Todorčević (Fund Math 150(1):55–66, 1996) shows that there is no Hausdorff gap (A, B) if A is analytic. In this note we extend the result by showing that the assertion “there is no Hausdorff gap (A, B) if A is coanalytic” is equivalent to “there is no Hausdorff gap (A, B) if A is ${{\bf \it{\Sigma}}^{1}_{2}}$ ”, and equivalent to ${\forall r \; (\aleph_1^{L[r]}\,< \aleph_1)}$ . We also consider real-valued games corresponding to Hausdorff gaps, and show (...) that ${\mathsf{AD}_\mathbb{R}}$ for pointclasses Γ implies that there are no Hausdorff gaps (A, B) if ${{\it{A}} \in {\bf \it{\Gamma}}}$. (shrink)

We explore an application of homological algebra to set theoretic objects by developing a cohomology theory for Hausdorff gaps. This leads to a natural equivalence notion for gaps about which we answer questions by constructing many simultaneous gaps. The first result is proved in ZFC while new combinatorial hypotheses generalizing ♣ are introduced to prove the second result. The cohomology theory is introduced with enough generality to be applicable to other questions in set theory. Additionally, the notion of an (...) incollapsible gap is introduced and the existence of such a gap is shown to be independent of ZFC. (shrink)

A Hausdorff space is called effectively Hausdorff if there exists a function F—called a Hausdorff operator—such that, for every with,, where U and V are disjoint open neighborhoods of x and y, respectively. Among other results, we establish the following in, i.e., in Zermelo–Fraenkel set theory without the Axiom of Choice (): is equivalent to “For every set X, the Cantor cube is effectively Hausdorff”. This enhances the result of Howard, Keremedis, Rubin and Rubin [13] that (...) is equivalent to “Hausdorff spaces are effectively Hausdorff” in. The Boolean Prime Ideal Theorem and the statement “For every infinite set X, the Stone space of the Boolean algebra is effectively Hausdorff” are mutually independent. In particular, the latter statement is not provable in. The Axiom of Choice for non‐empty subsets of () is equivalent to each of “Separable Hausdorff spaces are effectively Hausdorff” and “The Cantor cube is effectively Hausdorff”. The Principle of Dependent Choices in conjunction with the Axiom of Choice for continuum sized families of non‐empty subsets of does not imply the axiom of choice for partitions of. The latter independence result fills the gap in information in Howard and Rubin's book “Consequences of the Axiom of Choice”. The axiom of countable choice for non‐empty subsets of is equivalent to each of “Denumerable Hausdorff spaces are effectively Hausdorff”, “Denumerable T3 spaces are completely normal” and “Denumerable Tychonoff spaces are Urysohn”. (shrink)

Dan Talayco has recently defined the gap cohomology group of a tower in p/fin of height ω1. This group is isomorphic to the collection of gaps in the tower modulo the equivalence relation given by two gaps being equivalent if their levelwise symmetric difference is not a gap in the tower, the group operation being levelwise symmetric difference. Talayco showed that the size of this group is always at least 2N0 and that it attains its greatest possible size, 2N1, if (...) ⋄ holds and also in some generic extensions in which CH fails, for example on adding many Cohen or random reals. In this paper it is shown that there is always some tower whose gap cohomology group has size 2N1. It is still open as to whether there are models in which there are towers whose gap cohomology group has size less than 2ω1. (shrink)

For a stationary set $S \subseteq \omega_{1}$ and a ladder system C over S, a new type of gaps called C-Hausdorff is introduced and investigated. We describe a forcing model of ZFC in which, for some stationary set S, for every ladder C over S, every gap contains a subgap that is C-Hausdorff. But for every ladder E over \omega_{1} \ S$ there exists a gap with no subgap that is E-Hausdorff. A new type of chain condition, (...) called polarized chain condition, is introduced. We prove that the iteration with finite support of polarized c.c.c. posets is again a polarized c.c.c poset. (shrink)

The purpose of this study is to examine whether corporate social responsibility activities perceived by consumers affect brand trust and corporate sustainability management. In other words, this study tried to examine whether the compatibility and authenticity of CSR influences brand trust, thereby affecting CSM including economic viability, environmental soundness, and social responsibility. To measure this, an empirical analysis was conducted on 479 consumers who had experience purchasing products from cosmetic companies that are carrying out CSR. As a result of the (...) analysis, it was found that the compatibility and authenticity of CSR have a positive effect on brand trust. Also, it was found that brand trust had a positive effect on social responsibility among the sub-concepts of CSM, but did not affect economic viability and environmental soundness. The results of this study are expected to provide strategic implications for social responsibility performance and brand trust building necessary for cosmetics companies to grow continuously. (shrink)

