A Turing degree d is homogeneous bounding if every complete decidable (CD) theory has a d-decidable homogeneous model A, i.e., the elementary diagram De (A) has degree d. It follows from results of Macintyre and Marker that every PA degree (i.e., every degree of a complete extension of Peano Arithmetic) is homogeneous bounding. We prove that in fact a degree is homogeneous bounding if and only if it is a PA degree. We do this (...) by showing that there is a single CD theory T such that every homogeneous model of T has a PA degree. (shrink)

In the last five years there have been a number of results about the computable content of the prime, saturated, or homogeneous models of a complete decidable theory T in the spirit of Vaught's "Denumerable models of complete theories" combined with computability methods for degrees d ≤ 0′. First we recast older results by Goncharov, Peretyat'kin, and Millar in a more modern framework which we then apply. Then we survey recent results by Lange, "The degree spectra of (...) class='Hi'>homogeneous models," which generalize the older results and which include positive results on when a certain homogeneous model of T has an isomorphic copy of a given Turing degree. We then survey Lange's "A characterization of the 0-basis homogeneous bounding degrees" for negative results about when does not have such copies, generalizing negative results by Goncharov, Peretyat'kin, and Millar. Finally, we explain recent results by Csima, Harizanov, Hirschfeldt, and Soare, "Bounding homogeneous models," about degrees d that are homogeneous bounding and explain their relation to the PA degrees. (shrink)

Much previous study has been done on the degree spectra of prime models of a complete atomic decidable theory. Here we study the analogous questions for homogeneous models. We say a countable model A has a d-basis if the types realized in A are all computable and the Turing degree d can list $\Delta _{0}^{0}$ -indices for all types realized in A. We say A has a d-decidable copy if there exists a model B ≅ A (...) such that the elementary diagram of B is d-computable. Goncharov, Millar, and Peretyat'kin independently showed there exists a homogeneous A with a 0-basis but no decidable copy. We prove that any homogeneous A with a 0'-basis has a low decidable copy. This implies Csima's analogous result for prime models. In the case where all types of the theory T are computable and A is a homogeneous model with a 0-basis, we show A has copies decidable in every nonzero degree. A degree d is 0-homogeneous bounding if any automorphically nontrivial homogeneous A with a 0-basis has a d-decidable copy. We show that the nonlow₂ $\Delta _{2}^{0}$ degrees are 0-homogeneous bounding. (shrink)

Goncharov and Peretyat'kin independently gave necessary and sufficient conditions for when a set of types of a complete theory is the type spectrum of some homogeneous model of. Their result can be stated as a principle of second order arithmetic, which is called the Homogeneous Model Theorem (HMT), and analyzed from the points of view of computability theory and reverse mathematics. Previous computability theoretic results by Lange suggested a close connection between HMT and the (...) Atomic Model Theorem (AMT), which states that every complete atomic theory has an atomic model. The authors show that HMT and AMT are indeed equivalent in the sense of reverse mathematics, as well as in a strong computability theoretic sense and do the same for an analogous result of Peretyat'kin giving necessary and sufficient conditions for when a set of types is the type spectrum of some model. (shrink)

We generalize the result of non-finite axiomatizability of totally categorical first-order theories from elementary model theory to homogeneous model theory. In particular, we lift the theory of envelopes to homogeneous model theory and develope theory of imaginaries in the case of ω-stable homogeneous classes of finite U-rank.

We classify countable metrically homogeneous graphs of diameter 3.

We show that if M is a strongly minimal large homogeneous structure in a countable similarity type and the pregeometry of M is locally modular but not modular, then the pregeometry is affine over a division ring.

We classify countable metrically homogeneous graphs of diameter 3.

