In [6] Messmer and Wood proved quantifier elimination for separably closed fields of finite Ershov invariant e equipped with a (certain) Hasse derivation. We propose a variant of their theory, using a sequence of e commuting Hasse derivations. In contrast to [6] our Hasse derivations are iterative.

We define a complete theory SHF e of separably closed fields of finite invariant e (= degree of imperfection) which carry an infinite stack of Hasse-derivations. We show that SHF e has quantifier elimination and eliminates imaginaries.

We give a complete description of minimal groups infinitely definable in separably closed fields of finite degree of imperfection. In particular we answer positively the question of the existence of such a group with infinite transcendence degree (i.e., a minimal group with non thin generic).

The main scope of this short article is to provide a modification of the axioms given by Messmer and Wood for the theory of separably closed fields of positive characteristic and finite imperfectness degree. As their original axioms failed to meet natural expectations, a new axiomatization was given (i.e., Ziegler’s one), but the new axioms do not follow Messmer and Wood’s initial idea. Therefore, we aim to give a correct axiomatization that is more similar to the original one (...) and that, as with the original axioms, involves only one Hasse–Schmidt derivation, this time based on the iterativity conditions corresponding to the Witt group. (shrink)

We give an alternative proof of the stability of separably closed fields of fixed Éršov invariant to the one given in [W]. We show that in case the Éršov invariant is finite, the theory is in fact equational. We also characterize the independence relation in those theories.

The stability of each of the theories of separably closed fields is proved, in the manner of Shelah's proof of the corresponding result for differentially closed fields. These are at present the only known stable but not superstable theories of fields. We indicate in § 3 how each of the theories of separably closed fields can be associated with a model complete theory in the language of differential algebra. We assume familiarity with some basic facts (...) about model completeness [4], stability [7], separably closed fields [2] or [3], and (for § 3 only) differential fields [8]. (shrink)

We study the infinitely definable subgroups of the additive group in a separably closed field of finite positive imperfection degree. We give some constructions of families of such subgroups which confirm the diversity and the richness of this class of groups. We show in particular that there exists a locally modular minimal subgroup such that the division ring of its quasi-endomorphisms is not a fraction field of the ring of its definable endomorphisms, and that in contrast (...) there exist 20 pairwise orthogonal locally modular minimal subgroups whose induced structure is exactly that of an-vector space. We also show that there exist infinitely many pairwise orthogonal subgroups of infinite U-rank. Furthermore, these constructions are carried out in the additive group considered as a module over its ring of definable endomorphisms. (shrink)

We consider the reduct to the module language of certain theories of fields with a non surjective endomorphism. We show in some cases the existence of a model companion. We apply our results for axiomatizing the reduct to the theory of modules of non principal ultraproducts of separably closed fields of fixed but non zero imperfection degree.

In this note we show that groups with definable generics in a separably closed valued field K of finite imperfection degree can be embedded into groups definable in the algebraic closure of K.

Fix an algebraically closed field of characteristic zero and let G be its geometry of transcendence degree one extensions. Let X be a set of points of G. We show that X extends to a projective subgeometry of G exactly if the partial derivatives of the polynomials inducing dependence on its elements satisfy certain separability conditions. This analysis produces a concrete representation of the coordinatizing fields of maximal projective subgeometries of G.

We give a reduction of the function field Mordell–Lang conjecture to the function field Manin–Mumford conjecture, for abelian varieties, in all characteristics, via model theory, but avoiding recourse to the dichotomy theorems for (generalized) Zariski geometries. Additional ingredients include the “Theorem of the Kernel”, and a result of Wagner on commutative groups of finite Morley rank without proper infinite definable subgroups. In positive characteristic, where the main interest lies, there is one more crucial ingredient: “quantifier-elimination” for the corresponding (...) [Formula: see text] where [Formula: see text] is a saturated separably closed field. (shrink)

"Combining aspects of his acclaimed street work with an innovative approach to portraiture, Chicago-based photographer Jed Fielding has concentrated closely on these children's features and gestures, probing the enigmatic boundaries between surface and interior. Design, composition, and the play of light and shadow are central elements in these photographs, but the images are much more than formal experiments; they confront disability in a way that affirms life. Fielding's sightless subjects project a vitality that seems to extend beyond the limits of (...) self-consciousness. In collaborative, joyful participation with the children, he has made pictures that reveal essential gestures of absorption and the basic expressions of our creatureliness. Fielding's work achieves what only great art, and particularly great portraiture can: it launches and then complicates a process of identification across the barriers that separate us from each other. (shrink)

A first-order theory is equational if every definable set is a Boolean combination of instances of equations, that is, of formulae such that the family of finite intersections of instances has the descending chain condition. Equationality is a strengthening of stability. We show the equationality of the theory of proper extensions of algebraically closed fields and of the theory of separably closed fields of arbitrary imperfection degree.

