order to ensure effective protection of intellectual property rights. In other words, the purposes of compensation can be recognised: restoring (compensating) the infringed interests of the injured party; simplifying judicial proceedings, whereby faster proceedings (litigation) and easier substantiation is sought; and the preventive purpose (strong preventive effect). However, the Court’s argument regarding the criterion of compensation calculation, saying that the damages incurred by the intellectual property rights owner depend on the sale value of the product concerned is criticisable, as it is rarely the case that the licence fee or profit acquired in the result of unauthorised use of intellectual property objects is calculated with reference to the final sale price of the product. Notwithstanding the arguably weak argument of the Court, the Court’s position regarding the criterion of final sale price itself does not preclude the possibility of reaching a fair final decision. Finally, the Court acknowledged that trademark protection was limited to the economic functions of the trademark and the full protection thereof. (shrink)

We study the preservation of selective covering properties, including classic ones introduced by Menger, Hurewicz, Rothberger, Gerlits and Nagy, and others, under products with some major families of concentrated sets of reals.Our methods include the projection method introduced by the authors in an earlier work, as well as several new methods. Some special consequences of our main results are : Every product of a concentrated space with a Hurewicz S1S1 space satisfies S1S1. On the other hand, assuming the Continuum (...) Hypothesis, for each Sierpiński set S there is a Luzin set L such that L×SL×S can be mapped onto the real line by a Borel function. Assuming Semifilter Trichotomy, every concentrated space is productively Menger and productively Rothberger. Every scale set is productively Hurewicz, productively Menger, productively Scheepers, and productively Gerlits–Nagy. Assuming d=ℵ1d=ℵ1, every productively Lindelöf space is productively Hurewicz, productively Menger, and productively Scheepers.A notorious open problem asks whether the additivity of Rothberger's property may be strictly greater than addadd, the additivity of the ideal of Lebesgue-null sets of reals. We obtain a positive answer, modulo the consistency of Semifilter Trichotomy >ℵ1.Our results improve upon and unify a number of results, established earlier by many authors. (shrink)

An inaccessible cardinal $\kappa$ is supercompact when $(\kappa, \lambda)$-ITP holds for all $\lambda\geq \kappa$. We prove that if there is a model of ZFC with infinitely many supercompact cardinals, then there is a model of ZFC where for every $n\geq 2$ and $\mu\geq \aleph_n$, we have $(\aleph_n, \mu)$-ITP.

An inaccessible cardinal κ is supercompact when (κ, λ)-ITP holds for all λ ≥ κ. We prove that if there is a model of ZFC with two supercompact cardinals, then there is a model of ZFC where simultaneously \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(\aleph_2, \mu)}$$\end{document} -ITP and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(\aleph_3, \mu')}$$\end{document} -ITP hold, for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mu\geq \aleph_2}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} (...) \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mu'\geq \aleph_3}$$\end{document}. (shrink)

We prove that Martin’s Axiom implies the existence of a Cohen-indestructible mad family such that the Mathias forcing associated to its filter adds dominating reals, while \ is consistent with the negation of this statement as witnessed by the Laver model for the consistency of Borel’s conjecture.

The class DA of diagonalizable algebras enjoys the interpolation property . Using LDA-Gentzen System corresponding to DA, we show that it has the strong amalgamation property as well.

In set theory without the axiom of choice, we investigate the deductive strength of the principle “every topological space with the minimal cover property is compact”, and its relationship with certain notions of finite as well as with properties of linearly ordered sets and partially ordered sets.

The Covering Property Axiom, which attempts to capture some of the combinatorics of the Sacks model, the model obtained from by countable support iteration of length of the Sacks forcing, seems to miss a Suslin tree. We add a diamond polish to the axiom to remedy this.

We give topological characterizations of filters${\cal F}$onωsuch that the Mathias forcing${M_{\cal F}}$adds no dominating reals or preserves ground model unbounded families. This allows us to answer some questions of Brendle, Guzmán, Hrušák, Martínez, Minami, and Tsaban.

We establish a surprising connection between Menger's classical covering property and Blass-Laflamme's modern combinatorial notion of groupwise density. This connection implies a short proof of the groupwise density bound on the additivity number for Menger's property.

Fontanella :193–207, 2014) showed that if \ is an increasing sequence of supercompacts and \, then the strong tree property holds at \. Building on a proof by Neeman, we show that the strong tree property at \ is consistent with \, where \ is singular strong limit of countable cofinality.

