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103:, proved several independently. 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.
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The theory of any countably infinite structure which is homogeneous over a finite relational language is omega-categorical. More generally, the theory of the
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of any uniformly locally finite Fraïssé class is omega-categorical. Hence, the following theories are omega-categorical:
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Many conditions on a theory are equivalent to the property of omega-categoricity. In 1959
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Mathematical logic theory with exactly one countably infinite model up to isomorphism
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A Course in Model Theory: An
Introduction to Contemporary Mathematical Logic
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Macpherson, Dugald (2011), "A survey of homogeneous structures",
75:, and omega-categorical theories are also referred to as
44:. Omega-categoricity is the special case Îş =
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The theory of dense linear orders without endpoints (
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with infinite models, the following are equivalent:
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333:Rami Grossberg, José Iovino and Olivier Lessmann,
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308:The theory of infinite linear spaces over any
79:. The notion is most important for countable
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155:has a model which, for every natural number
133:(that is, there are finitely many orbits on
87:Equivalent conditions for omega-categoricity
218:there are only finitely many formulas with
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222:free variables, in other words, for every
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148:has an oligomorphic automorphism group.
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383:Hodges, Model theory, Thm. 7.4.1.
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401:Oligomorphic permutation groups
131:oligomorphic automorphism group
159:, realizes only finitely many
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347:Hodges, Model Theory, p. 341.
586:. You can help Knowledge by
547:Introduction to Model Theory
297:Cantor's isomorphism theorem
545:Rothmaler, Philipp (2000),
336:A primer of simple theories
263:has a countable atomic and
214:, up to equivalence modulo
64:{\displaystyle \aleph _{0}}
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508:10.1016/j.disc.2011.01.024
252:Every countable model of
232:Lindenbaum–Tarski algebra
210:For every natural number
195:For every natural number
180:For every natural number
125:Every countable model of
643:Mathematical logic stubs
144:Some countable model of
25:omega-categorical theory
314:The theory of atomless
188:has only finitely many
97:Czesław Ryll-Nardzewski
582:-related article is a
523:Poizat, Bruno (2000),
477:A shorter model theory
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638:Mathematical theorems
163:-types, that is, the
122:is omega-categorical.
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31:that has exactly one
495:Discrete Mathematics
374:Macpherson, p. 1607.
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425:Chang, Chen Chung;
365:Cameron (1990) p.30
110:first-order theory
580:mathematical logic
427:Keisler, H. Jerome
356:Rothmaler, p. 200.
302:The theory of the
106:Given a countable
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33:countably infinite
21:mathematical logic
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538:978-0-387-98655-5
501:(15): 1599–1634,
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440:978-0-7204-0692-4
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151:The theory
118:The theory
81:first-order
42:isomorphism
627:Categories
419:0813.20002
390:References
304:Rado graph
256:is atomic.
238:is finite.
137:for every
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429:(1989) ,
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283:Examples
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199:, every
108:complete
517:2800979
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