We investigate profinite structures in the sense of Newelski interpretable in fields. We show that profinite structures interpretable in separably closed fields are the same as profinite structures weakly interpretable in . We also find a strong connection with the inverse Galois problem. We give field theoretic constructions of profinite structures weakly interpretable in and satisfying some model theoretic properties, like smallness, m-normality, non-triviality, being -rank 1. For example we interpret in this way the profinite structure consisting (...) of the profinite group together with a distinguished Sylow p-subgroup of its standard structural group. (shrink)

We provide a general framework for studying the expansion of strongly minimal sets by adding additional relations in the style of Hrushovski. We introduce a notion of separation of quantifiers which is a condition on the class of expansions of finitely generated models for the expanded theory to have a countable ω-saturated model. We apply these results to construct for each sufficiently fast growing finite-to-one function μ from 'primitive extensions' to the natural numbers a theory T μ of an expansion (...) of an algebraically closed field which has Morley rank 2. Finally, we show that if μ is not finite-to-one the theory may not be ω-stable. (shrink)

Non-perfect separably closed fields are stable, and not superstable. As a result, not all types can be ranked. We develop here a new tool, a "semi-rank", which takes values in the non-negative reals, and gives a sufficient condition for forking of types. This semi-rank is built up from a transcendence function, analogous to the one considered by Kolchin in the context of differentially closed fields. It yields some orthogonality and stratification results. /// Un corps séparablement clos non (...) algébriquement clos est stable sans être superstable. Cela signifie que seuls certains de ses types sont rangés. Nous développons un autre outil, un "semi-rang" à valeurs réelles, qui donne un critère de déviation des types. Ce semi-rang est construit à partir d'un analogue de la fonction de Kolchin associée à un type au-dessus d'un corps différentiel. Il produit des résultats de stratification des modèles et des résultats d'orthogonalité. (shrink)

The open-domain Frame Problem is the problem of determining what features of an open task environment need to be updated following an action. Here we prove that the open-domain Frame Problem is equivalent to the Halting Problem and is therefore undecidable. We discuss two other open-domain problems closely related to the Frame Problem, the system identification problem and the symbol-grounding problem, and show that they are similarly undecidable. We then reformulate the Frame Problem as a quantum decision problem, and show (...) that it is undecidable by any finite quantum computer. (shrink)

In this review, I discuss the justifications for focussing on Hobbes's On the Citizen (De Cive), the middle recension of his political philosophy, separately from his better known Leviathan. I provide an overview of the collection's chapter contents, and I close by calling for further research regarding the impact of this text on later European political philosophy (such as Spinoza, Rousseau, Kant).

RésuméPour tout entier n, on construit des sous-groupes, infiniment définissables de rang de Lascar ωn, du groupe additif d’un corps séparablement clos.

Let K be an NIP field and let v be a Henselian valuation on K. We ask whether [Formula: see text] is NIP as a valued field. By a result of Shelah, we know that if v is externally definable, then [Formula: see text] is NIP. Using the definability of the canonical p-Henselian valuation, we show that whenever the residue field of v is not separably closed, then v is externally definable. In the case of (...) separably closed residue field, we show that [Formula: see text] is NIP as a pure valued field. (shrink)

We study the algebraic implications of the non-independence property and variants thereof on infinite fields, motivated by the conjecture that all such fields which are neither real closed nor separably closed admit a henselian valuation. Our results mainly focus on Hahn fields and build up on Will Johnson’s “The canonical topology on dp-minimal fields” :1850007, 2018).

