[Paper Review] Independence of Sets Without Stability

Adi Jarden, Alon Sitton|arXiv (Cornell University)|Jan 17, 2010

Advanced Topology and Set Theory1 references3 citations

TL;DR

This paper introduces an independence relation in abstract elementary classes (AECs) without assuming stability, enabling the definition of dimension via a forking notion that satisfies axioms of a 'good frame minus stability.' The key contribution is proving that finite independence implies full independence under uniqueness triples, extending model-theoretic dimension theory beyond stable contexts.

ABSTRACT

We presents an independence relation on sets, one can define dimension by it, assuming that we have an abstract elementary class with a forking notion that satisfies the axioms of a good frame minus stability.

Motivation & Objective

• To define an independence relation in abstract elementary classes (AECs) without requiring stability, extending dimension theory beyond stable settings.
• To establish that a forking notion satisfying 'good frame minus stability' axioms supports a well-behaved independence relation.
• To prove that finite independence implies full independence under the assumption of uniqueness triples, generalizing results from stable or successful AECs.
• To remove assumptions of stability, goodness+, or existence of uniqueness triples in earlier results, making the framework more widely applicable.
• To provide a foundation for dimension theory in AECs by constructing an independence relation that satisfies key properties like monotonicity, transitivity, and continuity.

Proposed method

• Define an abstract elementary class (AEC) using standard axioms including smoothness, coherence, and LST number, ensuring closure under isomorphisms and directed unions.
• Introduce a non-forking relation (NF) satisfying the axioms of a 'good frame minus stability,' including monotonicity, symmetry, and transitivity in the absence of stability.
• Construct an independence relation based on the non-forking notion, ensuring finite character and continuity under increasing continuous sequences.
• Use the existence of uniqueness triples (from [JrSh 875], [Sh 600], [Sh 705]) to prove that finite independence implies full independence.
• Apply long transitivity and monotonicity of the non-forking relation to transfer independence across chains of models.
• Prove continuity of finite independence via a construction of intermediate models that preserve independence across limit stages.

Experimental results

Research questions

• RQ1 Can an independence relation be defined in AECs without assuming stability, such that it supports a consistent dimension theory?
• RQ2 Does finite independence imply full independence in AECs when uniqueness triples exist, even without stability?
• RQ3 How can the axioms of a 'good frame minus stability' be used to derive structural properties like transitivity and continuity?
• RQ4 To what extent can the results of [Sh 705] be generalized by removing assumptions of stability, goodness+, and existence of uniqueness triples?
• RQ5 What is the role of uniqueness triples in ensuring that finite independence implies full independence in AECs?

Key findings

• An independence relation is successfully defined in AECs without assuming stability, satisfying key properties such as monotonicity, transitivity, and finite character.
• Finite independence implies full independence in the presence of uniqueness triples, even when stability is not assumed.
• The non-forking relation satisfies long transitivity and respects the frame structure, enabling transfer of independence across model chains.
• The paper proves that the independence relation is continuous at limit stages, ensuring that independence is preserved under increasing unions.
• The construction allows for the definition of dimension in AECs via the independence relation, generalizing linear dimension in fields to broader model-theoretic contexts.
• The results improve upon [Sh 705] by removing assumptions of stability, successfulness, and goodness+, while proving key propositions without requiring existence of uniqueness triples in $K^{3,uq}$.