We consider properties of sets in an intuitionistic setting corresponding to large cardinals in classical set theory. Adding such ‘large set axioms’ to intuitionistic ZF set theory does not violate well-know metamathematical properties of intuitionistic systems. Moreover, we consider statements in constructive analysis equivalent to the consistency of such ‘large set axioms’.

Stochastic forecasts in complex environments can benefit from combining the estimates of large groups of forecasters (“judges”). But aggregating multiple opinions faces several challenges. First, human judges are notoriously incoherent when their forecasts involve logically complex events. Second, individual judges may have specialized knowledge, so different judges may produce forecasts for different events. Third, the credibility of individual judges might vary, and one would like to pay greater attention to more trustworthy forecasts. These considerations limit the value of simple (...) aggregation methods like linear averaging. In this paper, a new algorithm is proposed for combining probabilistic assessments from a large pool of judges. Two measures of a judge’s likely credibility are introduced and used in the algorithm to determine the judge’s weight in aggregation. The algorithm was tested on a data set of nearly half a million probability estimates of events related to the 2008 U.S. presidential election (∼ 16000 judges). (shrink)

We extend some results of Adam Kolany to show that large sets of satisfiable sentences generally contain equally large subsets of mutually consistent sentences. In particular, this is always true for sets of uncountable cofinality, and remains true for sets of denumerable cofinality if we put appropriate bounding conditions on the sentences. The results apply to both the propositional and the predicate calculus. To obtain these results, we use delta sets for regular cardinals, and, (...) for singular cardinals, a generalization of delta sets. All of our results are theorems in ZFC. (shrink)

This volume is the first of its kind to explore the notion of untranslatability from a wide variety of interdisciplinary perspectives and its implications within the broader context of translation studies. Featuring contributions from both leading authorities and emerging scholars in the field, the book looks to go beyond traditional comparisons of target texts and their sources to more rigorously investigate the myriad ways in which the term untranslatability is both conceptualized and applied. The first half of the volume focuses (...) on untranslatability as a theoretical or philosophical construct, both to ground and extend the term's conceptual remit, while the second half is composed of case studies in which the term is applied and contextualized in a diverse set of literary text types and genres, including poetry, philosophical works, song lyrics, memoir, and scripture. A final chapter examines untranslatability in the real world and the challenges it brings in practical contexts. Extending the conversation in this burgeoning contemporary debate, this volume is key reading for graduate students and researchers in translation studies, comparative literature, gender studies, and philosophy of language. The editors are grateful to the University of East Anglia Faculty of Arts and Humanities, who supported the book with a publication grant. (shrink)

The Bayes Blind Spot of a Bayesian Agent is the set of probability measures on a Boolean algebra that are absolutely continuous with respect to the background probability measure of a Bayesian Agent on the algebra and which the Bayesian Agent cannot learn by conditionalizing no matter what evidence he has about the elements in the Boolean algebra. It is shown that if the Boolean algebra is finite, then the Bayes Blind Spot is a very large set: it has (...) the same cardinality as the set of all probability measures ; it has the same measure as the measure of the set of all probability measures ; and is a ``fat'' set in topological sense in the set of all probability measures taken with its natural topology. (shrink)

This long-overdue translation brings to the English-speaking world the work that set the tone for the post-structuralist reading of Nietzsche.

Is it ethical for a school to accept sponsorship from business, and if so under what conditions? Indeed, given the poor provision of many UK local schools for their pupils is it ethical to refuse business sponsorship? Where does responsibility lie? “To attract and persuade is not an appropriate behaviour in dealing with inexperienced parties.” The author completed her MBA at London Business School in July 1997 and is currently setting up a new family business, The Organic Food Company Ltd.

Continuing the earlier research from [T. Bigorajska, H. Kotlarski, Partitioning α-large sets: some lower bounds, Trans. Amer. Math. Soc. 358 4981–5001] we show that for the price of multiplying the number of parts by 3 we may construct partitions all of whose homogeneous sets are much smaller than in [T. Bigorajska, H. Kotlarski, Partitioning α-large sets: some lower bounds, Trans. Amer. Math. Soc. 358 4981–5001]. We also show that the Paris–Harrington independent statement remains unprovable if (...) the number of colors is restricted to 2, in fact, the statement is unprovable in IΣb. Other results concern some lower bounds for partitions of pairs. (shrink)