Gravitation is interpreted to be an “ontomathematical” force or interaction rather than an only physical one. That approach restores Newton’s original design of universal gravitation in the framework of “The Mathematical Principles of Natural Philosophy”, which allows for Einstein’s special and general relativity to be also reinterpreted ontomathematically. The entanglement theory of quantum gravitation is inherently involved also ontomathematically by virtue of the consideration of the qubit Hilbert space after entanglement as the Fourier counterpart of pseudo-Riemannian space. Gravitation can be (...) also interpreted as purely mathematical or logical “force” or “interaction” as a corollary from its ontomathematical (rather than physical) realization. The ontomathematical approach to gravitation is implicit in general relativity equating it to operators in pseudo-Riemannian space obeying the Einstein field equation and also well-known by the “geometrization of physics”. Quantum mechanics shares the same by the separable complex Hilbert space and defining “physical quantity” by the Hermitian operators on it. One can interpret special Minkowski space involved by special relativity and the qubit Hilbert space of quantum information as Fourier counterparts immediately noticing that general relativity means gravitation as the Fourier counterpart of non-Hermitian operators implying non-unitarity and the violation of energy conservation and thus destroying Pauli’s particle paradigm. Since the Standard model obeys it, this explains the impossibility of “quantum gravitation” in any framework conservatively generalizing the Standard model so that it would include gravitation along with electromagnetic, weak, and strong interactions. Einstein’s geometrization of gravitation can be continued into a purely mathematical theory of it following Euclid’s realization for geometry to be exhaustively built in a deductive and axiomatic way as well as Riemann’s parametrization of all the class of Euclidean and non-Euclidean geometries by “space curvature”, then being generalized to Minkowski space as the operators on pseudo-Riemannian space as the Einstein field equation means gravitation. The transition from mathematical gravitation to logical one can rely on the historical lesson of the pair of Lobachevski’s and Riemann’s approaches now “reversely”, i.e., from the latter to the former. Logical gravitation is linkable to Hegel’s dialectical logic and ontological dialectics abandoning their interpretations as a new zero logic substituting classical propionyl logic. The approach of ontomathematics generalizing that of ontology, traceable even to Aristotle’s reformation of Plato’s doctrine, needs Hegel’s doctrine to be formalized as a first-order logic naturally containing Boolean algebra, isomorphic to both classical propositional logic and set theory being the class of all first-order logics, as a sub-logic along with Peano arithmetic as another sub-logic. The first-order logic at issue is called Hilbert arithmetic and elaborated in detail in other papers. It allows for both self-foundation of mathematics to be internally proved as complete and furthermore, quantum mechanics reinterpreted as quantum information to be included by the qubit Hilbert space interpretable in turn as a dual and physical counterpart of Hilbert arithmetic in a narrow sense, that is, both counterparts constitute Hilbert arithmetic in a wide sense, being mathematical and physical simultaneously and thus overcoming the Cartesian dualism of “body” gapped from “mind” by an abyss. Then, the proper philosophical interpretation of gravitation to be the fundamental ontomathematical force or interaction overcomes the ridiculous belief of the Big Bang wrongly alleged to be a scientific theory. Ontomathematical gravitation suggests an omnipresent and omnitemporal medium of “God’s” creation “ex nihilo” following only the natural necessity of quantum-information conservation particularly and locally manifested as energy conservation. (shrink)

This paper contains a corrected proof that the statement “every non-empty closed subset of a compact complete separable metric space is separably closed” implies the arithmetical comprehension axiom of reverse mathematics.

We develop a point-free construction of the classical one- dimensional continuum, with an interval structure based on mereology and either a weak set theory or logic of plural quantification. In some respects this realizes ideas going back to Aristotle,although, unlike Aristotle, we make free use of classical "actual infinity". Also, in contrast to intuitionistic, Bishop, and smooth infinitesimal analysis, we follow classical analysis in allowing partitioning of our "gunky line" into mutually exclusive and exhaustive disjoint parts, thereby demonstrating the independence (...) of "indecomposability" from a non-punctiform conception. It is surprising that such simple axioms as ours already imply the Archimedean property and that they determine an isomorphism with the Dedekind-Cantor structure of R as a complete, separable, ordered field. We also present some simple topological models of our system, establishing consistency relative to classical analysis. Finally, after describing how to nominalize our theory, we close with comparisons with earlier efforts related to our own. (shrink)

Distinguishes two senses of 'judgment' on the one hand as meaning a state of 'conviction' or 'belief', and on the other hand as meaning an act of 'affirmation' or 'assertion'. Certainly conviction and assertion stand in close relation to each other, but they delineate two heterogeneous logical spheres, and thereby divide the total field of the theory of judgment into two neighbouring but separate sub-fields. Once this is done it is shown to have implications for our understanding especially of (...) the phenomenon of negation, and of the distinction between negation as affirmation of a negative state of affairs, and negation as act of contradicting the judgment someone else has made. (shrink)

