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  1. Superstable Groups.Ch Berline & D. Lascar - 1986 - Annals of Pure and Applied Logic 30 (1):1-43.
  • NIP for Some Pair-Like Theories.Gareth Boxall - 2011 - Archive for Mathematical Logic 50 (3-4):353-359.
    Generalising work of Berenstein, Dolich and Onshuus (Preprint 145 on MODNET Preprint server, 2008) and Günaydın and Hieronymi (Preprint 146 on MODNET Preprint server, 2010), we give sufficient conditions for a theory T P to inherit N I P from T, where T P is an expansion of the theory T by a unary predicate P. We apply our result to theories, studied by Belegradek and Zilber (J. Lond. Math. Soc. 78:563–579, 2008), of the real field with a subgroup of (...)
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  • Transfering Saturation, the Finite Cover Property, and Stability.John T. Baldwin, Rami Grossberg & Saharon Shelah - 1999 - Journal of Symbolic Logic 64 (2):678-684.
    $\underline{\text{Saturation is} (\mu, \kappa)-\text{transferable in} T}$ 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. (...)
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  • Applications of Vaught Sentences and the Covering Theorem.Victor Harnik & Michael Makkai - 1976 - Journal of Symbolic Logic 41 (1):171-187.
    We use a fundamental theorem of Vaught, called the covering theorem in [V] (cf. theorem 0.1 below) as well as a generalization of it (cf. Theorem $0.1^\ast$ below) to derive several known and a few new results related to the logic $L_{\omega_1\omega}$. Among others, we prove that if every countable model in a $PC_{\omega_1\omega}$ class has only countably many automorphisms, then the class has either $\leq\aleph_0$ or exactly $2^{\aleph_0}$ nonisomorphic countable members (cf. Theorem $4.3^\ast$) and that the class of countable (...)
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  • On the Number of Nonisomorphic Models of an Infinitary Theory Which has the Infinitary Order Property. Part A.Rami Grossberg & Saharon Shelah - 1986 - Journal of Symbolic Logic 51 (2):302-322.
    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 (...)
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  • Uniqueness and Characterization of Prime Models Over Sets for Totally Transcendental First-Order Theories.Saharon Shelah - 1972 - Journal of Symbolic Logic 37 (1):107-113.
  • The Stability Function of a Theory.H. Jerome Keisler - 1978 - Journal of Symbolic Logic 43 (3):481-486.
    Let T be a complete theory with infinite models in a countable language. The stability function g T (κ) is defined as the supremum of the number of types over models of T of power κ. It is proved that there are only six possible stability functions, namely $\kappa, \kappa + 2^\omega, \kappa^\omega, \operatorname{ded} \kappa, (\operatorname{ded} \kappa)^\omega, 2^\kappa$.
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  • End Extensions and Numbers of Countable Models.Saharon Shelah - 1978 - Journal of Symbolic Logic 43 (3):550-562.
    We prove that every model of $T = \mathrm{Th}(\omega, countable) has an end extension; and that every countable theory with an infinite order and Skolem functions has 2 ℵ 0 nonisomorphic countable models; and that if every model of T has an end extension, then every |T|-universal model of T has an end extension definable with parameters.
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  • Notes on the Stability of Separably Closed Fields.Carol Wood - 1979 - Journal of Symbolic Logic 44 (3):412-416.
    The stability of each of the theories of separably closed fields is proved, in the manner of Shelah's proof of the corresponding result for differentially closed fields. These are at present the only known stable but not superstable theories of fields. We indicate in § 3 how each of the theories of separably closed fields can be associated with a model complete theory in the language of differential algebra. We assume familiarity with some basic facts about model completeness [4], stability (...)
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  • One Theorem of Zil′Ber's on Strongly Minimal Sets.Steven Buechler - 1985 - Journal of Symbolic Logic 50 (4):1054-1061.
    Suppose $D \subset M$ is a strongly minimal set definable in M with parameters from C. We say D is locally modular if for all $X, Y \subset D$ , with $X = \operatorname{acl}(X \cup C) \cap D, Y = \operatorname{acl}(Y \cup C) \cap D$ and $X \cap Y \neq \varnothing$ , dim(X ∪ Y) + dim(X ∩ Y) = dim(X) + dim(Y). We prove the following theorems. Theorem 1. Suppose M is stable and $D \subset M$ is strongly minimal. (...)
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  • Uncountable Theories That Are Categorical in a Higher Power.Michael Chris Laskowski - 1988 - Journal of Symbolic Logic 53 (2):512-530.
    In this paper we prove three theorems about first-order theories that are categorical in a higher power. The first theorem asserts that such a theory either is totally categorical or there exist prime and minimal models over arbitrary base sets. The second theorem shows that such theories have a natural notion of dimension that determines the models of the theory up to isomorphism. From this we conclude that $I(T, \aleph_\alpha) = \aleph_0 +|\alpha|$ where ℵ α = the number of formulas (...)
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  • Partially Ordered Sets and the Independence Property.James H. Schmerl - 1989 - Journal of Symbolic Logic 54 (2):396-401.
    No theory of a partially ordered set of finite width has the independence property, generalizing Poizat's corresponding result for linearly ordered sets. In fact, a question of Poizat concerning linearly ordered sets is answered by showing, moreover, that no theory of a partially ordered set of finite width has the multi-order property. It then follows that a distributive lattice is not finite-dimensional $\operatorname{iff}$ its theory has the independence property $\operatorname{iff}$ its theory has the multi-order property.
