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Set theory is a branch of mathematics with a special subject matter, the infinite, but also a general framework for all modern mathematics, whose notions figure in every branch, pure and applied. This Element will offer a concise introduction, treating the origins of the subject, the basic notion of set, the axioms of set theory and immediate consequences, the set-theoretic reconstruction of mathematics, and the theory of the infinite, touching also on selected topics from higher set theory, controversial axioms and undecided questions, and philosophical issues raised by technical developments.
Modern logic emerged in the period from 1879 to the Second World War. In the post-war period what we know as classical first-order logic largely replaced traditional syllogistic logic in introductory textbooks, but the main development has been simply enormous growth: The publications of the Association for Symbolic Logic, the main professional organization for logicians, became ever thicker. While 1950 saw volume 15 of the Journal of Symbolic Logic, about 300 pages of articles and reviews and a six‑page member list, 2000 saw volume 65 of that journal, over 1,900 pages of articles, plus volume 6 of the Bulletin of Symbolic Logic, 570 pages of reviews and a sixty‑page member list. Of so large a field, the present survey will have to be ruthlessly selective, with no coverage of the history of informal or inductive logic, or of philosophy or historiography of logic, and slight coverage of applications. Remaining are five branches of pure, formal, deductive logic, four being the branches of mathematical logic recognized in Barwise 1977, first of many handbooks put out by academic publishers: set theory, model theory, recursion theory, proof theory. The fifth is philosophical logic, in one sense of that label, otherwise called non-classical logic, including extensions of and alternatives to textbook logic. For each branch, a brief review of pre‑war background will be followed by a few highlights of subsequent history. The references will be a mix of primary and secondary sources, landmark papers and survey articles.
The risk of living with dementia and, separately, cancer, increases exponentially with age. However, to date, there is a paucity of research investigating the experiences of people living with both these conditions. This study used semi-structured interviews to explore the decision-making and treatment options for people who live with both dementia and cancer. In total, ten people living with both dementia and cancer (aged 39–93 years) and nine family carers were interviewed. Braun and Clarke's approach to thematic analysis was used together with framework matrices to organise the data. In this article four sequential and descriptive themes are presented. ‘Reaching a diagnosis of cancer’ describes the vital role that family carers play in encouraging the person with dementia to seek an explanation for their presenting (undiagnosed cancer) symptoms to their general practitioner. ‘Adjusting to the cancer diagnosis when living with dementia’ outlines a variety of emotional and practical responses to receiving news of the diagnosis. ‘Weighing up the cancer treatment options’ highlights the different decisions and circumstances that family carers and people living with both dementia and cancer are faced with post-diagnosis. ‘Undergoing cancer treatment’ shares the finding that cancer treatment decision-making was not straightforward and that people living with both dementia and cancer would often forget about their cancer and what procedures they had been through.
Local isomorphism constitutes the regulatory, cognitive and normative profile of a host country. The regulatory institutional setting reflects the rules and legislation governing collective bargaining agreements, trade unions, local content laws and employment relationships. The cultural or cognitive dimension supports the widely held cultural and social knowledge and the normative profile acknowledges the influences of social groups and organizations on acceptable normative behaviour. Earlier literature lends support to the importance of institutional profile and its influence on the design and implementation of multinational enterprises’ human resource management policies and practices. This paper seeks to advance the concept of local isomorphism and highlight the implications of local isomorphism for future research on the transfer of multinational enterprises’ human resource management practices across and between subsidiaries.
Art centres fulfil many functions in remote regions as a source of Indigenous identity and creativity; as a link to the global art market; as centres for community engagement and participation; and as a source of social capital providing a range of services for local communities. They are dependent on funding from State and Federal authorities and they are identified as one of the success stories in remote community development. However, they face an uncertain future in the light of their multiple functions and their position as both a source of traditional identity and a link to an external art market. The article highlights the challenges faced by government in the evaluation of their effectiveness and contribution; and in particular discusses the suitability of the hybrid economy model as a representation of their functions.
