Morley's Categoricity Theorem - Examples

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 fields of a given 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 closures C_p 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 given characteristic is not categorical in ω (the countable infinite cardinal); there are models of transcendence degree 0, 1, 2, ..., ω.
• Vector spaces over a given countable field. This includes abelian groups of given prime exponent (essentially the same as vector spaces over a finite field) and divisible torsion-free abelian groups (essentially the same as vector spaces over the rationals).
• The theory of the set of natural numbers with a successor function.

There are also examples of theories that are categorical in ω but not categorical in uncountable cardinals. The simplest example is the theory of an equivalence relation with exactly two equivalence classes both of which are infinite. Another example is the theory of dense linear orders with no endpoints; Cantor proved that any such countable linear order is isomorphic to the rational numbers.

Any theory T categorical in some infinite cardinal κ is very close to being complete. More precisely, the Łoś–Vaught test states that if a theory has no finite models and is categorical in some infinite cardinal κ at least equal to the cardinality of its language, then the theory is complete. The reason is that all infinite models are equivalent to some model of cardinal κ by the Löwenheim–Skolem theorem, and so are all equivalent as the theory is categorical in κ. Therefore the theory is complete as all models are equivalent.