We construct an existentially undecidable complete discretely valued field of mixed characteristic with existentially decidable residue field and decidable algebraic part, answering a question by Anscombe–Fehm in a strong way. Along the way, we construct an existentially decidable field of positive characteristic with an existentially undecidable finite extension, modifying a construction due to Kesavan Thanagopal.

We first show that the projection image of a discrete definable set is again discrete for an arbitrary definably complete locally o‐minimal structure. This fact together with the results in a previous paper implies a tame dimension theory and a decomposition theorem into good‐shaped definable subsets called quasi‐special submanifolds. Using this fact, we investigate definably complete locally o‐minimal expansions of ordered groups when the restriction of multiplication to an arbitrary bounded open box is definable. Similarly to o‐minimal expansions (...) of ordered fields, Łojasiewicz's inequality, Tietze's extension theorem and affiness of pseudo‐definable spaces hold true for such structures under the extra assumption that the domains of definition and the pseudo‐definable spaces are definably compact. Here, a pseudo‐definable space is a topological space having finite definable atlases. We also demonstrate Michael's selection theorem for definable set‐valued functions with definably compact domains of definition. (shrink)

We prove that Hensel minimal expansions of finitely ramified Henselian valued fields admit spherically complete immediate elementary extensions. More precisely, the version of Hensel minimality we use is 0‐hmix‐minimality (which, in equi‐characteristic 0, amounts to 0‐h‐minimality).

This article is a logical continuation of the Henri Lombardi and Franz-Viktor Kuhlmann article [9]. We address some classical points of the theory of valued fields with an elementary and constructive point of view. We deal with Krull valuations, and not simply discrete valuations. First of all, we show how to construct the Henselization of a valued field; we restrict to fields in which one has at one's disposal algorithmic tools to test the nullity or the valuation (...) ring membership. It is therefore a work that differs as much in spirit as in field of application from that of Mines, Richman and Bridges , who address the framework of Heyting fields and discrete valuation. We show then in a constructive way a batch of classical results in Henselian fields, notably factorization criteria and Krasner's Lemma. We conclude by a construction of the inertia field of a valued field. (shrink)

The purpose of this paper is to study an analogue of Hilbert's seventeenth problem for functions over a valued field which are integral definite on some definable set; that is, that map the given set into the valuation ring. We use model theory to exhibit a uniform method, on various theories of valued fields, for deriving an algebraic characterization of such functions. As part of this method we refine the concept of a function being integral at a (...) point, and make it dependent on the relevant class of valued fields. We apply our framework to algebraically closed valued fields, model complete theories of difference and differential valued fields, and real closed valued fields. (shrink)

We extend the characterization of extremal valued fields given in [2] to the missing case of valued fields of mixed characteristic with perfect residue field. This leads to a complete characterization of the tame valued fields that are extremal. The key to the proof is a model theoretic result about tame valued fields in mixed characteristic. Further, we prove that in an extremal valued field of finitep-degree, the images of all additive polynomials (...) have the optimal approximation property. This fact can be used to improve the axiom system that is suggested in [8] for the elementary theory of Laurent series fields over finite fields. Finally we give examples that demonstrate the problems we are facing when we try to characterize the extremal valued fields with imperfect residue fields. To this end, we describe several ways of constructing extremal valued fields; in particular, we show that in every ℵ1saturated valued field the valuation is a composition of extremal valuations of rank 1. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Journal of the History of Philosophy 40.1 (2002) 134-135 [Access article in PDF] Book Review Merleau-Ponty's Last Vision: A Proposal for the Completion of 'The Visible and the Invisible Douglas Low. Merleau-Ponty's Last Vision: A Proposal for the Completion of 'The Visible and the Invisible.' Evanston, IL: Northwestern University Press, 2000. Pp. xv + 124. Cloth, $75.00. Paper, $19.95. Low sets himself an impossible task, that of completing the (...) uncompleted work of a philosopher whose untimely decease prevented him from doing so himself. What he does succeed in providing is a systematic commentary and a rich integration of Merleau-Ponty's later philosophy. This task itself is important since it is believed by many that Merleau-Ponty's premature death prevented the completion of a book that would have altered the philosophical scene of Paris in the 1960s, and that, moreover, even in its incomplete form, speaks to pressing contemporary philosophical debates on such issues as mind/body dualism, and our relation to nature.The main focus of Low's reading is Merleau-Ponty's philosophy of language, which Low appropriately situates as a mediation between a modernist philosophy of presence and a postmodern Derridean approach to language as the constant deferral of meaning. Language is not the repository for ideal essences, nor do existents have an immediate [End Page 134] coincidence or fusion with things or the world. Indeed, the strength of Merleau-Ponty's philosophy is that he begins with an understanding of human action as meaningfully orientated; to move in the world is to interpretatively encounter it. As Low points out, "an ontological priority [is] granted to the body and the world upon which it opens, for together in lived experience they sustain a field of stable meanings, meanings that are then sublimated in the bodily gesture called speech" (22). Low differs from most Merleau-Ponty scholars in his assumption that there is no significant break between the earlier Phenomenology of Perception, where Merleau-Ponty carefully worked out his descriptions of embodied perception, and the later work that lays out the ontology. Indeed, one accomplishment of Low's book is to show how they fit together, explaining how the insights of the first work are necessary for grounding the ontology Merleau-Ponty began to work out in the later one.Low's book is divided into two sections. The first section, "The Visible and Nature," focuses on how we encounter the world, concluding with a helpful summary and assessment of Merleau-Ponty's ontology. Drawing upon the late essay "Eye and Mind," Low lays out Merleau-Ponty's ontology through his intuition that modern art, unlike the representational focus of Renaissance art, reveals our contact with the world as a "system of exchanges between the sensing and the sensible" (56). Perceiving the artwork is not an intellectual computation of a second-order representation of the world; rather, we see according to the work. It is a bodily process that reveals a reversibility between the viewer and the world. The second section, "The Invisible and Logos," focuses on how these encounters are sublimated or taken up in language. Language is, for Merleau-Ponty, first speech since it relies upon a speaker to take it up, "to sustain and modify it." Signs, gestures, and meaning are folded into one another. As embodied beings, we literally grasp meaning (89).This book would be of enormous benefit to scholars embarking on a reading of Merleau-Ponty's works, and it is one I highly recommend for that purpose. Although it is not original in its interpretations, its value lies in its gathering and integration of complex themes that provide an important introduction to Merleau-Ponty's late philosophy. The book maps out, moreover, a chronology of these themes and the specifics of where to find them; it also shows how the fragments fit together, indicating directions in which they might lead. His writing style is accessible without being simplistic. By interweaving his interpretations with long quotes from... (shrink)

