We propose a criterion to regard a property of a theory (in first or second order logic) as virtuous: the property must have significant mathematical consequences for the theory (or its models). We then rehearse results of Ajtai, Marek, Magidor, H. Friedman and Solovay to argue that for second order logic, ‘categoricity’ has little virtue. For first order logic, categoricity is trivial; but ‘categoricity in power’ has enormous structural consequences for any of the theories satisfying it. The stability hierarchy extends (...) this virtue to other complete theories. The interaction of model theory and traditional mathematics is examined by considering the views of such as Bourbaki, Hrushovski, Kazhdan, and Shelah to flesh out the argument that the main impact of formal methods on mathematics is using formal definability to obtain results in ‘mainstream’ mathematics. Moreover, these methods (e.g., the stability hierarchy) provide an organization for much mathematics which gives specific content to dreams of Bourbaki about the architecture of mathematics. (shrink)

We introduce Lascar strong types in excellent classes and prove that they coincide with the orbits of the group generated by automorphisms fixing a model. We define a new independence relation using Lascar strong types and show that it is well-behaved over models, as well as over finite sets. We then develop simplicity and show that, under simplicity, the independence relation satisfies all the properties of nonforking in a stable first order theory. Further, simplicity for an excellent class, as well (...) as the independence relation itself, is uniquely determined. Finally, we show that an excellent class is simple if and only if it has extensible U-rank . We deduce that any excellent class of finite U-rank is simple, and that any uncountably categorical excellent class has an expansion with countably many constants which is simple. (shrink)

This paper is concerned with extensions of geometric stability theory to some nonelementary classes. We prove the following theorem:Theorem. Let [Formula: see text] be a large homogeneous model of a stable diagram D. Let p, q ∈ SD(A), where p is quasiminimal and q unbounded. Let [Formula: see text] and [Formula: see text]. Suppose that there exists an integer n < ω such that [Formula: see text] for any independent a1, …, an∈ P and finite subset C ⊆ Q, but (...) [Formula: see text] for some independent a1, …, an, an+1∈ P and some finite subset C ⊆ Q.Then [Formula: see text] interprets a group G which acts on the geometry P′ obtained from P. Furthermore, either [Formula: see text] interprets a non-classical group, or n = 1,2,3 and•If n = 1 then G is abelian and acts regularly on P′.•If n = 2 the action of G on P′ is isomorphic to the affine action of K ⋊ K* on the algebraically closed field K.•If n = 3 the action of G on P′ is isomorphic to the action of PGL2(K) on the projective line ℙ1(K) of the algebraically closed field K.We prove a similar result for excellent classes. (shrink)

The study of classes of models of a finite diagram was initiated by S. Shelah in 1969. A diagram D is a set of types over the empty set, and the class of models of the diagram D consists of the models of T which omit all the types not in D. In this work, we introduce a natural dependence relation on the subsets of the models for the 0-stable case which share many of the formal properties of forking. This (...) is achieved by considering a rank for this framework which is bounded when the diagram D is 0-stable. We can also obtain pregeometries with respect to this dependence relation. The dependence relation is the natural one induced by the rank, and the pregeometries exist on the set of realizations of types of minimal rank. Finally, these concepts are used to generalize many of the classical results for models of a totally transcendental first-order theory. In fact, strong analogies arise: models are determined by their pregeometries or their relationship with their pregeometries; however the proofs are different, as we do not have compactness. This is illustrated with positive results as well as negative results . We also give a proof of a Two Cardinal Theorem for this context. (shrink)