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.

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 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)

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.

As sophisticated artificial intelligence software becomes more ubiquitously and more intimately integrated within domains of traditionally human endeavor, many are raising questions over how responsibility (be it moral, legal, or causal) can be understood for an AI’s actions or influence on an outcome. So called “responsibility gaps” occur whenever there exists an apparent chasm in the ordinary attribution of moral blame or responsibility when an AI automates physical or cognitive labor otherwise performed by human beings and commits an error. Healthcare (...) administration is an industry ripe for responsibility gaps produced by these kinds of AI. The moral stakes of healthcare are often life and death, and the demand for reducing clinical uncertainty while standardizing care incentivizes the development and integration of AI diagnosticians and prognosticators. In this paper, we argue that (1) responsibility gaps are generated by “black box” healthcare AI, (2) the presence of responsibility gaps (if unaddressed) creates serious moral problems, (3) a suitable solution is for relevant stakeholders to voluntarily responsibilize the gaps, taking on some moral responsibility for things they are not, strictly speaking, blameworthy for, and (4) should this solution be taken, black box healthcare AI will be permissible in the provision of healthcare. (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.

Background According to some medical ethicists and professional guidelines, there is no ethical difference between withholding and withdrawing life-sustaining treatment. However, medical professionals do not always agree with this notion. Patients and their families may also not regard these decisions as equivalent. Perspectives on life-sustaining treatment potentially differ between cultures and countries. This study compares Japanese physicians’ and citizens’ attitudes toward hypothetical cases of withholding and withdrawing life-sustaining treatment.Methods Ten vignette cases were developed. A web-based questionnaire was administered to 457 (...) citizens and 284 physicians to determine whether they supported withholding or withdrawing treatment.Results In a case where a patient had an advance directive refusing ventilation, 77% of the physicians and 68% of the citizens chose to withhold treatment. In a case where there was an advance directive but the patient’s family requested treatment, 55% of the physicians and 45% of the citizens chose to withhold the ventilator. When a family requested withdrawal of the ventilator but patient wishes were unknown, 19% of the physicians and 48% of the citizens chose to withdraw the ventilator. However, when the patient had also indicated their wishes in writing, 49% of the physicians and 66% of the citizens chose to withdraw treatment. More physicians were prepared to withdraw dialysis (84%) and artificial nutrition (81%) at a patient’s request than mechanical ventilation (49%).Conclusions A significant proportion of Japanese physicians and citizens were reluctant to withhold or withdraw life-sustaining treatment, even in cases where the patient had indicated their wishes in writing. They were more likely to withhold than withdraw treatment, and more likely to withdraw artificial nutrition than mechanical ventilation. Japanese physicians gave significant weight to family views about treatment but were less likely to agree to withdraw treatment than citizens, indicating a potential source of conflict in clinical settings. (shrink)

Consider the following sentences: The neighbouring sentence is not true. The neighbouring sentence is not true. Call these the no-no sentences. Symmetry considerations dictate that the no-no sentences must both possess the same truth-value. Suppose they are both true. Given Tarski’s truth-schema—if a sentence S says that p then S is true iff p—and given what they say, they are both not true. Contradiction! Conclude: they are not both true. Suppose they are both false. Given Tarski’s falsity-schema—if a sentence S (...) says that p then S is false iff not-p—and given what they say, they are both true, and so not false. Contradiction! Conclude: they are not both false. Thus, despite their symmetry, the no-no sentences must differ in truth-value. Such is the no-no paradox.[1] Sorensen (2001, 2005a, 2005b) has argued that: (1) The no-no paradox is not a version of the liar but rather a cousin of the truth-teller paradox. (2) Even so, the no-no paradox is more paradoxical than the truth-teller. (3) The no-no and truth-teller sentences have groundless truthvalues—they are bivalent but give rise to “truthmaker gaps”. (4) It is metaphysically impossible to know these truth-values. (5) A truthmaker gap response to the no-no paradox provides reason to accept a version of epistemicism. In this paper it is shown that a truthmaker gap solution to the no-no and truth-teller paradoxes runs afoul of the dunno-dunno paradox, the strengthened no-no paradox, and the strengthened truth-teller paradox. In consequence, the no-no paradox is best seen as a form of the liar paradox. As such, it cannot provide a case for epistemicism. (shrink)

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)

Miranda Fricker’s insightful work on epistemic injustice discusses two forms of epistemic injustice—testimonial injustice and hermeneutical injustice. Hermeneutical injustice occurs when the victim lacks the interpretative resources to make sense of her experience, and this lacuna can be traced down to a structural injustice. In this paper, I provide one model of how to fill the conceptual gap in hermeneutical injustice. First, I argue that the victims possess conceptual resources to make sense of their experiences, namely phenomenal concepts. Second, I (...) show how one might work the way up in a two-step process from a subjective, phenomenal concept to a novel, public concept. Finally, I discuss the conditions that have to be met for this process to be successful. The resulting model shows a way how the victims might alleviate hermeneutical injustice by developing novel concepts, given that the dominant group does not care about their predicament. (shrink)

In some of the Aristotelian accounts, Parmenides’ thesis is construed in opposition to the philosophy of nature; on the other hand, he is also depicted, in a different context, as a cosmologist, to whom the Stagirite (and a long tradition afterwards, ending with Simplicius) ascribes a theory of becoming and its principles. In this paper, I exhibit and analyse the relevant passages from Physics I 1-3, Metaphysics I 3 and 5 and On generation and corruption I 3, providing an interpretation (...) that aims to solve the apparent paradox, making sense of the information we can gather from Aristotle’s and Simplicius’ testimonies. Eventually, I propose a construal of the Two Ways of fr. 2 with an emphasis on the predicative reading of einai, which could hint at the Parmenidean approach to cosmology that runs in parallel with the argument on Being. (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)

In "Beyond the Myth of the Myth: A Kantian Theory of Non-Conceptual Content", Robert Hanna argues for a very strong kind of non-conceptualism, and claims that this kind of non-conceptualism originally has been developed by Kant. But according to "Kant's Non-Conceptualism, Rogue Objects and the Gap in the B Deduction", Kant's non-conceptualism poses a serious problem for his argument for the objective validity of the categories, namely the problem that there is a gap in the B Deduction. This gap is (...) that the B Deduction goes through only if conceptualism is true, but Kant is a non-conceptualist. In this paper I argue, contrary to what Hanna claims, that there is not a gap in the B Deduction. (shrink)

For the last 40 years or so the is/ought gap, the fact/value distinction and the naturalistic fallacy have figured prominently in ethical debates. This longevity, however, has had an adverse side effect. So familiar have they become that they—and their respective rationales—have tended to become blurred. It is the purpose of this paper to explain why they should be kept distinct.

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.

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)

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)

We prove that the maximal order type of the wqo of linear orders of finite Hausdorff rank under embeddability is φ2, the first fixed point of the ε-function. We then show that Fraïssé’s conjecture restricted to linear orders of finite Hausdorff rank is provable in +“φ2 is well-ordered” and, over , implies +“φ2 is well-ordered”.

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.

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)