"Vaught's test" redirects here. Not to be confused with the Tarski–Vaught test.
Not to be confused with Category theory.
In mathematical logic, a theory is categorical if it has exactly one model (up to isomorphism).[a] 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. A theory is κ-categorical (or categorical in κ) if it has exactly one model of cardinality κ up to isomorphism. Morley's categoricity theorem is a theorem of Michael D. Morley (1965) stating that if a first-order theory in a countable language is categorical in some uncountable cardinality, then it is categorical in all uncountable cardinalities.
Saharon Shelah (1974) extended Morley's theorem to uncountable languages: if the language has cardinality κ and a theory is categorical in some uncountable cardinal greater than or equal to κ then it is categorical in all cardinalities greater than κ.
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and 21 Related for: Categorical theory information
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...
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{\displaystyle \aleph _{0}} -categoricaltheory, then it always has a model companion. A model completion for a theory T is a model companion T* such...
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in the proof of Morley's categoricity theorem and were extensively studied as part of Saharon Shelah's classification theory, which showed a dichotomy...
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isomorphic to another is called categorial (sometimes categorical). The property of categoriality (categoricity) ensures the completeness of a system, however...
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