Axiomatic Theory of Independence Relations in Model Theory
Christian d'Elbée
TL;DR
The work develops an axiomatic framework for independence relations in model theory, organizing them into set-theoretic, closure-based, and theory-based axiom classes and showing how closures and pregeometries underpin a robust calculus of independence. It applies the calculus to canonical theories such as algebraically closed fields and random graphs, deriving foundational results about forking, dividing, and symmetry (Adler’s theorem) and connecting them to tameness concepts like simplicity and NSOP$_4$. The text then extends the theory to general ambient theories, proving central theorems (Adler symmetry, Kim–Pillay) and introducing second NSOP$_4$ criteria (Conant’s framework), including hereditary and amalgamation techniques. Through these results, the authors unify and extend independence notions, offering tools to determine theory classification (simple vs NSOP$_4$) and to analyze tameness in familiar mathematical structures. The practical impact lies in providing a cohesive, exporter-ready toolkit for identifying and exploiting tameness in diverse theories, with clear pathways from combinatorial independence to model-theoretic stability phenomena.
Abstract
This course introduces the fruitful links between model theory and a combinatoric of sets given by independence relations. An independence relation on a set is a ternary relation between subsets. Chapter 1 should be considered as an introductory chapter. It does not mention first-order theories or formulas. It introduces independence relations in a naive set theory framework. Its goal is to get the reader familiar with basic axioms of independence relations (which do not need an ambient theory to be stated) as well as introduce closure operators and pregeometries. Chapter 2 introduces the model-theoretic context. The two main examples (algebraically closed fields and the random graph) are described as well as independence relations in those examples. Chapter 3 gives the axioms of independence relations in a model-theoretic context. It introduces the general toolbox of the model-theorists (indiscernible sequences, Ramsey/Erdos-Rado and compactness) and the independence relations of heirs/coheirs with two main applications: Adler's theorem of symmetry (how symmetry emerges from a weaker set of axioms, which is rooted in the work of Kim and Pillay) and a criterion for NSOP4 using stationary independence relations in the style of Conant. Independence relations satisfying Adler's theorem of symmetry are here called 'Adler independence relations' or AIR. Chapter 4 treats forking and dividing. It is proved that dividing independence is always stronger than any AIR (even though it is not an AIR in general) a connection between the independence theorem and forking independence, which holds in all generality and is based on Kim-Pillay's approach. Then, simplicity is defined and the interesting direction of the Kim-Pillay theorem (namely that the existence of an Adler independence relation satisfying the independence theorem yields simplicity) is deduced from earlier results.