Extending the language of rings to include predicates for Jacobson radical relations, we show that the theory of regular rings defined by Carson, Lipshitz and Saracino is the model completion of the theory of semisimple rings. Removing the requirement on the Jacobson radical (reduced to {0}), we prove that the theory of rings with no nilpotents does not admit a model companion relative to this augmented language.

We present two conditions which are equivalent to having an almost χ0-categorical model companion.

In an earlier paper we proved that a universal Horn class generated by finitely many finite structures has a model companion. If the language has only finitely many fundamental operations then the theory of the model companion admits a primitive recursive elimination of quantifiers and is primitive recursive. The theory of the model companion is ℵ 0 -categorical iff it is complete iff the universal Horn class has the joint embedding property iff the universal (...) Horn class is generated by a single finite structure. In the last section we look at structure theorems for the model companions of universal Horn classes generated by functionally complete algebras, in particular for the cases of rings and groups. (shrink)

We study model companions of theories extending the graph axioms. First we prove general results concerning the existence of the model companion. Then, by applying these results to the case of graphs, we give a series of companionable and non‐companionable examples.

For a theory T in L, T σ is the theory of the models of T with an automorphism σ. If T is an unstable model complete theory without the independence property, then T σ has no model companion. If T is an unstable model complete theory and T σ has the amalgamation property, then T σ has no model companion. If T is model complete and has the fcp, then T σ has (...) no model completion. (shrink)

We introduce the notion T does not omit obstructions. If a stable theory does not admit obstructions then it does not have the finite cover property . For any theory T, form a new theory $T_{\rm Aut}$ by adding a new unary function symbol and axioms asserting it is an automorphism. The main result of the paper asserts the following: If T is a stable theory, T does not admit obstructions if and only if $T_{\rm Aut}$ has a model (...) companion. The proof involves some interesting new consequences of the nfcp. (shrink)

We introduce the notion T does not omit obstructions. If a stable theory does not admit obstructions then it does not have the finite cover property (nfcp). For any theory T, form a new theory by adding a new unary function symbol and axioms asserting it is an automorphism. The main result of the paper asserts the following: If T is a stable theory, T does not admit obstructions if and only if has a model companion. The proof (...) involves some interesting new consequences of the nfcp. (shrink)

This is an introductory paper to a series of results linking generic absoluteness results for second and third order number theory to the model theoretic notion of model companionship. Specifically we develop here a general framework linking Woodin’s generic absoluteness results for second order number theory and the theory of universally Baire sets to model companionship and show that (with the required care in details) a $$\Pi _2$$ -property formalized in an appropriate language for second order number (...) theory is forcible from some $$T\supseteq \mathsf {ZFC}+$$ large cardinals if and only if it is consistent with the universal fragment of T if and only if it is realized in the model companion of T. In particular we show that the first order theory of $$H_{\omega _1}$$ is the model companion of the first order theory of the universe of sets assuming the existence of class many Woodin cardinals, and working in a signature with predicates for $$\Delta _0$$ -properties and for all universally Baire sets of reals. We will extend these results also to the theory of $$H_{\aleph _2}$$ in a follow up of this paper. (shrink)

This paper considers the ethical implications of applying three major ethical theories to the memory structure of an artificial companion that might have different embodiments such as a physical robot or a graphical character on a hand-held device. We start by proposing an ethical memory model and then make use of an action-centric framework to evaluate its ethical implications. The case that we discuss is that of digital artefacts that autonomously record and store user data, where this data (...) are used as a resource for future interaction with users. (shrink)

In his Ph.D. thesis [7], L. van den Dries studied the model theory of fields with finitely many orderings and valuations where all open sets according to the topology defined by an order or a valuation is globally dense according with all other orderings and valuations. Van den Dries proved that the theory of these fields is companionable and that the theory of the companion is decidable .In this paper we study the case where the fields are expanded (...) with finitely many orderings and an independent derivation. We show that the theory of these fields still admits a model companion in the language Lequation image = {+, –, ·, D, <1, …, model companion by CODFm and give a geometric axiomatization of this theory which uses basic notions of algebraic geometry and some generalized open subsets which appear naturally in this context. This axiomatization allows to recover the one given in [4] for the theory CODF of closed ordered differential fields. Most of the technics we use here are already present in [2] and [4].Finally, we prove that it is possible to describe the completions of CODFm and to obtain quantifier elimination in a slightly enriched language. This generalizes van den Dries' results in the “derivation free” case. (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)

