Editing Model theory (section)

== Categoricity ==
{{main|Categorical theory}}

A theory was originally called ''categorical'' if it determines a structure up to isomorphism. It turns out that this definition is not useful, due to serious restrictions in the expressivity of first-order logic. The Löwenheim–Skolem theorem implies that if a theory ''T'' has an infinite model for some infinite [[cardinal number]], then it has a model of size {{mvar|κ}} for any sufficiently large [[cardinal number]] {{mvar|κ}}. Since two models of different sizes cannot possibly be isomorphic, only finite structures can be described by a categorical theory.

However, the weaker notion of {{mvar|κ}}-categoricity for a cardinal {{mvar|κ}} has become a key concept in model theory. A theory ''T'' is called ''{{mvar|κ}}-categorical'' if any two models of ''T'' that are of cardinality {{mvar|κ}} are isomorphic. It turns out that the question of {{mvar|κ}}-categoricity depends critically on whether {{mvar|κ}} is bigger than the cardinality of the language (i.e. <math>\aleph_0 + |\sigma|</math>, where {{math|{{mabs|''σ''}}}} is the cardinality of the signature). For finite or countable signatures this means that there is a fundamental difference between {{mvar|&omega;}}-cardinality and {{mvar|κ}}-cardinality for uncountable {{mvar|κ}}.

=== {{mvar|ω}}-categoricity ===
[[Omega-categorical theory|{{mvar|&omega;}}-categorical theories]] can be characterised by properties of their type space:
:For a complete first-order theory ''T'' in a finite or countable signature the following conditions are equivalent:
:#''T'' is {{mvar|&omega;}}-categorical.
:#Every type in ''S<sub>n</sub>''(''T'') is isolated.
:#For every natural number ''n'', ''S<sub>n</sub>''(''T'') is finite.
:#For every natural number ''n'', the number of formulas ''&phi;''(''x''<sub>1</sub>, ..., ''x''<sub>n</sub>) in ''n'' free variables, up to equivalence modulo ''T'', is finite.

The theory of <math>(\mathbb{Q},<)</math>, which is also the theory of <math>(\mathbb{R},<)</math>, is {{mvar|&omega;}}-categorical, as every ''n''-type <math>p(x_1, \dots, x_n)</math> over the empty set is isolated by the pairwise order relation between the <math>x_i</math>.
This means that every countable [[dense linear order]] is order-isomorphic to the rational number line. On the other hand, the theories of {{mathbb|Q}}, {{mathbb|R}} and {{mathbb|C}} as fields are not <math>\omega</math>-categorical. This follows from the fact that in all those fields, any of the infinitely many natural numbers can be defined by a formula of the form <math> x = 1 + \dots + 1 </math>.

<math>\aleph_0</math>-categorical theories and their countable models also have strong ties with [[oligomorphic group]]s:
:A complete first-order theory ''T'' in a finite or countable signature  is {{mvar|&omega;}}-categorical if and only if its automorphism group is oligomorphic.

The equivalent characterisations of this subsection, due independently to [[Erwin Engeler|Engeler]], [[Czesław Ryll-Nardzewski|Ryll-Nardzewski]] and [[Lars Svenonius|Svenonius]], are sometimes referred to as the Ryll-Nardzewski theorem.

In combinatorial signatures, a common source of {{mvar|&omega;}}-categorical theories are [[Fraïssé limit]]s, which are obtained as the limit of amalgamating all possible configurations of a class of finite relational structures.

=== Uncountable categoricity ===

[[Michael D. Morley|Michael Morley]] showed in 1963 that there is only one notion of ''uncountable categoricity'' for theories in countable languages.<ref>{{Cite journal|last=Morley|first=Michael|date=1963|title=On theories categorical in uncountable powers|journal=[[Proceedings of the National Academy of Sciences of the United States of America]]|volume=49|issue=2|pages=213–216|doi=10.1073/pnas.49.2.213|pmid=16591050|pmc=299780|bibcode=1963PNAS...49..213M|doi-access=free}}</ref>
:[[Morley's categoricity theorem]]
:If a first-order theory ''T'' in a finite or countable signature is {{mvar|&kappa;}}-categorical for some uncountable cardinal {{mvar|&kappa;}}, then ''T'' is &kappa;-categorical for all uncountable cardinals {{mvar|&kappa;}}.

Morley's proof revealed deep connections between uncountable categoricity and the internal structure of the models,  which became the starting point of classification theory and stability theory. 
Uncountably categorical theories are from many points of view the most well-behaved theories.
In particular, complete strongly minimal theories are uncountably categorical. This shows that the theory of algebraically closed fields of a given characteristic is uncountably categorical, with the transcendence degree of the field determining its isomorphism type.

A theory that is both {{mvar|&omega;}}-categorical and uncountably categorical is called ''totally categorical''.