Omega-categorical theory explained

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.^[1] 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.^{[2] [3]}

Given a countable complete first-order theory T with infinite models, the following are equivalent:

• The theory T is omega-categorical.
• Every countable model of T has an oligomorphic automorphism group (that is, there are finitely many orbits on Mⁿ for every n).
• Some countable model of T has an oligomorphic automorphism group.^[4]
• The theory T has a model which, for every natural number n, realizes only finitely many n-types, that is, the Stone space S_n(T) is finite.
• For every natural number n, T has only finitely many n-types.
• For every natural number n, every n-type is isolated.
• For every natural number n, up to equivalence modulo T there are only finitely many formulas with n free variables, in other words, for every n, the nth Lindenbaum–Tarski algebra of T is finite.
• Every model of T is atomic.
• Every countable model of T is atomic.
• The theory T has a countable atomic and saturated model.
• The theory T has a saturated prime model.

Examples

The theory of any countably infinite structure which is homogeneous over a finite relational language is omega-categorical.^[5] More generally, the theory of the Fraïssé limit of any uniformly locally finite Fraïssé class is omega-categorical.^[6] Hence, the following theories are omega-categorical:

• The theory of dense linear orders without endpoints (Cantor's isomorphism theorem)
• The theory of the Rado graph
• The theory of infinite linear spaces over any finite field
• The theory of atomless Boolean algebras

Notes

1. Rami Grossberg, José Iovino and Olivier Lessmann, A primer of simple theories
2. Hodges, Model Theory, p. 341.
3. Rothmaler, p. 200.
4. Cameron (1990) p.30
5. Macpherson, p. 1607.
6. Hodges, Model theory, Thm. 7.4.1.