The riddle of the Cleitophon is a creature of nineteenth-century German scholarship which premised that Plato had developed a profound philosophical system. Thus, having no intrinsic purpose to serve in the context of the development of Plato's philosophy, Cleitophon was disallowed as spurious. Documenting the reception of this minor dialogue provides insights into the pluralism and the perplexities of modern Platonic exegesis. The more recent idea of a genre of literary fiction helps to restore cleitophon to its place in the (...) Platonic corpus. S.R. Slings's edition of this dialogue, with introduction, translation and commentary reconsiders both Plato's dialogical engagements and intellectual commitments, utilizes a two-dimensional approach to the text that avoids exaggerated conjectures, and effectively defends the authenticity of Cleitophon. (shrink)

From the early 1930s to the early 1960s many scholars, whether liberal-minded or socialist ideologues, Marxist or scientific positivists, classical scholars or political theorists and historians, have shown a widespread consensus in discrediting and assailing the man and political philosopher Plato. Such an extensive assault led the 'Platonic Legend' to an unprecedented crisis. Philosophically, it was a reaction to the undisguised Platonolatry coming from Oxford and the school of the British Idealists. Ideologically, the appropriation of Plato by Nazi apologists fostered (...) further this vehement indictment. But a lot of other causes worked to the same effect. The general anguish and humanistic anxiety on the eve of World War II and the postwar traumas led scholars to reconsider the meaning of history and historicism, the psychology of the masses and the ethical responsibility of the citizen, the role of propaganda and state education. Such complementary elements converged in sustained anti-Platonic polemics, which in turn provoked a vigorous defence. Here an attempt is made to offer a preliminary survey of this complex debate and to provide a general intellectual framework in terms of which that controversy can be further explored. (shrink)

From the early 1930s to the early 1960s many scholars, whether liberalminded or socialist ideologues, Marxist or scientific positivists, classical scholars or political theorists and historians, have shown a widespread consensus in discrediting and assailing the man and political philosopher Plato. Such an extensive assault led the ‘Platonic Legend’ to an unprecedented crisis. Philosophically, it was a reaction to the undisguised Platonolatry coming from Oxford and the school of the British Idealists. Ideologically, the appropriation of Plato by Nazi apologists fostered (...) further this vehement indictment. But a lot of other causes worked to the same effect. The general anguish and humanistic anxiety on the eve ofWorldWar II and the postwar traumas led scholars to reconsider the meaning of history and historicism, the psychology of the masses and the ethical responsibility of the citizen, the role of propaganda and state education. Such complementary elements converged in sustained anti-Platonic polemics, which in turn provoked a vigorous defence. Here an attempt is made to offer a preliminary survey of this complex debate and to provide a general intellectual framework in terms of which that controversy can be further explored. (shrink)

From the early 1930s to the early 1960s many scholars, whether liberalminded or socialist ideologues, Marxist or scientific positivists, classical scholars or political theorists and historians, have shown a widespread consensus in discrediting and assailing the man and political philosopher Plato. Such an extensive assault led the ‘Platonic Legend’ to an unprecedented crisis. Philosophically, it was a reaction to the undisguised Platonolatry coming from Oxford and the school of the British Idealists. Ideologically, the appropriation of Plato by Nazi apologists fostered (...) further this vehement indictment. But a lot of other causes worked to the same effect. The general anguish and humanistic anxiety on the eve ofWorldWar II and the postwar traumas led scholars to reconsider the meaning of history and historicism, the psychology of the masses and the ethical responsibility of the citizen, the role of propaganda and state education. Such complementary elements converged in sustained anti-Platonic polemics, which in turn provoked a vigorous defence. Here an attempt is made to offer a preliminary survey of this complex debate and to provide a general intellectual framework in terms of which that controversy can be further explored. (shrink)

This article is currently available as a free download on ingentaconnect.

Previous studies have offered indications that the way pre-adolescents structure their perceptions of their teacher’s interaction in terms of Agency and Communion differs from adolescents. The purpose of this study was to delineate previous findings by thoroughly examining the structure of pre-adolescents’ perceptions of their teacher’s interpersonal behaviour, and by investigating the extent to which this structure relates to pre-adolescents’ learning outcomes. A mixed methods research design was implemented including a qualitative instrument adaptation procedure followed by a quantitative large-scale administration. (...) Participants for the latter were a random sample of 504 pre-adolescents and 36 teachers. Results showed that participants placed significantly more emphasis on their teachers’ Communion behavioural dimension and this emphasis was illustrated on the association of perceived teacher interaction and students’ learning outcomes. Implications for theory, research and practice are discussed. (shrink)