Glaciers in Alaska are currently losing mass at a rate of ~−50 Gt a−1, one of the largest ice loss rates of any regional collection of mountain glaciers on Earth. Existing projections of Alaska's future sea-level contributions tend to be divergent and are not tied directly to regional observations. Here we develop a simple, regional observation-based projection of Alaska's future sea-level contribution. We compute a time series of recent Alaska glacier mass variability using monthly GRACE gravity fields from August 2002 through December 2014. We also construct a three-parameter model of Alaska glacier mass variability based on monthly ERA-Interim snowfall and temperature fields. When these three model parameters are fitted to the GRACE time series, the model explains 94% of the variance of the GRACE data. Using these parameter values, we then apply the model to simulated fields of monthly temperature and snowfall from the Community Earth System Model, to obtain predictions of mass variations through 2100. We conclude that mass loss rates may increase between −80 and −110 Gt a−1 by 2100, with a total sea-level rise contribution of 19 ± 4 mm during the 21st century.
Nations love to go to war, argue leftist pacifists in democratic Western societies. By this account, governments cannot resist the temptation to assert their selfish interests by violent means. Political leaders use the language of just war to hide their real intentions: imperialistic domination and economic profit. These pacifists charge that with the Iraq war, the disastrous consequences of such politics have become hideously clear again.
Saul Kripke has made fundamental contributions to a variety of areas of logic. The task of devising a model theory for modal logic was really a series of tasks, of devising a model theory for each of the various systems. Or rather, it was to devise a general type of model theory which, by varying certain conditions, could produce specific model theories for which the various systems would be sound and complete. Kripke's model theory for modal predicate logic is related to his model theory for modal sentential logic rather as the standard model theory for nonmodal predicate logic is related to the standard model theory for nonmodal sentential logic. There is more of a gap between Kripke's formal models for intuitionistic logic, and Brouwer's and Heyting's explanations of the intended meaning of intuitionistic negation and other logical operators.
Saul Kripke's "Outline of a Theory of Truth" has been the most influential publication on truth and paradox since Alfred Tarski's "The Concept of Truth in Formalized Languages". The liar paradox was introduced by Eubulides and much discussed by Chrysippus and others in ancient times, while it and related paradoxes, under the label insolubilia, were much discussed by Bradwardine and others in the Middle Ages. The kind of formal language Tarski considers has predicates and terms, from which may be formed atomic sentences, from which may be formed other sentences using negation, conjunction, disjunction, and universal and existential quantification. Different schemes of rules have been proposed for evaluating logical compounds some or all of whose logical components may lack truth value, with some schemes looking more plausible for some types of truth-value gap and others for others.
Over the past two decades, thousands of studies have demonstrated that Blacks receive lower quality medical care than Whites, independent of disease status, setting, insurance, and other clinically relevant factors. Despite this, there has been little progress towards eradicating these inequities. Almost a decade ago we proposed a conceptual model identifying mechanisms through which clinicians' behavior, cognition, and decision making might be influenced by implicit racial biases and explicit racial stereotypes, and thereby contribute to racial inequities in care. Empirical evidence has supported many of these hypothesized mechanisms, demonstrating that White medical care clinicians: (1) hold negative implicit racial biases and explicit racial stereotypes, (2) have implicit racial biases that persist independently of and in contrast to their explicit (conscious) racial attitudes, and (3) can be influenced by racial bias in their clinical decision making and behavior during encounters with Black patients. This paper applies evidence from several disciplines to further specify our original model and elaborate on the ways racism can interact with cognitive biases to affect clinicians' behavior and decisions and in turn, patient behavior and decisions. We then highlight avenues for intervention and make specific recommendations to medical care and grant-making organizations.
§1. I propose to address not so much Gödel's own philosophy of mathematics as the philosophical implications of his work, and especially of his incompleteness theorems. Now the phrase “philosophical implications of Gödel's theorem” suggests different things to different people. To professional logicians it may summon up thoughts of the impact of the incompleteness results on Hilbert's program. To the general public, if it calls up any thoughts at all, these are likely to be of the attempt by Lucas  and Penrose  to prove, if not the immortality of the soul, then at least the non-mechanical nature of mind. One goal of my present remarks will be simply to point out a significant connection between these two topics.
But let me consider each separately a bit first, starting with Hilbert. As is well known, though Brouwer's intuitionism was what provoked Hilbert's program, the real target of Hilbert's program was Kronecker's finitism, which had inspired objections to the Hilbert basis theorem early in Hilbert's career. (See the account in Reid .) But indeed Hilbert himself and his followers (and perhaps his opponents as well) did not initially perceive very clearly just how far Brouwer was willing go beyond anything that Kronecker would have accepted. Finitism being his target, Hilbert made it his aim to convince the finitist, for whom no mathematical statements more complex than universal generalizations whose every instance can be verified by computation are really meaningful, of the value of “meaningless” classical mathematics as an instrument for establishing such statements.