For differential equations see Stability theory
. In Mathematics, stability theory deals with the stability of solutions (or sets of solutions for Differential equations and Dynamical systems
In model theory, a complete theory is called stable if it does not have too many types. In Mathematics, model theory is the study of (classes of mathematical structures such as groups, Fields graphs or even models 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.
T will be a complete theory in some language.
- T is called κ-stable (for an infinite cardinal κ) if for every set A of cardinality κ the number of complete types over A has cardinality κ. In Model theory, a type is a set of first-order formulas in a language L with free variables x_1x_2\ldotsx_n which are true of a sequence of
- ω-stable is an alternative name for ℵ0-stable.
- T is called stable if it is κ-stable for some cardinal κ
- T is called unstable if it is not stable.
- T is called superstable if it is stable for all sufficiently large cardinals.
- Totally transcendental theories are those such that every formula has Morley rank less than ∞. In Mathematical logic, Morley rank, introduced by, is a means of measuring the size of a subset of a model of a theory generalizing the notion of dimension in
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.
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=x1,. . . ,xn and Y=y1,. . . ,yn defining a relation on Mn 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 2κ.
- 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. In Mathematical logic, a first-order theory is given by a set of axioms in somelanguage
- 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 plently of unstable nilpotent groups. One example is the infinite dimensional Heisenberg group over the integers: this is generated by elements xi, yi, z for all natual numbers i, with the relations that any of these two generators commute except that xi and yi have commutator z for any i. In Mathematics, the term Heisenberg group, named after Werner Heisenberg, refers to the group of 3×3 upper triangular matrices of the If ai is the element x0x1. . . xi−1yi then ai and aj 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. In Mathematics, a real closed field is a field F in which any of the following equivalent conditions are true There is a Total order
T is called stable if it is κ-stable for some cardinal κ. Examples:
- The theory of any module over a ring is stable.
- The theory of a countable number of equivalence relations En for n a natural number such that each equivalence relation has an infinite number of equivalence classes and each equivalence class of En is the union of an infinite number of different classes of En+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. In Mathematics, a Differential field K is differentially closed if every finite system of Differential equations with a solution in some differential If it has non-zero characteristic it is not superstable, and if it has zero characteristic it is totally transcendental.
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≥2ω. 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 κ-stable for all sufficiently large cardinals κ
- T is κ-stable 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, ℵ0, ℵ1, or 2ω. If the number of models is 1 the theory is totally transcendental. There are examples with 1, ℵ0 or 2ω models, and it is not known if there are examples with ℵ1 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 2κ.
- The additive group of integers is superstable, but not totally transcendental. It has 2ω countable models.
- The theory with a countable number of unary relations Pi with model the positive integers where Pi(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 pnA/pn+1A infinite.
Totally transcendental theories and ω-stable
- Totally transcendental theories are those such that every formula has Morley rank less than ∞. In Mathematical logic, Morley rank, introduced by, is a means of measuring the size of a subset of a model of a theory generalizing the notion of dimension in Totally transcendental theories are stable in λ whenever λ≥|T|, so they are always superstable. ω-stable is an alternative name for ℵ0-stable. ω-stable theories in a countable language are κ-stable for all infinite cardinals κ. If |T| is countable then T is totally transcendental if and only if it is ω-stable. More generally, T is totally transcendental if and only if every restriction of T to a countable language is ω-stable.
- Any ω-stable theory is totally transcendental.
- Any finite model is totally transcendental.
- An infinite field is totally transcendental if and only if it is algebraically closed. In Mathematics, a field F is said to be algebraically closed if every Polynomial in one Variable of degree at least 1 with Coefficients (Macintyre's theorem. )
- A differentially closed field in characteristic 0 is totally transcendental. In Mathematics, a Differential field K is differentially closed if every finite system of Differential equations with a solution in some differential
- Any theory with a countable language that is categorical for some uncountable cardinal is totally transcendental. In Model theory, a branch of Mathematical logic, a theory is &kappa- categorical (or categorical in &kappa) if it has exactly one model of
- 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. In Mathematics, a linear algebraic group is a Subgroup of the group of invertible n × n matrices (under Matrix multiplication
- Any group of finite Morley rank is totally transcendental. In Model theory, a stable group is a group that is stable in the sense of stability theory.
