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Stability theory, Permutations of Indiscernibles, and Embedded Finite Models
 Trans. Amer. Math. Soc
, 2000
"... Abstract. We show that the expressive power of firstorder logic over finite models embedded in a model M is determined by stabilitytheoretic properties of M. In particular, we show that if M is stable, then every class of finite structures that can be defined by embedding the structures in M, can ..."
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Cited by 15 (1 self)
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Abstract. We show that the expressive power of firstorder logic over finite models embedded in a model M is determined by stabilitytheoretic properties of M. In particular, we show that if M is stable, then every class of finite structures that can be defined by embedding the structures in M, can be defined in pure firstorder logic. We also show that if M does not have the independence property, then any class of finite structures that can be defined by embedding the structures in M, can be defined in firstorder logic over a dense linear order. This extends known results on the definability of classes of finite structures and ordered finite structures in the setting of embedded finite models. These results depend on several results in infinite model theory. Let I be a set of indiscernibles in a model M and suppose (M,I) is elementarily equivalent to (M1,I1) whereM1 is I1  +saturated. If M is stable and (M,I) is saturated, then every permutation of I extends to an automorphism of M and the theory of (M,I) isstable. LetI be a sequence of <indiscernibles in a model M, which does not have the independence property, and suppose (M,I) is elementarily equivalent to (M1,I1) where(I1,<) is a complete dense linear order and M1 is I1  +saturated. Then (M, I)types over I are orderdefinable and if (M, I) isℵ1saturated, every order preserving permutation of I can be extended to a backandforth system. 1.
Embedded Finite Models, Stability, and the Impact of Order
 In LICS'98
, 1998
"... We extend bounds on the expressive power of firstorder logic over finite structures and over ordered finite structures, by generalizing to the situation where the finite structures are embedded in an infinite structure M , where M satisfies some simple combinatorial properties studied in modeltheo ..."
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We extend bounds on the expressive power of firstorder logic over finite structures and over ordered finite structures, by generalizing to the situation where the finite structures are embedded in an infinite structure M , where M satisfies some simple combinatorial properties studied in modeltheoretic stability theory. We first consider firstorder logic over finite structures embedded in a stable structure, and show that it has the same generic expressive power as firstorder logic on unordered finite structures. It follows from this that having the additional structure of, for example, an abelian group or an equivalence relation, does not allow one to define any new generic queries. We also consider firstorder logic over finite structures living within any model M that lacks the independence property and show that its expressive power is bounded by firstorder logic over finite ordered structures. This latter result gives an enormous class of structures in which the expressive po...
The elementary theory of Dedekind cuts in polynomially bounded structures
, 2003
"... Abstract. Let M be a polynomially bounded, ominimal structure with archimedean prime model, for example if M is a real closed field. Let C be a convex and unbounded subset of M. We determine the first order theory of the structure M expanded by the set C. We do this also over any given set of param ..."
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Abstract. Let M be a polynomially bounded, ominimal structure with archimedean prime model, for example if M is a real closed field. Let C be a convex and unbounded subset of M. We determine the first order theory of the structure M expanded by the set C. We do this also over any given set of parameters from M, which yields a description of all subsets of Mn � �, definable in the expanded structure. This paper is a sequel to [Tr], where we began the model theoretic study of Dedekind cuts of ominimal expansions of fields. Before explaining what we do here, we recall some terminology from [Tr]. If X is a totally ordered set, then a (Dedekind) cut p of X is a tuple p = (p L, p R) with X = p L ∪p R and p L < p R. If Y ⊆ X then Y + denotes the cut p of X with p R = {x ∈ X  x> Y}. Y + is called the upper edge of Y. Similarly the lower edge Y − of Y is defined. We fix an ominimal expansion T of the theory of real closed fields in a language L. If M is a model of T and p is a cut of (the underlying set of) M, then by the model theoretic properties of p we understand the model theoretic properties of the structure M expanded by the set p L in the language L (D) extending L, which has an additional unary predicate
The elementary theory of Dedekind cuts in polynomially bounded structures
, 2003
"... Abstract. Let M be a polynomially bounded, ominimal structure with archimedean prime model, for example if M is a real closed field. Let C be a convex and unbounded subset of M. We determine the first order theory of the structure M expanded by the set C. We do this also over any given set of param ..."
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Abstract. Let M be a polynomially bounded, ominimal structure with archimedean prime model, for example if M is a real closed field. Let C be a convex and unbounded subset of M. We determine the first order theory of the structure M expanded by the set C. We do this also over any given set of parameters from M, which yields a description of all subsets of Mn � �, definable in the expanded structure. This paper is a sequel to [Tr], where we began the model theoretic study of Dedekind cuts of ominimal expansions of fields. Before explaining what we do here, we recall some terminology from [Tr]. If X is a totally ordered set, then a (Dedekind) cut p of X is a tuple p = (p L, p R) with X = p L ∪p R and p L < p R. If Y ⊆ X then Y + denotes the cut p of X with p R = {x ∈ X  x> Y}. Y + is called the upper edge of Y. Similarly the lower edge Y − of Y is defined. We fix an ominimal expansion T of the theory of real closed fields in a language L. If M is a model of T and p is a cut of (the underlying set of) M, then by the model theoretic properties of p we understand the model theoretic properties of the structure M expanded by the set p L in the language L (D) extending L, which has an additional unary predicate