Aristotle is oftentimes viewed through a strictly philosophical lens as heir to Plato and has having introduced logical rigor where an emphasis on the theory of Forms formerly prevailed. It must be appreciated that Aristotle was the son of a physician, and that his inculcation of the thought of other Greek philosophers addressing health and the natural elements led to an extremely broad set of biologically- and medically-related writings. As this article proposes, Aristotle deepened the fourfold theory of (...) the elements with anatomic and physiologic observations. In books like History of Animals, Parts of Animals, and the lost Anatomai, he actively dissected organisms and recorded his findings. The corpus of medically-related literature Aristotle developed had a direct influence on subsequent Greek thinkers, including Galen, and on Medieval Islamic and modern Western practitioners such as William Harvey. Aristotle's ideas continue to influence modern medical thinkers in Europe and America through both his interpretation by European philosophers gaining the attention of medical humanists, and his writings' enduring impact on medical researchers balancing scientific with more personalistic approaches to medicine. (shrink)

We introduce a new approach to the model theory of metric structures by defining the notion of a metric abstract elementary class (MAEC) closely resembling the notion of an abstract elementary class. Further we define the framework of a homogeneous MAEC were we additionally assume the existence of arbitrarily large models, joint embedding, amalgamation, homogeneity and a property which we call the perturbation property. We also assume that the Löwenheim-Skolem number, which in this setting refers to the (...) density character of the set instead of the cardinality, is ${\aleph_0}$ . In these settings we prove an analogue of Morley’s categoricity transfer theorem. We also give concrete examples of homogeneous MAECs. (shrink)

The estrangement between genetic scientists and theologians originating in the 1960s is reflected in novel combinations of human thought (subject) and genes (investigational object), paralleling each other through the universal process known in chaos theory as self-similarity. The clash and recombination of genes and knowledge captures what Philip Hefner refers to as irony, one of four voices he suggests transmit the knowledge and arguments of the religion-and-science debate. When viewed along a tangent connecting irony to leadership, journal dissemination, and (...) the activities of the "public intellectual" and the public at large, the sequence of voices is shown to resemble the passage of genetic information from DNA to mRNA, tRNA, and protein, and from cell nucleus to surrounding environment. In this light, Hefner's inquiry into the voices of Zygon is bound up with the very subject matter Zygon covers. (shrink)

A generalization of Ehrenfest''s urn model is suggested. This will allow usto treat a wide class of stochastic processes describing the changes ofmicroscopic objects. These processes are homogeneous Markov chains. Thegeneralization proposed is presented as an abstract conditional (relative)probability theory. The probability axioms of such a theory and some simpleadditional conditions, yield both transition probabilities and equilibriumdistributions. The resulting theory interpreted in terms of particles andsingle-particle states, leads to the usual formulae of quantum and classicalstatistical (...) mechanics; in terms of chromosomes and allelic types, it allowsthe deduction of many genetical models including the Ewens sampling formula;in terms of agents'' strategies, it gives a justification of the ``herdbehaviour'''' typical of a population of heterogeneous economic agents. (shrink)

We generalize tools and results from first order stable theories to groups inside a simple stable strongly homogeneous model.

We prove that every relation-universal model of an ω-stable theory is saturated. We also show there is a large class of ω-stable theories for which every resplendent model is homogeneous.

Wesley Salmon has recently proposed a new theory of scientific explanation based on a model which he calls the statistical-relevance model. It is intended primarily as an account of the structure of explanations of particular events--explanations which, according to Salmon, are very often motivated largely by practical concerns. Two important features of this account are the concepts of homogeneity and screening off. In this paper we argue that the employment of these two concepts (which, in fact, are (...) intimately connected) is fundamentally at odds with the desire to capture everyday explanations of the practical sort, and that it leads instead to an account in which the only satisfactory explanations are those expressed in the very deepest physical terms. (shrink)

Coherence and correspondence are classical contenders as theories of truth. In this paper we examine them instead as interacting factors in the dynamics of belief across epistemic networks. We construct an agent-based model of network contact in which agents are characterized not in terms of single beliefs but in terms of internal belief suites. Individuals update elements of their belief suites on input from other agents in order both to maximize internal belief coherence and to incorporate ‘trickled in’ elements (...) of truth as correspondence. Results, though often intuitive, prove more complex than in simpler models (Hegselmann & Krause 2002, 2006; Grim, Singer, Reade & Fisher 2015). The optimistic finding is that pressures toward internal coherence can exploit and expand on small elements of truth as correspondence is introduced into epistemic networks. Less optimistic results show that pressures for coherence can also work directly against the incorporation of truth, particularly when coherence is established first and new data is introduced later. (shrink)

