Stable theories are a class of theories generalizing strongly minimal theories, uncountably categorical theories, and totally transcendental theories. Stable theories are the primary focus of stability theory.

Definition of stability[]

In this article, β€œtype” will mean complete {\displaystyle n}-type for some {\displaystyle n}.

Fix some complete theory {\displaystyle T}, and let {\displaystyle \mathbb {U} } be a monster model of {\displaystyle T}. We say that {\displaystyle T} is {\displaystyle \kappa }-stable if {\displaystyle \kappa \geq |T|} and for every set {\displaystyle A\subset \mathbb {U} } of cardinality at most {\displaystyle \kappa }, there are at most {\displaystyle \kappa } types over {\displaystyle A}: {\displaystyle |S_{n}(A)|\leq \kappa } for every {\displaystyle n}.

{\displaystyle T} is said to be stable if it is {\displaystyle \kappa }-stable for some {\displaystyle \kappa \geq |T|}.

There are many equivalent definitions of stability; we will list some below.


Notable examples of stable theories include

Any theory in which an infinite total order is definable (or interpretable) will not be stable. Consequently, theories such as DLO, RCF, ACVF, and the true theory of arithmetic are all unstable.

Forking Calculus[]

Main article: Forking in stable theories

If {\displaystyle T} is stable and {\displaystyle A,B,C} are subsets of the monster, there is a canonical notion of what it means for {\displaystyle A} and {\displaystyle B} to be independent over {\displaystyle C}, denoted {\displaystyle A\downarrow _{C}B}. There is a closely related notion, that of a β€œnon-forking extension” of complete types: {\displaystyle \operatorname {tp} (a/BC)} is a non-forking extension of {\displaystyle \operatorname {tp} (a/C)} if {\displaystyle a\downarrow _{C}B}, and is a forking extension of {\displaystyle \operatorname {tp} (a/C)} otherwise. The non-forking extensions of a type {\displaystyle p} can be thought of as the β€œgeneric” or β€œcanonical” extensions of {\displaystyle p} to larger sets. The relation {\displaystyle A\downarrow _{C}B} can be recovered from the notion of non-forking: {\displaystyle A\downarrow _{C}B} holds if and only if {\displaystyle \operatorname {tp} (a/BC)} is a non-forking extension of {\displaystyle \operatorname {tp} (a/C)} for each finite tuple {\displaystyle a} from {\displaystyle A}.

These notions satisfy a number of basic axioms, such as

There are six or seven other axioms besides these. All of these can be expressed in terms of the relation {\displaystyle \downarrow }. The formal manipulation of {\displaystyle \downarrow } using these axioms is sometimes called forking calculus. The forking calculus actually characterizes stable theories, in the sense that if {\displaystyle T} is a complete theory for which there is a ternary relation {\displaystyle \downarrow ^{*}} satisfying the axioms of forking independence in a stable theory, then {\displaystyle T} is stable and {\displaystyle \downarrow ^{*}} is forking independence.

The ternary relation {\displaystyle \downarrow } is the basis for other fundamental definitions in stability theory, such as Lascar rank, stationarity, independence, weight, domination, orthogonality, and regularity.

Ranks in stable theories[]

Several notions of β€œrank” or β€œdimension” play an important role in stable theories (particularly superstable theories). Each kind of rank assigns an ordinal number to a partial type {\displaystyle \Sigma (x)}. This number should be thought of as the number of degrees of freedom which the variable {\displaystyle x} has, after being constrained by the statements in {\displaystyle \Sigma (x)}. If {\displaystyle \Sigma (x)} is a finite type, i.e., a formula {\displaystyle \phi (x)}, then the rank of {\displaystyle \phi (x)} can be thought of as the β€œdimension” of the definable set {\displaystyle \phi (\mathbb {U} )}. On the other hand, the rank of a complete type {\displaystyle \operatorname {tp} (a/B)} can be thought of as a generalization of {\displaystyle tr.deg(a/B)}.

