The stability spectrum theorem for countable theories

Statement

A complete first-order theory T is called totally transcendental if every formula has bounded Morley rank, i.e. if RM(φ) < ∞ for every formula φ(x) with parameters in a model of T, where x may be a tuple of variables. It is sufficient to check that RM(x=x) < ∞, where x is a single variable.

For countable theories total transcendence is equivalent to stability in ω, and therefore countable totally transcendental theories are often called ω-stable for brevity. A totally transcendental theory is stable in every λ ≥ |T|, hence a countable ω-stable theory is stable in all infinite cardinals.

Every uncountably categorical countable theory is totally transcendental. This includes complete theories of vector spaces or algebraically closed fields. The theories of groups of finite Morley rank are another important example of totally transcendental theories.

A complete first-order theory T is superstable if there is a rank function on complete types that has essentially the same properties as Morley rank in a totally transcendental theory. Every totally transcendental theory is superstable. A theory T is superstable if and only if it is stable in all cardinals λ ≥ 2^|T|.

A theory that is stable in one cardinal λ ≥ |T| is stable in all cardinals λ that satisfy λ = λ^|T|. Therefore a theory is stable if and only if it is stable in some cardinal λ ≥ |T|.

Most mathematically interesting theories fall into this category, including complicated theories such as any complete extension of ZF set theory, and relatively tame theories such as the theory of real closed fields. This shows that the stability spectrum is a relatively blunt tool. To get somewhat finer results one can look at the exact cardinalities of the Stone spaces over models of size ≤ λ, rather than just asking whether they are at most λ.