A structure (M, $ ,...) is called quasi-o-minimal if in any structure elementarily equivalent to it the definable subsets are exactly the Boolean combinations of 0-definable subsets and intervals. We give a series of natural examples of quasi-o-minimal structures which are not o-minimal; one of them is the ordered group of integers. We develop a technique to investigate quasi-o-minimality and use it to study quasi-o-minimal ordered groups (possibly with extra structure). Main results: (...) any quasi-o-minimal ordered group is abelian; any quasi-o-minimal ordered ring is a real closed field, or has zero multiplication; every quasi-o-minimal divisible ordered group is o-minimal; every quasi-o-minimal archimedian densely ordered group is divisible. We show that a counterpart of quasi-o-minimality in stability theory is the notion of theory of U-rank 1. (shrink)

We present a relatively simple description of binary, definable subsets of models of weakly quasi-o-minimal theories. In particular, we closely describe definable linear orders and prove a weak version of the monotonicity theorem. We also prove that weak quasi-o-minimality of a theory with respect to one definable linear order implies weak quasi-o-minimality with respect to any other such order.

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)

Hume has a special place in justifications of claims made for rational choice theory to offer a unified language and explanatory framework for the social sciences. He is invoked in support of the assumptions characterising the instrumental rationality of agents and the constancy of their motivations across different institutional settings. This paper explores the problems with the expansionary aims of rational choice theory through criticism of these appeals to Hume. First, Hume does not assume constancy. Moreover, Hume’s sensitivity (...) to the relationships between institutional setting and individual motivation owes something to the relative transparency of the plural language of vices and virtues that he employs. Second, rational choice theory’s minimal modification of Hume’s account of the relation of reason and the passions through the introduction of constraints of consistency on preferences is unstable. (shrink)

A minimal truthmaker for a given proposition is the smallest portion of reality which makes this proposition true. Minimal truthmakers are frequently mentioned in the literature, but there has been no systematic account of what they are or of their importance. In this article we shall clarify the notion of a minimal truthmaker and argue that there is reason to think that at least some propositions have minimal truthmakers. We shall then argue that the notion can (...) play a useful role in truthmaker theory, by helping to explain the truth of certain propositions as precisely as possible. (shrink)

Here we prove the existence of sheaf cohomology theory in arbitrary o-minimal structures.

Two-classes of phase-space spin quasi-distribution functions are introduced and discussed. The first class of these distributions is based on the delta function construction. It is shown that such a construction can be carried out for an arbitrary spin s and an arbitrary ordering of the spin operators. The second class of the spin distributions is constructed with the help of the spin coherent states. The connection of the spin coherent states to the Stratonovich formalism is established and discussed. It (...) is shown that the c-number phase-space description of quantum fluctuations provides a simple statistical picture of quantum fluctuations of spinoperators in terms of random directions on a unit sphere. For quantum states of the spin system the statistics of these random orientations is given by non-positive spin quasi-distribution functions. It is shown that the application of these spin quasi-distribution functions to the Einstein-Podolsky-Rosen correlations provide an insight into the quantum theory of measurement. (shrink)

A totally ordered group G is essentially periodic if for every definable non-trivial convex subgroup H of G every definable subset of G is equal to a finite union of cosets of subgroups of G on some interval containing an end segment of H; it is coset-minimal if all definable subsets are equal to a finite union of cosets, intersected with intervals. We study definable sets and functions in such groups, and relate them to the quasi-o-minimal groups (...) introduced in Belegradek et al. . Main results: An essentially periodic group G is abelian; if G is discrete, then definable functions in one variable are ultimately piecewise linear. A group such that every model elementarily equivalent to it is coset-minimal is quasi-o-minimal , and its definable functions in one variable are piecewise linear. (shrink)

Intersectionality theory explores the special sorts of disadvantage that arise as the result of occupying multiple disadvantaged demographic categories. One significant methodological problem for the quantitative study of intersectionality is the difficulty of acquiring data sets large enough to produce significant results when one is looking for intersectional effects. For this reason, we argue, simulation methods may be particularly useful to this branch of theorizing because they can generate precise predictions and causal dependencies in a relatively cheap way, and (...) can thus guide future empirical work. We illustrate this point through models which show that intersectional oppression can arise under conditions where social groups are disadvantaged in the emergence of bargaining norms. As we show, intersectional disadvantage can arise even when actors from all social categories are completely identical in terms of preferences and abilities. The conditions necessary to derive such disadvantage are relatively minimal. And when actors behave in ways that reflect stronger intersectional identities, the potential for disadvantage increases. (shrink)

We study the monoid of global invariant types modulo domination-equivalence in the context of o-minimal theories. We reduce its computation to the problem of proving that it is generated by classes...

