For countable structure, "Scott rank" provides a measure of internal, model-theoretic complexity. For a computable structure, the Scott rank is at most [Formula: see text]. There are familiar examples of computable structures of various computable ranks, and there is an old example of rank [Formula: see text]. In the present paper, we show that there is a computable structure of Scott rank [Formula: see text]. We give two different constructions. The first starts with an arithmetical example due to Makkai, and (...) codes it into a computable structure. The second re-works Makkai's construction, incorporating an idea of Sacks. (shrink)

For countable structure, "Scott rank" provides a measure of internal, model-theoretic complexity. For a computable structure, the Scott rank is at most [Formula: see text]. There are familiar examples of computable structures of various computable ranks, and there is an old example of rank [Formula: see text]. In the present paper, we show that there is a computable structure of Scott rank [Formula: see text]. We give two different constructions. The first starts with an arithmetical example due to Makkai, and (...) codes it into a computable structure. The second re-works Makkai's construction, incorporating an idea of Sacks. (shrink)

For a computable structure A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{A}}$$\end{document}, there may not be a computable infinitary Scott sentence. When there is a computable infinitary Scott sentence φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varphi}$$\end{document}, then the complexity of the index set I\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${I}$$\end{document} is bounded by that of φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varphi}$$\end{document}. There are results giving “optimal” Scott sentences for (...) structures of various familiar kinds. These results have been driven by the thesis that the complexity of the index set should match that of an optimal Scott sentence. In this note, it is shown that the thesis does not always hold. For a certain subgroup of Q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{Q}}$$\end{document}, there is no computable d-Σ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Sigma_2}$$\end{document} Scott sentence, even though the index set is d-Σ20\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Sigma^0_2}$$\end{document}. (shrink)

The 2010 Nagoya Protocol is an international framework for access and benefit sharing (ABS) of the use of genetic and biological resources, with particular focus on indigenous communities. This is especially important in Africa, where local communities have a close reliance on environmental resources and ecosystems. However, national legislation and policies commonly lag behind international agreements, and this poses challenges for legal compliance as well as practical applications. This study reviews the academic literature on the Nagoya Protocol and ABS applications, (...) and then considers the implications of this analysis for research in Africa. Results show that there is uneven engagement with the principles of the Nagoya Protocol across different academic disciplines; local communities are sometimes sidelined in these studies; and only 8% of researchers in the literature analysed are located in Africa. Future developments should focus on ensuring national compliance with the Nagoya Protocol, and that researchers and industry work in partnership with local African communities on ABS issues. (shrink)

In this paper, we give an example of a complete computable infinitary theory T with countable models ${\mathcal{M}}$ and ${\mathcal{N}}$ , where ${\mathcal{N}}$ is a proper computable infinitary extension of ${\mathcal{M}}$ and T has no uncountable model. In fact, ${\mathcal{M}}$ and ${\mathcal{N}}$ are (up to isomorphism) the only models of T. Moreover, for all computable ordinals α, the computable ${\Sigma_\alpha}$ part of T is hyperarithmetical. It follows from a theorem of Gregory (JSL 38:460–470, 1972; Not Am Math Soc 17:967–968, 1970) (...) that if T is a Π 1 1 set of computable infinitary sentences and T has a pair of models ${\mathcal{M}}$ and ${\mathcal{N}}$ , where ${\mathcal{N}}$ is a proper computable infinitary extension of ${\mathcal{M}}$ , then T would have an uncountable model. (shrink)

In this paper, we state a metatheorem for constructions involving coding. Using the metatheorem, we obtain results on coding a family of sets into a family of relations, or into a single relation. For a concrete example, we show that the set of limit points in a recursive ordering of type ω 2 can have arbitrary 2-REA degree.

