We define a rich model to be one which contains a proper elementary substructure isomorphic to itself. Existence, nonstructure, and categoricity theorems for rich models are proved. A theory T which has fewer than $\min(2^\lambda,\beth_2)$ rich models of cardinality $\lambda(\lambda > |T|)$ is totally transcendental. We show that a countable theory with a unique rich model in some uncountable cardinal is categorical in ℵ 1 and also has a unique countable rich model. We also consider a stronger notion of richness, (...) and use it to characterize superstable theories. (shrink)

Let L(Q) be first order logic with Keisler's quantifier, in the λ + interpretation (= the satisfaction is defined as follows: $M \models (\mathbf{Q}x)\varphi(x)$ means there are λ + many elements in M satisfying the formula φ(x)). Theorem 1. Let λ be a singular cardinal; assume □ λ and GCH. If T is a complete theory in L(Q) of cardinality at most λ, and p is an L(Q) 1-type so that T strongly omits $p (= p$ has no support, to (...) be defined in $\S1$ ), then T has a model of cardinality λ + in the λ + interpretation which omits p. Theorem 2. Let λ be a singular cardinal, and let T be a complete first order theory of cardinality λ at most. Assume □ λ and GCH. If Γ is a smallness notion then T has a model of cardinality λ + such that a formula φ(x) is realized by λ + elements of M iff φ(x) is not Γ-small. The theorem is proved also when λ is regular assuming $\lambda = \lambda^{ . It is new when λ is singular or when |T| = λ is regular. Theorem 3. Let λ be singular. If $\operatorname{Con}(ZFC + GCH + (\exists\kappa)$ [κ is a strongly compact cardinal]), then the following in consistent: ZFC + GCH + the conclusions of all above theorems are false. (shrink)

The new result of this paper is that for θ-stable we have S1[θ] = D[θ, L, ∞]. S1 is Hrushovski's rank. This is an improvement of a result of Kim and Pillay, who for simple theories under the assumption that either of the ranks be finite obtained the same identity. Only the first equality is new, the second equality is a result of Shelah from the seventies. We derive it by studying localizations of several rank functions, we get the followingMain (...) Theorem. Suppose that μ is regular satisfying μ ≥ |T|+, p is a finite type, and Δ is a set of formulas closed under Boolean operations. If either R[p, Δ, μ+] < ∞ or p is Δ-stable and μ satisfies “for every sequence {μi : i < |Δ| + ℵ0} of cardinals μi < μ we have that equation image holds”, then S[p, Δ, μ+] = D[p, Δ, μ+] = R[p, Δ, μ+].The S rank above is a localized version of Hrushovski's S1 rank. This rank, as well as our systematic use of local stability, allows us to get a more conceptual proof of the equality of D and R, which is an old result of Shelah. A particular case of the theorem offers a new sufficient condition for the equality of S1 and D[·, L, ∞]. We also manage, due to a more general approach, to avoid some combinatorial difficulties present in Shelah's original exposition. (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 prove the following main theorem: Let be an abstract elementary class satisfying the joint embedding and the amalgamation properties with no maximal models of cardinality μ. Let μ be a cardinal above the the Löwenheim‐Skolem number of the class. If is μ‐Galois‐stable, has no μ‐Vaughtian Pairs, does not have long splitting chains, and satisfies locality of splitting, then any two ‐limits over M, for, are isomorphic over M.

We introduce tame abstract elementary classes as a generalization of all cases of abstract elementary classes that are known to permit development of stability-like theory. In this paper, we explore stability results in this new context. We assume that [Formula: see text] is a tame abstract elementary class satisfying the amalgamation property with no maximal model. The main results include:. Theorem 0.1. Suppose that [Formula: see text] is not only tame, but [Formula: see text]-tame. If [Formula: see text] and [Formula: (...) see text] is Galois stable in μ, then [Formula: see text], where [Formula: see text] is a relative of κ from first order logic. [Formula: see text] is the Hanf number of the class [Formula: see text]. It is known that [Formula: see text]. The theorem generalizes a result from [17]. It is used to prove both the existence of Morley sequences for non-splitting and the following initial step towards a stability spectrum theorem for tame classes:. Theorem 0.2. If [Formula: see text] is Galois-stable in some [Formula: see text], then [Formula: see text] is stable in every κ with κμ=κ. For example, under GCH we have that [Formula: see text] Galois-stable in μ implies that [Formula: see text] is Galois-stable in μ+n for all n < ω. (shrink)

