{{short description|Mathematical logic theory with exactly one countably infinite model up to isomorphism}} 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 = <math>\aleph_0</math> = 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.<ref name=primer>Rami Grossberg, José Iovino and Olivier Lessmann, [https://doi.org/10.1007%2Fs001530100126 ''A primer of simple theories'']</ref> 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.<ref name=Hodges341>Hodges, Model Theory, p. 341.</ref><ref name=Rothmaler>Rothmaler, p. 200.</ref>
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 ''M<sup>n</sup>'' for every ''n''). * Some countable model of ''T'' has an oligomorphic automorphism group.<ref name=Cam30>Cameron (1990) p.30</ref> * The theory ''T'' has a model which, for every natural number ''n'', realizes only finitely many ''n''-types, that is, the Stone space ''S<sub>n</sub>''(''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 ''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.
==Examples==
The theory of any countably infinite structure which is homogeneous over a finite relational language is omega-categorical.<ref name=Macpherson1607>Macpherson, p. 1607.</ref> More generally, the theory of the Fraïssé limit of any uniformly locally finite Fraïssé class is omega-categorical.<ref>Hodges, Model theory, Thm. 7.4.1.</ref> 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== <references />
==References== * {{citation | last=Cameron | first=Peter J. | authorlink=Peter Cameron (mathematician) | title=Oligomorphic permutation groups | series=London Mathematical Society Lecture Note Series | volume=152 | location=Cambridge | publisher=Cambridge University Press | year=1990 | isbn=0-521-38836-8 | zbl=0813.20002 }} * {{Citation | last1=Chang | first1=Chen Chung | last2=Keisler | first2=H. Jerome | author2-link=Howard Jerome Keisler | title=Model Theory | orig-date=1973 | publisher=Elsevier | isbn=978-0-7204-0692-4 | year=1989}} * {{Citation | last1=Hodges | first1=Wilfrid | author1-link=Wilfrid Hodges | title=Model theory | publisher=Cambridge University Press | location=Cambridge | isbn=978-0-521-30442-9 | year=1993 | url-access=registration | url=https://archive.org/details/modeltheory0000hodg }} * {{Citation | last1=Hodges | first1=Wilfrid | author1-link=Wilfrid Hodges | title=A shorter model theory | publisher=Cambridge University Press | location=Cambridge | isbn=978-0-521-58713-6 | year=1997}} * {{Citation | last1=Macpherson | first1=Dugald | title=A survey of homogeneous structures | mr=2800979 | year=2011 | journal=Discrete Mathematics | volume=311 | pages=1599–1634 | doi=10.1016/j.disc.2011.01.024 | issue=15| doi-access=free }} * {{Citation | last1=Poizat | first1=Bruno | title=A Course in Model Theory: An Introduction to Contemporary Mathematical Logic | publisher=Springer-Verlag | location=Berlin, New York | isbn=978-0-387-98655-5 | year=2000 | url-access=registration | url=https://archive.org/details/courseinmodelthe0000poiz }} * {{Citation | last1=Rothmaler | first1=Philipp | title=Introduction to Model Theory | publisher=Taylor & Francis | location=New York | isbn=978-90-5699-313-9 | year=2000}}
Category:Model theory Category:Mathematical theorems
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