This paper canvasses the various dimensions along which theories of truth may disagree about the extent to which truth is minimal.

The article begins with a selective outline of social theories of crisis. Such crisis diagnosis is important for general, societal argumentation. The current article positions normative-critical theories and Luhmann’s own version of system theory on opposite sides of the societal argument about the future of Europe and, generally, postnational society. The former supports moral and ethical visions of egalitarian pluralism, and the latter emphasizes the need to conform to the functional, communication logics of self-organizing social systems. It is then (...) proposed that if rival social theoretical traditions, the above named and others, are to engage in meaningful normative argument with one another, they must develop a minimal, shared understanding of what should constitute the theory of society. In pursuit of this goal, the article advances a shared, minimal, theoretical framework that addresses both the full spectrum of the normative culture of modernity and its critique, no matter from which social standpoint or theoretical tradition the latter is advanced. In conclusion, asserting its own normative standpoint, it is argued that both a European cultural model and a constitutional order based on cosmopolitan solidarity are needed. Such a model should assert moral principles of justice, equality, responsibility, and legitimacy that would adequately respect and include the full variety of milieus and associational forms in contemporary European society, not just those of the present elite. Such a model should not simply be abstractly asserted, though that is an essential first step, but its conditions of possibility should be supported by appropriate social research that would investigate capacities for transnational democratic innovation and the formation of substantive moral principles among European publics. (shrink)

For this, the second of four volumes comprising the papers submitted for publication by the invited participants to the Fifth International Congress of Logic, Methodology, and Philosophy of Science, held at the University of Western Ontario in 1975, the editors have selected papers concerned with "foundational problems" in the physical sciences, biology, psychology, and the social sciences. In spite of the wide range of papers included in the volume, the reader never learns exactly what constitutes a foundational problem in the (...) philosophy of science. Would it be reasonable to suppose that foundational problems focus upon the basic logical, epistemological, and metaphysical issues posed by the special sciences, to the exclusion of purely methodological and strictly historical issues? In fact, the contributions to this volume cover the entire scope of metaphysical, epistemological, logical, methodological, historical, and quasi-historical questions that are conventionally classified under the philosophy of science in contemporary academic discourse. There are essays on the genesis of the universe, the methodology of physics, and quantum mechanics; essays on various topics in the philosophy of biology—including the ontological status of species, theories and observation in developmental biology, determinism and teleology, and genetic information; essays on problems in the philosophy of psychology-including consciousness and the brain, causality and action, and methodological problems in the investigation of human activity; three essays on learning theory; and several essays on problems in the philosophy of social science—including methodology, axiological problems, and four papers on the concept of rationality which are principally concerned with J. Harsanyi’s researches on the contribution of Bayesian decision theory and recent developments in game theory to the understanding of rational behavior. (shrink)

[John Dupré] This paper attacks some prominent contemporary attempts to provide reductive accounts of ever wider areas of human behaviour. In particular, I shall address the claims of sociobiology (or evolutionary psychology) to provide a universal account of human nature, and attempts to subsume ever wider domains of behaviour within the scope of economics. I shall also consider some recent suggestions as to how these approaches might be integrated. Having rejected the imperialistic ambitions of these approaches, I shall briefly advocate (...) a more pluralistic approach to the understanding of human behaviour, and one which leaves some space for the possibility of genuine human autonomy. /// [John O'Neill] One response to Dupré's criticism of rational choice theory's unifying aspirations is that it is aimed at over-ambitious versions of the theory. Immodesty about the scope of rational choice theory may look more plausible given suitable modesty in assumptions about the rational agent. The paper examines problems with one immodest version of the theory-public choice theory-and show how these shed light on problems in modest versions employing minimal assumptions about the preference structure of rational agents. However, while rational choice theory may fail in its unifying ambitions, I argue those aspirations are defensible. (shrink)