I undertake a metaphysical investigation of Saul Kripke's modern classic, Naming and Necessity . The general problem of my study may be expressed as follows: What is the metaphysical justification of the validity and existence of the pertinent classes of truths, the necessary a posteriori and the contingent a priori, according to the Kripke Paradigm? My approach is meant to disclose the logical and ontological principles underlying Kripke's arguments for the necessary a posteriori and the contingent a priori respectively. The (...) results of my study are to a certain extent negative: They attempt to show that the classes of the necessary a posteriori and the contingent a priori statements cannot possibly be valid. If the general argument of this thesis is sound, then, on this ground, the realist conception of metaphysical essentialism is rejected. The positive thesis of this study is the articulation of certain ways to frame the necessary a posteriori and the contingent a priori which avoid the problems of realistic essentialism and which suggest a certain transcendental reading of modality. According to the Kripke Paradigm, any de dicto modal status derives ultimately from the de re modality inherent in the object designated, as the object is characterised by contingent and necessary properties on the ontological level. Thus, the Kripke Paradigm is primarily a thesis in de re realist essentialism. The final verdict in this study is that the Kripke Paradigm cannot sustain the realistic conception of de re metaphysical essentialism. If we should adopt a transcendental reading of modality, then certain portions of the Kripke Paradigm are valid. I do not delineate or possess the details of a comprehensive doctrine of transcendental metaphysic. Nevertheless, the observations I make should suffice to bring about the rough orientation of how I conceive the notion of transcendental modality. (shrink)

Essays on the ethics of monuments tend to focus on their morality in relation to domestic populations. In this article we turn our attention to how the principles we favor for the ‘ingroup’ apply to various ‘outgroups’, including foreigners and foreign governments, guest workers, visiting scholars, forcibly annexed or colonized peoples, and migrant communities. It argues that nations have a prima facie moral right to erect and maintain monuments offensive to foreigners and foreign governments or (in the case of institutions) (...) rivals. Furthermore, we hold that whereas prospective immigrants have little standing to advocate for the removal of monuments they find offensive in their receiving countries, those who have been forcibly annexed have a much greater claim to inoffensive memorialization, contingent on their willingness to be countrymen with the descendants of their former oppressors and principles of reasonably charitable interpretation. Lastly, the article advocates for the protection of heritage landscapes from foreign influence and highlights the responsibilities of migrant populations to avoid memorializing their old grievances in ways that undermine the interests of their adopted nations. (shrink)

This is the second paper of an essay in two parts. The first paper (Part I) is a critical discussion of Mark Risjord's conception of nursing knowledge where I argued against the conception of nursing knowledge as a kind of nursing science. The aim of the present paper (Part II) is to explicate and substantiate the thesis of nursing as a kind of concrete philosophy. My strategy is to elaborate upon certain themes from Wittgenstein's Tractatus in order to canvass a (...) general scheme of philosophy based on a distinction between reality and the world. This distinction will be employed in the appropriation of certain significant features of nursing and nursing knowledge. By elaborating on the contrast between the abstract and the concrete, I will suggest that nursing may be seen as a kind of concrete philosophy, being primarily concerned with reality (and secondarily with the world). This thesis, I will argue, implies that philosophy is the kind of theory that is essential to nursing (which is not so much a theory than a certain kind of activity). (shrink)

This essay addresses the problem of the essentiality of nursing knowledge and what kind of theory, if any, is essential to nursing practice. The overarching aim of the essay was to argue for the thesis that nursing may be described as a kind of philosophical activity, and, consequently, that philosophy is the kind of “theory” that is essential to nursing practice and to the nursing discipline at large. The essay consists of two papers. The present paper, Part I, is a (...) critical examination of Mark Risjord's discussion of the problem of the theory‐practice gap in his Nursing Knowledge: Practice, Science, Philosophy, from 2010. According to Risjord, the cause of the theory‐practice gap originates in an erroneous conception of science (logical positivism) which had a decisive influence upon the way nursing scholars appropriated theoretical frameworks for the nursing discipline. This philosophical influence is considered in effect to have generated the theory–practice gap. In order to bridge the gap, Risjord suggests, the nursing discipline needs to adopt a standpoint epistemology conjoined with a postpositivist conception of scientific theory. In this way, a legitimate brand of nursing science may be developed and the theory–practice gap overcome. I will argue that neither Risjord's diagnosis of the problem, nor his recommended cure, may succeed in rescuing the nursing discipline from the theory–practice gap. Rather, the real cause of the theory–practice gap, I will claim, derives from an erroneous conception of nursing (not of science), namely the conception of nursing as a kind of science (roughly speaking). On my view, to overcome the gap, the nursing discipline needs to make salient the inherently philosophical character of nursing. In the second paper (Part II), I will continue the discussion of nursing knowledge and delineate the thesis of nursing as a kind of concrete philosophy. (shrink)