We develop an intersection theory for definable Cp-manifolds in an o-minimal expansion of a real closed field and we prove the invariance of the intersection numbers under definable Cp-homotopies . In particular we define the intersection number of two definable submanifolds of complementary dimensions, the Brouwer degree and the winding numbers. We illustrate the theory by deriving in the o-minimal context the Brouwer fixed point theorem, the Jordan-Brouwer separation theorem and the invariance of the Lefschetz numbers under definable (...) Cp-homotopies. A. Pillay has shown that any definable group admits an abstract manifold structure. We apply the intersection theory to definable groups after proving an embedding theorem for abstract definably compact Cp-manifolds. In particular using the Lefschetz fixed point theorem we show that the Lefschetz number of the identity map on a definably compact group, which in the classical case coincides with the Euler characteristic, is zero. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Anthony Close THE EMPIRICAL AUTHOR: SALMAN RUSHDIE'S THE SATANIC VERSES HOBBES, comparing the author ofan action to the owner ofgoods, asserts, "And as the right of possession, is called dominion; so the right of doing any action, is called authority" (Leviathan, Book I, chap. 16). My purpose in this essay is to apply this Hobbesian maxim to the relation Author/Text, expanding somewhat Hobbes's notion of authority. I presuppose that (...) in our day-to-day dealings with our fellow men, we envisage each others' acts as exhibiting purpose and commitment, which make them ours in more than the platitudinous sense that we performed them. By virtue of that, they confer on us a privileged, though clearly not unchallengeable, authority in saying what they are and what they are for. Such, in effect, is the position reached by Austin in How to Do Things with Words, and it broadly corresponds to that of various philosophical writings on human action—by Anscombe, Davidson, and Melden.1 To be sure, we may legitimately consider human behavior from other viewpoints (e.g., structuralist, anthropological, psychoanalytical, Marxist) trained on its unconscious presuppositions. Yet, insofar as such theorizing fails to acknowledge the priority of the Hobbesian perspective, it will seem perverse for a reason that has been well put by Charles Taylor. Precisely because history is made up of purposeful activity, Taylor argues, we need to explain historical patterns in a way that is "intelligible in relation to conscious action" even though they may not "issue from conscious action."2 By this criterion much of our modern literary theory is defective. A vivid measure of the deficiency is provided by the furor over Salman Rushdie's TL· Satanic Verses, culminatingin thefatwa pronounced against Philosophy and Literature, © 1990, 14: 248-267 Anthony Close249 him by the Ayatollah Khomeini in February 1989. I take it that for anyone vocationally committed to the practice or study ofliterature and to the cause of freedom of expression, the defense of Rushdie cannot be a matter of indifference. Nor can one be indifferent to the chasm of misunderstanding that separates Rushdie and his apologists from the novel's Muslim opponents. I believe that these two causes—defending Rushdie and bridging the chasm—can be reconciled by offering better explanations ofhis novel in place ofcrude denunciations or, from some non-Muslim quarters, smug and knowing dismissals. Yet, as we shall see, the task of explanation could be considered hamstrung by our modern poetics. I was reminded ofHobbes's textby Rushdie's lucid and cogent defense of his novel in TL· Independent (London, February 4, 1990). Against the accusations that he had deliberately reviled the Islamic faith, he argued as follows. Though TL· Satanic Verses, like many novels that Rushdie esteems, is a work of radical dissent, questioning, and reimagining, it is motivated by a spirit of serious inquiry rather than debunking derision. More particularly, it reflects a determination that underlies all Rushdie's work: "to create a literary language and literary forms in which the experience offormerly-colonised, still-disadvantaged peoples might find full expression." He continues: "It is written from the very experience of uprooting, disjuncture and metamorphosis... that is the migrant condition, and from which, I believe, can be derived a metaphor for all humanity.... [It] celebrates hybridity, impurity, intermingling, the transformation that comes of new and unexpected combinations of human beings, cultures, ideas, politics, movies, songs. It rejoices in mongrelisation and fears the absolutism of the Pure." Note the insistence on how these themes flow direcdy from Rushdie's own experience. Two days after the publication of this article, Harold Pinter gave on Rushdie's behalf a lecture on the relationship of literature to religion and society at the Institute of Contemporary Arts in London. In this lecture, Rushdie articulated a Bakhtinian view of the novelist's function: to dramatize ironically and noncommittally the clash of voices, languages, and ideologies in a world without gods and without traditionsanctioned truth. Rushdie referred specifically, I think, to a species of postmodernist novel which we may dub "palimpsest history";3 the species includes Rushdie's works, García Márquez's One Hundred Years of Solitude, Carlos Fuentes's Terra Nostra... (shrink)