The notion of a D-ring, generalizing that of a differential or a difference ring, is introduced. Quantifier elimination and a version of the Ax-Kochen-Ersov principle is proven for a theory of valued D-fields of residual characteristic zero.

The notion of a D-ring, generalizing that of a differential or a difference ring, is introduced. Quantifier elimination and a version of the Ax-Kochen-Eršov principle is proven for a theory of valued D-fields of residual characteristic zero.

Journal of Mathematical Logic, Volume 22, Issue 02, August 2022. We show that every definable nested family of closed and bounded subsets of a P-minimal field K has nonempty intersection. As an application we answer a question of Darnière and Halupczok showing that P-minimal fields satisfy the “extreme value property”: for every closed and bounded subset [math] and every interpretable continuous function [math] (where [math] denotes the value group), f(U) admits a maximal value. Two further corollaries are obtained as (...) a consequence of their work. The first one shows that every interpretable subset of [math] is already interpretable in the language of rings, answering a question of Cluckers and Halupczok. This implies in particular that every P-minimal field is polynomially bounded. The second one characterizes those P-minimal fields satisfying a classical cell preparation theorem as those having definable Skolem functions, generalizing a result of Mourgues. (shrink)

A nontrivial ring with unit eliminates imaginaries just in case its complete theory has the following property: every definable m-ary equivalence relation E may be defined by a formula f = f, where f is an m-ary definable function. We show that for certain natural expansions of the field of p-adic numbers, elimination of imaginaries fails or is independent of ZPC. Similar results hold for certain fields of formal power series.

In constructive mathematics it is of interest to consider a more general, but classically equivalent, notion of linear order, a so-called pseudo-order. The prime example is the order of the constructive real numbers. We examine two kinds of constructive completions of pseudo-orders: order completions of pseudo-orders and Cauchy completions of ordered groups and fields. It is shown how these can be predicatively defined in type theory, also when the underlying set is non-discrete. Provable choice principles, in particular a generalisation of (...) dependent choice, are used for showing set-representability of cuts. (shrink)

Let [Formula: see text] be an expansion of either an ordered field [Formula: see text], or a valued field [Formula: see text]. Given a definable set [Formula: see text] let [Formula: see text] be the ring of continuous definable functions from [Formula: see text] to [Formula: see text]. Under very mild assumptions on the geometry of [Formula: see text] and on the structure [Formula: see text], in particular when [Formula: see text] is [Formula: see text]-minimal or [Formula: (...) see text]-minimal, or an expansion of a local field, we prove that the ring of integers [Formula: see text] is interpretable in [Formula: see text]. If [Formula: see text] is [Formula: see text]-minimal and [Formula: see text] is definably connected of pure dimension [Formula: see text], then [Formula: see text] defines the subring [Formula: see text]. If [Formula: see text] is [Formula: see text]-minimal and [Formula: see text] has no isolated points, then there is a discrete ring [Formula: see text] contained in [Formula: see text] and naturally isomorphic to [Formula: see text], such that the ring of functions [Formula: see text] which take values in [Formula: see text] is definable in [Formula: see text]. (shrink)

The infinite-valued logic of ukasiewicz was originally defined by means of an infinite-valued matrix. ukasiewicz took special forms of negation and implication as basic connectives and proposed an axiom system that he conjectured would be sufficient to derive the valid formulas of the logic; this was eventually verified by M. Wajsberg. The algebraic counterparts of this logic have become know as Wajsberg algebras. In this paper we show that a Wajsberg algebra is complete and atomic (as a (...) lattice) if and only if it is a direct product of finite Wajsberg chains. The classical characterization of complete and atomic Boolean algebras as fields of sets is a particular case of this result. (shrink)

In this field questions arise which are certainly difficult; but as I listened last time to members of the group, I felt that the main difficulty perhaps lay in determining precisely what questions we are trying to answer. I have the conviction that if we could only say clearly what the questions are, the answers to them might not appear so elusive. So I have begun with a simple list of questions about discretion which in one form or another (...) were, as it seemed to me, expressed by the group last time. I may indeed have omitted something and inserted something useless: if so, no doubt I shall be informed of this later. The central questions then seem to me to be the following: 1. What is discretion, or what is the exercise of discretion? 2. Under what conditions and why do we in fact accept or tolerate discretion in a legal system? 3. Must we accept discretion or tolerate discretion, and if so, why? 4. What values does the use of discretion menace, and what values does it maintain or promote? 5. What can be done to maximize the beneficial operation of the use of discretion and to minimize any harm that it does? (shrink)

