Stability spectrum

Stability spectrum

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 Stone space of every model of "T" of size ≤ λ has itself size ≤ λ. "T" is called a stable theory if there is no upper bound for the cardinals κ such that "T" is stable in κ. The stability spectrum of "T" is the class of all cardinals κ such that "T" is stable in κ.

For countable theories there are only four possible stability spectra. The corresponding dividing lines are those for total transcendentality, superstability and stability. This result is due to Saharon Shelah, who also defined stability and superstability.

The stability spectrum theorem for countable theories

Theorem.Every countable complete first-order theory "T" falls into one of the following classes:
* "T" is stable in λ for all infinite cardinals λ. – "T" is totally transcendental.
* "T" is stable in λ exactly for all cardinals λ with λ ≥ 2ω. – "T" is superstable but not totally transcendental.
* "T" is stable in λ exactly for all cardinals λ that satisfy λ = λω. – "T" is stable but not superstable.
* "T" is not stable in any infinite cardinal λ. – "T" is unstable.

The condition on λ in the third case holds for cardinals of the form λ = κω, but not for cardinals λ of cofinality ω (because λ < λcof λ).

Totally transcendental theories

A complete first-order theory "T" is called totally transcendental if every formula has bounded Morley rank, i.e. if RM(φ) < ∞ for every formula φ("x") with parameters in a model of "T", where "x" may be a tuple of variables. It is sufficient to check that RM("x"="x") < ∞, where "x" is a single variable.

For countable theories total transcendence is equivalent to stability in ω, and therefore countable totally transcendental theories are often called ω-stable for brevity. A totally transcendental theory is stable in every λ ≥ |"T"|, hence a countable ω-stable theory is stable in all infinite cardinals.

Every uncountably categorical countable theory is totally transcendental. This includes complete theories of vector spaces or algebraically closed fields. The theories of groups of finite Morley rank are another important example of totally transcendental theories.

Superstable theories

A complete first-order theory "T" is superstable if there is a rank function on complete types that has essentially the same properties as Morley rank in a totally transcendental theory. Every totally transcendental theory is superstable. A theory "T" is superstable if and only if it is stable in all cardinals λ &ge; 2|"T"|.

Stable theories

A theory that is stable in one cardinal λ ≥ |"T"| is stable in all cardinals λ that satisfy λ = λ|"T"|. Therefore a theory is stable if and only if it is stable in some cardinal λ ≥ |"T"|.

Unstable theories

Most mathematically interesting theories fall into this category, including complicated theories such as any complete extension of ZF set theory, and relatively tame theories such as the theory of real closed fields. This shows that the stability spectrum is a relatively blunt tool. To get somewhat finer results one can look at the exact cardinalities of the Stone spaces over models of size ≤ λ, rather than just asking whether they are at most λ.

The uncountable case

For a general stable theory "T" in a possibly uncountable language, the stability spectrum is determined by two cardinals &kappa; and &lambda;0, such that "T" is stable in &lambda; exactly when &lambda; &ge; &lambda;0 and &lambda;&mu; = &lambda; for all &mu;<&kappa;. So &lambda;0 is the smallest infinite cardinal for which "T" is stable. These invariants satisfy the inequalities
*&kappa; &le; |"T"|+
*&kappa; &le; &lambda;0
*&lambda;0 &le; 2|"T"|
*If &lambda;0 > |"T"|, then &lambda;0 &ge; 2&omega;

When |"T"| is countable the 4 possibilities for its stability spectrum correspond to the following values of these cardinals:
*&kappa; and &lambda;0 are not defined: "T" is unstable.
*&lambda;0 is 2&omega;, &kappa; is &omega;1: "T" is stable but not superstable
*&lambda;0 is 2&omega;, &kappa; is &omega;: "T" is superstable but not &omega;-stable.
* &lambda;0 is &omega;, &kappa; is &omega;: "T" is totally transcendental (or &omega;-stable)

See also

* Spectrum of a theory


last=Poizat|first= Bruno
title=A course in model theory. An introduction to contemporary mathematical logic|series= Universitext|publisher=Springer|place= New York|year= 2000|pages=xxxii+443 |isbn= 0-387-98655-3
Translated from the French
*Citation | last1=Shelah | first1=Saharon | author1-link=Saharon Shelah | title=Classification theory and the number of nonisomorphic models | origyear=1978 | publisher=Elsevier | edition=2nd | series=Studies in Logic and the Foundations of Mathematics | isbn=978-0-444-70260-9 | year=1990

Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Stability radius — The stability radius of a continuous function f (in a functional space F ) with respect to an open stability domain D is the distance between f and the set of unstable functions (with respect to D ). We say that a function is stable with respect… …   Wikipedia

  • Spectrum Saloon Car (SSC) — The Spectrum Saloon Car (SSC), or Spectrum Patrol Car (SPC), is a fictional vehicle from Gerry Anderson s Science fiction television series Captain Scarlet and the Mysterons (1967). Appearance in Captain Scarlet and the Mysterons Accessible only… …   Wikipedia

  • Full Spectrum Warrior — North American cover Developer(s) Pandemic Studios Publisher(s) THQ …   Wikipedia

  • Political spectrum — A political spectrum (plural spectra) is a way of modeling different political positions by placing them upon one or more geometric axes symbolizing independent political dimensions.Most long standing spectra include a right wing and left wing,… …   Wikipedia

  • Emotional spectrum — The emotional spectrum is a fictional concept in the DC Comics shared universe, primarily within the Green Lantern mythos. Within the comics, it is the power source for the power rings of the Green Lantern Corps. The spectrum is divided into the… …   Wikipedia

  • USAF Stability and Control DATCOM — The United States Air Force Stability and Control DATCOM is a collection, correlation, codification, and recording of best knowledge, opinion, and judgment in the area of aerodynamic stability and control prediction methods. It presents… …   Wikipedia

  • List of mathematics articles (S) — NOTOC S S duality S matrix S plane S transform S unit S.O.S. Mathematics SA subgroup Saccheri quadrilateral Sacks spiral Sacred geometry Saddle node bifurcation Saddle point Saddle surface Sadleirian Professor of Pure Mathematics Safe prime Safe… …   Wikipedia

  • 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… …   Wikipedia

  • List of mathematical logic topics — Clicking on related changes shows a list of most recent edits of articles to which this page links. This page links to itself in order that recent changes to this page will also be included in related changes. This is a list of mathematical logic …   Wikipedia

  • Mathematics and Physical Sciences — ▪ 2003 Introduction Mathematics       Mathematics in 2002 was marked by two discoveries in number theory. The first may have practical implications; the second satisfied a 150 year old curiosity.       Computer scientist Manindra Agrawal of the… …   Universalium