It was questions about points on the real line that initiated the study of set theory. Points paved the way to point sets and these to ever more abstract sets. And there was more: Reflection on structural properties of point sets not only initiated the study of ordinary sets; it also supplied blueprints for defining extra-ordinary, “large” sets, transcending those provided by standard set theory. In return, the existence of such large sets (...) turned out critical to settling open conjectures about point sets.How to explain such action at a distance between the very large and the rather small? Rather than having an air of magic, could these results rest on deep structural similarities between the two superficially distant species of sets?In this essay I dissect one group of such two-way results. Their linchpin is the notion of measure.§1. Vitali's impossibility result. Our starting point is a problem in measure theory regarding the notion of “Lebesgue measure.” Before presenting the problem, I would like to review the notion of Lebesgue measure. Rather than listing its main properties, I would like to show how Lebesgue measure is born out of an attempt to generalize the notion of the length of an interval to arbitrary sets of reals. One tries to approximate arbitrary sets of reals by intervals, in the hope that the lengths of the intervals will induce a measure on these sets. (shrink)

The independence phenomenon in set theory, while pervasive, can be partially addressed through the use of large cardinal axioms. One idea sometimes alluded to is that maximality considerations speak in favour of large cardinal axioms consistent with ZFC, since it appears to be `possible' to continue the hierarchy far enough to generate the relevant transfinite number. In this paper, we argue against this idea based on a priority of subset formation under the iterative conception. In particular, we argue (...) that there are several conceptions of maximality that justify the consistency but falsity of large cardinal axioms. We argue that the arguments we provide are illuminating for the debate concerning the justification of new axioms in iteratively-founded set theory. (shrink)

Labels provide a quick and effective solution to obtain people interesting content from large-scale social network information. The current interest label extraction method based on the subgraph stream proves the feasibility of the subgraph stream for user label extraction. However, it is extremely time-consuming for constructing subgraphs. As an effective mathematical method to deal with fuzzy and uncertain information, rough set-based representations for subgraph stream construction are capable of capturing the uncertainties of the social network. Therefore, we propose an (...) effective approach called RS_UNITE_SS, which is suitable for large-scale social network user interest label extraction. Specifically, we first propose the subgraph division algorithm to construct a subgraph stream by incorporating a rough set. Then, the algorithm for user real-time interest label extraction based on upper approximation is proposed by using sequentially characteristics of the subgraph. Empirically, we evaluate RS_UNITE_SS over real-world datasets, and experimental results demonstrate that our proposed approach is more computationally efficient than existing methods while achieving higher precision value and MRR value. (shrink)

The notion of “ramifiability” , usually applied to cardinals, can be extended to directed sets and is put in relation here with familiar “large cardinal” properties.

We define a potentialist system of ‐structures, i.e., a collection of possible worlds in the language of connected by a binary accessibility relation, achieving a potentialist account of the full background set‐theoretic universe V. The definition involves Berkeley cardinals, the strongest known large cardinal axioms, inconsistent with the Axiom of Choice. In fact, as background theory we assume just. It turns out that the propositional modal assertions which are valid at every world of our system are exactly those in (...) the modal theory. Moreover, we characterize the worlds satisfying the potentialist maximality principle, and thus the modal theory, both for assertions in the language of and for assertions in the full potentialist language. (shrink)

“Small” large cardinal notions in the language of ZFC are those large cardinal notions that are consistent with V = L. Besides their original formulation in classical set theory, we have a variety of analogue notions in systems of admissible set theory, admissible recursion theory, constructive set theory, constructive type theory, explicit mathematics and recursive ordinal notations (as used in proof theory). On the face of it, it is surprising that such distinctively set-theoretical notions have analogues in such (...) disaparate and relatively constructive contexts. There must be an underlying reason why that is possible (and, incidentally, why “large” large cardinal notions have not led to comparable analogues). My long term aim is to develop a common language in which such notions can be expressed and can be interpreted both in their original classical form and in their analogue form in each of these special constructive and semi-constructive cases. This is a program in progress. What is done here, to begin with, is to show how that can be done to a considerable extent in the settings of classical and admissible set theory (and thence, admissible recursion theory). The approach taken here is to expand the language of set theory to allow us to talk about (possibly partial) operations applicable both to sets and to operations and to formulate the large cardinal notions in question in terms of operational closure conditions; at the same time only minimal existence axioms are posited for sets. The resulting system, called Operational Set Theory, is a partial adaptation to the set-theoretical framework of the explicit mathematics framework Feferman (1975). The.. (shrink)

We investigate measure and category in the projective hierarchie in the presence of large cardinals. Assuming a measurable larger than $n$ Woodin cardinals we construct a model where every $\Delta ^1_{n+4}$ -set is measurable, but some $\Delta ^1_{n+4}$ -set does not have Baire property. Moreover, from the same assumption plus a precipitous ideal on $\omega _1$ we show how a model can be forced where every $\Sigma ^1_{n+4}-$ set is measurable and has Baire property.