- J. In Model theory, a branch of Mathematical logic, the spectrum of a theory is given by the number of isomorphism classes of models in various cardinalities In Model theory, a branch of Mathematical logic, a theory is &kappa- categorical (or categorical in &kappa) if it has exactly one model of In Mathematical logic, a first-order theory is given by a set of axioms in somelanguage In Model theory, a branch of Mathematical logic, a complete first-order theory T is called stable in λ (an infinite Cardinal number) if the T. Baldwin, "Fundamentals of stability theory" , Springer (1988)
- Baldwin, J. T. (2001), “Stability theory (in logic)”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104
- Buechler, Steven (1996), Essential stability theory, Perspectives in Mathematical Logic, Berlin: Springer-Verlag, pp. The Encyclopaedia of Mathematics is a large reference work in Mathematics. xiv+355, MR1416106, ISBN 3-540-61011-1
- Hodges, Wilfrid (1993), Model theory, Cambridge University Press, ISBN 978-0-521-30442-9
- D. Mathematical Reviews is a journal and online database published by the American Mathematical Society that contains brief synopses (and occasionally evaluations of many Wilfrid Hodges (born 1941 is a British Mathematician, known for his work in Model theory. Cambridge University Press (known colloquially as CUP is a Publisher given a Royal Charter by Henry VIII in 1534 Lascar, "Stability in model theory" , Wiley (1987)
- Marker, David (2002), Model Theory: An Introduction, Graduate Texts in Mathematics, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98760-6
- Morley, Michael (1965), “Categoricity in Power”, Transactions of the American Mathematical Society 114 (2): 514–538, ISSN 0002-9947, DOI 10. Graduate Texts in Mathematics (GTM is a series of graduate-level Textbooks in Mathematics published by Springer-Verlag. Springer Science+Business Media or Springer (ˈʃpʁɪŋɐ is a worldwide Publishing company based in Germany, which publishes textbooks academic Michael Darwin Morley is an American Mathematician, currently professor emeritus at Cornell University. Transactions of the American Mathematical Society is a monthly mathematics journal published by the American Mathematical Society. An International Standard Serial Number ( ISSN) is a unique eight-digit number used to identify a print or electronic Periodical publication. 2307/1994188
- Palyutin, E. A. & Taitslin, M. A. (2001), “Stable and unstable theories”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104
- A. The Encyclopaedia of Mathematics is a large reference work in Mathematics. Pillay, "An introduction to stability theory" , Clarendon Press (1983)
- Poizat, Bruno (2001), Stable groups, vol. 87, Mathematical Surveys and Monographs, Providence, RI: American Mathematical Society, pp. xiv+129, MR1827833, ISBN 0-8218-2685-9 (Translated from the 1987 French original. Mathematical Reviews is a journal and online database published by the American Mathematical Society that contains brief synopses (and occasionally evaluations of many )
- Scanlon, Thomas (2002), “Review of "Stable groups"”, Bull. Amer. Math. Soc. 39: 573-579, <http://www.ams.org/bull/2002-39-04/S0273-0979-02-00953-9/>
- Sela, Z (2006), Diophantine Geometry over Groups VIII: Stability, <http://arxiv.org/abs/math/0609096>
- Shelah, S. (1969), “Stable theories.”, Israel J. Math. 7: 187-202, MR0253889, DOI 10. Mathematical Reviews is a journal and online database published by the American Mathematical Society that contains brief synopses (and occasionally evaluations of many 1007/BF02787611
- Shelah, Saharon (1990), Classification theory and the number of nonisomorphic models (2nd ed. Saharon Shelah (שהרן שלח born July 3, 1945 in Jerusalem) is an Israeli Mathematician. ), Studies in Logic and the Foundations of Mathematics, Elsevier, ISBN 978-0-444-70260-9
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