stability in model theory



Stability theory, also referred to a classification theory, is a means to determine whether the isomorphism types of a given sort of structure can be classified by means of intelligible invariants of the structure. It was largely created by Saharon Shelah.

The basic idea is that for quite general classes of algebraic objects, one can prove what Shelah calls a “structure/nonstructure theorem”: either the isomorphism types are classifiable by a smallish number of invariants, or they are hopelessly wild in some sense, e.g., an arbitrary structure can be encoded set-theoretically in some isomorphism type of the class. An example of the “structure” case is the theory of algebraically closed fields, whose isomorphism types can be classified according to characteristic and transcendence degree. An example of the “nonstructure” case is the family of linear orderings, where a proliferation of complicated linear orders can be constructed by various set-theoretic means.

In very rough outline, stability theory analyzes good (or “stable”) notions of “free amalgams” M 3=M 1 M 0M 2M_3 = M_1 \cup_{M_0} M_2 where M 0M 1M_0 \subset M_1, M 0M 2M_0 \subset M_2 are substructures. In the “good” (structure) case, it is possible to analyze models by a series of free amalgams of small models, with the series indexed by a well-founded tree. Otherwise, if the class of algebraic objects does not admit a suitably good notion of free amalgam, we have a “bad” (nonstructure) case which permits arbitrarily wild models to be constructed.


(For the time being we are recording some definitions without any effort to motivate them. That should come later.)

For now we will be interested in complete theories T\mathbf{T} over a countable signature. Let M\mathbf{M} be a model of T\mathbf{T} with underlying set MM. Let AMA \subseteq M be a subset, and let Def A(M n)Def_A(M^n) be the Boolean algebra of subsets of M nM^n that are definable by a formula in T\mathbf{T} with parameters in AA.

Recall that an ultrafilter in a Boolean algebra BB is the set of elements that get mapped to the top element 11 under some Boolean algebra homomorphism ϕ:B2\phi: B \to \mathbf{2}; alternatively, we could define an ultrafilter as such a homomorphism.


The space of complete nn-types over AA is the Stone space S n M(A)S_n^{\mathbf{M}}(A) of ultrafilters in the Boolean algebra Def A(M n)Def_A(M^n). Such an ultrafilter is called a complete nn-type. More generally, an nn-type is a filter in Def A(M n)Def_A(M^n).


Suppose i:MNi: M \to N is an elementary embedding from M\mathbf{M} to N\mathbf{N}. Then ii induces an isomorphism S n M(A)S n N(i(A))S_n^{\mathbf{M}}(A) \cong S_n^{\mathbf{N}}(i(A)).


It is enough to show ii induces an isomorphism Def A(M n)Def A(N n)Def_A(M^n) \to Def_A(N^n). As a Boolean algebra, Def A(M n)Def_A(M^n) is the Boolean quotient of formulas with nn free variables ϕ(x¯,a¯)\phi(\bar{x}, \bar{a}) modulo the equivalence relation E(ϕ,ψ)ϕψE(\phi, \psi) \coloneqq \phi \Leftrightarrow \psi is satisfied in M\mathbf{M}. By elementary equivalence, ME(ϕ,ψ)(x¯,a¯)\mathbf{M} \models E(\phi, \psi)(\bar{x}, \bar{a}) iff NE(ϕ,ψ)(x¯,i(a¯))\mathbf{N} \models E(\phi, \psi)(\bar{x}, i(\bar{a})), as desired.


A complete type is realized by a point a¯M n\bar{a} \in M^n if it is of the form

Def A(M n)P(M n)=2 M neval a¯2Def_A(M^n) \hookrightarrow P(M^n) = \mathbf{2}^{M^n} \stackrel{eval_{\bar{a}}}{\to} \mathbf{2}

by restricting a principal ultrafilter generated by a point a¯M n\bar{a} \in M^n. Such a type is denoted tp(a¯/A)tp(\bar{a}/A).


For an infinite cardinal κ\kappa, a model M\mathbf{M} is called κ\kappa-saturated if for every AMA \subseteq M with |A|<κ{|A|} \lt \kappa, any complete nn-type is realized in M\mathbf{M}.


For an infinite cardinal κ\kappa, the theory T\mathbf{T} is κ\kappa-stable if for every model MT\mathbf{M} \models \mathbf{T} and AMA \subseteq M with |A|=κ{|A|} = \kappa, we have |S n M(A)|=κ{|S_n^{\mathbf{M}}(A)|} = \kappa. A structure M\mathbf{M} of a countable language is called κ\kappa-stable if the complete theory Th(M)Th(\mathbf{M}) is κ\kappa-stable.