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16
A dividing line within simple unstable theories.
, 2013
"... We give the first (ZFC) dividing line in Keisler’s order among the unstable theories, specifically among the simple unstable theories. That is, for any infinite cardinal λ for which there is µ < λ ≤ 2 µ, we construct a regular ultrafilter D on λ so that (i) for any model M of a stable theory or ..."
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We give the first (ZFC) dividing line in Keisler’s order among the unstable theories, specifically among the simple unstable theories. That is, for any infinite cardinal λ for which there is µ < λ ≤ 2 µ, we construct a regular ultrafilter D on λ so that (i) for any model M of a stable theory or of the random graph, M λ /D is λ +saturated but (ii) if Th(N) is not simple or not low then N λ /D is not λ +saturated. The nonsaturation result relies on the notion of flexible ultrafilters. To prove the saturation result we develop a property of a class of simple theories, called Qr 1, generalizing the fact that whenever B is a set of parameters in some sufficiently saturated model of the random graph, B  = λ and µ < λ ≤ 2 µ, then there is a set A with A  = µ so that any nonalgebraic p ∈ S(B) is finitely realized in A. In addition to giving information about simple unstable theories, our proof reframes the problem of saturation of ultrapowers in several key ways. We give a new characterization of good filters in terms of “excellence, ” a measure of the accuracy of the quotient Boolean algebra. We introduce and develop the notion of moral ultrafilters on Boolean algebras. We prove a socalled “separation of variables ” result which shows how the problem of constructing ultrafilters to have a precise degree of saturation may be profitably separated into a more settheoretic stage, building an excellent filter, followed by a more modeltheoretic stage: building socalled moral ultrafilters on the quotient Boolean algebra, a process which highlights the complexity of certain patterns, arising from firstorder formulas, in certain Boolean algebras.
Saturating the random graph with an independent family of small range
, 2012
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Generalized amalgamation in simple theories and characterization of dependence in nonelementary classes
, 2004
"... We examine the properties of dependence relations in certain nonelementary classes and firstorder simple theories. There are two major parts. The goal of the first part is to identify the properties of dependence relations in certain nonelementary classes that, firstly, characterize the modelthe ..."
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We examine the properties of dependence relations in certain nonelementary classes and firstorder simple theories. There are two major parts. The goal of the first part is to identify the properties of dependence relations in certain nonelementary classes that, firstly, characterize the modeltheoretic properties of those classes; and secondly, allow to uniquely describe an abstract dependence itself in a very concrete way. I investigate totally transcendental atomic models and finite diagrams, stable finite diagrams, and a subclass of simple homogeneous models from this point of view. The second part deals with simple firstorder theories. The main topic of this part is investigation of generalized amalgamation properties for simple theories. Namely, we are trying to answer the question of when does a simple theory have the property of ndimensional amalgamation, where 2dimensional amalgamation is the Independence theorem for simple theories. We develop the notion of nsimplicity and strong nsimplicity for 1 ≤ n ≤ ω, where both “1simple ” and “strongly 1simple ” is the same as “simple. ” We present examples of simple unstable theories in each subclass and prove a characteristic property of nsimplicity in terms of ndividing, a strengthening of the dependence relation called dividing in simple theories. We prove 3dimensional amalgamation property for 2simple theories, and, under an additional assumption, a strong (n + 1)dimensional amalgamation property for strongly nsimple theories. Stable theories are strongly ωsimple, and the idea behind developing extra simplicity conditions is to show that, for instance, ωsimple theories are almost as nice as stable theories. The third part of the thesis contains an application of ωsimplicity to construct a Morley sequence without the construction of a long independent sequence. ii
AN INDEPENDENCE THEOREM FOR NTP2 THEORIES
"... Abstract. We establish several results regarding dividing and forking in NTP2 theories. We show that dividing is the same as arraydividing. Combining it with existence of strictly invariant sequences we deduce that forking satisfies the chain condition over extension bases (namely, the forking idea ..."
