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Concept
Stable theory
Summary
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 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.
A common goal in model theory is to study a first-order theory by analyzing the complexity of the Boolean algebras of (parameter) definable sets in its models. One can equivalently analyze the complexity of the Stone duals of these Boolean algebras, which are type spaces. Stability restricts the complexity of these type spaces by restricting their cardinalities. Since types represent the possible behaviors of elements in a theory's models, restricting the number of types restricts the complexity of these models.
Stability theory has its roots in Michael Morley's 1965 proof of Łoś's conjecture on categorical theories. In this proof, the key notion was that of a totally transcendental theory, defined by restricting the topological complexity of the type spaces. However, Morley showed that (for countable theories) this topological restriction is equivalent to a cardinality restriction, a strong form of stability now called -stability, and he made significant use of this equivalence.
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