During the Asian and Pacific War, the Japanese government mobilized approximately 200,000 Asian women to military brothels to sexually serve Japanese soldiers. The majority of these victims were unmarried young women from Korea, Japan’s colony at that time. In the early 1990s, Korean feminist leaders helped more than 200 Korean survivors of Japanese military sexual slavery to come forward to tell the truth, which has further accelerated the redress movement for the women. One major issue in the redress movement and (...) research relating to the so-called “comfort women” issue is whether Japan’s colonization of Korea or gender hierarchy was a more fundamental cause of the Korean women’s suffering. Using an intersectional perspective, this article analyzes how colonial power, gender hierarchy, and class were inseparably tied together to make the victims’ lives miserable. By doing so, it shows that a one-sided emphasis on colonization or gender hierarchy will misrepresent the feminist political issue and misinterpret the “comfort women’s” experiences. (shrink)

We explore an application of homological algebra by developing a cohomology theory for a class of Aronszajn trees. Properties of this class, called Todorevi trees, are examined. The system is compared to that for Hausdorff gaps introduced in the author's previous work and general results about both tree and gap systems are also proven.

We introduce the Hausdorff measure for definable sets in an o-minimal structure, and prove the Cauchy—Crofton and co-area formulae for the o-minimal Hausdorff measure. We also prove that every definable set can be partitioned into “basic rectifiable sets”, and that the Whitney arc property holds for basic rectifiable sets.

Nous présentons dans cet article la genèse du projet de l'Édition Hausdorff, ainsi que sa structure organisationnelle ; une discussion suit sur un des aspects centraux de l'œuvre de Hausdorff.

Nous présentons dans cet article la genèse du projet de l'Édition Hausdorff, ainsi que sa structure organisationnelle ; une discussion suit sur un des aspects centraux de l'œuvre de Hausdorff.

The topological arithmetical hierarchy is the effective version of the Borel hierarchy. Its class Δta 2 is just large enough to include several types of pointsets in Euclidean spaces ℝ k which are fundamental in computable analysis. As a crossbreed of Hausdorff's difference hierarchy in the Borel class ΔB 2 and Ershov's hierarchy in the class Δ0 2 of the arithmetical hierarchy, the Hausdorff-Ershov hierarchy introduced in this paper gives a powerful classification within Δta 2. This is based (...) on suitable characterizations of the sets from Δta 2 which are obtained in a close analogy to those of the ΔB 2 sets as well as those of the Δ0 2 sets. A helpful tool in dealing with resolvable sets is contributed by the technique of depth analysis. On this basis, the hierarchy properties, in particular the strict inclusions between classes of different levels, can be shown by direct constructions of witness sets. The Hausdorff-Ershov hierarchy runs properly over all constructive ordinals, in contrast to Ershov's hierarchy whose denotation-independent version collapses at level ω 2. Also, some new characterizations of concepts of decidability for pointsets in Euclidean spaces are presented. (shrink)

By analyzing how one obtains the Stone space of the reduced product of an indexed collection of Boolean algebras from the Stone spaces of those algebras, we derive a topological construction, the "reduced coproduct", which makes sense for indexed collections of arbitrary Tichonov spaces. When the filter in question is an ultrafilter, we show how the "ultracoproduct" can be obtained from the usual topological ultraproduct via a compactification process in the style of Wallman and Frink. We prove theorems dealing with (...) the topological structure of reduced coproducts (especially ultracoproducts) and show in addition how one may use this construction to gain information about the category of compact Hausdorff spaces. (shrink)

The article investigates the relations between Hausdorff and non-Hausdorff manifolds as objects of general relativity. We show that every non-Hausdorff manifold can be seen as a result of gluing together some Hausdorff manifolds. In the light of this result, we investigate a modal interpretation of a non-Hausdorff differential manifold, according to which it represents a bundle of alternative space-times, all of which are compatible with a given initial data set.

