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