We initiate a systematic investigation of the abstract elementary classes that have amalgamation, satisfy tameness, and are stable in some cardinal. Assuming the singular cardinal hypothesis, we prove a full characterization of the stability cardinals, and connect the stability spectrum with the behavior of saturated models.We deduce that if a class is stable on a tail of cardinals, then it has no long splitting chains. This indicates that there is a clear notion of superstability in this framework.We also present an (...) application to homogeneous model theory: for [Formula: see text] a homogeneous diagram in a first-order theory [Formula: see text], if [Formula: see text] is both stable in [Formula: see text] and categorical in [Formula: see text] then [Formula: see text] is stable in all [Formula: see text]. (shrink)

The phase field microelasticity theory of a three-dimensional, elastically anisotropic system of voids and cracks is proposed. The theory is based on the equation for the strain energy of the continuous elastically homogeneous body presented as a functional of the phase field, which is the effective stress-free strain. It is proved that the stress-free strain minimizing the strain energy of this homogeneous modulus body fully determines the elastic strain and displacement of the body with voids and/or (...) cracks. The proposed phase field integral equation describing the elasticity of an arbitrary system of voids and cracks is exact. The geometry and evolution of multiple voids and/or cracks are described by the phase field, which is the solution of the time-dependent Ginzburg-Landau equation. Other defects, such as dislocations and precipitates, are trivially integrated into this theory. The proposed model does not impose a priori constraints on possible void and crack configurations or their evolution paths. Examples of computations of elastic equilibrium of systems with voids and/or cracks and the evolution of cracks under applied stress are considered. (shrink)

The theory of constructive (recursive) models follows from works of Froehlich, Shepherdson, Mal'tsev, Kuznetsov, Rabin, and Vaught in the 50s. Within the framework of this theory, algorithmic properties of abstract models are investigated by constructing representations on the set of natural numbers and studying relations between algorithmic and structural properties of these models. This book is a very readable exposition of the modern theory of constructive models and describes methods and approaches developed by representatives of the Siberian (...) school of algebra and logic and some other researchers (in particular, Nerode and his colleagues). The main themes are the existence of recursive models and applications to fields, algebras, and ordered sets (Ershov), the existence of decidable prime models (Goncharov, Harrington), the existence of decidable saturated models (Morley), the existence of decidable homogeneous models (Goncharov and Peretyat'kin), properties of the Ehrenfeucht theories (Millar, Ash, and Reed), the theory of algorithmic dimension and conditions of autostability (Goncharov, Ash, Shore, Khusainov, Ventsov, and others), and the theory of computable classes of models with various properties. Future perspectives of the theory of constructive models are also discussed. Most of the results in the book are presented in monograph form for the first time. The theory of constructive models serves as a basis for recursive mathematics. It is also useful in computer science, in particular, in the study of programming languages, higher level languages of specification, abstract data types, and problems of synthesis and verification of programs. Therefore, the book will be useful for not only specialists in mathematical logic and the theory of algorithms but also for scientists interested in the mathematical fundamentals of computer science. The authors are eminent specialists in mathematical logic. They have established fundamental results on elementary theories, model theory, the theory of algorithms, field theory, group theory, applied logic, computable numberings, the theory of constructive models, and the theoretical computer science. (shrink)