I agree with nearly everything Martin Davies says. He has written an elegant and highly informative analysis of recent philosophical debates about the mind–brain relation. I particularly enjoyed Davies’ discussion of B.A. Farrell, his precursor in the Oxford Wilde Readership (now Professorship) in Mental Philosophy. It is intriguing to see how closely Farrell anticipated many of the moves made by more recent ‘type-A’ physicalists who seek to show that, upon analysis, claims about conscious states turn out to be nothing more (...) than complex third-personal claims about internal and external behaviour. Davies is also exemplary in his even-handed treatment of those contemporary ‘type-B’ physicalists who have turned away from the neo-logical-behaviourism of Farrell and his ilk. Davies explains how type-B physicalists recognize distinctive subjective ‘phenomenal concepts’ for thinking about conscious states and so deny that phenomenal claims can be deduced a priori from behavioural or other third-personal claims. However, type-B physicalists do not accept that these subjective phenomenal concepts refer to any distinct non-material reality. In their view, phenomenal concepts and third-personal scientific concepts are simply another example of the familiar circumstance where we have two different ways of referring to a single reality. Since I am persuaded by pretty much all Davies says about these matters, I shall not comment substantially on the dialectical points he covers. Instead I want to raise two rather wider issues. The first is the set of ideas associated with the phrase ‘the explanatory gap’. Davies specifies that he is using this phrase in a specific technical sense. But the phrase has further connotations, and this can lead to a distorted appreciation of the philosophical issues. The second issue is the methodological implications of the philosophical debate. I shall argue that the philosophical issues addressed by Davies suggest that there are unexpected limitations on what empirical brain research can achieve.. (shrink)

The concept of legal gap is tackled from a number of logical perspectives and semantic methods. After presenting our own goal (Section 1), a first introduction into legal logic refers to Bobbio’s works and his formalization of legal statements (Sections 2 and 3). Then Woleński’s contribution to the area is taken into account through his reference to the distinction between two juridical systems (viz. Common Law vs Civil Law) and the notion of conditional norms (Section 4). The notion of reason (...) is also highlighted in the case of Raz’s legal logic, thereby leading to a future connection with von Wright’s logic of truth and an analogy made with an anti-realist reading of truth-values and norms (Section 5). Our personal contribution is introduced through a reflection on how logic should deal with the logical form of norms (Section 6), before entering a number of crucial definitions and distinctions for the concepts of norm, legal statement, and promulgation (Section 7). The final point is a proposed semantics for legal statements, which is both many-valued and gap-friendly (Section 8). A distinction between a number of requirements for permission and forbiddance leads to a set of non-classical juridical systems in which non-permission and forbiddance are not equivalent with each other any more; this does justice to Woleński’s former distinction between Common Law and Civil Law, also leading ultimately to a non-classical square of legal oppositions in which several legal operators may collapse into other ones (Section 9). (shrink)

I present solutions to several questions of Paul Bankston [2] by means of another version of the ultracoproduct construction, and explain the relation of ultracoproduct of compact Hausdorff spaces to other constructions combining topology, algebra and logic.

Mathematicians often intentionally leave gaps in their proofs. Based on interviews with mathematicians about their refereeing practices, this paper examines the character of intentional gaps in published proofs. We observe that mathematicians’ refereeing practices limit the number of certain intentional gaps in published proofs. The results provide some new perspectives on the traditional philosophical questions of the nature of proof and of what grounds mathematical knowledge.

We say that a Hausdorff locale is compactly generated if it is the colimit of the diagram of its compact sublocales connected by inclusions. We show that this is the case if and only if the natural map of its frame of opens into the second Lawson dual is an isomorphism. More generally, for any Hausdorff locale, the second dual of the frame of opens gives the frame of opens of the colimit. In order to arrive at this (...) conclusion, we generalize the Hofmann–Mislove–Johnstone theorem and some results regarding the patch construction for stably locally compact locales. (shrink)

A paradox, it is claimed, is a radical form of contradiction, one that produces gaps in meaning. In order to approach this idea, two senses of “separation” are distinguished: separation by something and separation by nothing. The latter does not refer to nothing in an ordinary sense, however, since in that sense what’s intended is actually less than nothing. Numerous ordinary nothings in philosophy as well as in other fields are surveyed so as to clarify the contrast. Then follows the (...) suggestion that philosophies which one would expect to have room for paradoxes actually tend either to exclude them altogether, or to dull them. There is a clear alternative, however, one that fully recognizes paradoxes and yet also strives to overcome them. (shrink)

We define a direct translation from finite rooted trees to finite natural functions which shows that the Worm Principle introduced by Lev Beklemishev is equivalent to a very slight variant of the well-known Kirby-Paris' Hydra Game. We further show that the elements in a reduction sequence of the Worm Principle determine a bad sequence in the well-quasi-ordering of finite sequences of natural numbers with respect to Friedman's gapembeddability. A characterization of gap-embeddability in terms of provability logic due to Lev Beklemishev (...) is also presented. (shrink)

