Berkeley Model Theory Seminar

In the first installment of this series we explore the notion of Abstract Elementary Classes (AECs) as a natural context in which to study non-elementary classes. The definition, several examples and Shelah's Categoricity Conjecture will be introduced. We identify one of the fundamental differences between AECs and first order logic: Galois types versus first order types. After extracting the notion of amalgamation from the definition of Galois types, we explain its significance in work towards Shelah's Categoricity Conjecture.

In the following weeks we will present a few amalgamation theorems using Devlin-Shelah's weak diamond and Martin's Axiom which represent the set-theoretic aspects of some theorems about AECs. The necessary set theory will be reviewed.

Finally we will discuss a direction of work towards Shelah's Categoricity Conjecture in a very general context where the amalgamation property is not known to hold. In particular we outline the proof of the uniqueness of limit models.

We define a Lie differential field as a field K of characteristic 0 with an action, as derivations on K, of some given Lie algebra L. We assume that L is a finite-dimensional vector space over some sub-field F given in advance. As an example take the field of rational functions on a smooth algebraic variety, with L = { rational vector fields }.

For every simple extension of Lie differential fields we find a finite system of differential equations that characterizes it. We then define, using first-order conditions, a collection of allowed systems of differential equations s.t. the above characteristic systems are allowed. We prove that for every allowed system there exists a generic solution in some extension, and this solution is unique (up to isomorphism).

We construct the model completion of the theory of Lie differential fields by adding axioms stating that every allowed system has almost generic solutions. The construction is a generalization of Blum's axioms for DCF₀. We also show that this model completion is w-stable.

The independence notion obtained from forking in simple theories is characterized by a number of basic axioms (symmetry, automorphism invariance, transitivity, local character, etc.). The independence notion obtained from þ-forking in rosy theories satisfies all of these axioms with the exception of the independence theorem, which states that if A and B are two sets which are independent over a common model M, p Î S(A) and q Î S(B)are types which do not fork over M and which agree when restricted to M, then p(x) È q(x) does not fork over M. One sees easily that this version of the independence theorem fails in o-minimal theories even when ``fork'' is replaced by ``þ-fork.'' However, another version, which we call consistent amalgamation does hold: With the notation as above, if p(x) È q(x) is consistent, then it does not þ-fork over M.

Using a Hrushovski-Fraisse construction we produce a rosy theory in which consistent amalgamation fails.