We present recent results on the model companions of set theory, placing them in the context of a current debate in the philosophy of mathematics. We start by describing the dependence of the notion of model companionship on the signature, and then we analyze this dependence in the specific case of set theory. We argue that the most natural model companions of set theory describe (as the signature in which we axiomatize set theory varies) theories of $H_{\kappa (...) ^+}$, as $\kappa $ ranges among the infinite cardinals. We also single out $2^{\aleph _0}=\aleph _2$ as the unique solution of the continuum problem which can (and does) belong to some model companion of set theory (enriched with large cardinal axioms). While doing so we bring to light that set theory enriched by large cardinal axioms in the range of supercompactness has as its model companion (with respect to its first order axiomatization in certain natural signatures) the theory of $H_{\aleph _2}$ as given by a strong form of Woodin’s axiom $(*)$ (which holds assuming $\mathsf {MM}^{++}$ ). Finally this model-theoretic approach to set-theoretic validities is explained and justified in terms of a form of maximality inspired by Hilbert’s axiom of completeness. (shrink)

While we completely understood that a binary dwarf companion of the Sun has not been accepted by majority of astronomers, allow us to present some new arguments, along with our own serendipitous encounter with such a binary companion of the Sun. We hope that the present note will be found useful for further investigations, in relation to Planet Nine and such a dwarf star companion of the Sun (sometimes dubbed as Nemesis). Nonetheless, this article is not an (...) exhaustive review of such dwarf companion star theories. (shrink)

We study model theory of fields with actions of a fixed finite group scheme. We prove the existence and simplicity of a model companion of the theory of such actions, which generalizes our previous results about truncated iterative Hasse–Schmidt derivations [13] and about Galois actions [14]. As an application of our methods, we obtain a new model complete theory of actions of a finite group on fields of finite imperfection degree.

The institution of owning pets has been subjected to compelling criticism on moral grounds. Yet advocates of a reformed, guardian/dependent model may yet face an abolitionist conclusion. We argue that treating companion animals as dependents entails an indefensible moral priority for them in the face of their guardians’ competing moral demands. An abolitionist dilemma arises as a result: if the property and reformed models fail, a morally acceptable characterization of the moral relationship between humans and their companion (...) animals has yet to be articulated. We close by inviting others to develop plausible alternatives to the ownership and reformed models. (shrink)

In this paper, we study bounded versions of some model-theoretic notions and results. We apply these results to the context of models of bounded arithmetic theories as well as some related complexity questions. As an example, we show that if the theory \(\rm S_2 ^1(PV)\) has bounded model companion then \(\rm NP=coNP\). We also study bounded versions of some other related notions such as Stone topology.

_The_ _Routledge Companion to Philosophy of Medicine _is a comprehensive guide to topics in the fields of epistemology and metaphysics of medicine. It examines traditional topics such as the concept of disease, causality in medicine, the epistemology of the randomized controlled trial, the biopsychosocial model, explanation, clinical judgment and phenomenology of medicine and emerging topics, such as philosophy of epidemiology, measuring harms, the concept of disability, nursing perspectives, race and gender, the metaphysics of Chinese medicine, and narrative medicine. (...) Each of the 48 chapters is written especially for this volume and with a student audience in mind. For pedagogy and clarity, each chapter contains an extended example illustrating the ideas discussed. This text is intended for use as a reference for students in courses in philosophy of medicine and philosophy of science, and pairs well with _The_ _Routledge Companion to Bioethics_ for use in medical humanities and social science courses. (shrink)

We prove some results about the model theory of fields with a derivation of the Frobenius map, especially that the model companion of this theory is axiomatizable by axioms used by Wood in the case of the theory $\operatorname {DCF}_p$ and that it eliminates quantifiers after adding the inverse of the Frobenius map to the language. This strengthens the results from [4]. As a by-product, we get a new geometric axiomatization of this model companion. Along (...) the way we also prove a quantifier elimination result, which holds in a much more general context and we suggest a way of giving “one-dimensional” axiomatizations for model companions of some theories of fields with operators. (shrink)

We study the theory of a Hilbert space H as a module for a unital C*-algebra ${\mathcal{A}}$ from the point of view of continuous logic. We give an explicit axiomatization for this theory and describe the structure of all the representations which are elementary equivalent to it. Also, we show that this theory has quantifier elimination and we characterize the model companion of the incomplete theory of all non-degenerate representations of ${\mathcal{A}}$ . Finally, we show that there is (...) an homeomorphism between the space of types of norm less than 1 in this model companion, and the space of quasistates of the C*-algebra ${\mathcal{A}}$. (shrink)

The class of all Artinian local rings of length at most l is ∀ 2 -elementary, axiomatised by a finite set of axioms Art l . We show that its existentially closed models are Gorenstein, of length exactly l and their residue fields are algebraically closed, and, conversely, every existentially closed model is of this form. The theory Got l of all Artinian local Gorenstein rings of length l with algebraically closed residue field is model complete and the (...) theory Art l is companionable, with model-companion Got l. (shrink)

