On \omega-categorical simple theories

In the present paper we shall prove that countable \omega-categorical simple CM-trivial theories and countable \omega-categorical simple theories with strong stable forking are low. In addition, we observe that simple theories of bounded finite weight are low

Forking in simple theories and CM-triviality

[cat] Aquesta tesi té tres objectius. En primer lloc, estudiem generalitzacions de la jerarquia no ample relatives a una família de tipus parcials. Aquestes jerarquies en permeten classificar la complexitat del “forking” respecte a una família de tipus parcials. Si considerem la família de tipus algebraics, aquestes generalitzacions corresponen a la jerarquia ordinària, on el primer i el segon nivell corresponen a one-basedness i a CM-trivialitat, respectivament. Fixada la família de tipus regulars “no one-based”, el primer nivell d'una d'aquestes possibles jerarquies no ample ens diu que el tipus de la base canònica sobre una realització és analitzable en la família. Demostrem que tota teoria simple amb suficients tipus regulars pertany al primer nivell de la jerarquia dèbil relativa a la família de tipus regulars no one-based. Aquest resultat generalitza una versió dèbil de la “Canonical Base Property” estudiada per Chatzidakis i Pillay. En segon lloc, discutim problemes d'eliminació de hiperimaginaris assumint que la teoria és CM-trivial, en tal cas la independència del “forking” té un bon comportament. Més concretament, demostrem que tota teoria simple CM-trivial elimina els hiperimaginaris si elimina els hiperimaginaris finitaris. En particular, tota teoria petita simple CM-trivial elimina els hiperimaginaris. Cal remarcar que totes les teories omega-categòriques simples que es coneixen són CM-trivials; en particular, aquelles teories obtingudes mitjançant una construcció de Hrushovski. Finalment, tractem problemes de classificació en les teories simples. Estudiem la classe de les teories simples baixes; classe que inclou les teories estables i les teories supersimples de D-rang finit. Demostrem que les teories simples amb pes finit acotat també pertanyen a aquesta classe. A més, provem que tota teoria omega-categòrica simple CM-trivial és baixa. Aquest darrer fet resol parcialment una pregunta formulada per Casanovas i Wagner.[eng] The development of first-order stable theories required two crucial abstract notions: forking independence, and the related notion of canonical base. Forking independence generalizes the linear independence in vector spaces and the algebraic independence in algebraically closed fields. On the other hand, the concept of canonical base generalizes the field of definition of an algebraic variety. The general theory of independence adapted to simple theories, a class of first-order theories which includes all stable theories and other interesting examples such as algebraically closed fields with an automorphism and the random graph. Nevertheless, in order to obtain canonical bases for simple theories, the model-theoretic development of hyperimaginaries --equivalence classes of arbitrary tuple modulo a type-definable (without parameters) equivalence relation-- was required. In the present thesis we deal with topics around the geometry of forking in simple theories. Our first goal is to study generalizations of the non ample hierarchy which will code the complexity of forking with respect to a family of partial types. We introduce two hierarchies: the non (weak) ample hierarchy with respect to a fixed family of partial types. If we work with respect to the family of bounded types, these generalizations correspond to the ordinary non ample hierarchy. Recall that in the ordinary non ample hierarchy the first and the second level correspond to one-basedness and CM-triviality, respectively. The first level of the non weak ample hierarchy with respect to some fixed family of partial types states that the type of the canonical base over a realization is analysable in the family. Considering the family of regular non one-based types, the first level of the non weak ample hierarchy corresponds to the weak version of the Canonical Base Property studied by Chatzidakis and Pillay. We generalize Chatzidakis' result showing that in any simple theory with enough regular types, the canonical base of a type over a realization is analysable in the family of regular non one-based types. We hope that this result can be useful for the applications; for instance, the Canonical Base Property plays an essential role in the proof of Mordell-Lang for function fields in characteristic zero and Manin-Mumford due to Hrushovski. Our second aim is to use combinatorial properties of forking independence to solve elimination of hyperimaginaries problems. For this we assume the theory to be simple and CM-trivial. This implies that the forking independence is well-behaved. Our goal is to prove that any simple CM-trivial theory which eliminates finitary hyperimaginaries --hyperimaginaries which are definable over a finite tuple-- eliminates all hyperimaginaries. Using a result due to Kim, small simple CM-trivial theories eliminate hyperimaginaries. It is worth mentioning that all currently known omega-categorical simple theories are CM-trivial, even those obtained by an ab initio Hrushovski construction. To conclude, we study a classification problem inside simple theories. We study the class of simple low theories, which includes all stable theories and supersimple theories of finite D-rank. In addition, we prove that it also includes the class of simple theories of bounded finite weight. Moreover, we partially solve a question posed by Casanovas and Wagner: Are all omega-categorical simple theories low? We solve affirmatively this question under the assumption of CM-triviality. In fact, our proof exemplifies that the geometry of forking independence in a possible counterexample cannot come from finite sets

Exact saturation in pseudo-elementary classes for simple and stable theories

We study PC-exact saturation for stable and simple theories. Among other results, we show that PC-exact saturation characterizes the stability cardinals of size at least continuum of a countable stable theory and, additionally, that simple unstable theories have PC-exact saturation at singular cardinals, satisfying mild set-theoretic hypotheses, which had previously been open even for the random graph. We characterize supersimplicity of countable theories in terms of having PC-exact saturation at singular cardinals of countable cofinality. We also consider the local analogue of PC-exact saturation, showing that local PC-exact saturation for singular cardinals of countable cofinality characterizes supershort theories

Some Applications of Set Theory to Model Theory

We investigate set-theoretic dividing lines in model theory. In particular, we are interested in Keisler's order and Borel complexity. Keisler's order is a pre-order on complete countable theories $T$, measuring the saturation of ultrapowers of models of $T$. In Chapter~\ref{SurveyChapter}, we present a self-contained survey on Keisler's order. In Chapter~\ref{KeislerNew}, we uniformize and sharpen several ultrafilter constructions of Malliaris and Shelah. We also investigate the model-theoretic properties detected by Keisler's order among the simple unstable theories. Borel complexity is a pre-order on sentences of $\mathcal{L}_{\omega_1 \omega}$ measuring the complexity of countable models. In Chapter~\ref{ChapterURL}, we describe joint work with Richard Rast and Chris Laskowski on this order. In particular, we connect the Borel complexity of $\Phi \in \mathcal{L}_{\omega_1 \omega}$ with the number of potential canonical Scott sentences of $\Phi$. In Chapter~\ref{ChapterSB}, we introduce the notion of thickness; when $\Phi$ has class-many potential canonical Scott sentences, thickness is a measure of how quickly this class grows in size. In Chapter~\ref{ChapterTFAG}, we describe joint work with Saharon Shelah on the Borel complexity of torsion-free abelian groups