Scientific discourse is rife with passages that appear to be ordinary descriptions of systems of interest in a particular discipline. Equally, the pages of textbooks and journals are filled with discussions of the properties and the behavior of those systems. Students of mechanics investigate at length the dynamical properties of a system consisting of two or three spinning spheres with homogenous mass distributions gravitationally interacting only with each other. Population biologists study the evolution of one species procreating at a constant (...) rate in an isolated ecosystem. And when studying the exchange of goods, economists consider a situation in which there are only two goods, two perfectly rational agents, no restrictions on available information, no transaction costs, no money, and dealings are done immediately. Their surface structure notwithstanding, no competent scientist would mistake descriptions of such systems as descriptions of an actual system: we know very well that there are no such systems. These descriptions are descriptions of a model-system, and scientists use model-systems to represent parts or aspects of the world they are interested in. Following common practice, I refer to those parts or aspects as target-systems. What are we to make of this? Is discourse about such models merely a picturesque and ultimately dispensable façon de parler? This was the view of some early twentieth century philosophers. Duhem (1906) famously guarded against confusing model building with scientific theorizing and argued that model building has no real place in science, beyond a minor heuristic role. The aim of science was, instead, to construct theories, with theories understood as classificatory or representative structures systematically presented and formulated in precise symbolic.. (shrink)

Pixelism is the combination of three metaphysical thesis, namely a radical form of exdurantism, mereological nihilism and counterpart theory. Pixelism is a theory that evaluates all the metaphysical phenomena of persistence, composition and modality in a homogeneous and consistent manner. In a pixel world, there is no identity over time and over possible worlds and nothing persists over more than an instant or a world. Entities can be univocally identified by a five-coordinates system (the three spatial dimensions, (...) the temporal one and the possible worlds), and their relation is a counterpart relation both in different worlds and at different times or different regions of space. In this paper I will provide two models for pixelism: the first one takes pixels to be hypercubes, i. e. four-dimensional cubes, the acceptance of which is conditional on the acceptance of extended simples. The second one considers pixels as points in a four-dimensional space. (shrink)

Despite its ground-breaking character, Castoriadis’s theory of society remains, in some respects, caught up in conceptual difficulties common to social theory generally, particularly the problem of conceptualizing social unity without resort to an understanding of society that downplays heterogeneity and over-emphasizes the homogeneous. Unlike many other theoretical approaches and traditions, Castoriadis’s work also offers a possible path out of this dilemma in the form of philosophical innovations which could enable us to conceptualize social unity without flattening the (...) heterogeneous characteristics of the social. Castoriadis sometimes fails to recognize the full implications of his own rejection of ontological determinism, and so ends up proposing a far too deterministic and homogeneous model of social types. This essay aims to show how this might be remedied by drawing on Castoriadis’s own fundamental insights. (shrink)

We examine the computably enumerable (c.e.) degrees of prime models of complete atomic decidable (CAD) theories. A structure has degree d if d is the degree of its elementary diagram. We show that if a CAD theory T has a prime model of c.e. degree c, then T has a prime model of strictly lower c.e. degree b, where, in addition, b is low (b' = 0'). This extends Csima's result that every CAD theory has a (...) low prime model. We also prove a density result for c.e. degrees of prime models. In particular, if c and d are c.e. degrees with d < c and c not low₂ (c" > 0"), then for any CAD theory T, there exists a c.e. degree b with d < b < c such that T has a prime model of degree b, where b can be chosen so that b' is any degree c.e. in c with d' ≤ b'. As a corollary, we show that for any degree c with 0 < c < 0', every CAD theory has a prime model of low c.e. degree incomparable with c. We show also that every CAD theory has prime models of low c.e. degree that form a minimal pair, extending another result of Csima. We then discuss how these results apply to homogeneous models. (shrink)

