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138
A Survey on the Model Theory of Difference Fields
, 2000
"... We survey the model theory of difference fields, that is, fields with a distinguished automorphism σ. After introducing the theory ACFA and stating elementary results, we discuss independence and the various concepts of rank, the dichotomy theorems, and, as an application, the Manin–Mumford conject ..."
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Cited by 67 (9 self)
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We survey the model theory of difference fields, that is, fields with a distinguished automorphism σ. After introducing the theory ACFA and stating elementary results, we discuss independence and the various concepts of rank, the dichotomy theorems, and, as an application, the Manin–Mumford conjecture over a number field. We conclude with some other applications.
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 23 (8 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
FiniteModel Theory  A Personal Perspective
 Theoretical Computer Science
, 1993
"... Finitemodel theory is a study of the logical properties of finite mathematical structures. This paper is a very personalized view of finitemodel theory, where the author focuses on his own personal history, and results and problems of interest to him, especially those springing from work in his Ph ..."
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Cited by 20 (0 self)
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Finitemodel theory is a study of the logical properties of finite mathematical structures. This paper is a very personalized view of finitemodel theory, where the author focuses on his own personal history, and results and problems of interest to him, especially those springing from work in his Ph.D. thesis. Among the topics discussed are:
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 20 (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.
Extended OrderGeneric Queries
, 1998
"... We consider relational databases organized over an ordered domain with some additional relationsa typical example is the ordered domain of rational numbers together with the operation of addition. In the focus of our study are the firstorder (FO) queries that are invariant under orderpreser ..."
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Cited by 19 (2 self)
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We consider relational databases organized over an ordered domain with some additional relationsa typical example is the ordered domain of rational numbers together with the operation of addition. In the focus of our study are the firstorder (FO) queries that are invariant under orderpreserving "permutations"such queries are called ordergeneric. It has recently been discovered that for some domains ordergeneric FO queries fail to express more than pure order queries. For example, every ordergeneric FO query over rational numbers with + can be rewritten without +. For some other domains, however, this is not the case. We provide very general conditions on the FO theory of the domain that ensure the collapse of ordergeneric extended FO queries to pure order queries over this domain: the Pseudofinite Homogeneity Property and a stronger Isolation Property. We further distinguish one broad class of domains satisfying the Isolation Property, the socalled quasio ...
Nonexistence of Universal Orders in Many Cardinals
 Journal of Symbolic Logic
, 1992
"... Our theme is that not every interesting question in set theory is independent of ZF C. 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 ZF C a covering theorem from the hypothesis of the existence of a universal model ..."
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Cited by 19 (15 self)
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Our theme is that not every interesting question in set theory is independent of ZF C. 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 ZF C 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 ℵ1 < λ < 2 ℵ0). In fact, what we show is that if there is a universal linear order at a regular λ and its existence is not a result of a trivial cardinal arithmetical reason, then λ “resembles ” ℵ1 — a cardinal for which the consistency of having a universal order is known. As for singular cardinals, we show that for many singular cardinals, if they are not strong limits then they have no universal linear order. As a result of the non existence of a universal linear order, we show the nonexistence of universal models for all theories possessing the strict order property (for example, ordered fields and groups, Boolean algebras, padic rings and fields, partial orders, models of PA and so on).
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 18 (4 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.
Introduction to theories without the independence property
"... We present an updated exposition of the classical theory of complete first order theories ..."
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Cited by 17 (1 self)
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We present an updated exposition of the classical theory of complete first order theories
A new uncountably categorical group
 Trans. Amer. Math. Soc
, 1996
"... Abstract. We construct an uncountably categorical group with a geometry that is not locally modular. It is not possible to interpret a field in this group. We show the group is CMtrivial. 1. ..."
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Cited by 17 (3 self)
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Abstract. We construct an uncountably categorical group with a geometry that is not locally modular. It is not possible to interpret a field in this group. We show the group is CMtrivial. 1.