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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.〔Rami Grossberg, José Iovino and Olivier Lessmann, (''A primer of simple theories'' )〕 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.〔Hodges, Model Theory, p. 341.〕〔Rothmaler, p. 200.〕 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.〔Cameron (1990) p.30〕 * 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 ''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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Omega-categorical theory」の詳細全文を読む スポンサード リンク
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