The three most commonly used ranks in stability theory are Morley rank {\displaystyle RM(-)}, Lascar rank {\displaystyle U(-)}, and Shelah -rank {\displaystyle R^{\infty }(-)}. Each of these ranks is ordinal valued, but can take the failure value {\displaystyle \infty }, which is thought of as being greater than any ordinal number. The ranks are related as follows: {\displaystyle U(\Sigma )\leq R^{\infty }(\Sigma )\leq RM(\Sigma )} for any partial type {\displaystyle \Sigma }. The totally transcendental theories are exactly the stable theories for which Morley rank is never {\displaystyle \infty }. The superstable theories are exactly the stable theories for which Lascar rank is never {\displaystyle \infty }, and are also exactly the stable theories for which {\displaystyle R^{\infty }(-)<\infty }. (Superstable theories can also be characterized as the theories which are {\displaystyle \kappa }-stable for all sufficiently large {\displaystyle \kappa }.) In practice, many of the mathematically interesting examples of stable theories are superstable; but SCF is a notable counter-example.

If {\displaystyle R(-)} is one of these three ranks, it is customary to write {\displaystyle R(a/B)} for {\displaystyle R(\operatorname {tp} (a/B))}. If {\displaystyle D} is definable (or type-definable), it is customary to write {\displaystyle R(D)} for the rank of the partial type cutting out {\displaystyle D}.

A number of fundamental properties are shared by these ranks. Letting {\displaystyle R(-)} denote one of Morley rank, Lascar rank, or Shelah {\displaystyle \infty }-rank,

Each of these ranks governs forking, when less than {\displaystyle \infty }. That is, if {\displaystyle R(a/BC)<\infty }, then {\displaystyle a\downarrow _{C}B\iff R(a/BC)=R(a/C),} {\displaystyle a\not \downarrow _{C}B\iff R(a/BC)<R(a/C)} In other words, the non-forking extensions of a complete type are exactly the extensions having the same rank.

Morley rank and Shelah {\displaystyle \infty }-rank both possess the following continuity property: {\displaystyle R(\Sigma (x))} is the minimum of {\displaystyle R(\Sigma _{0}(x))}, for {\displaystyle \Sigma _{0}(x)} a finite subtype of {\displaystyle \Sigma (x)}. For a complete type {\displaystyle \operatorname {tp} (a/B)}, this means that given {\displaystyle a} and {\displaystyle B}, {\displaystyle R(a/B)=R(\phi (x))} for some {\displaystyle \phi (x)\in \operatorname {tp} (a/B)}. In particular, all types in a neighborhood of {\displaystyle \operatorname {tp} (a/B)} have rank no greater than {\displaystyle R(a/B)}. This means that the rank function on {\displaystyle S_{n}(B)} is semi-continuous.

Lascar rank fails to have this property in general. However, it satisfies the extremely useful Lascar inequality: {\displaystyle U(a/Cb)+U(b/C)\leq U(ab/C)\leq U(a/Cb)\oplus U(b/C).} Here {\displaystyle +} denotes the usual ordinal sum, while {\displaystyle \oplus } denotes the so-called β€œnatural sum.” For finite numbers, these agree, making the Lascar inequality into an equality. There is also an adjunct statement to the Lascar inequality, namely {\displaystyle U(ab/C)=U(a/C)\oplus U(b/C)\Leftarrow a\downarrow _{C}b.} From this, one can show that if {\displaystyle D_{1}} and {\displaystyle D_{2}} are non-empty definable sets, then {\displaystyle U(D_{1}\times D_{2})=U(D_{1})\oplus U(D_{2})}.

Stable Groups[]

Definable groups in stable structures have a number of nice properties, including the following:

In the case of groups of finite Morley rank, many structural results are known, using tools such as Zilber’s indecomposability theorem. For example, it is known that if {\displaystyle G} is a group of finite Morley rank, then {\displaystyle G} is simple if and only if {\displaystyle G} is definably simple. That is, if {\displaystyle G} has a normal subgroup, then {\displaystyle G} has a definable normal subgroup. Additionally, Morley rank in {\displaystyle G} agrees with Lascar rank and is definable. Conjecturally, all simple groups of finite Morley rank are (as abstract groups) algebraic groups over algebraically closed fields.

