# Omega-categorical theory

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 = = 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*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, every*n*th 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

- ↑ Rami Grossberg, José Iovino and Olivier Lessmann,
*A primer of simple theories* - ↑ Hodges, Model Theory, p. 341.
- ↑ Rothmaler, p. 200.
- ↑ Cameron (1990) p.30

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