We study conditions for a topological space to be metrizable, properties of metrizable spaces, and the role the axiom of choice plays in these matters.

This paper aims to argue for the thesis that it is not a priori morally justified that the first phase of space colonisation is based on sexual reproduction. We ground this position on the argument that, at least in the first colonisation settlements, those born in space may not have a good chance of having a good life. This problem does not depend on the fact that life on another planet would have to deal with issues such as (...) solar radiation or with the decrease or entire absence of the force of gravity. These issues could plausibly be addressed given that the planets or settlements we will feasibly colonise could be completely transformed through geoengineering processes. Likewise, the ability of humans to live in space could be enhanced through genetic modification interventions. Even if, however, the problems concerning survival in space were solved, we think that, at least in the first period of colonisation of space or other planets, giving birth to children in space could be a morally irresponsible choice since we argue, the life we ​​could give them might not be good enough. We contend that this is the case since when we decide to have a baby. We argue that it is not morally right to be content that our children have a minimally sufficient life worth living; before we give birth to children in space, we should make sure we can give them a reasonable chance of having a good life. This principle applies both on Earth - at least where you can choose - and for space travel. (shrink)

One central objection to the maximin payoff criterion is that it focuses on the state that yields the lowest payoffs regardless of how low these are. We allow different states to have different sets of possible outcomes and show that the original axioms of Milnor (1954) continue to characterize the maximin payoff criterion, provided that the sets of payoffs achievable across states overlap. If instead payoffs in some states are always lower than in all others then ignoring the “bad” states (...) is no longer inconsistent with these axioms. Similar dependence on overlap of outcome spaces across states holds for the minimax regret and maximin joy criteria. (shrink)

Transnational higher education (TNHE) has been characterised as a crude form of market-driven internationalisation, often targeting immobile student populations in countries with high demand for international academic degrees. In response to recent scholarship on the role of higher education internationalisation in facilitating and producing diasporic networks, this study examines its inverse: how TNHE services existing diasporic communities in situ by mobilising institutions across borders rather than student bodies. It specifically examines these dynamics within the United Arab Emirates (UAE), simultaneously host (...) to one of the largest concentrations of TNHE globally and a five million-member diasporic community of long-term, yet impermanent residents from South Asia. Drawing on interviews with South Asian students in undergraduate degree programmes at three British international branch campuses (IBCs), it explores how students perceive their IBCs as strategic sources of valuable degree capitals for enhancing employment opportunities in the UAE and securing against precarity and uncertain futures as perpetually impermanent residents. The research implicates the role of TNHE in diaspora policymaking, asking how IBCs function as an extension of a limited social contract between a diasporic community and its ‘host’ state. (shrink)

This is a continuation of [2]. We study the Tychonoff Compactness Theorem for various definitions of compactness and for various types of spaces . We also study well ordered Tychonoff products and the effect that the multiple choice axiom has on such products.

Under the axiom of choice, every first countable space is a Fréchet-Urysohn space. Although, in its absence even ℝ may fail to be a sequential space.Our goal in this paper is to discuss under which set-theoretic conditions some topological classes, such as the first countable spaces, the metric spaces, or the subspaces of ℝ, are classes of Fréchet-Urysohn or sequential spaces.In this context, it is seen that there are metric spaces which are not sequential spaces. This (...) fact raises the question of knowing if the completion of a metric space exists and it is unique. The answer depends on the definition of completion.Among other results it is shown that: every first countable space is a sequential space if and only if the axiom of countable choice holds, the sequential closure is idempotent in ℝ if and only if the axiom of countable choice holds for families of subsets of ℝ, and every metric space has a unique equation image-completion. (shrink)

We show that the countable axiom of choice CAC strictly implies the statements “Lindelöf metric spaces are second countable” “Lindelöf metric spaces are separable”. We also show that CAC is equivalent to the statement: “If is a Lindelöf topological space with respect to the base ℬ, then is Lindelöf”.

The axiom of multiple choice implies that metric spaces are paracompact but the reverse implication cannot be proved in set theory without the axiom of choice.

We find properties of topological spaces which are not shared by disjoint unions in the absence of some form of the Axiom of Choice.