We prove two sets of results concerning computational complexity classes. First, we propose a new variation of the random oracle hypothesis, originally posed by Bennett and Gill after they showed that relative to a randomly chosen oracle, with probability 1. Their original hypothesis was quickly disproven in several ways, most famously in 1992 with the result that, in spite of the classes being shown unequal with probability 1. Here we propose a variation of what it means to be “ (...) class='Hi'>large” using the Ellentuck topology. In this new context, we demonstrate that the set of oracles separating and is not small, and obtain similar results for the separation of from along with the separation of from. We also show that the set of oracles equating with is large in this new sense. We demonstrate that this version of the hypothesis provides a sufficient condition for unrelativized relationships, at least in the cases considered here. Second, we examine the descriptive complexity of the classes of oracles providing the separations for these various classes, and determine their exact placement in the Borel hierarchy. (shrink)

One result of this note is about the nonconstructivity of countably infinite lotteries: even if we impose very weak conditions on the assignment of probabilities to subsets of natural numbers we cannot prove the existence of such assignments constructively, i.e., without something such as the axiom of choice. This is a corollary to a more general theorem about large-small filters, a concept that extends the concept of free ultrafilters. The main theorem is that proving the existence of large-small (...) filters requires a nonconstructive axiom like AC. (shrink)

The independence phenomenon in set theory, while pervasive, can be partially addressed through the use of large cardinal axioms. A commonly assumed idea is that large cardinal axioms are species of maximality principles. In this paper, I argue that whether or not large cardinal axioms count as maximality principles depends on prior commitments concerning the richness of the subset forming operation. In particular I argue that there is a conception of maximality through absoluteness, on which large (...) cardinal axioms are restrictive. I argue, however, that large cardinals are still important axioms of set theory and can play many of their usual foundational roles. (shrink)

Contemporary analytic philosophy is dominated by metaphilosophical naturalism, the view that philosophy ought to be continuous with science. This naturalistic turn is for a significant part due to the work of W. V. Quine. Yet, the development and the reception of Quine’s naturalism have never been systematically studied. In this paper, I examine Quine’s evolving naturalism as well as the reception of his views. Scrutinizing a large set of unpublished notes, correspondence, drafts, papers, and lectures as well as published (...) responses to Quine’s work, I show how both internal tensions and external criticisms forced him to continuously develop, rebrand, and refine his metaphilosophy before he explicitly decided to label his view ‘naturalism’ in the late 1960s. (shrink)

Mathematicians justify axioms of set theory “intrinsically”, by reference to the universe of sets of their intuition, and “extrinsically”, for example, by considerations of simplicity or usefullness for mathematical practice. Here we apply the same kind of justifications to Nonstandard Analysis and argue for acceptance of BNST+ . BNST+ has nontrivial consequences for standard set theory; for example, it implies existence of inner models with measurable cardinals. We also consider how to practice Nonstandard Analysis in BNST+, and compare it (...) with other existing nonstandard set theories. (shrink)

We study some large cardinals in terms of reflection, establishing new connections between the model-theoretic and the set-theoretic approaches.

Six-month-old infants discriminate between large sets of objects on the basis of numerosity when other extraneous variables are controlled, provided that the sets to be discriminated differ by a large ratio (8 vs. 16 but not 8 vs. 12). The capacities to represent approximate numerosity found in adult animals and humans evidently develop in human infants prior to language and symbolic counting.