Introduction 11 PART I: THE HALLE YEARS Chapter One: The Rehabilitation of the Imagination in Husserl’s Early Thought. 17 §1. Brentano’s Rehabilitation of Intentionality and the Problem of Imagination. §2. Husserl and the Breakthrough of Phenomenology. §2.1 The Meaning-Bestowing Act as ‘the Peg from which Everything hangs.’ §2.2 Consciousness is not a Container. §2.3 ‘A Difference that cannot be Phenomenologically Reduced.’ §3. Imagination as an Authentic, Intuitive Intentionality. PART II: THE GÖTTINGEN YEARS Chapter Two: Irreconcilable Differences: Imagination and Image Consciousness. (...) 71 §1. Separations, Convergences, and Goodbyes §2. A nothing in the midst of things that sees me. §3. Wrestling with the angel of Imagination. §4. Images of Productive Phantasy. Chapter Three: The Garden of the Forking Paths: On Time Consciousness. 107 §1. Picking up an Ancient Cross. §2. Continuum of Continua. §3. In Search of Time Lost. Chapter Four: Absence and the Absolute: The Revisions of Phantasy and Time-Consciousness. 143 Recapitulation: Cutting loose the baggage of the ‘Content for an Apprehension and Apprehension’ Schema. §1. The Concept of an Absolute Consciousness and the Embedment of Contents. §2. Transcendental Phenomenology and the Opening Up of a Closed Field. §3. The Breakthrough on the Contents of Consciousness. §3.1. ‘Consciousness through and through’ §3.2. Ambiguities of Immanence. §4. Primary and Secondary Memory revisited. §5. The New Account of Internal Time Consciousness. Chapter Five: The Breakthrough of the Life of Phantasy: Recollection and Pure Phantasy. 187 §1. Introducing the Pure Ego. §2. The Pending Problem of Belief in Presentifications. §3. Recollection and Internal Time Consciousness. §3.1. The Characterization of Recollection as Consciousness ‘Once Again’ following the Revision of the Schema. §3.2. Recollection as a Modified Appearance and a Directedness toward the Past Object. §3.3. Recollection as the Reproductive Modification of the Internal Consciousness of a Past Act. §4. The Differentiation of Recollection and Pure Phantasy on the basis of Internal Time-Consciousness. PART III: THE FREIBURG YEARS AND BEYOND Chapter Six: The Subject in Image: Husserl’s Revisiting of Image-Consciousness. 239 §1. Introducing the Concept of Perceptual Phantasy. §2. The Subject in Image. §3. Is There Temporality in Image?. §4. Revisiting the Theory of Image-consciousness as Depiction. §5. Pictoriality and Phantasy Temporality. Chapter Seven: ‘Myth that is Truest Memory:’ Phantasy and the Literary Work of Art. 303 §1. Theater and Drama. §2. Is the Literary Work of Art Itself Imaginary? §3. The Literary Work as an Idea with a Temporality. §4. Phantasy and the Consciousness of Possibility. Parting Words on Poetry and Philosophy. Bibliography: 361. (shrink)

For decades, student ratings of university faculty have been used by administrators in high stakes faculty employment decisions such as tenure, promotion, contract renewal and reappointment, and merit pay. However, virtually no attention has been paid to the ethical questions of using ratings in employment decisions. Instead, the ratings literature is generally limited to psychometric issues such as whether a given student ratings instrument exhibits the statistical properties of reliability and validity. There is no consensus understanding of teaching effectiveness, the (...) very attribute that students are alleged to “evaluate.” What students are actually doing when they complete a ratings form—whether measuring, evaluating, reporting, judging, opining, etc.—remains unsettled in the ratings literature. If ratings are surveys of student satisfaction, they have no logical or ethical connection with teaching expertise. I argue that the administrative use of student ratings in faculty employment decisions violates basic moral principles including nonmaleficence, beneficence, professional autonomy and clinical independence, and multiple aspects of justice including due care, truthfulness, and equitable treatment. These ethical violations rule against any administrative use of student ratings in faculty employment decisions, including the “use with caution in conjunction with other evaluative methods” deployment of student ratings. My conclusion is that such use should be immediately and universally terminated. Formative use of student questionnaires as part of ordinary instructional communication and feedback between instructor and students is a separate issue and outside of the scope of this paper. (shrink)

