Editing Categorical theory (section)

==Examples==
There are not many natural examples of theories that are categorical in some uncountable cardinal. The known examples include:
* Pure identity theory (with no functions, constants, predicates other than "=", or axioms).
* The classic example is the theory of [[Algebraically closed field|algebraically closed]] [[Field (mathematics)|fields]] of a given [[Characteristic (algebra)|characteristic]]. Categoricity does ''not'' say that all algebraically closed fields of characteristic 0 as large as the [[complex numbers]] '''C''' are the same as '''C'''; it only asserts that they are isomorphic ''as fields'' to '''C'''. It follows that although the completed [[p-adic|''p''-adic]] closures '''C'''<sub>''p''</sub> are all isomorphic as fields to '''C''', they may (and in fact do) have completely different [[topological]] and analytic properties. The theory of algebraically closed fields of a given characteristic is '''not''' categorical in {{math|''ω''}} (the countable infinite cardinal); there are models of [[transcendence degree]] 0, 1, 2, ..., {{math|''ω''}}.
* [[Vector space]]s over a given countable field. This includes [[abelian group]]s of given [[Prime number|prime]] [[Torsion group|exponent]] (essentially the same as vector spaces over a finite field) and [[Divisible group|divisible]] [[torsion-free abelian group]]s (essentially the same as vector spaces over the [[Rational number|rationals]]).
* The theory of the set of [[natural number]]s with a successor function.

There are also examples of theories that are categorical in {{math|''ω''}} but not categorical in uncountable cardinals. 
The simplest example is the theory of an [[equivalence relation]] with exactly two [[equivalence class]]es, both of which are infinite. Another example is the theory of [[Dense order|dense]] [[linear order]]s with no endpoints; [[Georg Cantor|Cantor]] proved that any such countable linear order is isomorphic to the rational numbers: see [[Cantor's isomorphism theorem]].