The Karp complexity of unstable classes

Karp complexity and classes with the independence property

A class K of structures is controlled if for all cardinals lambda, the relation of L_{infty,lambda}-equivalence partitions K into a set of equivalence classes (as opposed to a proper class). We prove that no pseudo-elementary class with the independence property is controlled. By contrast, there is a pseudo-elementary class with the strict order property that is controlled.

Unstable classes of metric structures

We prove a strong non-structure theorem for a class of metric structures with an unstable pair of formulae. As a consequence, we show that weak categoricity (that is, categoricity up to isomorphisms and not isometries) implies several versions of stability. This is the first step in the direction of the investigation of weak categoricity and weak stability of metric classes.

Keisler's order has infinitely many classes

We prove, in ZFC, that there is an infinite strictly descending chain of classes of theories in Keisler's order. Thus Keisler's order is infinite and not a well order. Moreover, this chain occurs within the simple unstable theories, considered model-theoretically tame. Keisler's order is a central notion of the model theory of the 60s and 70s which compares first-order theories (and implicitly ultrafilters) according to saturation of ultrapowers. Prior to this paper, it was long thought to have finitely many classes, linearly ordered. The model-theoretic complexity we find is witnessed by a ve…

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 satura…

通过算法重新审视不稳定公式定理

The unstable formula theorem revisited via algorithms

本文提出PEC统计学习模型，建立等价定理，并基于此给出Shelah不稳定公式定理的完整算法类比，用算法性质刻画Littlestone类。

关于抽象素数类模的稳定性问题

On the problem of stability of abstract elementary classes of modules

本文研究了抽象素数类模的稳定性问题，通过构造不稳定例子否定了一个开放问题，并引入了几乎稳定性概念，证明在特定条件下抽象素数类模的稳定性。