In today’s multi-cultural world and global economy, attention is often focused on the diversity of cultural values and practices and the need for management approaches to take these differing cultural environments into account. While there is much to be valued in this approach, the focus is often on how to navigate through distinct cultural practices in order to achieve a singular business aim, which falls within the current neoliberal paradigm of global trade. In addition, by focusing on differences in (...) cultural practices, rather than on similarities in underlying values, this approach fails to utilize an important way to achieve a greater degree of global integration in management. Despite striking cultural differences, it is possible to identify some enduring values that underlie what often appear to be quite discrete value paradigms, especially by looking at value paradigms that have endured for centuries, even millennia. This paper proposes to explore the values embedded in two such value paradigms or religious traditions, zen buddhism and Judaism, describing crucial core values in each and comparing and contrasting their ethical frameworks, in order to be able to evaluate their utility in today’s world and especially their significance for creating a more holistic, inclusive, and responsible management framework. The paper will begin by explicating the set of foundational values in each religious tradition. For example, zen buddhism, especially as expressed by Vietnamese monk Thich Nhan Hanh, utilizes the concepts of engaged mindfulness and inter-being to demonstrate the ways in which all global issues are linked together, and the collective responsibility that all humans bear for the state of our planet and the beings who live on it. This approach is inherently holistic and inclusive, and requires a management approach that takes into account the moral responsibility that we all carry. Likewise, the Jewish tradition, while using quite different terminology, also lays out a holistic and inclusive moral vision. Key concepts include the values of Tikkum Olam, to repair the world, and Tsedakah, which literally means justice or righteousness, but is commonly used to indicate charitable giving. The first value, to repair the world, emphasizes the collective moral responsibility to heal whatever damage has been done to the world in order to make the it a better place for all. It does not matter who has done the damage; what matters is each person’s obligation to improve the status quo, under the assumption that all our fates are linked together. The second value, justice and charity, is also considered to be a moral obligation, not simply a voluntary act of charity. The emphasis here is on anonymous giving, done in order to help others and without any expectation of receiving some benefit, such as positive publicity. As a moral obligation, it is not something that one should benefit from. Thus, the concept of tsekakah also emphasizes the inter-connectedness of all humanity. In addition, in the Torah, there are a series of rules about how to deal with others in business as well as the obligation, during harvest-time, to leave some grain in the fields for the less fortunate. In all, we see a focus on being mindful of others, on seeing our fates as interconnected, as in zen buddhism. The last section of the paper will explore some of the ways in which these moral imperatives can be utilized to undergird and buttress a more holistic, inclusive and responsible management approach. Some management models already include moral imperatives, such as corporal social responsibility, and this section will explore in greater detail how the above concepts have been more fully integrated into current management models. (shrink)

A concise yet rigorous introduction to logic and discrete mathematics. This book features a unique combination of comprehensive coverage of logic with a solid exposition of the most important fields of discrete mathematics, presenting material that has been tested and refined by the authors in university courses taught over more than a decade. The chapters on logic - propositional and first-order - provide a robust toolkit for logical reasoning, emphasizing the conceptual understanding of the language and the semantics of classical (...) logic as well as practical applications through the easy to understand and use deductive systems of Semantic Tableaux and Resolution. The chapters on set theory, number theory, combinatorics and graph theory combine the necessary minimum of theory with numerous examples and selected applications. Written in a clear and reader-friendly style, each section ends with an extensive set of exercises, most of them provided with complete solutions which are available in the accompanying solutions manual. Key Features: Suitable for a variety of courses for students in both Mathematics and Computer Science. Extensive, in-depth coverage of classical logic, combined with a solid exposition of a selection of the most important fields of discrete mathematics Concise, clear and uncluttered presentation with numerous examples. Covers some applications including cryptographic systems, discrete probability and network algorithms. Logic and Discrete Mathematics: A Concise Introduction is aimed mainly at undergraduate courses for students in mathematics and computer science, but the book will also be a valuable resource for graduate modules and for self-study. (shrink)

The literature acknowledges a distinction between immoral, amoral and moral management. This paper makes a case for the employee (at any level) as a moral agent, even though the paper begins by highlighting a body of evidence which suggests that individual moral agency is sacrificed at work and is compromised in deference to other pressures. This leads to a discussion about the notion of discretion and an examination of a separate, contrary body of literature which indicates that some individuals in (...) corporations may use their discretion to behave in a socially entrepreneurial manner. My underlying assumption is that CSR isn’t solely driven by economics and that it may also be championed as a result of a personal morality, inspired by employees’ own socially oriented personal values. A conceptual framework is put forward and it is suggested that individuals may be categorized as Active or Frustrated Corporate Social Entrepreneurs; Conformists or Apathetics, distinguished by their individualistic or collectivist personal values. In a discussion of the nature of values, this paper highlights how values may act as drivers of our behavior and pays particular attention to the values of the entrepreneur, thereby linking the existing debate on moral agency with the field of corporate social responsibility. (shrink)

We consider first-order theories of topological fields admitting a model-completion and their expansion to differential fields . We give a criterion under which the expansion still admits a model-completion which we axiomatize. It generalizes previous results due to M. Singer for ordered differential fields and of C. Michaux for valued differential fields. As a corollary, we show a transfer result for the NIP property. We also give a geometrical axiomatization of that model-completion. Then, for certain differential valued fields, (...) we extend the positive answer of Hilbert’s seventeenth problem and we prove an Ax–Kochen–Ershov theorem. Similarly, we consider first-order theories of topological fields admitting a model-companion and their expansion to differential fields, and under a similar criterion as before, we show that the expansion still admits a model-companion. This last result can be compared with those of M. Tressl: on one hand we are only dealing with a single derivation whereas he is dealing with several, on the other hand we are not restricting ourselves to definable expansions of the ring language, taking advantage of our topological context. We apply our results to fields endowed with several valuations. (shrink)

We prove that a definably complete expansion of a field cannot be the image of a definable discrete set under a definable function.

