structural:summarization
The stability hierarchy

A fundamental result in stability theory is the stability spectrum theorem, which implies that every complete theory T in a countable signature falls in one of the following classes:

A theory of the first type is called unstable, a theory of the second type is called strictly stable and a theory of the third type is called superstable.Furthermore, if a theory is ω {\displaystyle \omega } -stable, it is stable in every infinite cardinal, so ω {\displaystyle \omega } -stability is stronger than superstability.

Many construction in model theory are easier when restricted to stable theories; for instance, every model of a stable theory has a saturated elementary extension, regardless of whether the generalised continuum hypothesis is true.

Shelah's original motivation for studying stable theories was to decide how many models a countable theory has of any uncountable cardinality. If a theory is uncountably categorical, then it is ω {\displaystyle \omega } -stable. More generally, the Main gap theorem implies that if there is an uncountable cardinal λ {\displaystyle \lambda } such that a theory T has less than 2 λ {\displaystyle 2^{\lambda }} models of cardinality λ {\displaystyle \lambda }, then T is superstable.