[John Dupré] This paper attacks some prominent contemporary attempts to provide reductive accounts of ever wider areas of human behaviour. In particular, I shall address the claims of sociobiology to provide a universal account of human nature, and attempts to subsume ever wider domains of behaviour within the scope of economics. I shall also consider some recent suggestions as to how these approaches might be integrated. Having rejected the imperialistic ambitions of these approaches, I shall briefly advocate a more pluralistic (...) approach to the understanding of human behaviour, and one which leaves some space for the possibility of genuine human autonomy. /// [John O'Neill] One response to Dupré's criticism of rational choice theory's unifying aspirations is that it is aimed at over-ambitious versions of the theory. Immodesty about the scope of rational choice theory may look more plausible given suitable modesty in assumptions about the rational agent. The paper examines problems with one immodest version of the theory-public choice theory-and show how these shed light on problems in modest versions employing minimal assumptions about the preference structure of rational agents. However, while rational choice theory may fail in its unifying ambitions, I argue those aspirations are defensible. (shrink)

Irish Travellers constitute a pre-demographic-shift population living among a post-demographic-shift one. Their socio-medico profile identifies them as largely on fast life-history trajectories. In addition, they are strongly religious, highly sexually behaviorally dimorphic, with strong traditions of male-male competition and quasi-symbolic bride capture. Their male-male competitions thus allow for the comparative testing of a number of interesting theories pertaining to the nature and function of types of violence in society. As a pilot study, we used expert raters to analyze a (...) number of real-life bare-knuckle competitions in terms of the support said spectacles offered to theories of this sort of violence as reinforcing ideas of antisociality, hierarchical promotion, intersexual signaling, or maintenance of within-group equality. We found good evidence to support theories of within-group, prosocial hierarchical functions for these contests. Limitations and implications for future research, such as direct measurement of fitness, are discussed. (shrink)

In this paper we describe, in a purely algebraic language, truth-complete finite-valued propositional logical calculi extending the classical Boolean calculus. We also give a new proof of the Completeness Theorem for such calculi. We investigate the quasi-varieties of algebras playing an analogous role in the theory of these finite-valued logics to the role played by the variety of Boolean algebras in classical logic.

A totally ordered group G is called coset-minimal if every definable subset of G is a finite union of cosets of definable subgroups intersected with intervals with endpoints in G{±∞}. Continuing work in Belegradek et al. 1115) and Point and Wagner 261), we study coset-minimality, as well as two weak versions of the notion: eventual and ultimate coset-minimality. These groups are abelian; an eventually coset-minimal group, as a pure ordered group, is an ordered abelian group of finite regular (...) rank. Any pure ordered abelian group of finite regular rank is ultimately coset-minimal and has the exchange property; moreover, every definable function in such a group is piecewise linear. Pure coset-minimal and eventually coset-minimal groups are classified. In a discrete coset-minimal group every definable unary function is piece-wise linear 261), where coset-minimality of the theory of the group was required). A dense coset-minimal group has the exchange property ); moreover, any definable unary function is piecewise linear, except possibly for finitely many cosets of the smallest definable convex nonzero subgroup. Finally, we give some examples and open questions. (shrink)

Machine generated contents note: Part 1 - The Constituent Disciplines of Cognitive Science -- Philosophical Epistemology -- Glossary -- 1.0 What is Philosophical Epistemology? -- 1.1 The reduced history of Philosophy Part I - The Classical Age -- 1.2 Mind and World - The problem of objectivity -- 1.3 The reduced history of Philosophy Part II - The twentieth century -- 1.4 The philosophy of Cognitive Science -- 1.5 Mind in Philosophy: summary -- 1.6 The Nolanian Framework (so far) -- (...) Psychology -- 2.0 Why is Psychology so difficult? -- 2.1 A brief history of Experimental Psychology -- 2.2 Methodologies in Psychology -- 2.3 Perception -- 2.4 Memory -- 2.5 Mind in Psychology -- Linguistics -- 3.0 Introduction -- 3.1 Why Linguistics? -- 3.2 Computation and Linguistics -- 3.3 The main grammatical theories -- 3.4 Language development and linguistics -- 3.5 Toward a definition of context -- 3.6 The multifarious uses of Language -- 3.7 Linguistics and Computational Linguistics -- 3.8 Language and other symbol systems -- 3.9 On the notion of context -- 3.10 Mind in Linguistics: summary -- Neuroscience -- 4.0 The constituent disciplines of Neuroscience -- 4.1 The methodology of Neuroscience -- 4.2 Gross Neuroanatomy -- 4.3 Some relevant findings -- 4.4 Connectionism (PDP) -- 4.5 The victory of Neuroscience? -- 4.6 Mind in Neuroscience: summary -- Artificial Intelligence -- 5.0 Introduction -- 5.1 Al and Cognitive Science -- 5.2 Skeptics and their techniques -- 5.3 Al as Computer Science -- 5.4 Al as software -- 5.5 The current methodological debate -- 5.6 Context, syntax and semantics -- 5.7 Mind in Al -- 5.8 Texts on Al -- Etholoqy and Ethnoscience -- 6.1 Etology -- 6.2 Ethnoscience -- 6.3 Mind in Ethology arid Ethnoscience -- Part II - A New Foundation for Cognitive Science -- - Symbol Systems -- 7.1 Characteristics of symbol systems -- 7.2 Context and the layers of symbol systems -- 7.3 Mind and symbol systems -- Consciousness and Selfhood -- 8.0 Introduction -- 8.1 Cognitive views -- 8.2 What is at stake? -- 8.3 Consciousness as treated in Philosophy -- 8.4 The Development of Selfhood -- 8.5 The minimal requirements for this theory -- 8.6 Self as a filter -- 8.7 Self and motivation -- 8.8 Conclusions -- 8.9 Recent developments -- Cognitive Science and the Search for Mind -- 9.1 Introduction -- 9.2 Review -- 9.3 A Theory of Mind anyone? -- 9.4 Foundational considerations -- 9.5 Coda: the Nolanian Framework. (shrink)

