It follows directly from Shelah’s structure theory that if T is a classifiable theory, then the isomorphism type of any model of T is determined by the theory of that model in the language L∞,ω1. Leo Harrington asked if one could improve this to the logic L∞, In [S. Shelah, Characterizing an -saturated model of superstable NDOP theories by its L∞,-theory, Israel Journal of Mathematics 140 61–111] Shelah gives a partial positive answer, showing that for T a countable superstable NDOP (...) theory, two -saturated models of T are isomorphic if and only if they have the same L∞,-theory. We give here a negative answer to the general question by constructing two classifiable theories, each with 21 pairwise non-isomorphic models of cardinality 1, which are all L∞,-equivalent, a shallow depth 3 ω-stable theory and a shallow NOTOP depth 1 superstable theory. In the other direction, we show that in the case of an ω-stable depth 2 theory, the L∞,-theory is enough to describe the isomorphism type of all models. (shrink)

We give some general criteria for the stable embeddedness of a definable set. We use these criteria to establish the stable embeddedness in algebraically closed valued fields of two definable sets: The set of balls of a given radius r < 1 contained in the valuation ring and the set of balls of a given multiplicative radius r < 1. We also show that in an algebraically closed valued field a 0-definable set is stably embedded if and only if its (...) algebraic closure is stably embedded. (shrink)

Reviewed Works:B. I. Zil'ber, L. Pacholski, J. Wierzejewski, A. J. Wilkie, Totally Categorical Theories: Structural Properties and the Non-Finite Axiomatizability.B. I. Zil'ber, Strongly Minimal Countably Categorical Theories.B. I. Zil'ber, Strongly Minimal Countably Categorical Theories. II.B. I. Zil'ber, Strongly Minimal Countably Categorical Theories. III.B. I. Zil'ber, E. Mendelson, Totally Categorical Structures and Combinatorial Geometries.B. I. Zil'ber, The Structure of Models of Uncountably Categorical Theories.

We give examples of (i) a simple theory with a formula (with parameters) which does not fork over [Formula: see text] but has [Formula: see text]-measure 0 for every automorphism invariant Keisler measure [Formula: see text] and (ii) a definable group [Formula: see text] in a simple theory such that [Formula: see text] is not definably amenable, i.e. there is no translation invariant Keisler measure on [Formula: see text]. We also discuss paradoxical decompositions both in the setting of discrete groups (...) and of definable groups, and prove some positive results about small theories, including the definable amenability of definable groups. (shrink)

A set of axioms is presented defining an ‘analytic Zariski structure’, as a generalisation of Hrushovski and Zilber’s Zariski structures. Some consequences of the axioms are explored. A simple example of a structure constructed using Hrushovski’s method of free amalgamation is shown to be a non-trivial example of an analytic Zariski structure. A number of ‘quasi-analytic’ results are derived for this example e.g. analogues of Chow’s theorem and the proper mapping theorem.

In [E. Hrushovski, D. Palacín and A. Pillay, On the canonical base property, Selecta Math. (N.S.) 19(4) (2013) 865–877], Hrushovski and the authors proved, in a certain finite rank environment, that rigidity of definable Galois groups implies that [Formula: see text] has the canonical base property in a strong form; “internality to” being replaced by “algebraicity in”. In the current paper, we give a reasonably robust definition of the “strong canonical base property” in a rather more general finite (...) rank context than [E. Hrushovski, D. Palacín and A. Pillay, On the canonical base property, Selecta Math. (N.S.) 19(4) (2013) 865–877], and prove its equivalence with rigidity of the relevant definable Galois groups. The new direction is an elaboration on the old result that [Formula: see text]-based groups are rigid. (shrink)

We present a self-contained exposition of the basic aspects of simple theories while developing the fundamentals of forking calculus. We expound also the deeper aspects of S. Shelah's 1980 paper Simple unstable theories. The concept of weak dividing has been replaced with that of forking. The exposition is from a contemporary perspective and takes into account contributions due to S. Buechler, E. Hrushovski, B. Kim, O. Lessmann, S. Shelah and A. Pillay.