This essay addresses the problem of the essentiality of nursing knowledge and what kind of theory, if any, is essential to nursing practice. The overarching aim of the essay was to argue for the thesis that nursing may be described as a kind of philosophical activity, and, consequently, that philosophy is the kind of “theory” that is essential to nursing practice and to the nursing discipline at large. The essay consists of two papers. The present paper, Part I, is a (...) critical examination of Mark Risjord's discussion of the problem of the theory‐practice gap in his Nursing Knowledge: Practice, Science, Philosophy, from 2010. According to Risjord, the cause of the theory‐practice gap originates in an erroneous conception of science (logical positivism) which had a decisive influence upon the way nursing scholars appropriated theoretical frameworks for the nursing discipline. This philosophical influence is considered in effect to have generated the theory–practice gap. In order to bridge the gap, Risjord suggests, the nursing discipline needs to adopt a standpoint epistemology conjoined with a postpositivist conception of scientific theory. In this way, a legitimate brand of nursing science may be developed and the theory–practice gap overcome. I will argue that neither Risjord's diagnosis of the problem, nor his recommended cure, may succeed in rescuing the nursing discipline from the theory–practice gap. Rather, the real cause of the theory–practice gap, I will claim, derives from an erroneous conception of nursing (not of science), namely the conception of nursing as a kind of science (roughly speaking). On my view, to overcome the gap, the nursing discipline needs to make salient the inherently philosophical character of nursing. In the second paper (Part II), I will continue the discussion of nursing knowledge and delineate the thesis of nursing as a kind of concrete philosophy. (shrink)

The British utilitarians are not generally considered explorers of classical Greek thought. This paper examines the contribution of James Mill, John Stuart Mill, and George Grote to the development of Platonic studies in nineteenth-century Britain. Their understanding of Platonic philosophy challenged prevalent interpretations, and caused a fruitful debate over long neglected aspects of Plato's thought. Grote's Platonic analysis, which comes last in order of time, cannot, of course, be considered in isolation from the relevant debates in Germany. Grote, the erudite (...) historian of ancient Greece, paid considerable attention to the arguments of the German classicists, put forward in many cases a new point of view, and prompted a radical revaluation of Platonic political thought. (shrink)

It is shown that in ZF Martin's $ \aleph_{0}^{}$-axiom together with the axiom of countable choice for finite sets imply that arbitrary powers 2X of a 2-point discrete space are Baire; and that the latter property implies the following: the axiom of countable choice for finite sets, power sets of infinite sets are Dedekind-infinite, there are no amorphous sets, and weak forms of the Kinna-Wagner principle.

We show that none of the following statements is provable in Zermelo-Fraenkel set theory (ZF) answering the corresponding open questions from Brunner in ``The axiom of choice in topology'':(i) For every T2 topological space (X, T) if X is well-ordered, then X has a well-ordered base,(ii) For every T2 topological space (X, T), if X is well-ordered, then there exists a function f : X × W T such that W is a well-ordered set and f ({x} × W) is (...) a neighborhood base at x for each x X,(iii) For every T2 topological space (X, T), if X has a well-ordered dense subset, then there exists a function f : X × W T such that W is a well-ordered set and {x} = f ({x} × W) for each x X. (shrink)

An overview is provided of how the concept of the thought experiment has developed and changed for the natural sciences in the course of the 20th century. First, we discuss the existing definitions of the term 'thought experiment' and the origin of the thought experimentation method, identifying it in Greek Presocratics epoch. Second, only in the end of the 19th century showed up the first systematic enquiry on thought experiments by Ernst Mach's work. After the Mach's work, a negative attitude (...) towards thought experiments came in the beginning of the 20th century, which went on until the Thomas Kuhn's and Karl Popper's work on thought experiments. Only from the mid-1980s did thought experiments begin to be considered relevant to scientific enterprise. Finally, we show the existing empirical and 'functional' theories which have developed about the nature and purpose of thought experiments. (shrink)