We consider the sets definable in the countable models of a weakly o-minimal theory T of totally ordered structures. We investigate under which conditions their Boolean algebras are isomorphic , in other words when each of these definable sets admits, if infinite, an infinite coinfinite definable subset. We show that this is true if and only if T has no infinite definable discrete subset. We examine the same problem among arbitrary theories of mere linear orders. Finally we prove (...) that, within expansions of Boolean lattices, every weakly o-minimal theory is p-ω-categorical. (shrink)

The aim of this paper is to develop the theory of groups definable in the p-adic field ${{\mathbb {Q}}_p}$, with “definable f-generics” in the sense of an ambient saturated elementary extension of ${{\mathbb {Q}}_p}$. We call such groups definable f-generic groups.So, by a “definable f-generic” or $dfg$ group we mean a definable group in a saturated model with a global f-generic type which is definable over a small model. In the present context the group is definable over ${{\mathbb {Q}}_p}$, (...) and the small model will be ${{\mathbb {Q}}_p}$ itself. The notion of a $\mathrm {dfg}$ group is dual, or rather opposite to that of an $\operatorname {\mathrm {fsg}}$ group (group with “finitely satisfiable generics”) and is a useful tool to describe the analogue of torsion-free o-minimal groups in the p-adic context.In the current paper our group will be definable over ${{\mathbb {Q}}_p}$ in an ambient saturated elementary extension $\mathbb {K}$ of ${{\mathbb {Q}}_p}$, so as to make sense of the notions of f-generic type, etc. In this paper we will show that every definable f-generic group definable in ${{\mathbb {Q}}_p}$ is virtually isomorphic to a finite index subgroup of a trigonalizable algebraic group over ${{\mathbb {Q}}_p}$. This is analogous to the o-minimal context, where every connected torsion-free group definable in $\mathbb {R}$ is isomorphic to a trigonalizable algebraic group [5, Lemma 3.4]. We will also show that every open definable f-generic subgroup of a definable f-generic group has finite index, and every f-generic type of a definable f-generic group is almost periodic, which gives a positive answer to the problem raised in [28] of whether f-generic types coincide with almost periodic types in the p-adic case. (shrink)

A subset A $\subseteq$ M of a totally ordered structure M is said to be convex, if for any a, b $\in A: [a . A complete theory of first order is weakly o-minimal (M. Dickmann [D]) if any model M is totally ordered by some $\emptyset$ -definable formula and any subset of M which is definable with parameters from M is a finite union of convex sets. We prove here that for any model M of a weakly (...) o-minimal theory T, any expansion M + of M by a family of unary predicates has a weakly o-minimal theory iff the set of all realizations of each predicate is a union of a finite number of convex sets (Theorem 63), that solves the Problem of Cherlin-Macpherson-Marker-Steinhorn [MMS] for the class of weakly o-minimal theories. (shrink)