It is shown that for compact metric spaces the following statements are pairwise equivalent: “X is Loeb”, “X is separable”, “X has a we ordered dense subset”, “X is second countable”, and “X has a dense set G = ∪{Gn : n ∈ ω}, ∣Gn∣ < ω, with limn→∞ diam = 0”. Further, it is shown that the statement: “Compact metric spaces are weakly Loeb” is not provable in ZF0 , the Zermelo-Fraenkel set theory without the axiom of regularity, and (...) that the countable axiom of choice for families of finite sets CACfin does not imply the statement “Compact metric spaces are separable”. (shrink)

We show that the assertion that every vector space is a projective module implies the axiom of multiple choice and that the reverse implication does not hold in set theory weakened to permit the existence of atoms.

In this paper, we describe a semiotic programme that proposes an alternative conceptual framework to understand the moral positionalities that people have in socio-material space. The study amalgamates moral character and signs and signification through a discussion of moral choice and value acts in an existential semiotic framework, as laid out by Eero Tarasti. The programme was triggered by a lived experience in a non-place, yielding the concept of semiotic space awareness – i.e., the value acts that (...) work as signs of moral character in people’s socio-material space. It is the moral positioning of a subject in the socio-material and semiotic space in relation to other subjects. People’s positionalities primarily take place in the socio-material space, and the dimensions we discuss focus on how value acts are produced and interpreted in space and place. Our aim is to take the approach used in, e.g., proxemics to a universal metalevel in terms of its key, undivided semiotic ingredients irrespective of cultural variation. We then extrapolate by discussing how these value acts trigger potential tensions and conflicts that can be approached using semiotics as a foundation for analysis. More specifically, the moral character that people portray in their value acts is theorized and applied as an explanatory tool to understand the semiosis and its repercussions in Dasein. We also introduce affordance as an additional dimension in the interplay and modalizations between moral character and moral choice to understand value acts and semiotic space awareness in subjects’ Dasein and Umwelt. (shrink)

We study the Tychonoff Compactness Theorem for several different definitions of a compact space.

We show that the axiom of choice AC is equivalent to the Vector Space Kinna-Wagner Principle, i.e., the assertion: “For every family [MATHEMATICAL SCRIPT CAPITAL V]= {Vi : i ∈ k} of non trivial vector spaces there is a family ℱ = {Fi : i ∈ k} such that for each i ∈ k, Fiis a non empty independent subset of Vi”. We also show that the statement “every vector space over ℚ has a basis” implies that (...) every infinite well ordered set of pairs has an infinite subset with a choice set, a fact which is known not to be a consequence of the axiom of multiple choice MC. (shrink)

We study theorems from Functional Analysis with regard to their relationship with various weak choice principles and prove several results about them: “Every infinite‐dimensional Banach space has a well‐orderable Hamel basis” is equivalent to ; “ can be well‐ordered” implies “no infinite‐dimensional Banach space has a Hamel basis of cardinality ”, thus the latter statement is true in every Fraenkel‐Mostowski model of ; “No infinite‐dimensional Banach space has a Hamel basis of cardinality ” is not provable (...) in ; “No infinite‐dimensional Banach space has a well‐orderable Hamel basis of cardinality ” is provable in ; (the Axiom of Choice for denumerable families of non‐empty finite sets) is equivalent to “no infinite‐dimensional Banach space has a Hamel basis which can be written as a denumerable union of finite sets”; Mazur's Lemma (“If X is an infinite‐dimensional Banach space, Y is a finite‐dimensional vector subspace of X, and, then there is a unit vector such that for all and all scalars α”) is provable in ; “A real normed vector space X is finite‐dimensional if and only if its closed unit ball is compact” is provable in ; (Principle of Dependent Choices) + “ can be well‐ordered” does not imply the Hahn‐Banach Theorem () in ; and “no infinite‐dimensional Banach space has a Hamel basis of cardinality ” are independent from each other in ; “No infinite‐dimensional Banach space can be written as a denumerable union of finite‐dimensional subspaces” lies in strength between (the Axiom of Countable Choice) and ; implies “No infinite‐dimensional Banach space can be written as a denumerable union of closed proper subspaces” which in turn implies ; “Every infinite‐dimensional Banach space has a denumerable linearly independent subset” is a theorem of, but not a theorem of ; and “Every infinite‐dimensional Banach space has a linearly independent subset of cardinality ” implies “every Dedekind‐finite set is finite”. (shrink)

