Saturated models constitute one of the powerful methods of conventional model theory, with many applications. Here we develop a categorical abstract model theoretic approach to saturated models within the theory of institutions. The most important consequence is that the method of saturated models becomes thus available to a multitude of logical systems from logic or from computing science. In this paper we define the concept of saturated model at an abstract institution-independent level and develop the fundamental (...) existence and uniqueness theorems. As an application we prove a general institution-independent version of the Keisler–Shelah isomorphism theorem “any two elementarily equivalent models have isomorphic ultrapowers” (assuming Generalized Continuum Hypothesis). (shrink)

A notion called Herbrand saturation is shown to provide the model-theoretic analogue of a proof-theoretic method, Herbrand analysis, yielding uniform model-theoretic proofs of a number of important conservation theorems. A constructive, algebraic variation of the method is described, providing yet a third approach, which is finitary but retains the semantic flavor of the model-theoretic version.

Given a regular cardinal κ > ω1 and a cardinal λ with κ ≤ cf (λ) < λ, we show that NSκ,λ | T is not λ+-saturated, where T is the set of all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${a\in P_\kappa (\lambda)}$$\end{document} such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${| a | = | a \cap \kappa|}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\rm cf} \big( {\rm sup} (...) (a\cap\kappa)\big) = {\rm cf} \big({\rm sup} (a)\big) = \omega}$$\end{document}. (shrink)

We put in print a classical result that states that for most purposes, there is no harm in assuming the existence of saturated models in model theory. The presentation is aimed for model theorists with only basic knowledge of axiomatic set theory.

Given a regular uncountable cardinal κ and a cardinal λ > κ of cofinality ω, we show that the restriction of the non-stationary ideal on Pκ to the set of all a with equation image is not λ++-saturated . We actually prove the stronger result that there is equation image with |Q| = λ++ such that A∩B is a non-cofinal subset of Pκ for any two distinct members A, B of Q, where NGκ, λ denotes the game ideal on (...) Pκ. We also remark that for κ > ω1, adding λ+3 Cohen subsets of ω1 to equation image makes NGκ, λ λ+3-saturated. (shrink)

We show that if κ is an infinite successor cardinal, and λ > κ a cardinal of cofinality less than κ satisfying certain conditions, then no ideal on Pκ is weakly λ+-saturated. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

Theorem 0.1LetK\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {K}$$\end{document}be an abstract elementary class with amalgamation and no maximal models. Letλ>LS\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda > {LS}$$\end{document}. IfK\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {K}$$\end{document}is categorical inλ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document}, then the model of cardinalityλ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document}is Galois-saturated.This answers a question asked independently by Baldwin (...) and Shelah. We deduce several corollaries: K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {K}$$\end{document} has a unique limit model in each cardinal below λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document}, K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {K}$$\end{document} is weakly tame below λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document}, and the thresholds of several existing categoricity transfers can be improved.We also prove a downward transfer of solvability :Corollary 0.2LetK\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {K}$$\end{document}be an AEC with amalgamation and no maximal models. Letλ>μ>LS\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda> \mu > {LS}$$\end{document}. IfK\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {K}$$\end{document}is solvable inλ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document}, thenK\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {K}$$\end{document}is solvable inμ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document}. (shrink)

We show that is a quasi-minimal torsion-free divisible abelian group. After discussing the axiomatization of the theory of this structure, we present its ω-saturated quasi-minimal model.

A method for constructing Boolean-valued models of some fragments of arithmetic was developed in Krajíček (Forcing with Random Variables and Proof Complexity, London Mathematical Society Lecture Notes Series, Cambridge University Press, Cambridge, 2011), with the intended applications in bounded arithmetic and proof complexity. Such a model is formed by a family of random variables defined on a pseudo-finite sample space. We show that under a fairly natural condition on the family [called compactness in Krajíček (Forcing with Random Variables and Proof (...) Complexity, London Mathematical Society Lecture Notes Series, Cambridge University Press, Cambridge, 2011)] the resulting structure has a property that is naturally interpreted as saturation for existential types. We also give an example showing that this cannot be extended to universal types. (shrink)

This paper is devoted to the problem of existence of saturated models for first-order many-valued logics. We consider a general notion of type as pairs of sets of formulas in one free variable that express properties that an element of a model should, respectively, satisfy and falsify. By means of an elementary chains construction, we prove that each model can be elementarily extended to a $\kappa $-saturated model, i.e. a model where as many types as possible are realized. (...) In order to prove this theorem we obtain, as by-products, some results on tableaux (understood as pairs of sets of formulas) and their consistency and satisfiability and a generalization of the Tarski–Vaught theorem on unions of elementary chains. Finally, we provide a structural characterization of $\kappa $-saturation in terms of the completion of a diagram representing a certain configuration of models and mappings. (shrink)

