Categorical theory

In model theory, Morley's categoricity theorem is a theorem of Michael D. Morley which states that if a first-order theory is complete in a countable language, then if it is categorical in some uncountable cardinality, it is categorical in all uncountable cardinalities.

The classic example is the theory of algebraically closed fields of given characteristic, which are categorical for fields of size the continuum, and hence in all uncountable cardinalities. It should be noted that categoricity is not saying that all algebraically closed fields as large as ${\mathbb {C}}$ are the same as ${\mathbb {C}}$, but that they are isomorphic as fields to ${\mathbb {C}}$; hence the completed p-adic closures ${\mathbb {C}}_{p}$ are all isomorphic as fields to ${\mathbb {C}}$, but have completely different topological and analytic properties.