We show that the Axiom of Countable Choice is necessary and sufficient to prove that the existence of a Borel measure on a pseudometric space such that the measure of open balls is positive and finite implies separability of the space. In this way a negative answer to an open problem formulated in Górka (Am Math Mon 128:84–86, 2020) is given. Moreover, we study existence of maximal $$\delta $$ δ -separated sets in metric and pseudometric spaces from (...) the point of view the Axiom of Choice and its weaker forms. (shrink)

We consider the problem of choosing the location of a public facility either (a) on a tree network or (b) in a Euclidean space. (a) (1996) characterize the class of target rules on a tree network by Pareto efficiency and population-monotonicity. Using Vohra's (1999) characterization of rules that satisfy Pareto efficiency and replacement-domination, we give a short proof of the previous characterization and show that it also holds on the domain of symmetric preferences. (b) The result obtained for model (...) (a) proves to be crucial for the analysis of the problem of choosing the location of a public facility in a Euclidean space. Our main result is the characterization of the class of coordinatewise target rules by unanimity, strategy-proofness, and either replacement-domination or population-monotonicity. (shrink)

As space travel and intentions to colonise other planets are becoming the norm in public debate and scholarship, we must also confront the technical and survival challenges that emerge from these hostile environments. This paper aims to evaluate the various arguments proposed to meet the challenges of human space travel and extraterrestrial planetary colonisation. In particular, two primary solutions have been present in the literature as the most straightforward solutions to the rigours of extraterrestrial survival and flourishing: (1) (...) geoengineering, where the environment is modified to become hospitable to its inhabitants, and (2) human (bio)enhancement where the genetic heritage of humans is modified to make them more resilient to the difficulties they may encounter as well as to permit them to thrive in non-terrestrial environments. Both positions have strong arguments supporting them but also severe philosophical and practical drawbacks when exposed to different circumstances. This paper aims to show that a principled stance where one position is accepted wholesale necessarily comes at the opportunity cost of the other where the other might be better suited, practically and morally. This paper concludes that case-by-case evaluations of the solutions to space travel and extraterrestrial colonisation are necessary to ensure moral congruency and the survival and flourishing of astronauts now and into the future. (shrink)

In the realm of metric spaces the role of choice principles is investigated.

Volume 19, Issue 6, June 2019, Page 76-78.

We study within the framework of Zermelo-Fraenkel set theory ZF the role that the axiom of choice plays in the theory of Lindelöf metric spaces. We show that in ZF the weak choice principles: Every Lindelöf metric space is separable and Every Lindelöf metric space is second countable are equivalent. We also prove that a Lindelöf metric space is hereditarily separable iff it is hereditarily Lindelöf iff it hold as well the axiom of choice (...) restricted to countable sets and to topologies of Lindelöf metric spaces as the countable union theorem restricted to Lindelöf metric spaces. (shrink)

We find some characterizations of the Axiom of Choice in terms of certain families of open sets in T1 spaces.

In set theory without the Axiom of Choice ( $\mathsf {AC}$ ), we investigate the open problem of the deductive strength of statements which concern the existence of almost disjoint and maximal almost disjoint (MAD) families of infinite-dimensional subspaces of a given infinite-dimensional vector space, as well as the extension of almost disjoint families in infinite-dimensional vector spaces to MAD families.

In this note we show that the Axiom of Countable Choice is equivalent to two statements from the theory of pseudometric spaces: the first of them is a well-known characterization of uniform continuity for functions between metric spaces, and the second declares that sequentially compact pseudometric spaces are \—meaning that all real valued, continuous functions defined on these spaces are necessarily uniformly continuous.