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  • A Many Permutation Group Result for Unstable Theories.Mark D. Schlatter - 1998 - Journal of Symbolic Logic 63 (2):694-708.
    We extend Shelah's first many model result to show that an unstable theory has 2 κ many non-permutation group isomorphic models of size κ, where κ is an uncountable regular cardinal.
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  • Finite Variable Logic, Stability and Finite Models.Marko Djordjević - 2001 - Journal of Symbolic Logic 66 (2):837-858.
  • Examples in Dependent Theories.Itay Kaplan & Saharon Shelah - 2014 - Journal of Symbolic Logic 79 (2):585-619.
  • Erdős and Set Theory.Akihiro Kanamori - 2014 - Bulletin of Symbolic Logic 20 (4):449-490,.
  • Stability Theory and Set Existence Axioms.Victor Harnik - 1985 - Journal of Symbolic Logic 50 (1):123-137.
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  • Magidor-Malitz Quantifiers in Modules.Andreas Baudisch - 1984 - Journal of Symbolic Logic 49 (1):1-8.
    We prove the elimination of Magidor-Malitz quantifiers for R-modules relative to certain Q 2 α -core sentences and positive primitive formulas. For complete extensions of the elementary theory of R-modules it follows that all Ramsey quantifiers (ℵ 0 -interpretation) are eliminable. By a result of Baldwin and Kueker [1] this implies that there is no R-module having the finite cover property.
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  • Preservation of Saturation and Stability in a Variety of Nilpotent Groups.Pat Rogers - 1981 - Journal of Symbolic Logic 46 (3):499-512.
  • ℵ0-Categorical Modules.Walter Baur - 1975 - Journal of Symbolic Logic 40 (2):213 - 220.
    It is shown that the first-order theory Th R (A) of a countable module over an arbitrary countable ring R is ℵ 0 -categorical if and only if $A \cong \bigoplus_{t finite, n ∈ ω, κ i ≤ ω. Furthermore, Th R (A) is ℵ 0 -categorical for all R-modules A if and only if R is finite and there exist only finitely many isomorphism classes of indecomposable R-modules.
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  • Weak Definability in Infinitary Languages.Saharon Shelah - 1973 - Journal of Symbolic Logic 38 (3):399-404.
    We shall prove that if a model of cardinality κ can be expanded to a model of a sentence ψ of Lλ+,ω by adding a suitable predicate in more than κ ways, then, it has a submodel of power μ which can be expanded to a model of ψ in $> \mu$ ways provided that λ,κ,μ satisfy suitable conditions.
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  • ℵ0-Categorical, ℵ0-Stable Structures.G. Cherlin, L. Harrington & A. H. Lachlan - 1985 - Annals of Pure and Applied Logic 28 (2):103-135.
  • Fundamentals of Forking.Victor Harnik & Leo Harrington - 1984 - Annals of Pure and Applied Logic 26 (3):245-286.
  • Toward Classifying Unstable Theories.Saharon Shelah - 1996 - Annals of Pure and Applied Logic 80 (3):229-255.
  • Models and Types of Peano's Arithmetic.Haim Gaifman - 1976 - Annals of Mathematical Logic 9 (3):223-306.
  • A Remark on the Strict Order Property.A. H. Lachlan - 1975 - Mathematical Logic Quarterly 21 (1):69-70.
  • Some EC∑ Classes of Rings.John T. Baldwin - 1978 - Mathematical Logic Quarterly 24 (31‐36):489-492.
  • Les Beaux Automorphismes.Daniel Lascar - 1991 - Archive for Mathematical Logic 31 (1):55-68.
    Assume that the class of partial automorphisms of the monster model of a complete theory has the amalgamation property. The beautiful automorphisms are the automorphisms of models ofT which: 1. are strong, i.e. leave the algebraic closure (inT eq) of the empty set pointwise fixed, 2. are obtained by the Fraïsse construction using the amalgamation property that we have just mentioned. We show that all the beautiful automorphisms have the same theory (in the language ofT plus one unary function symbol (...)
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  • Paires de Structures Stables.Bruno Poizat - 1983 - Journal of Symbolic Logic 48 (2):239-249.
  • Distal and Non-Distal Pairs.Philipp Hieronymi & Travis Nell - 2017 - Journal of Symbolic Logic 82 (1):375-383.
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  • Stability Theory and Algebra.John T. Baldwin - 1979 - Journal of Symbolic Logic 44 (4):599-608.
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  • Hanf Number of Omitting Type for Simple First-Order Theories.Saharon Shelah - 1979 - Journal of Symbolic Logic 44 (3):319-324.
    Let T be a complete countable first-order theory such that every ultrapower of a model of T is saturated. If T has a model omitting a type p in every cardinality $ then T has a model omitting p in every cardinality. There is also a related theorem, and an example showing the $\beth_\omega$ cannot be improved.
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  • On Models with Power-Like Ordering.Saharon Shelah - 1972 - Journal of Symbolic Logic 37 (2):247-267.
    We prove here theorems of the form: if T has a model M in which P 1 (M) is κ 1 -like ordered, P 2 (M) is κ 2 -like ordered ..., and Q 1 (M) if of power λ 1 , ..., then T has a model N in which P 1 (M) is κ 1 '-like ordered ..., Q 1 (N) is of power λ 1 ,.... (In this article κ is a strong-limit singular cardinal, and κ' is (...)
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  • Spectra of Ω-Stable Theories.A. H. Lachlan - 1978 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 24 (9-11):129-139.