We examine the sequences A that are low for dimension, i.e. those for which the effective dimension relative to A is the same as the unrelativized effective dimension. Lowness for dimension is a weakening of lowness for randomness, a central notion in effective randomness. By considering analogues of characterizations of lowness for randomness, we show that lowness for dimension can be characterized in several ways. It is equivalent to lowishness for randomness, namely, that every Martin-Löf random sequence has effective dimension (...) 1 relative to A, and lowishness for K, namely, that the limit of KA/K is 1. We show that there is a perfect [Formula: see text]-class of low for dimension sequences. Since there are only countably many low for random sequences, many more sequences are low for dimension. Finally, we prove that every low for dimension is jump-traceable in order nε, for any ε > 0. (shrink)

Since 1981 AIDS has illuminated, like a roving searchlight, a series of ethical questions, which extend far beyond the apparently narrow limits of one disease. It has revealed, one by one, human attitudes and behaviours that were previously unquestioned, or unobserved - based on unidentified but shaky pre-suppositions.This commentary offers two contrasting perspectives on the problems facing developing countries. In the first part, I comment on the preceding article, from the perspective of a clinician who has worked for many years (...) in Africa, and who witnessed the emergence of AIDS in the early 1980s and its medical, social, political and economic consequences.1 In the second, I comment on a conference in Lusaka in September 1999 that addressed many of the issues.Narrowing the gap?“Narrowing the gap” raises questions about truth or transparency in the activities of multinational companies, the ethics and economics of health care in global society and the distribution of responsibility for unwanted side effects of new treatments. The scale of HIV infection in developing countries draws attention to weaknesses or instability in theoretical and practical structures, which serve - or should serve - human wellbeing, whenever it is disturbed by disease.Truth and transparencyDuring the last 60 years, combined with improved standards of living, the products of pharmaceutical effort and ingenuity have resulted in large gains in life expectancy. These were most marked in rich northern societies, but even poor southern countries had seen it extended by 15 to 20 years, until recently.However, “the costs of drug development are not small”.2 A detailed study of costs of clinical trials found that the total cost of new drug development can be as high as $500 million per drug.3 Patent laws allow a temporary monopoly for the production and sale for a number of years. During this “protected” …. (shrink)

From the outset, critical social theory has sought to diagnose people’s participation in their own oppression, by revealing the roots of irrational and self-undermining choices in the complex interplay between human nature, social structures, and cultural beliefs. As part of this project, Ideologiekritik has aimed to expose faulty conceptions of this interplay, so that the objectively pathological character of what people are “freely” choosing could come more clearly into view. The challenge, however, has always been to find a way of (...) doing this without arrogantly assuming special access to what is good for people. And this danger of paternalism is one to which social theorists have all too often fallen prey. In this brief essay, I focus on contemporary instances of clearly self-defeating behavior in contexts of complex choices. I begin by discussing a recent attempt to diagnose and solve these failures of choices, namely the public policy recommendations of behavioral economist Richard Thaler and reform-minded legal theorist Cass Sunstein. Their influential “libertarian paternalist” approach is particularly interesting, both in what it includes (attention to the socially constructed nature of choice situations and the roots of the problems in human nature) and in what it leaves out (an understanding of the social construction of human nature and an adequate appreciation of the value of autonomy). After discussing it, I consider a broadly perfectionist alternative, to the effect that the problem lies in a failure to adequately appreciate the importance of developing autonomy. I then turn to sketching the outlines of a new approach, based on the concept of “autonomy gaps,” which approaches overly demanding policies in relational and action-theoretical terms. In the final section, I show how this provides the basis for an analysis both in terms of a critique of ideology and of social pathology. (shrink)

Using the Godel incompleteness result for leverage, Roger Penrose has argued that the mechanism for consciousness involves quantum gravitational phenomena, acting through microtubules in neurons. We show that this hypothesis is implausible. First the Godel result does not imply that human thought is in fact non-algorithmic. Second, whether or not non-algorithmic quantum gravitational phenomena actually exist, and if they did how that could conceivably implicate microtubules, and if microtubules were involved, how that could conceivably implicate consciousness, is entirely speculative. Third, (...) cytoplasmic ions such as calcium and sodium are almost certainly present in the microtubule pore, barring the quantum-mechanical effects Penrose envisages. Finally, physiological evidence indicates that consciousness does not directly depend on microtubule properties in any case, rendering doubtful any theory according to which consciousness is generated in the microtubules. (shrink)

You appeal for money to save the gorillas. Very laudable, no doubt. But it doesn't seem to have occurred to you that there are thousands of human children suffering on the very same continent of Africa. There'll be time enough to worry about gorillas when we've taken care of every last one of the kiddies. Let's get our priorities right, please!

