 Stable theory

For differential equations see Stability theory.
In model theory, a complete theory is called stable if it does not have too many types. One goal of classification theory is to divide all complete theories into those whose models can be classified and those whose models are too complicated to classify, and to classify all models in the cases where this can be done. Roughly speaking, if a theory is not stable then its models are too complicated and numerous to classify, while if a theory is stable there might be some hope of classifying its models, especially if the theory is superstable or totally transcendental.
Stability theory was started by Morley (1965), who introduced several of the fundamental concepts, such as totally transcendental theories and the Morley rank. Stable and superstable theories were first introduced by Shelah (1969), who is responsible for much of the development of stability theory. The definitive reference for stability theory is (Shelah 1990), though it is notoriously hard even for experts to read.
Contents
Definitions
T will be a complete theory in some language.
 T is called κstable (for an infinite cardinal κ) if for every set A of cardinality κ the number of complete types over A has cardinality κ.
 ωstable is an alternative name for ℵ_{0}stable.
 T is called stable if it is κstable for some infinite cardinal κ
 T is called unstable if it is not stable.
 T is called superstable if it is κstable for all sufficiently large cardinals κ.
 Totally transcendental theories are those such that every formula has Morley rank less than ∞.
As usual, a model of some language is said to have one of these properties if the complete theory of the model has that property.
An incomplete theory is defined to have one of these properties if every completion, or equivalently every model, has this property.
Unstable theories
Roughly speaking, a theory is unstable if one can use it to encode the ordered set of natural numbers. More precisely, if there is a model M and a formula Φ(X,Y) in 2n variables X=x_{1},...,x_{n} and Y=y_{1},...,y_{n} defining a relation on M^{n} with an infinite totally ordered subset then the theory is unstable. (Any infinite totally ordered set has a subset isomorphic to either the positive or negative integers under the usual order, so one can assume the totally ordered subset is ordered like the positive integers.) The totally ordered subset need not be definable in the theory.
The number of models of an unstable theory T of any uncountable cardinality κ≥T is the maximum possible number 2^{κ}.
Examples:
 Most sufficiently complicated theories, such as set theories and Peano arithmetic, are unstable.
 The theory of the rational numbers, considered as an ordered set, is unstable. Its theory is the theory of dense linear orders without endpoints.
 The theory of addition of the natural numbers is unstable.
 Any infinite Boolean algebra is unstable.
 Any monoid with cancellation that is not a group is unstable, because if a is an element that is not a unit then the powers of a form an infinite totally ordered set under the relation of divisibility. For a similar reason any integral domain that is not a field is unstable.
 There are many unstable nilpotent groups. One example is the infinite dimensional Heisenberg group over the integers: this is generated by elements x_{i}, y_{i}, z for all natural numbers i, with the relations that any of these two generators commute except that x_{i} and y_{i} have commutator z for any i. If a_{i} is the element x_{0}x_{1}...x_{i−1}y_{i} then a_{i} and a_{j} have commutator z exactly when i<j, so they form an infinite total order under a definable relation, so the group is unstable.
 Real closed fields are unstable, as they are infinite and have a definable total order.
Stable theories
T is called stable if it is κstable for some cardinal κ. Examples:
 The theory of any module over a ring is stable.
 The theory of a countable number of equivalence relations E_{n} for n a natural number such that each equivalence relation has an infinite number of equivalence classes and each equivalence class of E_{n} is the union of an infinite number of different classes of E_{n+1} is stable but not superstable.
 Sela (2006) showed that free groups, and more generally torsion free hyperbolic groups, are stable. Free groups on more than one generator are not superstable.
 A differentially closed field is stable. If it has nonzero characteristic it is not superstable, and if it has zero characteristic it is totally transcendental.
Superstable theories
T is called superstable if it is stable for all sufficiently large cardinals, so all superstable theories are stable. For countable T superstability is equivalent to stability for all κ≥2^{ω}. The following conditions on a theory T are equivalent:
 T is superstable.
 All types of T are ranked by at least one notion of rank.
 T is κstable for all sufficiently large cardinals κ
 T is κstable for all cardinals κ that are at least 2^{T}.
