In model theory

Model theory

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

, a branch of mathematical logic

Mathematical logic

Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...

, a complete first-order theory T is called stable in λ (an infinite cardinal number

Cardinal number

In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of sets. The cardinality of a finite set is a natural number – the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite...

), if the Stone space of every model

Structure (mathematical logic)

In universal algebra and in model theory, a structure consists of a set along with a collection of finitary operations and relations which are defined on it....

 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. The stability spectrum of T is the class of all cardinals κ such that T is stable in κ.

For countable theories there are only four possible stability spectra. The corresponding dividing lines are those for total transcendentality, superstability and stability

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...

. This result is due to Saharon Shelah

Saharon Shelah

Saharon Shelah is an Israeli mathematician. He is a professor of mathematics at the Hebrew University of Jerusalem and Rutgers University in New Jersey.-Biography:...

, who also defined stability and superstability.

The stability spectrum theorem for countable theories

Theorem.
Every countable complete first-order theory T falls into one of the following classes:

• T is stable in λ for all infinite cardinals λ. – T is totally transcendental.
• T is stable in λ exactly for all cardinals λ with λ ≥ 2^ω. – T is superstable but not totally transcendental.
• T is stable in λ exactly for all cardinals λ that satisfy λ = λ^ω. – T is stable but not superstable.
• T is not stable in any infinite cardinal λ. – T is unstable.

The condition on λ in the third case holds for cardinals of the form λ = κ^ω, but not for cardinals λ of cofinality ω (because λ < λ^cof λ).

Totally transcendental theories

A complete first-order theory T is called totally transcendental if every formula has bounded 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...

, 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

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....

 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.

Superstable 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|.

Stable theories

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|.

Unstable theories

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.

The uncountable case

For a general stable theory T in a possibly uncountable language, the stability spectrum is determined by two cardinals κ and λ₀, such that T is stable in λ exactly when λ ≥ λ₀ and λ^μ = λ for all μ<κ. So λ₀ is the smallest infinite cardinal for which T is stable. These invariants satisfy the inequalities

• κ ≤ |T|⁺
• κ ≤ λ₀
• λ₀ ≤ 2^|T|
• If λ₀ > |T|, then λ₀ ≥ 2^ω

When |T| is countable the 4 possibilities for its stability spectrum correspond to the following values of these cardinals:

• κ and λ₀ are not defined: T is unstable.
• λ₀ is 2^ω, κ is ω₁: T is stable but not superstable
• λ₀ is 2^ω, κ is ω: T is superstable but not ω-stable.
• λ₀ is ω, κ is ω: T is totally transcendental (or ω-stable)