This book is a ricochet against mainstream physics. It sprang out of the idea that outer symmetries of space-time are the same as inner symmetries of matter. In other words, the standard model of physics is a space-time group. This book is about structures and phenomena that are lying hidden underneath the surface of space-time. It begins with a few biographic events, Majoranas legacy, the philosophy of Gerhard Frey and some related anthropological topics which have to do (...) with high energy physics. It continues with a reconstruction of the theorem by Banach and Tarski in Minkowski space. We are making acquaintance with the standard model as a property of space-time. So we are challenging quite unusual actions such as penetration of quarks by a probe. We propose to apply a penetrating function D. Then, measure and basis are connected with the axiom of choice. (shrink)

The standard topological representation of a Boolean algebra via the clopen sets of a Stone space requires a nonconstructive choice principle, equivalent to the Boolean Prime Ideal Theorem. In this article, we describe a choice-free topological representation of Boolean algebras. This representation uses a subclass of the spectral spaces that Stone used in his representation of distributive lattices via compact open sets. It also takes advantage of Tarski’s observation that the regular open sets of any topological (...) class='Hi'>space form a Boolean algebra. We prove without choice principles that any Boolean algebra arises from a special spectral space X via the compact regular open sets of X; these sets may also be described as those that are both compact open in X and regular open in the upset topology of the specialization order of X, allowing one to apply to an arbitrary Boolean algebra simple reasoning about regular opens of a separative poset. Our representation is therefore a mix of Stone and Tarski, with the two connected by Vietoris: the relevant spectral spaces also arise as the hyperspace of nonempty closed sets of a Stone space endowed with the upper Vietoris topology. This connection makes clear the relation between our point-set topological approach to choice-free Stone duality, which may be called the hyperspace approach, and a point-free approach to choice-free Stone duality using Stone locales. Unlike Stone’s representation of Boolean algebras via Stone spaces, our choice-free topological representation of Boolean algebras does not show that every Boolean algebra can be represented as a field of sets; but like Stone’s representation, it provides the benefit of a topological perspective on Boolean algebras, only now without choice. In addition to representation, we establish a choice-free dual equivalence between the category of Boolean algebras with Boolean homomorphisms and a subcategory of the category of spectral spaces with spectral maps. We show how this duality can be used to prove some basic facts about Boolean algebras. (shrink)

Branching space-time is a simple blend of relativity and indeterminism. Postulates and definitions rigorously describe the causal order relation between possible point events. The key postulate is a version of everything has a causal origin; key defined terms include history and choice point. Some elementary but helpful facts are proved. Application is made to the status of causal contemporaries of indeterministic events, to how splitting of histories happens, to indeterminism without choice, and to Einstein-Podolsky-Rosen distant correlations.

Nondeterministic models of collective choice posit convergence among the outcomes of simple-majority decisions. The object of this research is to estimate the extent of convergence of majority choice under different procedural conditions. The paper reports results from a computer simulation of simple-majority decision making by committees. Simulation experiments generate distributions of majority-adopted proposals in two-dimensional space. These represent nondeterministic outcomes of majority choice by committees. The proposal distributions provide data for a quantitative evaluation of committee-choice (...) procedures in respect to outcome convergence. Experiments were run under general conditions, and under conditions that restrict committee choice to several game-theoretic solution sets. The findings are that, compared to distributions of voter ideal points, majority-adopted proposals confined to the solution sets demonstrate different degrees of convergence. Second, endogenous agenda formation is a more important obstacle to convergence than the inherent instability of simple-majority rule. Third, if members maximize preferences in respect to agenda formation, a committee choice that approximates the central tendency of the distribution of voter preferences is unlikely. The conclusion is that the most effective way to increase the convergence of majority choice is to restrict the role of individual preferences in agenda formation: identification of proposals to be voted up or down by a committee. (shrink)

Mirko Garasic (2021) argued that space travel and, by extension, the colonization of other planets could morally justify using technologies and interventions capable of profoundly modifying the characteristics of astronauts and future Martian generations. According to Garasic, however, the fact that space interventions such as human (bio)enhancement or reproductive technologies such as artificial wombs may be morally justified does not mean that they are morally acceptable technologies to be used on Earth as well. Garasic’s thesis is that we (...) should resist the temptation to establish or reinforce a continuity between the ethical standards that apply to space travel and those that apply on Earth because what applies in space (or on other planets) does not necessarily apply to our planet. Garasic argues that in space, genetic enhancement interventions are morally acceptable, as they are essential for survival. On Earth, however, we can survive even without any form of (bio)enhancement. In a previous article (2022), we presented several arguments against Garasic’s thesis regarding the exceptional morality of what is morally acceptable in space. In this article, we examine and reply to Garasic (2022), showing that he fails to defend his thesis from the objections we previously advanced. Garasic’s mistakes are: (1) assuming nature (and consequently what is “natural”) as a normative point of reference which allows establishing what is moral and what is not moral; (2) defending the choices concerning reproduction as an insufficiently demanding and therefore unsatisfactory moral principle (according to Garasic we should not bring into the world people who can have a quality of life or a condition of well-being superior to mere survival); and finally (3) of not explaining clearly what it means to defend a principle of moral exceptionalism for space travel (Garasic does not explain whether this position applies to any behavior, for which what on Earth is a virtue, in space is a vice or is not a general principle, but he makes this case only for enhancement and reproductive technologies). Through our analysis, we not only refute Garasic’s position but also show that any attempt to address space travel issues as “special” moral issues faces several difficulties and, ultimately, is doomed to failure. (shrink)

