Superstability

Superstability is a notion in model theory describing a strong form of stability for complete first-order theories. Introduced by Saharon Shelah, it provides a finer division inside stable theories and underpins a robust theory of forking and independence. A theory being superstable means its models exhibit a high level of tameness in how types can vary over sets.

Formally, a complete theory T is superstable if it is stable and, for every model M ⊨ T and every subset A ⊆ M, every complete type over A has finite U-rank. This condition implies that forking—a central notion of independence in stability theory—behaves well: there are no infinite sequences of strict forking over a single parameter set, and a local character property holds. Equivalently, T is stable in every infinite cardinal, and the rank theory (such as U-rank or Morley rank) assigns finite ordinal values to definable sets.

Examples and contrasts: The theory of algebraically closed fields is superstable (in fact ω-stable). The theory of vector spaces over a fixed field is also superstable. By contrast, a stable theory that is not superstable can be built by expanding an equivalence relation with infinitely many infinite classes; such a theory is stable but fails superstability because types can have infinite U-rank.

Relation and significance: Every superstable theory is stable, but not every stable theory is superstable. Superstability plays a central role in stability theory and the study of tame algebraic structures, including the analysis of groups and fields definable in superstable theories.