Large language models (LLMs) represent the currently most relevant incarnation of artificial intelligence with respect to the future fate of democratic governance. Considering their potential, this paper seeks to answer a pressing question: Could LLMs outperform humans as expert advisors to democratic assemblies? While bearing the promise of enhanced expertise availability and accessibility, they also present challenges of hallucinations, misalignment, or value imposition. Weighing LLMs’ benefits and drawbacks compared to their human counterparts, I argue for their careful integration to (...) augment democracy’s ability to address complex policy issues. The paper posits that time-tested democratic procedures like deliberation and aggregation by voting provide safeguards effective against both human and machine advisor imperfections. Additional protective measures include custom LLM training for the advisory role, boosting representatives’ competencies in query formulation, or implementation of adversarial proceedings in which LLM advisors could debate each other and provide dissenting opinions. These could further mitigate the risks that LLMs present in advisory roles and empower human decision-makers toward increased autonomy and quality of their collective choices. My conceptual exploration offers a roadmap for the co-evolution of AI and democratic institutions, setting the stage for an empirical research agenda to finetune the implementation specifics. (shrink)

We show that if κ is a weakly compact cardinal then the embeddability relation on trees of size κ is invariantly universal. This means that for every analytic quasi-order R on the generalized Cantor space View the MathML source there is an Lκ+κ-sentence φ such that the embeddability relation on its models of size κ, which are all trees, is Borel bi-reducible to R. In particular, this implies that the relation of embeddability on trees of size κ is complete for (...) analytic quasi-orders on View the MathML source. These facts generalize analogous results for κ=ω obtained in Louveau and Rosendal [17] and Friedman and Motto Ros [6], and it also partially extends a result from Baumgartner [3] concerning the structure of the embeddability relation on linear orders of size κ. (shrink)

We propose a new approach for tracing value change. Value change may lead to a mismatch between current value priorities in society and the values for which technologies were designed in the past, such as energy technologies based on fossil fuels, which were developed when sustainability was not considered a very important value. Better anticipating value change is essential to avoid a lack of social acceptance and moral acceptability of technologies. While value change can be studied historically and qualitatively, we (...) propose a more quantitative approach that uses large text corpora. It uses probabilistic topic models, which allow us to trace values that are latent. We demonstrate the approach for five types of value change in technology. Our approach is useful for testing hypotheses about value change, such as verifying whether value change has occurred and identifying patterns of value change. The approach can be used to trace value change for various technologies and text corpora, including scientific articles, newspaper articles, and policy documents. (shrink)

The following problem is studied: How large and how small can the Löwenheim and Hanf numbers of unbounded logics be in relation to the most common large cardinals? The main result is that the Löwenheim number of the logic with the Härtig-quantifier can be consistently put in between any two of the first weakly inaccessible, the first weakly Mahlo, the first weakly compact, the first Ramsey, the first measurable and the first supercompact cardinals.

ArgumentThis paper argues that the economics discipline is highly concentrated, which may inhibit scientific innovation and change in the future. The argument is based on an empirical investigation of six dimensions of concentration in economics between 1956 and 2016 using a large-scale data set. The results show that North America accounts for nearly half of all articles and three quarters of all citations. Twenty institutions reap a share of 42 percent of citations, five journals a share of 28.5 percent, (...) and 100 authors a share of 15.5 percent. A total of 2.8 percent of citations may be attributed to heterodox schools of thought. Also top articles are concentrated along these dimensions. Overall, concentration has strongly increased over the last six decades. (shrink)

BACKGROUND: There is a growing interest in aggregating more biomedical and patient data into large health data sets for research and public benefits. However, collecting and processing patient data raises new ethical issues regarding patient's rights, social justice and trust in public institutions. The aim of this empirical study is to gain an in-depth understanding of the awareness of possible ethical risks and corresponding obligations among those who are involved in projects using patient data, i.e. healthcare professionals, regulators (...) and policy makers. METHODS: We used a qualitative design to examine Swiss healthcare stakeholders' experiences and perceptions of ethical challenges with regard to patient data in real-life settings where clinical registries are sponsored, created and/or used. A semi-structured interview was carried out with 22 participants (11 physicians, 7 policy-makers, 4 ethical committee members) between July 2014 and January 2015. The interviews were audio-recorded, transcribed, coded and analysed using a thematic method derived from Grounded Theory. RESULTS: All interviewees were concerned as a matter of priority with the needs of legal and operating norms for the collection and use of data, whereas less interest was shown in issues regarding patient agency, the need for reciprocity, and shared governance in the management and use of clinical registries' patient data. This observed asymmetry highlights a possible tension between public and research interests on the one hand, and the recognition of patients' rights and citizens' involvement on the other. CONCLUSIONS: The advocation of further health-related data sharing on the grounds of research and public interest, without due regard for the perspective of patients and donors, could run the risk of fostering distrust towards healthcare data collections. Ultimately, this could diminish the expected social benefits. However, rather than setting patient rights against public interest, new ethical approaches could strengthen both concurrently. On a normative level, this study thus provides material from which to develop further ethical reflection towards a more cooperative approach involving patients and citizens in the governance of their health-related big data. (shrink)