We extend results of Harizanov and Barker. For a relation R on a recursive structure /oA, we give conditions guaranteeing that the image of R in a recursive copy of /oA can be made to have arbitrary ∑α0 degree over Δα0. We give stronger conditions under which the image of R can be made ∑α0 degree as well. The degrees over Δα0 can be replaced by certain more general classes. We also generalize the Friedberg-Muchnik Theorem, giving conditions on a pair (...) of relations R and S under which the images of R and S can be made ∑α0 and independent over Δα0 in a recursive copy of /oA. (shrink)

The first main result isolates some conditions which fail for the class of graphs and hold for the class of Abelian p-groups, the class of Abelian torsion groups, and the special class of "rank-homogeneous" trees. We consider these conditions as a possible definition of what it means for a class of structures to have "Ulm type". The result says that there can be no Turing computable embedding of a class not of Ulm type into one of Ulm type. We apply (...) this result to show that there is no Turing computable embedding of the class of graphs into the class of "rank-homogeneous" trees. The second main result says that there is a Turing computable embedding of the class of rank-homogeneous trees into the class of torsion-free Abelian groups. The third main result says that there is a "rank-preserving" Turing computable embedding of the class of rank-homogeneous trees into the class of Boolean algebras. Using this result, we show that there is a computable Boolean algebra of Scott rank ${\mathrm{\omega }}_{1}^{\mathrm{C}\mathrm{K}}$. (shrink)

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

Let be a recursive structure, and let R be a recursive relation on . Harizanov isolated a syntactical condition which is necessary and sufficient for to have recursive copies in which the image of R is r.e. of arbitrary r.e. degree. We had conjectured that a certain extension of Harizanov's syntactical condition would be necessary and sufficient for to have recursive copies in which the image of R is ∑α0 of arbitrary ∑α0 degree, but this is not the case. Here (...) we give examples illustrating some restrictions on the possible ∑α0 degrees. In these examples, the image of R cannot be ∑α0 of degree d unless d possesses an “α-table”. (shrink)

The Hanf number for a set S of sentences in \ is the least infinite cardinal \ such that for all \, if \ has models in all infinite cardinalities less than \, then it has models of all infinite cardinalities. Friedman asked what is the Hanf number for Scott sentences of computable structures. We show that the value is \. The same argument proves that \ is the Hanf number for Scott sentences of hyperarithmetical structures.

Here we prove that if T and T′ are strongly minimal theories, where T′ satisfies a certain property related to triviality and T does not, and T′ is model complete, then there is no computable embedding of Mod(T) into Mod(T′). Using this, we answer a question from [4], showing that there is no computable embedding of VS into ZS, where VS is the class of infinite vector spaces over Q, and ZS is the class of models of Th(Z, S). Similarly, (...) we show that there is no computable embedding of ACF into ZS, where ACF is the class of algebraically closed fields of characteristic 0. (shrink)

This paper calculates, in a precise way, the complexity of the index sets for three classes of computable structures: the class $K_{\omega _{1}^{\mathit{CK}}}$ of structures of Scott rank $\omega _{1}^{\mathit{CK}}$ , the class $K_{\omega _{1}^{\mathit{CK}}+1}$ of structures of Scott rank $\omega _{1}^{\mathit{CK}}+1$ , and the class K of all structures of non-computable Scott rank. We show that I(K) is m-complete $\Sigma _{1}^{1},\,I(K_{\omega _{1}^{\mathit{CK}}})$ is m-complete $\Pi _{2}^{0}$ relative to Kleen's O, and $I(K_{\omega _{1}^{\mathit{CK}}+1})$ is m-complete $\Sigma _{2}^{0}$ relative to O.

Computable structures of Scott rank ${\omega_1^{CK}}$ are an important boundary case for structural complexity. While every countable structure is determined, up to isomorphism, by a sentence of ${\mathcal{L}_{\omega_1 \omega}}$ , this sentence may not be computable. We give examples, in several familiar classes of structures, of computable structures with Scott rank ${\omega_1^{CK}}$ whose computable infinitary theories are each ${\aleph_0}$ -categorical. General conditions are given, covering many known methods for constructing computable structures with Scott rank ${\omega_1^{CK}}$ , which guarantee that the (...) resulting structure is a model of an ${\aleph_0}$ -categorical computable infinitary theory. (shrink)