In this paper we give a differential lifting principle which provides a general method to geometrically axiomatize the model companion (if it exists) of some theories of differential topological fields. The topological fields we consider here are in fact topological systems in the sense of van den Dries, and the lifting principle we develop is a generalization of the geometric axiomatization of the theory DCF₀ given by Pierce and Pillay. Moreover, it provides a geometric alternative to the axiomatizations obtained by (...) Tressl and Guzy/Point in separate papers where the authors also build general schemes of axioms for some model complete theories of differential fields. We first characterize the existentially closed models of a given theory of differential topological fields and then, under an additional hypothesis of largeness, we show how to modify this characterization to get a general scheme of first-order axioms for the model companion of any large theory of differential topological fields. We conclude with an application of this lifting principle proving that, in existentially closed models of a large theory of differential topological fields, the jet-spaces are dense in their ambient topological space. (shrink)

Abstract.In this paper we show that the theory of fields together with an integrally closed subring, the theory of formally real fields with a real holomorphy ring and the theory of formally \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $p$\end{document}-adic fields with a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $p$\end{document}-adic holomorphy ring have no model companions in the language of fields augmented by a unary predicate for the corresponding ring.

We study algebraic and model-theoretic properties of existentially closed fields with an action of a fixed finite group. Such fields turn out to be pseudo-algebraically closed in a rather strong sense. We place this work in a more general context of the model theory of fields with a group scheme action.

Shepherdson [14] showed that for a discrete ordered ring I, I is a model of IOpen iff I is an integer part of a real closed ordered field. In this paper, we consider integer parts satisfying PA. We show that if a real closed ordered field R has an integer part I that is a nonstandard model of PA (or even IΣ₄), then R must be recursively saturated. In particular, the real closure of I, RC (I), (...) is recursively saturated. We also show that if R is a countable recursively saturated real closed ordered field, then there is an integer part I such that R = RC(I) and I is a nonstandard model of PA. (shrink)

Management scholars and social scientists investigate dynamics of subjective fairness perceptions in the workplace under the umbrella term “organizational justice.” Philosophers and ethicists, on the other hand, think of justice as a normative requirement in societal relationships with conflicting interests. Both ways of looking at justice have neither remained fully separated nor been clearly integrated. It seems that much could be gained and learned by more closely integrating the ethical and the empirical fields of justice. On the other hand, it (...) may simply not be possible to bridge the divide between the subjective empirical and the normative prescriptive justice as both fields pose different questions and rely on different assumptions and methods. In this paper, we propose a “reconciliation” model, as a third way of considering justice in the workplace, taking into account normative and psychological issues pertaining to justice. Through applying a reconciliation model, we provide a new way of looking at the interconnections between justice philosophy and organizational justice that could advance future research in both fields. Our model also implies that justice researchers can and should be concerned with the moral implications of their own subject of research. (shrink)

Let us call an integer part of an ordered field any subring such that every element of the field lies at distance less than 1 from a unique element of the ring. We show that every real closed field has an integer part.

The following conjecture is due to Shelah–Hasson: Any infinite strongly NIP field is either real closed, algebraically closed, or admits a non‐trivial definable henselian valuation, in the language of rings. We specialise this conjecture to ordered fields in the language of ordered rings, which leads towards a systematic study of the class of strongly NIP almost real closed fields. As a result, we obtain a complete characterisation of this class.

In the field of philosophy, the common view of philosophy as an essentially Western discipline persists even today, while non-Western philosophy tends to be undervalued and not investigated seriously. In the field of Japanese studies, in turn, research on Japanese philosophy tends to be reduced to a matter of projecting existing stereotypes of alleged Japanese cultural uniqueness through the reading of texts. In Nothingness in the Heart of Empire: The Moral and Political Philosophy of the Kyoto School in (...) Imperial Japan, Harumi Osaki resists both these tendencies. She closely interprets the wartime discourses of the Kyoto School, a group of modern Japanese philosophers who drew upon East Asian traditions as well as Western philosophy. Her book lucidly delves into the non-Western forms of rationality articulated in such discourses, and reveals the problems inherent in them as the result of these philosophers' engagements in Japan's wartime situation, without cloaking these problems under the pretense of "Japanese cultural uniqueness." In addition, in a manner reminiscent of the controversy surrounding Martin Heidegger's involvement with Nazi Germany, the book elucidates the political implications of the morality upheld by the Kyoto School and its underlying metaphysics. As such, this book urges dialogue beyond the divide between Western and non-Western philosophies, and beyond the separation between "lofty" philosophy and "common" politics. (shrink)

