Stable and unstable theories

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

Let denote the cardinality of the set of formulas of signature . If is stable, then it is stable at all cardinalities that satisfy the equality . If is stable, then there exist a model of and an infinite set such that for any formula of signature and for any two sequences , of different elements of , the truth of in is equivalent to the truth of in ; the set is then called the set of indistinguishable elements in . A characteristic property of unstable theories is the existence of a set which has somehow opposite properties. Namely, the instability of a theory is equivalent to the existence of a formula of signature , of a model of and of a sequence of tuples of elements of , such that the truth of in is equivalent to the inequality . 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 is unstable, then the number of isomorphism types of models of at every uncountable cardinal number is equal to . A theory that is categorical at an uncountable cardinal number (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 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 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 above is totally transcendental. Totally transcendental theories can also be characterized in other terms. Let be a complete theory of finite or countable signature and let be an infinite model of . A formula of signature is given the rank if it is false on all elements of the model , and the rank ( is an ordinal number) if it does not have any rank lower than ; however, for every elementary extension of the system , and for every formula of signature , one of the formulas or is given a rank less than . A theory is totally transcendental if and only if for every model of , each formula of signature is given a certain rank.

References

 [1] S. Shelah, "Stability, the f.c.p., and superstability; model theoretic properties of formulas in first order theory" Ann. of Math. Logic , 3 : 3 (1971) pp. 271–362 [2] S. Shelah, "Classification theory and the number of non-isomorphic models" , North-Holland (1990)