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  1. The Prospects for Mathematical Logic in the Twenty-First Century.Samuel R. Buss, Alexander S. Kechris, Anand Pillay & Richard A. Shore - 2001 - Bulletin of Symbolic Logic 7 (2):169-196.
    The four authors present their speculations about the future developments of mathematical logic in the twenty-first century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently.
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  • Characterizing Model-Theoretic Dividing Lines Via Collapse of Generalized Indiscernibles.Vincent Guingona, Cameron Donnay Hill & Lynn Scow - 2017 - Annals of Pure and Applied Logic 168 (5):1091-1111.
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  • < I> N_-Simple Theories.Alexei S. Kolesnikov - 2005 - Annals of Pure and Applied Logic 131 (1):227-261.
  • 10th Asian Logic Conference: Sponsored by the Association for Symbolic Logic.Toshiyasu Arai - 2009 - Bulletin of Symbolic Logic 15 (2):246-265.
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  • Nonexistence of Universal Orders in Many Cardinals.Menachem Kojman & Saharon Shelah - 1992 - Journal of Symbolic Logic 57 (3):875-891.
    Our theme is that not every interesting question in set theory is independent of ZFC. We give an example of a first order theory T with countable D(T) which cannot have a universal model at ℵ1 without CH; we prove in ZFC a covering theorem from the hypothesis of the existence of a universal model for some theory; and we prove--again in ZFC--that for a large class of cardinals there is no universal linear order (e.g. in every regular $\aleph_1 < (...)
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  • From Stability to Simplicity.Byunghan Kim & Anand Pillay - 1998 - Bulletin of Symbolic Logic 4 (1):17-36.
  • Automorphism–Invariant Measures on ℵ0-Categorical Structures Without the Independence Property.Douglas E. Ensley - 1996 - Journal of Symbolic Logic 61 (2):640 - 652.
    We address the classification of the possible finitely-additive probability measures on the Boolean algebra of definable subsets of M which are invariant under the natural action of $\operatorname{Aut}(M)$ . This pursuit requires a generalization of Shelah's forking formulas [8] to "essentially measure zero" sets and an application of Myer's "rank diagram" [5] of the Boolean algebra under consideration. The classification is completed for a large class of ℵ 0 -categorical structures without the independence property including those which are stable.
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  • A Note on Lascar Strong Types in Simple Theories.Byunghan Kim - 1998 - Journal of Symbolic Logic 63 (3):926-936.
    Let T be a countable, small simple theory. In this paper, we prove that for such T, the notion of Lascar strong type coincides with the notion of strong type, over an arbitrary set.
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  • Definability and Definable Groups in Simple Theories.Anand Pillay - 1998 - Journal of Symbolic Logic 63 (3):788-796.
    We continue the study of simple theories begun in [3] and [5]. We first find the right analogue of definability of types. We then develop the theory of generic types and stabilizers for groups definable in simple theories. The general ideology is that the role of formulas (or definability) in stable theories is replaced by partial types (or ∞-definability) in simple theories.
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  • 1998–1999 Winter Meeting of the Association for Symbolic Logic.Peter Cholak - 1999 - Bulletin of Symbolic Logic 5 (2):273-283.
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  • Definability in Low Simple Theories.Ziv Shami - 2000 - Journal of Symbolic Logic 65 (4):1481-1490.
  • Simplicity, and Stability in There.Byunghan Kim - 2001 - Journal of Symbolic Logic 66 (2):822-836.
    Firstly, in this paper, we prove that the equivalence of simplicity and the symmetry of forking. Secondly, we attempt to recover definability part of stability theory to simplicity theory. In particular, using elimination of hyperimaginaries we prove that for any supersimple T, canonical base of an amalgamation class P is the union of names of ψ-definitions of P, ψ ranging over stationary L-formulas in P. Also, we prove that the same is true with stable formulas for an 1-based theory having (...)
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  • Coordinatisation by Binding Groups and Unidimensionality in Simple Theories.Ziv Shami - 2004 - Journal of Symbolic Logic 69 (4):1221-1242.
