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A primer of simple theories
 Archive Math. Logic
"... Abstract. We present a selfcontained 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. ..."
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Abstract. We present a selfcontained 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.
Shelah’s categoricity conjecture from a successor for tame abstract elementary classes
 The Journal of Symbolic Logic
, 2006
"... elementary classes. Theorem 0.1. Suppose that K is a χtame 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 ..."
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elementary classes. Theorem 0.1. Suppose that K is a χtame 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 [Sh 394], we derive a form of Shelah’s Categoricity Conjecture for tame abstract elementary classes: Corollary 0.2. Suppose K is a χtame abstract elementary class satisfying the amalgamation and joint embedding properties. Let µ0:= Hanf(K). If χ ≤ ℶ (2 µ 0) + and K is categorical in some λ +> ℶ (2 µ 0) +, then K is categorical in µ for all µ> ℶ (2 µ 0) +.
Spacetime foam dense singularities and de Rham cohomology,”
 Acta Applicandae Mathematicae,
, 2001
"... ..."
Categoricity in Abstract Elementary Classes with No Maximal Models
 ANNALS OF PURE AND APPLIED LOGIC
, 2005
"... The results in this paper are in a context of abstract elementary classes identified by Shelah and Villaveces in which the amalgamation property is not assumed. The longterm goal is to solve Shelah’s Categoricity Conjecture in this context. Here we tackle a problem of Shelah and Villaveces by prov ..."
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Cited by 14 (3 self)
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The results in this paper are in a context of abstract elementary classes identified by Shelah and Villaveces in which the amalgamation property is not assumed. The longterm goal is to solve Shelah’s Categoricity Conjecture in this context. Here we tackle a problem of Shelah and Villaveces by proving that in their context, the uniqueness of limit models follows from categoricity under the assumption that the subclass of amalgamation bases is closed under unions of bounded, ≺Kincreasing chains.
Differential algebras with dense singularities on manifolds
, 2006
"... Recently the spacetime foam differential algebras of generalized functions with dense singularities were introduced, motivated by the so called spacetime foam structures in General Relativity with dense singularities, and by Quantum Gravity. A variety of applications of these algebras has been p ..."
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Cited by 9 (5 self)
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Recently the spacetime foam differential algebras of generalized functions with dense singularities were introduced, motivated by the so called spacetime foam structures in General Relativity with dense singularities, and by Quantum Gravity. A variety of applications of these algebras has been presented, among them, a global CauchyKovalevskaia theorem, de Rham cohomology in abstract differential geometry, and so on. So far the spacetime foam algebras have only been constructed on Euclidean spaces. In this paper, owing to their relevance in General Relativity among others, the construction of these algebras is extended to arbitrary finite dimensional smooth manifolds. Since these algebras contain the Schwartz distributions, the extension of their construction to manifolds also solves the long outstanding problem of defining distributions on manifolds, and doing so in ways compatible with nonlinear operations. Earlier, similar attempts were made in the literature with respect to the extension of the Colombeau algebras to manifolds, algebras which also contain the distributions. These attempts have encountered significant technical difficulties, owing to the growth condition type limitations the elements of Colombeau algebras have to satisfy near singularities. Since in this paper no any type of such or other growth conditions are required in the construction of spacetime foam algebras, their extension to manifolds proceeds in a surprisingly easy and natural way. It is also shown that the spacetime foam algebras form a fine and flabby sheaf, properties which are important in securing a considerably large class of singularities which generalized functions can handle.
Categoricity and Stability in Abstract Elementary Classes
, 2002
"... 1This version includes changes made since the acceptance of the thesis. The main changes occur in Chapter II Section 9. This thesis is dedicated to my daughter Ariella Ronit. Dedication page ii ACKNOWLEDGEMENTS I would like to thank Rami Grossberg, my advisor and husband, for his patient support and ..."
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1This version includes changes made since the acceptance of the thesis. The main changes occur in Chapter II Section 9. This thesis is dedicated to my daughter Ariella Ronit. Dedication page ii ACKNOWLEDGEMENTS I would like to thank Rami Grossberg, my advisor and husband, for his patient support and guidance during my Ph.D. thesis. I am also indebted to John Baldwin for his unlimited willingness to comment on and discuss preliminary drafts of this thesis through email communication as well as professional visits at Carnegie Mellon University and at the Bogota Meeting in Model Theory at the National University of Colombia at Bogota. I would like to thank Andr¶es Villaveces for organizing the Bogota Meeting in Model Theory and for the invitation to give several talks on this thesis. Thanks also go to Andr¶es Villaveces and Olivier Lessmann for a critical reading of a preliminary version of Chapter II.