Three results in [14] and one in [8] are analyzed in Sections 3–6 in order to supply examples on Loeb probability spaces, which distinguish the different strength among three generalizations of k-saturation, as well to answer some questions in Section 7 of [15]. In Section 3 we show that not every automorphism of a Loeb algebra is induced by an internal permutation, in Section 4 we show that if the 1-special model axiom is true, then every automorphism of a Loeb (...) algebra is induced by a point-automorphism, in Section 5 we show that not every measure-preserving homomorphism from a small subalgebra to a Loeb algebra is induced by an internal permutation, without assuming full-saturation, in Section 6 we show that, under some cardinality assumptions, the 1-isomorphism property does not guarantee the compactness of a Loeb space, and in Section 7 an application of the 1-special model axiom is given on the existence of ergodic transformations of a Loeb space, which partially answers Problem 2.3 of [5]. (shrink)

$\underline{\text{Saturation is} (\mu, \kappa)-\text{transferable in} T}$ if and only if there is an expansion T 1 of T with ∣ T 1 ∣ = ∣ T ∣ such that if M is a μ-saturated model of T 1 and ∣ M ∣ ≥ κ then the reduct M ∣ L(T) is κ-saturated. We characterize theories which are superstable without f.c.p., or without f.c.p. as, respectively those where saturation is (ℵ 0 , λ)- transferable or (κ (T), λ)-transferable for all λ. (...) Further if for some $\mu \geq \mid T \mid, 2^\mu > \mu^+$ , stability is equivalent to for all μ ≥ ∣ T ∣, saturation is (μ, 2 μ )- transferable. (shrink)

We show that it is consistent, relative to ω many supercompact cardinals, that the super tree property holds at for all but there are weak square and a very good scale at.

The weak non-finite cover property (wnfcp) was introduced in [1] in connection with "axiomatizability" of lovely pairs of models of a simple theory. We find a combinatorial condition on a simple theory equivalent to the wnfcp, yielding a direct proof that the non-finite cover property implies the wnfcp, and that the wnfcp is preserved under reducts. We also study the question whether the wnfcp is preserved when passing from a simple theory T to the theory TP of lovely (...) pairs of models of T (true in the stable case). While the question remains open, we show, among other things, that if (for a T with the wnfcp) TP is low, then TP has the wnfcp. To study this question, we describe "double lovely pairs", and, along the way, we develop the notion of a "lovely n-tuple" of models of a simple theory, which is an analogue of the notion of a beautiful tuple of models of stable theories [2]. (shrink)

Assuming $V=L+AD$, using methods from inner model theory, we give a new proof of the strong partition property for ${\sim}{ \delta }^{2}_{1}$. The result was originally proved by Kechris et al.

Downey and Remmel have completely characterized the degrees of c.e. bases for c.e. vector spaces (and c.e. fields) in terms of weak truth table degrees. In this paper we obtain a structural result concerning the interaction between the c.e. Turing degrees and the c.e. weak truth table degrees, which by Downey and Remmel's classification, establishes the existence of c.e. vector spaces (and fields) with the strong antibasis property (a question which they raised). Namely, we construct c.e. sets $B<_{\rm (...) T}A$ such that the c.e. W-degrees below that of A are disjoint from the nonzero c.e. T-degrees below that of A and comparable to that of B. (shrink)

Let L be a finite relational language and α=(αR:R ∈ L) a tuple with 0 < αR ≤1 for each R ∈ L. Consider a dimension function $ \delta _\alpha (A) = \left| A \right| - \sum\limits_{R \in L} {\alpha {\mathop{\rm Re}\nolimits} R(A)} $ where each eR(A) is the number of realizations of R in A. Let $K_\alpha $ be the class of finite structures A such that $\delta _\alpha (X) \ge 0$ 0 for any substructure X of A. We (...) show that the theory of the generic model of $K_\alpha $ is AE-axiomatizable for any α. (shrink)

We analyze combinatorial properties of open covers of sets of real numbers by using filters on the natural numbers. In fact, the goal of this paper is to characterize known properties related to ω-covers of the space in terms of combinatorial properties of filters associated with these ω-covers. As an example, we show that all finite powers of a set R of real numbers have the covering property of Menger if, and only if, each filter on ω associated (...) with its countable ω-cover is a P + filter. (shrink)