Groups arise naturally in stability theory in several ways. In the analysis of uncountably categorical theories, or more generally when studying internality, binding groups appear and govern some of the structure. For example, in Poizat’s book on stable groups, there is a theorem saying that if {\displaystyle T} is uncountably categorical and has no definable groups, then {\displaystyle T} is almost strongly minimal.

Groups also arise from the group configuration theorem, which constructs a group from a certain configuration occurring in the pregeometry of a strongly minimal set (or more generally, a rank 1 type).

One-based groups also play an important role in the model-theoretic proofs of function field Mordell-Lang.

Stability Spectra[]

The set of {\displaystyle \kappa } such that {\displaystyle T} is {\displaystyle \kappa }-stable is called the stability spectrum of {\displaystyle T}. In Poizat’s introductory Model Theory book (in chapter 13{\displaystyle \pm }1?), the possible stability spectra are classified. It turns out that the stability spectrum of {\displaystyle T} depends on {\displaystyle \lambda }, the smallest cardinal such that {\displaystyle T} is {\displaystyle \lambda }-stable, and {\displaystyle \kappa }, the smallest cardinal {\displaystyle \kappa } which works in the local character property of forking. (That is, {\displaystyle \kappa } is the smallest cardinal with the property that if {\displaystyle a} is a finite tuple and {\displaystyle C} is a set, then there is some {\displaystyle C'\subseteq C} with {\displaystyle |C'|<\kappa } such that {\displaystyle a\downarrow _{C'}C}.) Somebody should add the correct statement of the theorem here.

For countable theories, it turns out that there are three possibilities:

Stability and Classification Theory[]

Stability theory was originally motivated by things such as Morley’s Theorem, and more generally, classification theory. The goal of this subject is to classify theories {\displaystyle T} according to their spectra, that is, according to how many models they have of each uncountable cardinality. A basic result in this direction is that if a countable theory {\displaystyle T} is not stable, then for any uncountable {\displaystyle \kappa }, {\displaystyle T} has {\displaystyle 2^{\kappa }} distinct models up to isomorphism, which is as large as possible. Such theories are thought of as "unclassifiable," because there is no hope of classifying their models. More generally, Shelah proved that if {\displaystyle T} is not superstable, then {\displaystyle T} is unclassifiable. The focus of classification theory is therefore on superstable theories. (Can someone verify the claims made in this paragraph?)

Equivalent Definitions of Stability[]

There are many equivalent definitions of stability, some of which are easier to check than {\displaystyle \kappa }-stability in various settings.

All of the following are equivalent, for a complete theory {\displaystyle T}:

Many of these properties can be checked on the level of 1-types. Specifically, the following conditions are also equivalent to stability:

These conditions are sometimes easier to check. For example, if {\displaystyle T} is a strongly minimal theory, it is easy to check that no formula {\displaystyle \phi (x;y)} with {\displaystyle x} a singleton has the order property. (If {\displaystyle \phi (x;y)} has the order property, then by compactness one can find {\displaystyle a_{i},b_{i}} for {\displaystyle i\in \mathbb {Z} } such that {\displaystyle \models \phi (a_{i};b_{j})} holds if and only if {\displaystyle i<j}. Then {\displaystyle \phi (\mathbb {U} ;b_{0})} contains {\displaystyle a_{-1},a_{-2},\ldots } and doesn’t contain {\displaystyle a_{1},a_{2},\ldots }. Therefore it is neither finite nor cofinite, contradicting strong minimality.)

If {\displaystyle T} is a stable theory, so is {\displaystyle T^{eq}}. Any definitional expansion of {\displaystyle T} is stable. Any reduct of a stable theory is a stable theory. We can name any number of elements from {\displaystyle \mathbb {U} } without losing stability.

Miscellaneous Facts about Stable Theories[]

Other noteworthy facts about stable theories are the following: