# Stable theory

In the mathematical field of 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).

## Definitions

T will be a complete theory in some language.

• T is called κ-stable (for an infinite cardinal κ) if for every set A of cardinality κ the set of complete types over A has cardinality κ.
• ω-stable is an alternative name for ℵ0-stable.
• T is called stable if it is κ-stable for some infinite cardinal κ.
• T is called unstable if it is not κ-stable for any infinite cardinal κ.
• T is called superstable if it is κ-stable for all sufficiently large cardinals κ.
• Totally transcendental theories are those such that every formula has Morley rank less than ∞.

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.

## Unstable theories

Roughly speaking, a theory is unstable if one can use it to encode the ordered set of natural numbers. More precisely, Saharon Shelah's unstable formula theorem in model theory characterizes the unstable theories by the nonexistence of countably infinite half graphs. Shelah defines a complete theory as having the order property if there exist a model $M$  of the theory, a formula $\phi ({\bar {x}},{\bar {y}})$  on two finite tuples of free variables ${\bar {x}}$  and ${\bar {y}}$ , and, a system of countably many values ${\bar {x}}_{i}$  and ${\bar {y}}_{i}$  for these variables such that the pairs ${\bigl \{}({\bar {x}}_{i},{\bar {y}}_{i})\mid M\models \phi ({\bar {x}}_{i},{\bar {y}}_{j}){\bigr \}}$  form the edges of a countable half graph on vertices ${\bar {x}}_{i}$  and ${\bar {y}}_{i}$ . Intuitively, the existence of these half graphs allows one to construct the comparison operation of an infinite ordered set within the model, via the equivalence $(i\leq j)\Leftrightarrow {\bigl (}M\models \phi ({\bar {x}}_{i},{\bar {y}}_{j}){\bigr )}$ . The unstable formula theorem of Shelah (1990, pp. 30–31) states that a complete theory is unstable if and only if it has the order property.

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

Examples:

• Most sufficiently complicated theories, such as set theories and Peano arithmetic, are unstable.
• The theory of the rational numbers, considered as an ordered set, is unstable. Its theory is the theory of dense total orders without endpoints. More generally, the theory of every infinite total order is unstable.
• The theory of addition of the natural numbers is unstable.
• Any infinite Boolean algebra is unstable.
• Any monoid with cancellation that is not a group is unstable, because if a is an element that is not a unit then the powers of a form an infinite totally ordered set under the relation of divisibility. For a similar reason any integral domain that is not a field is unstable.
• There are many unstable nilpotent groups. One example is the infinite dimensional Heisenberg group over the integers: this is generated by elements xi, yi, z for all natural numbers i, with the relations that any of these two generators commute except that xi and yi have commutator z for any i. If ai is the element x0x1...xi−1yi then ai and aj have commutator z exactly when i < j, so they form an infinite total order under a definable relation, so the group is unstable.
• Real closed fields are unstable, as they are infinite and have a definable total order.

## Stable theories

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

• The theory of any module over a ring is stable.
• The theory of a countable number of equivalence relations, (En)nN, such that each equivalence relation has an infinite number of equivalence classes and each equivalence class of En is the union of an infinite number of different classes of En+1 is stable but not superstable.
• Sela (2013) showed that free groups, and more generally torsion-free hyperbolic groups, are stable. Free groups on more than one generator are not superstable.
• A differentially closed field is stable. If it has non-zero characteristic it is not superstable, and if it has zero characteristic it is totally transcendental.

## Superstable theories

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 ≥ 2ω. The following conditions on a theory T are equivalent:

• T is superstable.
• All types of T are ranked by at least one notion of rank.
• T is κ-stable for all sufficiently large cardinals κ
• T is κ-stable for all cardinals κ that are at least 2|T|.

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

The number of countable models of a countable superstable theory must be 1, ℵ0, ℵ1, or 2ω. If the number of models is 1 the theory is totally transcendental. There are examples with 1, ℵ0 or 2ω models, and it is not known if there are examples with ℵ1 models if the continuum hypothesis does not hold. If a theory T is not superstable then the number of models of cardinality κ > |T| is 2κ.

Examples:

• The additive group of integers is superstable, but not totally transcendental. It has 2ω countable models.
• The theory with a countable number of unary relations Pi with model the positive integers where Pi(n) is interpreted as saying n is divisible by the ith prime is superstable but not totally transcendental.
• An abelian group A is superstable if and only if there are only finitely many pairs (p,n) with p prime, n a natural number, with pnA/pn+1A infinite.

## Totally transcendental theories and ω-stable

• Totally transcendental theories are those such that every formula has Morley rank less than ∞. Totally transcendental theories are stable in λ whenever λ ≥ |T|, so they are always superstable. ω-stable is an alternative name for ℵ0-stable. The ω-stable theories in a countable language are κ-stable for all infinite cardinals κ. If |T| is countable then T is totally transcendental if and only if it is ω-stable. More generally, T is totally transcendental if and only if every restriction of T to a countable language is ω-stable.

Examples: