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49
On what I do not understand (and have something to say), model theory
 Mathematica Japonica, submitted. [Sh:702]; math.LO/9910158
"... Abstract. This is a nonstandard paper, containing some problems in set theory I have in various degrees been interested in. Sometimes with a discussion on what I have to say; sometimes, of what makes them interesting to me, sometimes the problems are presented with a discussion of how I have tried ..."
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Cited by 29 (10 self)
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Abstract. This is a nonstandard paper, containing some problems in set theory I have in various degrees been interested in. Sometimes with a discussion on what I have to say; sometimes, of what makes them interesting to me, sometimes the problems are presented with a discussion of how I have tried to solve them, and sometimes with failed tries, anecdote and opinion. So the discussion is quite personal, in other words, egocentric and somewhat accidental. As we discuss many problems, history and side references are erratic, usually kept at a minimum (“see... ” means: see the references there and possibly the paper itself). The base were lectures in Rutgers Fall ’97 and reflect my knowledge then. The other half, [122], concentrating on model theory, will subsequently appear. I thank Andreas Blass and Andrzej Ros̷lanowski for many helpful comments. (666) revision:20011112 modified:20031118
Classification Theory for Abstract Elementary Classes
 In Logic and Algebra, Yi Zhang editor, Contemporary Mathematics 302, AMS,(2002), 165–203
, 2002
"... In this paper some of the basics of classification theory for abstract elementary classes are discussed. Instead of working with types which are sets of formulas (in the firstorder case) we deal instead with Galois types which are essentially orbits of automorphism groups acting on the structure. S ..."
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Cited by 23 (5 self)
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In this paper some of the basics of classification theory for abstract elementary classes are discussed. Instead of working with types which are sets of formulas (in the firstorder case) we deal instead with Galois types which are essentially orbits of automorphism groups acting on the structure. Some of the most basic results in classification theory for non elementary classes are presented. The motivating point of view is Shelah's categoricity conjecture for L# 1 ,# . While only very basic theorems are proved, an effort is made to present number of different technologies: Flavors of weak diamond, models of weak set theories, and commutative diagrams. We focus in issues involving existence of Galois types, extensions of types and Galoisstability.
Shelah’s stability spectrum and homogeneity spectrum in finite diagrams
 Arch. Math. Logic
"... 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 poin ..."
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Cited by 21 (16 self)
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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.
Categoricity from one successor cardinal in Tame Abstract Elementary Classes
, 2005
"... We prove that from categoricity in λ + we can get categoricity in all cardinals ≥ λ + in a χtame abstract elementary classes which has arbitrarily large models and satisfies the amalgamation and joint embedding properties, provided λ> LS(K) and λ ≥ χ. For the missing case when λ = LS(K), we pro ..."
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Cited by 18 (5 self)
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We prove that from categoricity in λ + we can get categoricity in all cardinals ≥ λ + in a χtame abstract elementary classes which has arbitrarily large models and satisfies the amalgamation and joint embedding properties, provided λ> LS(K) and λ ≥ χ. For the missing case when λ = LS(K), we prove that K is totally categorical provided that K is categorical in LS(K) and LS(K) +.
Covers of the multiplicative group of an algebraically closed field of characteristic
"... Consider the classical universal cover of the one dimensional complex torus C ∗ , which gives us the exact sequence 0 − → Z i ..."
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Cited by 18 (0 self)
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Consider the classical universal cover of the one dimensional complex torus C ∗ , which gives us the exact sequence 0 − → Z i
Categoricity in Abstract Elementary Classes with No Maximal Models
 Annals of Pure and Applied Logic
"... Abstract. 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 Villavece ..."
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Cited by 11 (3 self)
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Abstract. 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.
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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Cited by 11 (3 self)
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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) +.
Toward classifying unstable theories
 Annals of Pure and Applied Logic
, 1995
"... (500) revision:19960622 modified:19960622 Abstract. We prove a consistency results saying, that for a simple (first order) theory, it is easier to have a universal model in some cardinalities, than for the theory of linear order. We define additional properties of first order theories, the nstr ..."
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Cited by 11 (3 self)
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(500) revision:19960622 modified:19960622 Abstract. We prove a consistency results saying, that for a simple (first order) theory, it is easier to have a universal model in some cardinalities, than for the theory of linear order. We define additional properties of first order theories, the nstrong order property (SOPn in short). The main result is that a first order theory with the 4strong order property behaves like linear orders concerning existence of universal models. Key words and phrases. Model theory, classification thoery, stability theory, unstable theories, universal models, simple theories, Keisler’s order. Done: §1: with 457; section 2: 8/92: 2.2 + sufficiency for non existence of universal; 12/92 rest
Model theory, geometry and arithmetic of universal covers of a semiabelian variety
 In Model Theory and Applications, Quaterna di matematica
, 2003
"... latest misprint corrections 24.10.05 ..."
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