In this paper, we introduce an AEC framework for studying fields with commuting automorphisms. Fields with commuting automorphisms are closely related to difference fields. Some authors define a difference ring (or field) as a ring (or field) together with several commuting endomorphisms, while others only study one endomorphism. Z. Chatzidakis and E. Hrushovski have studied in depth the model theory of ACFA, the model companion of difference fields with one automorphism. Our fields with commuting automorphisms generalize this setting. We (...) have several automorphisms and they are required to commute. Hrushovski has proved that in the case of fields with two or more commuting automorphisms, the existentially closed models do not necessarily form a first order model class. In the present paper, we introduce FCA-classes, an AEC framework for studying the existentially closed models of the theory of fields with commuting automorphisms. We prove that an FCA-class has AP and JEP and thus a monster model, that Galois types coincide with existential types in existentially closed models, that the class is homogeneous, and that there is a version of type amalgamation theorem that allows to combine three types under certain conditions. Finally, we use these results to show that our monster model is a simple homogeneous structure in the sense of S. Buechler and O. Lessman (this is a non-elementary analogue for the classification theoretic notion of a simple first order theory). (shrink)

Generalising Hrushovski's fusion technique we construct the free fusion of two strongly minimal theories T₁, T₂ intersecting in a totally categorical sub-theory T₀. We show that if, e.g., T₀ is the theory of infinite vector spaces over a finite field then the fusion theory Tω exists, is complete and ω-stable of rank ω. We give a detailed geometrical analysis of Tω, proving that if both T₁, T₂ are 1-based then, Tω can be collapsed into a strongly minimal theory, if (...) some additional technical conditions hold—all trivially satisfied if T₀ is the theory of infinite vector spaces over a finite field Fq. (shrink)

We consider${\cal S}$, the class of finite semilattices;${\cal T}$, the class of finite treeable semilattices; and${{\cal T}_m}$, the subclass of${\cal T}$which contains trees with branching bounded bym. We prove that${\cal E}{\cal S}$, the class of finite lattices with linear extensions, is a Ramsey class. We calculate Ramsey degrees for structures in${\cal S}$,${\cal T}$, and${{\cal T}_m}$. In addition to this we give a topological interpretation of our results and we apply our result to canonization of linear orderings on finite semilattices. In (...) particular, we give an example of a Fraïssé class${\cal K}$which is not a Hrushovski class, and for which the automorphism group of the Fraïssé limit of${\cal K}$is not extremely amenable but is uniquely ergodic. (shrink)

We propose a criterion to regard a property of a theory (in first or second order logic) as virtuous: the property must have significant mathematical consequences for the theory (or its models). We then rehearse results of Ajtai, Marek, Magidor, H. Friedman and Solovay to argue that for second order logic, ‘categoricity’ has little virtue. For first order logic, categoricity is trivial; but ‘categoricity in power’ has enormous structural consequences for any of the theories satisfying it. The stability hierarchy extends (...) this virtue to other complete theories. The interaction of model theory and traditional mathematics is examined by considering the views of such as Bourbaki, Hrushovski, Kazhdan, and Shelah to flesh out the argument that the main impact of formal methods on mathematics is using formal definability to obtain results in ‘mainstream’ mathematics. Moreover, these methods (e.g., the stability hierarchy) provide an organization for much mathematics which gives specific content to dreams of Bourbaki about the architecture of mathematics. (shrink)

Several attempts have been done to distinguish “positive” information in an arbitrary first order theory, i.e., to find a well behaved class of closed sets among the definable sets. In many cases, a definable set is said to be closed if its conjugates are sufficiently distinct from each other. Each such definition yields a class of theories, namely those where all definable sets are constructible, i.e., boolean combinations of closed sets. Here are some examples, ordered by strength:Weak normality describes a (...) rather small class of theories which are well understood by now (see, for example, [P]). On the other hand, normalization is so weak that all theories, in a suitable context, are normalizable (see [HH]). Thus equational theories form an interesting intermediate class of theories. Little work has been done so far. The original work of Srour [S1, S2, S3] adopts a point of view that is closer to universal algebra than to stability theory. The fundamental definitions and model theoretic properties can be found in [PS], though some easy observations are missing there. Hrushovski's example of a stable non-equational theory, the first and only one so far, is described in the unfortunately unpublished manuscript [HS]. In fact, it is an expansion of the free pseudospace constructed independently by Baudisch and Pillay in [BP] as an example of a strictly 2-ample theory. Strong equationality, defined in [Hr], is also investigated in [HS]. (shrink)