“Big Data” in health and medicine in the 21st century differs from “Big Data” used in health and medicine in the 1700s and 1800s. However, the old data sets share one key component: large numbers. The term “Big Data” is not synonymous with large numbers. Large numbers are a key component of Big Data in health and medicine, both for understanding the full range of how a disease presents in a human for diagnosis, and for understanding (...) if one treatment of a disease is better than another treatment or better than just leaving the patient on his or her own without therapy. In this paper, we examine the first considerations of Big Data in medicine in Paris in the early 1800s when urologic surgeon Jean Civiale collected the first large numbers. Civiale collected the large numbers to defend the efficacy of his urologic instrument, the lithotrite, and the surgical procedure he developed, lithotrity, for the removal of bladder stones compared with earlier, more invasive surgical approaches. We examine how large numbers were adjudicated in social decision-making in the Académie des sciences, Paris, when a dispute arose among French urologic surgeons about the importance of large numbers in surgical science. After Civiale’s successful defense of his instrument and procedure in Paris, we examine how his approach to Big Data impacted data collection by George Buchanan in his use of the procedure at the Royal Hospital Infirmary in Glasgow. (shrink)

There is a growing interest in aggregating more biomedical and patient data into large health data sets for research and public benefits. However, collecting and processing patient data raises new ethical issues regarding patient’s rights, social justice and trust in public institutions. The aim of this empirical study is to gain an in-depth understanding of the awareness of possible ethical risks and corresponding obligations among those who are involved in projects using patient data, i.e. healthcare professionals, regulators and (...) policy makers. We used a qualitative design to examine Swiss healthcare stakeholders’ experiences and perceptions of ethical challenges with regard to patient data in real-life settings where clinical registries are sponsored, created and/or used. A semi-structured interview was carried out with 22 participants between July 2014 and January 2015. The interviews were audio-recorded, transcribed, coded and analysed using a thematic method derived from Grounded Theory. All interviewees were concerned as a matter of priority with the needs of legal and operating norms for the collection and use of data, whereas less interest was shown in issues regarding patient agency, the need for reciprocity, and shared governance in the management and use of clinical registries’ patient data. This observed asymmetry highlights a possible tension between public and research interests on the one hand, and the recognition of patients’ rights and citizens’ involvement on the other. The advocation of further health-related data sharing on the grounds of research and public interest, without due regard for the perspective of patients and donors, could run the risk of fostering distrust towards healthcare data collections. Ultimately, this could diminish the expected social benefits. However, rather than setting patient rights against public interest, new ethical approaches could strengthen both concurrently. On a normative level, this study thus provides material from which to develop further ethical reflection towards a more cooperative approach involving patients and citizens in the governance of their health-related big data. (shrink)

We show that many large cardinal notions up to measurability can be characterized through the existence of certain filters for small models of set theory. This correspondence will allow us to obtain a canonical way in which to assign ideals to many large cardinal notions. This assignment coincides with classical large cardinal ideals whenever such ideals had been defined before. Moreover, in many important cases, relations between these ideals reflect the ordering of the corresponding large cardinal (...) properties both under direct implication and consistency strength. (shrink)

The large-structure tools of cohomology including toposes and derived categories stay close to arithmetic in practice, yet published foundations for them go beyond ZFC in logical strength. We reduce the gap by founding all the theorems of Grothendieck’s SGA, plus derived categories, at the level of Finite-Order Arithmetic, far below ZFC. This is the weakest possible foundation for the large-structure tools because one elementary topos of sets with infinity is already this strong.

The interaction between large cardinals, determinacy of two-person perfect information games, and inner model theory has been a singularly powerful driving force in modern set theory during the last three decades. For the outsider the intellectual excitement is often tempered by the somewhat daunting technicalities, and the seeming length of study needed to understand the flow of ideas. The purpose of this article is to try and give a short, albeit rather rough, guide to the broad lines of development.

The independence phenomenon in set theory, while pervasive, can be partially addressed through the use of large cardinal axioms. A commonly assumed idea is that large cardinal axioms are species of maximality principles. In this paper I question this claim. I show that there is a kind of maximality (namely absoluteness) on which large cardinal axioms come out as restrictive relative to a formal notion of restrictiveness. Within this framework, I argue that large cardinal axioms can (...) still play many of their usual foundational roles. (shrink)