Stable theory

In model theory, a complete theory is called stable if it does not have too many types. One goal of classification theory is to divide all complete theories into those whose models can be classified and those whose models are too complicated to classify, and to classify all models in the cases where this can be done. Roughly speaking, if a theory is not stable then its models are too complicated and numerous to classify, while if a theory is stable there might be some hope of classifying its models, especially if the theory is superstable or totally transcendental.

Stability theory was started by Morley (1965), who introduced several of the fundamental concepts, such as totally transcendental theories and the Morley rank. Stable and superstable theories were first introduced by Shelah (1969), who is responsible for much of the development of stability theory. The definitive reference for stability theory is (Shelah 1990), though it is notoriously hard even for experts to read, as mentioned, e.g., in (Grossberg, Iovino & Lessmann 2002, p. 542).

T will be a complete theory in some language.

As usual, a model of some language is said to have one of these properties if the complete theory of the model has that property.

An incomplete theory is defined to have one of these properties if every completion, or equivalently every model, has this property.

Roughly speaking, a theory is unstable if one can use it to encode the ordered set of natural numbers. More precisely, if there is a model M and a formula Φ(X,Y) in 2n variables X = x₁,...,x_n and Y = y₁,...,y_n defining a relation on Mⁿ with an infinite totally ordered subset then the theory is unstable. (Any infinite totally ordered set has a subset isomorphic to either the positive or negative integers under the usual order, so one can assume the totally ordered subset is ordered like the positive integers.) The totally ordered subset need not be definable in the theory.

The number of models of an unstable theory T of any uncountable cardinality κ ≥ |T| is the maximum possible number 2^κ.

Examples:

T is called stable if it is κ-stable for some cardinal κ. Examples:

T is called superstable if it is stable for all sufficiently large cardinals, so all superstable theories are stable. For countable T superstability is equivalent to stability for all κ≥2^ω. The following conditions on a theory T are equivalent:

If a theory is superstable but not totally transcendental it is called strictly superstable.

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