In mathematical logic, an omega-categorical theory is a theory that has exactly one countably infinite model up to isomorphism. Omega-categoricity is the special case κ = \aleph_0 = ω of κ-categoricity, and omega-categorical theories are also referred to as ω-categorical. The notion is most important for countable first-order theories.

Equivalent conditions for omega-categoricity

Many conditions on a theory are equivalent to the property of omega-categoricity. In 1959 Erwin Engeler, Czesław Ryll-Nardzewski and Lars Svenonius, proved several independently. Despite this, the literature still widely refers to the Ryll-Nardzewski theorem as a name for these conditions. The conditions included with the theorem vary between authors. Given a countable complete first-order theory T with infinite models, the following are equivalent:

Examples

The theory of any countably infinite structure which is homogeneous over a finite relational language is omega-categorical. More generally, the theory of the Fraïssé limit of any uniformly locally finite Fraïssé class is omega-categorical. Hence, the following theories are omega-categorical: