It is shown that the theory of fields with an automorphism has a decidable model companion. Quantifier-elimination is established in a natural language. The theory is intimately connected to Ax's theory of pseudofinite fields, and analogues are obtained for most of Ax's classical results. Some indication is given of the connection to nonstandard Frobenius maps.

I see model theory as becoming increasingly detached from set theory, and the Tarskian notion of set-theoretic model being no longer central to model theory. In much of modern mathematics, the set-theoretic component is of minor interest, and basic notions are geometric or category-theoretic. In algebraic geometry, schemes or algebraic spaces are the basic notions, with the older “sets of points in affine or projective space” no more than restrictive special cases. The basic notions may be given sheaf-theoretically, or functorially. (...) To understand in depth the historically important affine cases, one does best to work with more general schemes. The resulting relativization and “transfer of structure” is incomparably more flexible and powerful than anything yet known in “set-theoretic model theory”.It seems to me now uncontroversial to see the fine structure of definitions as becoming the central concern of model theory, to the extent that one can easily imagine the subject being called “Definability Theory” in the near future.Tarski's set-theoretic foundational formulations are still favoured by the majority of model-theorists, and evolution towards a more suggestive language has been perplexingly slow. None of the main texts uses in any nontrivial way the language of category theory, far less sheaf theory or topos theory. Given that the most notable interactions of model theory with geometry are in areas of geometry where the language of sheaves is almost indispensable, this is a curious situation, and I find it hard to imagine that it will not change soon, and rapidly. (shrink)

We extend the field of Laurent series over the reals in a canonical way to an ordered differential field of “logarithmic-exponential series” , which is equipped with a well behaved exponentiation. We show that the LE-series with derivative 0 are exactly the real constants, and we invert operators to show that each LE-series has a formal integral. We give evidence for the conjecture that the field of LE-series is a universal domain for ordered differential algebra in Hardy fields. We define (...) composition of LE-series and establish its basic properties, including the existence of compositional inverses. Various interesting subfields of the field of LE-series are also considered. (shrink)

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

Let n be a positive integer and FAℓ be the free abelian lattice-ordered group on n generators. We prove that FAℓ and FAℓ do not satisfy the same first-order sentences in the language if m≠n. We also show that is decidable iff n{1,2}. Finally, we apply a similar analysis and get analogous results for the free finitely generated vector lattices.

We give foundational results for the model theory of AfinK, the ring of finite adeles over a number field, construed as a restricted product of local fields. In contrast to Weispfenning we work in the language of ring theory, and various sortings interpretable therein. In particular we give a systematic treatment of the product valuation and the valuation monoid. Deeper results are given for the adelic version of Krasner's hyperfields, relating them to the Basarab–Kuhlmann formalism.

Gauss used quadratic forms in his second proof of quadratic reciprocity. In this paper we begin to develop a theory of binary quadratic forms over weak fragments of Peano Arithmetic, with a view to reproducing Gauss’ proof in this setting.

Gauss used quadratic forms in his second proof of quadratic reciprocity. In this paper we begin to develop a theory of binary quadratic forms over weak fragments of Peano Arithmetic, with a view to reproducing Gauss’ proof in this setting.

In this second paper of the series we do a local analysis of quadratic forms over completions of a non-standard model of IΔ0+Ω1.

We give a definition, in the ring language, of Zp inside Qp and of Fp[[t]] inside Fp), which works uniformly for all p and all finite field extensions of these fields, and in many other Henselian valued fields as well. The formula can be taken existential-universal in the ring language, and in fact existential in a modification of the language of Macintyre. Furthermore, we show the negative result that in the language of rings there does not exist a uniform (...) definition by an existential formula and neither by a universal formula for the valuation rings of all the finite extensions of a given Henselian valued field. We also show that there is no existential formula of the ring language defining Zp inside Qp uniformly for all p. For any fixed finite extension of Qp, we give an existential formula and a universal formula in the ring language which define the valuation ring. (shrink)

Background:The debate over the ethical implications of care robots has raised a range of concerns, including the possibility that such technologies could disrupt caregiving as a core human moral activity. At the same time, academics in information ethics have argued that we should extend our ideas of moral agency and rights to include intelligent machines.Research objectives:This article explores issues of the moral status and limitations of machines in the context of care.Design:A conceptual argument is developed, through a four-part scheme derived (...) from the work of Alasdair MacIntyre. No empirical data are used.Ethical consideration:No primary data were gathered for this study. Secondary sources and authorship have been acknowledged throughout.Findings / discussion:Certain kinds of social experience, including the narrative unity of a life, and the giving and receiving of care, are essential for moral development. Machines, no matter how advanced, cannot participate in such experiences in key respects, and thus cannot develop as practical reasoners.Conclusion:It follows that they cannot be moral agents and that they cannot care. There are, it seems, no such things as care robots. In view of the institutional power of tech companies and commissioning bodies, care practitioners need to take more of a lead in developing new assistive technologies which are appropriate to their practice. (shrink)

Alasdair MacIntyre explores some central philosophical, political and moral claims of modernity and argues that a proper understanding of human goods requires a rejection of these claims. In a wide-ranging discussion, he considers how normative and evaluative judgments are to be understood, how desire and practical reasoning are to be characterized, what it is to have adequate self-knowledge, and what part narrative plays in our understanding of human lives. He asks, further, what it would be to understand the modern (...) condition from a neo-Aristotelian or Thomistic perspective, and argues that Thomistic Aristotelianism, informed by Marx's insights, provides us with resources for constructing a contemporary politics and ethics which both enable and require us to act against modernity from within modernity. This rich and important book builds on and advances MacIntyre's thinking in ethics and moral philosophy, and will be of great interest to readers in both fields. (shrink)

A biblical approach to reconciling God’s sovereignty with creaturely responsibility should avoid the extremes of global occasionalism and completely autonomous creatures. This paper evaluates the standard intermediary solutions offered by conservationists and concurrentists. It argues that while each contributes insights which a satisfactory account should retain, none is fully adequate. Even Leibniz’s sophisticated response, which accounts for providence, miracles, and moral responsibility, unacceptably abridges creaturely power to implement decisions. My alternative proposal seeks to explain how creatures can retain full responsibility (...) for their actions even though God reserves the right to override creaturely power with His greater power. (shrink)

This classic and controversial book examines the roots of the idea of virtue, diagnoses the reasons for its absence in modern life, and proposes a path for its recovery.