 Stable theory

A complete theory $T$ is $\kappa$-stable, for an infinite cardinal $\kappa \ge |T|$, when $|S_1(A)|\le \kappa$ for every parameter set $A$ of cardinality $\kappa$ in a model of $T$. Here $S_1(A)$ is the type space of complete one-variable types. The theory is stable if this holds for some such $\kappa$; it is omega-stable if it holds for ${\aleph }_0$.

For the theory of an infinite set with equality alone, a type over $A$ either names one element of $A$ or says that it differs from every member of $A$. There are only $|A|+1$ possibilities. By contrast, a dense linear order without endpoints has the order property, producing many types corresponding to cuts and hence instability. Stability measures the number of ways a new element can relate to existing parameters, rather than simply the number of models.

Stable theory is a branch of model theory, which is a field of mathematical logic. Introduced by Morley in the early 1960s, stable theory primarily concerns the study of structures that satisfy certain stability conditions. Stability, here, refers to a way of categorizing theories based on their behavior in terms of definability and the complexity of their types. A theory is said to be stable if its behavior can be well-controlled, especially in terms of the number of types over various sets.