Orthogonality of all families of pairwise weakly orthogonal 1-types for ℵ0-categorical weakly o-minimal theories of finite convexity rank has been proved in 6. Here we prove orthogonality of all such families for binary 1-types in an arbitrary ℵ0-categorical weakly o-minimal theory and give an extended criterion for binarity of ℵ0-categorical weakly o-minimal theories . © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

Objective: To provide an ethical analysis of whether the Ethiopian and Tanzanian national HIV/AIDS treatment guidelines can be considered legitimate and fair rationing tools.Method: Qualitative study and ethical analysis involving guideline documents and interviews with nine key members involved in the development of the guidelines. The analysis followed an editing organising style. The theoretical framework was a guideline-specific framework based on theories of just resource allocation in healthcare and conditions that ensure fair processes in guideline development. According to this framework, (...) legitimate rationing requires reasons for patient selection to be explicit, public and relevant, and decisions must be open to question and revision.Results: The only explicit rationing criteria that both guidelines recommended were clinical antiretroviral treatment indications. Explicit non-clinical rationing criteria were expressed in a separate Ethiopian implementation guideline. Neither of the guideline development processes fully satisfies minimal requirements of procedural fairness. There is a lack of transparency. The reasons for decisions are rarely given and are not publicly available. This reduces the opportunity for public questioning, debate and revisions. The guidelines were based on expert opinion and consensus. Recommendations from the WHO were copied without much discussion, disagreement or adjustment.Conclusions: The two national HIV treatment guidelines discussed are de facto mechanisms for rationing but were developed using methods that do not fully satisfy the requirements of fair processes. (shrink)

BackgroundSocio-cultural perceptions surrounding death have profoundly changed since the 1950s with development of modern intensive care and progress in solid organ transplantation. Despite broad support for organ transplantation, many fundamental concepts and practices including brain death, organ donation after circulatory death, and some antemortem interventions to prepare for transplantation continue to be challenged. Attitudes toward the ethical issues surrounding death and organ donation may influence support for and participation in organ donation but differences between and among diverse populations have not (...) been studied.ObjectivesIn order to clarify attitudes toward brain death, organ donation after circulatory death and antemortem interventions in the context of organ donation, we conducted a scoping review of international English-language quantitative surveys in various populations.Study appraisalA search of literature up to October 2020 was performed, using multiple databases. After screening, 45 studies were found to meet pre-specified inclusion criteria.Results32 studies examined attitudes to brain death, predominantly in healthcare professionals. In most, around 75% of respondents accepted brain death as equivalent to death of the person. Less common perspectives included equating death with irreversible coma and willingness to undertake organ donation even if it caused death. 14 studies examined attitudes to organ donation following circulatory death. Around half of respondents in most studies accepted that death could be confidently diagnosed after only 5 min of cardiorespiratory arrest. The predominant reason was lack of confidence in doctors or diagnostic procedures. Only 6 studies examined attitudes towards antemortem interventions in prospective organ donors. Most respondents supported minimally invasive procedures and only where specific consent was obtained.ConclusionsOur review suggests a considerable proportion of people, including healthcare professionals, have doubts about the medical and ethical validity of modern determinations of death. The prognosis of brain injury was a more common concern in the context of organ donation decision-making than certainty of death. (shrink)

Furley’s book deals with two disparate, if ultimately relatable, themes: ‘Invisible Magnitudes’ and ‘Aristotle and Epicurus on Voluntary Action’. Both topics feature in Epicureanism. The first is concerned with the minimal parts of atoms and discusses the origins of the notion in, for example, Eleatic puzzles about divisibility, the atomism of Democritus and Aristotle’s continuum. The second topic deals with the familiar problem of free will and the ‘swerve’: here the author in his interpretation parts company with Bailey and (...) Giussani. For him the Epicurean theory on this point is intended to be a correction of Aristotelianism. In brief in relation to both topics the author attempts to show that Epicureanism corrected Aristotle who had corrected the Atomists. The argument is well sustained and impressively documented. (shrink)

We examine two-cardinal problems for the class of O-minimal theories. We prove that an O-minimal theory which admits some (κ, λ) must admit every (κ , λ ). We also prove that every “reasonable” variant of Chang’s Conjecture is true for O-minimal structures. Finally, we generalize these results from the two-cardinal case to the δ-cardinal case for arbitrary ordinals δ.

We examine two-cardinal problems for the class of O-minimal theories. We prove that an O-minimal theory which admits some must admit every. We also prove that every "reasonable" variant of Chang's Conjecture is true for O-minimal structures. Finally, we generalize these results from the two-cardinal case to the $\delta$-cardinal case for arbitrary ordinals $\delta$.