We prove a categoricity transfer theorem for tame abstract elementary classes. Theorem 0.1. Suppose that K is a χ-tame abstract elementary class and satisfies the amalgamation and joint embedding properties and has arbitrarily large models. Let λ ≥ Max{χ.LS(K)⁺}. If K is categorical in λ and λ⁺, then K is categorical in λ⁺⁺. Combining this theorem with some results from [37], we derive a form of Shelah's Categoricity Conjecture for tame abstract elementary classes: Corollary 0.2. Suppose K is a χ-tame (...) abstract elementary class satisfying the amalgamation and joint embedding properties. Let μ₀:= Hanf(K). If χ ≤ ‮ב‬(2μ0)+ and K is categorical in some λ⁺ > ‮ב‬(2μ0)+, then K is categorical in μ for all μ > ‮ב‬(2μ0)+. (shrink)

Let κ and λ be infinite cardinals such that κ ≤ λ (we have new information for the case when $\kappa ). Let T be a theory in L κ +, ω of cardinality at most κ, let φ(x̄, ȳ) ∈ L λ +, ω . Now define $\mu^\ast_\varphi (\lambda, T) = \operatorname{Min} \{\mu^\ast:$ If T satisfies $(\forall\mu \kappa)(\exists M_\chi \models T)(\exists \{a_i: i Our main concept in this paper is $\mu^\ast_\varphi (\lambda, \kappa) = \operatorname{Sup}\{\mu^\ast(\lambda, T): T$ is a theory (...) in L κ +, ω of cardinality κ at most, and φ (x, y) ∈ L λ +, ω }. This concept is interesting because of THEOREM 1. Let $T \subseteq L_{\kappa^+,\omega}$ of cardinality ≤ κ, and φ (x̄, ȳ) ∈ L λ +, ω . If $(\forall\mu then $(\forall_\chi > \kappa) I(\chi, T) = 2^\chi$ (where I(χ, T) stands for the number of isomorphism types of models of T of cardinality χ). Many years ago the second author proved that $\mu^\ast (\lambda, \kappa) \leq \beth_{(2^\lambda)^+}$ . Here we continue that work by proving. THEOREM 2. $\mu^\ast (\lambda, \aleph_0) = \beth_{\lambda^+}$ . THEOREM 3. For every κ ≤ λ we have $\mu^\ast (\lambda, \kappa) \leq \beth)_{(\lambda^\kappa)}^+$ . For some κ or λ we have better bounds than in Theorem 3, and this is proved via a new two cardinal theorem. THEOREM 4. For every $\kappa \leq \lambda, T \subseteq L_{\kappa^+,\omega}$ , and any set of formulas $\Lambda \subseteq L_{\lambda^+,\omega}$ such that $\Lambda \subseteq L_{\kappa^+,\omega}$ , if T is (Λ,μ)-unstable for μ satisfying μ μ * (λ, κ) = μ then T is Λ-unstable (i.e. for every χ ≥ λ, T is (Λ, χ)-unstable). Moreover, T is L κ +, ω -unstable. In the second part of the paper, we show that always in the applications it is possible to replace the function I(χ, T) by the function IE(χ, T), and we give an application of the theorems to Boolean powers. (shrink)

We prove that from categoricity in λ+ we can get categoricity in all cardinals ≥ λ+ in a χ-tame abstract elementary classe [Formula: see text] which has arbitrarily large models and satisfies the amalgamation and joint embedding properties, provided [Formula: see text] and λ ≥ χ. For the missing case when [Formula: see text], we prove that [Formula: see text] is totally categorical provided that [Formula: see text] is categorical in [Formula: see text] and [Formula: see text].

. We study a local version of the order property in several frameworks, with an emphasis on frameworks where the compactness theorem fails: (1) Inside a fixed model, (2) for classes of models where the compactness theorem fails and (3) for the first order case. Appropriate localizations of the order property, the independence property, and the strict order property are introduced. We are able to generalize some of the results that were known in the case of local stability for the (...) first order theories, and for stability for nonelementary classes (existence of indiscernibles, existence of averages, stability spectrum, equivalence between order and instability). In the first order case, we also prove the local version of Shelah's Trichotomy Theorem. Finally, as an application, we give a new characterization of stable types when the ambient first order theory is simple. (shrink)

We present Saharon Shelah's Stability Spectrum and Homogeneity Spectrum theorems, as well as the equivalence between the order property and instability in the framework of Finite Diagrams. Finite Diagrams is a context which generalizes the first order case. Localized versions of these theorems are presented. Our presentation is based on several papers; the point of view is contemporary and some of the proofs are new. The treatment of local stability in Finite Diagrams is new.