    In a simple theory with elimination of finitary hyperimaginaries if tp is real and analysable over a definable set Q, then there exists a finite sequence ⊆ dcleq with an*=a such that for every i≤ n*, if pi=tp then Aut is type-definable with its action on pi.
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  • Coordinatisation by Binding Groups and Unidimensionality in Simple Theories.Ziv Shami - 2004 - Journal of Symbolic Logic 69 (4):1221 - 1242.
    In a simple theory with elimination of finitary hyperimaginaries if tp(a) is real and analysable over a definable set Q, then there exists a finite sequence ( $a_{i}|i \leq n^{*}$ ) $\subseteq dcl^{eq}$ (a) with $a_{n}*$ = a such that for every $i \leq n*$ , if $p_{i} = tp(a_{i}/{a_{i}|j < i}$ ) then $Aut(p_{i}/Q)$ is type-definable with its action on $p_{i}^{c}$ . A unidimensional simple theory eliminates the quantifier $\exists^{\infty}$ and either interprets (in $C^{eq}$ ) an infinite type-definable group (...)
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  • Universal Graphs at the Successor of a Singular Cardinal.Mirna Džamonja & Saharon Shelah - 2003 - Journal of Symbolic Logic 68 (2):366-388.
    The paper is concerned with the existence of a universal graph at the successor of a strong limit singular μ of cofinality ℵ0. Starting from the assumption of the existence of a supercompact cardinal, a model is built in which for some such μ there are $\mu^{++}$ graphs on μ+ that taken jointly are universal for the graphs on μ+, while $2^{\mu^+} \gg \mu^{++}$ . The paper also addresses the general problem of obtaining a framework for consistency results at the (...)
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  • 2005 Annual Meeting of the Association for Symbolic Logic.John Steel - 2006 - Bulletin of Symbolic Logic 12 (1):143-167.
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  • Simple Theories.Byunghan Kim & Anand Pillay - 1996 - Annals of Pure and Applied Logic 88 (2):149-164.
  • Mekler's Construction Preserves CM-Triviality.Andreas Baudisch - 2002 - Annals of Pure and Applied Logic 115 (1-3):115-173.
    For every structure M of finite signature Mekler 781) has constructed a group G such that for every κ the maximal number of n -types over an elementary equivalent model of cardinality κ is the same for M and G . These groups are nilpotent of class 2 and of exponent p , where p is a fixed prime greater than 2. We consider stable structures M only and show that M is CM -trivial if and only if G is (...)
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  • Hypergraph Sequences as a Tool for Saturation of Ultrapowers.M. E. Malliaris - 2012 - Journal of Symbolic Logic 77 (1):195-223.
    Let T 1 , T 2 be countable first-order theories, M i ⊨ T i , and any regular ultrafilter on λ ≥ $\aleph_{0}$ . A longstanding open problem of Keisler asks when T 2 is more complex than T 1 , as measured by the fact that for any such λ, , if the ultrapower (M 2 ) λ / realizes all types over sets of size ≤ λ, then so must the ultrapower (M 1 ) λ /. (...)
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  • Forking and Dividing in NTP₂ Theories.Artem Chernikov & Itay Kaplan - 2012 - Journal of Symbolic Logic 77 (1):1-20.
    We prove that in theories without the tree property of the second kind (which include dependent and simple theories) forking and dividing over models are the same, and in fact over any extension base. As an application we show that dependence is equivalent to bounded non-forking assuming NTP 2.
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  • The Manin–Mumford Conjecture and the Model Theory of Difference Fields.Ehud Hrushovski - 2001 - Annals of Pure and Applied Logic 112 (1):43-115.
    Using methods of geometric stability , we determine the structure of Abelian groups definable in ACFA, the model companion of fields with an automorphism. We also give general bounds on sets definable in ACFA. We show that these tools can be used to study torsion points on Abelian varieties; among other results, we deduce a fairly general case of a conjecture of Tate and Voloch on p-adic distances of torsion points from subvarieties.
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  • Completeness and Categoricity : Formalization Without Foundationalism.John T. Baldwin - 2014 - Bulletin of Symbolic Logic 20 (1):39-79.