We consider locally o-minimal structures possessing tame topological properties shared by models of DCTC and uniformly locally o-minimal expansions of the second kind of densely linearly ordered abelian groups. We derive basic properties of dimension of a set definable in the structures including the addition property, which is the dimension equality for definable maps whose fibers are equi-dimensional. A decomposition theorem into quasi-special submanifolds is also demonstrated.

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)

The main result of this paper is Theorem 3.1 which is a criterion for weak o-minimality of a linearly ordered structure in terms of realizations of 1-types. Here we also prove some other properties of weakly o-minimal structures. In particular, we characterize all weakly o-minimal linear orderings in the signature $\{ . Moreover, we present a criterion for density of isolated types of a weakly o-minimal theory. Lastly, at the end of the paper we present some (...) remarks on the Exchange Principle for algebraic closure in a weakly o-minimal structure. (shrink)

O-minimal geometry generalizes both semialgebraic and subanalytic geometries, and has been very successful in solving special cases of some problems in arithmetic geometry, such as André–Oort conjecture. Among the many tools developed in an o-minimal setting are cohomology theories for abstract-definable continuous manifolds such as singular cohomology, sheaf cohomology and Čech cohomology, which have been used for instance to prove Pillay’s conjecture concerning definably compact groups. In the present thesis we elaborate an o-minimal de Rham cohomology (...) class='Hi'>theory for abstract-definable $C^{\infty }$ manifolds in an o-minimal expansion of the real field which admits smooth cell decomposition and defines the exponential function. We can specify the o-minimal cohomology groups and attain some properties such as the existence of Mayer–Vietoris sequence and the invariance under abstract-definable $C^{\infty }$ diffeomorphisms. However, in order to obtain the invariance of our o-minimal cohomology under abstract-definable homotopy we must work in a tame context that defines sufficiently many primitives and assume the validity of a statement related to Bröcker’s question.Abstract prepared by Rodrigo Figueiredo.E-mail: [email protected]: https://doi.org/10.11606/T.45.2019.tde-28042019-181150. (shrink)

We give an example of two ordered structures $\mathcal {M},\mathcal {N}$ in the same language $\mathcal {L}$ with the same universe, the same order and admitting the same one-variable definable subsets such that $\mathcal {M}$ is a model of the common theory of o-minimal $\mathcal {L}$ -structures and $\mathcal {N}$ admits a definable, closed, bounded, and discrete subset and a definable injective self-mapping of that subset which is not surjective. This answers negatively two question by Schoutens; the first (...) being whether there is an axiomatization of the common theory of o-minimal structures in a given language by conditions on one-variable definable sets alone. The second being whether definable completeness and type completeness imply the pigeonhole principle. It also partially answers a question by Fornasiero asking whether definable completeness of an expansion of a real closed field implies the pigeonhole principle. (shrink)

We investigate small theories of Boolean ordered o-minimal structures. We prove that such theories are $\aleph_{0}-categorical$ . We give a complete characterization of their models up to bi-interpretability of the language. We investigate types over finite sets, formulas and the notions of definable and algebraic closure.

Given a weakly o-minimal theory T, the T-height of an element of a model of T is defined as a means of classifying the order of magnitude of the element. If T satisfies some easily met technical conditions, then this classification is coarse enough for a Wilkie-type inequality: given a set of elements of a model of T, each of which has a different T-height, the cardinality of this set is at most 1 plus the minimum cardinality of (...) a set that generates the structure. (shrink)

For a countable complete o-minimal theory T, we introduce the notion of a sequentially complete model of T. We show that a model M of T is sequentially complete if and only if $\mathscr{M} \prec \mathscr{N}$ for some Dedekind complete model N. We also prove that if T has a Dedekind complete model of power greater than 2 ℵ 0 , then T has Dedekind complete models of arbitrarily large powers. Lastly, we show that a dyadic theory--namely, (...) a theory relative to which every formula is equivalent to a Boolean combination of formulas in two variables--that has some Dedekind complete model has Dedekind complete models in arbitrarily large powers. (shrink)

We consider VC-minimal theories admitting unpackable generating families, and show that in such theories, forking of formulae over a model M is equivalent to containment in global types definable over M, generalizing a result of Dolich on o-minimal theories in [4].