This study interrogates the Gap campaign from a political economic perspective to determine whether it goes beyond merely touting the virtuous line of social responsibility. Critics cite the irony of capitalist-based solutions that perpetuate the inequities they are trying to address. Others suggest the aid generated is problematic in and of itself because it keeps Africa from becoming self-sufficient. This research contends the purpose of the Gap’s participation is genuine, going beyond window dressing and the surface level benefit of capitalistic (...) enterprise, and is in fact linked to a higher motive of propagating ethical trade which ultimately offers African beneficiaries a path to financial independence. However, because the message of the current Gap advertising campaign does convey a false sense of doing good, another advertising approach is warranted. (shrink)

This introductory research seeks to underline how a legal gap can be approached as an indispensable “negative moment” within the legal system. It has been supposed that this gap signals a lack in the legal system and, thereby, contributes to its improvement. The matter is much too complex to be managed in a limited space, so here it will not be given a complete explanation of the theory of legal gaps. The purpose will be rather to present a possible way (...) to tackle this issue by sketching how a legal gap could be interpreted not only as a Problem, but also as a Possibility given to a legal system to improve itself. (shrink)

Philosophers disagree about whether vagueness requires us to admit truth-value gaps, about whether there is a gap between the objects of which a given vague predicate is true and those of which it is false on an appropriately constructed sorites series for the predicate—a series involving small increments of change in a relevant respect between adjacent elements, but a large increment of change in that respect between the endpoints. There appears, however, to be widespread agreement that there is some sense (...) in which vague predicates are gappy which may be expressed neutrally by saying that on any appropriately constructed sorites series for a given vague predicate there will be a gap between the objects of which the predicate is definitely true and those of which it is definitely false. Taking as primitive the operator ‘it is definitely the case that’, abbreviated as ‘D’, we may stipulate that a predicate F is definitely true (or definitely false) of an object just in case ‘DF (a)’, where a is a name for the object, is true (or false) simpliciter.1 This yields the following conditional formulation of a ‘gap principle’: (DΦ(x) ∧ D¬Φ(y)) → ¬R(x, y). Here ‘Φ’ is to be replaced with a vague predicate, while ‘R’ is to stand for a sorites relation for that predicate: a relation that can be used to construct a sorites series for the predicate—such as the relation of being just one millimetre shorter than for the predicate ‘is tall’. Disagreements about the sense in which it is correct to say that vague predicates are gappy can then be recast as disagreements about how to understand the definitely operator. One might give it, for example, a pragmatic construal such as ‘it would not be misleading to assert that’; or an epistemic construal such as ‘it is known that’ or ‘it is knowable that’; or a semantic construal such as ‘it is true that’. (shrink)

1 IntroductionNumerous solutions have been proposed to the semantic paradoxes. Two that are frequently singled out and compared are the following. The first is that according to which paradoxical sentences are neither true nor false — as it is sometimes put, they are semantic gaps. The second is that according to which paradoxical sentences are both true and false — as it is sometimes put, they are semantic gluts. Calling the first of these a solution is, in fact, somewhat misleading: (...) it is rather like calling an opening gambit a game of chess. For the solution runs into severe problems almost immediately, and so can be only the first of a series of moves made to defend the original weak opening. Nonetheless, the symmetry involved in the gap and glut solutions is obvious enough to make the comparison a natural one. (shrink)

Danaher (2016) has argued that increasing robotization can lead to retribution gaps: Situation in which the normative fact that nobody can be justly held responsible for a harmful outcome stands in conflict with our retributivist moral dispositions. In this paper, we report a cross-cultural empirical study based on Sparrow’s (2007) famous example of an autonomous weapon system committing a war crime, which was conducted with participants from the US, Japan and Germany. We find that (i) people manifest a considerable willingness (...) to hold autonomous systems morally responsible, (ii) partially exculpate human agents when interacting with such systems, and that more generally (iii) the possibility of normative responsibility gaps is indeed at odds with people’s pronounced retributivist inclinations. We discuss what these results mean for potential implications of the retribution gap and other positions in the responsibility gap literature. (shrink)

In “God of the Gaps: A Neglected Reply to God’s Stone Problem”, Jc Beall and A. J. Cotnoir offer a gappy solution to the paradox of (unrestricted) omnipotence that is typified by the classic stone problem. Andrew Tedder and Guillermo Badia, however, have recently argued that this solution could not be extended to a more serious Curry-like version of the paradox. In this paper, we show that such a gappy solution does extend to it.