This paper was motivated by the following two questions which arise in the theory of complexity for computation over ordered rings in the now famous computational model introduced by Blum, Shub and Smale: 1. is the answer to the question P = ?NP the same in every real-closed field?2. if P ≠ NP for , does there exist a problem of which is NP but neither P nor NP-complete ?Some unclassical complexity classes arise naturally in the study of these questions. (...) They are still open, but we could obtain unconditional results of independent interest.Michaux introduced/const complexity classes in an effort to attack question . We show that , answering a question of his. Here A is the class of real problems which are algorithmic in bounded time. We also prove the stronger result: , where PAR stands for parallel polynomial time. In our terminology, we say that is A-saturated and PAR-saturated. We also prove, at the nonuniform level, the above results for every real-closed field. It is not known whether is P-saturated. In the case of the reals with addition and order we obtain P-saturation ). More generally, we show that an ordered -vector space is P-saturated at the nonuniform level ).We also study a stronger notion that we call P-stability. Blum, Cucker, Shub and Smale have shown that fields of characteristic 0 are P-stable. We show that the reals with addition and order are P-stable, but real-closed fields are not.Questions and and the/const complexity classes have some model theoretic flavor. This leads to the theory of complexity over “arbitrary” structures as introduced by Poizat. We give a summary of this theory with a special emphasis on the connections with model theory and we study/const complexity classes from this point of view. Note also that our proof of the PAR-saturation of shows that an o-minimal structure which admits quantifier elimination is A-saturated at the nonuniform level. (shrink)

$\underline{\text{Saturation is} (\mu, \kappa)-\text{transferable in} T}$ if and only if there is an expansion T 1 of T with ∣ T 1 ∣ = ∣ T ∣ such that if M is a μ-saturated model of T 1 and ∣ M ∣ ≥ κ then the reduct M ∣ L(T) is κ-saturated. We characterize theories which are superstable without f.c.p., or without f.c.p. as, respectively those where saturation is (ℵ 0 , λ)- transferable or (κ (T), λ)-transferable for (...) all λ. Further if for some $\mu \geq \mid T \mid, 2^\mu > \mu^+$ , stability is equivalent to for all μ ≥ ∣ T ∣, saturation is (μ, 2 μ )- transferable. (shrink)

We show that under the assumption of the existence of the canonical inner model with one Woodin cardinal$M_1$, there is a model of$\mathsf {ZFC}$in which$\mbox {NS}_{\omega _{1}}$is$\aleph _2$-saturated and${\Delta }_{1}$-definable with$\omega _1$as a parameter which answers a question of S. D. Friedman and L. Wu. We also show that starting from an arbitrary universe with a Woodin cardinal, there is a model with$\mbox {NS}_{\omega _{1}}$saturated and${\Delta }_{1}$-definable with a ladder system$\vec {C}$and a full Suslin treeTas parameters. Both results (...) rely on a new coding technique whose presentation is the main goal of this article. (shrink)

The paper presents an outline of the general theory of countable arithmetically saturated models of PA and some of its applications. We consider questions concerning the automorphism group of a countable recursively saturated model of PA. We prove new results concerning fixed point sets, open subgroups, and the cofinality of the automorphism group. We also prove that the standard system of a countable arithmetically saturated model of PA is determined by the lattice of its elementary substructures.

Our aim was to try to generalize some theorems about the saturation of ultrapowers to reduced powers. Naturally, we deal with saturation for types consisting of atomic formulas. We succeed to generalize “the theory of dense linear order (or T with the strict order property) is maximal and so is any which is SOP3”, (where Δ consists of atomic or conjunction of atomic formulas). However, the theorem on “it is enough to deal with symmetric pre‐cuts” (so the theorem) cannot be (...) generalized in this case. Similarly the uniqueness of the dual cofinality fails in this context. (shrink)

We use exact saturation to study the complexity of unstable theories, showing that a variant of this notion called pseudo-elementary class (PC)-exact saturation meaningfully reflects combinatorial dividing lines. We study PC-exact saturation for stable and simple theories. Among other results, we show that PC-exact saturation characterizes the stability cardinals of size at least continuum of a countable stable theory and, additionally, that simple unstable theories have PC-exact saturation at singular cardinals satisfying mild set-theoretic hypotheses. This had previously been open even (...) for the random graph. We characterize supersimplicity of countable theories in terms of having PC-exact saturation at singular cardinals of countable cofinality. We also consider the local analog of PC-exact saturation, showing that local PC-exact saturation for singular cardinals of countable cofinality characterizes supershort theories. (shrink)

