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) |
Comments
See also Stability theory (in logic).
References
| [a1] | J.T. Baldwin, "Fundamentals of stability theory" , Springer (1988) |
| [a2] | D. Lascar, "Stability in model theory" , Wiley (1987) |
| [a3] | A. Pillay, "An introduction to stability theory" , Clarendon Press (1983) |
| [a4] | C.C. Chang, H.J. Keisler, "Model theory" , North-Holland (1990) |
Stable and unstable theories. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stable_and_unstable_theories&oldid=16041