Next SectionThe principle of respect for autonomy has shaped much of the bioethics' discourse over the last 50 years, and is now most commonly used in the meaning of respecting autonomous choice. This is probably related to the influential concept of informed consent, which originated in research ethics and was soon also applied to the field of clinical medicine. But while available choices in medical research are well defined, this is rarely the case in healthcare. Consideration of ordinary medical (...) practice reveals that the focus on patient choice does not properly grasp the moral aspects involved in healthcare. Medical decisions are often portrayed as if doctors and patients in confidence confront specific decisions about examinations or treatment, yet the reality often involves many different participants, with decisions being made over time and space. Indeed, most of the decisions are never even presented to patients, as it would be unethical to suggest something that is not medically justifiable. The options patients do confront are somewhat arbitrarily constructed within the narrow framework of both what is deemed to be medically appropriate and how the healthcare system is organised practically. While the autonomy discourse has proven valuable, a failure to distinguish between the fields of medical research and clinical medicine has generated a focus on patient choice that does not reflect what is really at stake in healthcare settings. This is alarming, because the current discourse misrepresents medical practice in a way that actually contributes to bioethical self-delusion. (shrink)

The concepts of choice, negation, and infinity are considered jointly. The link is the quantity of information interpreted as the quantity of choices measured in units of elementary choice: a bit is an elementary choice between two equally probable alternatives. “Negation” supposes a choice between it and confirmation. Thus quantity of information can be also interpreted as quantity of negations. The disjunctive choice between confirmation and negation as to infinity can be chosen or not in (...) turn: This corresponds to set-theory or intuitionist approach to the foundation of mathematics and to Peano or Heyting arithmetic. Quantum mechanics can be reformulated in terms of information introducing the concept and quantity of quantum information. A qubit can be equivalently interpreted as that generalization of “bit” where the choice is among an infinite set or series of alternatives. The complex Hilbert space can be represented as both series of qubits and value of quantum information. The complex Hilbert space is that generalization of Peano arithmetic where any natural number is substituted by a qubit. “Negation”, “choice”, and “infinity” can be inherently linked to each other both in the foundation of mathematics and quantum mechanics by the meditation of “information” and “quantum information”. (shrink)

This article is devoted to the problem of ontological foundations of three-dimensional Euclidean geometry. Starting from Bertrand Russell’s intuitions concerning the sensual world we try to show that it is possible to build a foundation for pure geometry by means of the so called regions of space. It is not our intention to present mathematically developed theory, but rather demonstrate basic assumptions, tools and techniques that are used in construction of systems of point-free geometry and topology by means of (...) mereology and Whitehead-like connection structures. We list and briefly analyze axioms for mereological structures, as well as those for connection structures. We argue that mereology is a good tool to model so called spatial relations. We also try to justify our choice of axioms for connection relation. Finally, we briefly discuss two theories: Grzegorczyk’s point-free topology and Tarski’s geometry of solids. (shrink)

The emergence of personal assistants in the form of smart speakers has begun to significantly alter people’s everyday experiences with technology. The rate at which household Intelligent Personal Assistants such as Amazon’s Echo and Google Home emerged in household spaces has been rapid. They have begun to move human–computer interaction from text-based to voice-activated input, offering a multiplicity of features through speech. The supporting infrastructure connects with artificial intelligence and the internet of things, allowing digital interfaces with domestic appliances, lighting (...) systems, thermostats, media devices and more. Yet this also constitutes a significant new production of situated and sensitive data. This study focuses on how users negotiate and make choices about household Intelligent Personal Assistant use in connection with their data. This study is based on empirical research in Europe with early adopters in Germany and potential users in the Netherlands. This examination of users’ early stage technology acceptance considerations highlights particular practices and choices of users to either preserve their privacy or determine what is acceptable use for their data. Drawing on a simplified version of Unified Theory of Acceptance and Use of Technology 2, a quantitative model for technology acceptance, we demonstrate how acceptance of a household Intelligent Personal Assistants does not imply access to all household data, how users see usefulness in relation to a proliferation of devices, and note the recognition by users regarding the efforts needed for full use and acceptance. The study highlights the complexity of data production at a household level and how these devices produce myopic views of users for platforms. (shrink)

