Editing Model theory (section)

===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.