Some arguments, based on the possible spontaneous violation of the cosmological principle (represented by the observed large-scale structures of galaxies), on the Cartan geometry of simple spinors, and on the Fock formulation of hydrogen atom wave equation in momentum space, are presented in favor of the hypothesis that space-time and momentum space should be both conformally compactified and should both originate from the two four-dimensional homogeneous spaces of the conformai group, both isomorphic (S 3 ×S 1)/Z 2 and correlated (...) by conformal inversion, but should not necessarily be identified with them. Within this framework, the possible common origin for the SO(4) symmetry underlying the geometrical structure of the Universe, of Kepler orbits, and of the H atom is discussed. One of the consequences of the proposed hypothesis could be that any quantum field theory should be naturally free from both infrared and ultraviolet divergences. But then physical spaces defined as those where physical phenomena may be best described through some set of fields, could be different from those homogeneous spaces. A simple, exactly soluble, toy model, valid for a two-dimensional spacetime, is presented where the conjectured conformally compactified homogeneous spaces are both isomorphic to (S 1 ×S 1)/Z 2,while the possible physical spaces could be two finite lattices which are dual since Fourier transforms, represented by finite, discrete, sums, may be well defined on them. (shrink)

This book contains the material for a first course in pure model theory with applications to differentially closed fields.

The usual construction of models of NFU (New Foundations with urelements, introduced by Jensen) is due to Maurice Boffa. A Boffa model is obtained from a model of (a fragment of) Zermelo–Fraenkel with Choice (ZFC) with an automorphism which moves a rank: the domain of the Boffa model is a rank that is moved. “Most” elements of the domain of the Boffa model are urelements in terms of the interpreted NFU. The main result of this paper (...) is that the restriction of the membership relation of the original model of set theory with automorphism to the domain of the Boffa model is first-order definable in the language of NFU. In particular, all information about the extensions in the original model of the urelements of the model of NFU is definable in terms of NFU. A corollary (answering a question of Thomas Forster) is that the urelements in a Boffa model are not homogeneous. (shrink)

CONTINUOUS MODEL THEORY CHAPTER I TOPOLOGICAL PRELIMINARIES. Notation Throughout the monograph our mathematical notation does not differ drastically from ...

Provability, Computability and Reflection.

In its most common use, the term ‘model’ refers to a simplified and stylised version of the socalled target system, the part or aspect of the world that we are interested in. For instance, in order to determine the orbit of a planet moving around the sun we model the planet and the sun as perfect homogenous spheres that gravitationally interact with each other but nothing else in the universe, and then apply Newtonian mechanics to this system, which (...) reveals that the planet moves on an elliptical orbit. Views diverge about what sort of entity such a model is. Those focussing on the formal aspects of models regard them either as equations or settheoretical structures, while those opposed to such an approach take them to be descriptions or abstract (yet non-mathematical) entities. A further question concerns the relation of models and theories. In some cases models can be derived from theory simply by specifying the relevant determinables in a theory’s general equations. But many models cannot be obtained from theory in this straightforward way, and some even involve assumptions that contradict the fundamental theory. The relation of models to their respective target systems is equally complex and fraught with controversy. Two influential proposals take the relation between a model and its target to be isomorphism or similarity, respectively. This, however, has been criticised as too restrictive as many models do not seem to fit this mould. (shrink)

The paper addresses the formation of striking patterns within originally near-homogenous tissue, the process prototypical for embryology, and represented in particularly purist form by cut sections of hydra regenerating, by internal reorganisation of the pre-existing tissue, a complete animal with head and foot. The essential requirements are autocatalytic, self-enhancing activation, combined with inhibitory or depletion effects of wider range – “lateral inhibition”. Not only de-novo-pattern formation, but also well known, striking features of developmental regulation such as induction, inhibition, and proportion (...) regulation can be explained on this basis. The theory provides a mathematical recipe for the construction of molecular models with criteria for the necessary non-linear interactions. It has since been widely applied to different developmental processes. (shrink)

Several important physical implications left out in The Five Dimension Space-Time Universe: A creation and grand unified field theory model. Book, are presented under rigorous mathematical theorems. It was found that Temperature, a classical variable, must be added as an imaginary component to time, under the Quantum uncertainty dt∙dE = h/2π, so that the Gell-Mann Quark model can be verified, with gauge invariance, to form hadrons at the Bethe Fusion Temperature. Accordingly from the corresponding uncertainty dp∙dr = (...) h/2π. Pairs of Diagonal Long Range Ordered gravitons, with continuous frequency spectrum together with those represented by magnetic monopoles must be formed within the space r, of the homogenous 5D manifold, without the presents of photons, thus defines the 5D as a Black Hole. Then from which we can derive the classical Newtonian Law of attractive Gravity, as the 5D manifold is mapped by Perelmann Ricci-flow entropy mapping and the DLRO graviton pair symmetry is broken and converted into two masses, with motions satisfying Special Relativity in the doughnut shape Lorentz manifold, thus indirectly verifies the principle of Covariant Riemannian curvatures and General Relativity theory. (shrink)

