We give an overview of the role of equicontinuity of sequences of real-valued functions on [0,1] and related notions in classical mathematics, intuitionistic mathematics, Bishop’s constructive mathematics, and Russian recursive mathematics. We then study the logical strength of theorems concerning these notions within the programme of Constructive Reverse Mathematics. It appears that many of these theorems, like a version of Ascoli’s Lemma, are equivalent to fan-theoretic principles.
We prove constructively that, in order to derive the uniform continuity theorem for pointwise continuous mappings from a compact metric space into a metric space, it is necessary and sufficient to prove any of a number of equivalent conditions, such as that every pointwise continuous mapping of [0, 1] into R is bounded. The proofs are analytic, making no use of, for example, fan-theoretic ideas.
Working within Bishop-style constructive mathematics, we examine some of the consequences of the anti-Specker property, known to be equivalent to a version of Brouwer's fan theorem. The work is a contribution to constructive reverse mathematics.
This paper is dedicated to Prof. Dr. Günter Asser, whose work in founding this journal and maintaining it over many difficult years has been a major contribution to the activities of the mathematical logic community.At first sight it appears highly unlikely that Urysohn's Lemma has any significant constructive content. However, working in the context of an apartness space and using functions whose values are a generalisation of the reals, rather than real numbers, enables us to produce a significant constructive version (...) of that lemma. (shrink)
Working within the framework of Bishop's constructive mathematics, we will show that it is possible to define compactness in a more general setting than that of uniform spaces. It is also shown that it is not possible to do this in a topological space.
It is shown that a principle, which can be seen as a constructivised version of sequential compactness, is equivalent to a form of Brouwer’s fan theorem. The complexity of the latter depends on the geometry of the spaces involved in the former.
Varieties of the Fan Theorem have recently been developed in reverse constructive mathematics, corresponding to different continuity principles. They form a natural implicational hierarchy. Some of the implications have been shown to be strict, others strict in a weak context, and yet others not at all, using disparate techniques. Here we present a family of related Kripke models which separates all of the as yet identified fan theorems.
This book draws together the latest work from scholars around the world using subjective well-being data to understand and compare well-being across countries and cultures. Starting from many different vantage points, the authors reached a consensus that many measures of subjective well-being, ranging from life evaluations through emotional states, based on memories and current evaluations, merit broader collection and analysis. Using data from the Gallup World Poll, the World Values Survey, and other internationally comparable surveys, the authors document wide divergences (...) among countries in all measures of subjective well-being, The international differences are greater for life evaluations than for emotions. Despite the well-documented differences in the ways in which subjective evaluations change through time and across cultures, the bulk of the very large international differences in life evaluations are due to differences in life circumstances rather than differences in the way these differences are evaluated. (shrink)
For valid informed consent, it is crucial that patients or research participants fully understand all that their consent entails. Testing and revising informed consent documents with the assistance of their addressees can improve their understandability. In this study we aimed at further developing a method for testing and improving informed consent documents with regard to readability and test-readers’ understanding and reactions. We tested, revised, and retested template informed consent documents for biobank research by means of 11 focus group interviews with (...) members from the documents’ target population. For the analysis of focus group excerpts we used qualitative content analysis. Revisions were made based on focus group feedback in an iterative process. Focus group participants gave substantial feedback on the original and on the revised version of the tested documents. Revisions included adding and clarifying explanations, including an info-box summarizing the main points of the text and an illustrative graphic. Our results indicate positive effects on the tested and revised informed consent documents in regard to general readability and test-readers’ understanding and reactions. Participatory methods for improving informed consent should be more often applied and further evaluated for both, medical interventions and clinical research. Particular conceptual and methodological challenges need to be addressed in the future. (shrink)
It is easy to prove in ZF− that a relation R satisfies the maximal condition if and only if its transitive hull R* does; equivalently: R is well-founded if and only if R* is. We will show in the following that, if the maximal condition is replaced by the chain condition, as is often the case in Algebra, the resulting statement is not provable in ZF− anymore . More precisely, we will prove that this statement is equivalent in ZF− to (...) the countable axiom of choice ACω. Moreover, applying this result we will prove that the axiom of dependent choices, restricted to partial orders as used in Algebra, already implies the general form for arbitrary relations as formulated first by Teichmüller and, independently, some time later by Bernays and Tarski. (shrink)