Poorly saturated colors are closer to a pure grey than strongly saturated ones and, therefore, appear less “colorful”. Color saturation is effectively manipulated in the visual arts for balancing conflicting sensations and moods and for inducing the perception of relative distance in the pictorial plane. While perceptual science has proven quite clearly that the luminance contrast of any hue acts as a self-sufficient cue to relative depth in visual images, the role of color saturation in such figure-ground organization (...) has remained unclear. We presented configurations of colored inducers on grey ‘test’ backgrounds to human observers. Luminance and saturation of the inducers was uniform on each trial, but varied across trials. We ran two separate experimental tasks. In the relative background brightness task, perceptual judgments indicated whether the apparent brightness of the grey test background contrasted with, assimilated to, or appeared equal (no effect) to that of a comparison background with the same luminance contrast. Contrast polarity and its interaction with color saturation affected response proportions for contrast, assimilation and no effect. In the figure-ground task, perceptual judgments indicated whether the inducers appeared to lie in front of, behind, or in the same depth with the background. Strongly saturated inducers produced significantly larger proportions of foreground effects indicating that these inducers stand out as figure against the background. Weakly saturated inducers produced significantly larger proportions of background effects, indicating that these inducers are perceived as lying behind the backgrounds. We infer that color saturation modulates figure-ground organization, both directly by determining relative inducer depth, and indirectly, and in interaction with contrast polarity, by affecting apparent background brightness. The results point towards a hitherto undocumented functional role of color saturation in the genesis of form, and in particular figure-ground percepts in the absence of chromatostereopsis. (shrink)

We show that it is consistent relative to a huge cardinal that for all infinite cardinals [Formula: see text], [Formula: see text] holds and there is a stationary [Formula: see text] such that [Formula: see text] is [Formula: see text]-saturated.

We prove that it is consistent that there exists a saturated tail club guessing ideal on ω1 which is not a restriction of the non-stationary ideal. Two proofs are presented. The first one uses a new forcing axiom whose consistency can be proved from a supercompact cardinal. The resulting model can satisfy either CH or 20=2. The second one is a direct proof from a Woodin cardinal, which gives a witnessing model with CH.

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

Suppose that λ is the successor of a singular cardinal μ whose cofinality is an uncountable cardinal κ. We give a sufficient condition that the club filter of λ concentrating on the points of cofinality κ is not λ+-saturated.1 The condition is phrased in terms of a notion that we call weak reflection. We discuss various properties of weak reflection. We introduce a weak version of the ♣-principle, which we call ♣*−, and show that if it holds on a (...) stationary subset S of λ, then no normal filter on S is λ+-saturated. Under the above assumptions, ♣*− is true for any stationary subset S of λ which does not contain points of cofinality κ. For stationary sets S which concentrate on points of cofinality κ, we show that ♣*− holds modulo an ideal obtained through the weak reflection. (shrink)

Starting with a λ-supercompact cardinal κ, where λ is a regular cardinal greater than or equal to κ, we produce a model with a stationary subset S of such that , the ideal generated by the non-stationary ideal over together with , is λ+-saturated. Using this model we prove the consistency of the existence of such a stationary set together with the Generalized Continuum Hypothesis . We also show that in our model we can make -saturated, where S (...) is the set of all such that , the order type of x, is a regular cardinal and x is stationary in sup. Furthermore we construct a model where is κ+-saturated but GCH fails. We show that if SS is stationary in , then S can be split into λ many disjoint stationary subsets. (shrink)

Individuals suffering from depression consistently report experiencing a lack of connectedness with others. David Karp (2017, 73), in his memoir and study of depression, has gone so far to describe depression as “an illness of isolation, a disease of disconnectedness”. It has become common, in phenomenological circles, to attribute this social impairment to the depressed individual experiencing their body as corporealized, acting as a barrier between them and the world around them (Fuchs 2005, 2016). In this paper, I offer an (...) alternative view of the experience of social disconnectedness in depression, suggesting that rather than necessarily experiencing their body as object-like, the depressed individual’s bodily is saturated with experiences of lethargy, tiredness, heaviness, sadness, hopelessness and so on, to the exclusion of being able to bodily connect to others. I suggest that depression does not involve a complete social impairment but a specific impairment of affective forms of interpersonal experience. (shrink)

The paper presents four open problems concerning recursively saturated models of Peano Arithmetic. One problems concerns a possible converse to Tarski's undefinability of truth theorem. The other concern elementary cuts in countable recursively saturated models, extending automorphisms of countable recursively saturated models, and Jonsson models of PA. Some partial answers are given.