We introduce the subject of modal model theory, where one studies a mathematical structure within a class of similar structures under an extension concept that gives rise to mathematically natural notions of possibility and necessity. A statement φ is possible in a structure (written φ) if φ is true in some extension of that structure, and φ is necessary (written φ) if it is true in all extensions of the structure. A principal case for us will be the (...) class Mod(T) of all models of a given theory T—all graphs, all groups, all fields, or what have you—considered under the substructure relation. In this article, we aim to develop the resulting modal model theory. The class of all graphs is a particularly insightful case illustrating the remarkable power of the modal vocabulary, for the modal language of graph theory can express connectedness, k-colorability, finiteness, countability, size continuum, size ℵ1, ℵ2, ℵω, ℶω, first ℶ-fixed point, first ℶ-hyper-fixed point, and much more. A graph obeys the maximality principle φ(a)→φ(a) with parameters if and only if it satisfies the theory of the countable random graph, and it satisfies the maximality principle for sentences if and only if it is universal for finite graphs. (shrink)

The model theoretic `back and forth' construction of isomorphisms and automorphisms is based on the proof by Cantor that the theory of dense linear orderings without endpoints is ℵ 0 -categorical. However, Cantor's method is slightly different and for many other structures it yields an injection which is not surjective. The purpose here is to examine Cantor's method (here called `going forth') and to determine when it works and when it fails. Partial answers to this question are found, (...) extending those earlier given by Cameron. We also give fuller characterisations of when forth suffices for model theoretic classes such as structures containing Jordan sets for the automorphism group, and ℵ 0 -categorical ω-stable structures. The work is based on the author's Ph.D. thesis. (shrink)

Computer models and simulations have become, since the 1960s, an essential instrument for scientific inquiry and political decision making in several fields, from climate to life and social sciences. Philosophical reflection has mainly focused on the ontological status of the computational modeling, on its epistemological validity and on the research practices it entails. But in computational sciences, the work on models and simulations are only two steps of a longer and richer process where operations on data are as important as, (...) and even more time and energy-consuming than modeling itself. Drawing on two study cases—computational embryology and computational epidemiology—this article contributes to filling the gap by focusing on the operations of producing and re-using data in computational sciences. The different phases of the scientific and artisanal work of modelers include data collection, aggregation, homogenization, assemblage, analysis and visualization. The article deconstructs the ideas that data are self-evident informational aggregates and that data-driven approaches are exempted from theoretical work. More importantly, the paper stresses the fact that data are constructed and theory laden not only in their fabrication, but also in their reusing. (shrink)

A Dedekind Algebra is an ordered pair (B,h) where B is a non-empty set and h is an injective unary function on B. Each Dedekind algebra can be decomposed into a family of disjoint, countable subalgebras called configurations of the Dedekind algebra. There are N0 isomorphism types of configurations. Each Dedekind algebra is associated with a cardinal-valued function on omega called its configuration signature. The configuration signature of a Dedekind algebra counts the number of configurations in the decomposition of the (...) algebra in each isomorphism type.The configuration signature of a Dedekind algebra encodes the structure of that algebra in the sense that two Dedekind algebras are isomorphic iff their configuration signatures are identical. Configuration signatures are used to establish various results in the first-order model theory of Dedekind algebras. These include categoricity results for the first-order theories of Dedekind algebras and existence and uniqueness results for homogeneous, universal and saturated Dedekind algebras. Fundamental to these results is a condition on configuration signatures that is necessary and sufficient for elementary equivalence. (shrink)