Editing Model theory (section)

== Stability theory==
{{main|Stable theory}}

A key factor in the structure of the class of models of a first-order theory is its place in the ''stability hierarchy''. 
:A complete theory ''T'' is called ''<math>\lambda</math>-stable'' for a cardinal <math>\lambda</math> if for any model <math>\mathcal{M}</math> of ''T'' and any parameter set <math>A \subset \mathcal{M}</math> of cardinality not exceeding <math>\lambda</math>, there are at most <math>\lambda</math> complete ''T''-types over ''A''.
A theory is called ''stable'' if it is <math>\lambda</math>-stable for some infinite cardinal <math>\lambda</math>. Traditionally, theories that are <math>\aleph_0</math>-stable are called ''<math>\omega</math>-stable''.{{sfn|Marker|2002|p=135}}

===The stability hierarchy===
A fundamental result in stability theory is the ''[[Stability spectrum|stability spectrum theorem]]'',{{sfn|Marker|2002|p=172}} which implies that every complete theory ''T'' in a countable signature falls in one of the following classes:

# There are no cardinals <math>\lambda</math> such that ''T'' is <math>\lambda</math>-stable.
# ''T'' is <math>\lambda</math>-stable if and only if <math>\lambda^{\aleph_0} = \lambda</math>  (see [[Cardinal exponentiation]] for an explanation of <math>\lambda^{\aleph_0}</math>).
# ''T'' is <math>\lambda</math>-stable for any <math>\lambda \geq 2^{\aleph_0}</math> (where <math>2^{\aleph_0}</math> is the cardinality of the [[Continuum (set theory)|continuum]]).

A theory of the first type is called ''unstable'', a theory of the second type is called ''strictly stable'' and a theory of the third type is called ''superstable''.
Furthermore, if a theory is <math>\omega</math>-stable, it is stable in every infinite cardinal,{{sfn|Marker|2002|p=136}} so <math>\omega</math>-stability is stronger than superstability.

Many construction in model theory are easier when restricted to stable theories; for instance, every model of a stable theory has a saturated elementary extension, regardless of whether the generalised continuum hypothesis is true.{{sfn|Hodges|1993|p=494}}

Shelah's original motivation for studying stable theories was to decide how many models a countable theory has of any uncountable cardinality.<ref>{{Cite book|last=Saharon.|first=Shelah|url=http://worldcat.org/oclc/800472113|title=Classification theory and the number of non-isomorphic models|date=1990|publisher=North-Holland|isbn=0-444-70260-1|oclc=800472113}}</ref> 
If a theory is uncountably categorical, then it is <math>\omega</math>-stable. More generally, the ''[[Spectrum of a theory|Main gap theorem]]'' implies that if there is an uncountable cardinal <math>\lambda</math> such that a theory ''T'' has less than <math>2^{\lambda}</math> models of cardinality <math>\lambda</math>, then ''T'' is superstable.

===Geometric stability theory===
The stability hierarchy is also crucial for analysing the geometry of definable sets within a model of a theory.  
In <math>\omega</math>-stable theories, ''[[Morley rank]]'' is an important dimension notion for definable sets ''S'' within a model. It is defined by [[transfinite induction]]:

*The Morley rank is at least 0 if ''S'' is non-empty.
*For ''α'' a [[successor ordinal]], the Morley rank is at least ''α'' if in some [[elementary extension]] ''N'' of ''M'', the set ''S'' has infinitely many disjoint definable subsets, each of rank at least ''α''&nbsp;&minus;&nbsp;1.
*For ''α'' a non-zero [[limit ordinal]], the Morley rank is at least ''α'' if it is at least ''β'' for all ''β'' less than ''α''.

A theory ''T'' in which every definable set has well-defined Morley rank is called ''totally transcendental''; if ''T'' is countable, then ''T'' is totally transcendental if and only if ''T'' is <math>\omega</math>-stable.
Morley Rank can be extended to types by setting the Morley rank of a type to be the minimum of the Morley ranks of the formulas in the type. Thus, one can also speak of the Morley rank of an element ''a'' over a parameter set ''A'', defined as the Morley rank of the type of ''a'' over ''A''.  
There are also analogues of Morley rank which are well-defined if and only if a theory is superstable ([[U-rank]]) or merely stable (Shelah's <math>\infty</math>-rank).
Those dimension notions can be used to define notions of independence and of generic extensions.

More recently, stability has been decomposed into simplicity and "not the independence property" (NIP). 
[[Simple theory|Simple theories]] are those theories in which a well-behaved notion of independence can be defined, while [[NIP (model theory)|NIP theories]] generalise o-minimal structures.
They are related to stability since a theory is stable if and only if it is NIP and simple,<ref>{{Cite book|last=Wagner|first=Frank|url=https://link.springer.com/book/10.1007/978-94-017-3002-0|title=Simple theories|date=2011|publisher=Springer|doi=10.1007/978-94-017-3002-0|isbn=978-90-481-5417-3}}</ref> and various aspects of stability theory have been generalised to theories in one of these classes.