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FORKING IN VCMINIMAL THEORIES
"... Abstract. We consider VCminimal theories admitting unpackable generating families, and show that in such theories, forking of formulae over a model M is equivalent to containment in global types definable over M, generalizing a result of Dolich on ominimal theories in [4]. 1. ..."
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Abstract. We consider VCminimal theories admitting unpackable generating families, and show that in such theories, forking of formulae over a model M is equivalent to containment in global types definable over M, generalizing a result of Dolich on ominimal theories in [4]. 1.
VCminimality: Examples and Observations
"... Abstract. VCminimality is a recent notion in model theory which generalizes strong minimality, ominimality, weak ominimality and cminimality, but is strong enough to imply various other properties of interest such as NIP and dpminimality. In this paper, we answer several questions posed by Adl ..."
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Abstract. VCminimality is a recent notion in model theory which generalizes strong minimality, ominimality, weak ominimality and cminimality, but is strong enough to imply various other properties of interest such as NIP and dpminimality. In this paper, we answer several questions posed by Adler about the relationship of VCminimality and older stabilitytheoretic notions in model theory. We also give a proof that Presberger arithmetic is not VCminimal and an example which separates local VCminimality from convex orderability, answering a question posed by Guingona.
ON DEFINABILITY OF TYPES IN DEPENDENT THEORIES
, 2011
"... Using definability of types for stable formulas, one develops the powerful tools of stability theory, such as canonical bases, a nice forking calculus, and stable embeddability. When one passes to the class of dependent formulas, this notion of definability of types is lost. However, as this disser ..."
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Using definability of types for stable formulas, one develops the powerful tools of stability theory, such as canonical bases, a nice forking calculus, and stable embeddability. When one passes to the class of dependent formulas, this notion of definability of types is lost. However, as this dissertation shows, we can recover suitable alternatives to definability of types for some dependent theories. Using these alternatives, we can recover some of the power of stability theory. One alternative is uniform definability of types over finite sets (UDTFS). We show that all formulas in dpminimal theories have UDTFS, as well as formulas with VCdensity < 2. We also show that certain Henselian valued fields have UDTFS. Another alternative is isolated extensions. We show that dependent formulas are characterized by the existence of isolated extensions, and show how this gives a weak stable embeddability result. We also explore the idea of UDTFS rank and show how it relates to VCdensity. Finally, we use the machinery developed in this dissertation to show that VCminimal theories satisfy the Kueker Conjecture.