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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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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.
The elementary theory of Dedekind cuts in polynomially bounded structures
, 2003
"... 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 fr ..."
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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.
Heirs of box types in polynomially bounded structures; preprint, submitted for publication
"... 2. Heirs of cuts in polynomially bounded structures 3. Box types 4. The box type associated to a cut ..."
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2. Heirs of cuts in polynomially bounded structures 3. Box types 4. The box type associated to a cut
The Elementary Theory of Dedekind Cuts
"... this article we do the model theoretic groundwork of the first order theory of cuts, relative to a given ominimal structure expanding real closed fields. This means the following. Let M be an ominimal structure in the language L , expanding a real closed field. Let D be a new unary predicate and l ..."
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this article we do the model theoretic groundwork of the first order theory of cuts, relative to a given ominimal structure expanding real closed fields. This means the following. Let M be an ominimal structure in the language L , expanding a real closed field. Let D be a new unary predicate and let p be a cut of (the underlying ordered set of) M . Then we study the L (D)structure (M; p