Model theory is concerned mainly, although not exclusively, with infinite structures. In recent years, finite structures have risen to greater prominence, both within the context of mainstream model theory, e.g., in work of Lachlan, Cherlin, Hrushovski, and others, and with the advent of finite model theory, which incorporates elements of classical model theory, combinatorics, and complexity theory. The purpose of this survey is to provide an overview of what might be called the model theory of finite structures. Some topics (...) in finite model theory have strong connections to theoretical computer science, especially descriptive complexity theory. In fact, it has been suggested that finite model theory really is, or should be, logic for computer science. These connections with computer science will, however, not be treated here.It is well-known that many classical results of ‘infinite model theory’ fail over the class of finite structures, including the compactness and completeness theorems, as well as many preservation and interpolation theorems. The failure of compactness in the finite, in particular, means that the standard proofs of many theorems are no longer valid in this context. At present, there is no known example of a classical theorem that remains true over finite structures, yet must be proved by substantially different methods. It is generally concluded that first-order logic is ‘badly behaved’ over finite structures.From the perspective of expressive power, first-order logic also behaves badly: it is both too weak and too strong. Too weak because many natural properties, such as the size of a structure being even or a graph being connected, cannot be defined by a single sentence. Too strong, because every class of finite structures with a finite signature can be defined by an infinite set of sentences. Even worse, every finite structure is defined up to isomorphism by a single sentence. In fact, it is perhaps because of this last point more than anything else that model theorists have not been very interested in finite structures. Modern model theory is concerned largely with complete first-order theories, which are completely trivial here. (shrink)

The thesis pseudofinite structures and counting dimensions is about the model theory of pseudofinite structures with the focus on groups and fields. The aim is to deepen our understanding of how pseudofinite counting dimensions can interact with the algebraic properties of underlying structures and how we could classify certain classes of structures according to their counting dimensions. Our approach is by studying examples. We treat three classes of structures: The first one is the class of H-structures, which are generic expansions (...) of existing structures. We give an explicit construction of pseudofinite H-structures as ultraproducts of finite structures. The second one is the class of finite difference fields. We study properties of coarse pseudofinite dimension in this class, show that it is definable and integer-valued and build a partial connection between this dimension and transformal transcendence degree. The third example is the class of pseudofinite primitive permutation groups. We generalise Hrushovski’s classical classification theorem for stable permutation groups acting on a strongly minimal set to the case where there exists an abstract notion of dimension, which includes both the classical model theoretic ranks and pseudofinite counting dimensions. In this thesis, we also generalise Schlichting’s theorem for groups to the case of approximate subgroups with a notion of commensurability.prepared by Tingxiang Zou.E-mail: [email protected]: https://tel.archives-ouvertes.fr/tel-02283810/document. (shrink)

Generalizedn-gons are certain geometric structures (incidence geometries) that generalize the concept of projective planes (the nontrivial generalized 3-gons are exactly the projective planes).In a simplified world, every generalizedn-gon of finite Morley rank would be an algebraic one, i.e., one of the three families described in [9] for example. To our horror, John Baldwin [2], using methods discovered by Hrushovski [7], constructed ℵ1-categorical projective planes which are not algebraic. The projective planes that Baldwin constructed fail to be algebraic in a (...) dramatic way.Indeed, every algebraic projective plane over an algebraically closed field is Desarguesian [12]. In particular, an algebraically closed field (isomorphic to the base field) can be interpreted in every one of them. However, in the projective planes that Baldwin constructed, one cannot even interpret an infinite group.In this article we show that the same phenomenon occurs for the generalizedn-gons ifn≥ 3 is an odd integer. For each suchnwe constructmany nonisomorphic generalizedn-gons of finite Morley rank that do not interpret an infinite group. As one may expect, our method is inspired by Hrushovski and Baldwin, and we follow Baldwin's line of approach. Quite often our proofs are a verification of the fact that the proofs of Baldwin [2] forn= 3 carry over to an arbitrary positive odd integern(which is sometimes far from being obvious). As in [2], we begin by defining a certain collection of finite graphsK* and a binary relation ≤ on these graphs. We show that (K*, ≤) satisfies the amalgamation property. (shrink)

Background Journal editors are responsible for what they publish and therefore have a duty to correct the record if published work is found to be unreliable. One method for such correction is retraction of an article. Anecdotal evidence suggested a lack of consistency in journal policies and practices regarding retraction. In order to develop guidelines, we reviewed retractions in Medline to discover how and why articles were retracted. Methods We retrieved all available Medline retractions from 2005 to 2008 and a (...) one-in-three random selection of those from 1988 to 2004. This yielded 312 retractions (from a total of 870). Details of the retraction including the reason for retraction were recorded by two investigators. Results Medline retractions have increased sharply since 1980 and currently represent 0.02% of included articles. Retractions were issued by authors (63%), editors (21%), journals (6%), publishers (2%) and institutions (1%). Reasons for retraction included honest error or non-replicable findings (40%), research misconduct (28%), redundant publication (17%) and unstated/unclear (5%). Some of the stated reasons might have been addressed by corrections. Conclusions Journals' retraction practices are not uniform. Some retractions fail to state the reason, and therefore fail to distinguish error from misconduct. We have used our findings to inform guidelines on retractions. (shrink)