Some researchers have recognized that privileged communities dominate the discourse on AI Ethics, and other voices need to be heard. As such, we identify the current ethics milieu as arising from WEIRD (Western, Educated, Industrialized, Rich, Democratic) contexts, and aim to expand the discussion to non-WEIRD global communities, who are also stakeholders in global sociotechnical systems. We argue that accounting for honor, along with its values and related concepts, would better approximate a global ethical perspective. This complex concept already underlies (...) some of the WEIRD discourse on AI ethics, but certain cultural forms of honor also bring overlooked issues and perspectives to light. We first describe honor according to recent empirical and philosophical scholarship. We then review “consensus” principles for AI ethics framed from an honor-based perspective, grounding comparisons and contrasts via example settings such as content moderation, job hiring, and genomics databases. A better appreciation of the marginalized concept of honor could, we hope, lead to more productive AI value alignment discussions, and to AI systems that better reflect the needs and values of users around the globe. (shrink)

In this chapter we focus on the debate over publicly-maintained racist monuments as it manifests in the mid-2010s Anglosphere, primarily in the US (chiefly regarding the over 700 monuments devoted to the Confederacy), but to some degree also in Britain and Commonwealth countries, especially South Africa (chiefly regarding monuments devoted to figures and events associated with colonialism and apartheid). After pointing to some representative examples of racist monuments, we discuss ways a monument can be thought racist, and neutrally categorize removalist (...) and preservationist arguments heard in the monument debate. We suggest that both extremist and moderate removalist goals are likely to be self-defeating, and that when concerns of civic sustainability are put on moral par with those of fairness and justice, something like a Mandela-era preservationist policy is best: one which removes the most offensive of the minor racist monuments, but which focuses on closing the monumentary gap between peoples and reframing existing racist monuments. (shrink)

We investigate the set theoretical strength of some properties of normality, including Urysohn's Lemma, Tietze-Urysohn Extension Theorem, normality of disjoint unions of normal spaces, and normality of Fσ subsets of normal spaces.

Let X be an infinite set and let and denote the propositions “every filter on X can be extended to an ultrafilter” and “X has a free ultrafilter”, respectively. We denote by the Stone space of the Boolean algebra of all subsets of X. We show: For every well‐ordered cardinal number ℵ, (ℵ) iff (2ℵ). iff “ is a continuous image of ” iff “ has a free open ultrafilter ” iff “every countably infinite subset of has a limit point”. (...) implies “every open filter on extends to an open ultrafilter” implies “has an open ultrafilter” implies It is relatively consistent with that (ω) holds, whereas (ω) fails. In particular, none of the statements given in (2) implies (ω). (shrink)

In the realm of metric spaces the role of choice principles is investigated.

In this chapter, we focus on the debate over publicly maintained racist monuments as it manifests in the mid-2010s Anglosphere, primarily in the United States and South Africa. After pointing to some representative examples of racist monuments, we discuss ways a monument can be thought racist and neutrally categorize removalist and preservationist arguments heard in the monument debate. We suggest that both extremist and moderate removalist goals are likely to be self-defeating and that when concerns of civic sustainability are put (...) on moral par with those of fairness and justice, something like a Mandela-era preservationist policy is best: one which removes the most offensive of the minor racist monuments, but which focuses on closing the monumentary gap between peoples and reframing existing racist monuments. (shrink)

We show that some well known theorems in topology may not be true without the axiom of choice.

Derk Pereboom's Four-Case Argument is among the most famous and resilient manipulation arguments against compatibilism. I contend that its resilience is not a function of the argument's soundness but, rather, the ill-gotten gain from an ambiguity in the description of the causal relations found in the argument's foundational case. I expose this crucial ambiguity and suggest that a dilemma faces anyone hoping to resolve it. After a thorough search for an interpretation which avoids both horns of this dilemma, I conclude (...) that none is available. Rather, every metaphysically coherent interpretation invites either a hard- or soft-line reply to Pereboom's argument. I then consider a recharacterization of the dilemma which seems to clear the way for the defence of a revised Four-Case Argument. I address this rejoinder by identifying a still more fundamental problem shared by all viable interpretations of the manipulation cases, showing that each involves a type of manipulation which undermines the victim's agency. Because this diagnosis supports a soft-line reply to every viable interpretation of the argument and can be endorsed by any compatibilist, I consider it the final piece of the Soft-line Solution to the Four-Case Argument. Finally, I suggest a new taxonomy of manipulation arguments, arguing that none that employs the suppressive variety of manipulation found in Pereboom's argument offers a threat to compatibilism. -/- (For direct download option, search Kristin M. Mickelson, Soft-line Solution). (shrink)