The mathematical model of physical problems interprets physical phenomena closely. This research work is focused on numerical solution of a nonlinear mathematical model of fractional Maxwell nanofluid with the finite difference element method. Addition of nanoparticles in base fluids such as water, sodium alginate, kerosene oil, and engine oil is observed, and velocity profile and heat transfer energy profile of solutions are investigated. The finite difference method involving the discretization of time and distance parameters is applied for numerical results by (...) using the Caputo time fractional operator. These results are plotted against different physical parameters under the effects of magnetic field. These results depicts that a slight decrease occurs for velocity for a high value of Reynolds number, while a small value of Re provides more dominant effects on velocity and temperature profile. It is observed that fractional parameters α and β show inverse behavior against u y, t and θ y, t. An increase in volumetric fraction of nanoparticles in base fluids decreases the temperature profile of fractional Maxwell nanofluids. Using mathematical software of MAPLE, codes are developed and executed to obtain these results. (shrink)

We examine fields in which model theoretic algebraic closure coincides with relative field theoretic algebraic closure. These are perfect fields with nice model theoretic behaviour. For example, they are exactly the fields in which algebraic independence is an abstract independence relation in the sense of Kim and Pillay. Classes of examples are perfect PAC fields, model complete large fields and henselian valued fields of characteristic 0.

Studying social phenomena is often assumed to be inherently different from studying natural science phenomena. In psychology, this assumption has led to a division of the field into social and experimental domains. The same kind of division has carried over into ecological psychology, despite the fact that Gibson clearly intended his theory for both social and natural phenomena. In this paper, we argue that the social/natural science dichotomy can be derived from a distinction between hermeneutics and science that is (...) deeply rooted in the atomistic, structuralist ontological tradition. We show that, from a process-based perspective, the central questions of hermeneutics (action of an individual within a context of possible actions), ecological psychology (behavior of an organism in an ecological niche) and physics (motion of a particle in a field) share a similar structure. Building on these ideas, we propose a common, process-based methodology for psychology that integrates field theory with insights from quantum mechanics to accommodate traditionally problematic concepts in natural science such as teleology and values. To demonstrate the feasibility of this approach, empirical findings on the paradigmatic problem of prospective control (such as gaze control in automobile driving in relation to perceptual tuning) are presented. (shrink)

The physical, mathematical, and philosophical foundations of the quantum theory of free Bose fields in fixed general relativistic spacetimes are examined. It is argued that the theory is logically and mathematically consistent whereas semiclassical prescriptions for incorporating the back-reaction of the quantum field on the geometry lead to inconsistencies. Still, the relations and heuristic value of the semiclassical approach to canonical and covariant schemes of quantum gravity-plus-matter are assessed. Both conventional and rigorous formulations of the theory and of its (...) principal predictions, cosmological particle creation and horizon radiation, are expounded and compared. Special attention is devoted to spacetime properties needed for the existence or uniqueness of the relevant theoretical elements , renormalization of the stress tensor). The emergence of unitarily inequivalent representations in a single dynamical context is used as motivation for the introduction of the abstract $\rm C\sp{\*}$-algebraic axiomatic formalism. The operationalist and conventionalist claims of the original abstract algebraic program are criticized in favor of its tempered outgrowth, local quantum physics. The interpretation of the theory as a wave mechanics of classical field configurations, deriving from the Schrodinger representations of the abstract algebra, is discussed and is found superior, at least on the level of analogy, to particle or harmonic oscillator interpretations. Further, it is argued that the various detector results and the Fulling nonuniqueness problem do not undermine the particle concept in the ways commonly claimed. In particular, arguments are offered against the attribution of particle status to the Rindler quanta, against the physical realizability of the Rindler vacuum, and against the more general notion of observer-dependence as to the definition of 'particle' or 'vacuum'. However, the question of the ontological status of particles is raised in terms of the consistency of quantum field theory with non-reductive realism about particles, the latter being conceived as entities exhibiting attributes of discreteness and localizability. Two arguments against non-reductive realism about particles, one from axiomatic algebraic local quantum theory in Minkowski spacetime and one from quantum field theory in curved spacetime, are developed. (shrink)

The derivation on the differential-valued field Tlogof logarithmic transseries induces on its value group${{\rm{\Gamma }}_{{\rm{log}}}}$a certain mapψ. The structure${\rm{\Gamma }} = \left$is a divisible asymptotic couple. In [7] we began a study of the first-order theory of$\left$where, among other things, we proved that the theory$T_{{\rm{log}}} = Th\left$has a universal axiomatization, is model complete and admits elimination of quantifiers in a natural first-order language. In that paper we posed the question whetherTloghas NIP. In this paper, we answer that (...) question in the affirmative:Tlogdoes have NIP. Our method of proof relies on a complete survey of the 1-types ofTlog, which, in the presence of QE, is equivalent to a characterization of all simple extensions${\rm{\Gamma }}\left\langle \alpha \right\rangle$of${\rm{\Gamma }}$. We also show thatTlogdoes not have the Steinitz exchange property and we weigh in on the relationship between models ofTlogand the so-calledprecontraction groupsof [9]. (shrink)

In this paper we are concerned with definably, with or without parameters, complete expansions of ordered fields, i. e. those with no definable gaps. We present several axiomatizations, like being definably connected, in each of the two cases. As a corollary, when parameters are allowed, expansions of ordered fields are o-minimal if and only if all their definable subsets are finite disjoint unions of definably connected subsets. We pay attention to how simply a definable gap in an expansion is (...) so. Next we prove that over parametrically definably complete expansions of ordered fields, all one-to-one definable continuous functions are monotone and open. Moreover, in both parameter and parameter-free cases again, definably complete expansions of ordered fields satisfy definable versions of the Heine-Borel and Extreme Value theorems and also Bounded Intersection Property for definable families of closed bounded subsets. (shrink)