Elementary Recursive Nonstandard Analysis, in short ERNA, is a constructive system of nonstandard analysis with a PRA consistency proof, proposed around 1995 by Patrick Suppes and Richard Sommer. It is built on a previous system by Rolando Chuaqui and Patrick Suppes, which was recently reconsidered by Michal Rössler and Emil Jeřábek. A $\Pi _1 $ -transfer principle has already been added to ERNA and the consistency of the resulting theory proved in PRA. Here, we equip ERNA with Σ₂-transfer and a (...) saturation principle, while keeping the consistency proof inside PRA. We show that the extended theory allows for generalized transfer, a basic tool of nonstandard analysis, and interprets several strong theories, like BΣ₂ and IΣ₂. (shrink)

In this article, I will argue that while Marion’s criticism of Heidegger’s project of fundamental ontology is in many ways sound, Marion remains bound to the conceptual opposition that existential phenomenology has successfully overcome. Namely, I will argue that Marion remains dependent upon the transcendental dilemma according to which we must rely on the strict differentiation between explanans and explanandum. Marion sees no way of departing from Heidegger’s project other than reversing the order of explanation and switching the places of (...) the explanans and explanandum, which means that phenomena start appearing as explanans. I will demonstrate how existential phenomenology has overcome this conceptual dilemma, and then I will argue that we could make a much better account of saturated phenomena, if we ground our insights in the idea of being-in-the-world. (shrink)

A theory [Formula: see text] is said to have exact saturation at a singular cardinal [Formula: see text] if it has a [Formula: see text]-saturated model which is not [Formula: see text]-saturated. We show, under some set-theoretic assumptions, that any simple theory has exact saturation. Also, an NIP theory has exact saturation if and only if it is not distal. This gives a new characterization of distality.

Multiple-universe theories (involving the saturation of a state space of alternative universes) enjoy growing popularity as proposed counters to the design-suggestiveness of increasingly impressive cosmic fine-tuning cases. In this paper I explore this type of counter, and argue that (i) multiple-universe theoriesdo not necessarily undercut the design relevance of cosmic fine-tuning, that (ii) both the required saturation and the hypothesized mechanisms supposed to generate such saturation face seriously non-trivial constraints, and that (iii) in some cases the specific characteristics of such (...) saturations may themselves be suggestive of design. (shrink)

We use the language of categorical logic to construct generic saturated models of intuitionistic theories. Our main technique is the thorough study of the filter construction on categories with finite limits, which is the completion of subobject lattices under filtered meets. When restricted to coherent or Heyting categories, classifying categories of intuitionistic first-order theories, the resulting categories are filtered meet coherent categories, coherent categories with complete subobject lattices such that both finite disjunctions and existential quantification distribute over filtered meets. (...) Such categories naturally embed into Grothendieck toposes which then contain saturated models of the theory we started with. (shrink)

The philosophical work of Jean-Luc Marion has opened new ways of speaking about religious convictions and experiences. In this exploration of Marion’s philosophy and theology, Christina M. Gschwandtner presents a comprehensive and critical analysis of the ideas of saturated phenomena and the phenomenology of givenness. She claims that these phenomena do not always appear in the excessive mode that Marion describes and suggests instead that we consider degrees of saturation. Gschwandtner covers major themes in Marion’s work—the historical event, art, (...) nature, love, gift and sacrifice, prayer, and the Eucharist. She works within the phenomenology of givenness, but suggests that Marion himself has not considered important aspects of his philosophy. (shrink)

This article aims to explore the philosophical approach to faith after deconstruction as manifested in the work of Jean-Luc Marion and Jean-Luc Nancy. By taking the saturated phenomenon as its focus, the analysis seeks to demonstrate that whilst Marion’s thinking proves to be an innovative re-imagining of the possibilities of phenomenology, its problematic recourse to a supplementary hermeneutic means that saturation can never be adequately applied to faith without simultaneously compromising the excessive intuition upon which it relies. The article (...) then explores whether Nancy’s suggestion that saturation be re-framed as faith can offer a viable alternative approach. Whilst the post-phenomenological modality within which Nancy operates means it may be problematic to retain the term ‘saturation’ in the exact sense Marion gives it, it is argued that Nancy’s version of saturated faith allows us to approach the binary divide between philosophy and theology from a different direction, resulting in a vision of faith that cuts across theism and atheism, destabilising them from within. Although Nancy’s thought in this area certainly does nothing to respond to persistent questions surrounding the place of institutionalized religion within secular modernity, it nevertheless serves as a powerful tool for thinking the possibilities of faith in the twenty-first century. (shrink)

The paper is a continuation of the investigations of [3]. We assume that the reader is familiar with the notion of an !-saturated model and the notation applied in the theory of models. We refer to [2] for all model- theoretic terms which appear in this note.