We present a self-contained exposition of the basic aspects of simple theories while developing the fundamentals of forking calculus. We expound also the deeper aspects of S. Shelah's 1980 paper Simple unstable theories. The concept of weak dividing has been replaced with that of forking. The exposition is from a contemporary perspective and takes into account contributions due to S. Buechler, E. Hrushovski, B. Kim, O. Lessmann, S. Shelah and A. Pillay.

We present a downward Löwenheim-Skolem theorem which transfers downward formulas from L ∞,ω to L κ +, ω . The simplest instance is: Theorem 1. Let $\lambda > \kappa$ be infinite cardinals, and let L be a similarity type of cardinality κ at most. For every L-structure M of cardinality λ and every $X \subseteq M$ there exists a model $N \prec M$ containing the set X of power |X| · κ such that for every pair of finite sequences a, (...) b ∈ N $\langle N, \mathbf{a}\rangle \equiv_{\| N \|^+,\omega} \langle N, \mathbf{b}\rangle \Leftrightarrow \langle M, \mathbf{a}\rangle \equiv_{\infty,\omega} \langle M, \mathbf{b}\rangle.$ The following theorem is an application: Theorem 2. Let $\lambda , and suppose χ is a Ramsey cardinal greater than λ. If T has the (χ, L κ +, ω -unsuperstability property, then T has the (χ, L λ +, ω )-unsuperstability property. (shrink)

Let M be a given model with similarity type L = L(M), and let L' be any fragment of L |L(M)| +, ω of cardinality |L(M)|. We call $N \prec M L'$ -relatively saturated $\operatorname{iff}$ for every $B \subseteq N$ of cardinality less than | N | every L'-type over B which is realized in M is realized in M is realized in N. We discuss the existence of such submodels. The following are corollaries of the existence theorems. (1) If (...) M is of cardinality at least $\beth_{\omega_1}$ , and fails to have the ω order property, then there exists $N \prec M$ which is relatively saturated in M of cardinality $\beth_{\omega_1}$ . (2) Assume GCH. Let ψ ∈ L_{ω_1, ω, and let $L' \subseteq L_{\omega 1, \omega$ be a countable fragment containing ψ. If $\exists \chi > \aleph_0$ such that $I(\chi, \psi) , then for every $M \models \psi$ and every cardinal $\lambda of uncountable cofinality, M has an L'-relatively saturated submodel of cardinality λ. (shrink)

Basic results on the model theory of substructures of a fixed model are presented. The main point is to avoid the use of the compactness theorem, so this work can easily be applied to the model theory of L ω 1 ,ω and its relatives. Among other things we prove the following theorem: Let M be a model, and let λ be a cardinal satisfying λ |L(M)| = λ. If M does not have the ω-order property, then for every $A (...) \subseteq M, |A| \leq \lambda$ , and every $\mathbf{I} \subseteq M$ of cardinality λ + there exists $\mathbf{J} \subseteq \mathbf{I}$ of cardinality λ + which is an indiscernible set over A. This is an improvement of a result of S. Shelah. (shrink)

In this paper we argue that autobiographical memory can be conceptualized as a mental state resulting from the interplay of a set of psychological capacities?self-reflection, self-agency, self-ownership and personal temporality?that transform a memorial representation into an autobiographical personal experience. We first review evidence from a variety of clinical domains?for example, amnesia, autism, frontal lobe pathology, schizophrenia?showing that breakdowns in any of the proposed components can produce impairments in autobiographical recollection, and conclude that the self-reflection, agency, ownership, and personal temporality are (...) individually necessary and jointly sufficient for autobiographical memorial experience. We then suggest a taxonomy of amnesic disorders derived from consideration of the consequences of breakdown in each of the individual component processes that contribute to the experience of autobiographical recollection. (shrink)

Plamondon & Alimi (P&A) have unified much data on speed/accuracy trade-offs during reaching movements using a delta-lognormal form factor that describes notably neuromuscular systems. Their approach raises questions about whether a large number of systems is needed, whether they are linear, and whether the results disclose the neural design principles that control reaching behaviors. The authors admit that (sect. 6, para. 4).

In this paper I am concerned with the semantic analysis of sentences of the form 'It is true that p'. I will compare different proposals that have been made to analyse such sentences and will defend a view that treats this sentences as a mere sytactic variation of sentences of the form 'That p is true'.

To understand schizophrenia, a linking hypothesis is needed that shows how brain mechanisms lead to behavioral functions in normals, and also how breakdowns in these mechanisms lead to behavioral symptoms of schizophrenia. Such a linking hypothesis is now available that complements the discussion offered by Phillips & Silverstein (P&S).