The relative reinforcing value of food measures how hard someone will work for a high-energy-dense food when an alternative reward is concurrently available. Higher RRV for HED food has been linked to obesity, yet this association has not been examined in low-income preschool-age children. Further, the development of individual differences in the RRV of food in early childhood is poorly understood. This cross-sectional study tested the hypothesis that the RRV of HED to low-energy-dense food would be greater in children with (...) obesity compared to children without obesity in a sample of 130 low-income 3- to 5-year-olds enrolled in Head Start classrooms in Central Pennsylvania. In addition, we examined individual differences in the RRV of food by child characteristics and food security status. The RRV of food was measured on concurrent progressive-ratio schedules of reinforcement. RRV outcomes included the last schedule reached for cookies and fruit, the breakpoint for cookies in proportion to the total breakpoint for cookies and fruit combined, and response rates. Parents completed the 18-item food security module to assess household food security status and the Behavioral Activation System scale to assess reward sensitivity. Pearson’s correlations and mixed models assessed associations between continuous and discrete child characteristics with RRV outcomes, respectively. Two-way mixed effects interaction models examined age and sex as moderators of the association between RRV and Body Mass Index z-scores. Statistical significance was defined as p < 0.05. Children with obesity had a greater cookie Pmax [F = 4.95, p = 0.03], higher RRV cookie [F = 4.28, p = 0.04], and responded at a faster rate for cookies [F = 17.27, p < 0.001] compared to children without obesity. Children with higher cookie response rates had higher BMIZ ; and RRV cookie was positively associated with BMIZ for older children and boys, but not younger children or girls. The RRV of food did not differ by household food security status. Low-income children with obesity showed greater motivation to work for cookies than fruit compared to their peers without obesity. The RRV of HED food may be an important contributor to increased weight status in boys and future research is needed to better understand developmental trajectories of the RRV of food across childhood. (shrink)

In spite of the analogies between Q p and F p ((t)) which became evident through the work of Ax and Kochen, an adaptation of the complete recursive axiom system given by them for Q p to the case of F p ((t)) does not render a complete axiom system. We show the independence of elementary properties which express the action of additive polynomials as maps on F p ((t)). We formulate an elementary property expressing this action and (...) show that it holds for all maximal valued fields. We also derive an example of a rather simple immediate valued function field over a henselian defectless ground field which is not a henselian rational function field. This example is of special interest in connection with the open problem of local uniformization in positive characteristic. (shrink)

Given a theory T of a polynomially bounded o-minimal expansion R of $${\bar{\mathbb{R}} = \langle\mathbb{R}, +,., 0, 1, < \rangle}$$ with field of exponents $${\mathbb{Q}}$$, we introduce a theory $${\mathbb{T}}$$ whose models are expansions of dense pairs of models of T by a discrete multiplicative group. We prove that $${\mathbb{T}}$$ is complete and admits quantifier elimination when predicates are added for certain existential formulas. In particular, if T = RCF then $${\mathbb{T}}$$ axiomatises $${\langle\bar{\mathbb{R}}, \mathbb{R}_{alg}, 2^{\mathbb{Z}}\rangle}$$, where $${\mathbb{R}_{alg}}$$ denotes (...) the real algebraic numbers. We describe types and definable sets in our models and prove that $${\mathbb{T}}$$ is dependent. (shrink)

In conventional generalization of the main results of classical measure theory to Stone algebra valued measures, the values that measures and functions can take are Booleanized, while the classical notion of a σ-field is retained. The main purpose of this paper is to show by abundace of illustrations that if we agree to Booleanize the notion of a σ-field as well, then all the glorious legacy of classical measure theory is preserved completely.

The theory of immediate pairs of Henselian valued fields, with a given residual theory (of characteristic zero) and a given theory of valuation group (nonzero), is undecidable and has 2ℵ0 completions.

Adaptive dynamic programming, which belongs to the field of computational intelligence, is a powerful tool to address optimal control problems. To overcome the bottleneck of solving Hamilton–Jacobi–Bellman equations, several state-of-the-art ADP approaches are reviewed in this paper. First, two model-based offline iterative ADP methods including policy iteration and value iteration are given, and their respective advantages and shortcomings are discussed in detail. Second, the multistep heuristic dynamic programming method is introduced, which avoids the requirement of initial admissible control and (...) achieves fast convergence. This method successfully utilizes the advantages of PI and VI and overcomes their drawbacks at the same time. Finally, the discrete-time optimal control strategy is tested on a power system. (shrink)

-/- Continuous first-order logic has found interest among model theorists who wish to extend the classical analysis of “algebraic” structures (such as fields, group, and graphs) to various natural classes of complete metric structures (such as probability algebras, Hilbert spaces, and Banach spaces). With research in continuous first-order logic preoccupied with studying the model theory of this framework, we find a natural question calls for attention. Is there an interesting set of axioms yielding a completeness result? -/- The primary (...) purpose of this article is to show that a certain, interesting set of axioms does indeed yield a completeness result for continuous first-order logic. In particular, we show that in continuous first-order logic a set of formulae is (completely) satisfiable if (and only if) it is consistent. From this result it follows that continuous first-order logic also satisfies an approximated form of strong completeness, whereby Σ⊧φ (if and) only if Σ⊢φ ∸2−n for all n < ω. This approximated form of strong completeness asserts that if Σ⊧φ, then proofs from Σ, being finite, can provide arbitrarily better approximations of the truth of φ. -/- Additionally, we consider a different kind of question traditionally arising in model theory—that of decidability. When is the set of all consequences of a theory (in a countable, recursive language) recursive? Say that a complete theory T is decidable if for every sentence φ, the value φ T is a recursive real, and moreover, uniformly computable from φ. If T is incomplete, we say it is decidable if for every sentence φ the real number φ T o is uniformly recursive from φ, where φ T o is the maximal value of φ consistent with T. As in classical first-order logic, it follows from the completeness theorem of continuous first-order logic that if a complete theory admits a recursive (or even recursively enumerable) axiomatization then it is decidable. (shrink)

