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 = = 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. 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. 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. 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. 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, 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. The theory of any countably infinite structure which is homogeneous over a finite relational language is omega-categorical.

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Saturated model

In mathematical logic, and particularly in its subfield model theory, a saturated model M is one that realizes as many complete types as may be "reasonably expected" given its size. For example, an ultrapower model of the hyperreals is -saturated, meaning that every descending nested sequence of internal sets has a nonempty intersection. Let κ be a finite or infinite cardinal number and M a model in some first-order language. Then M is called κ-saturated if for all subsets A ⊆ M of cardinality less than κ, the model M realizes all complete types over A.

Categorical theory

In mathematical logic, a theory is categorical if it has exactly one model (up to isomorphism). Such a theory can be viewed as defining its model, uniquely characterizing the model's structure. In first-order logic, only theories with a finite model can be categorical. Higher-order logic contains categorical theories with an infinite model. For example, the second-order Peano axioms are categorical, having a unique model whose domain is the set of natural numbers In model theory, the notion of a categorical theory is refined with respect to cardinality.

Omega-categorical theory

MATH-381: Mathematical logic

Branche des mathématiques en lien avec le fondement des mathématiques et l'informatique théorique. Le cours est centré sur la logique du 1er ordre et l'articulation entre syntaxe et sémantique.