Richard Nisbett and Dov Cohen’s (1996) influential account of “cultures of honor” speculates that honor norms are a socially-adaptive deterrence strategy. This theory has been appealed to by multiple empirically-minded philosophers, and plays an important role in John Doris and Alexandra Plakias’ (2008) antirealist argument from disagreement. In this essay, I raise four objections to the Nisbett-Cohen deterrence thesis, and offer another theory of honor in its place that sees honor as an agonistic normative system regulating prestige competitions. Since my (...) account portrays honor norms as radically different from liberal ones, it actually strengthens Doris and Plakias’ case in some respects: cultures of honor are not merely superficially different from Western liberal ones. Nonetheless, the persistent appeal of honor’s principles, and their moral plausibility in certain contexts, suggests not antirealism, but pluralism—a reply on behalf of realism that itself has considerable empirical support. (shrink)

[Updated 2/23/21: complete chapter scan] In this chapter I sketch a rightist approach to monumentary policy in a diverse polity beleaguered by old ethnic grievances. I begin by noting the importance of tribalism, memorialization, and social trust. I then suggest a policy which 1) gradually narrows the gap between peoples in the heritage landscape, 2) conserves all but the most offensive of the least beloved racist monuments, 3) avoids recrimination (i.e., “keeps it positive”) and eschews ideological commentary in new monuments (...) or revisions to old ones, 4) as much as politically feasible, recognizes only the offense of willing tribemates, and 5) responds to aesthetic and other “irrational” offenses more than to “objective” historical or philosophical critiques. (shrink)

According to John Mackie, moral talk is representational but its metaphysical presuppositions are wildly implausible. This is the basis of Mackie's now famous error theory: that moral judgments are cognitively meaningful but systematically false. Of course, Mackie went on to recommend various substantive moral judgments, and, in the light of his error theory, that has seemed odd to a lot of folk. Richard Joyce has argued that Mackie's approach can be vindicated by a fictionalist account of moral discourse. And Mark (...) Kalderon has argued that moral fictionalism is attractive quite independently of Mackie's error-theory. Kalderon argues that the Frege-Geach problem shows that we need moral propositions, but that a fictionalist can and should embrace propositional content together with a non-cognitivist account of acceptance of a moral proposition. Indeed, it is clear that any fictionalist is going to have to postulate more than one kind of acceptance attitude. We argue that this doubleapproach to acceptance generates a new problem -a descendent of Frege-Geach -which we call the acceptance-transfer problem. Although we develop the problem in the context of Kalderon's version of non-cognitivist fictionalism, we show that it is not the noncognitivist aspect of Kalderon's account that generates the problem. A closely related problem surfaces for the more typical variants of fictionalism according to which accepting a moral proposition is believing some closely related non-moral proposition. Fictionalists of both stripes thus have an attitude problem. (shrink)

Faced with the choice between supporting industrial plant agriculture and hunting, Tom Regan’s rights view can be plausibly developed in a way that permits a form of hunting we call “dignitarian.” To motivate this claim, we begin by showing how the empirical literature on animal deaths in plant agriculture suggests that a non-trivial amount of hunting would not add to animal harm. We discuss how Tom Regan’s miniride principle appears to morally permit hunting in that case, and we address recent (...) objections by Jason Hanna to environmentally-based culling that may be seen to speak against this conclusion. We then turn to dignity, which is especially salient in scenarios where harm is necessary or justifiable. We situate “dignitarian” hunting within a larger framework of adversarial ethics, and argue that dignitarian hunting gives animals a more dignified death than the alternatives endemic to large-scale plant agriculture, and so is permissible based on the kinds of principles that Regan endorses. Indeed, dignitarian hunting may actually fit better with Regan’s widely endorsed animal rights framework than the practice of many vegans, and should only be rejected if we’re just as willing to condemn supporting conventional plant agriculture. (shrink)

We formulate an abstract version of the finite injury method in the form of the Baire category theorem. The theorem has the following corollaries: The Friedberg-Muchnik pair of recursively enumerable degrees, the Sacks splitting theorem, the existence of a minimal degree below 0′ and the Shoenfield jump theorem.