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Abstract. We establish several results regarding dividing and forking in NTP2 theories. We show that dividing is the same as arraydividing. Combining it with existence of strictly invariant sequences we deduce that forking satisfies the chain condition over extension bases (namely, the forking ideal is S1, in Hrushovski’s terminology). Using it we prove an independence theorem over extension bases (which, in the case of simple theories, specializes to the ordinary independence theorem). As an application we show that Lascar strong type and compact strong type coincide over extension bases in an NTP2 theory. We also define the dividing order of a theory – a generalization of Poizat’s fundamental order from stable theories – and give some equivalent characterizations under the assumption of NTP2. The last section is devoted to a refinement of the class of strong theories and its place in the classification hierarchy. hal00713494, version 2 13 Aug 2013
MAIN GAP FOR LOCALLY SATURATED ELEMENTARY SUBMODELS OF A HOMOGENEOUS STRUCTURE
"... We prove a main gap theorem for locally saturated submodels of a homogeneous structure. We also study the number of locally saturated models, which are not elementarily embeddable into each other. 676 revision:20000413 modified:20000905 Hard experience has indicated that before we speak on this ..."
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We prove a main gap theorem for locally saturated submodels of a homogeneous structure. We also study the number of locally saturated models, which are not elementarily embeddable into each other. 676 revision:20000413 modified:20000905 Hard experience has indicated that before we speak on this particular paper, we should say something on classification theory for nonelementary classes and of the specific context chosen here. Classification theory for first order theories is so established now that many tend to forget that there are other possibilities. There are some good reasons to consider these other possibilities: first, it is better to understand a more general context, we like to classify more; second, concerning applications many classes arising in ’nature ’ are not first order; third, understanding more general contexts may shed light on the first order one. Of course, we may suspect that applying to a wider context will leave us with less content, but only trying will teach us if there are enough interesting things to discover.
TAMENESS AND FRAMES REVISITED
"... Abstract. We combine tameness for types of singleton with the existence of a good frame to obtain some amount of tameness for types of longer sequences. We use this to show how to use tameness to extend a good frame in one cardinality to a good frame in all cardinalities, improving a theorem of Bone ..."
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Abstract. We combine tameness for types of singleton with the existence of a good frame to obtain some amount of tameness for types of longer sequences. We use this to show how to use tameness to extend a good frame in one cardinality to a good frame in all cardinalities, improving a theorem of Boney. Along the way, we prove many general results on the independent sequences induced by the good frame. In particular, we show that tameness and a good frame imply Shelah’s notion of dimension is wellbehaved,
Ya’acov Peterzil
, 2006
"... We discuss measures, invariant measures on definable groups, and genericity, often in an NIP (failure of the independence property) environment. We complete the proof of the third author’s conjectures relating definably compact groups G in saturated ominimal structures to compact Lie groups. We als ..."
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We discuss measures, invariant measures on definable groups, and genericity, often in an NIP (failure of the independence property) environment. We complete the proof of the third author’s conjectures relating definably compact groups G in saturated ominimal structures to compact Lie groups. We also prove some other structural results about such G, for example the existence of a left invariant finitely additive probability measure on definable subsets of G. We finally introduce a new notion “compact domination ” (domination of a definable set by a compact space) and raise some new conjectures in the ominimal case. 1
Forking in short and tame AECs
, 2013
"... We develop a notion of forking for Galoistypes in the context of AECs. Under the hypotheses of tameness, type shortness, and few models, we show that this nonforking is a well behaved notion of independence and has a corresponding Urank, and we find conditions for local character. Finally, we sh ..."
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We develop a notion of forking for Galoistypes in the context of AECs. Under the hypotheses of tameness, type shortness, and few models, we show that this nonforking is a well behaved notion of independence and has a corresponding Urank, and we find conditions for local character. Finally, we show that the proofs are simpler and the nonforking is more powerful when we
EXISTENCE OF OPTIMAL ULTRAFILTERS AND THE FUNDAMENTAL COMPLEXITY OF SIMPLE THEORIES
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