Dependence and Isolated Extensions

TL;DR

This paper characterizes dependent formulas via extensions to isolated types, showing that such extensions add limited new elements, and explores implications and parallels in stable theories.

Contribution

It provides a new characterization of dependent formulas through elementary isolated extensions and quantifies the domain expansion involved.

Findings

Dependent formulas are characterized by the existence of elementary isolated extensions.

Extensions add at most twice the independence dimension in new elements.

Corollaries and parallels to stable theories are discussed.

Abstract

In this paper, we show that \phi is a dependent formula if and only if all \phi-types have an extension to a \phi-isolated \phi-type that is an "elementary \phi-extension" (see Definition 2.3 in the paper). Moreover, we show that the domain of this extension adds at most 2 times the independence dimension of \phi new elements to the domain of the original \phi-type. We give corollaries to this theorem and discuss parallels to the stable setting.