Omega-categorical theory

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 = ${\displaystyle \aleph _{0}}$ = 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.

References

• Cameron, Peter J. (1990), Oligomorphic permutation groups, London Mathematical Society Lecture Note Series, vol. 152, Cambridge: Cambridge University Press, ISBN 0-521-38836-8, Zbl 0813.20002
• Chang, Chen Chung; Keisler, H. Jerome (1989) [1973], Model Theory, Elsevier, ISBN 978-0-7204-0692-4
• Hodges, Wilfrid (1993), Model theory, Cambridge: Cambridge University Press, ISBN 978-0-521-30442-9
• Hodges, Wilfrid (1997), A shorter model theory, Cambridge: Cambridge University Press, ISBN 978-0-521-58713-6
• Macpherson, Dugald (2011), "A survey of homogeneous structures", Discrete Mathematics, 311 (15): 1599–1634, doi:10.1016/j.disc.2011.01.024, MR 2800979
• Poizat, Bruno (2000), A Course in Model Theory: An Introduction to Contemporary Mathematical Logic, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98655-5
• Rothmaler, Philipp (2000), Introduction to Model Theory, New York: Taylor & Francis, ISBN 978-90-5699-313-9