We give an exposition of results of Baldwin–Shelah [2] on saturated free algebras, at the level of generality of complete first order theoriesTwith a saturated modelMwhich is in the algebraic closure of an indiscernible set. We then make some new observations whenM isa saturated free algebra, analogous to (more difficult) results for the free group, such as a description of forking.

We examine the relationship between two different notions of a structure being Scott set saturated and identify sufficient conditions which guarantee that a structure is uniquely Scott set saturated. We also consider theories representing Scott sets; in particular, we identify a sufficient condition on a theory T so that for any given countable Scott set there exists a completion of T that is saturated with respect to the given Scott set. These results extend Scott's characterization of countable (...) Scott sets via models and completions of Peano arithmetic. (shrink)

We prove that WRP and saturation of the ideal NSω1together imply$\left\{ {a \in [\lambda ]^{\omega _1 } :{\text{cof}}\left( {{\text{sup}}\left( a \right)} \right) = \omega _1 } \right\}$, for every cardinalλwith cof(λ)≥ω2.

A key topic within the field of the phenomenology of medicine has been the relationship between body and self in illness, including discussions about the otherness and mineness of the body. The aim of this article is to distinguish between different meanings of bodily otherness and mineness in illness with reference to the interpretation of the body as “saturated phenomenon,” inspired by the phenomenology of Jean-Luc Marion. With the help of Marion’s ideas it is possible to distinguish between two (...) meanings of bodily otherness and of mineness (objective and non-objective forms). These distinctions support and elaborate on ideas already found in the phenomenology of medicine and offer further insights into the nature of the experience of illness. (shrink)

We introduce the class of unshreddable theories, which contains the simple and NIP theories, and prove that such theories have exactly saturated models in singular cardinals, satisfying certain set-theoretic hypotheses. We also give criteria for a theory to have singular compactness.

During the last few decades, many thinkers have advocated for the importance of the phenomenological approach in developing the understanding of the lived experience of illness. In their attempts, they have referred to ideas found in the history of phenomenology, most notably, in the works of Edmund Husserl, Martin Heidegger, Maurice Merleau-Ponty and Jean-Paul Sartre. The aim of this paper is to sketch out an interpretation of illness based on a yet unexplored conceptual framework of the phenomenology of French thinker (...) Jean-Luc Marion. Focusing on concepts of the saturated phenomenon and flesh, the paper develops an interpretation of illness as the saturated phenomenon, which highlights a variety of dimensions of illness already elaborated within the phenomenology of medicine, such as the affective dimension of illness, the disruptive dimension of illness, the transformed perception of the self in illness, mineness of flesh in illness and the inexpressible and hermeneutical dimension of illness. In addition to that, the paper explores some of the consequences the proposed interpretation of illness offers regarding the nature of illness and health. It is argued that illness in its essence is very similar to the experience of other saturated phenomena, suggesting that the difference between them does not lie within the character of the affective givenness, but rather within the dynamic relationship between the affective givenness and its conceptualization. It is also shown that the experience of health is compatible with the experience of saturation and thus is not limited to the tacit and harmonious background state. (shrink)

Letκbe the cardinality of some saturated model of Peano Arithmetic. There is a set of${2^{{\aleph _0}}}$saturated models of PA, each having cardinalityκ, such that wheneverMandNare two distinct models from this set, then Aut(${\cal M}$) ≇ Aut ($${\cal N}$$).

In this paper we shall discuss properties of saturation in monotonic neighbourhood models and study some applications, like a characterization of compact and modally saturated monotonic models and a characterization of the maximal Hennessy-Milner classes. We shall also show that our notion of modal saturation for monotonic models naturally extends the notion of modal saturation for Kripke models.

We discuss automorphisms of saturated models of PA and boundedly saturated models of PA. We show that Smoryński's Lemma and Kaye's Theorem are not only true for countable recursively saturated models of PA but also true for all boundedly saturated models of PA with slight modifications.

We study the two model-theoretic concepts of weak saturation and weak amalgamation property in the context of accessible categories. We relate these two concepts providing sufficient conditions for existence and uniqueness of weakly saturated objects of an accessible category ${\cal K}$. We discuss the implications of this fact in classical model theory.

The notion of quasi-minimal structures was defined by B. Zil'ber as a natural generalization of minimal structures. Inspired by his work, we study here basic model theoretic properties of quasiminimal structures. Main result is the construction of ω-saturated quasi-minimal models under ω-stability assumption.