Monadic second order logic and linear temporal logic are two logical formalisms that can be used to describe classes of infinite words, i.e., first-order models based on the natural numbers with order, successor, and finitely many unary predicate symbols.Monadic second order logic over infinite words can alternatively be described as a first-order logic interpreted in${\cal P}\left$, the power set Boolean algebra of the natural numbers, equipped with modal operators for ‘initial’, ‘next’, and ‘future’ states. We prove that the first-order theory (...) of this structure is the model companion of a class of algebras corresponding to a version of linear temporal logic without until.The proof makes crucial use of two classical, nontrivial results from the literature, namely the completeness of LTL with respect to the natural numbers, and the correspondence between S1S-formulas and Büchi automata. (shrink)

This paper examines one approach to the ethics of companion animals, which emerges from the dominant historical tradition and is increasingly familiar in everyday life as well as in work on companion animals in the social sciences. I label it the “utilization with welfare-safeguards” model, or, more gently worded, “seeking benefits while ensuring welfare.” Some of the “benefits” considered are complex ones (like guiding the sight impaired) and others simpler (like reducing stress or providing affection). I explore (...) several problems involved in this approach (including the sometimes jarring inappropriateness of “benefit” terminology). I then offer an alternative account where the primary moral obligation toward companion animals is to develop, nurture, respect, and protect the loving relationship between them and their human companions, since thriving in such a relationship, I claim, has become part of their evolved telos (to use Bernard Rollin’s term)or evolved nature. This priority naturally leads to ensuring welfare, but the highly pro-active approach involved takes the obligation beyond standard welfare provision and “TLC” (“tender loving care”). Some implications of this position are explored. (shrink)

This is the second of four online Companions to Velmans, M. (ed.) (2018) Consciousness (Critical Concepts in Psychology), a 4-volume collection of Major Works on Consciousness commissioned by Routledge, London. -/- The Companion to Volume 2 Part 1 focuses on the detailed relationship of phenomenal consciousness to mental processing described either functionally (as human information processing) or in terms of neural activity, in the ways typically explored by cognitive psychology and cognitive neuroscience. Beginning with reviews of functional differences between (...) unconscious, preconscious and conscious processing and the different senses in which mental processing can be said to be “conscious”, the readings turn to seminal work on blindsight and related phenomena, experimental evidence of perception without awareness, ways to combine “subjective” and “objective” measures of awareness, evidence for distinct neural pathways governing visual awareness and visual control of motor acts, and the complex ways in which consciousness of action relates to both voluntary and involuntary acts. The readings then focus on the intimate links between attention and consciousness, starting with the writings of William James and surveying the extensive work on this subject over the last 60 years that explores the relationships between and attended and non-attended processing, preconscious and conscious processing, and attention, primary (working, short-term) memory and consciousness. The readings go on to survey the evidence that attention contributes to the “neural binding” required for an integrated conscious experience and evidence that attention (in humans) is dissociable from consciousness, necessary for consciousness, but not sufficient for consciousness. The readings then turn to research on “inattentional blindness”, and conclude with a review of Baars’ global workspace theory, which integrates many theories and findings in this area into a coherent, global model. The Companion then summarises seminal readings on learning, memory and consciousness; the extensive research on mental imagery, long thought to epitomize private, subjective conscious experience—and consequently ruled out of science by reductive, materialist philosophies of mind; the complex relations of consciousness to sleep and dreaming; the development of consciousness in human infants; and the debates surrounding, and extensive evidence for consciousness in non-human animals, with attendant consequences for their humane treatment. -/- As with the other Companions to these Volumes there are many links to background resources (marked in pink) and to the selected readings themselves (marked in blue). (shrink)

We examine topological dynamical systems on the Cantor set from the point of view of the continuous model theory of commutative C*-algebras. After some general remarks, we focus our attention on the generic homeomorphism of the Cantor set, as constructed by Akin, Glasner, and Weiss. We show that this homeomorphism is the prime model of its theory. We also show that the notion of “generic” used by Akin, Glasner, and Weiss is distinct from the notion of “generic” encountered (...) in Fraïssé theory. (shrink)

In this paper, we develop the beginning of Lie-differential algebra, in the sense of Kolchin by using tools introduced by Hubert in [E. Hubert, Differential algebra for derivations with nontrivial commutation rules, J. Pure Appl. Algebra 200 163–190]. In particular it allows us to adapt the results of Tressl 3933–3951]) by showing the existence of a theory of Lie-differential fields of characteristic zero. This theory will serve as a model companion for every theory of large and Lie-differential fields (...) extending a model complete theory of pure fields. As an application, we introduce the Lie counterpart of classical theories of differential fields in several commuting derivations. (shrink)

Using methods of geometric stability , we determine the structure of Abelian groups definable in ACFA, the model companion of fields with an automorphism. We also give general bounds on sets definable in ACFA. We show that these tools can be used to study torsion points on Abelian varieties; among other results, we deduce a fairly general case of a conjecture of Tate and Voloch on p-adic distances of torsion points from subvarieties.