BackgroundIn Sweden, individuals with visual field loss have their driving license withdrawn. The literature clearly indicates that individuals with VFL are unsafe drivers on a group level. However, many drivers with VFL can be safe on an individual level. The literature also suggests that self-perception, beliefs, and insights of one’s own capabilities are related to driving performance. This study had three aims: To investigate self-perceived driving capability ratings for individuals with VFL; to compare these ratings between groups with different (...) medical conditions associated with VFL ; and to relate the self-perception ratings to actual driving performance in an advanced driving simulator.MethodsParticipants comprised 723 individuals whose driver’s license had been withdrawn because of VFL and 92 normally sighted elderly individuals. All participants completed a background survey, rated difficulties with different traffic situations, rated their strengths and weaknesses as drivers, and rated aspects that were important for causing traffic accidents. Of the VFL group participants, 264 also completed a simulator-based driving test that they knew could lead to renewal of their driving license. VFL participants and normally sighted was at the same age when they completed the simulator driving test.ResultsOverall, individuals with VFL rated their capabilities as high on all instruments and scales used, even higher than the elderly normally sighted control group. The only VFL etiology group that rated lower than other groups was the diabetes group. Safety orientation and internal control orientation values were best at discriminating between VFL participants in terms of self-perception of driving performance. Participants categorized as “high” in terms of safety skills and internal control were more modest in their ratings. Finally, participants who passed the simulated driving test did not differ from those who failed, in any of the self-perception measures.ConclusionSelf-perception ratings among individuals with VFL were higher than those of normally sighted elderly individuals. Self-assessed skills did not predict driving performance. Groups with different VFL etiologies rated similarly. Self-ratings of driving abilities cannot be used to assess actual driving performance. Actual driving tests are necessary to discriminate between safe and unsafe drivers with VFL. (shrink)

For ten years, 1971–1981, the Institute onHuman Values in Medicine (IHVM) played a keyrole in the development of Bioethics as afield. We have written this history andanalysis to bring to new generations ofBioethicists information about the developmentof their field within both the humanitiesdisciplines and the health professions. Thepioneers in medical humanities and ethics cametogether with medical professionals in thedecade of the 1960s. By the 1980s Bioethics wasa fully recognized discipline. We show the rolethat IHVM programs played in defining thefield, (...) training faculty and helping schools todevelop programs. We review the beginnings ofthe IHVM in the crucible of social andtechnological change that led to theestablishment of the IHVM's parentorganization, the Society for Health and HumanValues. We then turn to the IHVM programsthrough which Faculty members receivedfellowships to explore new crossovers betweenthe humanities and the health professions. Wehave not only described the Fellows Program asit existed in 1973–1980, but have completed asurvey of the fellows a quarter of a centuryafter they held their fellowships. We describeother IHVM programs designed to facilitate theinitiation and development of new humanitiesprograms, to explore conceptual issues betweenmedicine and five humanities fields, to conductissue driven or educational method conferencesand to advance humanities programs intograduate education through the Directors ofMedical Education. (shrink)

Although philosophers have considered some of the implications of the nature of quantum statistics of many-particle systems for the interpretation problem, e.g., Reichenbach, they have not produced a complete analysis of the relationship between aspects of quantum statistics and complications and/or possible solutions of the interpretation problem. While the present work by no means provides a complete account, it does explore some heretofore uncharted regions. One of the latter is an analysis of a situation that I call 'The (...) Paradox of Identical Particles ,' which arises in Bose-Einstein statistics. PIP refers to the fact that Elementary Quantum Mechanics seems to be committed to the thesis that systems of particles of the same species, "identical" particles, exhibit behavior differing from that of systems of particles of different species, simply in virtue of the fact that the former contain entities of the same species while the latter do not. Taken at face value this is tantamount to the Empedoclean "axiom" that like attracts like, while like and unlike repel, a somewhat embarassingly naive proposition for modern physicists to embrace, and one which, furthermore, poses a host of perplexing problems, e.g., how do the particles "know" whether they are of the same species or not? A result proven in the present work is that any "pluralistic" interpretation of quantum theory is unable to provide a satisfactory way of ridding quantum physicists of this appeal to what I call "species sensitivity." Arguing that Quantum Field Theory is most naturally and plausibly interpreted as a "monistic" theory--which means, inter alia, that for a system of n identical particles to exist is not for n distinct objects of any kind to exist, but rather for one unified underlying entity, a quantum field, to be in a certain kind of state or condition--I show that from this point of view PIP can be dissolved. On the basis of this and other considerations, I argue that the interpretation problem is more amenable to solution if we take the QFT point of view as the fundamental expression of quantum theory. (shrink)

This paper collects and presents unpublished notes of Kurt Gödel concerning the field of many-valued logic. In order to get a picture as complete as possible, both formal and philosophical notes, transcribed from the Gabelsberger shorthand system, are included.

The notion of a continuously variable quantity can be regarded as a generalization of that of a particular quantity, and the properties of such quantities are then akin to, and derived from, the properties of constants. For example, the continuous, real-valued functions on a topological space behave like the field of real numbers in many ways, but instead form a ring. Topos theory permits one to apply this same idea to logic, and to consider continuously variable sets . (...) In this expository paper, such applications are explained to the non-specialist. Some recent results are mentioned, including a new completeness theorem for higher-order logic. (shrink)

In the present paper the concept of a covering is presented and developed. The relationship between cover schemes, frames (complete Heyting algebras), Kripke models, and frame-valued set theory is discussed. Finally cover schemes and framevalued set theory are applied in the context of Markopoulou’s account of discrete spacetime as sets “evolving” over a causal set. We observe that Markopoulou’s proposal may be effectively realized by working within an appropriate frame-valued model of set theory. We go on to (...) show that, within this framework, cover schemes may be used to force certain conditions to prevail in the associated models: for example, rendering the universe timeless, obliterating a given event or forcing it to become the universe’s “beginning”. (shrink)

A Dirac-type equation on R×S 3 topology is derived. It is a generalization of the previously obtained Klein-Gordon-type, Schrödinger-type, and Weyl-type equations, and reduces to the latter in the appropriate limit. The (discrete) energy spectrum is found and the corresponding complete set of solutions is given as expansions in terms of the matrix elements of the irreducible representations of the group SU 2 . Finally, the properties of the solutions are discussed.

