Explanation of Independence

TL;DR

This paper provides an axiomatic framework for understanding independence relations in model theory, focusing on forking and thorn-forking, and explores their properties and interrelations.

Contribution

It introduces a unified axiomatic approach to independence relations, characterizes thorn-forking via modular pairs, and examines their duality and connections to canonical bases.

Findings

Forking is the finest independence relation.

Thorn-forking is the coarsest meaningful independence relation.

Axiomatic characterization unifies forking and thorn-forking analysis.

Abstract

An axiomatic treatment of `independence relations' (notions of independence) for complete first-order theories is presented, the principal examples being forking (due to Shelah) and thorn-forking (due to Onshuus). Thorn-forking is characterised in terms of modular pairs in the lattice of algebraically closed sets. Wherever possible, forking and thorn-forking are treated in a uniform way. They are dual in the sense that forking is the finest (most restrictive) and thorn-forking the coarsest independence relation worth examining. We finish by defining the kernel of a sequence of indiscernibles and studying its relation to canonical bases.