If a theory is superstable but not totally transcendental it is called strictly superstable.
The number of countable models of a countable superstable theory must be 1, ℵ_{0}, ℵ_{1}, or 2^{ω}. If the number of models is 1 the theory is totally transcendental. There are examples with 1, ℵ_{0} or 2^{ω} models, and it is not known if there are examples with ℵ_{1} models if the continuum hypothesis does not hold. If a theory T is not superstable then the number of models of cardinality κ>T is 2^{κ}.
Examples:
 The additive group of integers is superstable, but not totally transcendental. It has 2^{ω} countable models.
 The theory with a countable number of unary relations P_{i} with model the positive integers where P_{i}(n) is interpreted as saying n is divisible by the nth prime is superstable but not totally transcendental.
 An abelian group A is superstable if and only if there are only finitely many pairs (p,n) with p prime, n a natural number, with p^{n}A/p^{n+1}A infinite.
Totally transcendental theories and ωstable
 Totally transcendental theories are those such that every formula has Morley rank less than ∞. Totally transcendental theories are stable in λ whenever λ≥T, so they are always superstable. ωstable is an alternative name for ℵ_{0}stable. ωstable theories in a countable language are κstable for all infinite cardinals κ. If T is countable then T is totally transcendental if and only if it is ωstable. More generally, T is totally transcendental if and only if every restriction of T to a countable language is ωstable.
Examples:
 Any ωstable theory is totally transcendental.
 Any finite model is totally transcendental.
 An infinite field is totally transcendental if and only if it is algebraically closed. (Macintyre's theorem.)
 A differentially closed field in characteristic 0 is totally transcendental.
 Any theory with a countable language that is categorical for some uncountable cardinal is totally transcendental.
 An abelian group is totally transcendental if and only if it is the direct sum of a divisible group and a group of bounded exponent.
 Any linear algebraic group over an algebraically closed field is totally transcendental.
 Any group of finite Morley rank is totally transcendental.
See also
References
 J.T. Baldwin, "Fundamentals of stability theory" , Springer (1988)
 Baldwin, J. T. (2001), "Stability theory (in logic)", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 9781556080104, http://eom.springer.de/s/s087080.htm
 Buechler, Steven (1996), Essential stability theory, Perspectives in Mathematical Logic, Berlin: SpringerVerlag, pp. xiv+355, ISBN 3540610111, MR1416106
 Hodges, Wilfrid (1993), Model theory, Cambridge University Press, ISBN 9780521304429
 D. Lascar, "Stability in model theory" , Wiley (1987)
 Marker, David (2002), Model Theory: An Introduction, Graduate Texts in Mathematics, Berlin, New York: SpringerVerlag, ISBN 9780387987606
 Morley, Michael (1965), "Categoricity in Power", Transactions of the American Mathematical Society (American Mathematical Society) 114 (2): 514–538, doi:10.2307/1994188, JSTOR 1994188
 Palyutin, E.A.; Taitslin, M.A. (2001), "Stable and unstable theories", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 9781556080104, http://eom.springer.de/S/s087100.htm
 A. Pillay, "An introduction to stability theory" , Clarendon Press (1983)
 Poizat, Bruno (2001), Stable groups, Mathematical Surveys and Monographs, 87, Providence, RI: American Mathematical Society, pp. xiv+129, ISBN 0821826859, MR1827833 (Translated from the 1987 French original.)
 Scanlon, Thomas (2002), "Review of "Stable groups"", Bull. Amer. Math. Soc. 39 (04): 573–579, doi:10.1090/S0273097902009539
 Sela, Z (2006). "Diophantine Geometry over Groups VIII: Stability". arXiv:math/0609096 [math.GR].
 Shelah, S. (1969), "Stable theories", Israel J. Math. 7 (3): 187–202, doi:10.1007/BF02787611, MR0253889
 Shelah, Saharon (1990) [1978], Classification theory and the number of nonisomorphic models, Studies in Logic and the Foundations of Mathematics (2nd ed.), Elsevier, ISBN 9780444702609
External links
 A. Pillay, Lecture notes on model theory
 A. Pillay, Lecture notes on stability theory
 A. Pillay, Lecture notes on applied stability theory
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