Let T be a model-complete theory that eliminates the quantifier $\exists^\infty x$ . For T we construct a theory T+ such that any element in a model of T+ determines a model of T. We show that T+ has a model companion T1. We can iterate the construction. The produced theories are investigated.

This chapter has two aims. The first aim is to compare and contrast three different conceptual-explanatory models for thinking about mental illness with an eye towards identifying the assumptions upon which each model is based, and exploring the model’s advantages and limitations in clinical contexts. Major Depressive Disorder is used as an example to illustrate these points. The second aim is to address the question of what conceptual-theoretical framework for thinking about mental illness is most likely to facilitate (...) the discovery of causes and treatments of mental illness in research contexts. To this end, the National Institute of Mental Health’s Research Domain Criteria (RDoC) Project is briefly considered. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:The Wiley Blackwell Companion to Religion and Ecology ed. by John HartDannis M. MattesonThe Wiley Blackwell Companion to Religion and Ecology Edited by John Hart OXFORD: JOHN WILEY & SONS LTD, 2017. 560 pp. $195.00If ecology is the study of "relationships in a place," as John Hart reminds readers in the preface of the Wiley Blackwell Companion to Religion and Ecology, it is fitting that (...) this volume centers its thrust relationally (xxiv). Throughout its pages, a diverse community of authors explores the relationships among religious and spiritual traditions, human life, and care for the Earth. Defining religion as a language with which to grapple with meanings assigned to cultural values (xxi), this volume turns to the meanings that humans have historically placed on their relationship to the Earth within spiritual and religious frameworks. It further ponders how our religions and spiritual traditions reflect a relationship to the Earth that is conscious of the status of our planet's health today. Offering insights from Buddhism, Hinduism, Islam, Judaism, Eastern and Western Christianity, and indigenous perspectives, the authors reflect on new and old insights from within their tradition that speak to specific ecological issues. The volume is split into four parts, flowing thematically from conceptual and descriptive, to concrete visions for practical steps toward healing the Earth. [End Page 199]Part 1 describes the various ways religious and spiritual traditions enhance ecological consciousness. Seyyed Hossein Nasr shows how the Qur'an depicts the created order as a reflection of God's face, suggesting that we are literally defacing the face of God in our industrial recklessness. David Mevorach Seidenberg presents a Jewish eco-theology, which decenters the human as the pinnacle of creation. In part 2, the authors further draw out the socioecological implications of religious teachings. Melanie Harris argues that religious communities must open their eyes to the link between structural violence against black women to structural violence against the Earth. She maintains that this analysis is key to finding solutions for ecological reparations. When Harris argues that social justice is Earth justice, she is claiming that the dismantling of both white supremacy and colonial ecology go hand in hand.Part 3, titled "Ecological Commitment," highlights the specific contexts in which the relationship between religion and ecology are embedded. Myrna Perez Sheldon and Naomi Oreskes trace the entanglement of evangelical identity with the politics of scientific doubt. In response, they suggest a new framework for evangelical participation in environmental restoration that moves beyond a political identity that is chained to right-wing doubt about climate change. In asking how "Korean Christianity can become more Nature-conscious, Nature-related, and Nature-integrated," Yong Bum Park points to Chondogyo, a Korean indigenous religion (333). This tradition is shown to take seriously the ecological crisis as a "crisis of consciousness and conduct," encouraging religious people to foster a socioecological vision of restoration (333).Part 4, which provides examples of concrete visions and projects on the ground, ends with John Hart's reimagining of the Earth as a "common commons." The notion that the Earth is a sacred "common commons" of "interdependent, integral, evolving relational community" requires conversion in light of how religious people relate to the Earth today (437). According to this commons ethic, land should be "commonized and communalized" rather than rendered private property (476).I highly recommend this volume for teaching undergraduate and graduate courses that are exploring the intersection of religion and ecology, especially courses focusing on religious ethics. Though the breadth of this work might strike the reader as boundless, herein lies its strength, especially for teachers of Christian ethics. This resource broadens the conversation recently highlighted by Pope Francis in Laudato Si' beyond the Christian perspective on ecology. The Companion models interreligious dialogue, opening to the diverse ways many different religious and spiritual traditions demand action for environmental restoration. [End Page 200]Dannis M. MattesonLoyola University ChicagoCopyright © 2018 Society of Christian Ethics... (shrink)