It is well known that the validity of Choice Principles is problematic in non-standard Set Theories which do not abide by the Limitation of Size Principle. In this paper we discuss the consistency of various Choice Principles with respect to the Generalized Positive Comprehension Principle . The Principle GPC allows to take as sets those classes which can be specified by Generalized Positive Formulae, e.g. the universe. In particular we give a complete characterization of which choice principles (...) hold in Hyperuniverses. Hyperuniverses are structures which arose independently in Non-well-founded Set Theory and in Mathematical Semantics of Concurrent Programming Languages and are hitherto the only existing models of GPC. Hyperuniverses are naturally endowed with a κ-compact uniform κ-topology and are uniformly isomorphic to their exponential space, i.e. the space of their closed subsets endowed with the Exponential Uniformity. (shrink)

Impossibility theorems for 2-person and majority continuous games on the unit circle are presented. The emphasis is on simple methods, albeit generating new results, to offer insights into the sophisticated results of theorists in topological social choice.

We systematically study the completion of choice problems in the Weihrauch lattice. Choice problems play a pivotal rôle in Weihrauch complexity. For one, they can be used as landmarks that characterize important equivalences classes in the Weihrauch lattice. On the other hand, choice problems also characterize several natural classes of computable problems, such as finite mind change computable problems, non-deterministically computable problems, Las Vegas computable problems and effectively Borel measurable functions. The closure operator of completion generates the (...) concept of total Weihrauch reducibility, which is a variant of Weihrauch reducibility with total realizers. Logically speaking, the completion of a problem is a version of the problem that is independent of its premise. Hence, studying the completion of choice problems allows us to study simultaneously choice problems in the total Weihrauch lattice, as well as the question which choice problems can be made independent of their premises in the usual Weihrauch lattice. The outcome shows that many important choice problems that are related to compact spaces are complete, whereas choice problems for unbounded spaces or closed sets of positive measure are typically not complete. (shrink)

We study whether it is possible to generalise Seidenfeld et al.’s representation result for coherent choice functions in terms of sets of probability/utility pairs when we let go of Archimedeanity. We show that the convexity property is necessary but not sufficient for a choice function to be an infimum of a class of lexicographic ones. For the special case of two-dimensional option spaces, we determine the necessary and sufficient conditions by weakening the Archimedean axiom.

The general framework of this paper is a reformulation of Hilbert’s program using the theory of locales, also known as formal or point-free topology [P.T. Johnstone, Stone Spaces, in: Cambridge Studies in Advanced Mathematics, vol. 3, 1982; Th. Coquand, G. Sambin, J. Smith, S. Valentini, Inductively generated formal topologies, Ann. Pure Appl. Logic 124 71–106; G. Sambin, Intuitionistic formal spaces–a first communication, in: D. Skordev , Mathematical Logic and its Applications, Plenum, New York, 1987, pp. 187–204]. Formal topology presents a (...) topological space, not as a set of points, but as a logical theory which describes the lattice of open sets. The application to Hilbert’s program is then the following. Hilbert’s ideal objects are represented by points of such a formal space. There are general methods to “eliminate” the use of points, close to the notion of forcing and to the “elimination of choice sequences” in intuitionist mathematics, which correspond to Hilbert’s required elimination of ideal objects. This paper illustrates further this general program on the notion of valuations. They were introduced by Dedekind and Weber [R. Dedekind, H. Weber, Theorie des algebraischen Funktionen einer Veränderlichen, J. de Crelle t. XCII 181–290] to give a rigorous presentation of Riemann surfaces. It can be argued that it is one of the first example in mathematics of point-free representation of spaces [N. Bourbaki, Eléments de Mathématique. Algèbre commutative, Hermann, Paris, 1965, Chapitre 7]. It is thus of historical and conceptual interest to be able to represent this notion in formal topology. (shrink)

In many decision scenarios, one has to choose an element from a set S given some reference point e. For the case where S is a subset of a Euclidean space, we axiomatize the choice method that selects the point in S that is closest to e. © 1999 Elsevier Science B.V. All rights reserved.

