# Omega-categorical theory

(Redirected from Ryll-Nardzewski theorem)

In mathematical logic, an omega-categorical theory is a theory that has only one countable 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.
• 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 Sn(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, every 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.

## 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