Stable and unstable theories

A branch of model theory dealing with the stability of elementary theories (cf. Elementary theory). Let $ T $ be a complete theory of the first order, of signature (language) $ \Omega $, let $ A $ be a model of $ T $ and let $ X \subseteq | A | $. The signature $ \langle \Omega , X\rangle $ is obtained from $ \Omega $ by adding isolated element symbols $ c _ {a} $ for all $ a \in X $. The system $ \langle A, X\rangle $ has signature $ \langle \Omega , X\rangle $ and is an enrichment (simple expansion) of the model $ A $, in which $ c _ {a} $ is interpreted as $ a $ for all $ a \in X $. The theory $ T( A, X) $ is the totality of formulas of signature $ \langle \Omega , X\rangle $ that are true in $ \langle A, X\rangle $. A set $ \tau ( x) $ of formulas $ \phi ( x) $ in the language $ \langle \Omega , X\rangle $ with one free variable $ x $ is a type of $ \langle A, X\rangle $ if $ \tau ( x) \cup T( A, X) $ is satisfiable. $ S( A, X) $ is the collection of all maximal types of $ \langle A , X\rangle $. The theory $ T $ is said to be stable at cardinality $ \lambda $ if for any model $ A $ of $ T $ and any $ X \subseteq | A | $ of cardinality not exceeding $ \lambda $, the cardinality of $ S( A, X) $ also does not exceed $ \lambda $. A theory is called stable if it is stable at even one infinite cardinality.

Let $ | T | $ denote the cardinality of the set of formulas of signature $ \Omega $. If $ T $ is stable, then it is stable at all cardinalities that satisfy the equality $ \lambda = \lambda ^ {| T | } $. If $ T $ is stable, then there exist a model $ A $ of $ T $ and an infinite set $ Y \subseteq | A | $ such that for any formula $ \phi ( v _ {1} \dots v _ {n} ) $ of signature $ \Omega $ and for any two sequences $ \langle a _ {1} \dots a _ {n} \rangle $, $ \langle b _ {1} \dots b _ {n} \rangle $ of different elements of $ Y $, the truth of $ \phi ( a _ {1} \dots a _ {n} ) $ in $ A $ is equivalent to the truth of $ \phi ( b _ {1} \dots b _ {n} ) $ in $ A $; the set $ Y $ is then called the set of indistinguishable elements in $ T $. A characteristic property of unstable theories is the existence of a set which has somehow opposite properties. Namely, the instability of a theory $ T $ is equivalent to the existence of a formula $ \phi ( v _ {1} \dots v _ {n} ; u _ {1} \dots u _ {n} ) $ of signature $ \Omega $, of a model $ A $ of $ T $ and of a sequence $ \langle a _ {1} ^ {0} \dots a _ {n} ^ {0} \rangle , \langle a _ {1} ^ {1} \dots a _ {n} ^ {1} \rangle \dots $ of tuples of elements of $ A $, such that the truth of $ \phi ( a _ {1} ^ {i} \dots a _ {n} ^ {i} ; a _ {1} ^ {j} \dots a _ {n} ^ {j} ) $ in $ A $ is equivalent to the inequality $ i < j $. For this reason, complete extensions of the theory of totally ordered sets with infinite models, as well as the theory of any infinite Boolean algebra, are unstable. In particular, the theory of natural numbers with addition and the theory of the field of real numbers are unstable. If a theory $ T $ is unstable, then the number of isomorphism types of models of $ T $ at every uncountable cardinal number $ \lambda > | T | $ is equal to $ 2 ^ \lambda $. A theory $ T $ that is categorical at an uncountable cardinal number $ \lambda > | T | $( cf. Categoricity in cardinality) is therefore stable. There do exist stable theories, however, that are not categorical at any infinite cardinality. Such an example is the theory $ T _ {1} $ whose signature consists of a one-place predicate and a countable set of isolated elements. The axioms of this theory state that a predicate is true on the isolated elements, divides every model of $ T _ {1} $ into two infinite sets, and that the isolated elements are not equal to each other.

Theories of finite or countable signature that are stable at a countable cardinality are also said to be totally transcendental. Every totally transcendental theory is stable at all infinite cardinalities. Every categorical theory of finite or countable signature at an uncountable cardinality is totally transcendental. The theory $ T _ {1} $ above is totally transcendental. Totally transcendental theories can also be characterized in other terms. Let $ T $ be a complete theory of finite or countable signature $ \Omega $ and let $ A $ be an infinite model of $ T $. A formula $ \phi ( v _ {0} ) $ of signature $ \langle \Omega , | A | \rangle $ is given the rank $ - 1 $ if it is false on all elements of the model $ \langle A, | A | \rangle $, and the rank $ \alpha $( $ \alpha $ is an ordinal number) if it does not have any rank lower than $ \alpha $; however, for every elementary extension $ B $ of the system $ A $, and for every formula $ \psi ( v _ {0} ) $ of signature $ \langle \Omega , | B | \rangle $, one of the formulas $ \psi ( v _ {0} ) \& \psi ( v _ {0} ) $ or $ \neg \psi ( v _ {0} ) \& \phi ( v _ {0} ) $ is given a rank less than $ \alpha $. A theory $ T $ is totally transcendental if and only if for every model $ A $ of $ T $, each formula $ \phi $ of signature $ \langle \Omega , | A | \rangle $ is given a certain rank.

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See also Stability theory (in logic).

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