We work in set theory without the Axiom of Choice ZF. We prove that the Principle of Dependent Choices (DC) implies that the closed unit ball of a uniformly convex Banach space is weakly compact and, in particular, that the closed unit ball of a Hilbert space is weakly compact. These statements are not provable in ZF and the latter statement does not imply DC. Furthermore, DC does not imply that the closed unit ball of a reflexive (...) space is weakly compact. (shrink)

In this introduction, we explain our choice to approach the topic of space from a cosmological perspective, that is, by studying the conceptions of space that were implicitly or explicitly entailed by ancient, medieval and early modern representations of the cosmos, and the role that imagination played in those conceptions. We compare our approach with those of Alexandre Koyré and Edward Grant, and we present the two important issues this book intends to shed light on, namely the (...) continuity and discontinuity between ancient, medieval, and early modern conceptions of space and the cosmos; and the role that metaphysical, cosmological, and theological considerations played in the elaboration of new theories of space in the course of history. This chapter also presents the main, recurring themes of this book: the relation between place and space; the notion of imaginary spaces; the role played by thought experiments in discussions concerning the nature of space and the structure of the cosmos; the impact of the condemnation of 1277 on subsequent theories of space; and the relation between God’s immensity and the infinity of space. (shrink)

Solution concepts for game play implicitly assume what information is relevant for choosing moves. Maximin and Eliminating Dominated Strategies use payoff order; mixed strategies and Harsanyi's risk dominance use payoff values. "Cooperative" strategies use previous choices, ignoring payoffs altogether. We first define the 12 symmetric 2 × 2 games as a continuous payoff space then use this space to evaluate strategies based on different types of information. Strategic success is shown to be sensitive to actual payoffs and seldom (...) consistent for all games of an ordinally defined category such as Chicken. A new hybrid strategy using both payoff and choice information is introduced and shown to produce efficient outcomes in all symmetric 2×2 games when employed by both players. It does well in evolutionary tournaments against strategies of other types, especially in the difficult distribution games including Battle-of-the-Sexes. (shrink)

Can we design a perfect democratic decision procedure? Condorcet famously observed that majority rule, our paradigmatic democratic procedure, has some desirable properties, but sometimes produces inconsistent outcomes. Revisiting Condorcet’s insights in light of recent work on the aggregation of judgments, I show that there is a conflict between three initially plausible requirements of democracy: “robustness to pluralism”, “basic majoritarianism”, and “collective rationality”. For all but the simplest collective decision problems, no decision procedure meets these three requirements at once; at most (...) two can be met together. This “democratic trilemma” raises the question of which requirement to give up. Since different answers correspond to different views about what matters most in a democracy, the trilemma suggests a map of the “logical space” in which different conceptions of democracy are located. It also sharpens our thinking about other impossibility problems of social choice and how to avoid them, by capturing a core structure many of these problems have in common. More broadly, it raises the idea of “cartography of logical space” in relation to contested political concepts. (shrink)

The objective of this essay is to provide the beginning of a principled classification of some of the ways space is intelligently used. Studies of planning have typically focused on the temporal ordering of action, leaving as unaddressed questions of where to lay down instruments, ingredients, work-in-progress, and the like. But, in having a body, we are spatially located creatures: we must always be facing some direction, have only certain objects in view, be within reach of certain others. How (...) we manage the spatial arrangement of items around us is not an afterthought: it is an integral part of the way we think, plan, and behave. The proposed classification has three main categories: spatial arrangements that simplify choice; spatial arrangements that simplify perception; and spatial dynamics that simplify internal computation. The data for such a classification is drawn from videos of cooking, assembly and packing, everyday observations in supermarkets, workshops and playrooms, and experimental studies of subjects playing Tetris, the computer game. This study, therefore, focuses on interactive processes in the medium and short term: on how agents set up their workplace for particular tasks, and how they continuously manage that workplace. (shrink)