Background: Breaches of publication ethics such as plagiarism, data fabrication and redundant publication are recognised as forms of research misconduct that can undermine the scientific literature. We surveyed journal editors to determine their views about a range of publication ethics issues. Methods: Questionnaire sent to 524 editors-in-chief of Wiley-Blackwell science journals asking about the severity and frequency of 16 ethical issues at their journals, their confidence in handling such issues, and their awareness and use of guidelines. Results: Responses were obtained (...) from 231 editors (44%), of whom 48% edited healthcare journals. The general level of concern about the 16 issues was low, with mean severity scores of <1 (on a scale of 0–3) for all but one. The issue of greatest concern (mean score 1.19) was redundant publication. Most editors felt confident in handling the issues, with <15% feeling “not at all confident” for all but one of the issues (gift authorship, 22% not confident). Most editors believed such problems occurred less than once a year and >20% of the editors stated that 12 of the 16 items never occurred at their journal. However, 13%–47% did not know the frequency of the problems. Awareness and use of guidelines was generally low. Most editors were unaware of all except other journals’ instructions. Conclusions: Most editors of science journals seem not very concerned about publication ethics and believe that misconduct occurs only rarely in their journals. Many editors are unfamiliar with available guidelines but would welcome more guidance or training. (shrink)

We show that every infinite computable partial ordering has either an infinite Δ 0 2 chain or an infinite Π 0 2 antichain. Our main result is that this cannot be improved: We construct an infinite computable partial ordering that has neither an infinite Δ 0 2 chain nor an infinite Δ 0 2 antichain.

This paper assesses several prominent recent attacks on the view that epistemic justification is conceptually prior to knowledge. I argue that this view—call it the Received View (RV)—emerges from these attacks unscathed. I start with Timothy Williamson’s two strongest arguments for the claim that all evidence is knowledge (E>K), which impugns RV when combined with the claim that justification depends on evidence. One of Williamson’s arguments assumes a false epistemic closure principle; the other misses some alternative (to E>K) explanations of (...) a putative fact about the evidence a particular subject has. Next, I neutralize each of Jonathan Sutton’s three recent arguments to the conclusion that any justified belief constitutes knowledge. Finally, I consider a recent analysis of justification due to Alexander Bird, according to which justified belief is possible knowledge. I argue that Bird’s analysis delivers neither a sufficient nor (more importantly) a necessary condition for justification. [Word count: 149]. (shrink)

Studies examining the ways in which the training of engineers and scientists shapes their research strategies and scientific identities.

Institutional investors are increasingly becoming active owners through voting their shares and engaging in dialogue with investee companies to improve corporate environmental, social and corporate governance (ESG) performance. This article applies a model of stakeholder salience to the shareholder context, analysing the attributes of power, legitimacy and urgency, to determine the factors that are likely to enhance shareholder salience. It is found that a strong business case and the values of the managers of investee companies are likely to be the (...) most important contributors to shareholder salience. (shrink)

Clinical diagnostic medicine is an experimental science based on observation, hypothesis making, and testing. It is an use dynamic process that involves observation and summary, diagnostic conjectures, testing, review, observation and summary, new or revised conjectures, i.e. it is an iterative process. It can then be said that diagnostic hypotheses are also ‘observation-laden’. My aim is to enlarge on the strategies of medical diagnosis as these are meshed in training and clinical experience—that is, to describe the patterns of reasoning used (...) by experienced clinicians under different diagnostic circumstances and how these patterns of inquiry allow further insight into the evaluation and treatment of patients. I do not aim to present a theory and illustrate it with examples; I wish rather am to let a realistic example, similar to actual clinical scenarios, direct the exposition. To this end, I introduce an account of medical diagnosis—briefly comparing and contrasting it to other accounts—in order to focus on discussing the process of diagnosis through a detailed clinical case. (shrink)