Nietzsche's opinions on philosophy and aesthetics developed under strong and lasting impulses from classical antiquity. These were not always the same, for at various periods in his life Nietzsche placed Heraclitus, Empedocles, Aeschylus, and even Socrates and Plato on the highest summit of wisdom. In his so-called first stage of development the pre-Socratics were generally his favourite thinkers, and in the third and last stage these same figures tend to come into prominence again. On the other hand, in the works (...) of Nietzsche's second, rationalistic period, when he was particularly influenced by Comte, Voltaire, and Darwin, Socrates and Plato—usually so hated and despised—are mentioned with affection, with gratitude or even with warm enthusiasm; and so, over and over again, is Epicurus. (shrink)

Dr. Cyril Bailey and Dr. C. M. Bowra have most recently analysed Virgil's method of using the expressions of Lucretius and Ennius respectively, and Mile A.-M. Guillemin has lately added significant considerations to Father F.-X. M. J. Roiron's long examination of Virgil's method of using again his own former expressions. Since then other work has been done with the purpose of clarifying the less rational part of Virgil's self-repetition; it might be called complementary to the well-known researches of Mr. John (...) Sparrow. With such a purpose I have tried to suggest how Virgil reached the strange phrase in the first Eclogue, rapidum Cretae, or cretae, Oaxen, by a not unparalleled confusion of the names of several different rivers and at least one town. (shrink)

This handbook presents a much-needed and comprehensive exploration of the rapidly growing fields of animal welfare and law. In recent years there has been increasing attention paid to our complex, multifaceted relationships with other animals, and in particular, the depth and breadth of various societal uses of animals. This has led to a reconsideration of their moral and social status, which has sometimes challenged the interests of those who use animals. In such a contested domain, sound evidence and reasoning become (...) particularly important. Through firm commitment to such principles, this book explores the biological foundations for the moral consideration of animals, and for evolving conceptualisations of animal welfare. It reviews in detail the welfare concerns associated with numerous forms of animal use. The inclusion of key recent developments such as climate change, pandemics and antimicrobial resistance, ensure this text is among the most current in its field. The ethical implications of the various uses of animals by society are considered, and chapters provide important recommendations for reforms of practice, law or policy. The status of animal law internationally, and in major world regions, is reviewed. Finally, the book considers human behavioural change, and strategies for improving stakeholder communication and education. The handbook is essential reading for students and scholars of animal welfare, animal law and animal ethics everywhere, and for policymakers and other professionals working in the animal welfare sector. (shrink)

Introduction -- Systems of evidence -- Science in practice -- Risk -- Pesticides -- Genetic engineering in agriculture -- Climate change -- Nuclear power -- The intersection of policy, science and risk.

For about a hundred years there has been an intermittent but sometimes vigorous debate1 on the question whether Quintus Smyrnaeus and Tryphiodorus directly used the Second Aeneid as a source for their epic descriptions “of the capture and destruction of Troy. Heyne thought that they did not; but towards the end of the nineteenth century it appeared more likely that they did. Heinze opposed the general belief: but it was reaffirmed for Quintus by Paschal and Becker4 and for Tryphiodorus by (...) Dr. E. Cesareo on internal evidence. Meanwhile Professor Samuel E. Bassett had concluded that Quintus was not after all dependent on Vergil: and still more recently Dr. J. W. Mackail, reviewing Cesareo's monograph, has maintained with emphasis that the Aeneid is certainly not the source of the late epic. Accordingly the controversy cannot even now be considered closed. (shrink)

It is sufficiently probable that quantitative scansion in Latin, imposed on a language in which accentuation by stress was alone significant originally, not only gave way to the earlier principle in the decline of Latin literature, but scarcely tended to suppress it at any time in common speech and in familiar writing. It is also probable therefore that even in literature dominated by quantity stress-accentuation was not obliterated altogether. In fact the incidences of it, in Vergilian verse at least, seemed (...) so characteristic that in late Antiquity the Vergilian stress-rhythms were apparently copied without any knowledge of the prosody of the hexameter. In modern expositions, the variety of Vergil's rhythms is generally referred, not only to scansion, but also to stress-accentuation: and it has recently been shown that lines in which stress and metrical ictus predominantly coincide are different in quality and expressiveness from lines in which they do not. Since however no general explanation or definition of this difference in quality seemed to have been offered, I lately attempted a simple theory which would account for the different aesthetic values of lines of these two classes, and which would define, if possible, what these different values are. (shrink)