Stable theories are rooted in the proof of Morley's categoricity theorem and were extensively studied as part of Saharon Shelah's classification theory, which showed a dichotomy that either the models of a theory admit a nice classification or the models are too numerous to have any hope of a reasonable classification.
A first step of this program was showing that if a theory is not stable then its models are too numerous to classify.
A common goal in model theory is to study a first-order theory by analyzing the complexity of the Boolean algebras of (parameter) definable sets in its models.
One can equivalently analyze the complexity of the Stone duals of these Boolean algebras, which are type spaces.
Since types represent the possible behaviors of elements in a theory's models, restricting the number of types restricts the complexity of these models.
In this proof, the key notion was that of a totally transcendental theory, defined by restricting the topological complexity of the type spaces.
The main goal of this program was to show a dichotomy that either the models of a first-order theory can be nicely classified up to isomorphism using a tree of cardinal-invariants (generalizing, for example, the classification of vector spaces over a fixed field by their dimension), or are so complicated that no reasonable classification is possible.
[a] Shelah's approach was to identify a series of "dividing lines" for theories.
Stability was the first such dividing line in the classification theory program, and since its failure was shown to rule out any reasonable classification, all further work could assume the theory to be stable.
These local dimensions then give rise to the cardinal-invariants classifying models up to isomorphism.
-stable are described by the stability spectrum,[7] which singles out the even tamer subset of superstable theories.
A common alternate definition of stable theories is that they do not have the order property.
As with Morley's totally transcendental theories, the cardinality restrictions of stability are equivalent to bounding the topological complexity of type spaces in terms of Cantor-Bendixson rank.
[12] Another characterization is via the properties that non-forking independence has in stable theories, such as being symmetric.
Geometric stability theory is concerned with the fine analysis of local geometries in models and how their properties influence global structure.
This line of results was later key in various applications of stability theory, for example to Diophantine geometry.
It is usually taken to start in the late 1970s with Boris Zilber's analysis of totally categorical theories, eventually showing that they are not finitely axiomatizble.
Every model of a totally categorical theory is controlled by (i.e. is prime and minimal over) a strongly minimal set, which carries a matroid structure[d] determined by (model-theoretic) algebraic closure that gives notions of independence and dimension.
[24] The second question is answered by Zilber's Ladder Theorem, showing every model of a totally categorical theory is built up by a finite sequence of something like "definable fiber bundles" over the strongly minimal set.
[25] For the first question, Zilber's Trichotomy Conjecture was that the geometry of a strongly minimal set must be either like that of a set with no structure, or the set must essentially carry the structure of a vector space, or the structure of an algebraically closed field, with the first two cases called locally modular.
First, that (local) modularity serves to divide combinatorial or linear behavior from nonlinear, geometric complexity as in algebraic geometry.
[27] Second, that complicated combinatorial geometry necessarily comes from algebraic objects;[28] this is akin to the classical problem of finding a coordinate ring for an abstract projective plane defined by incidences, and further examples are the group configuration theorems showing certain combinatorial dependencies among elements must arise from multiplication in a definable group.
[29] By developing analogues of parts of algebraic geometry in strongly minimal sets, such as intersection theory, Zilber proved a weak form of the Trichotomy Conjecture for uncountably categorical theories.
[30] Although Ehud Hrushovski developed the Hrushovski construction to disprove the full conjecture, it was later proved with additional hypotheses in the setting of "Zariski geometries".
[31] Notions from Shelah's classification program, such as regular types, forking, and orthogonality, allowed these ideas to be carried to greater generality, especially in superstable theories.
Here, sets defined by regular types play the role of strongly minimal sets, with their local geometry determined by forking dependence rather than algebraic dependence.
In place of the single strongly minimal set controlling models of a totally categorical theory, there may be many such local geometries defined by regular types, and orthogonality describes when these types have no interaction.
For about twenty years after its introduction, stability was the main subject of pure model theory.
[44] Two notable examples of such broader classes are simple and NIP theories.
[45] In particular, simple theories can be characterized by non-forking independence being symmetric,[46] while NIP can be characterized by bounding the number of types realized over either finite[47] or infinite[48] sets.