In the mathematical field of model theory, a theory is called stable if it satisfies certain combinatorial restrictions on its complexity. Stable theories are rooted in the proof of Morley's categoricity theorem and were extensively studied as part of Saharon Shelah's classification theory, which showed a dichotomy that either the models of a theory admit a nice classification or the models are too numerous to have any hope of a reasonable classification. A first step of this program was showing that if a theory is not stable then its models are too numerous to classify.
Stable theories were the predominant subject of pure model theory from the 1970s through the 1990s, so their study shaped modern model theory[1] and there is a rich framework and set of tools to analyze them. A major direction in model theory is "neostability theory," which tries to generalize the concepts of stability theory to broader contexts, such as simple and NIP theories.
- ^ Baldwin, John (2021). "The dividing line methodology: Model theory motivating set theory" (PDF). Theoria. 87 (2): 1. doi:10.1111/theo.12297. S2CID 211239082.