Recall that a group G has finitely satisfiable generics (fsg) or definable f-generics (dfg) if there is a global type p on G and a small model $M_0$ such that every left translate of p is finitely satisfiable in $M_0$ or definable over $M_0$, respectively. We show that any abelian group definable in a p-adically closed field is an extension of a definably compact fsg definable group by a dfg definable group. We discuss an approach which might prove (...) a similar statement for interpretable abelian groups. In the case where G is an abelian group definable in the standard model $\mathbb {Q}_p$, we show that $G^0 = G^{00}$, and that G is an open subgroup of an algebraic group, up to finite factors. This latter result can be seen as a rough classification of abelian definable groups in $\mathbb {Q}_p$. (shrink)

Jan Krajíček posed the following problem: Is there is a generalization result in the theory of real closed fields of the form: If A is provable in length k for all n ϵ ω , then A is provable? It is argued that the answer to this question depends on the particular formulation of the “theory of real closed fields.” Four distinct formulations are investigated with respect to their generalization behavior. It is shown that there is a positive (...) answer to Krajíček's question for 1. the axiom system RCF of Artin-Schreier with Gentzen's LK as underlying logical calculus, 2. RCF with the variant LK B of LK allowing introduction of several quantifiers of the same type in one step, 3. LK B and the first-order schemata corresponding to Dedekind cuts and the supremum principle. A negative answer is given for 4. any system containing the schema of extensionality. (shrink)

Somewhat in contrast to their proposal of two separate somatosensory streams, Dijkerman & de Haan (D&dH) propose that tactile recognition involves active manual exploration, and therefore involves parietal cortex. I argue that interactions from perception for action to object recognition can be found also in vision. Furthermore, there is evidence that perception for action and perception for recognition rely on similar processing principles.

In this paper, we consider reducts of p-adically closed fields. We introduce a notion of shadows: sets Mf={∈K2∣|y|=|f|}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${M_f = \{ \in K^2 \mid |y| = |f|\}}$$\end{document}, where f is a semi-algebraic function. Adding symbols for such sets to a reduct of the ring language, we obtain expansions of the semi-affine language where multiplication is nowhere definable, thus giving a negative answer to a question posed by Marker, Peterzil and Pillay. The (...) second main result of this paper is the fact that in p-adic fields, full multiplication becomes definable if we add a rational function to the semi-affine language. (shrink)

We give an intuitionistic axiomatisation of real closed fields which has the constructive reals as a model. The main result is that this axiomatisation together with just the decidability of the order relation gives the classical theory of real closed fields. To establish this we rely on the quantifier elimination theorem for real closed fields due to Tarski, and a conservation theorem of classical logic over intuitionistic logic for geometric theories.

We consider interpretable topological spaces and topological groups in a p-adically closed field K. We identify a special class of “admissible topologies” with topological tameness properties like generic continuity, similar to the topology on definable subsets of Kn. We show that every interpretable set has at least one admissible topology, and that every interpretable group has a unique admissible group topology. We then consider definable compactness (in the sense of Fornasiero) on interpretable groups. We show that an interpretable (...) group is definably compact if and only if it has finitely satisfiable generics (fsg), generalizing an earlier result on definable groups. As a consequence, we see that fsg is a definable property in definable families of interpretable groups, and that any fsg interpretable group defined over Qp is definably isomorphic to a definable group. (shrink)

We construct and study structures imitating the field of complex numbers with exponentiation. We give a natural, albeit non first-order, axiomatisation for the corresponding class of structures and prove that the class has a unique model in every uncountable cardinality. This gives grounds to conjecture that the unique model of cardinality continuum is isomorphic to the field of complex numbers with exponentiation.

We introduce a new construction of real closed fields by using an elementary extension of an ordered field with an integer part satisfying. This method can be extend to a finite extension of an ordered field with an integer part satisfying. In general, a field obtained from our construction is either real closed or algebraically closed, so an analogy of Ostrowski's dichotomy holds. Moreover we investigate recursive saturation of an o‐minimal extension of a real (...) closed field by finitely many function symbols. (shrink)