  • Generic Pairs of SU-Rank 1 Structures.Evgueni Vassiliev - 2003 - Annals of Pure and Applied Logic 120 (1-3):103-149.
    For a supersimple SU-rank 1 theory T we introduce the notion of a generic elementary pair of models of T . We show that the theory T* of all generic T-pairs is complete and supersimple. In the strongly minimal case, T* coincides with the theory of infinite dimensional pairs, which was used in 1184–1194) to study the geometric properties of T. In our SU-rank 1 setting, we use T* for the same purpose. In particular, we obtain a characterization of linearity (...)
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  • Toward Classifying Unstable Theories.Saharon Shelah - 1996 - Annals of Pure and Applied Logic 80 (3):229-255.
  • Model Theory of Finite Fields and Pseudo-Finite Fields.Zoé Chatzidakis - 1997 - Annals of Pure and Applied Logic 88 (2-3):95-108.
    We give a survey of results obtained in the model theory of finite and pseudo-finite fields.
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  • Generic Structures and Simple Theories.Z. Chatzidakis & A. Pillay - 1998 - Annals of Pure and Applied Logic 95 (1-3):71-92.
    We study structures equipped with generic predicates and/or automorphisms, and show that in many cases we obtain simple theories. We also show that a bounded PAC field is simple. 1998 Published by Elsevier Science B.V. All rights reserved.
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  • Strongly Determined Types.Alexandre A. Ivanov & Dugald Macpherson - 1999 - Annals of Pure and Applied Logic 99 (1-3):197-230.
    The notion of a strongly determined type over A extending p is introduced, where p .S. A strongly determined extension of p over A assigns, for any model M )- A, a type q S extending p such that, if realises q, then any elementary partial map M → M which fixes acleq pointwise is elementary over . This gives a crude notion of independence which arises very frequently. Examples are provided of many different kinds of theories with strongly determined (...)
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  • The Number of Types in Simple Theories.Enrique Casanovas - 1999 - Annals of Pure and Applied Logic 98 (1-3):69-86.
    We continue work of Shelah on the cardinality of families of pairwise incompatible types in simple theories obtaining characterizations of simple and supersimple theories. We develop a local analysis of the number of types in simple theories and we find a new example of a simple unstable theory.
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  • Simple Generic Structures.Massoud Pourmahdian - 2003 - Annals of Pure and Applied Logic 121 (2-3):227-260.
    A study of smooth classes whose generic structures have simple theory is carried out in a spirit similar to Hrushovski 147; Simplicity and the Lascar group, preprint, 1997) and Baldwin–Shi 1). We attach to a smooth class K0, of finite -structures a canonical inductive theory TNat, in an extension-by-definition of the language . Here TNat and the class of existentially closed models of =T+,EX, play an important role in description of the theory of the K0,-generic. We show that if M (...)
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  • Internality and Interpretable Automorphism Groups in Simple Theories.Ziv Shami - 2004 - Annals of Pure and Applied Logic 129 (1-3):149-162.
    The binding group theorem for stable theories is partially extended to the simple context. Some results concerning internality are proved. We also introduce a ‘small’ normal subgroup G0+ of the automorphism group and show that if p is Q-internal then it has a finite exponent and G/G0+ is interpretable.
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  • Neostability-Properties of Fraïssé Limits of 2-Nilpotent Groups of Exponent $${P > 2}$$ P > 2.Andreas Baudisch - 2016 - Archive for Mathematical Logic 55 (3-4):397-403.
  • On the Existence of Universal Models.Mirna Džamonja & Saharon Shelah - 2004 - Archive for Mathematical Logic 43 (7):901-936.
    Suppose that λ=λ <λ ≥ℵ0, and we are considering a theory T. We give a criterion on T which is sufficient for the consistent existence of λ++ universal models of T of size λ+ for models of T of size ≤λ+, and is meaningful when 2λ +>λ++. In fact, we work more generally with abstract elementary classes. The criterion for the consistent existence of universals applies to various well known theories, such as triangle-free graphs and simple theories. Having in mind (...)
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  • N-Simple Theories.Alexei S. Kolesnikov - 2005 - Annals of Pure and Applied Logic 131 (1-3):227-261.