It is shown that AC, the axiom of choice for families of non-empty subsets of the real line ℝ, does not imply the statement PW, the powerset of ℝ can be well ordered. It is also shown that the statement “the set of all denumerable subsets of ℝ has size 2math image” is strictly weaker than AC and each of the statements “if every member of an infinite set of cardinality 2math image has power 2math image, then the union has (...) power 2math image” and “ℵ ≠ ℵω” is Hartogs' aleph, the least ℵ not ≤ 2math image), is strictly weaker than the full axiom of choice AC. (shrink)

Call the ethos understanding rightness in terms of spiritual purity and piety, and wrongness in terms of corruption and sacrilege, the “fetish ethic.” Jonathan Haidt and his colleagues suggest that this ethos is particularly salient to political conservatives and non-liberal cultures around the globe. In this essay, I point to numerous examples of moral fetishism in mainstream academic ethics. Once we see how deeply “infected” our ethical reasoning is by fetishistic intuitions, we can respond by 1) repudiating the fetishistic impulse, (...) by 2) “sublimating” our fetishism into liberal rationales, or by 3) accepting the fetishism on its own terms. Of these options, I argue that sublimating our fetishism is not advisable, and that embracing our ethical fetishism isn’t as obviously misguided as some suggest. (shrink)

In \(\textbf{ZF}\), the well-known Fichtenholz–Kantorovich–Hausdorff theorem concerning the existence of independent families of _X_ of size \(|{\mathcal {P}} (X)|\) is equivalent to the following portion of the equally well-known Hewitt–Marczewski–Pondiczery theorem concerning the density of product spaces: “The product \({\textbf{2}}^{{\mathcal {P}}(X)}\) has a dense subset of size |_X_|”. However, the latter statement turns out to be strictly weaker than \(\textbf{AC}\) while the full Hewitt–Marczewski–Pondiczery theorem is equivalent to \(\textbf{AC}\). We study the relative strengths in \(\textbf{ZF}\) between the statement “_X_ has (...) no independent family of size \(|{\mathcal {P}}(X)|\) ” and some of the definitions of “_X_ is finite” studied in Levy’s classic paper, observing that the former statement implies one such definition, is implied by another, and incomparable with some others. (shrink)

We establish the following results: 1. In ZF (i.e., Zermelo-Fraenkel set theory minus the Axiom of Choice AC), for every set I and for every ordinal number α ≥ ω, the following statements are equivalent: (a) The Tychonoff product of| α| many non-empty finite discrete subsets of I is compact. (b) The union of| α| many non-empty finite subsets of I is well orderable. 2. The statement: For every infinite set I, every closed subset of the Tychonoff product [0, 1] (...) I which consists of functions with finite support is compact, is not provable in ZF set theory. 3. The statement: For every set I, the principle of dependent choices relativised to I implies the Tychonoff product of countably many non-empty finite discrete subsets of I is compact, is not provable in ZF⁰ (i.e., ZF minus the Axiom of Regularity). 4. The statement: For every set I, every ℵ₀-sized family of non-empty finite subsets of I has a choice function implies the Tychonoff product of ℵ₀ many non-empty finite discrete subsets of I is compact, is not provable in ZF⁰. (shrink)

We prove that, in any fine structural extender model with Jensen’s λ-indexing, there is a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\square(\kappa^{+})}$$\end{document} -sequence if and only if there is a pair of stationary subsets of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\kappa^{+} \cap {\rm {cof}}( < \kappa)}$$\end{document} without common reflection point of cofinality \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${ < \kappa}$$\end{document} which, in turn, is equivalent to the existence of a (...) family of size \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${ < \kappa}$$\end{document} of stationary subsets of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\kappa^{+} \cap {\rm {cof}}( < \kappa)}$$\end{document} without common reflection point of cofinality \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${ < \kappa}$$\end{document}. By a result of Burke/jensen, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\square_\kappa}$$\end{document} fails whenever \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\kappa}$$\end{document} is a subcompact cardinal. Our result shows that in extender models, it is still possible to construct a canonical \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\square(\kappa^{+})}$$\end{document} -sequence where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\kappa}$$\end{document} is the first subcompact. (shrink)

In the realm of Lindelöf metric spaces the following results are obtained in : (i) If is a Lindelöf metric space then it is both densely Lindelöf and almost Lindelöf. In addition, under the countable axiom of choice, the three notions coincide. (ii) The statement “every separable metric space is almost Lindelöf” implies that every infinite subset of has a countably infinite subset). (iii) The statement “every almost Lindelöf metric space is quasi totally bounded implies. (iv) The proposition “every quasi (...) totally bounded metric space is separable” lies, in the deductive hierarchy of choice principles, strictly between the countable union theorem and. Likewise, the statement “every pre‐Lindelöf (or Lindelöf) metric space is separable” lies strictly between and. (shrink)