Notes on quasiminimality and excellence
 Bulletin of Symbolic Logic
"... Zilber’s proposes [60] to prove ‘canonicity results for pseudoanalytic ’ structures. Informally, ‘canonical means the theory of the structure in a suitable possibly infinitary language (see Section 2) has one model in each uncountable power ’ while ‘pseudoanalytic means the model of power 2 ℵ0 can ..."
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Zilber’s proposes [60] to prove ‘canonicity results for pseudoanalytic ’ structures. Informally, ‘canonical means the theory of the structure in a suitable possibly infinitary language (see Section 2) has one model in each uncountable power ’ while ‘pseudoanalytic means the model of power 2 ℵ0 can be taken as a reduct of an expansion of the complex numbers by analytic functions’. This program interacts with two other lines of research. First is the general study of categoricity theorems in infinitary languages. After initial results by Keisler, reported in [31], this line was taken up in a long series of works by Shelah. We place Zilber’s work in this context. The second direction stems from Hrushovski’s construction of a counterexample to Zilber’s conjecture that every strongly minimal set is ‘trivial’, ‘vector spacelike’, or ‘fieldlike’. This construction turns out to be very concrete example of the Abstract Elementary Classes which arose in Shelah’s analysis. This paper examines the intertwining of these three themes. The study of (C, +, ·, exp) leads one immediately to some extension of first order logic; the integers with all their arithmetic are first order definable in (C, +, ·, exp). Thus, the first order theory of complex exponentiation is horribly complicated; it is certainly unstable and so can’t be first order categorical. One solution is to use infinitary logic to pin down the pathology. Insist that the kernel of the exponential map is fixed as a single copy of the integers while allowing the rest of the structure to grow. We describe in Section 5 Zilber’s program to
Categoricity Without Equality
"... We study categoricity in power for reduced models of first order logic without equality. 1 Introduction The object of this paper is to study categoricity in power for theories in first order logic without equality. Our results will reveal some surprising differences between the model theory for l ..."
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We study categoricity in power for reduced models of first order logic without equality. 1 Introduction The object of this paper is to study categoricity in power for theories in first order logic without equality. Our results will reveal some surprising differences between the model theory for logic without equality and for logic with equality. When we consider categoricity, it is natural to identify elements which are indistinguishable from each other. We will do this by confining our attention to reduced models, that is, models M such that any pair of elements which satisfy the same formulas with parameters in M are equal. We also confine our attention to complete theories T in a countable language such that all models of T are infinite. T is said to be categorical if T has exactly one reduced model of cardinality up to isomorphism. The classical result about !categoricity for logic with equality is the RyllNardzewski theorem, which says that T is !categorical if and only i...
c○1995 American Mathematical Society 02730979/95 $1.00 + $.25 per page BOOK
"... Mathematical logic began as the general study of mathematical reasoning. Several specializations have developed: recursion theory studies abstract computation; set theory studies the foundations of mathematics as formalized in ZermeloFraenkel set theory; proof theory studies systems of formal proof ..."
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Mathematical logic began as the general study of mathematical reasoning. Several specializations have developed: recursion theory studies abstract computation; set theory studies the foundations of mathematics as formalized in ZermeloFraenkel set theory; proof theory studies systems of formal proof; model theory, says Hodges, is the study of the construction and classification of structures within specified classes of structures. Despite this somewhat jargonladen definition, model theory is the part of logic that over the last half century has developed the deepest contacts with core mathematics. I would modify Hodges’s definition to emphasize a central concern of model theory: the discussion of classes of classes of structures. In this review we consider several developments, most covered by Hodges, to clarify the distinction between studying a single structure, a class of structures (a theory), or a class of theories. We will relate these modeltheoretic concepts to specific developments in the theory of fields. Asignature L is a collection of relation and function symbols. A structure for that signature (Lstructure) is a set with an interpretation for each of those