    The main topic of this paper is the investigation of generalized amalgamation properties for simple theories. That is, we are trying to answer the question of when a simple theory has the property of n-dimensional amalgamation, where two-dimensional amalgamation is the Independence Theorem for simple theories. We develop the notions of strong n-simplicity and n-simplicity for 1≤n≤ω, where both “1-simple” and “strongly 1-simple” are the same as “simple”. For strong n-simplicity, we present examples of simple unstable theories in each subclass (...)
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  • Independence, Order, and the Interaction of Ultrafilters and Theories.M. E. Malliaris - 2012 - Annals of Pure and Applied Logic 163 (11):1580-1595.
    We consider the question, of longstanding interest, of realizing types in regular ultrapowers. In particular, this is a question about the interaction of ultrafilters and theories, which is both coarse and subtle. By our prior work it suffices to consider types given by instances of a single formula. In this article, we analyze a class of formulas φ whose associated characteristic sequence of hypergraphs can be seen as describing realization of first- and second-order types in ultrapowers on one hand, and (...)
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  • Supersimplicity and Quadratic Extensions.A. Martin-Pizarro & F. O. Wagner - 2009 - Archive for Mathematical Logic 48 (1):55-61.
    An elliptic curve over a supersimple field with exactly one extension of degree 2 has an s-generic point.
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  • XI Latin American Symposium on Mathematical Logic.Carlos Augusto Di Prisco - 1999 - Bulletin of Symbolic Logic 5 (4):495-524.
  • Theories Without the Tree Property of the Second Kind.Artem Chernikov - 2014 - Annals of Pure and Applied Logic 165 (2):695-723.
    We initiate a systematic study of the class of theories without the tree property of the second kind — NTP2. Most importantly, we show: the burden is “sub-multiplicative” in arbitrary theories ; NTP2 is equivalent to the generalized Kimʼs lemma and to the boundedness of ist-weight; the dp-rank of a type in an arbitrary theory is witnessed by mutually indiscernible sequences of realizations of the type, after adding some parameters — so the dp-rank of a 1-type in any theory is (...)
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  • Notions Around Tree Property 1.Byunghan Kim & Hyeung-Joon Kim - 2011 - Annals of Pure and Applied Logic 162 (9):698-709.
    In this paper, we study the notions related to tree property 1 , or, equivalently, SOP2. Among others, we supply a type-counting criterion for TP1 and show the equivalence of TP1 and k- TP1. Then we introduce the notions of weak k- TP1 for k≥2, and also supply type-counting criteria for those. We do not know whether weak k- TP1 implies TP1, but at least we prove that each weak k- TP1 implies SOP1. Our generalization of the tree-indiscernibility results in (...)
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  • A Note on Weak Dividing.Byunghan Kim & Niandong Shi - 2007 - Archive for Mathematical Logic 46 (2):51-60.
    We study the notion of weak dividing introduced by S. Shelah. In particular we prove that T is stable iff weak dividing is symmetric.
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  • Strict Independence.Itay Kaplan & Alexander Usvyatsov - 2014 - Journal of Mathematical Logic 14 (2):1450008.
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  • The Amalgamation Spectrum.John T. Baldwin, Alexei Kolesnikov & Saharon Shelah - 2009 - Journal of Symbolic Logic 74 (3):914-928.
    We study when classes can have the disjoint amalgamation property for a proper initial segment of cardinals. Theorem A For every natural number k, there is a class $K_k $ defined by a sentence in $L_{\omega 1.\omega } $ that has no models of cardinality greater than $ \supset _{k - 1} $ , but $K_k $ has the disjoint amalgamation property on models of cardinality less than or equal to $\mathfrak{N}_{k - 3} $ and has models of cardinality $\mathfrak{N}_{k (...)
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  • An Independence Theorem for Ntp2 Theories.Itaï Ben Yaacov & Artem Chernikov - 2014 - Journal of Symbolic Logic 79 (1):135-153.
  • Espaces Ultramétriques.Françoise Delon - 1984 - Journal of Symbolic Logic 49 (2):405-424.
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