We present the bundle (Aff(3)⊗ℂ⊗Λ)(ℝ3), with a geometric Dirac equation on it, as a three-dimensional geometric interpretation of the SM fermions. Each (ℂ⊗Λ)(ℝ3) describes an electroweak doublet. The Dirac equation has a doubler-free staggered spatial discretization on the lattice space (Aff(3)⊗ℂ)(ℤ3). This space allows a simple physical interpretation as a phase space of a lattice of cells.We find the SM SU(3) c ×SU(2) L ×U(1) Y action on (Aff(3)⊗ℂ⊗Λ)(ℝ3) to be a maximal anomaly-free gauge action preserving E(3) symmetry and symplectic (...) structure, which can be constructed using two simple types of gauge-like lattice fields: Wilson gauge fields and correction terms for lattice deformations.The lattice fermion fields we propose to quantize as low energy states of a canonical quantum theory with ℤ2-degenerated vacuum state. We construct anticommuting fermion operators for the resulting ℤ2-valued (spin) field theory.A metric theory of gravity compatible with this model is presented too. (shrink)

Formal approaches that aim at representing human reasoning should be evaluated based on how humans actually reason. One way of doing so is to investigate whether psychological findings of human reasoning patterns are represented in the theoretical model. The computational logic approach discussed here is the so-called weak completion semantics which is based on the three-valued ᴌukasiewicz logic. We explain how this approach adequately models Byrne’s suppression task, a psychological study where the experimental results show that participants’ conclusions systematically (...) deviate from the classical logically correct answers. As weak completion semantics is a novel technique in the field of computational logic, it is important to examine how it corresponds to other already established non-monotonic approaches. For this purpose we investigate the relation of weak completion with respect to completion and three-valued stable model semantics. In particular, we show that well-founded semantics, a widely accepted approach in the field of non-monotonic reasoning, corresponds to weak completion semantics for a specific class of modified programs. (shrink)

I propose a version of quantum mechanics featuring a discrete and finite number of states that is plausibly a model of the real world. The model is based on standard unitary quantum theory of a closed system with a finite-dimensional Hilbert space. Given certain simple conditions on the spectrum of the Hamiltonian, Schrödinger evolution is periodic, and it is straightforward to replace continuous time with a discrete version, with the result that the system only visits a discrete and finite set (...) of state vectors. The biggest challenges to the viability of such a model come from cosmological considerations. The theory may have implications for questions of mathematical realism and finitism. (shrink)

The total ideology of the revolutionary-political themes of Ukrainian consciousness of the twentieth century, the poetization of the absurdly inverted hierarchy of values, was opposed by the new generation of artists with their philosophical and ethical orientation of their creativity. As I.Franko notes, they "... sought a completely modern European way to portray the peculiarity of the life of the Ukrainian people," revealing the unique collisionality of human existence, the diversity of psychological types, ideological orientations, and the human experience of (...) its existence on the background of this difficult time. However, Ukrainian writersphilosophers who talently laid the foundations of national culture, but are now returning to the generations of the Shattered Renaissance, filling black and bloody spots of troubled history of Ukraine. Among them are V. Pidmogylny, B. Antonovenko-Davidovich, G. Kosinka, D. Falkivsky, M.Ivchenko, T.Osmachka, E.Pluzhnik. This is a non-exhaustive list of the leading figures of the post-revolutionary period who were removed from the creative field, whose works are the golden fund of Ukrainian culture. (shrink)

International migration of work force is presently a high amplitude phenomenon. Romanian people have emigrated for work around the world, being engaged both in the physically hardest jobs and in activities that require completion of specialized courses and certification in a particular field. This last category includes social workers who, following schooling and certification and even having a minimal experience in the home country, apply for jobs in the field of social assistance. These recruiters aim to distribute social (...) workers at the workplace of interest, according to certain well-defined criteria. In the present paper we describe international migration of work force, and we illustrate it by adding case studies of life experience of social workers who underwent the process of emigration to the United Kingdom. Throughout the emigration process, these workers experienced certain changes in their life principles, mentality, values, and were marked by professional dilemmas in extreme situations. (shrink)