Modern societies depend on a growing production of scientific knowledge, which is based on the functional differentiation of science into still more specialised scientific disciplines and subdisciplines. This is the basis for the paradox of scientific expertise: The growth of science leads to a fragmentation of scientific expertise. To resolve this paradox, the present paper investigates three hypotheses: 1) All scientific knowledge is perspectival. 2) The perspectival structure of science leads to specific forms of knowledge asymmetries. 3) Such perspectival knowledge (...) asymmetries must be handled through second order perspectives. We substantiate these hypotheses on the basis of a perspectivist philosophy of science grounded in Peircean semiotics and autopoietic systems theory. Perspectival knowledge asymmetries are an unavoidable and necessary part of the growth of scientific knowledge, and more awareness of this fact can help avoid blind and futile struggles between scientific perspectives, and direct efforts toward more appropriate ways of handling these fundamental knowledge asymmetries. Concretely, we show how different kinds of scientific knowledge, expertise, disagreement and learning can be correlated to the perspectival structure of science, and propose how polyocular communication based on observations of the observations made by specialised perspectives can be used to handle such perspectival knowledge asymmetries. This can help overcome the observed problems in carrying out cross-disciplinary research and in the collective use of different kinds of scientific expertise, and thereby make society better able to solve complex, real-world problems. (shrink)

I have stressed here and elsewhere that perspective cannot and need not claim to represent the world "as we see it." The perceptual constancies which make us underrate the degree of objective diminutions with distance, it turns out, constitute only one of the factors refuting this claim. The selectivity of vision can now be seen to be another. There are many ways of "seeing the world," but obviously the claim would have to relate to the "snapshot vision" of the stationary (...) single eye. To ask, as it has so often been asked, whether this eye sees the world in the form of a hollow sphere or of a projection plane makes little sense, for it sees neither. The one point in focus can hardly be said to be either curved or flat, and the remainder of the field of vision is too indistinct to permit a decision. True, we can shift the point of focus at will, but in doing so we lose the previous perception, and all that remains is its memory. Can we, and do we, compare the exact extension of these changing percepts in scanning a row of columns extended at right angles from the central line of vision—to mention the most recalcitrant of the posers of perspectival theory?1 I very much doubt it. The question refers to the convenient choice of projection planes, not to the experience of vision.· 1. I now prefer this formulation to my somewhat laboured discussion in Art and Illusion, chap. 8, sec. 4.E.H. Gombrich was director of the Warburg Institute and Professor of the History of the Classical Tradition at the University of London from 1959 to 1976. His many influential works include The Story of Art, Art and Illusion, Meditations on a Hobby Horse, The Sense of Order, and Ideals and Idols. An early version of "Standards of Truth" was presented at Swarthmore College in October 1976 at a symposium to mark the retirement of Professor Hans Wallach. His contributions to Critical Inquiry include "The Museum: Past, Present, and Future" , "Notes and Exchanges" , and, with Quentin Bell, "Canons and Values in the Visual Arts: A Correspondence". (shrink)

Louis Althusser remained until his death in 1990 the most controversial of the “master thinkers” who emerged from the turbulent Parisian intellectual scene of the 1960s. The publication of his bestselling posthumous “autobiography”, L'avenir dure longtemps, has now refueled some of these controversies. Hugely influential, whether lauded or vilified, Althusser occupies a unique place in contemporary philosophy. What is certain is that Althusserian themes and motifs continue to constitute a vital region in materialist thought. The Althusserian Legacy is the first (...) collective attempt to draw up a balance sheet, not on Althusser alone but on the questions that his work helped to bring to the forefront of Marxist theory. The volume brings together work in history, philosophy, economics, sociology, and literary criticism, all of it derived from or significantly inflected by Althusser. Taken together, the essays assess soberly, critically, but always generously, the full extent of his legacy. The volume contains a lengthy interview with Jacques Derrida, a long-time friend and colleague of Althusser at the Ecole Normale in Paris, and concludes with obituaries by Derrida and Gregory Elliott. Perhaps only now, more than a decade after his active intellectual life has come to a close, is it possible to render sound, just judgments on the meaning and significance of this much-debated body of work. The Althusserian Legacy is a rich beginning to that important task. (shrink)

In this paper I report a public discussion with E.P. Wigner that took place in 1987 at a conference on fundamental problems in Quantum Mechanics. In it Wigner clarified an idea that was widely attributed to him about consciousness playing a direct role in the quantum measurement process. He significantly revised that idea, and distanced himself from the earlier notion that consciousness plays a direct role.

We discuss the extent to which the visibility of the heavens was a necessary condition for the development of science, with particular reference to the measurement of time. Our conclusion is that while astronomy had significant importance, the growth of most areas of science was more heavily influenced by the accuracy of scientific instruments, and hence by current technology.