In model theory

Model theory

In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....

, 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 , who introduced several of the fundamental concepts, such as totally transcendental theories and the Morley rank.
Stable and superstable theories were first introduced by , who is responsible for much of the development of stability theory. The definitive reference for stability theory is , though it is notoriously hard even for experts to read.

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 number 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 stable.
• 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
  Morley rank
  In mathematical logic, Morley rank, introduced by , is a means of measuring the size of a subset of a model of a theory, generalizing the notion of dimension in algebraic geometry.-Definition:Fix a theory T with a model M...
  
   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, 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:

• 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 linear orders without endpoints.
• 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 x_i, y_i, z for all natural numbers i, with the relations that any of these two generators commute except that x_i and y_i have commutator z for any i. If a_i is the element x₀x₁...x_i−1y_i then a_i and a_j 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 field
  Real closed field
  In mathematics, a real closed field is a field F that has the same first-order properties as the field of real numbers. Some examples are the field of real numbers, the field of real algebraic numbers, and the field of hyperreal numbers.-Definitions:...
  
  s 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 E_n for n a natural number such that each equivalence relation has an infinite number of equivalence classes and each equivalence class of E_n is the union of an infinite number of different classes of E_n+1 is stable but not superstable. 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
  Differentially closed field
  In mathematics, a differential field K is differentially closed if every finite system of differential equations with a solution in some differential field extending K already has a solution in K. This concept was introduced by...
  
   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≥2^ω.
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 2^ω. If the number of models is 1 the theory is totally transcendental. There are examples with 1, ℵ₀ or 2^ω 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 2^κ.

Examples:

• The additive group of integers is superstable, but not totally transcendental. It has 2^ω countable models.
• The theory with a countable number of unary relations P_i with model the positive integers where P_i(n) is interpreted as saying n is divisible by the nth 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
  Morley rank
  In mathematical logic, Morley rank, introduced by , is a means of measuring the size of a subset of a model of a theory, generalizing the notion of dimension in algebraic geometry.-Definition:Fix a theory T with a model M...
  
   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. 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:

• Any ω-stable theory is totally transcendental.
• Any finite model is totally transcendental.
• An infinite field is totally transcendental if and only if it is algebraically closed. (Macintyre's theorem.)
• A differentially closed field
  Differentially closed field
  In mathematics, a differential field K is differentially closed if every finite system of differential equations with a solution in some differential field extending K already has a solution in K. This concept was introduced by...
  
   in characteristic 0 is totally transcendental.
• Any theory with a countable language that is categorical for some uncountable cardinal is totally transcendental.
• An abelian group is totally transcendental if and only if it is the direct sum of a divisible group and a group of bounded exponent.
• Any linear algebraic group
  Linear algebraic group
  In mathematics, a linear algebraic group is a subgroup of the group of invertible n×n matrices that is defined by polynomial equations...
  
   over an algebraically closed field is totally transcendental.
• Any group of finite Morley rank is totally transcendental.

See also

• Spectrum of a theory
  Spectrum of a theory
  In model theory, a branch of mathematical logic, the spectrum of a theoryis given by the number of isomorphism classes of models in various cardinalities. More precisely,...
  
  
• Morley's categoricity theorem
  Morley's categoricity theorem
  In model theory, a branch of mathematical logic, a theory is κ-categorical if it has exactly one model of cardinality κ up to isomorphism....
  
  
• List of first-order theories
• Stability spectrum
  Stability spectrum
  In model theory, a branch of mathematical logic, a complete first-order theory T is called stable in λ , if the Stone space of every model of T of size ≤ λ has itself size ≤ λ. T is called a stable theory if there is no upper bound for the cardinals κ such that T is stable in κ...
  
  

External links

• A. Pillay, Lecture notes on model theory
• A. Pillay, Lecture notes on stability theory
• A. Pillay, Lecture notes on applied stability theory