I address an intuition commonly endorsed by metaphysicians, that there must be a fundamental layer of reality, i.e., that chains of ontological dependence must terminate: there cannot be turtles all the way down. I discuss applications of this intuition with reference to Bradley’s regress, composition, realism about the mental and the cosmological argument. I discuss some arguments for the intui- tion, but argue that they are unconvincing. I conclude by making some suggestions for how the intuition should be argued for, (...) and discussing the ramifications of giving the justification I think best. (shrink)

We give an answer to the question as to whether quantifier elimination is possible in some infinite algebraic extensions of Qp (‘infinite p-adic fields’) using a natural language extension. The present paper deals with those infinite p-adic fields which admit only tamely ramified algebraic extensions (so-called tame fields). In the case of tame fields whose residue fields satisfy Kaplansky’s condition of having no extension of p-divisible degree quantifier elimination is possible when the language of valued fields is extended by the (...) power predicates Pn introduced by Macintyre and, for the residue field, further predicates and constants. For tame infinite p-adic fields with algebraically closed residue fields an extension by Pn predicates is sufficient. (shrink)

In my paper I review the main strategies adopted by two leading schools in cognitive science, symbolic artificial intelligence (AI) and connectionism, in modeling time-dependent phenomena such as learning. In particular, I briefly mention shortcomings of non-monotonic logic approach that dominates symbolic AI. I also discuss the problems that beset the recurrent networks approach advocated by connectionists (e.g., S. Grossberg) and philosophers (e.g., P. Churchland), who focus their attention on oscillatory behavior of such networks. I point out that neither (...) approach adequately captures essential features of the dynamics of human temporal awareness. Finally, I develop certain ideas borrowed from neurophysiologists (e.g., W. Freeman) and system theorists and describe the brain-mind, the entity that is capable of producing temporal awareness, as a self-organizing system that exhibits chaotic dynamics and generates a dynamic structure that resembles the chaotic hierarchy o Rossler. (shrink)

The desire to see a sign phenomenon in different ways always has a reason to interpret it in an unbiased, panoramic way, and in some places - even allowing for contradictions in its understanding by different participants in the interpretative process. For modern humanities, this disposition is quite understandable, since it follows from its very postmodern nature, thereby defining the semantic semantic fields of the leading humanities. True, it is not so wide-spread, but instead, it is evident that Ukrainian religious (...) thinkers of the Middle Ages and Baroque have interpreted the iconic phenomena more deeply. In the story of Petro Mohyla "On the Wonderful Elder Grigory Mezhyhirsky" the metropolitan Mohyla clearly expresses the need to build before the thinking gaze of the reader his own, immanently religious-philosophical conception of the Divine, which every time would present him not just as an act of manifestation, otherwise he appears before the person, while remaining alone. The contemporary study of this issue in order to integrate its leading concepts in the religious-philosophical and general cultural discourse of the postmodern era has largely determined the relevance of the proposed article. The study of the phenomenon of "leaving" by itself in changing forms of the phenomenon of the Divine is also one of the key ones in the proposed studio, because it expresses the very meaning of the Divine as a religious, and therefore a cultural phenomenon. (shrink)

Given a physical system, one knows that there is a logical duality between its properties and its states. In this paper, we choose its states as the undefined notions of our axiomatic construction. In fact, by means of well-motivated assumptions expressed in terms of a transition probability function defined on the set of all pure states of the system, we construct a system of elementary propositions, i.e., a complete orthomodular atomic lattice satisfying the covering law. We also study in this (...) framework the important notion of compatibility of propositions, and we define the superpositions and the mixtures of the states of the physical system. (shrink)

Richard Taylor was born in Charlotte, Michigan on 5 November 1919. He received his A. B. from the University of illinois in 1941, his M. A. from Oberlin College in 1947, and his Ph. D. from Brown University in 1951. He has been William H. P. Faunce Professor of Philosophy at Brown University, Professor of Philosophy (Graduate Faculties) at Columbia University, and Professor of Philosophy at the University of Rochester. He is the author of about fifty articles and of five (...) philosophical books. This volume consists of essays presented to Richard Taylor on the occa sion of his sixtieth birthday. Some of the contributors have been Taylor'S students; some have been his colleagues; and all have been, and continue to be, his admirers. I have made several attempts to articulate what it is I (I would not presume to speak for anyone else) admire about Richard Taylor: (1) There is a particular 'flavor' to Taylor's philosophical writing and con versation that is wholly delightful. Like any other flavor, it can be tasted and enjoyed and remembered but never adequately described. (If there should be someone who has picked up this book who does not know what I mean, I recommend that he read the chapter on 'God' in Taylor's Metaphysics. ) (2) Taylor is a masterful dialectician. (shrink)