The purpose of this Article is to consider a novel framework for institutional shareholders’ activism in the United States. This new activism framework would be aimed at improving, at minimal costs, the performance of the portfolio companies in which institutional shareholders invest. The Article begins by laying out this new activism framework and then compares the proposed framework with the prevalent mode of activism through hedge funds. The Article concludes with a discussion of certain implementation challenges, and calls for future (...) research into the proposed activism framework. (shrink)
We will prove in Zermelo-Fraenkel set theory without axiom of choice that the transitive hull R* of a relation R is not much “bigger” than R itself. As a measure for the size of a relation we introduce the notion of κ+-narrowness using surjective Hartogs numbers rather than the usul injective Hartogs values. The main theorem of this paper states that the transitive hull of a κ+-narrow relation is κ+-narrow. As an immediate corollary we obtain that, for every infinite cardinal (...) κ, the class HCκ of all κ-hereditary sets is a set with von Neumann rank ϱ ≤ κ+. Moreover, ϱ = κ+ if and only if κ is singular, otherwise ϱ = κ. The statements of the corollary are well known in the presence of the axiom of choice . To prove them without AC - as carried through here - is, however, much harder. A special case of the corollary has been treated independently by T. JECH. (shrink)
PET studies using classical conditioning paradigms are reported. It is emphasized that PET studies show and not in learning paradigms. The importance of dissociating motor performance and learning deficits in human lesions studies is demonstrated in two exemplary studies. The different role of the cerebellum in adaptation of postural reflexes and learning of complex voluntary arm movements is discussed, [THACH].
Throughout his works Kierkegaard repeatedly claims that the modern age has subverted authentic Christianity. While interpretations of Kierkegaard’s critique of the modern approach to religion abound, they generally agree that the critique is based on various conceptual distinctions regarding the limits of human reason, the epistemological differences between subjective and objective truth, or the nature of religious faith. Very little attention, however, has been paid to the prominent role authority plays in the critique or to the fact that according to (...) Kierkegaard it is disobedience to authority, and not any conceptual confusion, that is the primary fault of the flawed modern approach. This paper argues that Kierkegaard’s most fundamental indictment of the modern approach to religion is its disobedience to the authority of the Christian command. Given this, the paper then examines what, for Kierkegaard, is the basis on which a person should choose to submit to an alleged religious authority. (shrink)
We will prove that some so-called union theorems are equivalent in ZF0 to statements about the transitive closure of relations. The special case of “bounded” union theorems dealing with κ-hereditary sets yields equivalents to statements about the transitive closure of κ-narrow relations. The instance κ = ω1 yields an equivalent to Howard-Rubin's Form 172 of every hereditarily countable set x is countable). In particular, the countable union theorem and, a fortiori, the axiom of countable choice imply Form 172.
It would be unkind but not inaccurate to say that most experimental philosophy is just psychology with worse methods and better theories. In Experimental Ethics: Towards an Empirical Moral Philosophy, Christoph Luetge, Hannes Rusch, and Matthias Uhl set out to make this comparison less invidious and more flattering. Their book has 16 chapters, organized into five sections and bookended by the editors’ own introduction and prospectus. Contributors hail from four countries (Germany, USA, Spain, and the United Kingdom) and five (...) disciplines (philosophy, psychology, cognitive science, economics, and sociology). While the chapters are of mixed quality and originality, there are several fine contributions to the field. These especially include Stephan Wolf and Alexander Lenger’s sophisticated attempt to operationalize the Rawlsian notion of a veil of ignorance, Nina Strohminger et al.’s survey of the methods available to experimental ethicists for studying implicit morality, Fernando Aguiar et al.’s exploration of the possibility of operationalizing reflective equilibrium in the lab, and Nikil Mukerji’s careful defusing of three debunking arguments about the reliability of philosophical intuitions. (shrink)
This article will discuss the ongoing development of a Marxist theory of international relations. Examining the work of Hannes Lacher and that of the contributors toMarxism and World Politicsreveals an overarching concern amongst this group of scholars to engage with the central concerns of the discipline of International Relations – the nature of the state, anarchy, and war. Their analysis provides an excellent starting point for the development of a Marxist approach to international relations.