In the history of İslāmic thought, Māturīdī is a famous scholar both in the field of kalām and tafsir. Being approved by Māturīdī, the distinction of tafsir and ta’wīl, which makes possible to take the comments made about the verses into sistematic framework, is quite important. There is an important information both about content of the distinction approved by Māturīdī and the main reasons that necessiated this distinction in the introduction of Samarqandī’s Sharh at Ta’wīlāt. From this information, it (...) is understood that the distinction of tafsir and ta’wīl had dated back to older time than Māturīdī and previous scholars dealed with the problem that necessiated this distinction and Māturīdī inherited this distinction from his predecessors. Information in the Sharh at Ta’wīlāt reveals that according to Māturīdī the companions had made interpretation (ta’wīl) in addition to tafsir and the explanations made by certain evidences is tafsir too. Therefore this information gives us the opportunity flexing the acceptance of the idea “tafsir, is dedicated to the companions, ta'wīl is dedicated to fuqaha (jurists)” and asserting that companions and scholars made both tafsir and ta’wīl. In this paper, we will research whether the distinction tafsir and ta’wīl regarded by Māturīdī completely belonged to his or not and content of the distinction and its reasons. We will also analyze this distinction according to examples in the Ta’wîlāt. Summary: An important distinction is made between the concepts of tafsir and ta’wīl in the short preface of Māturīdī's tafsir, and this distinction is particularly prominent in contemporary tafsir studies. Because from the point of view of the value they carry, the explanations for these differences and concepts express the legitimacy of the following mufassirūn (writers of a commentary on the Qur’an) to be able to perform tafsir, as well as the value of the explanations found in tafsir literature in terms of certainty. In Ta'wīlāt, tafsir is described as the precise determination of what is intended. Determination of what is exactly intended in verse/wording is bound to certainly know the conditions under which it was revealed. Because they are the first to be addressed directly and they witnessed the context, the tafsir is only for the companions (ṣaḥābah pl. aṣḥāb). On the other hand, ta’wīl is not a definite determination of what is intended, but rather it is explained as putting the possibilities forward. In this case, ta’wīl is dedicated to the scholars after the companions. When we look at the other tafsir sources that include his views, it can be understood that this brief information in the preface of the tafsir which survived to our day is partially incomplete. Samarqandī's Sharhu’t-ta'wīlāt is the most important of these sources. According to the information here, Māturīdī states that some scholars distinguish tafsir and ta’wīl from each other. In these distinctions, tafsir is either described in relation to a definitive statement or in relation to being familiar with the conditions of revelation, while ta’wīl is described as extracting meanings from the verses and making possible meanings without claiming certainty. These explanations show that the distinction between tafsir and ta’wīl in Ta'wīlāt's preface was mainly based on the period before Māturīdī. These statements, which Māturīdī transmitted from some scholars in the beginning of Sharhu’t-ta'wīlāt, were then transmitted by Omar al-Nasafī, Mahāyimī and Molla Husrev. In some sources, it was also transmitted that similar explanations were made by scholars such as an-Nadr b. Shumayl, al-Hussein b. Fadl al-Bacelī and Ibn Durayd before Māturīdī.Some explanations of Māturīdī in Ta'wīlāt enables the theory of tafsir-ta’wīl to be understood at a wider level. When the examples are examined, it is observed that Māturīdī exhibited two types of attitude against the tafsir of the companions. He states that one of these is that if the transmission is true, what is intended is exactly what is stated in the narrated. The second is that he sometimes does not accept these explanations and suggests additional opinions. The reason he refuses is that sometimes he does not find the transmission reliable, and sometimes he thinks that companions performed ta'wīl. If this is the case, the dedication of ta’wīl to those coming later should not be absolutized. Because the companions can perform both tafsir and ta’wīl. The examples in Ta'wīlāt clearly reveal what areas tafsir, which is defined as a definite statement, covers. According to this, first, tafsir is related to the field of “invisible” about the past and the future. Secondly, its meaning and practice is only about the information of the “statement” about the issues that can be revealed by the Prophet (pbuh). In the third aspect, it is related to “revelation” information reporting the context in which the verses descended. In the beginning of Sharhu’t-ta'wīlāt, Māturīdī says that the explanations made based on the definitive evidences such as mutawātir (repetition; narration reported by large number reporters) and ijmā (consensus) also fall under the category of tafsir. However, it is also necessary to add intellectual evidence to these evidences. Because intellectual evidence is also important in the epistemology of Māturīdī. In fact, the truthfulness of nass (text) is primarily based on intellectual evidence. First, the issues of theology are proven by certain intellectual evidence, and then, on the basis of this, the meanings expressed by the nass acquire the value of certainty and truthfulness. With the definitive acceptance of reason in theology, what some nass express confirm each other and are used as evidence in the process of interpretation. Therefore, since some of the intellectual evidence, ijmā and mutawātir news provide definitive information, explanations based on them are also seen as tafsir. Considering the fact that the above-mentioned evidence was also used by those coming after the companions, it can be concluded that tafsir can be performed by not only the companions but also those coming after them. In this case, it should be accepted that tafsir is not performed only by the companions, and that the succeeding mufassirūn can also perform tafsir as well as ta’wīl. When we look at the examples based on these evidence, there are two different situations in Ta'wīlāt. The first is that Māturīdī sometimes has views on evidence based on mutawātir and ijmā and sometimes suggests different opinions from the assumptions that he said there was ijmā in particular. Because these evidences are certain, it is understandable that he remains bound to them. However, as a possible reason for his opposition to ijmā, it can be said that he thinks only the following generations enter the scope of ijmā. Because, according to Samarqandī, Māturīdī's objection to such information does not seem possible, since ijmā of the companions in a matter is accepted. Although he sometimes said that there was ijmā, the reason why no such ijmā was mentioned in other tafsir sources and sometimes, it may be considered that he did not respect other views in the face of ijmā as a reason for including different views or it may be thought that he was unaware of the different views that were transmitted. The second remarkable situation is that according to Māturīdī, intellectual evidence, in particular, can reveal only the “knowledge of existence” of the issues related to theology. Māturīdī thinks that their nature cannot be certainly known, and that the explanations made regarding their nature are only ta’wīl. For this reason, he firmly rejects the interpretation of the khabarī attributes about Allah in accordance with the idea of tashbīh and tajsīm (anthropomorphism). (shrink)

Giving an account of the relation between evolution and consciousness is painted as posing a dilemma between panpsychism, with minimal consciousness in every grain of matter, and radical emergence, with consciousness appearing as from nowhere in living structures. Panpsychism has been seen as suffering from a combination problem and radical emergence as unjustified in physics. The underpinning of physics now lies in field theory, which may provide a way out on both sides. Only, and always, in a field (...) theory account do influences at different points in spacetime combine in the same indivisible event. Radical emergence is also inherent to field theory. Moreover, by providing rich patterns of influence involving both discrete identities and quantitative values, field theory might provide a basis for sensed propositional meaning with subjects and predicates. Ordered condensed matter within living tissue may support unusual emergent dynamic units uniquely suited to building representations of the world with sensed meaning. The evolution of consciousness may then be seen as a tractable biological problem centred on increasingly sophisticated ways for external world dynamics to be mirrored by internal representations with semantic content, based in field relations within condensed matter with genetically encoded complex order. (shrink)